content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
How to Calculate Reactions on a Truss
What is the first step in calculating reactions on a truss? The first step in calculating reactions on a truss is to calculate the reactions on the entire truss using a free body diagram.
Trusses are commonly used in engineering and architecture to support loads. When analyzing a truss structure, one of the key steps is to calculate the reactions at the supports. These reactions are
crucial in determining whether the truss is stable and can safely support the applied loads.
Free Body Diagram
A free body diagram is a visual representation that shows all the forces acting on a body. In the case of a truss, the body is the entire truss structure. By drawing a free body diagram of the truss,
you can identify the reactions at the supports.
To calculate the reactions on a truss, start by drawing a free body diagram of the entire truss structure. Include all the applied loads and the reactions at the supports in the diagram. Once you
have a clear representation of the forces acting on the truss, you can proceed to solve for the reaction forces.
By following the steps of calculating reactions on a truss, you can ensure that the structure is capable of withstanding the applied loads and is safe for use. Understanding how to analyze truss
structures is essential for engineers and architects in designing robust and reliable buildings and bridges. | {"url":"https://www.brundtlandnet.com/engineering/how-to-calculate-reactions-on-a-truss.html","timestamp":"2024-11-07T15:37:07Z","content_type":"text/html","content_length":"20909","record_id":"<urn:uuid:cb2f8ca1-c92d-4af2-97ac-f9a2189d5f4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00271.warc.gz"} |
FAQ: What's the difference between the two Simulink Precision Loss ...
Accepted Answer
Edited: Andy Bartlett on 20 Nov 2023
Edited: Andy Bartlett on 20 Nov 2023
What is the difference between the Detect precision loss that applies to fixed-point constants and the Detect precision loss for Parameters from the Diagnostics -> Data validity?
8 views (last 30 days)
FAQ: What's the difference between the two Simulink Precision Loss Diagnostics for Parameters and Fixed-Point Constants?
Simulink's detect parameter precision loss diagnostic applys to run-time parameter values. For example, a Gain blocks gain parameter value or a Saturation blocks upper and lower limit parameter
The diagnostic is about precision loss when converting from the data type used when entering the parameter value on the dialog and value obtained when quantizing the dialog value to the data type
used by the run-time parameter in simulation, code generation, etc.
For example, suppose the parameter entered on the dialog was the double precision floating-point approximation of pi
Now suppose the run-time data type for the parameter is single
paramRunTimeValueSingle = single(paramDialogValue)
quantErrSingle = double(paramRunTimeValueSingle) - double(paramDialogValue)
As second example, suppose the run-time data type is int8.
paramRunTimeValueInt8 = int8(paramDialogValue)
quantErrInt8 = double(paramRunTimeValueInt8) - double(paramDialogValue)
As third example, suppose the run-time data type is a 32 bit fixed-point type
paramRunTimeValueUfix32En30 = fi(paramDialogValue,0,32,30)
quantErrUfix32En30 = double(paramRunTimeValueUfix32En30) - double(paramDialogValue)
All three examples of quantizing the run-time parameter pi involve precision loss. 32-bit fixed-point has the least precision loss. Single's precision loss is in the middle. Int8 gives the most
precision loss. Simulink's diagnostic for run-time parameter precision loss would apply to all three of these cases.
Simulink's diagnostic for detecting precision loss in fixed-point Net-Scaling constants only applies to fixed-point math involving Slope-Bias scaling.
For example, suppost the ideal net-slope for a fixed-point cast operation was
Now let's assume the fixed-point cast was handled in generated C code as follows
y = (int8_T) ( ((int32_T) uStoredInteger * 11439) >> 21 );
The net-slope is represented in the C code by
multiplication by 11439
arithmetic shift right 21 bits
Arithmetic shifts right are mathematically equivalent to division by 2^nbits.
So effectively the quantized net-slope is
quantizedNetSlope = 11439 / 2^21
quantErrNetSlope = double(quantizedNetSlope) - double(netSlope)
quantRelErrNetSlope = abs(quantErrNetSlope)/double(netSlope)
There is a small difference between the ideal net-slope and the quantized net slope. Small net-scaling differences like this are very common when slope-bias scaling is used. These small errors do not
occur if all the scaling is restricted to binary point scaling which occurs when Slopes are exact powers of two and Biases are all zero.
Simulink's diagnostic for detecting precision loss in fixed-point Net-Scaling constants is about these small errors.
Specific example
A specific example where net-scaling example occurs is a Cast from
Even though the scaling bias for the two signals were not zero, the ideal net-bias for this cast does become zero. Zero can be perfectly quantized to zero, so there is no precision loss. In generated
code, the zero bias would be completely optimized away, i.e. no code needed.
The ideal net-slope is the rational 3/800. To avoid the cost of division, this is approximated with a integer multiplication followed by a shift right. This approximation effectively gives a
net-scaling constant precision loss that the diagnostic would apply to.
In this example, the precision loss due to going from an input scaling of 0.0003 to an output scaling of 0.08 is orders of magnitude bigger than the small precision loss of quantizing 3/800 to 31457
/ 2^23.
The ideal net slope 3/800 = 0.0037499999999999999 = 0.95999999999999996 * 2^-8.
Quantization of this net slope produces a diagnostic that only mentions the mantissa 0.95999999999999996.
Net scaling quantization caused precision loss. The original value for NetSlope was 0.95999999999999985. The quantized value was 0.959991455078125 with an error of 8.54492187485345E-6.
The full quantization error needs to account for the exponent too. 8.54492187485345E-6 * 2^-8
quantErrorNetSlope = 8.54492187485345E-6 * 2^-8
When multiplied by the biggest input stored integers, the amplified error is
worstCaseError = quantErrorNetSlope * [-2^15, 2^15-1]
In terms of the output scaling of 0.08, the worst case error do to Net Slope quantization is quite small.
errorInOutputBits = worstCaseError / 0.08
Not even 1.4% of a bit value.
If we round toward Floor, then ideally we expect an average rounding error 0.04 (half a bit) and a worst case error of 0.08 (1 bit).
If we round toward Nearest, then ideally we expect an average rounding error 0.02 (quarter of a bit) and a worst case error of 0.04 (half a bit).
Figure 1: Box and wisker plots showing quantization error statistics for casts. Top wisker is maximum error. Bottom wisker is smallest error. Red line is median. Top and bottom of blue boxes are 75th
and 25th percentiles, respectively. Round to Nearest gives half the error of Round to Floor.
Figure 1 compares the error statistics of the fixed-point casts Simulink would provide including quantization of net scaling vs a more costly "luxury" cast that used very precise representations of
the net scaling. For Floor rounding, the fixed-point cast and the cast using high precision net slope the box plots are indistiguishable. Likewise, for Nearest rounding, the fixed-point cast and the
cast using high precision net slope the box plots are indistiguishable. What dominates the casts are the Slope of the output data type 0.08, and the type of rounding selected Floor vs Nearest.
One of the reasons the impact of Net Slope quantization is small is because "Best Precision Scaling" is applied when quantizing the scalar value. This will minimize the quantization error for the
number of bits used to represent the net slope. In contrast, depending on how run-time parameters have been configured they might be scalars, vectors, ... n-d arrays and they might get scaling that
is a great fit or a poor fit for the individual values within the run-parameter. Best precision scaling is one of the reasons quantization of a net slope constants is unlikely to have significant
impact on a design.
Let's look at an example showing the huge contrast between best precisions scaling and bad scaling.
quantNetSlopeBestPrecisionScaling = fi( idealNetScaling, 1, 16 ) % No scaling means use best
relativeErrorNetSlope = abs( (double(quantNetSlopeBestPrecisionScaling) - idealNetScaling))/idealNetScaling
Notice that with best precision scaling, quantization of the Net Slope gives a relative error of a meer 0.00089 percent.
idealRunTimeParam = 3/800
quantRunTimeParamPoorScaling = fi( idealRunTimeParam, 1, 16, 5)
relativeErrorRunTimeParam = abs( (double(quantRunTimeParamPoorScaling) - idealRunTimeParam) )/idealRunTimeParam
In contrast, the same value as a run-time parameter could get very bad scaling assigned. In this case, the quantization error of the run-time parameter lead to a 100% relative error. In other words,
in relative terms, all value information was completely lost.
Given the expectation of precision losses in fixed-point designs, Simulink's diagnostic for detecting precision loss in fixed-point Net-Scaling constants is usually not important to investigate
directly. It is better to invest in testing the fixed-point design at the system level to make sure system level behavior is within system design tolerances. In the example given in this section, we
saw that the slope of the output data type and the rounding mode selected had over 30 times larger impact on accuracy than Net Slope quantization did. Testing to make sure an output slope of 0.08 met
system requirements is much more important than directly investigating a net slope quantization error with a worst case impact less than 0.0011 which is 70 times smaller than the output slope.
More Answers (0) | {"url":"https://ch.mathworks.com/matlabcentral/answers/2048147-faq-what-s-the-difference-between-the-two-simulink-precision-loss-diagnostics-for-parameters-and-fi","timestamp":"2024-11-04T18:46:02Z","content_type":"text/html","content_length":"148605","record_id":"<urn:uuid:31fb267c-b6aa-44e5-ab99-42a848a9cb31>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00724.warc.gz"} |
Understanding these algorithms is essential if you want to #BreakIntoAI
Dear friends,
Years ago, I had to choose between a neural network and a decision tree learning algorithm. It was necessary to pick an efficient one, because we planned to apply the algorithm to a very large set of
users on a limited compute budget. I went with a neural network. I hadn’t used boosted decision trees in a while, and I thought they required more computation than they actually do — so I made a bad
call. Fortunately, my team quickly revised my decision, and the project was successful.
This experience was a lesson in the importance of learning, and continually refreshing, foundational knowledge. If I had refreshed my familiarity with boosted trees, I would have made a better
Machine learning, like many technical fields, evolves as the community of researchers builds on top of one another's work. Some contributions have staying power and become the basis of further
developments. Consequently, everything from a housing-price predictor to a text-to-image generator is built on core ideas that include algorithms (linear and logistic regression, decision trees, and
so on) and concepts (regularization, optimizing a loss function, bias/variance, and the like).
A solid, up-to-date foundation is one key to being a productive machine learning engineer. Many teams draw on these ideas in their day-to-day work, and blog posts and research papers often assume
that you’re familiar with them. This shared base of knowledge is essential to the rapid progress we've seen in machine learning in recent years.
That's why I’m updating my original machine learning class as the new Machine Learning Specialization, which will be available in a few weeks.
My team spent many hours debating the most important concepts to teach. We developed extensive syllabi for various topics and prototyped course units in them. Sometimes this process helped us realize
that a different topic was more important, so we cut material we had developed to focus on something else. The result, I hope, is an accessible set of courses that will help anyone master the most
important algorithms and concepts in machine learning today — including deep learning but also a lot of other things — and to build effective learning systems.
In that spirit, this week’s issue of The Batch explores some of our field’s most important algorithms, explaining how they work and describing some of their surprising origins. If you’re just
starting out, I hope it will demystify some of the approaches at the heart of machine learning. For those who are more advanced, you’ll find lesser-known perspectives on familiar territory. Either
way, I hope this special issue will help you build your intuition and give you fun facts about machine learning’s foundations that you can share with friends.
Keep learning!
Essential Algorithms
Machine learning offers a deep toolbox for solving all kinds of problems, but which tool is best for which task? When is the open-ended wrench better than the adjustable kind? Who invented these
things anyway? In this special issue of The Batch, we survey six of the most useful algorithms in the kit: where they came from, what they do, and how they’re evolving as AI advances into every facet
of society. If you want to learn more, the Machine Learning Specialization provides a simple, practical introduction to these algorithms and more. Join the waitlist to be notified when it’s
Linear Regression: Straight & Narrow
Linear regression may be the key statistical method in machine learning, but it didn’t get to be that way without a fight. Two eminent mathematicians claimed credit for it, and 200 years later the
matter remains unresolved. The longstanding dispute attests not only to the algorithm’s extraordinary utility but also to its essential simplicity.
Whose algorithm is it anyway? In 1805, French mathematician Adrien-Marie Legendre published the method of fitting a line to a set of points while trying to predict the location of a comet (celestial
navigation being the science most valuable in global commerce at the time, much like AI is today — the new electricity, if you will, two decades before the electric motor). Four years later, the
24-year-old German wunderkind Carl Friedrich Gauss insisted that he had been using it since 1795 but had deemed it too trivial to write about. Gauss’ claim prompted Legendre to publish an addendum
anonymously observing that “a very celebrated geometer has not hesitated to appropriate this method.”
Slopes and biases: Linear regression is useful any time the relationship between an outcome and a variable that influences it follows a straight line. For instance, a car’s fuel consumption bears a
linear relationship to its weight.
• The relationship between a car’s fuel consumption y and its weight x depends on the line’s slope w (how steeply fuel consumption rises with weight) and bias term b (fuel consumption at zero
weight): y=w*x+b.
• During training, given a car’s weight, the algorithm predicts the expected fuel consumption. It compares expected and actual fuel consumption. Then it minimizes the squared difference, typically
via the technique of ordinary least squares, which hones the values of w and b.
• Taking the car’s drag into account makes it possible to generate more precise predictions. The additional variable extends the line into a plane. In this way, linear regression can accommodate
any number of variables/dimensions.
Two steps to ubiquity: The algorithm immediately helped navigators to follow the stars, and later biologists (notably Charles Darwin’s cousin Francis Galton) to identify heritable traits in plants
and animals. Two further developments unlocked its broad potential. In 1922, English statisticians Ronald Fisher and Karl Pearson showed how linear regression fit into the general statistical
framework of correlation and distribution, making it useful throughout all sciences. And, nearly a century later, the advent of computers provided the data and processing power to take far greater
advantage of it.
Coping with ambiguity: Of course, data is never perfectly measured, and some variables are more important than others. These facts of life have spurred more sophisticated variants. For instance,
linear regression with regularization (also called ridge regression) encourages a linear regression model to not depend too much on any one variable, or rather to rely evenly on the most important
variables. If you’re going for simplicity, a different form of regularization (L1 instead of L2) results in lasso, which encourages as many coefficients as possible to be zero. In other words, it
learns to select variables with high prediction power and ignores the rest. Elastic net combines both types of regularization. It’s useful when data is sparse or features appear to be correlated.
In every neuron: Still, the simple version is enormously useful. The most common sort of neuron in a neural network is a linear regression model followed by a nonlinear activation function, making
linear regression a fundamental building block of deep learning.
Logistic Regression: Follow the Curve
There was a moment when logistic regression was used to classify just one thing: If you drink a vial of poison, are you likely to be labeled “living” or “deceased”? Times have changed, and today not
only does calling emergency services provide a better answer to that question, but logistic regression is at the very heart of deep learning.
Poison control: The logistic function dates to the 1830s, when the Belgian statistician P.F. Verhulst invented it to describe population dynamics: Over time, an initial explosion of exponential
growth flattens as it consumes available resources, resulting in the characteristic logistic curve. More than a century passed before American statistician E. B. Wilson and his student Jane Worcester
devised logistic regression to figure out how much of a given hazardous substance would be fatal. How they gathered their training data is a subject for another essay.
Fitting the function: Logistic regression fits the logistic function to a dataset in order to predict the probability, given an event (say, ingesting strychnine), that a particular outcome will occur
(say, an untimely demise).
• Training adjusts the curve’s center location horizontally and its middle vertically to minimize error between the function’s output and the data.
• Adjusting the center to the right or the left means that it would take more or less poison to kill the average person. A steep slope signifies certainty: Before the halfway point, most people
survive; beyond the halfway point, sayonara. A gentle slope is more forgiving: lower than midway up the curve, more than half survive; farther up, less than half.
• Set a threshold of, say, 0.5 between one outcome and another, and the curve becomes a classifier. Just enter the dose into the model, and you’ll know whether you should be planning a party or a
More outcomes: Verhulst’s work found the probabilities of binary outcomes, ignoring further possibilities like which side of the afterlife a poison victim might land in. His successors extended the
• Working independently in the late 1960s, British statistician David Cox and Dutch statistician Henri Theil adapted logistic regression for situations that have more than two possible outcomes.
• Further work yielded ordered logistic regression, in which the outcomes are ordered values.
• To deal with sparse or high-dimensional data, logistic regression can take advantage of the same regularization techniques as linear regression.
Versatile curve: The logistic function describes a wide range of phenomena with fair accuracy, so logistic regression provides serviceable baseline predictions in many situations. In medicine, it
estimates mortality and risk of disease. In political science, it predicts winners and losers of elections. In economics, it forecasts business prospects. More important, it drives a portion of the
neurons, in which the nonlinearity is a sigmoid, in a wide variety of neural networks.
Gradient Descent: It’s All Downhill
Imagine hiking in the mountains past dusk and finding that you can’t see much beyond your feet. And your phone’s battery died so you can’t use a GPS app to find your way home. You might find the
quickest path down via gradient descent. Just be careful not to walk off a cliff.
Suns and rugs: Gradient descent is good for more than descending through precipitous terrain. In 1847, French mathematician Augustin-Louis Cauchy invented the algorithm to approximate the orbits of
stars. Sixty years later, his compatriot Jacques Hadamard independently developed it to describe deformations of thin, flexible objects like throw rugs that might make a downward hike easier on the
knees. In machine learning, though, its most common use is to find the lowest point in the landscape of a learning algorithm’s loss function.
Downward climb: A trained neural network provides a function that, given an input, computes a desired output. One way to train the network is to minimize the loss, or error in its output, by
iteratively computing the difference between the actual output and the desired output and then changing the network’s parameter values to narrow that difference. Gradient descent narrows the
difference, minimizing the function that computes the loss. The network’s parameter values are tantamount to a position on the landscape, and the loss is the current altitude. As you descend, you
improve the network’s ability to compute outputs close to the desired one. Visibility is limited because, in a typical supervised learning situation, the algorithm relies solely on the network’s
parameter values and the gradient, or slope of the loss function — that is, your position on the hill and the slope immediately beneath your feet.
• The basic method is to move in the direction where the terrain descends most steeply. The trick is to calibrate your stride. Too small, and it takes ages to make any progress. Too large, and you
leap into the unknown, possibly heading uphill instead of downward.
• Given the current position, the algorithm estimates the direction of steepest descent by computing the gradient of the loss function. The gradient points uphill, so the algorithm steps in the
opposite direction by subtracting a fraction of the gradient. The fraction α, which is called the learning rate, determines the size of the step before measuring the gradient again.
• Apply this iteratively, and hopefully you’ll arrive at a valley. Congratulations!
Stuck in the valley: Too bad your phone is out of juice, because the algorithm may not have propelled you to the bottom of a convex mountain. You may be stuck in a nonconvex landscape of multiple
valleys (local minima), peaks (local maxima), saddles (saddle points), and plateaus. In fact, tasks like image recognition, text generation, and speech recognition are nonconvex, and many variations
on gradient descent have emerged to handle such situations. For example, the algorithm may have momentum that helps it zoom over small rises and dips, giving it a better chance at arriving at the
bottom. Researchers have devised so many variants that it may seem as though there are as many optimizers as there are local minima. Luckily, local and global minima tend to be roughly equivalent.
Optimal optimizer: Gradient descent is the clear choice for finding the minimum of any function. In cases where an exact solution can be computed directly — say, a linear regression task with lots of
variables — it can approximate one, often faster and more cheaply. But it really comes into its own in complex, nonlinear tasks. Armed with gradient descent and an adventurous spirit, you might just
make it out of the mountains in time for dinner.
No advanced math required! The new Machine Learning Specialization balances intuition, practice, and theory to create a powerful learning experience for beginners. Enroll now and achieve your career
Neural Networks: Find the Function
Let’s get this out of the way: A brain is not a cluster of graphics processing units, and if it were, it would run software far more complex than the typical artificial neural network. Yet neural
networks were inspired by the brain’s architecture: layers of interconnected neurons, each of which computes its own output depending on the states of its neighbors. The resulting cascade of activity
forms an idea — or recognizes a picture of a cat.
From biological to artificial: The insight that the brain learns through interactions among neurons dates back to 1873, but it wasn’t until 1943 that American neuroscientists Warren McCulloch and
Walter Pitts modeled biological neural networks using simple mathematical rules. In 1958, American psychologist Frank Rosenblatt developed the perceptron, a single-layer vision network implemented on
punch cards with the intention of building a hardware version for the United States Navy.
Bigger is better: Rosenblatt’s invention recognized only classes that could be separated by a line. Ukrainian mathematicians Alexey Ivakhnenko and Valentin Lapa overcame this limitation by stacking
networks of neurons in any number of layers. In 1985, working independently, French computer scientist Yann LeCun, David Parker, and American psychologist David Rumelhart and his colleagues described
using backpropagation to train such networks efficiently. In the first decade of the new millennium, researchers including Kumar Chellapilla, Dave Steinkraus, and Rajat Raina (with Andrew Ng)
accelerated neural networks using graphical processing units, which has enabled ever-larger neural networks to learn from the immense amounts of data generated by the internet.
Fit for every task: The idea behind a neural network is simple: For any task, there’s a function that can perform it. A neural network constitutes a trainable function by combining many simple
functions, each executed by a single neuron. A neuron’s function is determined by adjustable parameters called weights. Given random values for those weights and examples of inputs and their desired
outputs, it’s possible to alter the weights iteratively until the trainable function performs the task at hand.
• A neuron accepts various inputs (for example, numbers representing a pixel or word, or the outputs of the previous layer), multiplies them by its weights, adds the products, and feeds the sum
through a nonlinear function, or activation function, chosen by the developer. Consider it linear regression plus an activation function.
• Training modifies the weights. For every example input, the network computes an output and compares it to the expected output. Backpropagation uses gradient descent to change the weights to
reduce the difference between actual and expected outputs. Repeat this process enough times with enough (good) examples, and the network should learn to perform the task.
Black box: While a trained network may perform its task, good luck determining how. You can read the final function, but often it’s so complex — with thousands of variables and nested activation
functions — that it’s exceedingly difficult to explain how the network succeeded at its task. Moreover, a trained network is only as good as the data it learned from. For instance, if the dataset was
skewed, the network’s output will be skewed. If it included only high-resolution pictures of cats, there’s no telling how it would respond to lower-resolution images.
Toward common sense: Reporting on Rosenblatt’s Perceptron in 1958, The New York Times blazed the trail for AI hype by calling it “the embryo of an electronic computer that the United States Navy
expects will be able to walk, talk, see, write, reproduce itself and be conscious of its existence.” While it didn’t live up to that billing, it begot a host of impressive models: convolutional
neural networks for images; recurrent neural networks for text; and transformers for images, text, speech, video, protein structures, and more. They’ve done amazing things, exceeding human-level
performance in playing Go and approaching it in practical tasks like diagnosing x-ray images. Yet they still have a hard time with common sense and logical reasoning. Ask GPT-3, “When counting, what
number comes before a million?” and it may respond, “Nine hundred thousand and ninety-nine comes before a million.” To which we reply: Keep learning!
Decision Trees: From Root to Leaves
What kind of beast was Aristotle? The philosopher's follower Porphyry, who lived in Syria during the third century, came up with a logical way to answer the question. He organized Aristotle’s
proposed “categories of being” from general to specific and assigned Aristotle himself to each category in turn: Aristotle’s substance occupied space rather than being conceptual or spiritual; his
body was animate not inanimate; his mind was rational not irrational. Thus his classification was human. Medieval teachers of logic drew the sequence as a vertical flowchart: An early decision tree.
The digital difference: Fast forward to 1963, when University of Michigan sociologist John Sonquist and economist James Morgan, dividing survey respondents into groups, first implemented decision
trees in a computer. Such work became commonplace with the advent of software that automates training the algorithm, now implemented in a variety of machine learning libraries including scikit-learn.
The code took a quartet of statisticians at Stanford and UC Berkeley 10 years to develop. Today, coding a decision tree from scratch is a homework assignment in Machine Learning 101.
Roots in the sky: A decision tree can perform classification or regression. It grows downward, from root to canopy, in a hierarchy of decisions that sort input examples into two (or more) groups.
Consider the task of Johann Blumenbach, the German physician and anthropologist who first distinguished monkeys from apes (setting aside humans) circa 1776, before which they had been categorized
together. The classification depends on various criteria such as presence or absence of a tail, narrow or broad chest, upright versus crouched posture, and lesser or greater intelligence. A decision
tree trained to label such animals would consider each criterion one by one, ultimately separating the two groups.
• The tree starts with a root node that can be viewed as containing all examples in a dataset of creatures — chimpanzees, gorillas, and orangutans as well as capuchins, baboons, and marmosets. The
root presents a choice between examples that exhibit a particular feature or not, leading to two child nodes that contain examples with and without that feature. Each child poses yet another
choice (with or without a different feature) leading to two more children, and so on. The process ends with any number of leaf nodes, each of which contains examples that belong — mostly or
wholly — to one class.
• To grow, the tree must find the root decision. To choose, it considers all features and their values — posterior appendage, barrel chest, and so on — and chooses the one that maximizes the purity
of the split, optimal purity being defined as 100 percent of examples of one class going to a particular child node and none going to the other node. Splits are rarely 100 percent pure after just
one decision and may never get there, so the process continues, producing level after level of child nodes, until purity won’t rise much by considering further features. At this point, the tree
is fully trained.
• At inference, a fresh example traverses the tree from top to bottom evaluating a different decision at each level. It takes the label of the data contained by the leaf node it lands in.
Top 10 hit: Given Blumenbach’s conclusion (later overturned by Charles Darwin) that humans are distinguished from apes by a broad pelvis, hands, and close-set teeth, what if we wanted to extend the
decision tree to classify not just apes and monkeys but humans as well? Australian computer scientist John Ross Quinlan made this possible in 1986 with ID3, which extended decision trees to support
nonbinary outcomes. In 2008, a further refinement called C4.5 capped a list of Top 10 Algorithms in Data Mining curated by the IEEE International Conference on Data Mining. In a world of rampant
innovation, that’s staying power.
Raking leaves: Decision trees do have some drawbacks. They can easily overfit the data by growing so many levels that leaf nodes include as few as one example. Worse, they’re prone to the butterfly
effect: Change one example, and the tree that grows could look dramatically different.
Into the woods: Turning this trait into an advantage, American statistician Leo Breiman and New Zealander statistician Adele Cutler in 2001 developed the random forest, an ensemble of decision trees,
each of which processes a different, overlapping selection of examples that vote on a final decision. Random forest and its cousin XGBoost are less prone to overfitting, which helps make them among
the most popular machine learning algorithms. It’s like having Aristotle, Porphyry, Blumenbach, Darwin, Jane Goodall, Dian Fossey, and 1,000 other zoologists in the room together, all making sure
your classifications are the best they can be.
K-Means Clustering: Group Think
If you’re standing close to others at a party, it’s likely that you have something in common. This is the idea behind using k-means clustering to split data points into groups. Whether the groups
formed via human agency or some other force, this algorithm will find them.
From detonations to dial tones: American physicist Stuart Lloyd, an alumnus of both Bell Labs’ iconic innovation factory and the Manhattan Project that invented the atomic bomb, first proposed
k-means clustering in 1957 to distribute information within digital signals. He didn’t publish it until 1982. Meanwhile, American statistician Edward Forgy described a similar method in 1965, leading
to its alternative name, the Lloyd-Forgy algorithm.
Finding the center: Consider breaking up the party into like-minded working groups. Given the positions of attendees in the room and the number of groups to be formed, k-means clustering can divide
the attendees into groups of roughly equal size, each gathered around a central point, or centroid.
• During training, the algorithm initially designates k centroids by randomly choosing k people. (K must be chosen manually, and finding an optimal value is not always trivial.) Then it grows k
clusters by associating each person to the closest centroid.
• For each cluster, it computes the mean position of all people assigned to the group and designates the mean position as the new centroid. Each new centroid probably isn’t occupied by a person,
but so what? People tend to gather around the chocolate fondue.
• Having calculated new centroids, the algorithm reassigns individuals to the centroid closest to them. Then it computes new centroids, adjusts clusters, and so on, until the centroids (and the
groups around them) no longer shift. From there, assigning newcomers to the right cluster is easy. Let them take their place in the room and look for the nearest centroid.
• Be forewarned: Given the initial random centroid assignments, you may not end up in the same group as that cute data-centric AI specialist you were hoping to be with. The algorithm does a good
job, but it’s not guaranteed to find the best solution. Better luck at the next party!
Different distances: Of course the distance between clustered objects doesn’t need to be spatial. Any measure between two vectors will do. For instance, rather than grouping partygoers according to
physical proximity, k-means clustering can divide them by their outfits, occupations, or other attributes. Online shops use it to partition customers based on their preferences or behavior, and
astronomers to group stars of the same type.
Power to the data points: The idea has spawned a few notable variations:
• K-medoids use actual data points as centroids rather than mean positions in a given cluster. The medoids are points that minimize the distance to all other points in their cluster. This variation
is more interpretable because the centroids are always data points.
• Fuzzy C-Means Clustering enables the data points to participate in multiple clusters to varying degrees. It replaces hard cluster assignments with degrees of membership depending on distance from
the centroids.
Revelry in n dimensions: Nonetheless, the algorithm in its original form remains widely useful — especially because, as an unsupervised algorithm, it doesn’t require gathering potentially expensive
labeled data. It’s also ever faster to use. For instance, machine learning libraries including scikit-learn benefit from the 2002 addition of kd-trees that partition high-dimensional data extremely
quickly. By the way, if you throw any high-dimensional parties, we’d love to be on the guest list. | {"url":"https://www.deeplearning.ai/the-batch/issue-146/?ref=dl-staging-website.ghost.io","timestamp":"2024-11-11T20:10:12Z","content_type":"text/html","content_length":"148429","record_id":"<urn:uuid:521ba6b9-e17f-4a0b-a427-fe23e3b15ae3>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00789.warc.gz"} |
a decimal
1/290 converted into 0.003 begins with understanding long division and which variation brings more clarity to a situation. Both are used to handle numbers less than one or between whole numbers,
known as integers. Choosing which to use starts with the real life scenario. Fractions are clearer representation of objects (half of a cake, 1/3 of our time) while decimals represent comparison
numbers a better (.333 batting average, pricing: $1.50 USD). So let’s dive into how and why you can convert 1/290 into a decimal.
1/290 is 1 divided by 290
Converting fractions to decimals is as simple as long division. 1 is being divided by 290. For some, this could be mental math. For others, we should set the equation. Fractions have two parts:
Numerators and Denominators. This creates an equation. Now we divide 1 (the numerator) into 290 (the denominator) to discover how many whole parts we have. Here's how our equation is set up:
Numerator: 1
• Numerators are the parts to the equation, represented above the fraction bar or vinculum. Comparatively, 1 is a small number meaning you will have less parts to your equation. 1 is an odd number
so it might be harder to convert without a calculator. Ultimately, having a small value may not make your fraction easier to convert. Now let's explore X, the denominator.
Denominator: 290
• Denominators differ from numerators because they represent the total number of parts which can be found below the vinculum. 290 is a large number which means you should probably use a calculator.
And it is nice having an even denominator like 290. It simplifies some equations for us. Have no fear, large two-digit denominators are all bark no bite. Let's start converting!
How to convert 1/290 to 0.003
Step 1: Set your long division bracket: denominator / numerator
$$ \require{enclose} 290 \enclose{longdiv}{ 1 } $$
We will be using the left-to-right method of calculation. This is the same method we all learned in school when dividing any number against itself and we will use the same process for number
conversion as well.
Step 2: Extend your division problem
$$ \require{enclose} 00. \\ 290 \enclose{longdiv}{ 1.0 } $$
Because 290 into 1 will equal less than one, we can’t divide less than a whole number. So we will have to extend our division problem. Add a decimal point to 1, your numerator, and add an additional
zero. Now 290 will be able to divide into 10.
Step 3: Solve for how many whole groups you can divide 290 into 10
$$ \require{enclose} 00.0 \\ 290 \enclose{longdiv}{ 1.0 } $$
Since we've extended our equation we can now divide our numbers, 290 into 10 (remember, we inserted a decimal point into our equation so we we're not accidentally increasing our solution) Multiply
this number by 290, the denominator to get the first part of your answer!
Step 4: Subtract the remainder
$$ \require{enclose} 00.0 \\ 290 \enclose{longdiv}{ 1.0 } \\ \underline{ 0 \phantom{00} } \\ 10 \phantom{0} $$
If you don't have a remainder, congrats! You've solved the problem and converted 1/290 into 0.003 If there is a remainder, extend 290 again and pull down the zero
Step 5: Repeat step 4 until you have no remainder or reach a decimal point you feel comfortable stopping. Then round to the nearest digit.
Remember, sometimes you won't get a remainder of zero and that's okay. Round to the nearest digit and complete the conversion. There you have it! Converting 1/290 fraction into a decimal is long
division just as you learned in school.
Why should you convert between fractions, decimals, and percentages?
Converting between fractions and decimals depend on the life situation you need to represent numbers. Remember, fractions and decimals are both representations of whole numbers to determine more
specific parts of a number. This is also true for percentages. Though we sometimes overlook the importance of when and how they are used and think they are reserved for passing a math quiz. But 1/290
and 0.003 bring clarity and value to numbers in every day life. Without them, we’re stuck rounding and guessing. Here are real life examples:
When you should convert 1/290 into a decimal
Investments - Comparing currency, especially on the stock market are great examples of using decimals over fractions.
When to convert 0.003 to 1/290 as a fraction
Progress - If we were writing an essay and the teacher asked how close we are to done. We wouldn't say .5 of the way there. We'd say we're half-way there. A fraction here would be more clear and
Practice Decimal Conversion with your Classroom
• If 1/290 = 0.003 what would it be as a percentage?
• What is 1 + 1/290 in decimal form?
• What is 1 - 1/290 in decimal form?
• If we switched the numerator and denominator, what would be our new fraction?
• What is 0.003 + 1/2?
Convert more fractions to decimals
From 1 Numerator From 290 Denominator
What is 1/291 as a decimal? What is 2/290 as a decimal?
What is 1/292 as a decimal? What is 3/290 as a decimal?
What is 1/293 as a decimal? What is 4/290 as a decimal?
What is 1/294 as a decimal? What is 5/290 as a decimal?
What is 1/295 as a decimal? What is 6/290 as a decimal?
What is 1/296 as a decimal? What is 7/290 as a decimal?
What is 1/297 as a decimal? What is 8/290 as a decimal?
What is 1/298 as a decimal? What is 9/290 as a decimal?
What is 1/299 as a decimal? What is 10/290 as a decimal?
What is 1/300 as a decimal? What is 11/290 as a decimal?
What is 1/301 as a decimal? What is 12/290 as a decimal?
What is 1/302 as a decimal? What is 13/290 as a decimal?
What is 1/303 as a decimal? What is 14/290 as a decimal?
What is 1/304 as a decimal? What is 15/290 as a decimal?
What is 1/305 as a decimal? What is 16/290 as a decimal?
What is 1/306 as a decimal? What is 17/290 as a decimal?
What is 1/307 as a decimal? What is 18/290 as a decimal?
What is 1/308 as a decimal? What is 19/290 as a decimal?
What is 1/309 as a decimal? What is 20/290 as a decimal?
What is 1/310 as a decimal? What is 21/290 as a decimal?
Convert similar fractions to percentages
From 1 Numerator From 290 Denominator
2/290 as a percentage 1/291 as a percentage
3/290 as a percentage 1/292 as a percentage
4/290 as a percentage 1/293 as a percentage
5/290 as a percentage 1/294 as a percentage
6/290 as a percentage 1/295 as a percentage
7/290 as a percentage 1/296 as a percentage
8/290 as a percentage 1/297 as a percentage
9/290 as a percentage 1/298 as a percentage
10/290 as a percentage 1/299 as a percentage
11/290 as a percentage 1/300 as a percentage | {"url":"https://www.mathlearnit.com/fraction-as-decimal/what-is-1-290-as-a-decimal","timestamp":"2024-11-07T06:02:54Z","content_type":"text/html","content_length":"33376","record_id":"<urn:uuid:63f784f8-debd-4fb4-b644-d5943b61dc54>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00398.warc.gz"} |
1,576 research outputs found
We have studied the transport properties of a molecular device composed of donor and acceptor moieties between two electrodes on either side. The device is considered to be one-dimensional with
different on-site energies and the non-equilibrium properties are calculated using Landauer's formalism. The current-voltage characteristics is found to be asymmetric with a sharp Negative
Differential Resistance at a critical bias on one side and very small current on the other side. The NDR arises primarily due to the bias driven electronic structure change from one kind of
insulating phase to another through a highly delocalized conducting phase. Our model can be considered to be the simplest to explain the experimental current-voltage characteristics observed in many
molecular devices.Comment: 5 pages, 4 figures (accepted for publication in Physical Review B
By requiring the lower limit for the lightest right-handed neutrino mass, obtained in the baryogenesis from leptogenesis scenario, and a Dirac neutrino mass matrix similar to the up-quark mass matrix
we predict small values for the $u_e$ mass and for the matrix element $m_{ee}$ responsible of the neutrinoless double beta decay, $m_{u_e}$ around $5\cdot10^{-3}$ eV and $m_{ee}$ smaller than $10^
{-3}$ eV, respectively. The allowed range for the mass of the heaviest right-handed neutrino is centered around the value of the scale of B - L breaking in the SO(10) gauge theory with Pati-Salam
intermediate symmetry.Comment: 9 pages, RevTex4. Revised, title change
By invoking the existence of a general custodial O(2) symmetry, a minimal Left-Right symmetric model based on the gauge group G=SU(2)L SU(2)R U(1)BL is shown to require the existence of only two
physical Higgs bosons. The lighter Higgs is predicted to have a small mass which could be evaluated by standard perturbation theory. The fermionic mass matrices are recovered by insertion of ad hoc
fermion-Higgs interactions. The model is shown to be undistinguishable from the standard model at the currently reachable energies.Comment: 1 figure in a separate ps fil
For neutrons bound inside nuclei, baryon instability can manifest itself as a decay into undetectable particles (e.g., $\it n \to u u \bar{u}$), i.e., as a disappearance of a neutron from its nuclear
state. If electric charge is conserved, a similar disappearance is impossible for a proton. The existing experimental lifetime limit for neutron disappearance is 4-7 orders of magnitude lower than
the lifetime limits with detectable nucleon decay products in the final state [PDG2000]. In this paper we calculated the spectrum of nuclear de-excitations that would result from the disappearance of
a neutron or two neutrons from $^{12}$C. We found that some de-excitation modes have signatures that are advantageous for detection in the modern high-mass, low-background, and low-threshold
underground detectors, where neutron disappearance would result in a characteristic sequence of time- and space-correlated events. Thus, in the KamLAND detector [Kamland], a time-correlated triple
coincidence of a prompt signal, a captured neutron, and a $\beta^{+}$ decay of the residual nucleus, all originating from the same point in the detector, will be a unique signal of neutron
disappearance allowing searches for baryon instability with sensitivity 3-4 orders of magnitude beyond the present experimental limits.Comment: 13 pages including 6 figures, revised version, to be
published in Phys.Rev.
The theory of Lie systems has recently been applied to Quantum Mechanics and additionally some integrability conditions for Lie systems of differential equations have also recently been analysed from
a geometric perspective. In this paper we use both developments to obtain a geometric theory of integrability in Quantum Mechanics and we use it to provide a series of non-trivial integrable quantum
mechanical models and to recover some known results from our unifying point of view
The left-right symmetric Pati-Salam model of the unification of quarks and leptons is based on SU(4) and SU(2)xSU(2) groups. These groups are naturally extended to include the classification of
families of quarks and leptons. We assume that the family group (the group which unites the families) is also the SU(4) group. The properties of the 4-th generation of fermions are the same as that
of the ordinary-matter fermions in first three generations except for the family charge of the SU(4)_F group: F=(1/3,1/3,1/3,-1), where F=1/3 for fermions of ordinary matter and F=-1 for the 4-th
generation. The difference in F does not allow the mixing between ordinary and fourth-generation fermions. Because of the conservation of the F charge, the creation of baryons and leptons in the
process of electroweak baryogenesis must be accompanied by the creation of fermions of the 4-th generation. As a result the excess n_B of baryons over antibaryons leads to the excess n_{\nu 4}=N-\bar
N=n_B of neutrinos over antineutrinos in the 4-th generation. This massive fourth-generation neutrino may form the non-baryonic dark matter. In principle their mass density n_{\nu 4}m_N in the
Universe can give the main contribution to the dark matter, since the lower bound on neutrino mass m_N from the data on decay of the Z-bosons is m_N > m_Z/2. The straightforward prediction of this
model leads to the amount of cold dark matter relative to baryons, which is an order of magnitude bigger than allowed by observations. This inconsistency may be avoided by non-conservation of the
F-charge.Comment: 9 pages, 2 figures, version accepted in JETP Letters, corrected after referee reports, references are adde
We give the analytic expressions of maximal probabilities of successfully controlled teleportating an unknown qubit via every kind of tripartite states. Besides, another kind of localizable
entanglement is also determined. Furthermore, we give the sufficient and necessary condition that a three-qubit state can be collapsed to an EPR pair by a measurement on one qubit, and characterize
the three-qubit states that can be used as quantum channel for controlled teleporting a qubit of unknown information with unit probability and with unit fidelity.Comment: 4 page
An investigation of an optimal universal unitary Controlled-NOT gate that performs a specific operation on two unknown states of qubits taken from a great circle of the Bloch sphere is presented. The
deep analogy between the optimal universal C-NOT gate and the `equatorial' quantum cloning machine (QCM) is shown. In addition, possible applications of the universal C-NOT gate are briefly
discussed.Comment: 18 reference
We explore in a model independent way the possibility of achieving the non supersymmetric gauge coupling unification within left-right symmetric models, with the minimal particle content at the
left-right mass scale which could be as low as 1 TeV in a variety of models, and with a unification scale M in the range $10^5$ GeV $< M< 10^{17.7}$ GeV.Comment: 18 pages, Latex file, uses epsf
style, four figures. Submitted for publication to Phys. Rev. D on Oct. 13, 199
This research aimed to find out the improvement of the students motivation learning physic by applying REACT model of Contextual learning.This pre-experimental research used one group pretest and
posttest design. There were 35 students of clas XI IPA I SMA Negeri 7 Pekanbaru have participated in this research. We used modification ARCS question one of motivation as instrument in collecting
data, before and after treatment. The data were analyzed descriptive by using N-Gain. From the data, we found that N-Gain score in high category for all indicators. And, we can conclude that REACT
model have improved students motivation in learning physis especially for class XI IPA I students of SMA Negeri 7 Pekanbaru | {"url":"https://core.ac.uk/search/?q=authors%3A(Pati%2C%20F)","timestamp":"2024-11-06T21:12:52Z","content_type":"text/html","content_length":"185734","record_id":"<urn:uuid:3444aec3-5874-40e3-a29a-b670c0a90c51>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00071.warc.gz"} |
Brownie Points I
Problem B
Brownie Points I
Stan and Ollie play the game of Odd Brownie Points. Some brownie points are located in the plane, at integer coordinates. Stan plays first and places a vertical line in the plane. The line must go
through a brownie point and may cross many (with the same $x$-coordinate). Then Ollie places a horizontal line that must cross a brownie point already crossed by the vertical line.
Those lines divide the plane into four quadrants. The quadrant containing points with arbitrarily large positive coordinates is the top-right quadrant.
The players score according to the number of brownie points in the quadrants. If a brownie point is crossed by a line, it does not count. Stan gets a point for each (uncrossed) brownie point in the
top-right and bottom-left quadrants. Ollie gets a point for each (uncrossed) brownie point in the top-left and bottom-right quadrants.
Your task is to compute the scores of Stan and Ollie given the point through which they draw their lines.
Input contains a number of test cases, at most $15$. The data of each test case appear on a sequence of input lines. The first line of each test case contains a positive odd integer $1 \le n \le 20\,
000$ which is the number of brownie points. Each of the following $n$ lines contains two integers, the horizontal ($x$) and vertical ($y$) coordinates of a brownie point. All coordinates are at most
$10^6$ in absolute value. No two brownie points occupy the same place. The input ends with a line containing $0$ (instead of the $n$ of a test).
For each test case of input, print a line with two numbers separated by a single space. The first number is Stan’s score, the second is the score of Ollie when their lines cross the point whose
coordinates are given at the center of the input sequence of points for this case.
Sample Input 1 Sample Output 1
2 -2
1 -3 6 3
-3 -3
-3 -2
-3 -4
3 -7 | {"url":"https://open.kattis.com/contests/cupsex/problems/browniepoints","timestamp":"2024-11-06T18:14:21Z","content_type":"text/html","content_length":"31162","record_id":"<urn:uuid:ed385f39-291e-4c9f-928f-035947db58a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00361.warc.gz"} |
Кафедра Физики Высоких Энергий и Элементарных Частиц Отправка письма
There are three different formulations of Quantum mechanics:
canonical quantization (or geometrical quantization), star product
quantization and path integral quantization. It is well known that the
first one manifests the ordering ambiguity: to properly define the
model it is not sufficient to produce a Hamiltonian, also one must fix
the so-called normal ordering, the way to pass from coordinates at the
phase space to the operators. Different choice of ordering can be
compensated by quantum corrections to the Hamiltonian, therefore a
kind of "gauge symmetry" present. This symmetry also appears when one
uses the star-product approach. However, in the path integral approach
it is hidden.
Our suggestion is that this ambiguity appears in the definition of the
classical action. Usually the Lagrangian is a (polynomial) function of
coordinates and velocities. But functions we integrate on are
non-differentiable (w.r.t. time). Therefore we need to define a proper
extension of the classical action. It can be done with the help of
stochastic calculus and is not unique - this is here where the
ordering ambiguity appears.
We consider several examples which come from financial word: the
stochastic calculus is conventional tool to study the financial market
properties. We translate this toolkit to the path integral language
and will see in details how ordering ambiguity (known in the
stochastic calculus community as Ito and Stratonovich integrals)
appears. Also we will see why different ordering schemes are natural
for Quantum Mechanics and Probability Theory (Wick rotation is not,
therefore, enough to pass from quantum mechanical to thermodynamical | {"url":"http://hep.phys.spbu.ru/science/seminars/annotation_r.php?0019_e.txt","timestamp":"2024-11-06T05:35:22Z","content_type":"text/html","content_length":"7713","record_id":"<urn:uuid:99c78caa-2c99-4a56-8cb9-25fab02c9f55>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00735.warc.gz"} |
Regular polygon
Rom Pinchasi
Let F be a family of n pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by F is at most 2n - 2. This bound is tight.
Furthermore, if no two circles in F touch, then the geometr ...
Electronic Journal Of Combinatorics | {"url":"https://graphsearch.epfl.ch/en/concept/333306","timestamp":"2024-11-14T04:32:11Z","content_type":"text/html","content_length":"126580","record_id":"<urn:uuid:73fd562f-98d8-4646-8fd9-ac29e96ab91f>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00526.warc.gz"} |
How do you want to input your function?
The Boolean Bot can build truth tables, minimize logical expressions and simulate boolean functions. It uses the Quine-McCluskey algorithm and Petrick's method to find the optimal sum of products
form of the function.
Enter the expression:
Grouping Not And Xor Or
(ab) a' ab a^b a+b
Enter a comma separated list of min terms:
Enter a comma separated list of don't cares:
Tip: You can specify a range (e.g. 0, 2, 5-10, 13)
Enter a comma separated list of input variables:
Example: a, b, c, d
Click a row to toggle between 0, 1 and X:
Min Terms
Minimized Expression(s) | {"url":"http://booleanbot.com/","timestamp":"2024-11-14T10:32:25Z","content_type":"application/xhtml+xml","content_length":"3929","record_id":"<urn:uuid:e246bb13-fe95-4815-8c76-e4b250f4e3a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00140.warc.gz"} |
The wrong way of benchmarking the most efficient integer comparison function - The Old New Thing
On StackOverflow, there’s a question about the most efficient way to compare two integers and produce a result suitable for a comparison function, where a negative value means that the first value is
smaller than the second, a positive value means that the first value is greater than the second, and zero means that they are equal.
There was much microbenchmarking of various options, ranging from the straightforward
int compare1(int a, int b)
if (a < b) return -1;
if (a > b) return 1;
return 0;
to the clever
int compare2(int a, int b)
return (a > b) - (a < b);
to the hybrid
int compare3(int a, int b)
return (a < b) ? -1 : (a > b);
to inline assembly
int compare4(int a, int b)
__asm__ __volatile__ (
"sub %1, %0 \n\t"
"jno 1f \n\t"
"cmc \n\t"
"rcr %0 \n\t"
"1: "
: "+r"(a)
: "r"(b)
: "cc");
return a;
The benchmark pitted the comparison functions against each other by comparing random pairs of numbers and adding up the results to prevent the code from being optimized out.
But here’s the thing: Adding up the results is completely unrealistic.
There are no meaningful semantics that could be applied to a sum of numbers for which only the sign is significant. No program that uses a comparison function will add the results. The only thing you
can do with the result is compare it against zero and take one of three actions based on the sign.
Adding up all the results means that you’re not using the function in a realistic way, which means that your benchmark isn’t realistic.
Let’s try to fix that. Here’s my alternative test:
// Looks for "key" in sorted range [first, last) using the
// specified comparison function. Returns iterator to found item,
// or last if not found.
template<typename It, typename T, typename Comp>
It binarySearch(It first, It last, const T& key, Comp compare)
// invariant: if key exists, it is in the range [first, first+length)
// This binary search avoids the integer overflow problem
// by operating on lengths rather than ranges.
auto length = last - first;
while (length > 0) {
auto step = length / 2;
It it = first + step;
auto result = compare(*it, key);
if (result < 0) {
first = it + 1;
length -= step + 1;
} else if (result == 0) {
return it;
} else {
length = step;
return last;
int main(int argc, char **argv)
// initialize the array with sorted even numbers
int a[8192];
for (int i = 0; i < 8192; i++) a[i] = i * 2;
for (int iterations = 0; iterations < 1000; iterations++) {
int correct = 0;
for (int j = -1; j < 16383; j++) {
auto it = binarySearch(a, a+8192, j, COMPARE);
if (j < 0 || j > 16382 || j % 2) correct += it == a+8192;
else correct += it == a + (j / 2);
// if correct != 16384, then we have a bug somewhere
if (correct != 16384) return 1;
return 0;
Let’s look at the code generation for the various comparison functions. I used gcc.godbolt.org with x86-64 gcc 7.2 and optimization -O3.
If we try compare1, then the binary search looks like this:
; on entry, esi is the value to search for
lea rdi, [rsp-120] ; rdi = first
mov edx, 8192 ; edx = length
jmp .L9
.L25: ; was greater than
mov rdx, rax ; length = step
test rdx, rdx ; while (length > 0)
jle .L19
mov rax, rdx ;
sar rax, 1 ; eax = step = length / 2
lea rcx, [rdi+rax*4] ; it = first + step
; result = compare(*it, key), and then test the result
cmp dword ptr [rcx], esi ; compare(*it, key)
jl .L11 ; if less than
jne .L25 ; if not equal (therefore if greater than)
... return value in rcx ; if equal, answer is in rcx
.L11: ; was less than
add rax, 1 ; step + 1
lea rdi, [rcx+4] ; first = it + 1
sub rdx, rax ; length -= step + 1
test rdx, rdx ; while (length > 0)
jg .L9
lea rcx, [rsp+32648] ; rcx = last
... return value in rcx
Exercise: Why is rsp - 120 the start of the array?
Observe that despite using the lamest, least-optimized comparison function, we got the comparison-and-test code that is much what we would have written if we had done it in assembly language
ourselves: We compare the two values, and then follow up with two branches based on the same shared flags. The comparison is still there, but the calculation and testing of the return value are gone.
In other words, not only was compare1 optimized down to one cmp instruction, but it also managed to delete instructions from the binarySearch function too. It had a net cost of negative instructions!
What happened here? How did the compiler manage to optimize out all our code and leave us with the shortest possible assembly language equivalent?
Simple: First, the compiler did some constant propagation. After inlining the compare1 function, the compiler saw this:
int result;
if (*it < key) result = -1;
else if (*it > key) result = 1;
else result = 0;
if (result < 0) {
... less than ...
} else if (result == 0) {
... equal to ...
} else {
... greater than ...
The compiler realized that it already knew whether constants were greater than, less than, or equal to zero, so it could remove the test against result and jump straight to the answer:
int result;
if (*it < key) { result = -1; goto less_than; }
else if (*it > key) { result = 1; goto greater_than; }
else { result = 0; goto equal_to; }
if (result < 0) {
... less than ...
} else if (result == 0) {
... equal to ...
} else {
... greater than ...
And then it saw that all of the tests against result were unreachable code, so it deleted them.
int result;
if (*it < key) { result = -1; goto less_than; }
else if (*it > key) { result = 1; goto greater_than; }
else { result = 0; goto equal_to; }
... less than ...
goto done;
... equal to ...
goto done;
... greater than ...
That then left result as a write-only variable, so it too could be deleted:
if (*it < key) { goto less_than; }
else if (*it > key) { goto greater_than; }
else { goto equal_to; }
... less than ...
goto done;
... equal to ...
goto done;
... greater than ...
Which is equivalent to the code we wanted all along:
if (*it < key) {
... less than ...
} else if (*it > key) {
... greater than ...
} else {
... equal to ...
The last optimization is realizing that the test in the else if could use the flags left over by the if, so all that was left was the conditional jump.
Some very straightforward optimizations took our very unoptimized (but easy-to-analyze) code and turned it into something much more efficient.
On the other hand, let’s look at what happens with, say, the second comparison function:
; on entry, edi is the value to search for
lea r9, [rsp-120] ; r9 = first
mov ecx, 8192 ; ecx = length
jmp .L9
.L11: ;
test eax, eax ; result == 0?
je .L10 ; Y: found it
; was greater than
mov rcx, rdx ; length = step
test rcx, rcx ; while (length > 0)
jle .L19
mov rdx, rcx
xor eax, eax ; return value of compare2
sar rdx, 1 ; rdx = step = length / 2
lea r8, [r9+rdx*4] ; it = first + step
; result = compare(*it, key), and then test the result
mov esi, dword ptr [r8] ; esi = *it
cmp esi, edi ; compare *it with key
setl sil ; sil = 1 if less than
setg al ; al = 1 if greater than
; eax = 1 if greater than
movzx esi, sil ; esi = 1 if less than
sub eax, esi ; result = (greater than) - (less than)
cmp eax, -1 ; less than zero?
jne .L11 ; N: Try zero or positive
; was less than
add rdx, 1 ; step + 1
lea r9, [r8+4] ; first = it + 1
sub rcx, rdx ; length -= step + 1
test rcx, rcx ; while (length > 0)
jg .L9
lea r8, [rsp+32648] ; r8 = last
... return value in r8
The second comparison function compare2 uses the relational comparison operators to generate exactly 0 or 1. This is a clever way of generating -1, 0, or +1, but unfortunately, that was not our goal
in the grand scheme of things. It was merely a step toward that goal. The way that compare2 calculates the result is too complicated for the optimizer to understand, so it just does its best at
calculating the formal return value from compare2 and testing its sign. (The compiler does realize that the only possible negative value is -1, but that’s not enough insight to let it optimize the
entire expression away.)
If we try compare3, we get this:
; on entry, esi is the value to search for
lea rdi, [rsp-120] ; rdi = first
mov ecx, 8192 ; ecx = length
jmp .L12
.L28: ; was greater than
mov rcx, rax ; length = step
mov rax, rcx
sar rax, 1 ; rax = step = length / 2
lea rdx, [rdi+rax*4] ; it = first + step
; result = compare(*it, key), and then test the result
cmp dword ptr [rdx], esi ; compare(*it, key)
jl .L14 ; if less than
jle .L13 ; if less than or equal (therefore equal)
; "length" is in eax now
.L15: ; was greater than
test eax, eax ; length == 0?
jg .L28 ; N: continue looping
lea rdx, [rsp+32648] ; rdx = last
... return value in rdx
.L14: ; was less than
add rax, 1 ; step + 1
lea rdi, [rdx+4] ; first = it + 1
sub rcx, rax ; length -= step + 1
mov rax, rcx ; rax = length
jmp .L15
The compiler was able to understand this version of the comparison function: It observed that if a < b, then the result of compare3 is always negative, so it jumped straight to the less-than case.
Otherwise, it observed that the result was zero if a is not greater than b and jumped straight to that case too. The compiler did have some room for improvement with the placement of the basic
blocks, since there is an unconditional jump in the inner loop, but overall it did a pretty good job.
The last case is the inline assembly with compare4. As you might expect, the compiler had the most trouble with this.
; on entry, edi is the value to search for
lea r8, [rsp-120] ; r8 = first
mov ecx, 8192 ; ecx = length
jmp .L12
.L14: ; zero or positive
je .L13 ; zero - done
; was greater than
mov rcx, rdx ; length = step
test rcx, rcx ; while (length > 0)
jle .L22
mov rdx, rcx
sar rdx, 1 ; rdx = step = length / 2
lea rsi, [r8+rdx*4] ; it = first + step
; result = compare(*it, key), and then test the result
mov eax, dword ptr [rsi] ; eax = *it
sub eax, edi
jno 1f
rcr eax, 1
test eax, eax ; less than zero?
jne .L14 ; N: Try zero or positive
; was less than
add rdx, 1 ; step + 1
lea r8, [rsi+4] ; first = it + 1
sub rcx, rdx ; length -= step + 1
test rcx, rcx ; while (length > 0)
jg .L12
lea rsi, [rsp+32648] ; rsi = last
... return value in rsi
This is pretty much the same as compare2: The compiler has no insight at all into the inline assembly, so it just dumps it into the code like a black box, and then once control exits the black box,
it checks the sign in a fairly efficient way. But it had no real optimization opportunities because you can’t really optimize inline assembly.
The conclusion of all this is that optimizing the instruction count in your finely-tuned comparison function is a fun little exercise, but it doesn’t necessarily translate into real-world
improvements. In our case, we focused on optimizing the code that encodes the result of the comparison without regard for how the caller is going to decode that result. The contract between the two
functions is that one function needs to package some result, and the other function needs to unpack it. But we discovered that the more obtusely we wrote the code for the packing side, the less
likely the compiler would be able to see how to optimize out the entire hassle of packing and unpacking in the first place. In the specific case of comparison functions, it means that you may want to
return +1, 0, and -1 explicitly rather than calculating those values in a fancy way, because it turns out compilers are really good at optimizing “compare a constant with zero”.
You have to see how your attempted optimizations fit into the bigger picture because you may have hyper-optimized one part of the solution to the point that it prevents deeper optimizations in other
parts of the solution.
Bonus chatter: If the comparison function is not inlined, then all of these optimization opportunities disappear. But I personally wouldn’t worry about it too much, because if the comparison function
is not inlined, then the entire operation is going to be dominated by the function call overhead: Setting up the registers for the call, making the call, returning from the call, testing the result,
and most importantly, the lost register optimization opportunities not only because the compiler loses opportunities to enregister values across the call, but also because the compiler has to protect
against the possibility that the comparison function will mutate global state and consequently create aliasing issues.
0 comments
Discussion are closed. | {"url":"https://devblogs.microsoft.com/oldnewthing/20171117-00/?p=97416","timestamp":"2024-11-14T21:17:38Z","content_type":"text/html","content_length":"190574","record_id":"<urn:uuid:202f823e-f4aa-492a-848b-38ade07977bf>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00309.warc.gz"} |
Numerical Analysis of the Influence of Geometry on a Large Scale Onshore Oscillating Water Column Device with Associated Seabed Ramp
Initially, it was numerically analyzed the performance of the reference case in which there is no ramp below the OWC chamber, being A[2] = H[2]/L[2] = 0 (see Figure 1). It was evaluated the H[1]/L[1]
effect over the available power of the device with H[3] = 2.5 m. Results are shown for the mass flow rate (Figure 7a) and pressure variation (Figure 7b) of air along time. Figure 7 presents three
values of the H[1]/L[1] ratio, representing the geometric extremes (H[1]/L[1] = 0.2 and 5.0) and the geometric configuration that achieved the maximum available power ((H[1]/L[1])[o] = 0.4).
From Figure 7a, a general symmetric oscillating behavior can be identified in mass flow rate for three H[1]/L[1] presented conditions with straight convergence in time, i.e., the results are in
phase. It is also observed that the H[1]/L[1] = 5.0 case achieved the worse results once the mass flow rate is integrally inferior to two other cases, for example, almost two times less than the best
case at instant t = 57.5 s. In this way, it is noticed that H[1]/L[1] = 0.2 and 0.4 presented similar comportment. However, H[1]/L[1] = 0.4 has a specific performance superiority. Therefore, results
indicate that higher chambers hinder that the airflow achieves the turbine-duct coupled to the hydro-pneumatic chamber.
(a) Mass flow rate
(b) Pressure variation
Figure 7. Transient results of the reference case (H[2]/L[2] = 0) with H[1]/L[1] = 0.2, 0.4, and 5.0 for H[3] = 2.5 m
Regarding the pressure variation analysis, Figure 7b shows an asymmetric oscillating behavior with more significant magnitudes when the mass flow rate is positive (flowing in the positive x
-direction, outward the chamber). That probably occurs because the airflow meets atmospheric pressure when flowing in the positive x-direction (outward), which offers less resistance; since the
airflow in the negative x-direction (flow back) is submitted to the more elevated internal chamber pressure. Besides, Figure 7 indicates that the mass flow rate and pressure variation seem to present
a similar response to geometry changes. It is observed that geometric configuration with (H[1]/L[1])[o] = 0.4 also offers more considerable pressure variation inside the duct after the first incident
wave (after t = 47 s).
Figure 8a presents RMS available power (P[RMS]) concerning time for the same cases of Figure 7, still considering the no ramp reference case. Hence, the behavior follows the pressure variation aspect
above mentioned (see Figure 7b), with higher peaks according to outward mass flow rate, as well as geometry (H[1]/L[1])[o] = 0.4 achieving the best performance while H[1]/L[1] = 0.5 worsened the
performance of the OWC converter.
In its turn, Figure 8b plots average available power of all studied H[1]/L[1] ratios (0.2; 0.4; 0.8; 1.0; 1.5; 2.0; 3.0; and 5.0). One can note that (H[1]/L[1])[o] = 0.4 achieved the best
performance, generating a maximized available power of P[m] = 240.88 W; while geometry H[1]/L[1] = 0.2 achieved P = 206.13 W (17% difference) and geometry H[1]/L[1] = 5.0 got P = 31.34 W being the
worst performance.
(a) Transient RMS available power
(b) Time-averaged RMS available power
Figure 8. Results of the reference case (H[2]/L[2] = 0) for different ratios of H[1]/L[1] and H[3] = 2.5 m
To evaluate the effect of setting a ramp below the OWC chamber, Figures 9a and 9b compare mass flow rate and pressure variation between reference case (without ramp, A[2] = H[2]/L[2] = 0) and Figure
1 described case (with A[2] = 40 m² and H[2]/L[2] = 0.8). The other degrees of freedom adopted for this comparison are H[1]/L[1] = 0.2 and H[3] = 2.5 m.
From Figure 9, it is possible to observe that ramp usage promotes a magnitude increase in both operational parameters (m ̇and ∆p). Besides, general graphs aspects are very similar to those presented
in Figure 7, where higher magnitudes were found at outward airflow instants. After that, the ramp's effect over the converter RMS available power along time can be viewed in Figure 10. Figure 10
indicates an improvement of the RMS available power due to ramp usage. Moreover, more significant peaks are identified at outward airflow instants as well.
Based on these preliminary results, it is relevant to investigate the influence of OWC chamber geometry (H[1]/L[1]) over its average available power when associated with different seabed ramp
configurations (H[2]/L[2]). These results are plotted in Figure 11, considering H[3] = 2.5 m. The reference case results, earlier presented in Figure 8b, are also plotted in Figure 11, aiming to help
the discussion.
One can infer in Figure 11 that the maximum values for the OWC available power were always reached for a specific value of H[1]/L[1] ratio, independently of H[2]/L[2]. Hence, its once optimized value
is (H[1]/L[1])[o] = 0.4, being the same configuration that optimizes the reference case. The H[2]/L[2] effect analysis of Figure 11 shows that ramp insertion benefits the device with more meaningful
average power for all H[1]/L[1] cases compared with the reference case. Additionally, power increment for H[2]/L[2] > 0.2 occurs significantly around optimal geometry, i.e., (H[1]/L[1])[o] ≈ 0.4. It
is also noticed that higher H[1]/L[1] values (H[1]/L[1] > 1.5) present very close performances regardless of the H[2]/L[2] ratio. Therefore, for H[3] = 2.5 m, the ramp showed more effectiveness in
the optimal region of the ratio H[1]/L[1].
(a) Mass flow rate
(b) Pressure variation
Figure 9. Comparison between transient results of reference case and the case with H[2]/L[2] = 0.8, considering H[1]/L[1] = 0.2 and H[3] = 2.5 m
Figure 10. Comparison between transient results of the reference case and the case with H[2]/L[2] = 0.8, considering H[1]/L[1] = 0.2 and H[3] = 2.5 m for RMS available power
Figure 11. Effect of H[1]/L[1] over available power for different H[2]/L[2] ratios and H[3] = 2.5 m
Still looking at Figure 11, in a general way, the increase of the H[2]/L[2] ratio promotes an augmentation in OWC available power. Therefore, it is possible to define the twice optimized chamber
geometry and the once optimized ramp geometry given, respectively, by (H[1]/L[1])[oo] = 0.4 and (H[2]/L[2])[o] = 0.8; reaching a twice maximized average power of P[mm]= 331.57 W. If this value is
compared with the maximum average power for the reference case (P[m] = 240.88 W), an improvement of 37.7% was achieved.
Beyond evaluated operational work parameters, it was also presented phase fraction and velocity fields. It was adopted the time instant of 68 s, the optimized OWC chamber geometry of (H[1]/L[1])[o] =
0.4, and the submergence of frontal plate chamber of H[3] = 2.5 m. In addition, three values for the seabed ramp geometry were considered: H[2]/L[2] = 0.05 (Figure 12), H[2]/L[2] = 0.2 (Figure 13),
and (H[2]/L[2])[oo] = 0.8 (Figure 14).
(a) Phase fraction field (b) Velocity field
Figure 12. Geometric configuration with (H[1]/L[1])[o] = 0.4, H[2]/L[2] = 0.05, and H[3] = 2.5 m for t = 68.0 s
Figures 12a, 13a, and 14a (where red and blue colors represent, respectively, water and air phases) show a wave crest interacting with the optimized OWC configuration, causing the water level
elevation inside the hydro-pneumatic chamber, which forces airflow passage outward the turbine-duct. One can observe this airflow in the velocity fields of Figures 12b, 13b, and 14b, respectively. It
is also possible to notice that the H[2]/L[2] ratio increases promote an increase of the velocity magnitude, showing a consistent behavior with previous analyzes about power performance.
The presented results so far were obtained for H[3] = 2.5 m. Now, these same analyses were carried out using an OWC frontal wall with H[3] = 5.0 m, aiming to identify how this geometric parameter
influences the fluid-dynamic behavior of the converter associated with the seabed ramp. Therefore, Figure 15 plots the H[1]/L[1] ratio's effect over average available power for all tested H[2]/L[2]
ratios, considering H[3] = 5.0 m.
As depicted in Figure 15, for the reference case (H[2]/L[2] = 0), the H[1]/L[1] effect over P is similar to the effect over the reference case with H[3] = 2.5 m (see Figure 11), but here a lower
value for the maximized available power is obtained. Therefore, the optimal chamber configuration suffers the influence of the front wall magnitude, being observed that the reference case achieves a
maximum power of 145.10 W, i.e., 39.7% lower than the best case with H[3] = 2.5 m. Moreover, the optimized H[1]/L[1] ratio is changed from 0.4 (for H[3] = 2.5 m) to 1.0 when H[3] = 5.0 m is applied.
(a) Phase fraction field (b) Velocity field
Figure 13. Geometric configuration with (H[1]/L[1])[o] = 0.4, H[2]/L[2] = 0.2, and H[3] = 2.5 m for t = 68.0 s
(a) Phase fraction field (b) Velocity field
Figure 14. Geometric configuration with (H[1]/L[1])[o] = 0.4, (H[2]/L[2])[oo] = 0.8, and H[3] = 2.5 m for t = 68.0 s
Figure 15. Effect of H[1]/L[1] over available power for different H[2]/L[2] ratios and H[3] = 5.0 m
(a) Phase fraction field (b) Velocity field
Figure 16. Geometric configuration with H[1]/L[1] = 0.8, H[2]/L[2] = 0.05, and H[3] = 5.0 m for t = 68.0 s
Figure 15 also indicates that most of the analyzed cases with a ramp obtained worse available power response than the reference case without a ramp. Considering the maximum available power of
reference case, one can note that the only configurations that improve the OWC performance due to seabed ramp usage are composed of H[2]/L[2] = 0.8 in the range 0.2 £ H[1]/L[1] £ 0.8. It is important
to mention that the H[4] < (h-H[3]) constraint (as described in section 2.3) restricted interval values of H[1]/L[1] for H[2]/L[2] > 0.1. Moreover, it is observed that there is no global H[1]/L[1]
optimal ratio. However, twice maximized power (P[m m] = 163.42 W) is achieved for (H[1]/L[1])[oo] = 0.4 and (H[2]/L[2])[o] = 0.8 and, being nearly three times superior to obtained power with the
worst geometry. Lastly, the twice maximized power obtained with H[3] = 5.0 m is 50.7 % lower than the previously obtained twice maximized available power considering H[3] = 2.5 m (see Figure 11),
what means that increasing the vertical dimension of the OWC frontal wall conducted to poor performance for the presented conditions.
To understand the fluid-dynamic behavior of the OWC converter having a frontal plate with H[3] = 5.0 m, the phase fraction and velocity fields were illustrated. Again, the time instant of 68.0 s was
adopted. Figure 16 presents the geometric configuration with H[1]/L[1] = 0.8 and H[2]/L[2] = 0.05; while Figure 17 shows the optimized geometry with (H[1]/L[1])[o] = 0.4 and (H[2]/L[2])[oo] = 0.8.
(a) Phase fraction field (b) Velocity field
Figure 17. Geometric configuration with (H[1]/L[1])[o] = 0.4, (H[2]/L[2])[oo] = 0.8, and H[3] = 5.0 m for t = 68.0 s
(a) H[3] = 2.5 m
(b) H[3] = 5.0 m
Figure 18. Effect of H[2]/L[2] ratio over once maximized available power (P[m]) and once optimized height/length chamber ratio (H[1]/L[1])[o]
The same trend is perceived in Figures 12, 13, and 14, and it can also be observed in Figures 16 and 17. Therefore, in a general way, the results show that longer ramps do not benefit the OWC
purpose, causing hydro-pneumatic effect loss (see Figure 16). On the other hand, when the seabed ramp is smaller than the device chamber, the water level elevations are intensified, and airflow
velocity magnitude increases inside the turbine-duct (see Figure 17).
Optimal results plotted in Figures 11 and 15 for both front wall submersions (H3 = 2.5 m and H3 = 5.0 m) are summarized in Figures 18a and 18b, respectively. More precisely, Figure 18a depicts the
effect of H2/L2 ratio over once maximized power (Pm) and once optimized (H1/L1)o ratio for H3 = 2.5 m, while Fig 16b depicts the same effect for H3 = 5.0 m.
From Figure 18a, it is observed that optimal chamber configuration is not affected by the ramp's configuration, i.e., the optimal H[1]/L[1] ratio is constant ((H[1]/L[1])[o ]= 0.4) for all tested
values of H[2]/L[2]. Moreover, as earlier mentioned, the twice maximized average available power of P[mm]= 331.57 W can be identified with twice optimized OWC chamber geometry of (H[1]/L[1])[oo] =
0.4 and once optimized ramp geometry of (H[2]/L[2])[o] = 0.8 for H[3] = 2.5 m.
Otherwise, Figure 18b shows that optimal chamber geometry varies significantly due to ramp configuration, achieving maximum local performance for the reference case (H[2]/L[2] = 0.0). However, for H
[3] = 5.0 m the maximum global performance of P[mm]= 163.42 W is also obtained when (H[1]/L[1])[oo] = 0.4 and (H[2]/L[2])[o] = 0.8.
As in the present study, only two values of H[3] were adopted, it is not possible to indicate an optimized value for this degree of freedom. However, results obtained here indicated that the wall
length has a strong influence over the device performance and the effect of other degrees of freedom (H[1]/L[1] and H[2]/L[2]) over the available power. | {"url":"https://iieta.org/journals/ijdne/paper/10.18280/ijdne.150613","timestamp":"2024-11-03T17:04:39Z","content_type":"text/html","content_length":"135273","record_id":"<urn:uuid:73f41839-aee7-42e2-b4cb-789289c64c03>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00631.warc.gz"} |
Measures of Spread
An important concept in statistics is measures of spread or variation. A basic numerical description of a data set requires both measures of central tendency and spread. For example, say that you
know that the average, or mean, income in a certain area is $30,000. Taking this number alone, it could just as easily describe an area where incomes are spread out fairly evenly between $25,000 and
$35,000, as an area where most of the incomes are $10,000 and a few exceptionally hard-working and/or very lucky people have an income of $1,000,000. This is where measures of spread come in.
Just as there are several measures (mean, median, mode) used to measure central tendency, there are also several measures of spread, including:
• Range, the difference between the lowest and highest values in a sample,
• Quartiles, representing values where 25%, 50%, or 75% of the data are below that value
• Variance, which describes how great a spread exists in the data
• Standard deviation, which describes the spread of data in relation to the mean. | {"url":"https://mathlair.allfunandgames.ca/variation.php","timestamp":"2024-11-12T16:56:17Z","content_type":"text/html","content_length":"3190","record_id":"<urn:uuid:28691f91-e1bf-4617-bce8-15d3073e0bbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00678.warc.gz"} |
How Widely Should We Draw The Circle?
For fifteen years, popular-science readers have gotten used to breathless claims about commercial quantum computers just around the corner. As far as I can tell, though, 2015 marked a turning point.
For the first time, the most hard-nosed experimentalists are talking about integrating 40 or more high-quality quantum bits (“qubits”) into a small programmable quantum computer—not in the remote
future, but in the next few years. If built, such a device will probably still be too small to do anything useful, but I honestly don’t care.
The point is, forty qubits are enough to do something that computer scientists are pretty sure would take trillions of steps to simulate using today’s computers. They’ll suffice to disprove the
skeptics, to show that nature really does put this immense computing power at our disposal—just as the physics textbooks implicitly predicted since the late 1920s. (And if quantum computing turns out
not be possible, for some deep reason? To me that’s unlikely, but even more exciting, since it would mean a revolution in physics.)
So then, is imminent quantum supremacy the “most interesting recent [scientific] news”? I can’t say that with any confidence. The trouble is, which news we find interesting depends on how widely we
draw the circle about our own hobbyhorses. And some days, quantum computing seems to me to fade into irrelevance next to the precarious state of the earth. Perhaps when people look back a century
from now, they’ll say that the most important science news of 2015 was that the West Antarctic Ice Sheet was found to be closer to collapse than even the alarmists predicted. Or, just possibly,
they’ll say the most important news was that in 2015, the “AI risk” movement finally went mainstream.
This movement posits that superhuman artificial intelligence is likely to be built within the next century, and that the biggest problem facing humanity today is to ensure that, when the AI arrives,
it will be “friendly” to human values (rather, than, say, razing the solar system for more computing power to serve its inscrutable ends). I like to tease my AI-risk friends that I’ll be more worried
about the impending AI singularity when my Wi-Fi stays working for more than a week. But who knows? At least this scenario, if it panned out, would render the melting glaciers pretty much irrelevant.
Instead of expanding my “circle of interest” to encompass the future of civilization, I could also contract it more tightly, around my fellow theoretical computer scientists. In that case, 2015 was
the year that Lazslo Babai of the University of Chicago announced the first “provably fast” algorithm for one of the central problems in computing: graph isomorphism. This problem is to determine
whether two networks of nodes and links are “isomorphic” (that is, whether they become the same if you relabel the nodes). For networks with N nodes, the best previous algorithm—which Babai also
helped to discover, thirty years ago—took a number of steps that grew exponentially with the square root of N.
The new algorithm takes a number of steps that grows exponentially with a power of log(N) (a rate that’s called “quasi-polynomial”). Babai’s breakthrough probably has no applications, since the
existing algorithms were already perfectly fast for any networks that would ever arise in practice. But for those who are motivated by an unquenchable thirst to know the ultimate limits of
computation, this is arguably the biggest news so far of the twenty-first century.
Drawing the circle even more tightly, in “quantum query complexity”—a tiny subfield of quantum computing that I cut my teeth on as a student—it was discovered this year that there are Boolean
functions that a quantum computer can evaluate in less than the square root of the number of input accesses that a classical computer needs, a gap that had stood as the record since 1996. Even if
useful quantum computers are built, this result will have zero applications, since the functions that achieve this separation are artificial monstrosities, constructed only to prove the point. But it
excited me: it told me that progress is possible, that the seemingly-eternal puzzles that drew me into research as a teenager do occasionally get solved. So damned if I’m not going to tell you about
At a time when the glaciers are melting, how can I justify getting excited about a new type of computer that will be faster for certain specific problems—let alone about an artificial function for
which the new type of computer gives you a slightly bigger advantage? The “obvious” answer is that basic research could give us new tools with which to tackle the woes of civilization, as it’s done
many times before. Indeed, we don’t need to go as far as an AI singularity to imagine how.
By letting us simulate quantum physics and chemistry, quantum computers might spark a renaissance in materials science, and allow (for example) the design of higher-efficiency solar panels. For me,
though, the point goes beyond that, and has to do with the dignity of the human race. If, in millions of years, aliens come across the ruins of our civilization and dig up our digital archives, I’d
like them to know that before humans killed ourselves off, we at least managed to figure out that the graph isomorphism problem is solvable in quasi-polynomial time, and that there exist Boolean
functions with super-quadratic quantum speedups. So I’m glad to say that they will know these things, and that now you do too. | {"url":"https://stage.edge.org/response-detail/26637","timestamp":"2024-11-11T15:03:49Z","content_type":"application/xhtml+xml","content_length":"53606","record_id":"<urn:uuid:cf6e30a3-1f11-41c9-9c05-63412c894e06>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00508.warc.gz"} |
Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after
Patrice Lumumba (RUDN University)2752910.22363/2658-4670-2021-29-3-251-259Research ArticleAsymptotically accurate error estimates of exponential convergence for the trapezoidal ruleBelovAleksandr A.
<p>Candidate of Physical and Mathematical Sciences, Assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University);
Researcher of Faculty of Physics, M.V. Lomonosov Moscow State University</p>aa.belov@physics.msu.ruhttps://orcid.org/0000-0002-0918-9263KhokhlachevValentin S.<p>Master’s degree student of
Faculty of Physics</p>valentin.mycroft@yandex.ruhttps://orcid.org/0000-0002-6590-5914M.V. Lomonosov Moscow State UniversityPeoples’ Friendship University of Russia (RUDN University)
3009202129325125930092021Copyright © 2021, Belov A.A., Khokhlachev V.S.2021<p style="text-align: justify;">In many applied problems, efficient calculation of quadratures with high accuracy is
required. The examples are: calculation of special functions of mathematical physics, calculation of Fourier coefficients of a given function, Fourier and Laplace transformations, numerical solution
of integral equations, solution of boundary value problems for partial differential equations in integral form, etc. For grid calculation of quadratures, the trapezoidal, the mean and the Simpson
methods are usually used. Commonly, the error of these methods depends quadratically on the grid step, and a large number of steps are required to obtain good accuracy. However, there are some cases
when the error of the trapezoidal method depends on the step value not quadratically, but exponentially. Such cases are integral of a periodic function over the full period and the integral over the
entire real axis of a function that decreases rapidly enough at infinity. If the integrand has poles of the first order on the complex plane, then the Trefethen-Weidemann majorant accuracy estimates
are valid for such quadratures. In the present paper, new error estimates of exponentially converging quadratures from periodic functions over the full period are constructed. The integrand function
can have an arbitrary number of poles of an integer order on the complex plane. If the grid is sufficiently detailed, i.e., it resolves the profile of the integrand function, then the proposed
estimates are not majorant, but asymptotically sharp. Extrapolating, i.e., excluding this error from the numerical quadrature, it is possible to calculate the integrals of these classes with the
accuracy of rounding errors already on extremely coarse grids containing only 10 steps.</p>trapezoidal ruleexponential convergenceerror estimateasymptotically sharp estimatesформула
трапецийэкспоненциальная сходимостьоценки точностиасимптотически точные оценки<p>1. Introduction Applied tasks. In many physical problems it is needed to calculate integrals, that cannot be obtained
in terms of elementary functions. Here are some examples: Belov A.A., Khokhlachev V.S., 2021 This work is licensed under a Creative Commons Attribution 4.0 International License http://
creativecommons.org/licenses/by/4.0/ 1) Calculation of special functions of mathematical physics: the Fermi-Dirac functions, which are equal to the moments of the Fermi distribution, the Gamma
function, cylindrical functions and a number of others. 2) Calculation of the Fourier coefficients of a given function, Fourier and Laplace transform. 3) Numerical solution of integral equations,
both well-posed and ill-posed. 4) Solving boundary value problems for partial differential equations (including eigenvalue problems) written in integral form, etc. Calculation of quadratures.
Trapezoidal rule, rectangle rule and Simpsons rule are commonly used for grid computation of quadratures. Usually the error of these methods quadratically depends on the grid step, and a large number
of steps is needed to obtain good accuracy. However, there are a number of cases when the error of the trapezoidal rule depends on the grid step exponentially, i.e. when the step is reduced by half,
the number of correct signs of the numerical result is approximately doubled. This rate of convergence is similar to that of Newtons method. Two such cases are known. These are: the integral of the
periodic function over the full period and the improper integral of a function that decreases rapidly enough at infinity. If the integrand has first order poles on the complex plane, then for such
quadratures there are majorant error estimates of Trefethen and Weidemann [1], see also [2]-[10]. In [11], [12] the generalization of Trefethen and Weidemann estimates is built for the case when the
nearest pole of an integrand function is multiple. In this paper, new error estimates of exponentially convergent quadratures of periodic functions over the full period are described. Integrand
function can have an arbitrary number of poles of an integer order on the complex plane. If the mesh is detailed enough and the profile of the integrand resolved well, then the proposed estimates are
not majorant, but asymptotically accurate. It is possible to calculate the integrals of the indicated classes with the accuracy of round-off errors even on extremely coarse grids containing only 10
steps by extrapolation (i.e., subtraction) of this error from the numerical value of the quadrature. 2. Exponentially convergent quadratures One of the classes of exponentially convergent quadratures
are integrals of periodic functions over the full period. By replacing</p>L. N. Trefethen and J. A. C. Weideman, “The exponentially convergent trapezoidal rule,” SIAM Review, vol. 56, no. 3, pp.
385-458, 2014. DOI: 10.1137/130932132.J. Mohsin and L. N. Trefethen, “A trapezoidal rule error bound unifying the Euler-Maclaurin formula and geometric convergence for periodic functions,” in
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 470, 2014, p. 20130571. DOI: 10.1098/rspa.2013.0571.J. A. C. Weideman, “Numerical integration of periodic
functions: A few examples,” The American Mathematical Monthly, vol. 109, no. 1, pp. 21- 36, 2002. DOI: 10.2307/2695765.N. Eggert and J. Lund, “The trapezoidal rule for analytic functions of rapid
decrease,” Journal of Computational and Applied Mathematics, vol. 27, no. 3, pp. 389-406, 1989. DOI: 10.1016/0377-0427(89)90024-1.H. Al Kafri, D. J. Jeffrey, and R. M. Corless, “Rapidly convergent
integrals and function evaluation,” Lecture Notes in Computer Science, vol. 10693, pp. 270-274, 2017. DOI: 10.1007/978-3-319-72453-9_20.J. Waldvogel, “Towards a general error theory of the
trapezoidal rule,” in Springer Optimization and Its Applications. 2010, vol. 42, pp. 267- 282. DOI: 10.1007/978-1-4419-6594-3_17.E. T. Goodwin, “The evaluation of integrals of the form f(x)e-x2dx,”
Mathematical Proceedings of the Cambridge Philosophical Society, vol. 45, no. 2, pp. 241–245, 1949. DOI: 10.1017/ S0305004100024786.N. N. Kalitkin and S. A. Kolganov, “Quadrature formulas with
exponential convergence and calculation of the Fermi–Dirac integrals,” Doklady Mathematics, vol. 95, no. 2, pp. 157–160, 2017. DOI: 10.1134/S1064562417020156.N. N. Kalitkin and S. A. Kolganov,
“Refinements of precision approximations of Fermi–Dirak functions of integer indices,” Mathematical Models and Computer Simulations, vol. 9, no. 5, pp. 554–560, 2017. DOI: 10.1134/
S2070048217050052.N. N. Kalitkin and S. A. Kolganov, “Computing the Fermi–Dirac functions by exponentially convergent quadratures,” Mathematical Models and Computer Simulations, vol. 10, no. 4, pp.
472–482, 2018. DOI: 10.1134/S2070048218040063.A. A. Belov, N. N. Kalitkin, and V. S. Khokhlachev, “Improved error estimates for an exponentially convergent quadratures [Uluchshennyye otsenki
pogreshnosti dlya eksponentsial’no skhodyashchikhsya kvadratur],” Preprints of IPM im. M.V. Keldysh, no. 75, 2020, in Russian. DOI: 10.20948/prepr-2020-75.V. S. Khokhlachev, A. A. Belov, and N. N.
Kalitkin, “Improvement of error estimates for exponentially convergent quadratures [Uluchsheniye otsenok pogreshnosti dlya eksponentsial’no skhodyashchikhsya kvadratur],” Izv. RAN. Ser. fiz., vol.
85, no. 2, pp. 282–288, 2021, in Russian. DOI: 10.31857/S0367676521010166. | {"url":"https://journals.rudn.ru/miph/article/xml/27529/en_US","timestamp":"2024-11-03T22:41:02Z","content_type":"application/xml","content_length":"12039","record_id":"<urn:uuid:f6add747-02a8-4ef3-8ed1-a671aa5f07b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00825.warc.gz"} |
How to convert joules to volts
How to convert energy in joules (J) to electrical voltage in volts (V).
You can calculate volts from joules and coulombs, but you can't convert joules to volts since volt and joule units represent different quantities.
Joules to volts calculation formula
The voltage V in volts (V) is equal to the energy E in joules (J), divided by the charge Q in coulombs (C):
V[(V)] = E[(J)] / Q[(C)]
volt = joule / coulomb
V = J / C
What is the voltage supply of an electrical circuit with energy consumption of 60 joules and charge flow of 4 coulombs?
V = 60J / 4C = 15V
How to convert volts to joules ►
See also | {"url":"https://jobsvacancy.in/convert/electric/Joule_to_Volt.html","timestamp":"2024-11-04T05:31:13Z","content_type":"text/html","content_length":"6956","record_id":"<urn:uuid:7ca53106-b859-4a82-86f2-2eaa962dd4ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00511.warc.gz"} |
In this application, we show a 1D calculation of quantum and optical properties of a AlGaN/GaN LED diode with three GaN quantum wells.
The simulated structure is the following:
After a buffer n-doped AlGaN layer, a Al[0.78]InN barrier layer is present, then a series of three AlGaN/GaN quantum wells, each 2 nm-wide, followed by a p-doped AlGaN layer.
In this wurtzite nitride heterostructure, strain and strain induced piezoelectric polarization play a fundamental role in the description of the electronic properties. In fact, on one hand, effects
of strain on the conduction and band profiles have to be taken in account through the appropriate k.p model and, on the other hand, the piezoelectric polarization term, together with the spontaneous
polarization one, have to be included in the Poisson-Drift-Diffusion calculations. | {"url":"http://www.tibercad.org/sample_applications/article/","timestamp":"2024-11-05T06:18:03Z","content_type":"application/xhtml+xml","content_length":"13944","record_id":"<urn:uuid:8bd706a1-c3f8-4732-9a4b-46c7eea39777>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00277.warc.gz"} |
Task 1 • How many cupcakes?
An array can be partitioned to form smaller arrays. Adding the products of the smaller arrays gives the total in the original array.
Whole class
Trays of Arrays PowerPoint
Each group
Cupcake array picture
Blank A3 paper
Each student
Zachary and Maddie’s strategies Student sheet
Isabella and Archie’s Student sheet
Post-it notes for the gallery walk
Use Trays of arrays PowerPoint to introduce the context of the reSolve Bakery. Show students the illustration on slide 4.
Discuss: What multiplication do you see? Some examples of multiplication represented in the picture include the arrays of cupcakes and bread rolls in the cabinet, and the bags of bread rolls sitting
on the counter.
Show the picture of the cupcake array on slide 5.
Each day, six different flavours of cupcakes are made in the reSolve Bakery.
Pose the task: How many cupcakes are there altogether?
Estimated time
10 minutes
Choosing numbers and the design of the array
The array used in this question has been carefully designed and the numbers purposefully chosen.
Choosing numbers and the design of the array
The array used in this question has been carefully designed and the numbers purposefully chosen.
Why has the array been designed this way?
6 x 15 can be seen in two different ways which lead to mathematically similar solutions.
1. There are 6 smaller arrays of 15 cakes. These smaller arrays can easily be partitioned into a group of 10 and a group of 5. Partitioning the smaller arrays in this way creates 6 groups of 10 and
6 groups of 5, or (6 x 10) + (6 x 5).
2. The larger array of cakes is arranged in 6 rows of 15. This larger array can be partitioned into a 6 x 10 array and a 6 x 5 array, which can also be solved using (6 x 15) + (6 x 5).
Why were the numbers 15 and 6 chosen?
15 was selected as it is typically an easier number for students to work with. Students can add together the arrays of 15 cakes across the rows or down the columns to find the total in the
collection. 15 can also be arranged as a 5 x 3 array, which can then be partitioned into a 2 x 5 array to make 10 and a 1 x 5 array to make 5.
As a multiple of 2, doubling strategies can be used to multiply by 6. Students can calculate 3 x 15 using multiplication or repeated addition. They then just need to double this answer to find out 6
x 15.
Choosing numbers and the design of the array
The array used in this question has been carefully designed and the numbers purposefully chosen.
Why has the array been designed this way?
6 x 15 can be seen in two different ways which lead to mathematically similar solutions.
1. There are 6 smaller arrays of 15 cakes. These smaller arrays can easily be partitioned into a group of 10 and a group of 5. Partitioning the smaller arrays in this way creates 6 groups of 10 and
6 groups of 5, or (6 x 10) + (6 x 5).
2. The larger array of cakes is arranged in 6 rows of 15. This larger array can be partitioned into a 6 x 10 array and a 6 x 5 array, which can also be solved using (6 x 15) + (6 x 5).
Why were the numbers 15 and 6 chosen?
15 was selected as it is typically an easier number for students to work with. Students can add together the arrays of 15 cakes across the rows or down the columns to find the total in the
collection. 15 can also be arranged as a 5 x 3 array, which can then be partitioned into a 2 x 5 array to make 10 and a 1 x 5 array to make 5.
As a multiple of 2, doubling strategies can be used to multiply by 6. Students can calculate 3 x 15 using multiplication or repeated addition. They then just need to double this answer to find out 6
x 15.
Ask students to work in pairs to solve the problem. Provide each pair with Cupcake array picture and a sheet of A3 paper. Ask students to use the A3 paper to create a poster of their solution method.
This task serves as a helpful pre-assessment task. The strategies that students use indicate their existing understandings of multiplication. Pose questions or prompts that help you to make sense of
student thinking, for example:
• Explain your strategy to me.
• Why have you partitioned the numbers in that way?
• You have created smaller arrays from the larger array. Will the total number of cakes in all the smaller arrays be the same as the total number of cakes in the large array? How do you know?
Students will use a range of strategies to determine the total number of cupcakes. The purpose at this early stage in the sequence is not to point students to using a particular strategy but rather
to take note of the strategy and what this indicates about students’ thinking and understanding.
• Do students use additive or multiplicative thinking to solve the problem?
• Do students use of strategies demonstrate an understanding of the multiplicative properties of associativity or distributivity?
Watch the Distributive and associative properties of multiplication professional learning video embedded in this step to learn more about strategies that use these properties.
Estimated time
25 minutes
Student work samples
Look at how these students have solved the problem.
Student work samples
Look at these work samples and see how these students have solved the problem.
Discuss with colleagues: What do these students know and what have they shown that they can do? Give evidence from the work samples for your statements.
Look at these work samples and see how these students have solved the problem.
Discuss with colleagues: What do these students know and what have they shown that they can do? Give evidence from the work samples for your statements.
The importance of mathematical talk
Mathematical talk is fundamental to both knowing and doing mathematics.
The importance of mathematical talk
We have suggested working in pairs or small groups during the Explore phase of this task to promote mathematical discourse. Mathematical talk in the classroom is fundamental to both knowing and doing
mathematics. Students should have regular opportunities to work on and talk about solving problems in community with peers. It is likely that each student in the group will approach the task in
slightly different ways. Through discourse, students put forth claims and justify them as well as listening to and critiquing claims of others. Solving problems in community provides a venue for more
talking and listening than is available when working individually or in a teacher-led lesson.
The importance of mathematical talk
We have suggested working in pairs or small groups during the Explore phase of this task to promote mathematical discourse. Mathematical talk in the classroom is fundamental to both knowing and doing
mathematics. Students should have regular opportunities to work on and talk about solving problems in community with peers. It is likely that each student in the group will approach the task in
slightly different ways. Through discourse, students put forth claims and justify them as well as listening to and critiquing claims of others. Solving problems in community provides a venue for more
talking and listening than is available when working individually or in a teacher-led lesson.
Display students’ work around the classroom in preparation for a gallery walk.
Review the original task that students were asked to solve and ask students to think about what they expect to see as they complete the gallery walk.
Ask students to consider the following questions as they look at others’ work:
• Look at how other students have solved the problems. What do you notice?
• Which strategies were particularly helpful for working out the total number of cupcakes? Why?
Conduct the class gallery walk.
At the end of the class gallery walk, allow students time to read and reflect on any post-it notes left on their work. They may choose to adjust or change to their solution strategies and/or
recording methods.
Estimated time
15 minutes
Gallery walk
In a gallery walk the role of the students is to critically view and review others’ mathematical activity.
Gallery walk
In a gallery walk the role of the student is to critically view and review others’ mathematical activity. They need to think more broadly than the strategy they have personally used, as they consider
how their thinking fits with the representations of thinking used by others in the class.
In this gallery walk, students are asked to look at how others have solved the problem and consider which strategy/strategies are the most helpful when determining the total number of cakes in the
array. As students look at others’ strategies, they are able to reflect on and refine their own approach to solving the problem.
In a gallery walk the role of the student is to critically view and review others’ mathematical activity. They need to think more broadly than the strategy they have personally used, as they consider
how their thinking fits with the representations of thinking used by others in the class.
In this gallery walk, students are asked to look at how others have solved the problem and consider which strategy/strategies are the most helpful when determining the total number of cakes in the
array. As students look at others’ strategies, they are able to reflect on and refine their own approach to solving the problem.
What is a gallery walk?
In a gallery walk, students move around the classroom like they are in an art gallery.
What is a gallery walk?
In a gallery walk, students move around the classroom like they are in an art gallery, in silence or whispering with a partner. Students use post-it notes to post comments and questions about the
mathematics they see. They should be encouraged to take their time to respectfully read and respond to the work of others.
At the end of the gallery walk students should be given time to read and reflect on the post-it notes that have been left for them. Students should be allowed time to modify their work if they would
like to. The questions and feedback on these notes will help refine students’ thinking and the manner of their mathematical recording. It is also likely that the students will have developed new
thinking as they critically viewed and reflected on others’ work, and it is important that students have the opportunity to act on this new learning.
In a gallery walk, students move around the classroom like they are in an art gallery, in silence or whispering with a partner. Students use post-it notes to post comments and questions about the
mathematics they see. They should be encouraged to take their time to respectfully read and respond to the work of others.
At the end of the gallery walk students should be given time to read and reflect on the post-it notes that have been left for them. Students should be allowed time to modify their work if they would
like to. The questions and feedback on these notes will help refine students’ thinking and the manner of their mathematical recording. It is also likely that the students will have developed new
thinking as they critically viewed and reflected on others’ work, and it is important that students have the opportunity to act on this new learning.
Provide students with Zachary and Maddie’s strategies Student sheet. Explain that these two students solved the problem in different yet similar ways.
Ask: How are these strategies similar and how are they different?
Allow students time to explore the similarities and differences. Have students record their noticings on their student sheet.
Provide students with Isabella and Archie’s strategies Student sheet. Explain that these two students also solved the problem in different yet similar ways.
Ask: How are these strategies similar and how are they different?
Allow students time to explore the similarities and differences. Have students record their noticings on their student sheet.
Estimated time
30 minutes
Comparing strategies
In this video, we look at the different strategies that are compared in the Connect phase of this task.
Transcript not yet available
This sequence is designed to build students’ understanding of the distributive and associative properties of multiplication and how these properties can be applied to solve multiplication problems.
In this video, we illustrate how the distributive and associative properties of multiplication are linked to the four strategies that students compare in the Connect phase of this task.
Similarities and differences
The similarities that students notice between the different strategies help build a generalised mathematical idea.
Similarities and differences
Students are shown four different strategies in this Connect phase, and they discuss the similarities and differences that they notice. The key similarity that we want students to notice is that the
array can be partitioned into smaller parts to aid computation. It is important that students see that the array must be partitioned fully, no parts can be left out. The total is then calculated by
adding together the products of the smaller arrays.
Similarities and differences
Students are shown four different strategies in this Connect phase, and they discuss the similarities and differences that they notice. The key similarity that we want students to notice is that the
array can be partitioned into smaller parts to aid computation. It is important that students see that the array must be partitioned fully, no parts can be left out. The total is then calculated by
adding together the products of the smaller arrays.
Use the Trays of arrays PowerPoint to support the discussion. Share Zachary and Maddie’s strategy on slide 6.
• How are these strategies different?
□ Zachary doubles 15 and Maddie adds 15 three times.
• How are these strategies similar?
□ Both strategies partition the array into groups of 15 and these groups of 15 are then added together.
Share Isabella and Archie’s strategy on slide 7.
• How are these strategies different?
□ Isabella partitions each small cupcake array of 15 into a group of 10 and a group of 5, making 6 groups of 10 and 6 groups of 5.
□ Archie partitions the full cupcake array into a 6 x 10 array and a 5 x 6 array.
• How are these strategies similar?
□ Both strategies can be solved using the expression (6 x 10) + (6 x 5).
Look at Strategies 1-4 as a group on slide 8.
• What is similar about each of these strategies?
□ In each case, the larger array has been fully partitioned to form smaller arrays based on known multiplication facts. All the partial products formed are then added together to find the total
in the whole collection.
Estimated time
15 minutes
Whole class discussion
Whole class discussions make the mathematics visible to the students.
Whole class discussion
The purpose of whole class discussion is to make the mathematics visible, so that students can develop their thinking beyond what they are able to do on their own or in small groups. Through
discussions, teachers can build a thoughtful community of mathematical thinkers, where students actively participate in sharing their ideas and listening to those of others.
In making the mathematics visible, you guide students in noticing and building connections between mathematical concepts that they may have seen as disconnected. It is through this deeper
understanding of the mathematics that students can begin to form generalisations.
The purpose of whole class discussion is to make the mathematics visible, so that students can develop their thinking beyond what they are able to do on their own or in small groups. Through
discussions, teachers can build a thoughtful community of mathematical thinkers, where students actively participate in sharing their ideas and listening to those of others.
In making the mathematics visible, you guide students in noticing and building connections between mathematical concepts that they may have seen as disconnected. It is through this deeper
understanding of the mathematics that students can begin to form generalisations.
Arrays can be partitioned
Ask the students to consider whether the strategy that they used was most similar to Zachary, Maddie, Isabella, or Archie’s strategy. Discuss how in each instance, what was known was used to work out
what was unknown.
Explain: We can calculate the total number of objects in an array by partitioning the large array into smaller arrays. The total number of objects in the smaller arrays are then added together to
find the total number of objects in the large array.
As a class, look at some of the ways that students partitioned the larger array to create smaller arrays using known multiplication facts.
Create a class display using the students' posters, using the summary statement from above as a title for the display. Read the Class display professional learning embedded in this step to learn how
this display can be used to build a shared understanding amongst the students.
Class display
Create a class display to help build a shared understanding.
Class display
In the Summarise phase, the following statement is presented to students:
We can calculate the total number of objects in an array by partitioning the large array into smaller arrays. The total number of objects in the smaller arrays are then added together to find the
total number of objects in the large array.
This statement reflects the learning goal for the task and is the understanding that we want all in the class to share. As the sequence progresses, this shared understanding is built on.
Create a banner or poster of this shared understanding and display the students’ posters of solutions strategies underneath or around the shared understanding. This provides the opportunity for
students to revisit and reflect on this understanding and the ways that that it is reflected in the different solutions strategies used by students. The display is a way to help ensure the
understanding is truly shared by all in the class.
Presenting everyone’s work with this statement communicates that everyone has contributed to and participated in developing this understanding as shared in the class, regardless of how complex or
sophisticated their thinking may or may not be.
In the Summarise phase, the following statement is presented to students:
We can calculate the total number of objects in an array by partitioning the large array into smaller arrays. The total number of objects in the smaller arrays are then added together to find the
total number of objects in the large array.
This statement reflects the learning goal for the task and is the understanding that we want all in the class to share. As the sequence progresses, this shared understanding is built on.
Create a banner or poster of this shared understanding and display the students’ posters of solutions strategies underneath or around the shared understanding. This provides the opportunity for
students to revisit and reflect on this understanding and the ways that that it is reflected in the different solutions strategies used by students. The display is a way to help ensure the
understanding is truly shared by all in the class.
Presenting everyone’s work with this statement communicates that everyone has contributed to and participated in developing this understanding as shared in the class, regardless of how complex or
sophisticated their thinking may or may not be.
For you
Download all of the task steps in this sequence, in a single editable document.
All of the resources included in the list of materials. | {"url":"https://resolve.edu.au/teaching-sequences/year-4/multiplication-trays-arrays/task-1-how-many-cupcakes","timestamp":"2024-11-14T22:17:13Z","content_type":"text/html","content_length":"253531","record_id":"<urn:uuid:5fa771b0-3d31-4903-b618-959498b3199d>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00759.warc.gz"} |
of truncated cone
Radar cross section of truncated cone
Since R2021a
rcspat = rcstruncone(r1,r2,height,c,fc) returns the radar cross section pattern of a truncated cone. r1 is the radius of the small end of the cone, r2 is the radius of the large end, and height is
the cone height. The radar cross section is a function of signal frequency, fc, and signal propagation speed, c. You can create a non-truncated cone by setting r1 to zero. The cone points downward
towards the xy-plane. The origin is located at the apex of a the non-truncated cone constructed by extending the truncated cone to an apex.
rcspat = rcstruncone(r1,r2,height,c,fc,az,el) also specifies the azimuth angles, az, and elevation angles, el, at which to compute the radar cross section.
[rcspat,azout,elout] = rcstruncone(___) also returns the azimuth angles, azout, and elevation angles, elout, at which the radar cross sections are computed. You can use these output arguments with
any of the previous syntaxes.
Radar Cross Section of Truncated Cone
Display the radar cross section (RCS) pattern of a truncated cone as a function of azimuth angle and elevation. The truncated cone has a bottom radius of 9.0 cm and a top radius of 12.5 cm. The cone
height is 1 m. The operating frequency is 4.5 GHz.
Define the truncated cone geometry and signal parameters.
c = physconst('Lightspeed');
fc = 4.5e9;
radbot = 0.090;
radtop = 0.125;
hgt = 1;
Compute the RCS for all directions using the default direction values.
[rcspat,azresp,elresp] = rcstruncone(radbot,radtop,hgt,c,fc);
xlabel('Azimuth Angle (deg)')
ylabel('Elevation Angle (deg)')
title('Truncated Cone RCS (dBsm)')
Radar Cross Section of Truncated Cone as Function of Elevation
Plot the radar cross section (RCS) pattern of a truncated cone as a function of elevation for a fixed azimuth angle of 5 degrees. The cone has a bottom radius of 9.0 cm and a top radius of 12.5 cm.
The truncated cone height is 1 m. The operating frequency is 4.5.
Define the truncated cone geometry and signal parameters.
c = physconst('Lightspeed');
fc = 4.5e9;
radbot = 0.090;
radtop = 0.125;
hgt = 1;
Compute the RCS at an azimuth angle of 5 degrees.
az = 5.0;
el = -90:90;
[rcspat,azresp,elresp] = rcstruncone(radbot,radtop,hgt,c,fc,az,el);
xlabel('Elevation Angle (deg)')
ylabel('RCS (dBsm)')
title('Truncated Cone RCS as Function of Elevation')
grid on
Radar Cross Section of Truncated Cone as Function of Frequency
Plot the radar cross section (RCS) pattern of a truncated cone as a function of frequency for a single direction. The cone has a bottom radius of 9.0 cm and a top radius of 12.5 cm. The truncated
cone height is 1 m.
Specify the truncated cone geometry and signal parameters.
c = physconst('Lightspeed');
radbot = 0.090;
radtop = 0.125;
hgt = 1;
Compute the RCS over a range of frequencies for a single direction.
az = 5.0;
el = 20.0;
fc = (100:100:4000)*1e6;
[rcspat,azpat,elpat] = rcstruncone(radbot,radtop,hgt,c,fc,az,el);
xlabel('Frequency (MHz)')
ylabel('RCS (dBsm)')
title('Truncated Cone RCS as Function of Frequency')
grid on
Radar Cross Section of Full Cone as Function of Elevation
Plot the radar cross section (RCS) pattern of a full cone as a function of elevation for a fixed azimuth angle. To define a full cone set the bottom radius to zero. Set the top radius to 20.0 cm and
the cone height to 50 cm. Assume the operating frequency is 4.5 GHz and the azimuth angle is 5 degrees.
Define the cone geometry and signal parameters.
c = physconst('Lightspeed');
fc = 4.5e9;
radsmall = 0.0;
radlarge = 0.20;
hgt = 0.5;
Compute the RCS for a fixed azimuth angle of 5 degrees.
az = 5.0;
el = -89:0.1:89;
[rcspat,azresp,elresp] = rcstruncone(radsmall,radlarge,hgt,c,fc,az,el);
xlabel('Elevation Angle (deg)')
ylabel('RCS (dBsm)')
title('Full Cone RCS as Function of Elevation')
grid on
Input Arguments
r1 — Radius of small end of truncated cone
nonnegative scalar
Radius of small end of truncated cone, specified as a nonnegative scalar. Units are in meters.
Example: 5.5
Data Types: double
r2 — Radius of large end of truncated cone
positive scalar
Radius of large end of truncated cone, specified as a positive scalar. Units are in meters.
Example: 5.5
Data Types: double
height — Height of truncated cone
positive scalar
Height of truncated cone, specified as a positive scalar. Units are in meters.
Example: 3.0
Data Types: double
fc — Frequency for computing radar cross section
positive scalar | positive, real-valued, 1-by-L row vector
Frequency for computing radar cross section, specified as a positive scalar or positive, real-valued, 1-by-L row vector. Frequency units are in Hz.
Example: [100e6 200e6]
Data Types: double
Output Arguments
rcspat — Radar cross section pattern
real-valued N-by-M-by-L array
Radar cross section pattern, returned as a real-valued N-by-M-by-L array. N is the length of the vector returned in the elout argument. M is the length of the vector returned in the azout argument. L
is the length of the fc vector. Units are in meters-squared.
Data Types: double
azout — Azimuth angles
real-valued 1-by-M row vector
Azimuth angles for computing directivity and pattern, returned as a real-valued 1-by-M row vector where M is the number of azimuth angles specified by the az input argument. Angle units are in
The azimuth angle is the angle between the x-axis and the projection of the direction vector onto the xy-plane. The azimuth angle is positive when measured from the x-axis toward the y-axis.
Data Types: double
elout — Elevation angles
real-valued 1-by-N row vector
Elevation angles for computing directivity and pattern, returned as a real-valued 1-by-N row vector where N is the number of elevation angles specified in el output argument. Angle units are in
The elevation angle is the angle between the direction vector and xy-plane. The elevation angle is positive when measured towards the z-axis.
Data Types: double
More About
Azimuth and Elevation
This section describes the convention used to define azimuth and elevation angles.
The azimuth angle of a vector is the angle between the x-axis and its orthogonal projection onto the xy-plane. The angle is positive when going from the x-axis toward the y-axis. Azimuth angles lie
between –180° and 180° degrees, inclusive. The elevation angle is the angle between the vector and its orthogonal projection onto the xy-plane. The angle is positive when going toward the positive z
-axis from the xy-plane. Elevation angles lie between –90° and 90° degrees, inclusive.
[1] Mahafza, Bassem. Radar Systems Analysis and Design Using MATLAB, 2nd Ed. Boca Raton, FL: Chapman & Hall/CRC, 2005.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Version History
Introduced in R2021a | {"url":"https://de.mathworks.com/help/radar/ref/rcstruncone.html","timestamp":"2024-11-05T03:37:41Z","content_type":"text/html","content_length":"119590","record_id":"<urn:uuid:9849b136-b9d0-4ee2-bc4b-af8f6de43b2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00317.warc.gz"} |
A force F depends on θ as F=sinθ+ncosθK, where K and n are (+v... | Filo
A force depends on as , where and are (+ve) constants. Find the minimum value of ( is acute)
Not the question you're searching for?
+ Ask your question
to be minimum,
Was this solution helpful?
Found 3 tutors discussing this question
Discuss this question LIVE for FREE
5 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Practice more questions from Motion in Straight Line
View more
Practice questions on similar concepts asked by Filo students
View more
Stuck on the question or explanation?
Connect with our Physics tutors online and get step by step solution of this question.
231 students are taking LIVE classes
Question Text A force depends on as , where and are (+ve) constants. Find the minimum value of ( is acute)
Updated On Jul 8, 2022
Topic Motion in Straight Line
Subject Physics
Class Class 11
Answer Type Text solution:1 Video solution: 1
Upvotes 140
Avg. Video Duration 3 min | {"url":"https://askfilo.com/physics-question-answers/a-force-f-depends-on-theta-as-f-frac-k-sin-theta-n-cos-theta","timestamp":"2024-11-06T18:49:04Z","content_type":"text/html","content_length":"328761","record_id":"<urn:uuid:89860b15-f355-4558-b3c3-8822c9955046>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00126.warc.gz"} |
gcc/testsuite/gnat.dg/aspect1_vectors_2d.ads - gcc - Git at Google
type T_horizontal is new float;
-- Declaration of types, constants, and common functions on 3D vectors.
-- Corresponds to PVS theory vectors/vectors_2D
package Aspect1_Vectors_2D is
-- A 2D vector, represented by an x and a y coordinate.
type Vect2 is record
x: T_horizontal;
y: T_horizontal;
end record;
subtype Nz_vect2 is Vect2
with Predicate => (Nz_vect2.x /= 0.0 and then Nz_Vect2.y /= 0.0);
end Aspect1_Vectors_2D; | {"url":"https://gnu.googlesource.com/gcc/+/a8404c07e7fca388c02c39077865f7d5fa928430/gcc/testsuite/gnat.dg/aspect1_vectors_2d.ads","timestamp":"2024-11-15T03:54:39Z","content_type":"text/html","content_length":"9514","record_id":"<urn:uuid:e54ba3a1-3745-4488-bc31-e67eb16e9c40>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00041.warc.gz"} |
List of Currently Registered Institutions
List of Currently Registered Institutions []
New Institution
1. American Mathematical Society, AMS Prizes and Awards
2. American Mathematical Society, Fellows of the AMS
3. American Mathematical Society, Mathematics Research Communities (1)
4. American Mathematical Society, Meetings and Conferences (4)
5. American Mathematical Society, Programs and Travel Grants (3)
6. Association for Women in Mathematics (1)
7. Baruch College (CUNY), Mathematics (1)
8. Boston University, PROMYS
9. Brin Mathematics Research Center
10. Brown University, Applied Math
11. Carnegie Mellon University, Mathematical Sciences (1)
12. Central Washington University, Department of Mathematics
13. Clemson University
14. Columbia University, Department of Mathematics
15. Cornell University, Department of Mathematics
16. Duke University, Department of Mathematics (2)
17. The EDGE Foundation (1)
18. Georgia Institute of Technology, School of Mathematics
19. IAS/Park City Mathematics Institute (5)
20. ICM 2026 Satellite Events Committee (1)
21. Indiana University, Mathematics
22. Institute for Advanced Study, School of Mathematics (2)
23. International Mathematical Union
24. Iowa State University, Department of Mathematics
25. James Madison University, Mathematics and Statistics (1)
26. Joint Mathematics Meetings, Special Programs
27. Lafayette College, Mathematics Department
28. Massachusetts Institute of Technology, MIT PRIMES (Program for Research in Mathematics, Engineering and Science for High School Students) (3)
29. Massachusetts Institute of Technology, Summer Geometry Initiative
30. Mathematical Association of America, MAA Project NExT
31. Mathematical Congress of the Americas (1)
32. North Carolina State University, Mathematics
33. Northwestern University, Mathematics Department
34. The Ohio State University, Department of Mathematics
35. Oregon State University, Mathematics
36. PZ Math, PZ Math - PZMC
37. Ross Mathematics Foundation, Ross Mathematics Program
38. Simons Laufer Mathematical Sciences Institute (formerly MSRI) (2)
39. Texas A&M University-Commerce, Mathematics
40. Towson University, Department of Mathematics
41. University of California, Los Angeles, Department of Mathematics
42. University of California, Los Angeles, Institute for Pure and Applied Mathematics (2)
43. University of Chicago, Math/PSD
44. University of Connecticut, Department of Mathematics
45. University of Illinois at Chicago, Mathematics, Statistics and Computer Science
46. University of Maryland, Department of Mathematics
47. University of Minnesota Duluth, Department of Mathematics and Statistics
48. University of Minnesota Twin Cities, School of Mathematics (1)
49. University of Notre Dame, Department of Mathematics
50. University of Utah, Department of Mathematics
51. University of Virginia, Mathematics
52. University of Virginia, University of Virginia Topology REU
53. Vanderbilt University, Department of Mathematics
54. Wayne State University, electronic Computational Homotopy Theory
55. West Virginia University, School of Mathematical and Data Sciences/Research Experiences for Undergraduates
56. Williams College, Mathematics and Statistics (5)
57. Worcester Polytechnic Institute, Center for Industrial Mathematics and Statistics REU
58. Worcester Polytechnic Institute, Mathematical Sciences Department
59. Yale University - Mathematics Department, SUMRY Program
60. York College (CUNY), Mathematics & Computer Science
(‡ 37 active program ads)
© 2024 MathPrograms.Org, American Mathematical Society. All Rights Reserved. | {"url":"https://www.mathprograms.org/db/employers","timestamp":"2024-11-02T23:35:27Z","content_type":"application/xhtml+xml","content_length":"8970","record_id":"<urn:uuid:6bb55f2d-7425-4bf3-946e-d2f4e12be510>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00042.warc.gz"} |
VTU 1st Year Elements of Civil Engineering [SET-2] Solved Model Question paper
On this website, you will find all subjects solved model question paper with answers and VTU 1st Year 21CIV14/24 Elements of Civil Engineering and Mechanics Solved Model Question paper with answers.
VTU 1st Year Elements of Civil Engineering [SET-2] Solved Model Question paper
1.A] Explain briefly the scope of civil engineering in
i) Environmental and sanitary engineering
ii) Construction engineering
Get Answer
1.B] Explain briefly the role of civil engineers in the development of the nation
Get Answer
1.C] What are the requirements of a good brick?
Get Answer
2.A] Explain briefly the scope of civil engineering in
i) Geotechnical engineering
ii) Earthquake engineering
Get Answer
2.B] Explain briefly different types of cement.
Get Answer
2.C] Explain the classification of steel
Get Answer
3.A] State and prove parallelogram law of forces
Get Answer
3.B] Determine the resultant of the force system shown in figure.
3.C] Three forces of magnitude 200N each are acting along the sides of an equilateral triangle as shown in fig Q 3 (c). Determine the resultant in magnitude and direction with reference to point B.
Get Answer
4.A] State and Prove Lami’s theorem
Get Answer
4.B] A vertical load of 30kN is supported at A by a system of cables as shown in the figure. Determine the force in each cable for equilibrium.
4.C] Knowing that W[A] = 80N and θ = 40^0, determine the smallest and largest value of W[B] for which the system is in equilibrium, refer fig. Q4 (c). Take µ[s] = 0.35 and µ[k] = 0.25.
5.A] Find the centroid of the area enclosed by a right-angled triangle from the first principle.
Get Answer
5.B] Locate the centroid of the shaded area as shown in fig. Q 5(b)
6.A] State and prove parallel axes theorem
Get Answer
6.B] Derive an expression for moment of inertia of a rectangle from first principle about its vertical centroidal axis.
Get Answer
6.C] Find the polar radius of gyration for the area as shown in fig. Q 6 (c)
7.A] Explain different types of supports and reactions.
Get Answer
7.B] Analyse the truss as shown in fig Q 7(b), by methods of joints.
8.A] What are the assumptions made in the analysis of a truss?
Get Answer
8.B] Find the support reactions for the beam as shown in fig Q 8 (b).
8.C] A floor truss is loaded as shown in fig Q 8 (c), Determine the forces in members CF, EF and EG.
9.A] Define i) Displacement ii) Velocity iii) Acceleration iv) Speed
Get Answer
9.B] A stone is released from top of the tower, during the last second of its motion, it covers 1/4^th of the height of the tower. Find the height of the tower.
Get Answer
9.C] A target is fired with an initial velocity of 180m/s at a target located 500m above the gun and at a horizontal distance of 2100m. Neglecting air resistance, Determine the value of the firing
Get Answer
10.A] State and explain D’ Alembert’s principle.
Get Answer
10.B] A fly wheel rotates at 200rpm and after 10 seconds it rotating at 160rpm. If the retardation is uniform determine number of revolutions made and time taken by flywheel before it-.comes to rest
from the speed of 200 rpm.
Get Answer
10.C] A particle of mass 100N is acted upon by a force F = (20t2 – 40)N, where ‘t’ is time in second. At t = 0s, velocity of the particle is 5m/s and position x = 0. Find velocity and position of the
particle at t = 2s.
Get Answer | {"url":"https://vtuupdates.com/solved-model-papers/elements-of-civil-engineering-set-2-solved-model-paper/","timestamp":"2024-11-05T04:35:06Z","content_type":"text/html","content_length":"175780","record_id":"<urn:uuid:9fa29895-631a-4b09-bde7-320bc9213472>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00140.warc.gz"} |
Mastering the Least Common Denominator (LCD) of Exponents: A Key to Algebraic Expression Simplification
Mastering The Least Common Denominator (Lcd) Of Exponents: A Key To Algebraic Expression Simplification
The least common denominator of exponents (LCD) is the lowest common multiple of the exponents in a given expression. It enables the combination of algebraic expressions involving exponents by
creating a common base for all terms. To find the LCD, identify the exponents with the highest power and multiply them to find the common denominator. Understanding the LCD of exponents is crucial
for simplifying and combining algebraic expressions, as it provides a standardized base for operations involving exponents.
Unveiling the Enigma of Algebraic Expressions
Embark on an adventure into the enigmatic realm of algebraic expressions, enigmatic equations that govern the world around us. Think of these expressions as powerful tools, like magic wands that
transform numbers and variables into mathematical wonders.
At the heart of these expressions lie variables, mysterious placeholders that represent unknown values. Much like actors in a play, they take on different roles, allowing us to explore countless
possibilities. And then we have mathematical operations, the building blocks of algebraic expressions. Like master craftsmen, they shape and transform these expressions, bringing them to life.
Addition, subtraction, multiplication, and division all dance together to create a mesmerizing symphony of numbers.
The Intriguing World of Exponents: Unraveling the Power of Numbers
What are Exponents and Why Do They Matter?
Exponents, the tiny superscript numbers that dance above mathematical expressions, hold a profound significance in the realm of mathematics. They represent a shorthand notation for repeated
multiplication, making them indispensable tools for simplifying complex computations.
Exponents: The Mathematical Magnifier
Exponents are symbolized by small numbers positioned slightly above and to the right of a base number. For instance, 5³ represents the base number 5 multiplied by itself three times: 5 x 5 x 5. This
concise notation allows us to express large numerical values in a compact and elegant way.
The Relationship Between Exponents and the Base Number
The relationship between exponents and base numbers is integral to understanding their power. Theexponent indicates how many times the base number is multiplied by itself. For example, in 10², the
exponent 2 indicates that 10 is multiplied by itself twice, resulting in the value 100.
Exponents as Multiplication Simplifiers
Exponents shine when it comes to simplifying multiplication. Instead of writing out the repetitive process of multiplying a number by itself multiple times, we can simply employ exponents. For
instance, instead of writing 3 x 3 x 3 x 3 x 3, we can use the more concise notation 3⁵, indicating that 3 is multiplied by itself five times.
Least Common Denominator of Exponents:
• Define the least common denominator (LCD) of exponents.
• Explain the importance of the LCD in combining algebraic expressions.
• Outline the process of finding the LCD of exponents.
The Least Common Denominator of Exponents: A Gateway to Simplifying Algebraic Expressions
In the realm of mathematics, algebraic expressions often take center stage. These expressions are made up of variables, constants, and mathematical operations, offering a concise representation of
mathematical concepts. To manipulate and combine these expressions effectively, the least common denominator (LCD) of exponents plays a crucial role.
What is the LCD of Exponents?
Imagine a fraction where the denominators are different. To add or subtract these fractions, you need a common denominator that allows you to compare and combine the numbers. Similarly, when working
with algebraic expressions involving exponents, the LCD of exponents is the lowest common multiple that aligns the exponents of the variables in the expression.
Why is the LCD Important?
The LCD is the key to unlocking the ability to simplify and combine algebraic expressions. By finding the LCD, we can ensure that the variables have the same exponent, making it possible to add,
subtract, multiply, or divide the expressions without losing any valuable information.
Finding the LCD of Exponents
Determining the LCD of exponents is a straightforward process. Here’s how it’s done:
• Step 1: Identify the variables with exponents in the expression.
• Step 2: Prime factorize each variable’s base number.
• Step 3: Select the largest exponent for each common prime factor.
• Step 4: Multiply the highest exponents together to find the LCD.
Understanding the LCD of exponents empowers us to conquer even the most complex algebraic expressions. It’s a tool that unlocks the door to efficient calculations and simplifies the intricate world
of mathematics.
Practical Applications: The Power of the LCD
Simplify and Combine with Ease
In the realm of algebra, the Least Common Denominator (LCD) of exponents plays a pivotal role in simplifying and combining algebraic expressions. Just as finding the lowest common denominator of
fractions helps in adding and subtracting them, the LCD of exponents streamlines the manipulation of algebraic terms.
Solving Complex Problems with Confidence
The LCD’s significance extends beyond mere simplification. It becomes an indispensable tool for tackling complex mathematical problems. By identifying the LCD of exponents, we can uncover common
factors and simplify expressions, allowing us to solve equations and manipulate complex algebraic terms with greater ease.
Examples of LCD in Action
Let’s consider a practical example to illustrate the power of the LCD. Suppose we have the following expression:
(2x^3 + 5x^2) - (x^3 + 3x^2)
To simplify this expression, we first identify the LCD of the exponents, which is x^3.
= 2x^3 + 5x^2 - x^3 - 3x^2
Note: When subtracting terms with different exponents, we must align them using the LCD.
= (2x^3 - x^3) + (5x^2 - 3x^2)
= x^3 + 2x^2
The concept of the Least Common Denominator of exponents is a fundamental principle in algebra. Understanding and applying the LCD allows us to simplify expressions, combine terms effectively, and
solve complex mathematical problems with greater confidence. Whether you’re a seasoned mathematician or a budding algebraist, the LCD is an essential tool in your problem-solving arsenal.
Common Multiples and Related Concepts:
• Define common multiples and their role in finding the LCD.
• Discuss the relationship between common multiples and the LCD of exponents.
• Emphasize the importance of understanding multiples and common multiples in mathematical operations.
Common Multiples and Their Importance in Exponent Operations
In the mathematical realm, the concept of exponents plays a pivotal role in simplifying and understanding algebraic expressions. The ability to find the least common denominator (LCD) of exponents is
crucial for combining and manipulating these expressions effectively.
Enter the notion of common multiples, which are pivotal in this process. A common multiple is any number that is divisible by two or more other numbers. In the context of exponents, common multiples
serve as the building blocks for finding the LCD.
Consider, for instance, the exponents 2 and 3. Their common multiples are 6, 12, 18, and so on. Among these, the least common multiple (LCM) is 6, which is the smallest number divisible by both 2 and
The significance of common multiples lies in their ability to form the LCD of exponents. The LCD is the smallest positive number that can be divided evenly by all the exponents in an expression. By
finding the LCD, we can combine like terms and simplify the expression.
For example, consider the expression 2x²y³ / 3xy². To combine the terms, we need to find the LCD of the exponents:
• The common multiples of 2 and 3 are 6, 12, 18, …
• The LCD of the exponents is 6
Therefore, we can rewrite the expression as:
(2x²y³) / (3xy²) = (2/3) * (x²y³ / xy²) = (2/3) * (x * y) = 2/3xy
The understanding of common multiples and the LCD of exponents is fundamental to manipulating and solving complex algebraic expressions. It provides a systematic approach to combining like terms and
simplifies the process, ultimately leading to more efficient and accurate solutions.
Leave a Comment | {"url":"https://www.bootstrep.org/lcd-of-exponents-algebraic-expression-simplification/","timestamp":"2024-11-03T22:08:33Z","content_type":"text/html","content_length":"149619","record_id":"<urn:uuid:9c7363af-9609-452f-88cf-db01b58a8c9b>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00247.warc.gz"} |
Mobius Strip
Chiralkine counting behaves like rotation of a Mobius strip divided up into three equal pairs of parts. A Mobius strip has one global side perceived as two local sides. As the strip is rotated, each
part comes into view in order on one of the local sides while its opposite pair member comes into position on the other.
The following diagram depicts the construction of a Mobius strip conceptually using the letters a, b and c as in chiralkine counting to mark the parts, but when constructing a real model it is better
to use the primary colours and their opposites to mark the opposed parts, because letters are asymmetric. | {"url":"http://chiralkine.com/mobius-strip/","timestamp":"2024-11-02T04:53:31Z","content_type":"text/html","content_length":"27102","record_id":"<urn:uuid:5a72f7c9-8670-4790-b052-4d2079a21c5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00204.warc.gz"} |
Pre Algebra 2 Step Equations - Tessshebaylo
Pre Algebra 2 Step Equations
Algebra with 2 step equations worksheets k5 learning prealgebra 5 solving multi you two equation printable answers examples one l beyond integers kuta addition subtraction set homeschool books math
workbooks and free worksheet lovely monks
Algebra With 2 Step Equations Worksheets K5 Learning
Prealgebra 2 5 Solving Multi Step Equations You
Two Step Equations
Two Step Equation Worksheets Printable Answers Examples
Solving One Step Equations Worksheets L Beyond
Two Step Equations With Integers Kuta
Two Step Equations
Algebra 2 Step Addition Subtraction Equations Set Homeschool Books Math Workbooks And Free Printable Worksheets
Two Step Equations Worksheet Lovely Math Worksheets Free Algebra
Two Step Equation Worksheets Printable Answers Examples
One Step Equations Worksheets Math Monks
Solve One Step Equations With Smaller Values A Algebra Worksheet Multi Worksheets Two
15 Awesome Activities To Learn Two Step Equations Teaching Expertise
Two Step Equations With Decimals Worksheets
Two Step Equations With Fractions Worksheets
Two Step Equation Worksheets Printable Answers Examples
Multi Step Equations
Solving Multi Step Equations Educational Resources K12 Learning Algebra I Expressions And Pre Math Lesson Plans Activities Experiments Homeschool Help
Solved Kuta Infinite Pre Algebra Two Step Equation Chegg Com
Algebra Equations Two Step
Kuta Prealgebra Two Step Equations With Integers You
Two Step Equations Worksheets Math Monks
One Step Equations Addition And Subtraction Worksheet Teach Starter
2 step equations worksheets k5 learning solving multi two equation printable one with integers kuta addition subtraction worksheet lovely math monks
Trending Posts
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://www.tessshebaylo.com/pre-algebra-2-step-equations/","timestamp":"2024-11-12T16:49:46Z","content_type":"text/html","content_length":"58687","record_id":"<urn:uuid:9c5ba642-997a-4da9-adb5-e24840b422f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00308.warc.gz"} |
Mother Nature’s Units & The Metric System
It is generally underappreciated in the history of science that many new scientific discoveries and theories have been realized from the study of engineering problems. Lord Kelvin (1824-1907)
famously understood this when he stated:
The steam engine has done much more for science than science has done for the steam engine.
In the early 1900s, the light bulb was of great importance, but as Sven has said to me: “a light bulb is a heater that produces light as a byproduct.” German industry and the German government wanted
a light bulb that would more efficiently produce light from the amount of electricity it dissipated. Max Planck (1858-1947), a physicist, was tasked with looking into black-body radiation. Classical
physics expected that higher frequencies would be produced by black-bodies, and an infinite amount of energy could be generated. This was known as the Ultraviolet Catastrophe. Clearly it didn’t
happen, but why?
Planck made an assumption that electromagnetic radiation (light) could only be absorbed or emitted in discreet packets of energy, called quanta. The energy of these quanta were directly proportional
to the wavelength of the electromagnetic radiation. He introduced a proportionality constant h to related the energy E in each quanta, and its frequency v. E(quanta) = hv. The constant h is now
appropriately known as Planck’s constant.
Planck noted in his 1899 paper that:
… it is possible to set up units for length, mass, time and temperature, which are independent of special bodies or substances, necessarily retaining their meaning for all times and for all
civilizations, including extraterrestrial and non-human ones, which can be called “natural units of measure”.
Wikipedia gives the original values put forward by Planck as:
Name Dimension Expression Value (SI units)
Planck length length (L) 1.616255(18)×10^−35 m^[7]
Planck mass mass (M) 2.176434(24)×10^−8 kg^[8]
Planck time time (T) 5.391247(60)×10^−44 s^[9]
Planck temperature temperature (Θ) 1.416784(16)×10^32 K^[10]
The value for Planck length is very small, outside of even the recently expanded prefixes for the metric system. The new prefix quecto is 10^-30 so the best we can do with metric prefixes for the
Planck Length is 0.000 016 162 quectometers. Wow, that’s really small.
The value for mass can be written in a more familiar manner as 21.764 micrograms. This seems downright large and relatable compared with the Planck length.
It is the definition of the Kilogram that has been of great interest in recent years. The Kilogram has been defined in terms of fundamental constants as:
Kg = (299792458)^2/(6.62607015×10^−34)(9192631770)hΔν[Cs]/c^2
The Kilogram is defined in terms of these three fundamental physical constants:
• a specific atomic transition frequency Δν[Cs], which defines the duration of the second,
• the speed of light c, which when combined with the second, defines the length of the metre,
• and the Planck constant h. which when combined with the metre and second, defines the mass of the kilogram.
One can see that a value other than those originally used by Planck is introduced. If one goes through and cancels the different dimensions they will find they all cancel, with only a Kilogram left
as the final unit. What we are doing is multiplying to get unity in terms of Mother Nature’s units:
Kg = c^2/(h Δν[Cs])hΔν[Cs]/c^2 = 1 Kilogram
The value hΔν[Cs]/c^2 is Planck’s constant, which is (Kg m^2/s) multiplied by Δν[Cs] which is (1/s) all divided by the speed of light squared which is (m/s)^2. So we have:
((Kg m^2/s)(1/s))/(m/s)^2 = Kg (m^2/s^2) (s^2/m^2) = 1 Kg
The meters cancel, the seconds cancel and we are solely left with a Kilogram. Sorry for throwing a small amount of math in, but I thought is might make how the Kilogram appears from the constants as
they are defined in SI a bit more clear. We normalize it back to obtain the Kilogram in terms of Planck Units.
The table of Planck Units given above also contains a unit of time. Physicists often are asked what happened before the Big Bang. The value they give is the Planck time:
5.391247(60)×10^−44 s
Our current models of the universe only allow us to compute backwards to this time following the Big Bang. At this point, Mother Nature does not allow us to peer behind this minimum amount of time.
The good news is, She does allow us to define a Kilogram without the use of barleycorns, or other grain. Using grain may seem natural to humans, but it’s not using the very basic values that mother
nature has offered us, so we should use them, because it’s not nice to fool with Mother Nature.
If you liked this essay and wish to support the work of The Metric Maven, please visit his Patreon Page and contribute. Also purchase his books about the metric system:
The first book is titled: Our Crumbling Invisible Infrastructure. It is a succinct set of essays that explain why the absence of the metric system in the US is detrimental to our personal heath and
our economy. These essays are separately available for free on my website, but the book has them all in one place in print. The book may be purchased from Amazon here.
The second book is titled The Dimensions of the Cosmos. It takes the metric prefixes from yotta to Yocto and uses each metric prefix to describe a metric world. The book has a considerable number of
color images to compliment the prose. It has been receiving good reviews. I think would be a great reference for US science teachers. It has a considerable number of scientific factoids and anecdotes
that I believe would be of considerable educational use. It is available from Amazon here.
The third book is called Death By A Thousand Cuts, A Secret History of the Metric System in The United States. This monograph explains how we have been unable to legally deal with weights and
measures in the United States from George Washington, to our current day. This book is also available on Amazon here. | {"url":"https://themetricmaven.com/mother-natures-units-the-metric-system/","timestamp":"2024-11-02T09:24:15Z","content_type":"text/html","content_length":"36289","record_id":"<urn:uuid:a9bd4c1d-f62b-413d-b67b-581b5c2992e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00834.warc.gz"} |
First CFD Analysis - Results Question
May 19,
2019, First CFD Analysis - Results Question #1
Member Hello,
Join Date: I did my first CFD Analysis with Ansys CFX and it worked pretty well. I have different shaped Bodys (different simulations) and I want to compare how they affect the flow (Mach 0,3 - Mach
May 2019 0,7) behind the body. I want to find out, which one has the at least bad effect on a rotor, which will be placed 200 mm behind the body. (the rotor is not part of the CFD analysis) What
would be a good approach to analyze the results I've got from the CFD Analysis?
Posts: 30
I hope you can help me, I'm not experienced (still studying) and if you need some information which I forgot, please ask me.
Rep Power:
Best Regards | {"url":"https://www.cfd-online.com/Forums/main/217623-first-cfd-analysis-results-question.html","timestamp":"2024-11-07T00:45:28Z","content_type":"application/xhtml+xml","content_length":"140356","record_id":"<urn:uuid:2e671b44-21e3-4bfa-b00b-8034861d8212>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00147.warc.gz"} |
What is 284 Celsius to Fahrenheit? - ConvertTemperatureintoCelsius.info
Converting 284 Celsius to Fahrenheit is a common task for many people who need to convert temperatures from one scale to another. Whether you’re a student, scientist, or simply someone who wants to
know what a certain temperature in Celsius would be in Fahrenheit, understanding how to make this conversion is important.
To begin the conversion, it’s important to understand the difference between Celsius and Fahrenheit. Celsius is a unit of measurement for temperature that is widely used in most countries around the
world. It is based on the freezing and boiling points of water, with 0 degrees Celsius being the freezing point and 100 degrees Celsius being the boiling point at standard atmospheric pressure.
Fahrenheit, on the other hand, is a temperature scale commonly used in the United States. It was developed by Daniel Gabriel Fahrenheit and is based on a scale where the freezing point of water is 32
degrees and the boiling point is 212 degrees at standard atmospheric pressure.
Now, let’s move on to the actual conversion. The formula to convert Celsius to Fahrenheit is as follows: (°C × 9/5) + 32 = °F. Using this formula, we can easily convert 284 degrees Celsius to
First, we’ll plug in 284 for °C in the formula: (284 × 9/5) + 32 = °F. Then we’ll simplify the equation: (568/5) + 32 = °F. After simplifying, we get: 113.6 + 32 = °F. The final step is to add 113.6
and 32, which gives us 145.6 °F. Therefore, 284 degrees Celsius is equivalent to 145.6 degrees Fahrenheit.
It’s important to note that understanding conversions between Celsius and Fahrenheit can be useful in various scenarios. For example, when traveling to a country that uses a different temperature
scale, understanding how to convert temperatures can help you dress appropriately and stay comfortable. Additionally, in scientific research and experiments, it’s crucial to be able to make accurate
temperature conversions.
In conclusion, being able to convert temperatures from Celsius to Fahrenheit is a valuable skill. By using the simple formula (°C × 9/5) + 32 = °F, we can easily convert 284 degrees Celsius to 145.6
degrees Fahrenheit. Whether for academic, scientific, or practical reasons, having this knowledge can be beneficial in many different situations. | {"url":"https://converttemperatureintocelsius.info/what-is-284celsius-in-fahrenheit/","timestamp":"2024-11-05T21:40:14Z","content_type":"text/html","content_length":"72744","record_id":"<urn:uuid:e3fe98bf-797c-4ed7-a13a-8429960d87fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00520.warc.gz"} |
Find last row and select cells
My worksheet has an undefined number of rows (different for each time).
I want to find the last row with data in it, and then put a formula (vlookup something...) into coloumn A, and stretch/copy this to all cells in Auntil I reach the last A-cell containing data.
Anyone who feel the call to help me out here??
Excel Facts
How to show all formulas in Excel?
Press Ctrl+` to show all formulas. Press it again to toggle back to numbers. The grave accent is often under the tilde on US keyboards.
Sep 22, 2002
In your vlookup formula, you are able to use a range name for Table_array once you have defined the name of the table in Insert|Name |Define.
A limitation to name ranges is the inability to use the same range name more than once in different workbooks on the same Excel file.
I need to perform the same task myself, but how might you do this using VBA. I also need to allow the code insert a new column where the formula will be inserted.
Mar 23, 2002
first you need to name the range that contains the data you are looking up. for example if cols a and b contain the following:
adam 23
bob 45
chris 24
david 34
go to insert,name then define. In the formula bit put:
=offset('Sheet1'!$A$1,0,0,counta('Sheet1'!$A:$A),2) name the thing info or something.
what this does is create a dynamic range that increases or decreases as you enter more or less data.
Suppose your inputs are in column D
You can use the following macro to see how many inputs you have and copy the vlookup formula down in column e.
sub test()
dim i as integer
i= range("D1").end(xldown).row
range("e1").value = "=vlookup(D1,info,2,false)"
range("e1:e" & i).select
end sub
P.S. this only works if there are no gaps in the inputs.
Hope that all makes sense and does what you want.Let us know how you get on.
This message was edited by bolo on 2002-10-05 08:12
THat one didn't work for me... I'll have to go to work and pick up my VBA macro there, and I'll come back to this forum with more details about my problem.
Thanks so far - seems that people are very helpfull and has great skills at this forum.
I'm really not following you guys (and/or girls) here
Lets take it from scratch:
In my MAIN.xls I have column A with i.e. these numbers:
Then I have another worksheet in another Excel-document called DATALIST.XLS.
In column A and B is the following info:
943827 Dan
5453 Eric
345 John
65456 Melissa
11111 David
43234 Ron
4343 Carrie
3 Max
34545 Jamie
725487 Bob
Then in MAIN.XLS I'd like a formula that searches the whole column A from my DATALIST.XLS, starting with cell A1 in MAIN.XLS. Then onto cell A2 and so on....
When it finds a identical number, I want the formula to put the info from column B (in DATALIST.XLS) into column C in MAIN.xls.
So in my example I would have the name David to appear in cell C1 in MAIN.XLS.
Actually I've done all this using a macro in VBA.
I could of course put the formula into the cells A1 to A10000 in MAIN.XLS, but the data in MAIN.XLS is different from each time I do this task with this macro of mine. Because I import data from a
text file to my MAIN.XLS, and the number of rows in the text file are never the same. Sometimes it could be 163 rows, the next time it could be 6932 rows. See ??
So I only need the VBA code for telling my macro that: After putting in the formula in C1 in MAIN.XLS, I want the same formula to be put into cell C2, C3, C4 and so on.
The formula should of course refer to the data in cell A at the same row as the formula is.
Did that make any more sense or what??
My other problem is that I have the Norwegian version of Excel at work, and are not that familiar with the english trems and words in VBA and Excel.
At home I have english version of Excel, so I just have to put my language skills up for a test now
Appreciate any help on this.
Mar 23, 2002
OK in your datalist.xls goto insert then tools then define. In the formula box put
in the name bit write info and click add
(this is assuming the data is in the sheet called Sheet1)
Now in your main.xls write the code into a module:
Private sub obtain_info()
dim i as integer
i= range("a1").end(xldown).row
range("c1").value = "=vlookup(a1,datalist!info,2,false)"
range("c1:c" & i).select
end sub
Now for each sheet that needs this information you can run this macro to copy the formula into the rows that require it. Hope this works for you
This message was edited by bolo on 2002-10-05 20:16
YEEHAA !!!
That sure did the trick.
Thank you very much, now I'm almost done with this little project of mine.
Viking S6 | {"url":"https://www.mrexcel.com/board/threads/find-last-row-and-select-cells.23452/","timestamp":"2024-11-04T08:36:44Z","content_type":"text/html","content_length":"132841","record_id":"<urn:uuid:a863f954-3281-49a6-a563-e3f3eca2df33>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00809.warc.gz"} |
Data Science Lab
In supervised learning, the distance or similarity measure is widely used in a lot of classification algorithms. When calculating the categorical data similarity, the strategy used by the traditional
classifiers often overlooks the inter relationship between different data attributes and assumes that they are independent to each other, for example, the overlap similarity and the frequency based
similarity. While for the numerical data, the most used Euclidean distance or Minkowski distance are all restricted in each single feature and assume the features in the data set have no outer
connections. That will cause some problem in expressing the real similarity or distance between instances and may give an incorrect result if the inter-relationship between attributes is ignored. The
same problems exist in other supervised learning, such as the classification tasks of class-imbalance or multi-label. In order to solve these research limitations and challenges, this thesis proposes
an insightful analysis on coupled similarity in supervised learning, not only used on categorical-feature data sets, but also on numerical-feature data sets and mixed type data, to give an expression
of similarity which is more closely related to the real nature of problem.
Firstly, we proposed a coupled fuzzy kNN to classify imbalanced categorical data which has strong relationships between objects, attributes and classes in Chapter 3. It incorporates the size
membership of a class with attribute weight into a coupled similarity measure, which effectively extracts the intercoupling and intra-coupling relationships from categorical attributes. As it reveals
the inner true inner-relationship between attributes, the similarity strategy we used can make the instances of each class more compact to each other when expressed by the distance. That will bring
us some goodness in dealing with the class imbalance data. The experiment results show that our supposed method has a more stable and higher average performance than the classic algorithms, such as
kNN, kENN, CCWkNN, SMOTE-based kNN, Decision Tree and NaiveBayes, especially when applied on class-imbalanced categorical data.
We also introduced a similar coupled distance for continuous features, by considering the intra-coupled relationship and inter-coupled relationship between the numerical attributes and their
corresponding extends. As detailed in Chapter 4, we present the coupling distance using the Pearson’s correlations between attributes and their square root and square. Substantial experiments have
verified that our coupled distance outperforms the original distance, and this is also supported by statistical analysis. The experiment result demonstrates that our coupling strategy on continuous
features has a great improvement in expressing the real distance between objects with numerical features.
When we consider the similarity concept, people may only relate to the categorical data, while for the distance concept, people may only think about the numerical data. Seldom have methods taken into
account both of them, especially when considering the coupling relationship between features. In Chapter 5, we proposed a new method which integrated our coupling concept for mixed type data, that is
to say, data that contains both numerical and categorical features. In our method, we first do discretization on numerical attributes to transfer such continuous values into separate groups, so as to
adopt the inter-coupling distance as we do on categorical features (coupling similarity), then we combine this new coupled distance to the original distance (Euclidean distance), to overcome the
shortcoming of the previous algorithms. The experiment results show some improvement when compared to the basic and some variants of kNN algorithms.
We also extended our coupling concept to multi-label classification tasks. The traditional single-label classifiers are known to be not suitable for multilabel tasks anymore, due to the overlap
concept of the class labels. The most used classifier in multi-label problems, ML-kNN, learns a single classifier for each label independently, so it is actually a binary relevance classifier. In
other words, it does not consider the correlations between different labels. This algorithm is often criticized for such drawbacks. To overcome this problem, we introduced a coupled label similarity,
which explores the inner relationship between different labels in multi-label classification according to their natural co-occupance. This similarity reflects the distance of the different classes.
By integrating this similarity with the multi-label kNN algorithm, we overcome the ML-kNN’s shortcoming and improve the performance significantly. Evaluated over three commonly-used verification
criteria (the Hamming Loss, One Error and Average Precision) for multi-label classifiers, our proposed coupled multi-label classifier outperforms the ML-KNN, BR- kNN and even IBLR. The result
indicates that our supposed coupled label similarity is appropriate for multi-label learning problems and can work more effectively compared to other methods.
All the classifiers analyzed in this thesis are based on our coupling similarity (or distance), and applied to different tasks in supervised learning. The performance of these models are examined by
the widely used verification criteria, such as ROC, Accuracy Rate, Average Precision and Hamming Loss. This analysis provide insightful knowledge for investors to find the inner relationship between
features in supervised learning tasks. | {"url":"https://datasciences.org/chunming-liu-coupled-similarity-analysis-in-supervised-learning/","timestamp":"2024-11-07T06:19:51Z","content_type":"text/html","content_length":"32916","record_id":"<urn:uuid:0fc16be9-9fe7-42f4-b76c-823e0cc8ded3>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00457.warc.gz"} |
eport #301
Defect Report #301
Previous Defect Report < - > Next Defect Report
Submitter: Fred Tydeman (USA)
Submission Date: 2004-08-27
Source: WG 14
Reference Document:
Version: 1.1
Date: 2006-03-05
Subject: Meaning of FE_* macros in <fenv.h>
Exactly WHERE are the MEANINGS of any of the FE_* macros defined in cases where <fenv.h> applies to an environment that is not IEEE-754 (IEC 60559)?
5.1.2.3p2 Program execution says:
Accessing a volatile object, modifying an object, modifying a file, or calling a function that does any of those operations are all side effects,^11) which are changes in the state of the
execution environment. Evaluation of an expression may produce side effects.
11) The IEC 60559 standard for binary floating-point arithmetic requires certain user-accessible status flags and control modes. Floating-point operations implicitly set the status flags; modes
affect result values of floating-point operations. Implementations that support such floating-point state are required to regard changes to it as side effects see annex F for details. The
floating-point environment library <fenv.h> provides a programming facility for indicating when these side effects matter, freeing the implementations in other cases.
The above footnote is the closest I can find to a requirement that there is any relationship between floating-point operations, status flags, and modes. But, it is a footnote, and only for IEC 60559.
5.2.4.2.2p6 Characteristics of floating types <float.h> has:
The rounding mode for floating-point addition is characterized by the implementation-defined value of FLT_ROUNDS:^18)
-1 indeterminable
0 toward zero
1 to nearest
2 toward positive infinity
3 toward negative infinity
All other values for FLT_ROUNDS characterize implementation-defined rounding behavior.
18) Evaluation of FLT_ROUNDS correctly reflects any execution-time change of rounding mode through the function fesetround in <fenv.h>.
The above mentions, but does not define, some rounding modes.
7.6p5 Floating-point environment <fenv.h> has:
Each of the macros
is defined if and only if the implementation supports the floating-point exception by means of the functions in 7.6.2. ^175) Additional implementation-defined floating-point exceptions, with
macro definitions beginning with FE_ and an uppercase letter, may also be specified by the implementation.
175) The implementation supports an exception if there are circumstances where a call to at least one of the functions in 7.6.2, using the macro as the appropriate argument, will succeed. It is
not necessary for all the functions to succeed all the time.
The above mentions, but does not define, some floating-point exceptions.
If an implementation defines a new floating-point exception, FE_BLUEMOON, such that:
• feraiseexcept(FE_BLUEMOON) succeeds,
• fetestexcept(FE_BLUEMOON) returns the current status of that "exception",
• feclearexcept(FE_BLUEMOON) succeeds,
but FE_BLUEMOON is NOT tied to any floating-point operation, is this valid "support"?
7.6p7 Floating-point environment <fenv.h> has:
Each of the macros
is defined if and only if the implementation supports getting and setting the represented rounding direction by means of the fegetround and fesetround functions. Additional implementation-defined
rounding directions, with macro definitions beginning with FE_ and an uppercase letter, may also be specified by the implementation. The defined macros expand to integer constant expressions
whose values are distinct nonnegative values.^176)
176) Even though the rounding direction macros may expand to constants corresponding to the values of FLT_ROUNDS, they are not required to do so.
The above mentions, but does not define, some rounding modes.
F.8.1p1 Global transformations says:
Floating-point arithmetic operations and external function calls may entail side effects which optimization shall honor, at least where the state of the FENV_ACCESS pragma is "on". The flags and
modes in the floating-point environment may be regarded as global variables; floating-point operations (+, *, etc.) implicitly read the modes and write the flags.
The above is a clear description of how modes and flags interact with operations, but it applies only to IEEE-754.
Suggested Technical Corrigendum
7.6 Floating-point environment <fenv.h>: Add to paragraph 5:
A necessary condition for an implementation to support a given FE_* exception is that it implicitly occur as a side effect of at least one floating-point operation. Just having feraiseexcept(),
fetestexcept() and feclearexcept() succeed for a given FE_* exception is not sufficient.
FE_INVALID should be a side-effect of:
□ operations on signaling NaN or trap representation,
□ adding infinities with different signs,
□ subtracting infinities with the same signs,
□ multipling zero by infinity,
□ dividing zero by zero and infinity by infinity,
□ remainder (x REM y), where x is infinite or y is zero,
□ square root of a negative number (excluding -0.0),
□ converting a too large to represent floating value to an integer [both signed and unsigned], e.g., int i = INFINITY; unsigned int ui = -1.0;
□ comparison with a relational operator (<, <=, >=, >) when (at least) one of the operands is a NaN.
FE_DIVBYZERO should be a side-effect of dividing a non-zero finite number by zero, e.g., 1.0/0.0. There should be no exception when dividing an infinity by zero, nor when dividing a NaN by zero.
It is implementation defined as to whether FE_INVALID, FE_DIVBYZERO, or no exception is raised for zero / zero.
FE_OVERFLOW should be a side-effect of producing a rounded floating-point result (assuming an unbounded exponent range) larger in magnitude than the largest finite number.
FE_UNDERFLOW should be a side-effect of producing a rounded floating-point result (assuming an unbounded exponent range) smaller in magnitude than the smallest non-zero finite number, or an
inexact denormal number smaller than the smallest non-zero normalized number.
FE_INEXACT should be a side-effect of producing a rounded floating-point result that differs from the mathematical (or infinitely precise) result.
Also in 7.6, change footnote 175 from "The implementation supports an exception if ..." to "The implementation supports an exception if that exception happens as a side-effect of at least one
floating-point operation and if ..."
5.2.4.2.2 Characteristics of floating types <float.h>: Add to paragraph 6:
See 7.6 Floating-point environment paragraph 7 for meaning of these rounding modes.
7.6 Floating-point environment <fenv.h>: Add to paragraph 7:
A necessary condition for an implementation to support these rounding control modes is that they can be set explicitly and that they affect result values of floating-point operations. Just having
fegetround() and fesetround() succeed for a given FE_* rounding direction is not sufficient.
FE_TOWARDZERO means the result shall be the format's value closest to and no greater in magnitude than the infinitely precise result. For example, if rounding to integer value in floating-point
format, +3.7 rounds to +3.0 and -3.7 rounds to -3.0.
FE_UPWARD means the result shall be the format's value closest to and no less than the infinitely precise result. For example, if rounding to integer value in floating-point format, +3.1 rounds
to +4.0 and -3.7 rounds to -3.0.
FE_DOWNWARD means the result shall be the format's value closest to and no greater than the infinitely precise result. For example, if rounding to integer value in floating-point format, +3.7
rounds to +3.0 and -3.1 rounds to -4.0.
FE_TONEAREST means the result shall be the format's value closest to the infinitely precise result. It is implementation defined as to what happens when the two nearest representable values are
equally near. For example, if rounding to integer value in floating-point format, +3.1 rounds to +3.0 and +3.7 rounds to +4.0, and +3.5 rounds to either +3.0 or +4.0.
Add to J.3.6 Floating point:
-- to nearest rounding result when the two nearest representable values are equally near.
-- whether FE_INVALID, FE_DIVBYZERO, or no exception is raised for zero / zero.
Add 7.6 to the index entry for floating-point rounding mode.
Committee Discussion
Footnote 173 in 7.6 paragraph 1 also describes intent of <fenv.h>. Footnote 180 in 7.6.2.3 paragraph 2 mentions exceptions as raised by floating-point operations.
Some members would like FE_BLUEMOON to be a valid macro (even though none of the basic floating-point operations would raise it). Hence, they do not want to require the FE_* macros to be side-effects
of floating-point operations.
The current FE_* macros are unspecified as that was the best compromise that could be agreed to by the various committee members when C99 was being developed.
Not really a defect, but a deficiency.
Two Heads of Delegations would like LIA-1 added as a normative reference by C99 as a way to define floating-point in C.
Several members believe that nailing down floating-point would be a good thing, but that the DR process is not the way to do it. Perhaps an amendment (similar to how wide characters were added to
C90) should be done to C99 as a way to "clean up" floating-point. Several members would like 2.0+3.0 being 5.0 to be true.
Most of the proposed TC material should be added to the C Rationale.
This material could be added to C99 as Recommended Practice.
Committee Response
This is not really a defect, but an area which could be addressed in a future revision of the C Standard.
Previous Defect Report < - > Next Defect Report | {"url":"https://www.open-std.org/JTC1/SC22/WG14/www/docs/dr_301.htm","timestamp":"2024-11-14T18:14:53Z","content_type":"text/html","content_length":"13572","record_id":"<urn:uuid:16eba85c-ab3f-4277-8f50-9b9a50ae17ca>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00456.warc.gz"} |
Fraction Divided By Whole Number Calculator
Welcome to our Fraction Divided By Whole Number Calculator. Here you can enter a fraction and a whole number (integer). We will show you step-by-step how to divide the fraction by the whole number.
Please enter your math problem below so we can show you the solution with explanation:
Copyright | Privacy Policy | Disclaimer | Contact | {"url":"https://divisible.info/FractionDividedByWholeNumber/Fraction-Divided-By-Whole-Number-Calculator.html","timestamp":"2024-11-09T14:11:24Z","content_type":"text/html","content_length":"8133","record_id":"<urn:uuid:9dcf0eba-7a96-4048-b444-94a38bd308d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00357.warc.gz"} |
What is definite and indefinite integral?
What is definite and indefinite integral?
A definite integral represents a number when the lower and upper limits are constants. The indefinite integral represents a family of functions whose derivatives are f. The difference between any two
functions in the family is a constant.
How do you write a definite integral?
After the Integral Symbol we put the function we want to find the integral of (called the Integrand).
1. And then finish with dx to mean the slices go in the x direction (and approach zero in width).
2. A Definite Integral has start and end values: in other words there is an interval [a, b].
How do you read a definite integral?
The definite integral of a positive function f(x) over an interval [a, b] is the area between f, the x-axis, x = a and x = b. The definite integral of a positive function f(x) from a to b is the area
under the curve between a and b.
What does definite mean in math?
DEFINITE NUMBER. An ascertained number; the term is usually applied in opposition to an indefinite number. 2.
What is the meaning of definite in chemistry?
Definite article, the essential law of chemical combination that every definite compound always contains the same elements in the same proportions by weight; and, if two or more elements form more
than one compound with each other, the relative proportions of each are fixed. …
How do you write a definite integral in latex?
Integral expression can be added using the \int_{lower}^{upper} command. Note, that integral expression may seems a little different in inline and display math mode.
Do definite integrals have C?
5 Answers. For any C, f(x)+C is an antiderivative of f′(x). These are two different things, so there is no reason to include C in a definite integral.
Is definite integral accurate?
accurate to an infinite number of decimal places — for the area under the smooth, curving function, x2 + 1, based on the areas of flat-topped rectangles that run along the curve in a jagged,
saw-tooth fashion. Finding the exact area of 12 by using the limit of a Riemann sum is a lot of work.
What are the properties of a definite integral?
Zero rule and Reverse Limits. The applet shows a graph of an exponential function,with the area under the curve from ato bin green.
Constant multiple rule. Select the second example from the drop down menu.
Addition rule . Select the third example.
Internal addition.
Min – Max Inequality.
Area between curves.
Crossing curves
How would we evaluate the definite integral?
So, to evaluate a definite integral the first thing that we’re going to do is evaluate the indefinite integral for the function. This should explain the similarity in the notations for the indefinite
and definite integrals. Also notice that we require the function to be continuous in the interval of integration.
How to find the indefinite integral?
The process of finding the indefinite integral is also called integration or integrating f(x). f ( x).
The above definition says that if a function F F is an antiderivative of f,f,then∫f(x)dx = F(x)+C∫f ( x) d x = F
Unlike the definite integral,the indefinite integral is a function.
What does it mean to find the integral?
Integration is the algebraic method of finding the integral for a function at any point on the graph. Finding the integral of a function with respect to x means finding the area to the x axis from
the curve. The integral is usually called the anti-derivative, because integrating is the reverse process of differentiating. | {"url":"https://yoursageinformation.com/what-is-definite-and-indefinite-integral/","timestamp":"2024-11-12T15:58:13Z","content_type":"text/html","content_length":"67964","record_id":"<urn:uuid:d9fa9d3a-d79b-4938-8f9b-70286b1547f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00481.warc.gz"} |
Learning How To Learn
I have given a pitch to middle school children a couple times where I go through various math classes that I have taken over time and explain how each of them work.
For example, I talk about algebra and how it makes the transition from arithmetic, where things can be counted, to an abstract world where operations are performed on variables. I talk about geometry
and learning how to do proofs, trigonometry and the triangle. Then I talk about calculus and how it is yet another abstraction layer from algebra, where we look at how functions perform with respect
to different variables. I also explain linear algebra, differential equations, statistics, computational mathematics, and other forms of math.
After getting the kid’s heads spinning with all this information, I tell them that I have used almost none of this math, even when I was doing serious, complex design work. The plain fact was that
the point of the education was not to have a collection of facts and ability to solve the homework problems.
The point that I try to drive home is that my education was really about learning how to learn. It was not about the details.
I show the kids how solving geometric proofs teaches how to construct arguments that are used in politics, and how the various layers of abstraction in algebra and calculus teach the ability to
analyze social interaction problems on different levels and with different layers. In each case, even though I do not solve differential equations at work, I use many of the skills acquired in those
classes in many different ways.
After talking with the kids for an hour or so, I ask them to tell me which class they think was the single most valuable class I ever took. After many different answers, they never get the correct
The most important and single most useful class I have ever taken in any form and any level of education was Typing. I tell the kids that in today’s society, if you never learn to type, it will be
like trying to run with a limp for the rest of your life.
BTW, when I took Typing in 1979 on the manual typewriters, there were only a couple boys in the class, and about 30 girls. Admittedly, I wanted to take it so I could run the brand new Digital PDP 11
that was the pride and joy of the newly formed computer club, even though it only had one CRT terminal and three dot matrix printer terminals. I fondly remember all of us huddling around the CRT
playing Adventure after school. About then, my dad bought a Z-80 based Northstar computer system for home, running CP/M with 48K of RAM. That is where I cut my teeth writing code. Even though Typing
was the most useful class in high school, it was the only one in which I got a C. | {"url":"http://krajec.com/learning-how-to-learn/","timestamp":"2024-11-03T22:18:46Z","content_type":"text/html","content_length":"48798","record_id":"<urn:uuid:fea192d4-aeff-4f12-894f-81f92f4ad091>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00142.warc.gz"} |
Online calculators
Side length of the regular polygon
Calculates the side length of the regular polygon circumscribed or inscribed to a circle. Created by user's request
Circle formulas
Circle formulas
Counting pipes from the end face
The approach to the calculation of the number of nested circles of smaller radius at a known length of the describing circle of a larger radius. Created by user request
Circumcircle of a triangle
This online calculator determines the radius and area of the circumcircle of a triangle given the three sides
Triangle Incircle Calculator: Radius, Area, and Ratio
The Triangle Incircle Calculator is a tool that allows you to determine the properties of the incircle of a triangle based on its side lengths. By entering the lengths of the three sides&
comma; this calculator calculates the radius and area of the incircle, which is the largest circle that can fit inside the triangle.
Regular Polygon Incircle and Circumcircle Calculator
This page presents two online calculators: the Incircle of a regular polygon calculator and the circumcircle of a regular polygon calculator.
Circular segment
Here you can find the set of calculators related to circular segment: segment area calculator, arc length calculator, chord length calculator, height and perimeter of circular
segment by radius and angle calculator.
Regular polygon, number of sides and length of side from incircle and circumcircle radii
This online calculator finds number of sides and length of side of regular polygon given the radii of incircle and circumcircle
How many circles of radius r fit in a bigger circle of radius R
This calculator estimates how many circles of radius r can be placed inside another circle of radius R.
Find the intersection of two circles
This online calculator finds the intersection points of two circles given the center point and radius of each circle. It also plots them on the graph.
Equation of a circle calculator
This circle equation calculator displays a standard form equation of a circle, a parametric form equation of a circle, and a general form equation of a circle given the center and radius
of the circle. Formulas can be found below the calculator.
Equation of a circle passing through 3 given points
This online calculator finds a circle passing through three given points. It outputs the center and radius of a circle, circle equations and draws a circle on a graph. The method
used to find a circle center and radius is described below the calculator.
Arc length calculator
This universal online calculator can find arc length of circular segment by radius and angle, by chord and height and by radius and height.
Area-to-Radius Calculator
This online calculator calculates the radius of a circle from the given area of a circle.
Cutting a circle
Two ways to cut a circle into equal parts : sector cuts and parallel cuts.
General to Standard Form Circle Converter
The calculator takes the equation of a circle in general form, with variables for x, y, and constants a, b, c, d and e, and converts it to the standard form
equation for a circle with variables h, k, and r. It then calculates the center of the circle (h, k) and its radius r. If the equation cannot be converted to
standard form, the calculator reports an error message. | {"url":"https://embed.planetcalc.com/search/?tag=1103","timestamp":"2024-11-12T19:58:14Z","content_type":"text/html","content_length":"20789","record_id":"<urn:uuid:989d4a79-6a80-4c0a-83e5-f38f08c36b35>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00464.warc.gz"} |
Theory group
• Dmitri SEMIKOZ (responsible)
• Eric HUGUET (deputy responsible)
The theorists at APC have a very broad range of interests. Their research is both closely linked with observations and focused on fundamental theories. A summary of the main research interest is
given below.
For more information, see the
lab's activity report (2017-2021)
Main scientific interests
String theory and holography
String theory is a theoretical framework that seeks to unify all known physical forces, including gravity, at the quantum level. One of its key insights is the Gauge/Gravity duality, also known as
the Holographic correspondence. This conjecture states that a quantum field theory in lower dimensions is equivalent to a gravitational theory in higher dimensions. This has important implications
for understanding the relationship between quantum field theory, string theory, and general relativity. Additionally, it can be used as a tool to study strongly coupled field theories, such as QCD,
and has been applied to other areas such as high-temperature superconductors and strongly correlated quantum systems. Holographic correspondence has also led to new connections between fluid dynamics
and gravity, as well as information theory and entanglement. It has also played a significant role in developing theories of gravity beyond Einstein's General Relativity.
• Holographic description of QCD and strongly coupled phases of matter
The gauge/gravity duality, also known as holographic correspondence, is a conjecture in string theory that equates a quantum field theory with a gravitational theory in higher dimensions. It has
been used to study strongly coupled field theories such as QCD, and has potential applications in condensed-matter systems and neutron stars. It has also been used to engineer theories that match
the properties of QCD, like confinement, mass gap, and running of the coupling constant, allowing the understanding of their phase diagrams at finite temperature and baryon-number density. This
approach may have interesting implications for the study of neutron stars and gravitational waves.
• Emergent gravity and the Self-tuning of the cosmological constant
The gauge/gravity correspondence suggests that gravity is an effective low-energy theory of a high-energy ordinary QFT, with the graviton being a composite of high-energy degrees of freedom. This
connection allows researchers to attack the cosmological constant problem, in which the observed small value of the cosmological constant today is in contrast with the large value expected from
effective field theory. Researchers have proposed a framework, based on holography, which suggests that our observed 4D universe is a defect embedded in a 5D curved spacetime, with the fields of
the Standard Model confined to the defect and gravity able to propagate in the bulk. This framework leads to a stabilization of the defect, resulting in a flat spacetime geometry, regardless of
the localized vacuum energy.
• Holographic Renormalization Group of QFTs on curved manifolds
In gauge/gravity duality, the QFT renormalization group is given a geometric representation as evolution along a radial dimension of the higher-dimensional dual spacetime. We have been
investigating the nature of these holographic RG flows, and have developed a general framework for analyzing holographic field theories on curved spacetimes.
Quantum Field Theory
• Quantum Field Theory in curved spacetime
The study of quantum effects in strong gravitational backgrounds is a subject of topical interest in cosmology and astrophysics. Paradigm examples are the gravitational amplification of
primordial density perturbations in inflationary cosmology or the Unruh-Hawking radiation from black holes which are cornerstones of modern cosmology and of the fundamental understanding of black
hole physics. More generally, understanding situations where both quantum and gravitational effects come into play sheds a new light onto the laws of physics at work and may give some insight
concerning more fundamental laws.
• Conformal methods for fields in curved geometries
The group studies quantization and quantum field theory on manifolds, focusing on conformal scalar and Maxwell fields in de Sitter and Robertson Walker spaces. The goal is to obtain exact and
explicit expressions for objects such as two-point functions, allowing for practical calculations and interpretations in curved backgrounds. A formalism based on differential geometry has been
developed, which generalizes the Dirac's six-cone formalism and converts Maxwell equations in a conformal gauge to a set of conformal scalar equations. The group is currently working on applying
this formalism and generalizing the use of conformal transformations to non conformally-invariant equations.
• Interacting fields in de Sitter space
The group studies interacting scalar fields in de Sitter space, focusing on light fields in units of the inverse curvature, which are of interest for inflation. Perturbative techniques often
result in divergent contributions, which must be resummed to obtain reliable answers. The group develops and applies resummation methods, such as the p-representation of correlators, large-N
techniques, 2PI techniques and renormalization group techniques, to questions of physical interest in the context of inflationary physics. Recent original results include the observation that
quantum contributions to non-Gaussian correlators can contribute the same as tree-level contributions, and the fact that amplified quantum fluctuations can lead to the restoration of symmetries
in O(N) theories in any space-time dimension.
• Integral quantization of geometries
Integral quantization is a method of giving a classical object a quantum version by quantizing various geometries like symplectic manifolds. It is based on operator-valued measures and has
probabilistic aspects at each stage of the procedure. Various types of integral quantization exist, like Berezin or Klauder quantization, and coherent state quantization. It is well established
that the Weyl-Heisenberg group underlies the canonical commutation rule. The approach also includes less familiar quantization based on the affine group of the real line, which is useful for
dealing with gravitational singularities on a quantum level in quantum cosmology. The main issue of this approach is the appearance of a quantum centrifugal potential that allows for
regularization of the singularity, self-adjointness of the Hamiltonian, and unambiguous quantum dynamics.
• Nonabelian gauge theories and confinement
The understanding of the long distance aspects of the theory of strong interactions is a major open question in high-energy physics, particularly the quantitative description of confinement in
non-abelian gauge theories. Numerical calculations on the lattice are currently the main tool to address this regime, but are limited in their application. The Theory group studies various
aspects of Yang-Mills theories and QCD related to the infrared regime, confinement, and the quark-gluon plasma. They also propose a new approach to gauge-fixing in Yang-Mills theories, which
takes into account Gribov copies, and an alternative approach using holographic Gauge/Gravity duality.
• Early-universe cosmology and primordial black holes
Primordial Black Holes (PBHs) are attracting increasing attention due to recent observations of gravitational waves from black-hole mergers and unresolved questions in Cosmology. PBHs could form
from large quantum fluctuations in the early universe, offering an opportunity to study the physics during cosmic inflation. This theory group made a number of contribution on this important
question. The "stochastic-δN formalism" was developed to understand quantum backreaction in shaping the early universe and shows that PBHs could have a heavy exponential tail, deviating from
Gaussian statistics. The theory also shows that "metric preheating" could produce ultra-light PBHs, which could have dominated the universe before evaporating. The standard cosmological paradigm
raises fundamental issues and new approaches are being proposed to better understand the quantum origin of cosmological structures. Extending tools developed in quantum information theory to the
realm of quantum cosmological fields, new approaches have thus been proposed to better describe (and hopefully reveal) genuine quantum properties of the primordial fluids.
• Dark energy and modified gravity
Dark energy models are often based on scalar-tensor theories, and the most general framework for these theories is known as Degenerate Higher-Order Scalar-Tensor (DHOST) theories. However, a
constraint on the speed of gravitational waves from the binary neutron star merger GW170817 places severe restrictions on DHOST theories. Some researchers argue that this constraint does not
necessarily apply to dark energy models, as it corresponds to a scale much smaller than cosmological scales. Relaxing this constraint allows for a much richer phenomenology in dark energy models.
An important aspect of these models is their linear stability, and researchers have used the Effective Theory of Dark Energy to examine this aspect within DHOST theories, providing a powerful
tool for studying cosmological perturbations in a wide range of dark energy models.
• Inflation and cosmological perturbations
Inflation is a proposed phase of accelerated expansion in the early universe that explains the origin of primordial fluctuations observed in the CMB. The nature of the inflaton field driving
inflation is still unknown. The group has studied various models of inflation, focusing on those involving multiple scalar fields and their potential signatures in present or future data,
including non-Gaussianities and isocurvature perturbations. The group also examines the link between inflation and approximate conformal invariance and scaling in Quantum Field Theory through the
gauge/gravity duality. This approach offers a new perspective on the standard problems of inflation.
Gravity and cosmology
• Gravitational waves
In 2016, the LIGO interferometers directly detected a gravitational wave signal for the first time. The LISA Pathfinder satellite also demonstrated the technology needed to detect gravitational
waves from space. The detection of gravitational waves opens a new window on the universe and allows us to detect objects that are invisible through electromagnetic radiation. The gravitational
wave detection also provides us with a unique opportunity to test the universe through a new messenger and the theory group is exploring this opportunity in particular concerning the potential of
the LISA mission to probe cosmology.
• General relativity and modified gravity theories
The study of possible deformations of general relativity is of major theoretical and phenomenological importance. The group has worked on several theories of modified gravity, including f(R) and
chameleon theories, with application to the structures of stars and spherical collapse, as well as Galileon models. Group members worked on the ghost-free formulation of massive gravity, its
cosmological solutions, and the Vainshtein mechanism. The dynamics and problems of massive gravity are also investigated by using the holographic link to Quantum Field Theory. The dynamics and
problems of massive gravity are also investigated by using the holographic link to Quantum Field Theory. This same link makes massive gravity a model for studying momentum dissipation in finite
density strongly coupled systems with potential applications to condensed matter physics.
• Gauge gravity
In Teleparallel Equivalent to General Relativity (TEGR) theory, gravity is encoded in torsion rather than curvature, through the Weitzenböck connection. The theory leads to the same predictions
as General Relativity. With collaborators, we proposed a formulation of TEGR using a Cartan connection, showing how the usual gauge theory formalism must be modified to interpret it as a gauge
theory for the translation group. This allows for a coherent mathematical framework for describing TEGR, including matter coupling.
• Neutrino physics
Neutrinos tell us stories from far away in space and time, and have key unknown properties to reveal. They include their absolute mass and mass ordering, leptonic CP violation, the neutrino
Majorana or Dirac nature and the existence of a fourth sterile neutrino. These measurements will bring fundamental bricks for physics beyond the Standard Model of particles and interactions. The
theory group works at the forefront of this research, investigating fundamental aspects and conceiving new avenues in close connection with experiments.
• UHE neutrinos
Ultra-High Energy (UHE) cosmic rays produce secondary charge pions on background fields in the sources or in intergalactic space. Neutrinos produced during propagation are called "cosmogenic
neutrinos". Flux of those neutrinos depends on the unknown distribution of sources and on the initial proton spectrum produced at those sources. Experiments like ANITA, AUGER and Icecube will be
able to study this flux. For direct neutrino flux from the sources even backgrounds are unknown or at least model dependent. This make predictions of neutrino flux from sources even more
difficult. We are developing theoretical models of the UHE neutrino sources.
• Cosmological neutrinos at the epoch of big-bang nucleosynthesis
Big Bang Nucleosynthesis (BBN) is one of key stones of cosmology. When the Universe cools down to sub-MeV temperatures, the plasma is not hot enough anymore to destroy light nuclei, produced from
protons and remaining neutrons. The abundances of light elements is then governed by the neutron-to-proton ratio, which in turn depends on reactions with electron neutrinos and anti-neutrinos as
well as neutron decay, and on the total energy density of Universe at that time. The observed abundances of light elements strongly restrict any new physics connected with MeV scales. This offers
a poweful tool to constrain the parameter spaces of exotic models or novel particles such as sterile neutrinos.
• Low energy weak interaction and neutrino physics
Low energy weak interaction and neutrino physics has brought milestone in the build up of the Standard Model. Nowadays it is a powerful tool to search for new physics beyond it. The group has
been strongly contributing to this domain by predicting neutrino-nucleus cross sections crucial for the interpretation of oscillation experiments, exploring the connection with the lepton-flavor
violating neutrinoless doble-beta decay, proposing experiments nearby existing facilities such as spallation source ones (ESS), investigating other low energy weak processes. The group is also
renown for the proposal of a novel neutrino facility in the 100 MeV energy range based on a new concept : the low energy beta-beam. This facility is of great interest for nuclear physics,
neutrino and supernova physics.
• Cosmic Rays
Despite cosmic rays was discovered more than 100 years ago, their origin is still unknown. We suggested that local supernovae of 2 Myr can explain the number of anomalies in cosmic ray data. The
same supernovae can be a reason for the evolution of the climate and life diversity on Earth. The group developed a theory of galactic and extragalactic cosmic ray propagation. We discovered new
types of cosmic ray diffusion around their sources. We developed a theory of anisotropic cosmic ray diffusion in galaxy.
• Intergalactic magnetic fields
The signal excess recently discovered by several Pulsar Timing Arrays experiments can be explained in terms of a stochastic gravitational-wave background due to a primordial magnetic field at the
QCD scale. The Intergalactic Magnetic Field (IMF) in the voids of large scale structure is dominated by the contribution of primordial magnetic fields, possibly created during inflation or at
phase transitions in the Early Universe. Primordial magnetic fields can be probed via measurements of secondary gamma-ray emission from gamma-ray interactions with extragalactic background light.
Lower bounds on the magnetic field in the voids were derived from the non-detection of this emission.
• Multimessenger astrophysics
Astrophysical neutrinos have been discovered about 10 years ago, but the nature of their source is still unknown. We worked on multimessenger astrophysics which neutrinos and gamma-rays at
energies above TeV. In particular we constructed models of galactic and extralactic neutrino sources and compared their predictions with gamma-ray and neutrino observations. | {"url":"https://apc.u-paris.fr/FACe/en/theory","timestamp":"2024-11-03T03:29:23Z","content_type":"text/html","content_length":"47924","record_id":"<urn:uuid:960e9e99-e964-4188-a1bf-1643071dba67>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00835.warc.gz"} |
Sublattices and subrings of Z^n and random finite abelian groups (Nathan Kaplan, UC Irvine) - Claremont Center for the Mathematical Sciences
Sublattices and subrings of Z^n and random finite abelian groups (Nathan Kaplan, UC Irvine)
March 26 @ 12:15 pm - 1:10 pm
How many sublattices of Z^n have index at most X? If we choose such a lattice L at random, what is the probability that Z^n/L is cyclic? What is the probability that its order is odd? Now let R be
a random subring of Z^n. What is the probability that Z^n/R is cyclic? We will see how these questions fit into the study of random groups in number theory and combinatorics. We will discuss
connections to Cohen-Lenstra heuristics for class groups of number fields, sandpile groups of random graphs, and cokernels of random matrices over the integers.
Related Events | {"url":"https://colleges.claremont.edu/ccms/event/antc-seminar-nathan-kaplan-uc-irvine/","timestamp":"2024-11-10T21:04:06Z","content_type":"text/html","content_length":"205743","record_id":"<urn:uuid:987bd6c0-46bc-4e80-a908-89d84680bb27>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00411.warc.gz"} |
Express the cross product of given vectors in terms of the unit coordinate vectors - Stumbling Robot
Express the cross product of given vectors in terms of the unit coordinate vectors
First, we have
So, the cross product is given by
Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): | {"url":"https://www.stumblingrobot.com/2016/05/01/express-cross-product-given-vectors-terms-unit-coordinate-vectors/","timestamp":"2024-11-10T09:08:53Z","content_type":"text/html","content_length":"58365","record_id":"<urn:uuid:b214538e-6941-45a5-bb2d-f19c407f1494>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00436.warc.gz"} |
Math for challenging brains - Rama Manikanta
Math for challenging brains
here are few mathematical puzzles to challenge your bdrain
#C6CEEA #CCD5F4 #E2E3E7 #F07F0B
1. In a ten-digit number, first represents number of “1” present in the number, second digit represents no of “2” in that number and third digit represents no of “3” in that number and so on ninth
digit represents no of nines in that number and tenth digit represents no of “0” in that number. Then what is that ten digit number?
We start with the smallest possible ten digit number is 1000000009. Now, since last digit represents number of zeros, it should be 9.
Hence, the number becomes 1000000009.
Since, we have only 8 zeros so the last digit should be 8. Hence, the number becomes 1000000008.
Since, number of “8” present in the above case is one, next modification will be 1000000108.
Now, the number of “0” is only seven. Hence, the
number is modified to 1000000107.
Since, the number of “7” present is one so next modification will be 10000001007.
Now, since the number of “1” present is two, the next modification will be 2000001007.
Number of “2” present is to be one so the next modification will be 2100001007.
Here, since number of zeros present is six, the next modification will be 2100001006.
Now, since number of “6” present is 1, so the next modification will be 2100010006.
The number 2100010006 satisfies all the condition so this is the desired result.
2.without actually calculating x, show that \\ \text{If } x+1/x=1,\enskip \text{then prove } x^7+1/x^7=1
\text{Since } x+1/x=1, \\\text{then multiplying through by x gives } x^2+1=x \text{ or, rearranging }, x^2=x-1 \\ \text{Multiplying through by x again,} \\ x^3=x^2-x \\ \text{or, substituting our
expression for } \\ x^2, x^3=x-1-x=-1 \\ \text{Squaring this gives }\\ x^6=1,\\ \text{and then multiplying through by x we have } \\ x^7=x \\ \text{So, substituting }x^7 \text{ for x in the original
} \\ x+1/x=1 \\ \text{we have } \\ x^7+1/x^7=1 \\ \text {and we are done}. | {"url":"https://ramamanikanta.com/math-puzzles-for-adults/","timestamp":"2024-11-08T12:22:07Z","content_type":"text/html","content_length":"118898","record_id":"<urn:uuid:034433b0-ce76-46a8-8608-4e93342f53a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00174.warc.gz"} |
NEB Class 11 Mathematics Notes | All Chapter Notes - WebNotee
NEB Class 11 Mathematics Notes | All Chapter Notes
Class 11 mathematics solution PDF collection of all chapters in one place. You can scroll down and click on "Get Notes" button of the respective chapters to get the complete solutions.
NEB Class 11 Mathematics Notes. Every student studying in Grade 11 needs help when solving mathematics questions. You are not the only one who faces difficulties while solving mathematics questions.
When we get stuck on certain questions, we need a mathematics guide. To be more precise, we need a Class 11 Mathematics solution PDF.
NEB Class 11 Mathematics Notes [UPDATED]
In class 11 we study Basic Mathematics and most of the school uses the book by Sukunda Pustak Vawan. It is considered as one of the suitable book for grade 11 as it contains exercises, MCQs as well
as additional questions which can help students to score good marks in mathematics.
Note: We may not have the solutions of some chapters. If you have then you can submit us through G-mail.
I. Algebra
II. Trigonometry
8 Inverse Circular Functions and Trigonometric Equation Get Notes
III. Analytic Geometry
9 Analytical Geometry Get Notes
10 Pair of Straight Line Get Notes
11 Coordinates in Space Get Notes
IV. Vectors
V. Statistics and Probability
VI. Calculus
15 Limits and Continuity Get Notes
16 The Derivatives Get Notes
17 Application of Derivatives Get Notes
18 Antiderivatives Get Notes
VII. Computational Method
Class 11 Basic Mathematics Text Book
Basic Mathematics Text Book For Class 11 by Sukunda Pustak Bhawan according to the updated syllabus is now available at cheap price.
Why NEB Class 11 Mathematics Notes Matter?
Class 11 Mathematics can be challenging for students and having access to reliable solutions is very important for students. Solutions can play a vital role by providing step-by-step explanations on
how to solve mathematical problems. They serve as helpful guides which can help you in understanding the concepts and techniques involved.
The importance of class 11 mathematics solutions goes beyond simply obtaining correct answers. It can help students by enhancing their overall knowledge of the respective subject. Furthermore, these
solutions serve as valuable resources for students, equipping them with strategies to tackle similar problems in the future.
With the help of class 11 mathematics solution, you can easily solve any mathematics problems of any chapter of gade 11. I’m telling that because it contains the notes of almost all chapters that
you have to study in grade 11.
Strategies for Effective Use of Class 11 Mathematics Notes PDFs
Class 11 mathematics solution PDFs contains all questions solutions of the entire chapter taught in grade 11. But, you shouldn’t just copy each and every solution from the PDF. If you do that then
you will get weaker in mathematics day by day. I strongly recommend you to use this PDFs wisely. Don’t overuse it because it has negative sides too.
You should look at the PDFs solutions only when you get stuck while solving the questions given in your exercise. Just look the process of solving the specific questions and close the PDF and try
solving that question yourself. If you use the PDFs in right way then it will be beneficial for you.
At last I want to suggest you to give your 100% best while solving mathematics questions. It is because if you solve the questions yourself then you won’t forget it for a long time. So at least try
2-3 times before looking at the PDF for the solution of the questions.
About NEB Class 11 Mathematics Notes
Welcome to our Class 11 Mathematics Notes – an incredible collection tailored for each chapter covered in your mathematics curriculum. These notes act as your friendly guide, simplifying complex
concepts into easily understandable bits.
Mathematics can sometimes feel like decoding a puzzle, right? But worry not! These notes are here to demystify theories, acting as your mentor explaining each concept step by step.
Finding what you need is a breeze! Just click on the chapter’s name, and presto! The notes appear, at your service. It’s like having a virtual repository where each chapter is neatly summarized.
But hey, these notes aren’t just for reading! They’re your ally during practice sessions and exam prep. They cover everything comprehensively, ensuring you grasp every aspect.
With these notes, mastering Class 11 mathematics becomes much simpler. They’re not just notes; they’re your trustworthy study companion, aiding you through the world of mathematics. Clear, easy to
understand, and accessible whenever you need them. So, with these notes by your side, navigating through Class 11 mathematics will be an exhilarating journey into the realm of numbers!
2 Comments 2 Comments
• Trigonometry ra Curve sketching ko xaina ta 🥲
• if you have then you can send me
Leave a Reply Cancel reply
You Might Also Like
TAGGED:Class 11 GuideGuides
By Sanjeev Senior Editor
Hello, I'm Sanjeev Mangrati. Writing is my way of sharing thoughts, perspectives, and ideas that empower me. I thoroughly enjoy writing and have published many informative articles.I believe
knowledge and understanding can put you one step ahead in the clamorous race of life! | {"url":"https://webnotee.com/neb-class-11-mathematics-notes-all-chapter/","timestamp":"2024-11-07T06:00:40Z","content_type":"text/html","content_length":"133875","record_id":"<urn:uuid:36a90ea3-9a0c-44ac-897a-68409864f661>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00294.warc.gz"} |
Multiple Bar Chart Questions 2024 - Multiplication Chart Printable
Multiple Bar Chart Questions
Multiple Bar Chart Questions – You may create a Multiplication Chart Club by marking the posts. The kept line ought to say “1” and signify the total amount multiplied by 1. Around the right-hand
aspect of your dinner table, label the columns as “2, 4, 6 and 8 and 9”. Multiple Bar Chart Questions.
Tricks to understand the 9 instances multiplication table
Learning the nine times multiplication desk is just not a simple task. Counting down is one of the easiest, although there are several ways to memorize it. In this trick, you place the hands around
the desk and variety your fingers one by one in one to ten. Fold your 7th finger to enable you to begin to see the ones and tens upon it. Then count up the number of fingertips on the left and
correct of your folded away finger.
When understanding the table, kids could be afraid of larger amounts. This is because introducing larger amounts frequently turns into a laborious task. However, you can exploit the hidden patterns
to make learning the nine times table easy. One of the ways would be to compose the nine instances table on a cheat sheet, read it noisy, or exercise creating it down frequently. This process can
make the kitchen table a lot more remarkable.
Habits to find on a multiplication graph
Multiplication graph cafes are perfect for memorizing multiplication specifics. You will discover the merchandise of two amounts by exploring the columns and rows of your multiplication chart. For
instance, a column which is all twos and a row that’s all eights should meet up with at 56. Habits to look for over a multiplication graph or chart bar are exactly like those who work in a
multiplication table.
A design to consider over a multiplication chart will be the distributive house. This residence might be observed in all of the posts. By way of example, something by two is the same as five
(instances) c. This very same residence relates to any line; the sum of two columns is equal to the need for one other line. As a result, a strange amount instances a much number is surely an even
amount. Exactly the same pertains to the items of two unusual amounts.
Creating a multiplication chart from recollection
Creating a multiplication graph or chart from memory can help kids discover the distinct amounts from the times dining tables. This straightforward workout will permit your child to remember the
amounts and find out the way to increase them, which can help them in the future after they get more information difficult math concepts. For a enjoyable and easy way to commit to memory the amounts,
you can organize coloured control keys to ensure that every one corresponds to particular instances desk amount. Ensure that you label each and every row “1” and “” so that you can rapidly establish
which amount will come first.
After children have learned the multiplication graph pub from memory, they should commit them selves on the job. For this reason it is far better to use a worksheet instead of a classic laptop
computer to train. Vibrant and animated persona templates can attract the senses of your own young children. Let them color every correct answer before they move on to the next step. Then, exhibit
the graph in their study location or bed rooms to work as a reminder.
Employing a multiplication graph or chart in your everyday living
A multiplication chart demonstrates how to multiply phone numbers, a person to twenty. It also demonstrates the product of two amounts. It could be valuable in everyday life, like when splitting up
money or collecting data on men and women. These are the approaches you can use a multiplication chart. Rely on them to help your child be aware of the principle. We now have described just some of
the most frequent ways to use multiplication furniture.
Use a multiplication chart to assist your child discover ways to minimize fractions. The secret is always to follow the numerator and denominator on the left. By doing this, they will likely notice
that a small percentage like 4/6 could be decreased to a small fraction of 2/3. Multiplication maps are specifically ideal for youngsters mainly because they assist them to acknowledge amount
patterns. You will discover Totally free printable variations of multiplication chart night clubs online.
Gallery of Multiple Bar Chart Questions
Bar Graphs Sheet 2D Travel To School Survey Answers Bar Graph
Pin By Taufan Lubis On Matplotlib Bar Graphs Chart Graphing
Double Bar Graph Examples With Questions Free Table Bar Chart
Leave a Comment | {"url":"https://www.multiplicationchartprintable.com/multiple-bar-chart-questions/","timestamp":"2024-11-06T11:58:53Z","content_type":"text/html","content_length":"54679","record_id":"<urn:uuid:0f61d7db-73de-4119-952c-6997648eb658>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00870.warc.gz"} |
06-30, 12:00–12:30 (Europe/Tirane), UBT E / N209 - Floor 3
This paper presents different approaches to map bark beetle infested forests in Croatia. Bark beetle infestation presents threat to forest ecosystems and due to large unapproachable area presents
difficulties in mapping infested areas. This paper will analyse available machine learning options in open-source software such as QGIS and SAGA GIS. All options will be performed on Copernicus data,
Sentinel 2 satellite imagery. Machine learning and classification options that will be explored are maximum likelihood classifier, minimum distance, artificial neural network, decision tree, K
Nearest Neighbour, random forest, support vector machine, spectral angle mapper and Normal Bayes. Maximum likelihood algorithm is considered the most accurate classification scheme with high
precision and accuracy, and because of that it is widely used for classifying remotely sensed data.
Maximum likelihood classification is method for determining a known class of distributions as the maximum for a given statistic. An assumption of normality is made for the training samples. During
classifications all unclassified pixels are assigned to each class based on relative probability (likelihood) of that pixel occurring within each category’s probability density function.
Minimum distance classification is probably the oldest and simplest approach to pattern recognition, namely template matching. In a template matching we choose class or pattern to be recognized, such
as healthy vegetation. Unknown pattern is then classified into the pattern class whose template fits best the unknown pattern. Unknown distribution is classified into the class whose distribution
function is nearest (minimum distance) to the unknown distribution in terms of some predetermined distance measure.
A decision tree is a decision support tool that uses a decision tree model and its possible consequences, including the outcomes of random events, resource costs, and benefits. It's a way of
representing an algorithm that contains only conditional control statements. Decision trees are commonly used in operations research, particularly in decision analysis to identify the strategy most
likely to achieve a goal, but they are also a popular tool in machine learning.
K Nearest Neighbour is a simple algorithm that stores all the available cases and classifies the new data or case based on a similarity measure. It is mostly used to classifies a data point based on
how its neighbours are classified.
Random forests or random decision forests is an ensemble learning method for classification, regression and other tasks that operates by constructing a multitude of decision trees at training time.
For classification tasks, the output of the random forest is the class selected by most trees. For regression tasks, the mean or average prediction of the individual trees is returned.
Support vector machines (SVM) are supervised learning models with associated learning algorithms that analyse data for classification and regression analysis. SVMs are one of the most robust
prediction methods, being based on statistical learning frameworks. Given a set of training examples, each marked as belonging to one of two categories, a SVM training algorithm builds a model that
assigns new examples to one category or the other, making it a non-probabilistic binary linear classifier.
Spectral image mapper is a spectral classifier that can determine spectral similarity between image spectra and reference spectra by calculating the angle between the spectra, treating them as
vectors in a space with dimensionality equal to the number of bands used each time. Small angles between the two spectrums indicate high similarity, and high angles indicate low similarity.
Bayesian networks (normal Bayes) are a type of probabilistic graphical model that uses Bayesian inference to calculate probability. Bayesian networks aim to model condition dependence by representing
conditional dependence by edges in directed graph. Bayesian networks are designed for taking an event that occurred and predicting the likelihood that any one of possible known causes was a factor.
Copernicus, also known as Global Monitoring for Environment and Security (GMES) is a European program for the establishment of European capacity for Earth observation. European Space Agency is
developing satellite missions called Sentinels where every mission is based on constellation of two satellites. Main objective of Sentinel-2 mission is land monitoring and it is performed using
multispectral instrument. Sentinel-2 mission is active since 2015. Sentinel-2 mission carries multispectral imager (MSI) covering 13 spectral bands. Sentinel 2 mission produces two main products,
level-1C and level-2A. Level-1C products are tiles with radiometric and geometric correction applied. Geometric correction includes orthorectification. Level-1C products are projected combining UTM
projection and WGS84 ellipsoid. Level-2A products are considered as the mission Analysis Ready Data.
Each method is evaluated with error matrix and each method is compared to each other. A confusion matrix, also known as an error matrix, is a specific table layout that allows visualization of the
performance of an algorithm, typically a supervised learning one (in unsupervised learning it is usually called a matching matrix). Each row of the matrix represents the instances in an actual class
while each column represents the instances in a predicted class, or vice versa – both variants are found in the literature. The name stems from the fact that it makes it easy to see whether the
system is confusing two classes. Each error matrix contains Kappa value. Kappa coefficient is a statistic that is used to measure inter-rater reliability for qualitative (categorical) items. It is
generally thought to be a more robust measure than simple percent agreement calculation, as κ considers the possibility of the agreement occurring by chance.
All analyses are performed on data located in Republic of Croatia, Primorsko-goranska county.
Assistant professor at Faculty of Geotechnical Engineering, University of Zagreb. | {"url":"https://talks.osgeo.org/foss4g-2023/talk/PBV3F7/","timestamp":"2024-11-10T22:05:11Z","content_type":"text/html","content_length":"25630","record_id":"<urn:uuid:391cc462-70ff-4fd5-9fa2-032a45057dad>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00610.warc.gz"} |
Drishyam Day 6 (Wednesday) Box Office Collections
Drishyam is sustaining well at the domestic box office, as the film has added Rs 4.1 crore to its total on Wednesday. The trend is good, but the film still has a long way to go before it can recover
The landing costs of the Ajay Devgn starrer was around Rs 65 crore and the theatrical share at the end of Week 1 will be around the 24 crore mark.
Drishyam has to do well in Week 2 to recover costs and go on to be an average grosser at the box office.
Day Drishyam Growth / Drop Percentage
Preview 0.5 crore -
Day 1 (Friday) 5.3 crore -
Day 2 (Saturday) 7.5 crore + 29%
Day 3 (Sunday) 9.75 crore + 30%
Day 4 (Monday) 3.65 crore - 63%
Day 5 (Tuesday) 3.5 crore - 4%
Day 6 (Wednesday) 3.3 crore - 6%
Day 7 (Thursday) 3.15 crore - 5%
Day 8 (2nd Friday) 2.75 crore - 13%
Day 9 (2nd Saturday) 4.05 crore + 47%
Day 10 (2nd Sunday) 5.04 crore + 24%
Day 11 (2nd Monday) 1.60 crore - 68%
Day 12 (2nd Tuesday) 1.65 crore + 3%
Day 13 (2nd Wednesday) 1.5 crore - 9%
Day 14 (2nd Thursday) 1.30 crore - 13%
Third Weekend 4.50 crore
Day 18 (Mon) To Day 21 (Thu) 3.05 crore
Fourth Weekend 2.10 crore
Day 25 (Mon) To Day 28 (Thu) 0.84 crore
Fifth Weekend 0.84 crore
Day 32 (Mon) To Day 35 (Thu) 0.45 crore
India Net Collections 66.32 crore
46 Comments
• Even after 10 crore manipulation using Ajay Devgn calculator.. the superstar’s Drishyam is struggling to be even an average grosser.. most acclaimed word of mouth film can’t do even 70 crore
lifetime? LOL
Drishyam entire run = 3 days collection of Brothers. Wait and watch.
• Last scene was awesome……..
• @ indicine- how much it will be need to declare clean hit
□ @Yuvraj, considering the costs, it would need a minimum of 80-85 crore to be a clean hit.
• There some nice movies in a decade and drishyam is one of them.
• Indicine please tell me the budget of driysham movie
• Indicine…how can a movie shot in less than 2 months in Goa have a budget of 65cr? Please clarify.
• top movies tell july.
This movies must watch |
• 65 cr budget??
Oh Come on guys, everyone s saying the budget is 40-45 cr.
Btw, I’m really sure Drishyam will recover its cost whatever its and it will emerge a clean hit.
• PRDP Will near 280 if songs are outstanding then 350 is possible.
Dilwale Max 210 Cr with clash BM.
BM should be 150 Cr + because of bahubali air.
Driyshm will be semi hit.
Bajrangi will remain ATB this year.
• Good WOM , Good reviews, good manipulation too & still An above average …!
65-75 Lifetime
Kuch Nhi ho Skta Kuch Actors Ka :P
• @Neeraj I have had enough of you. Can you say anything good or positive? Stop bashing ajay. He is one of the most respected actors. If you don’t like him then just stay away from here.
• Very good movie.
• @Rew1… says someone whose entire life is dedicated to bashing SRK. 90% of your comments are against SRK.. why don’t you stay away from his articles?
• @ neeraj- ajay devgan got his acting fee so he don’t need for manipulation and it is not his home production movie. one thing i will waiting that brothers will trending like drishyam or not.
• After all, since the super success of bajrangi it was always difficult for a intense film like drishyam to sustain.a common man cannot spent 300-500 bucks in every 2 weeks.even if it were
brothers,its fate would have been same as drishyam.producers should use their common sense and try to maintain at least a month gap between big budgeted films.trade expected people to spent 600
rs for bajrangi another 600 for drishyam and will expect brothers to earn 30 crore day one,its logically not possible
• It Needs 60 plus to Be in a Safe Zone BT real 60 CR’s not the Viacom fake figures ..! It will be a losing film .
Hads hogyi yarr :P
Good quality film struggling to recover costs.
• Don’t wory indicine.Drishyam lifetime will be 80cr+
Its 2nd week will leave fools like neeraj and sunny chechi speechless
• Will be a clean hit
• @Neeraj
Good reply to @ REW1
He can’t digest anything good related to akki & srk ..!
• Drishyam bo prediction
Saturday -5.6
2nd week-22cr
Rest of the weeks-7cr
LIFETIME-80cr hit
• @Rew1 just ignore the Neeraj he is Chunky pandey Fan. what we can excpect from him is just joke. Ha ha ha.
• ajay devgan’s hit without rohit Shetty
phool aur kaante – superhit
jigar – hit
dilWALE – superhit
jaan – hit
ishq – superhit
pyaar to hona hi ta – superhit
kache daag – hit
bhoot – hit
masti – hit
SOS – hit
OUATIM – Hit
perhaps I missed few more. how many of those hits Shetty directed??? he was launched by ajay’s production in raju chacha, and gave him his breakthrough. Shetty is yet to giv a hit without a big
star, because of srk Chennai express collected 200cr net, but why he could he not do that with singham 2??? if he was really the only one responsible for giving hits??? ajay survived for 16 years
without Shetty, if he was a flop actor how did he managed to survive???
• The release date played the spoilsport on this occasion.BTW amazing reply to the stupid Rew1..lol.
• @Neeraj bhai, i see you before a month against about AJAY DEVGAN’s Drishyam. Your jadoo boy can’t act like Ajay but your jadoo more manipulated bo collection than ajay, movie like Jadoo3 and bong
bing manipulated 60cr, 40cr respectively…..!!!
• Drishyam showed amazing jump over the weekend aand Tuesday Wednesday collections are more than Monday yet some are saying it’s lifetime won’t be more than 65cr.lol
MARK MY WORDS DRISHYAM WILL SCORE 80CR LIFETIME
• Someone needs to get a life and stop hating others.
• I love Ajay! Always give foolish morons a chance to talk their trash more n more. So inquisitive to know about it’s collection??? Stay tuned and hope to see your comments on his Friday
collections. Devgn Sahab keep on killing the haters silently without even knowing u r really giving them a CALCULATED MANIPULATED HEARTACHE…they just have to COMMENT!!
SOME EVEN 1ST TO COMMENT on his article?? Wow! Some Indian desi guys are so so……??
I stay mum…….
Love you Sultan Mirza always always alwaysss
• BoxofficeIndia is gonna take an halt for few more weeks, Its been great 2 months with films like TWMR, ABCD, Bahubali and BB striking gold. Common man would have spent almost 5000 re watching all
these movies. They would not afford to do the same for next set of films like Bangistan, Brothers, Phanton(might work due to Kabhir Khan factor at metros). Next boxoffice bonaza would be taken
over by PRDP without a second thought. And the year gonna end with another Biggie Dilwale
• @Neeraj first of all you are not srk fan so that’s non of your business. What I hate is you are bashing an actor like ajay devgan whose caliber as an actor is way above any so called good looking
actors and so called superstars. You need to show some respect. Movies not doing well has many reasons and one of them is release period. The do called superstars like akshay and hrithik and srk
too would struggle to give big opening on non holiday period if they do such off beat thriller like drishyam. And I bash srk because his insecure loser fans’ entire life is dedicated in bashing
Aamir and Salman who are much bigger stars than srk and way more popular than srk.still they bash them cheaply so I have every single right to bash srk. It’s non of your business.
• And he wants to release his next movie on diwali, after 2 back to back flops and now average grosser how he still dares to occupy one of three biggest holidays in country?
Leave a Comment | {"url":"https://www.indicine.com/movies/bollywood/drishyam-day-6-wednesday-box-office-collections/","timestamp":"2024-11-05T13:54:39Z","content_type":"text/html","content_length":"96209","record_id":"<urn:uuid:cb8a8688-3a82-47f4-8907-7e166b2358db>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00078.warc.gz"} |
What is: K-Test
What is K-Test?
K-Test, also known as the Kullback-Leibler Divergence test, is a statistical method used to measure how one probability distribution diverges from a second, expected probability distribution. This
test is particularly useful in the fields of statistics, data analysis, and data science, where understanding the differences between distributions can provide insights into the underlying data. The
K-Test quantifies the information lost when one distribution is used to approximate another, making it a valuable tool for model evaluation and selection.
Understanding the Kullback-Leibler Divergence
The Kullback-Leibler Divergence (KLD) is a non-symmetric measure that quantifies the difference between two probability distributions. Given two probability distributions, P and Q, the KLD is defined
mathematically as D_KL(P || Q) = Σ P(x) * log(P(x) / Q(x)), where the sum is taken over all possible events x. This formula highlights how much information is lost when Q is used to approximate P.
The K-Test leverages this concept to assess the fit of a statistical model against the actual observed data.
Applications of K-Test in Data Science
In data science, the K-Test is widely applied in various scenarios, including model validation, anomaly detection, and feature selection. For instance, when developing predictive models, data
scientists can use the K-Test to compare the predicted probability distribution of outcomes against the actual distribution observed in the data. This comparison helps in identifying whether the
model is accurately capturing the underlying patterns in the data or if adjustments are needed to improve its performance.
K-Test vs. Other Statistical Tests
While the K-Test is a powerful tool, it is essential to understand how it compares to other statistical tests, such as the Chi-Square test or the Kolmogorov-Smirnov test. Unlike the Chi-Square test,
which assesses the goodness of fit for categorical data, the K-Test is more suited for continuous probability distributions. The Kolmogorov-Smirnov test, on the other hand, compares the cumulative
distribution functions of two samples, while the K-Test focuses on the divergence between probability distributions, making it a unique approach in statistical analysis.
Interpreting K-Test Results
Interpreting the results of a K-Test involves understanding the Kullback-Leibler Divergence value obtained from the analysis. A KLD value of zero indicates that the two distributions are identical,
while higher values signify greater divergence. However, it is crucial to note that the KLD is not bounded, meaning that there is no upper limit to the divergence value. Therefore, when interpreting
results, it is essential to consider the context of the data and the specific distributions being analyzed.
Limitations of the K-Test
Despite its usefulness, the K-Test has limitations that practitioners should be aware of. One significant limitation is its sensitivity to the choice of the reference distribution. If the reference
distribution is poorly chosen, the KLD may yield misleading results. Additionally, the K-Test is not symmetric; thus, D_KL(P || Q) is not equal to D_KL(Q || P). This non-symmetry can lead to
different interpretations depending on which distribution is considered the reference, necessitating careful consideration in its application.
Implementing K-Test in Python
Implementing the K-Test in Python can be accomplished using libraries such as SciPy or NumPy. The Kullback-Leibler Divergence can be calculated using the `scipy.special.kl_div` function, which
computes the KLD between two distributions. Data scientists can easily integrate this functionality into their data analysis workflows, allowing for efficient evaluation of model performance and
distribution comparisons.
Real-World Examples of K-Test Usage
In practice, the K-Test has been employed in various real-world scenarios, such as in natural language processing for comparing language models, in finance for assessing the performance of trading
algorithms, and in healthcare for evaluating diagnostic models. By quantifying the divergence between expected and observed distributions, practitioners can make informed decisions about model
adjustments and improvements, ultimately leading to better outcomes in their respective fields.
Conclusion on K-Test in Statistical Analysis
The K-Test serves as a critical tool in the arsenal of statisticians and data scientists, enabling them to quantify the divergence between probability distributions effectively. By understanding its
applications, limitations, and implementation techniques, professionals can leverage the K-Test to enhance their data analysis capabilities, leading to more accurate models and deeper insights into
their data. | {"url":"https://statisticseasily.com/glossario/what-is-k-test/","timestamp":"2024-11-02T11:15:33Z","content_type":"text/html","content_length":"138334","record_id":"<urn:uuid:aa619ddc-6cfe-477f-8e30-9f9e535994a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00241.warc.gz"} |
Wolfram|Alpha Examples: Step-by-Step Proofs
Examples for
Step-by-Step Proofs
Mathematical Induction
Prove a sum or product identity using induction step by step:
Prove divisibility by induction:
Derive a proof by induction of various inequalities step by step:
Prove a sum identity involving the binomial coefficient using induction:
Approximate the bounds of harmonic numbers using induction:
Series Convergence
Show how to prove if a sum of infinite terms diverges or converges with different tests: limit test, ratio test, root test, integral test, p-series test or geometric series test.
Investigate the convergence or divergence of an infinite sum step by step:
Trigonometric Identities
See the steps toward proving a trigonometric identity:
Prove equalities between tangent, sine and cosine:
Show the proof for equalities between trigonometric functions like tangent, arctangent, cosecant and secant: | {"url":"https://www3.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-proofs","timestamp":"2024-11-13T20:54:19Z","content_type":"text/html","content_length":"78830","record_id":"<urn:uuid:4adcedd7-bb51-4613-88d6-c69331f7117c>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00622.warc.gz"} |
An Introduction to Plasma Physics
by Alan Wootton
Publisher: Institute for High Energy Density Science 1997
This text is intended for final year undergraduate students and first year graduate students, who want to find out what plasma physics is about. It will tell you how to apply the ideas and techniques
of this branch of physics to many different situations. No prior knowledge of the subject is required.
Download or read it online for free here:
Download link
(multiple PDF files)
Similar books
Spectral Line Shapes in Plasmas
Evgeny Stambulchik (ed.)
MDPI AGLine-shape analysis is one of the most important tools for diagnostics of both laboratory and space plasmas. Its implementation requires accurate calculations, which imply the use of analytic
methods and computer codes of varying complexity.
Theoretical Plasma Physics
Allan N. Kaufman, Bruce I. Cohen
arXiv.orgThe series of lectures are divided into four major parts: (1) collisionless Vlasov plasmas; (2) nonlinear Vlasov plasmas and miscellaneous topics; (3) plasma collisional and discreteness
phenomena; (4) nonuniform plasmas.
Fundamentals of Plasma Physics
James D. Callen
University of Wisconsin,The book presents the fundamentals and applications of plasma physics. The emphasis is on high-temperature plasma physics in which the plasma is nearly fully ionized. The
level is suitable for senior undergraduates, graduates and researchers.
Fundamentals of Plasma Physics and Controlled Fusion
Kenro MiyamotoPrimary objective of this text is to provide a basic text for the students to study plasma physics and controlled fusion researches. Secondary objective is to offer a reference book
describing analytical methods of plasma physics for the researchers. | {"url":"https://www.e-booksdirectory.com/details.php?ebook=6366","timestamp":"2024-11-12T11:46:58Z","content_type":"text/html","content_length":"10970","record_id":"<urn:uuid:d3d6501a-03fc-4917-97c9-95f781edaf84>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00697.warc.gz"} |
How to get a Similarity Matrix
In article An Introduction to Similarity Analysis Using SAS by Leonard et al., similarity matrices are introduced in these terms :
Similarity measures can be used to compare several
time sequences to form a similarity matrix. This situation usually
arises in time series clustering. For example, given K time
sequences, a (KxK) symmetric matrix can be constructed whose ijth
element contains the similarity measure between the ith and jth
That's a neet idea. However, Proc Similarity (in SAS/ETS 9.3) doesn't accept the same series to be listed as an input and a target sequence. What is the best way to get a similarity matrix with Proc
06-25-2013 12:55 PM | {"url":"https://communities.sas.com/t5/SAS-Forecasting-and-Econometrics/How-to-get-a-Similarity-Matrix/td-p/118502","timestamp":"2024-11-14T15:48:08Z","content_type":"text/html","content_length":"207878","record_id":"<urn:uuid:abc1d991-bb0a-42d4-bde2-bbd43cfec3b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00181.warc.gz"} |
Initial value problem of the wave equation
• MHB
• Thread starter evinda
• Start date
In summary, the initial data have compact support and so does the solution of the initial value problem.
Hello! (Wave)
I want to prove that if for the initial value problem of the wave equation
$$u_{tt}=u_{xx}+f(x,t), x \in \mathbb{R}, 0<t<\infty$$
the data (i.e. the initial data and the non-homogeneous $f$) have compact support, then, at each time, the solution has compact support.
I have thought the following.
Suppose that we have the initial data $u(x,0)=\phi(x)$ and $u_t(x,0)=\psi(x)$.
The functions $f, \phi, \psi$ have compact support, meaning that the functions are zero outside a bounded set $[a,b]$.
The solution of the initial value problem is given by
$$u(x,t)=\frac{1}{2}[\phi(x+t)+\phi(x-t)]+\frac{1}{2}\int_{x-t}^{x+t} \psi(y) dy+\frac{1}{2} \int_0^t \int_{x-(t-s)}^{x+(t-s)} f(y,s)dy ds$$
Let $t=T$ arbitrary.
$$u(x,T)=\frac{1}{2}[\phi(x+T)+\phi(x-T)]+\frac{1}{2}\int_{x-T}^{x+T} \psi(y) dy+\frac{1}{2} \int_0^T \int_{x-(T-s)}^{x+(T-s)} f(y,s)dy ds$$
We check when $u(x,T)=0$.
We have $u(x,T)=0$ when
1. $x+T, x-T \in \mathbb{R} \setminus{[a,b]}$,
2. $x-T,x+T<a$ or $x-T,x+T>b$,
3. $x-(T-s)<a$ and $x+(T-s)<a$ or $x-(T-s)>b$ and $x+(T-s)>b$,
The second and third point holds for $x<a-T$ and $x>b+T$.
Thus $u$ is non-zero outside $[a-T,b+T]$ and $u$ has compact support. Is everything right? (Thinking)
evinda said:
Hello! (Wave)
I want to prove that if for the initial value problem of the wave equation
$$u_{tt}=u_{xx}+f(x,t), x \in \mathbb{R}, 0<t<\infty$$
the data (i.e. the initial data and the non-homogeneous $f$) have compact support, then, at each time, the solution has compact support.
I have thought the following.
Suppose that we have the initial data $u(x,0)=\phi(x)$ and $u_t(x,0)=\psi(x)$.
The functions $f, \phi, \psi$ have compact support, meaning that the functions are zero outside a bounded set $[a,b]$.
Hey evinda! (Wave)
I haven't figured everything out yet, but... how could you tell that $\phi$ and $\psi$ have compact support? (Wondering)
I like Serena said:
Hey evinda! (Wave)
I haven't figured everything out yet, but... how could you tell that $\phi$ and $\psi$ have compact support? (Wondering)
This is given that the initial data have compact support... (Thinking)
evinda said:
This is given that the initial data have compact support...
Ah okay.
Then it looks right to me! (Happy)
You may want to clarify that the 3 bulleted numbers correspond to the 3 terms in the solution though. (Nerd)
FAQ: Initial value problem of the wave equation
1. What is the wave equation?
The wave equation is a mathematical formula that describes how waves, such as sound waves or light waves, propagate through a medium. It is a second-order partial differential equation that
represents the relationship between the displacement of the wave and its rate of change over time and distance.
2. What is an initial value problem?
An initial value problem is a type of mathematical problem that involves finding the solution to a differential equation given initial conditions. In the context of the wave equation, this means
determining the displacement of the wave at a specific point in time and space.
3. How is the wave equation solved?
The wave equation is typically solved using mathematical techniques such as separation of variables, Fourier series, or Laplace transforms. These methods allow for the determination of the general
solution to the equation, which can then be modified to fit specific initial conditions.
4. What are the applications of the wave equation?
The wave equation has a wide range of applications in physics, engineering, and other fields. It is commonly used to model and understand the behavior of waves in various mediums, such as sound waves
in air, electromagnetic waves in space, and water waves in the ocean. It also has applications in areas such as signal processing, optics, and seismology.
5. What are the limitations of the wave equation?
The wave equation is a simplified model that does not take into account factors such as friction, dispersion, and non-linear effects. As a result, it may not accurately describe the behavior of all
types of waves in all situations. Additionally, the wave equation assumes a continuous medium, so it may not be applicable to systems with discrete particles or structures. | {"url":"https://www.physicsforums.com/threads/initial-value-problem-of-the-wave-equation.1039844/","timestamp":"2024-11-03T10:30:09Z","content_type":"text/html","content_length":"95444","record_id":"<urn:uuid:baffe446-9b3f-44e3-bbc2-03ba7fddaaef>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00421.warc.gz"} |
Reasoning Ability Quiz For Bank Foundation 2023 -15th May
Directions (1-2): Answer the questions based on the information given below.
A family consists of six members A, B, X, Z, C and Y. There is no single parent in the family. There are only two generations in the family. Y is the only daughter of Z. B is the mother of X. C is
the maternal aunt of X. A is the daughter-in-law of Z.
Q1. How is X related to Z?
(a) Son
(b) Daughter
(c) Brother
(d) Sister
(e) Can’t be determined
Q2. How will be the husband of B related to C?
(a) Brother-in-law
(b) Son-in-law
(c) Father-in-law
(d) Father
(e) None of these
Directions (3-5): Answer the questions based on the information given below.
A person starts walking in south direction from point A. After walking 6m, he reached at point B. From point B, he takes a left turn and walks for 4m to reach point C. Then, he turns right and walks
for 4m towards point D. Now, he again turns right from point D and walks for 8m to reach point E. He turns right from point E and walks for 8m till point F. Finally, he again turns right from point F
and walks for 12m till point G.
Q3. In which direction is Point A with respect to Point G?
(a) North-East
(b) North-west
(c) North
(d) West
(e) None of these
Q4. What is the shortest distance between point C and point G?
(a) √32m
(b) √38m
(c) √36m
(d) √40m
(e) None of these
Q5. In which direction is Point E with respect to Point A?
(a) North-East
(b) North-west
(c) North
(d) South-west
(e) None of these
Directions (6-10): Study the given arrangement of numbers, symbols & alphabets and answer the questions based on it.
1 A B 2 C D 4 ! F G H I J K L M N ^ O 6 E 5 $ & P Q 7 8 % 9 R S T # U * V
Q6. How many even numbers are in between the element which is 6th from left and element which is 11th from right end?
(a) One
(b) Two
(c) Three
(d) Four
(e) None of the above
Q7. If we remove all the vowels from the given arrangement then which is exactly in the middle of ! and 6 in new arrangement?
(a) J
(b) H
(c) K
(d) L
(e) None of the above
Q8. If all the symbols are dropped, then which element will be eighth to the right of eighth element from left end?
(a) N
(b) M
(c) O
(d) 6
(e) None of the above
Q9. How many symbols are in between 2nd number from left end and 2nd number from right end?
(a) 2
(b) 3
(c) 4
(d) 5
(e) None of the above
Q10. If all the numbers are dropped, then which element will be 2nd to the left of 19th element from left end?
(a) M
(b) E
(c) O
(d) ^
(e) None of the above
Solution (6-10):
S6. Ans. (b)
Sol. There are only two even numbers (4 and 6) between the element which is 6th from left end and 11th from right end.
S7. Ans. (c)
Sol. After removing all the vowels, we get K is exactly middle element between ! and 6.
S8. Ans. (a)
Sol. After dropping symbols: 1 A B 2 C D 4 F G H I J K L M N O 6 E 5 P Q 7 8 9 R S T U V
So, N is eighth to the right of eighth element from left end.
S9. Ans. (c)
Sol. There are 4 symbols (!, ^, $, &) between 2nd number from left end and 2nd number from right end.
S10. Ans. (b)
Sol. After dropping numbers: A B C D ! F G H I J K L M N ^ O E $ & P Q % R S T # U * V
So, E is second to the left of 19th element from left end. | {"url":"https://www.bankersadda.com/reasoning-ability-quiz-for-bank-foundation-2023-15th-may/","timestamp":"2024-11-10T02:15:00Z","content_type":"text/html","content_length":"604450","record_id":"<urn:uuid:a5578e77-03dd-4950-aae3-5422ac49ef29>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00015.warc.gz"} |
Lump Sum Repayment Calculator
Note: The information provided by the calculator is intended to provide illustrative examples based on stated assumptions and your inputs. Calculations are meant as estimates only and it is advised
that you consult with a mortgage broker about your specific circumstances. Financial Calculators © VisionAbacus Pty Ltd | {"url":"http://module50.visionabacus.com/Tools/B3/SuiteA/A300/Lump_Sum_Repayment_Calculator/ChapterFG","timestamp":"2024-11-13T12:41:22Z","content_type":"text/html","content_length":"13931","record_id":"<urn:uuid:5ce33721-c3a0-4483-8c00-98dcc26ebf17>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00368.warc.gz"} |
Fibonacci Sequence: Tips and Samples
Getting Through What Is Fibonacci Sequence and Why You Need It
Getting Through What Is Fibonacci Sequence and Why You Need It
Math models are widely used today. You can find them in your computer science assignments, for example, when getting your C++ assignment along with the examples of these models in real-life working
routines. Yet the use of these models is not today’s invention. They were widely used for centuries. The Fibonacci sequence is an obvious sample of that fact. Are you familiar with it? Let’s talk
briefly and discover the logic of the Fibonacci series and where it is used in computer science today.
In our Programming Assignment Help blog, we have collected several practical examples of where to use that sequence.
The Essence of Fibonacci Sequence and Where Is It Used
The main definition used in math is the sequence of numbers where the first two are 0 and 1 and the next ones is equal to the sum of previous pair. Moreover, when you divide each number of that
sequence by the previous one, you get something close to 1.618 which is also known as the so-called Number of God. That puzzling order was discovered by a Medieval mathematician Leonardo of Pisa
(Fibonacci) yet the palm belongs to Indian mathematics who went ahead of Fibonacci for almost a half millennium.
The main evidence of our ancestors successfully implemented the Fibonacci sequence is as follows:
• An Indian poet Acharya Pingala used its principles in his Sanskrit poetry samples.
• Egyptian pyramids were built using the Fibonacci series for their construction calculations.
• Legendary artist Da Vinci also was interested in what is Fibonacci series as he had noticed that order in a man’s body structure as well as in other living beings. He called it the golden ratio
and implied it in his arts and inquiries.
But that’s not all, it turns out that the laws of the nature of the Earth and the Cosmos somehow inexplicably obey the strict mathematical laws of the sequence of Fibonacci numbers.
Taking an example, both a shell on Earth and a galaxy in Space are built using the Fibonacci sequence. The vast majority of flowers have 5, 8, or 13 petals. In sunflowers, on plant stems, in swirling
clouds, in whirlpools, and even in Forex exchange rate charts, Fibonacci numbers work everywhere.
For example, both a shell on Earth and a galaxy in Space are built using the Fibonacci sequence. The vast majority of flowers have 5, 8, or 13 petals. In sunflowers, on plant stems, in swirling
clouds, in whirlpools, and even in Forex exchange rate charts, Fibonacci numbers work everywhere.
In brief, you can see that sequence like a row of numbers starting with 0 and 1. The following row looks like the picture below.
By the way, do you know what assignment Leonardo of Pisa was solving when he found out the Fibonacci sequence? The task was funny enough, well. He wondered about breeding rabbits. The essence of the
task: “A couple of rabbits are settled in a clearing. How many couples will live in this place in a year? For the solution, simplifications were introduced like rabbits do not die during the year,
they reach puberty a month after birth, and their offspring appear only a month after conception. Thus, in this assignment, the sequence is defined as follows:
– the first month is 1 couple;
– the second month is 1+1=2 couples;
– the third month is 2 + 1 = 3 couples. Here the first couple of rabbits give birth since the second has only reached puberty in the third month;
– the fourth month is 3+2=5 couples, both the first pair and the first offspring are already giving birth here, i.e. 2 couples are born.
At the end of the year, there will be 144+233=377 couples of rabbits in the clearing.
Various Coding Languages and Fibonacci Sequence in Them
Of course, the use of that mathematical consistency didn’t pass by the IT industry. It is used as one of the math models in various coding technologies. Let’s enclose the Fibonacci sequence meaning
in several coding languages.
When you use Java, there are two main methods on how to get the Fibonacci series program in Java.
The first option is using Stream. iterate for Fibonacci numbers generation. In general, it looks like that.
The second method is based on recursion for getting the Fibonacci sequence. It is slow enough but it works.
For C++ technology, the sample of the Fibonacci sequence is also available to perform via a special function.
In general, the Fibonacci series in C++ can be performed in the following way.
There is a difference whether the data type is int or long int: in the first case, to calculate and display the entire series of Fibonacci numbers up to and including the 46th, it takes about 40
minutes, and in the second, it is about a third less.
For the Fibonacci series in JavaScript, there is an algorithm that can be listed as follows.
One more example we can list there is the Fibonacci series on Python. You can see there is quite a simple solution for defining a row of numbers that are parts of the Fibonacci sequence.
The Help for Assignments With Fibonacci Sequence
Even though we have already listed the Fibonacci code for various technologies, it often takes time and experience to make such an algorithm. For students, it could be messy as quite oftenly that
mathematical order is a part of an assignment they get. Let’s make it easier for you! On our website, you can get programming assignment help from real pros who deal with these technologies every
day. Even the most complex tasks that concern Fibonacci code can be performed custom using various approaches and ideas to create a unique piece of code. Just place an order on the website and get
your homework done brilliantly. | {"url":"https://www.programmingassignment.net/blog/what-is-fibonacci-sequence/","timestamp":"2024-11-08T20:12:48Z","content_type":"text/html","content_length":"45847","record_id":"<urn:uuid:b9ef8ead-d97a-458b-b1c2-1d85f6d1e7ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00241.warc.gz"} |
Which of the following describes the time value of money? - Jobs For Felons Now
Which of the following describes the time value of money?
The Time Value of Money: A Crucial Concept in Finance
The time value of money is a fundamental concept in finance that refers to the idea that a dollar today is worth more than a dollar in the future. This concept is based on the premise that money
received today can be invested to generate interest, dividends, or other returns, making it more valuable than the same amount of money received in the future. In this article, we will explore the
time value of money in detail, discussing its definition, importance, and applications.
The time value of money can be defined as the idea that a dollar today is worth more than a dollar in the future because of the potential to earn interest or returns on investment. This concept is
based on the assumption that people prefer to receive money today rather than later, as it can be used to generate returns or invested to grow wealth.
The time value of money is crucial in finance because it helps individuals and organizations make informed decisions about investments, savings, and financial planning. By understanding the time
value of money, individuals can:
• Make informed decisions about investments and savings
• Determine the present value of future cash flows
• Calculate the future value of current investments
• Plan for retirement and other long-term financial goals
Key Concepts
Several key concepts are essential to understanding the time value of money:
• Present Value (PV): The current value of a future cash flow or investment.
• Future Value (FV): The value of an investment or cash flow at a future date.
• Discount Rate: The rate at which future cash flows are discounted to their present value.
• Compound Interest: The interest earned on both the principal amount and any accrued interest.
Several formulae are used to calculate the time value of money:
• Present Value (PV) Formula: PV = FV / (1 + r)^n
• Future Value (FV) Formula: FV = PV x (1 + r)^n
• Discount Rate Formula: r = (FV – PV) / PV x n
The time value of money has numerous applications in finance, including:
• Investment Analysis: Investors use the time value of money to determine the present value of future cash flows and make informed decisions about investments.
• Savings: Individuals use the time value of money to plan for retirement and other long-term financial goals by saving and investing their money.
• Budgeting: Businesses use the time value of money to determine the present value of future expenses and make informed decisions about budgeting and financial planning.
Here are some examples of how the time value of money is applied in real-life scenarios:
• Investment: An investor purchases a stock that pays a 5% annual dividend. The stock is expected to increase in value by 10% per year. If the investor holds the stock for 5 years, what will be the
future value of their investment? (Answer: Using the formula for future value, we can calculate that the future value of the investment will be $1,276.19)
• Savings: An individual saves $1,000 per year for 10 years, earning a 5% annual interest rate. What will be the future value of their savings? (Answer: Using the formula for future value, we can
calculate that the future value of their savings will be $16,919.34)
The time value of money is a fundamental concept in finance that helps individuals and organizations make informed decisions about investments, savings, and financial planning. By understanding the
time value of money, individuals can make the most of their financial resources and achieve their long-term financial goals.
Leave a Comment | {"url":"https://jobsforfelonsnow.com/which-of-the-following-describes-the-time-value-of-money/","timestamp":"2024-11-13T16:22:53Z","content_type":"text/html","content_length":"137564","record_id":"<urn:uuid:0a8de286-ae70-4a12-8e71-7d910e2de3c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00608.warc.gz"} |
Indefinite Articles
Indefinite Articles by Ángel Aguilar
1. In English, the two indefinite articles are a and an. Like other articles, indefinite articles are invariable. You use one or the other, depending on the first letter of the word following the
article, for pronunciation reasons.
2. "A" goes before words that begin with consonants.
2.1. a cat a dog a purple onion a buffalo a big apple
3. Examples
3.1. We use a/an for nouns that are not specific, when we can refer to any one of a certain kind of thing. For example, imagine your uncle just made cookies. You say:
3.2. I want a cookie!
3.2.1. The word “cookie” begins with a consonant sound, so we use a. If the word begins with a vowel sound, we use an. For example:
3.3. The scientist had an idea.
3.3.1. We do not know anything about the idea, so it is not specific. The word “idea” starts with a vowel sound, so we use an.
4. The Indefinite Article Video
5. "An" goes before words that begin with vowels.
5.1. an egg an Indian an orbit an hour
6. Exceptions
6.1. Use "an" before unsounded "h." Because the "h" hasn't any phonetic representation and has no audible sound, the sound that follows the article is a vowel; consequently, "an" is used.
6.1.1. an honorable peace an honest error
6.2. When "u" makes the same sound as the "y" in "you," or "o" makes the same sound as "w" in "won," then a is used. The word-initial "y" sound ("unicorn") is actually a glide [j] phonetically, which
has consonantal properties; consequently, it is treated as a consonant, requiring "a."
6.2.1. a union a united front a unicorn a used napkin a U.S. ship a one-legged man | {"url":"https://www.mindmeister.com/849357954/indefinite-articles","timestamp":"2024-11-12T10:52:13Z","content_type":"application/xhtml+xml","content_length":"47874","record_id":"<urn:uuid:d5a41f63-249b-47cb-9bda-69c6a15f334a>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00188.warc.gz"} |
My Maple Worksheets (not Maple Documents) have lots of explanetory Text blocks [..... surounding executable Math blocks ([> ..... I often insert mathematical symbols, most commonly subscripted
variables, in these Text blocks. For a simple example, consider the text block entered as
[This is a test of a subscripted variable "CTL-R" h__0 "CTL-T" in a text block.
The "CTL-R" (quotes are not actually entered) is the short cut to go into math mode, and "CTL-T" exits math mode and returns to text mode and the double underscore produces an atomic subscripted
The text block actually will look like
[This is a test of a subscripted variable h[0] in a text block.
The problem occurs when I reexecute the worksheet. The Text block actually produces output labeled with an equation number. For my simple example above the Text block becomes
[This is a test of a subscripted variable h[0] in a text block.
[ h[0] (1)
where the two lines started by [ are actually merged with one expanded [ for the Text block with its output. To get rid of the unwanted output, I have to put my curser over the h[0] that is in the
Text body (not the output h[0]) and hit "Shift-F5". The output h[0] with its equation number disappears. If there are a number of simple math expressions in a text block, I have to process them one
at a time with "Shift-F5". This takes up a lot of time. With earlier Maple versions (~2015 or earlier) I used to fly through Text blocks using the shortcuts "Ctl-R" and "Ctl-T" and these Text blocks
produced no output when the worksheet was reexecuted.
Starting with Maple 2016 I could enter math expressions in Text blocks using the shortcuts, but I could not copy and paste a Text block with inline math expressions without the expressions becoming
"live" in the copied block. Starting with Maple 2017 all my Text boxes with math expressions began executing the math and producing output.
I gave up on Maple 2017 and 2018. I have finally made the jump from Maple 2016 to Maple 2019, in part, because I finally discovered the "Shift-F5" trick to make math expressions in a Text block
Does anyone know how to make the default behaviour of Maple with math expressions in a Text block to be "Don't execute the math and produce output in the Text block"?
I would post an actual example worksheet, except I have never been successful whenever I have tried to upload a worksheet. I hope my description above is adequate.
Any help will be greatly appreciated. Neill Smith
I am using the LinearAlgebra package to do dynamics between a rotating Cartesian coordinate system and a fixed Cartesian coordinate system. The VectorCalculus package is not what I need.
Since I can't seem to get my test worksheet to paste into this post, I will manually enter an "approximation" to it. I assume that the notation [x, y, z] represents a column vector. I also assume
that x represents the cross product operator from the operator pallete.
I just want to get any one of the three ways of doing a vector cross product (see below) to simply display in math notation as R x V. What I get from the three methods below for an unevaluated cross
product is "ugly".
Any help or advice will be greatly appreciated.
> restart
> with(LinearAlgebra):
> R := Vector(3, [x, y, z])
R := [x, y, z]
> V := Vector(3, [u, v, w])
V := [u, v, w]
>R x V
[-vz + wy, uz - wx, -uy + vx]
>'R x V'
>CrossProduct(R, V)
[-vz + wy, uz - wx, -uy + vx]
> R &x V
[-vz + wy, uz - wx, -uy + vx]
'R &x V'
I often run into this problem. Say I have defined a new variable in terms of some old variables:
Eq1 := NewVar = OldVar1 + OldVar2
Then suppose I have an expression in terms of the old variables:
Eq2 := OldVar6 * OldVar5 = OldVar3 * (OldVar1 + OldVar2)
If I want to substitute Eq1 into Eq2 I have to either type
subs(OldVar1 + OldVar2 = NewVar1, Eq2)
or I can do this
I have a depressed quartic equation with the quadratic term also removed. It has at least one positive real root which is the solution that I am looking for. There are three cases to consider. Two of
the cases have trivial solutions. Maple gives its usual RootOf solution for the general case. My worksheet is below.
When I integrate the partial derivative of v(x,y) with respect to y from y = 0 to y = Y using
int(diff(v(x,y),y), y=0..Y)
Maple simple echos the integral in typeset format. I expected to get
v(x,Y) - v(x,0)
What am I missing here? This integral comes from integrating the 2D continuity equation in boundary layer. Any help would be greatly appreciated.
Neill Smith | {"url":"https://beta.mapleprimes.com/users/NeillSmith/questions?page=2","timestamp":"2024-11-08T10:48:22Z","content_type":"text/html","content_length":"101714","record_id":"<urn:uuid:8bba0f71-82a8-4292-bcd9-0bd4146c4d9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00292.warc.gz"} |
A Free Trial That Lets You Build Big!
Start building with 50+ products and up to 12 months usage for Elastic Compute Service
• Sales Support
1 on 1 presale consultation
• After-Sales Support
24/7 Technical Support 6 Free Tickets per Quarter Faster Response
• Alibaba Cloud offers highly flexible support services tailored to meet your exact needs. | {"url":"https://topic.alibabacloud.com/zqpop/scientific-notation-checker_418344.html","timestamp":"2024-11-13T08:48:38Z","content_type":"text/html","content_length":"100885","record_id":"<urn:uuid:c8ca7318-425e-4865-b086-a24b0b5b708d>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00103.warc.gz"} |
Custom Auto-Number?
Question on Auto-Number column capabilities. I want to add a column to my portfolio sheet that will assign a unique project ID, like the below example, to each project row. I have columns on the same
sheet that designate project class & start date. I'd like to use the auto-number feature to look at the start date and project class of a project, and assign a unique id to it. Is getting to the
below example, or something close to it possible? Anyone know how to do this?
Example of Unique Project ID: R-12-20
R = Renovation (Project Class)
12 = Project No. 12
20 = Year when project was approved (so that each year we could restart project numbers)
• It is possible. Are you able to provide a screenshot with sensitive/confidential data removed, blocked, and/or replaced with "dummy data" as needed?
• I created a quick grid with dummy data below. Thank you for your help!
• Ok. So I can tell that the leading letter is the first letter from the [Project Class] column.
I am assuming that the last portion of the ID is the year, but for R and I you have 2 digits and for C you have all 4. I want to confirm that this is correct?
Finally... How exactly are you determining the number in the middle section? I can't find a pattern/reference.
• Sorry I had a typo!
Yes: the leading letter is the first letter from the [Project Class] column.
Yes: the last portion of the ID is the year, only need two digits
Middle section is where i want the auto number to come in, assigning each project its own number. Does that need to be a separate, hidden column, then combine all fields with a formula? or is
there another way? Want the final [Project ID] result a unique ID for the project field, or row, on my portfolio sheet.
Thank you!
• It doesn't need to be a separate column. I just want to make sure I understand the logic. I don't notice a pattern for it though.
You have
How exactly did you determine those numbers?
• No pattern or logic. I was just typing in random dummy numbers. Was assuming the same number could be used more than once if its associated to a different Project Class. The middle number can be
any number, as long as the end result is a Unique Project ID.
• Ok. So how about this...
=LEFT([Project Class]@row + "-" + COUNTIFS([Project Class]$1:[Project Class]@row, [Project Class]@row, [Start Date]$1:[Start Date]@row, YEAR(@cell) = YEAR([Start Date]@row)) + "-" + RIGHT(YEAR
([Start Date]@row), 2)
This will pull the first letter from the [Project Class]. Then it will give a sequential count of how many previous rows had that same [Project Class] in that same year. Finally it will pull the
right two digits of the year.
So an example output of the above would be...
Does that work, or would you like something different for the middle portion?
• That is perfect. Thank you so much for your help!
• Hello, I would like to add an auto custom number column to a sheet. The format is YY-MM-sequence in 3 numerical digits. We start the sequence over at 001 at the beginning of each month which is a
piece to this function that I can't seem to get. Here's how far I've gotten:
• @Paul Newcome you're such a formula/function guru, are you able to help with the above?
• @Akellu Try something like this...
RIGTH(YEAR(Created@row), 2), + "-" + IF(MONTH(Created@row) < 10, "0") + MONTH(Created@row) + "-" + IF(COUNTIFS(Created:Created, AND(IFERROR(MONTH(@cell), 0) = MONTH(Created@row), IFERROR(YEAR
(@cell), 0) = YEAR(Created@row), @cell <= Created@row)) < 10, "00", IF(COUNTIFS(Created:Created, AND(IFERROR(MONTH(@cell), 0) = MONTH(Created@row), IFERROR(YEAR(@cell), 0) = YEAR(Created@row),
@cell <= Created@row)) < 100, "0")) + COUNTIFS(Created:Created, AND(IFERROR(MONTH(@cell), 0) = MONTH(Created@row), IFERROR(YEAR(@cell), 0) = YEAR(Created@row), @cell <= Created@row))
• Hi @Paul Newcome, thanks for your help. I copied your formula and still got Unparsable. I updated the column references, indicated by yellow dot so you can easily see. It's still not working. Any
additional help or insight is greatly appreciated.
• My apologies. I had a comma that snuck in where it didn't belong. Try this...
=RIGHT(YEAR(Created@row), 2) + "-" + IF(MONTH(Created@row) < 10, "0") + MONTH(Created@row) + "-" + IF(COUNTIFS(Created:Created, AND(IFERROR(MONTH(@cell), 0) = MONTH(Created@row), IFERROR(YEAR
(@cell), 0) = YEAR(Created@row), @cell <= Created@row)) < 10, "00", IF(COUNTIFS(Created:Created, AND(IFERROR(MONTH(@cell), 0) = MONTH(Created@row), IFERROR(YEAR(@cell), 0) = YEAR(Created@row),
@cell <= Created@row)) < 100, "0")) + COUNTIFS(Created:Created, AND(IFERROR(MONTH(@cell), 0) = MONTH(Created@row), IFERROR(YEAR(@cell), 0) = YEAR(Created@row), @cell <= Created@row))
• Paul maybe you can help me. I am trying to do the same autogenerating reference numbers using a column that it is like the project class (What is it?) and the reference date will be the created
column date that automatically assigned by system when the item is added.
I followed the formula that you put it upper but it is giving #unparseable error. I appreciate as usual your help
=LEFT([What is it?]@row + "-" + COUNTIFS([What is it?]$1:[ What is it?]@row, [What is it?]@row, [Created]$1:[ Created]@row, YEAR(@cell) = YEAR([Created]@row)) + "-" + RIGHT(YEAR([Created]@row),
Then what i am trying to get in that Auto Ref Test is R-XXX-21, if the What is it? is Issue the initial letter should be I-XXX-21 and if it is Audit should be A-XXX-21
Thanks in advance
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/69133/custom-auto-number","timestamp":"2024-11-09T13:16:58Z","content_type":"text/html","content_length":"447625","record_id":"<urn:uuid:0d6c6d2a-bd28-4ea9-b680-f09bfa9b4d5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00758.warc.gz"} |
What Do The Odds In Horse Racing Mean - Marthalake.org
what do the odds in horse racing mean
Odds in horse racing represent the probability of a horse winning a race and are expressed as a ratio of the amount wagered to the amount returned if the horse wins. For example, odds of 2:1 mean
that for every $2 wagered, $1 is returned if the horse wins, plus the original $2 wagered. Odds are determined by various factors including the horse’s past performance, the jockey’s record, the
trainer’s reputation, and the track conditions. Lower odds indicate a higher probability of winning, while higher odds indicate a lower probability. Bettors use odds to determine which horses to
wager on and how much to wager, with higher odds offering potentially higher payouts but also greater risk.
Understanding Horse Racing Odds
Horse racing odds are numbers assigned to each horse in a race that represent the likelihood of that horse winning. These odds are determined by factors such as the horse’s previous performances, the
track conditions, and the perceived strengths of the other horses in the race.
Odds Comparison and Conversion
Odds are typically expressed in two main formats: fractional odds and decimal odds.
Fractional Odds
• Represented as a ratio of two numbers, separated by a hyphen or slash (e.g., 5/1 or 5-1)
• The first number (numerator) represents the amount you would win for every unit you wager, and the second number (denominator) represents the amount you would need to wager
• For example, 5/1 odds mean that for every $1 you bet on the horse, you would win $5 if it wins
• The lower the fraction, the more likely the horse is to win
Decimal Odds
• Represented as a single number with a decimal point (e.g., 6.00)
• This number represents the amount you would win for every unit you wager
• For example, 6.00 odds mean that for every $1 you bet on the horse, you would win $6 if it wins
• The higher the number, the more likely the horse is to win
The following table provides a conversion between fractional and decimal odds:
Fractional Odds Decimal Odds
2/1 3.00
5/2 3.50
3/1 4.00
10/1 11.00
20/1 21.00
Fractional Odds
Fractional odds are the traditional way of expressing odds in horse racing. They are written as a fraction, with the numerator representing the amount you will win for every £1 you bet, and the
denominator representing the amount you must bet to win £1.
For example, odds of 2/1 mean that you will win £2 for every £1 you bet, and you must bet £1 to win £1. Odds of 3/1 mean that you will win £3 for every £1 you bet, and you must bet £1 to win £1, and
so on.
Decimal Odds
Decimal odds are a more modern way of expressing odds in horse racing. They are written as a single number, which represents the total amount you will win for every £1 you bet.
For example, odds of 3.0 mean that you will win £3 for every £1 you bet. Odds of 4.0 mean that you will win £4 for every £1 you bet, and so on.
Fractional Odds Decimal Odds
1/1 2.0
2/1 3.0
3/1 4.0
4/1 5.0
5/1 6.0
: FontFamily:ssy Somme sommige some: acute accentuateds accentuated precisely. ✔️
Odds Fluctuations and Implications
Odds in horse racing are simply a way of expressing the chances of a particular horse winning. The odds are calculated by bookmakers, who take into account a number of factors when setting the
prices, including the horse’s past performance, the distance of the race, and the level of competition. The odds are expressed as a fraction, such as 4/1 or 7/2. The first number represents the
amount you would win if you bet $1 on the horse, while the second number represents the amount you would have to bet to win $1. For example, a horse with odds of 4/1 would pay out $4 for every $1 you
bet on it, while a horse with odds of 7/2 would pay out $7 for every $2 you bet on it.
The odds in horse racing can fluctuate significantly in the lead-up to a race. This is because new information becomes available, such as the condition of the horse, the weather forecast, or the
jockey who will be riding it. As a result, it is important to pay attention to the odds fluctuations and adjust your betting strategy accordingly.
• If the odds on a horse are shortening, it means that more people are betting on it and the bookmakers believe it is more likely to win.
• If the odds on a horse are lengthening, it means that fewer people are betting on it and the bookmakers believe it is less likely to win.
The following table shows how the odds on a horse can fluctuate in the lead-up to a race:
Time Odds
1 week before the race 10/1
3 days before the race 7/1
1 day before the race 4/1
On the day of the race 2/1
As you can see from the table, the odds on the horse have shortened significantly in the lead-up to the race. This is because more people are betting on it, and the bookmakers believe it is more
likely to win.
When betting on horse races, it is important to consider the odds fluctuations and adjust your betting strategy accordingly. If you are betting on a horse with long odds, you should be aware that the
odds could lengthen further, and you could potentially lose your bet. However, if you are betting on a horse with short odds, you should be aware that the odds could shorten further, and you could
potentially win more money than you expected.
And there you have it, folks! Understanding the odds in horse racing is like being able to decode a secret language. It’s not rocket science, but it sure can make watching the races a whole lot more
exciting. So, whether you’re a seasoned bettor or just starting out, I hope this article has shed some light on the mysterious world of horse racing odds. Thanks for reading! Be sure to drop by again
soon for more tips, tricks, and all the latest on the tracks. | {"url":"https://marthalake.org/what-do-the-odds-in-horse-racing-mean/","timestamp":"2024-11-10T18:10:06Z","content_type":"text/html","content_length":"115112","record_id":"<urn:uuid:a6f9c6b2-7355-48d3-aa11-6299690b64e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00602.warc.gz"} |
Trapped modes in thin and infinite ladder like domains. Part1: Existence results
june, 2017
Publication type:
Paper in peer-reviewed journals
Asymptotic Analysis, vol. 103(3), pp. 103-134
Keywords :
Spectral theory, periodic media, quantum graphs, trapped modes
The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder
like domain, namely a thin periodic structure (the thickness being characterized by a small parameter ε>0) whose limit (as ε tends to 0) is a periodic graph. The localized perturbation consists in
changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we
wonder whether such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps.
We first find the formal limit of the eigenvalue problem as the ε tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the
periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit
operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover,
in that case, it is possible to create as many eigenvalues as one wants, provided that ε is small enough. Numerical experiments illustrate the theoretical results.
author={Bérangère Delourme and Sonia Fliss and Patrick Joly and
Elizaveta Vasilevskaya },
title={Trapped modes in thin and infinite ladder like domains. Part1:
Existence results },
doi={10.3233/ASY-171422 },
journal={Asymptotic Analysis },
year={2017 },
volume={103(3) }, | {"url":"https://uma.ensta.fr/uma2/publis/show.html?id=1660","timestamp":"2024-11-02T21:35:41Z","content_type":"text/html","content_length":"15549","record_id":"<urn:uuid:7c7a1b3b-5387-4c6c-8b4b-9deeb61ae223>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00863.warc.gz"} |
Lesson 11
Percentage Contexts
Let’s learn about more situations that involve percentages.
11.1: Leaving a Tip
Which of these expressions represent a 15% tip on a $20 meal? Which represent the total bill?
\(15 \boldcdot 20\)
\(20 + 0.15 \boldcdot 20\)
\(1.15 \boldcdot 20\)
\(\frac{15}{100} \boldcdot 20\)
11.2: A Car Dealership
A car dealership pays a wholesale price of $12,000 to purchase a vehicle.
1. The car dealership wants to make a 32% profit.
1. By how much will they mark up the price of the vehicle?
2. After the markup, what is the retail price of the vehicle?
2. During a special sales event, the dealership offers a 10% discount off of the retail price. After the discount, how much will a customer pay for this vehicle?
This car dealership pays the salesperson a bonus for selling the car equal to 6.5% of the sale price. How much commission did the salesperson lose when they decided to offer a 10% discount on the
price of the car?
11.3: Commission at a Gym
1. For each gym membership sold, the gym keeps $42 and the employee who sold it gets $8. What is the commission the employee earned as a percentage of the total cost of the gym membership?
2. If an employee sells a family pass for $135, what is the amount of the commission they get to keep?
11.4: Card Sort: Percentage Situations
Your teacher will give you a set of cards. Take turns with your partner matching a situation with a descriptor. For each match, explain your reasoning to your partner. If you disagree, work to reach
an agreement.
There are many everyday situations where a percentage of an amount of money is added to or subtracted from that amount, in order to be paid to some other person or organization:
│ │ goes to │ how it works │
│ sales tax │the government │added to the price of the item │
│ gratuity (tip) │the server │added to the cost of the meal │
│ interest │the lender (or account holder)│added to the balance of the loan, credit card, or bank account │
│ markup │the seller │added to the price of an item so the seller can make a profit │
│markdown (discount)│the customer │subtracted from the price of an item to encourage the customer to buy it │
│ commission │the salesperson │subtracted from the payment that is collected │
For example,
• If a restaurant bill is \$34 and the customer pays \$40, they left \$6 dollars as a tip for the server. That is 18% of $34, so they left an 18% tip. From the customer's perspective, we can think
of this as an 18% increase of the restaurant bill.
• If a realtor helps a family sell their home for \$200,000 and earns a 3% commission, then the realtor makes \$6,000, because \((0.03) \boldcdot 200,\!000 = 6,\!000\), and the family gets \
$194,000, because \(200,\!000 - 6,\!000 = 194,\!000\). From the family's perspective, we can think of this as a 3% decrease on the sale price of the home. | {"url":"https://curriculum.illustrativemathematics.org/MS/students/2/4/11/index.html","timestamp":"2024-11-02T20:45:40Z","content_type":"text/html","content_length":"73255","record_id":"<urn:uuid:82934dcf-cd58-4e23-9429-7187a5229f5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00374.warc.gz"} |
Introduction to Plotting
15.1 Introduction to Plotting
Earlier versions of Octave provided plotting through the use of gnuplot. This capability is still available. But, newer versions of Octave offer more modern plotting capabilities using OpenGL. Which
plotting system is used is controlled by the graphics_toolkit function. See Graphics Toolkits.
The function call graphics_toolkit ("qt") selects the Qt/OpenGL system, graphics_toolkit ("fltk") selects the FLTK/OpenGL system, and graphics_toolkit ("gnuplot") selects the gnuplot system. The
three systems may be used selectively through the use of the graphics_toolkit property of the graphics handle for each figure. This is explained in Graphics Data Structures.
Caution: The OpenGL-based toolkits use single precision variables internally which limits the maximum value that can be displayed to approximately 10^{38}. If your data contains larger values you
must use the gnuplot toolkit which supports values up to 10^{308}. Similarly, single precision variables can accurately represent only 6-9 base10 digits. If your data contains very fine differences
(approximately 1e-8) these cannot be resolved with the OpenGL-based graphics toolkits and the gnuplot toolkit is required.
Note: The gnuplot graphics toolkit uses the third party program gnuplot for plotting. The communication from Octave to gnuplot is done via a one-way pipe. This has implications for performance and
functionality. Performance is significantly slower because the entire data set, which could be many megabytes, must be passed to gnuplot over the pipe. Functionality is negatively affected because
the pipe is one-way from Octave to gnuplot. Octave has no way of knowing about user interactions with the plot window (be it resizing, moving, closing, or anything else). It is recommended not to
interact with (or close) a gnuplot window if you will access the figure from Octave later on. | {"url":"https://docs.octave.org/v6.1.0/Introduction-to-Plotting.html","timestamp":"2024-11-05T12:42:22Z","content_type":"text/html","content_length":"5786","record_id":"<urn:uuid:70f2ce78-0a53-4370-a7b1-d9a9f95c6426>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00003.warc.gz"} |
Quantum interference in time
I discuss the interpretation of a recent experiment showing quantum interference in time. It is pointed out that the standard nonrelativistic quantum theory does not have the property of coherence in
time, and hence cannot account for the results found. Therefore, this experiment has fundamental importance beyond the technical advances it represents. Some theoretical structures which consider the
time as an observable, and thus could, in principle, have the required coherence in time, are discussed briefly, and the application of Floquet theory and the manifestly covariant quantum theory of
Stueckelberg are treated in some detail. In particular, the latter is shown to account for the results in a simple and consistent way.
• Interference
• Quantum observables
• Time
Dive into the research topics of 'Quantum interference in time'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/quantum-interference-in-time","timestamp":"2024-11-03T13:25:35Z","content_type":"text/html","content_length":"46260","record_id":"<urn:uuid:d1eab1b5-79c6-47e2-b67e-fcf1c33a2f9c>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00693.warc.gz"} |
exponential Archives - Page 2 of 3 - Stumbling Robot
Evaluate the limit.
First, we write the expression using the definition of the exponential,
Now, considering the expression in the exponent and using the expansion of
Therefore, we have
Find the limit as x goes to 0 of ((1+x)^1/x – e) / x
Evalue the limit.
First, we have
From this we see that
Hence, using the expansion for
Therefore, we have
Find the limit as x goes to 0 of (x + e^2x)^1/x
Evaluate the limit.
From the definition of the exponential we have
So, first we use the expansion of
Therefore, as
Now, since
Therefore, as
So, getting back to the expression we started with,
But, as in the previous exercise (Section 7.11, Exercise #23) we know
Find the limit as x goes to 1 of x^1 / (1-x)
Evaluate the limit.
First, we write
From this exercise (Section 7.11, Exercise #4) we know that as
Therefore, as
So, we then have
(Here we could say that since the exponential is a continuous function we can bring the limit inside and so this becomes
Find the limit as x goes to 0 of (a^x – a^sin x) / x^3
Evaluate the limit.
First, we want to get expansions for
Next, for
Now, we need use the expansion for
and substitute this into our expansion of
(Again, this is the really nice part of little
So, now we have expansions for
Find the limit as x goes to 0 of (a^x – 1) / (b^x – 1)
Evaluate the limit for
First we write
Therefore, we have
Find all x satisfying equations given in terms of sinh
1. We are given
Then, from the given equation we have
So, then we have
Therefore we have
2. There can be no
Next, we use the equation given in the problem to write,
Furthermore, we can obtain an expression for
Putting these expressions for
But this implies
which is impossible. Hence, there can be no real
Derive some properties of the product of e^x with a polynomial
1. Prove that
2. Do part (a) in the case that
3. Find a similar formula and prove it in the case that
For all of these we recall from a previous exercise (Section 5.11, Exercise #4) that by Leibniz’s formula if
So, in the case at hand we have
(Since the
1. Proof. From the formula above we have
But, since
Hence, we have
2. If
Claim: If
Proof. We follow the exact same procedure as part (a) except now we have the derivatives of
Therefore, we now have
3. Claim: Let
Proof. Using Leibniz’s formula again, we have
But for the degree
Prove some properties of a differentiable function satisfying a given functional equation
1. Prove that
2. Conjecture and prove a formula for
3. Prove that
4. Prove that there exists a constant
for all
1. Proof. We can compute this using the functional equation:
Next, we conjecture
Proof. Again, we compute using the functional equation, and the above formula for
2. We conjecture
Proof. The proof is by induction. We have already established the cases
Thus, if the formula holds for
3. Proof. Using the functional equation
Then, since the derivative
4. Proof. Since the derivative
must exist for all
But then, from part (c) we know this exercise (Section 6.17, Exercise #38) we know
Therefore, we have
Prove that a function satisfying given properties must be e^x
Given a function
Prove the following:
1. The derivative
2. We must have
This problem is quite similar to two previous exercises here and here (Section 6.17, Exercises #39 and #40).
1. Proof. To show that the derivative
exists for all
2. Proof. From part (a) we know | {"url":"https://www.stumblingrobot.com/tag/exponential/page/2/","timestamp":"2024-11-07T09:23:29Z","content_type":"text/html","content_length":"195993","record_id":"<urn:uuid:a20f1b2a-8fd0-4869-8435-3194a1e55458>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00588.warc.gz"} |
Impact of Fast Sodium Channel Inactivation on Spike Threshold Dynamics and Synaptic Integration
Neurons spike when their membrane potential exceeds a threshold value. In central neurons, the spike threshold is not constant but depends on the stimulation. Thus, input-output properties of neurons
depend both on the effect of presynaptic spikes on the membrane potential and on the dynamics of the spike threshold. Among the possible mechanisms that may modulate the threshold, one strong
candidate is Na channel inactivation, because it specifically impacts spike initiation without affecting the membrane potential. We collected voltage-clamp data from the literature and we found,
based on a theoretical criterion, that the properties of Na inactivation could indeed cause substantial threshold variability by itself. By analyzing simple neuron models with fast Na inactivation
(one channel subtype), we found that the spike threshold is correlated with the mean membrane potential and negatively correlated with the preceding depolarization slope, consistent with experiments.
We then analyzed the impact of threshold dynamics on synaptic integration. The difference between the postsynaptic potential (PSP) and the dynamic threshold in response to a presynaptic spike defines
an effective PSP. When the neuron is sufficiently depolarized, this effective PSP is briefer than the PSP. This mechanism regulates the temporal window of synaptic integration in an adaptive way.
Finally, we discuss the role of other potential mechanisms. Distal spike initiation, channel noise and Na activation dynamics cannot account for the observed negative slope-threshold relationship,
while adaptive conductances (e.g. K+) and Na inactivation can. We conclude that Na inactivation is a metabolically efficient mechanism to control the temporal resolution of synaptic integration.
Author Summary
Neurons spike when their combined inputs exceed a threshold value, but recent experimental findings have shown that this value also depends on the inputs. Thus, to understand how neurons respond to
input spikes, it is important to know how inputs modify the spike threshold. Spikes are generated by sodium channels, which inactivate when the neuron is depolarized, raising the threshold for spike
initiation. We found that inactivation properties of sodium channels could indeed cause substantial threshold variability in central neurons. We then analyzed in models the implications of this form
of threshold modulation on neuronal function. We found that this mechanism makes neurons more sensitive to coincident spikes and provides them with an energetically efficient form of gain control.
Citation: Platkiewicz J, Brette R (2011) Impact of Fast Sodium Channel Inactivation on Spike Threshold Dynamics and Synaptic Integration. PLoS Comput Biol 7(5): e1001129. https://doi.org/10.1371/
Editor: Lyle J. Graham, Université Paris Descartes, Centre National de la Recherche Scientifique, France
Received: July 4, 2010; Accepted: March 31, 2011; Published: May 5, 2011
Copyright: © 2011 Platkiewicz et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the European Research Council (ERC StG 240132, http://erc.europa.eu/). The funders had no role in study design, data collection and analysis, decision to publish,
or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Action potentials are initiated when the membrane potential exceeds a threshold value, but this value depends on the stimulation and can be very variable in vivo [1]–[4], which has triggered a recent
controversy about the origin of this variability [5]–[7]. This phenomenon has been observed in many areas of the nervous system: visual cortex [1]–[3], somatosensory cortex [4]; prefrontal cortex [8]
; neostriatum [9], neocortex [10], [11], hippocampus [12], [13], and auditory brainstem [14]–[17]. Experimental studies have shown that the spike threshold is correlated with the average membrane
potential [2], [8], inversely correlated with the preceding rate of depolarization [1]–[4], [9], [12], [14] and inversely correlated with the preceding interspike interval [13], [18]. Thus, threshold
dynamics participate in the input-output properties of neurons: it enhances coincidence detection and gain modulation properties [1], [2], it contributes to feature selectivity in sensory processing
[2], [4], [19], contrast invariance [2], [20] and temporal coding [17], [21], [22].
Among the mechanisms that can modulate the spike threshold [23], two are thought to be especially relevant: inactivation of sodium channels [1], [2], [4], [8], [12], [17] and activation of potassium
channels [2], [10]–[12], [14]–[16]. In this study, we chose to focus on the role of sodium channel inactivation because it specifically impacts spike initiation without changing the membrane
potential, and because of the extensive voltage-clamp data available for Na channels. Our first goal was to check whether Na channel inactivation, given their measured properties, can account for
significant threshold variability and for the qualitative properties of the spike threshold dynamics, as listed above. Our second goal was to evaluate the consequences of threshold dynamics on the
integration of postsynaptic potentials (PSPs).
We analyzed the influence of Na inactivation on spike threshold in a model, in which we were able to express the spike threshold as a function of Na channel properties and variables [23]. We
collected previously published voltage clamp measurements of Na channel properties and found that Na inactivation by itself can account for substantial threshold variability, with the same
qualitative properties as experimentally observed. To investigate the implications for synaptic integration, we derived a dynamical equation for the spike threshold and defined effective PSPs as the
difference between the PSP and the threshold. We found that, with threshold adaptation as implied by Na inactivation, effective PSPs are briefer than PSPs and that their shape depends on membrane
depolarization. Finally, we discuss the potential contribution of other mechanisms of threshold modulation.
The threshold equation
We previously derived a formula, the threshold equation, which relates the instantaneous value of the spike threshold to ionic channels properties [23]:where V[a] is the half-activation voltage of Na
channels, k[a] is the activation slope factor, g[Na] is the total Na conductance, g[L] is the leak conductance, E[Na] is the Na reversal potential, h is the inactivation variable (1-h is the fraction
of inactivated Na channels). Here the spike threshold is defined as the voltage value at the minimum of the current-voltage function in the membrane equation (we compared various threshold
definitions in [23]). This formula is derived from the assumption that the Na activation curve is well described by a Boltzmann function, which implies that the Na current below spike initiation is
close to an exponential function of voltage (see Text S1 for the derivation). This approximation of the Na current is the basis of the exponential integrate-and-fire model (EIF) [24]. In this paper,
we focus on the impact of Na inactivation and therefore we ignore the last term of the threshold equation, which simplifies to:where V[T] is a constant term, corresponding to the minimum spike
threshold (when Na channels are not inactivated). We call the EIF model with Na inactivation the inactivating exponential integrate-and-fire model (iEIF; see Methods). After a spike, the voltage is
reset to the resting potential E[L], and h is unchanged. Thus, when the neuron is depolarized, Na channels inactivate (h decreases) and the threshold increases: the threshold adapts to the membrane
Steady-state threshold and threshold variability
We start by studying the steady-state threshold, which is the value of the spike threshold for a fixed voltage V[0]. It corresponds to the threshold measured with the following experiment. The cell
is clamped at a voltage V[0] (Figure 1A), and a fraction of Na channels inactivates. In the Hodgkin-Huxley formalism, this fraction is , where is the steady-state inactivation function (h is the
fraction of non-inactivated channels). If the clamp is relaxed and a current is injected, the neuron may produce a spike if the current is large enough (Figure 1A). The steady-state threshold
corresponds to the maximum voltage that can be reached without triggering an action potential, and it depends on the fraction (1-h) of inactivated Na channels: when the membrane is depolarized, Na
channels inactivate, which raises the spike threshold.
A, The membrane potential is clamped at a given voltage , then a constant current I is injected (iEIF model). The steady-state threshold is defined as the maximum voltage that can be reached without
triggering an action potential. B, Two excitability curves dV/dt=F(V,V[0])/C are shown in the phase plane , for two different initial clamp values V[0] (solid lines; V[0]=−80 mV and −26 mV). The
steady-state threshold is the voltage at the minimum of the excitability curve for the initial voltage V[0]. C, Steady-state threshold (red lines) of a cortical neuron model [63] for the original
maximal Na conductance (solid line) and for a higher and lower Na conductance (resp. bottom and top dashed line). When the cell is slowly depolarized, it spikes when , i.e., the spike threshold is
the intersection of the red and black dashed curves. If there is no intersection, the neuron cannot spike with slow depolarization. The top dashed line (low Na conductance) is interrupted because the
threshold is infinite at high voltages (i.e., the cell is no longer excitable).
One way to understand threshold adaptation is to look at how the excitability curve changes with h (and therefore with depolarization). The excitability curve (Figure 1B) shows the value of dV/dt vs.
V for a fixed value of h, as given by the membrane equation (which is equivalent to the I-V curve, if the current is scaled by the membrane capacitance). When h decreases (Na channels inactivate),
the entire excitability curve shifts towards higher voltages and the threshold shifts accordingly. As in [23], we define the threshold as the voltage at the minimum of the excitability curve, but
since the entire curve is shifted by Na inactivation, other definitions would produce similar results.
The membrane potential V is always below threshold, unless the cell spikes. Therefore the observable threshold values cannot be larger than the intersection between the threshold curve and the
diagonal line , if these two curves intersect (Figure 1C). Thus, the spike threshold may vary between the minimum steady-state threshold V[T] and the solution of . When there is no such solution, the
threshold can be arbitrarily large, meaning that a very slow depolarization would not elicit a spike (Figure 1C, top dashed curve). Thus, the range of threshold variability can be derived from the
steady-state threshold curve.
Using the threshold equation, we can calculate the steady-state threshold as a function of V: , where is the Na inactivation curve, which is generally well fitted by a Boltzmann function [25]:where
is the half-inactivation voltage, and is the inactivation slope factor. When we substitute this function in the threshold equation, we find that the steady-state threshold has a horizontal asymptote
(V[T]) for large negative potentials and a linear asymptote for large positive potentials, because the inactivation function is close to exponential (Figure 2A). Thus, the steady-state threshold can
be approximated by a piecewise linear function (see Text S1):
A, The steady-state threshold curve (red curve) is well approximated by a piecewise linear curve determined by Na channel properties (top dashed black curve), where V[i] is the half-inactivation
voltage and V[T] is the non-inactivated threshold. The slope of the linear asymptote is k[a]/k[i] (resp. activation and inactivation slope parameters). Na channel properties in this figure were taken
from Kuba et al. (2009). The spike threshold is variable only when , and very variable when (additionally) . B, The non-inactivated threshold V[T] is determined by the maximum Na conductance g[Na],
relative to the leak conductance g[L]. As the ratio increases, the steady-state threshold curve shifts downward (red curves; r=0.4; 2; 10) and threshold variability is reduced. C, Trajectory of the
model in the phase plane (blue), superimposed on the steady-state threshold curve (red). Spikes are initiated when (dashed line: ), but the empirical measurement overestimates the threshold. The
spike threshold is highly variable in this example (−50 to −10 mV). D, Trajectory of the model in the phase plane (blue), superimposed on the Na inactivation curve (black). The threshold is very
variable when most Na channels are inactivated. E, Voltage trace (black curve) and spike threshold (red curve; ) in the inactivating exponential model driven by a fluctuating input (see Methods),
where black dots represent empirical measurement of spike onsets (first derivative method, k[th]=5 mV/ms). Note that the membrane potential can exceed threshold without triggering a spike because
the threshold is soft (unlike in integrate-and-fire models).
In other words, the minimum threshold is V[T], which is determined by the maximum Na conductance (Figure 2B), the threshold increases above the half-inactivation voltage V[i], and the slope is the
ratio of activation and inactivation slope factors. Regarding threshold variability, we can distinguish three cases, depending on Na channel properties:
1. if then the spike threshold is constant
2. if and , then the threshold varies between and ;
3. if and , then the threshold can be arbitrarily large (that is, the neuron can be continuously depolarized without triggering spikes, as observed in some preparations [26]).
Figure 2C-E illustrates case 2 in a single-compartment model with fluctuating inputs (note that the membrane potential can exceed the threshold without triggering a spike because spike initiation is
not sharp, unlike in real cortical neurons and in multicompartmental models; see the discussion in [23]). We started by examining these conditions in the dataset collected in the literature by
Angelino and Brenner [25] about the properties of the 9 Nav1 channel types. These properties were obtained from voltage clamp measurements of Na channels expressed in exogenous systems. Figure 3A
shows the distribution of V[i] in this dataset, which is rather wide (−90 mV to −25 mV). Central neuron channel types, i.e., Nav1. [1], [2], [3], [6],[27], are shown in red. Since the minimum
threshold V[T] depends on the maximal Na conductance, it cannot be deduced from channel properties alone. Considering that V[T] should lie between −55 and −45 mV [28], a substantial part of the
channels fall into the first case, i.e., constant threshold, while the rest can fall into the second (moderate threshold variability) or third case (unbounded variability), depending on whether k[a]>
k[i]. Figure 3B shows that, while this latter condition is never met for channel types expressed in sensory neurons (blue dots), about half of those expressed in central neurons (red) and muscles
(green) satisfy k[a]>k[i]. Thus, it seems that all three cases occur in similar proportions for channel types expressed in central neurons.
A, Distribution of half-inactivation voltage (V[i]) of Na channels expressed in exogenous systems (from a database of 40 Na channels reported in Angelino and Brenner, 2007 [25]), including central
neuron channel types (red), sensory neuron channel types (blue) and muscular channel types (green). Assuming a minimum spike threshold between −55 mV and −45 mV (dashed lines), channels on the left
have variable threshold while channels of the right have a constant threshold. B, Inactivation (k[i]) vs. activation slope (k[a]) for the same dataset. Channels with V[i]<−50 mV (variable threshold)
are indicated by a black contour. These channels have high threshold variability when k[a]>k[i] (right of the dashed line). C, Distribution of V[i] for Na channels expressed in central neurons in
situ (see Table S1). The threshold should be variable in most cases. D, Inactivation (k[i]) vs. activation slope (k[a]) for the same dataset. High threshold variability is predicted in about half
However, not all Na channels are involved in spike initiation. In particular, in central neurons, spike initiation is mediated by Nav1.6 channels while Nav1.2 channels are involved in axonal
backpropagation [8]. This first dataset contained only 4 Nav1.6 channels, for which V[i]<−50 mV in all cases (−61±8.4 mV), suggesting significant threshold variability, but this is a small sample.
Besides, this first dataset was somewhat artificial, because channels, some of which had mutations, were artificially expressed in an exogenous system, which might alter their properties. Therefore
we looked at a second dataset, consisting of in situ measurements in intact central neurons that we collected in the literature (see Table S1). These measurements may combine the properties of
several channel types expressed at the same site, e.g. Nav1.1, Nav1.2, or Nav1.6. In some of these studies, the threshold was also measured and found to be variable [8], [17], [29], [30]. In this
dataset, as shown in Figure 3C, the half-inactivation voltage was always lower than −50 mV, which implies that most channels induce threshold variability (cases 2 and 3). About half of them met the
condition k[a]>k[i] (Figure 3D). Thus, in this dataset, Na inactivation induces unbounded threshold variability in about half cases and moderate variability in the other half.
Threshold dynamics
We have shown that Na channel properties, i.e., parameters , , ,, allow us to determine whether Na inactivation can make the spike threshold variable and we found that the answer is positive in
central neurons. While this analysis gives an estimate of potential threshold variability, the observed variability and its properties depend on the stimulation. The instantaneous value of the spike
threshold depends on the value of the inactivation variable h through the following formula [23]: . We now assume that h evolves according to a standard Hodgkin-Huxley equation with first order
kinetics:where is the inactivation time constant. By differentiating the threshold equation and substituting the differential equation for h, we obtain a differential equation for as function of the
membrane potential (see Text S1 A), which can be approximated by:with . To simplify the calculations, we assume in the following that the inactivation time constant does not vary significantly with
V, but we examine the effect of this voltage-dependence later. This equation describes how the threshold changes with the membrane potential, and therefore with the stimulation, and is entirely
determined by Na channel properties. Since the steady-state threshold increases with V (Figure 2), it appears that the threshold adapts to the membrane potential with characteristic time . Thus, we
readily see that 1) the threshold increases with the membrane potential and 2) the threshold is lower for faster depolarization, because it has less time to adapt to the membrane potential.
Before we describe threshold dynamics in more details, we need to make an important remark. As is seen in Figure 2E, which describes the dynamics of an iEIF model with fluctuating inputs, the
membrane potential can exceed the threshold without triggering a spike, if the fluctuation is fast enough. This reflects the fact that spike initiation in this model, as in any biophysical
single-compartment model, is not sharp: since there is no well-defined voltage threshold, what we describe as threshold variations are more accurately described as voltage shifts of the excitability
curve. This makes the definition of a dynamic threshold a little ambiguous. However, spike initiation in cortical neurons is much sharper than in single-compartment models [5], because of the active
backpropagation of spikes from the initiation site [6]. A direct in vitro measurement of the slope factor in cortical neurons (characterizing spike sharpness) gave Δ[T]≈1 mV [18] (compared to k[a] ≈
6 mV), meaning that spike initiation is almost as sharp as in an integrate-and-fire model. This phenomenon is well captured by multicompartmental models [8], [23] and it affects spike sharpness
independently of threshold variability: in Figure 7H of ref. [23], spikes are initiated as soon as the membrane potential exceeds the dynamic threshold, which is determined according to the threshold
equation. This motivates us to introduce a new model, the inactivating integrate-and-fire model (iLIF, see Methods), which is simply an integrate-and-fire model with an adaptive threshold given by
the differential equation above (after a spike, the voltage is reset to the resting potential E[L], and the threshold is increased - see Methods). This phenomenological model is not only simpler, but
also seemingly more realistic than the iEIF model for the present problem, in that it reproduces both the sharpness of spike initiation and the variability of spike threshold. We use this model in
the remainder of this paper.
The threshold also increases with each action potential [23] (see also Text S1 A), as was recently demonstrated in vitro [18]. This can be described as simple additive shift: , where is the average
value of the time constant during the action potential and is the spike duration (typically, a few ms). If the inactivation time constant is short compared to the typical interspike interval, then
this shift results in a relative refractory period, but has negligible influence on the subsequent dynamics of the model. If it is long, it results in spike-frequency adaptation and explains in vivo
observations where the threshold was found to be inversely correlated with the previous interspike interval [13]. This phenonemon can be seen in the noise-driven iLIF model when Na inactivation is
slow (not shown). In the following, we focus on the impact of fast Na inactivation.
Quantitatively, the relationship between average membrane potential and threshold depends on the steady-state threshold function . Figure 4 shows this relationship in a neuron model with adaptive
threshold (defined by the dynamical equation above) and fluctuating inputs of varying mean. As expected, the average threshold increases with the average membrane potential, and the slope is steeper
above half-inactivation voltage V[i]. In these simulations, the slope of the steady-state threshold curve was k[a]/k[i]=1, close to experimental values, but we note that the average threshold only
increases as about 2/3 the average membrane potential in the depolarized region. This is because the membrane potential is very variable (about 6 mV in this figure) and therefore the threshold is not
constantly in the sensitive region (V>V[i]). This is consistent with previous measurements in the visual cortex in vivo, where Azouz and Gray (2003) found a linear correlation with a slope of 0.5.
To calculate the relationship between the slope of depolarization and the threshold, we consider a linear depolarization with slope s (i.e., V(t)=V[0]+st) and calculate the intersection with the
threshold (Figure 5A). By linearizing the steady-state threshold as previously described, we find that the slope s and the threshold are related by the following equation (see Methods):
We simulated the iLIF model (see Methods) with a fluctuating input current. The standard deviation was fixed while the mean current was varied between trials. The mean spike threshold () is plotted
as a function of the mean membrane potential (). The slope of the curve is larger above half-inactivation voltage V[i] (0.64 from linear regression, red line) than below (0.23).
A, The neuron is linearly depolarized with a given slope s (V(t)=E[L]+st) until the membrane potential (black) reaches threshold (red) and the neuron spikes. The intersection of the black and red
traces (red dots) can be calculated (see Results). B, Threshold vs. depolarization slope (solid line) and analytical formula when k[a]=k[i] (dashed line). C, Slope-threshold relationship for
different values of the half-inactivation voltage V[i] (V[i]=−63 mV in panels A,B). D, Slope-threshold relationship for different values of the inactivation time constant ( in panels A,B). E, The
iLIF model is driven by a fluctuating current and we measure the slope of depolarization before each spike over a duration by linear regression. F, Slope-threshold relationship measured with linear
regression in the noise-driven iLIF model (red dots), superimposed on the calculated relationship from panel B.
Unfortunately, this implicit equation does not give a closed formula for as a function of s, except when :
In this particular case, the threshold diverges to infinity at , i.e., no spike is produced if the depolarization is slower than s* (Figure 5B, dashed line). This phenomenon can occur more generally
when (unbounded variability, case 3) and has been observed in neurons of the cochlear nucleus [16] (where it is described as a "rate threshold"). In all cases, for large s (fast depolarization), the
threshold tends to V[T], i.e., to the lowest possible threshold, and it increases for smaller s, i.e., slow depolarization (Figure 5B, solid line). The equations show that the slope-threshold
relationship depends on the half-inactivation voltage V[i] and on the threshold time constant (=). The relationship is more pronounced when V[i] is low compared to the minimum threshold V[T] (
Figure 5C; V[T] was −55 mV). The role of the threshold time constant can be seen as a scaling factor for slopes, i.e., the threshold depends on the product of the slope and threshold time constant.
The slope-threshold relationship is more pronounced when the threshold time constant is short (Figure 5D). In experiments in vivo, the slope-threshold relationship was measured using linear
regression on the membrane potential preceding each spike [2], [4]. We simulated the adaptive threshold model with a fluctuating input (Figure 5E) and performed a similar analysis, by calculating the
depolarization slopes over a duration equal to the threshold time constant. The resulting slope-threshold relationship matches our previous calculation (which only uses Na channel properties), but
with more variability (Figure 5F), as is also observed in experiments. Finally, we measured the slope-relationship in the multicompartmental model of Hu et al. [8] with fluctuating inputs, for which
we previously showed that the threshold equation accurately predicted the measured threshold [23]. The slope-threshold relationship also matched our prediction (Figure S1).
Threshold variability with fluctuating inputs
These dynamical properties of the threshold imply that the threshold should be variable for fluctuating inputs (typical of in vivo regimes) but not for constant DC inputs (typical of in vitro
stimulations). More generally, it implies that the threshold distribution depends on the membrane potential distribution, as shown in Figure 6 with a neuron model with adaptive threshold driven by
fluctuating inputs with different statistics. The average threshold depends mainly on the average membrane potential (Figure 6A), but the standard deviation is correlated with both the average and
the standard deviation of the membrane potential (Figure 6B). This could underlie the observed difference in threshold variability between spontaneous activity (<σ>=1.4 mV) and visual responses
(<σ>=2.3 mV) [1], because in visual responses the membrane potential is presumably both more depolarized and more variable. Interestingly, fast spiking cells showed lower threshold variability
together with a lower mean threshold, which is also consistent with our results.
An iLIF model was stimulated by fluctuating inputs with different means and standard deviations and the threshold distribution was measured. A, Average threshold (color-coded) as a function of the
mean (<V>) and standard deviation (σ[V]) of the membrane potential. The average threshold depends primarily on the average membrane potential. White areas correspond to parameter values that were not
tested (top) or that elicited no spike (bottom). B, Standard deviation of the threshold as a function of membrane potential statistics. Threshold variability depends on both the average and the
standard deviation of the membrane potential.
Implications for synaptic integration
These results have two main implications for synaptic integration: 1) threshold adaptation reduces the impact of the input mean, relative to its variance, and 2) the negative correlation between
threshold and depolarization rate shortens the timescale of synaptic integration.
Sensitivity to the mean and variance of inputs.
When V>V[i], the steady-state threshold increases with the voltage (Figure 2A), with a slope close to 1. As a result, when the neuron is driven by a fluctuating input (such as a sum of random
synaptic currents), the average threshold increases with the average membrane potential, as shown in Figure 4. Because the slope of this relationship is close to 1 (), the average difference between
the instantaneous value of the threshold and the membrane potential should be nearly constant above V[i]: . Thus, we expect that the mean of the input should have little impact on postsynaptic
firing, while it should be more sensitive to its variance. Figure 7 shows the results of simulations where fluctuating currents with varying mean and variance were injected into a neuron model with
adaptive threshold. When the threshold does not adapt, the output firing rate is sensitive both to the mean and the variance of the input (Figure 7A, mixed line, and Figure 7B). When the mean is
above threshold (−55 mV in Figure 7), the firing rate is mostly determined by the mean. However, as threshold adaptation is increased (Figure 7A, dashed and solid lines, and Figure 7C,D), the firing
rate becomes less and less sensitive to the input mean and relatively more sensitive to the variance. When threshold adaptation parameters correspond to experimentally measured properties of Na
channels (), the firing rate is mostly sensitive to the input variance, although the mean input still plays a role. Thus, by maintaining a constant difference between average potential and threshold,
Na channel inactivation acts as a homeostatic mechanism.
An iLIF model was simulated in the same way as in Figure 6, but with different values for the parameter k[a]/k[i], which controls threshold adaptation. A, Output firing rate vs. mean input with
threshold adaptation (solid line, k[a]/k[i]=1), with mild threshold adaptation (dashed line, k[a]/k[i]=0.5) and without threshold adaptation (mixed line, k[a]/k[i]=0). The horizontal axis is
the input resistance R times the mean input <I>, i.e., the mean depolarization in the absence of spikes. The input standard deviation was chosen so that the neuron fires at 10 Hz when the mean
depolarization is 10 mV. B, Firing rate (color-coded) vs. mean and standard deviation of the input, without adaptation (k[a]/k[i]=0). The standard deviation is shown in voltage units to represent
the standard deviation of the membrane potential in the absence of spikes, i.e., , where σ[I] is the input standard deviation (in current units) and is the input time constant. The horizontal mixed
line corresponds to the mixed line shown in panel A, and the vertical dashed line corresponds to the threshold for constant currents. C, Same as B, but with mild threshold adaptation (k[a]/k[i]=
0.5). D, Same as B, but with normal threshold adaptation (k[a]/k[i]=1).
Timescale of synaptic integration.
It was remarked in previous studies that the negative relationship between threshold and depolarization rate should make the neuron more sensitive to coincidences [2], [4], because depolarization is
faster and thus threshold is lower for coincident inputs. We make this remark more precise by looking at effective PSPs, defined as the difference between the PSP and the dynamic threshold (Figure 8
). Consider a neuron model in which the membrane potential is described by a sum of PSPs:where is the PSP at synapse i and is the timing of the spike received at synapse i. If we approximate
threshold dynamics by a linear differential equation (when V>V[i]), then the threshold is a low-pass filtered version of :where L is a first-order low-pass filter with time constant (i.e., cutoff
frequency ), i.e.:where . This model with adaptive threshold is equivalent to a model with fixed threshold , where the voltage is defined by , i.e., relatively to the threshold. In this equivalent
model, the voltage reads:
A, Top: Normalized postsynaptic potential (PSP, solid line) and threshold PSP, i.e., effect of the PSP on the threshold (dashed line). Bottom: The effective PSP is the difference between the PSP and
the threshold PSP. It is briefer and can change sign. B, The effect of the PSP on spike threshold depends on how the threshold changes with voltage (dθ/dV, bottom), which depends on the membrane
potential V and is determined by the Na inactivation curve (top; dashed line: half-inactivation). At high voltage, dθ/dV= k[a]/k[i] (=1 here). C, Half-width of the effective PSP (color-coded) as
a function of threshold sensitivity dθ/dV and the threshold time constant . The black cross corresponds to the situation shown in panel A. The membrane time constant () is shown by a horizontal solid
line. D, Zero crossing time of the effective PSP as a function of threshold sensitivity and threshold time constant. The white triangle corresponds to parameter values where the effective PSP is
always positive.
Thus, it is a linear superposition of effective PSPs (ePSPs), defined as the difference between the PSP and the threshold PSP (effect of PSP on threshold):where is the effective PSP at synapse i.
This equivalent model has exactly the same form as the initial model (superposition of PSPs), the only difference being that PSPs are replaced by effective PSPs with a different shape. This is
illustrated in Figure 8A.
In other words, threshold adaptation acts as a simultaneous inhibition with slower time constant (than the excitatory PSP), or as a simultaneous excitation for inhibitory PSPs. As a result, the
temporal width of effective PSPs is smaller than that of PSPs, so that the timescale of synaptic integration is shorter (Figure 8A,C; see also Text S1 B for analytical calculations). Far from V[i],
i.e., when the threshold varies linearly with the membrane potential, the threshold PSP is proportional to k[a]/k[i], which is close to 1 in experimental measurements. Closer to V[i], the threshold
PSP is proportional to dθ/dV, which lies between 0 and k[a]/k[i] (Figure 8B). This means that threshold adaptation increases when the neuron is more depolarized, so that effective PSPs become
sharper. This property is shown in Figure 8C, where the half-width of effective PSPs is seen to depend on the threshold time constant (sharper effective PSPs for shorter time constants) and on
threshold sensitivity dθ/dV, i.e., indirectly on depolarization. In all cases, effective PSPs are always sharper than PSPs. For example, when the threshold time constant equals the PSP time constant
and the neuron is depolarized well above V[i] (with k[a]=k[i]), threshold adaptation reduces the half-width of the PSP by a factor greater than 2 (intersection of the two lines in Figure 8C). In
some cases, the effective PSP may change sign, as shown in Figure 8A (bottom). This occurs when the threshold time constant or the threshold sensitivity is large (Figure 8D). In the case of
exponentially decaying PSPs, this condition can be analytically calculated (see Text S1 B): . This property implies that inhibitory PSPs may trigger delayed spikes because of threshold adaptation,
which we discuss below.
Similar properties are seen when synaptic filtering is taken into account, that is, when the synaptic current is an exponentially decaying function rather than an instantaneous pulse (Dirac), giving
biexponential PSPs (Figure 9A). As previously, effective PSPs are briefer and can change sign (Figure 9B). A new property can be observed: the peak time is shorter for ePSPs than for PSPs. This could
not be seen with exponential PSPs since in that case both the PSP and the ePSP peak at 0 ms. With synaptic filtering, ePSPs peak earlier and at a smaller value. The peak time of the PSP increases
with the time constant of synaptic filtering, but threshold adaptation makes ePSPs not only briefer but also less sensitive to the filtering time constant (Figure 9C,D). This phenomenon was recently
demonstrated in neurons of the medial superior olive (MSO), a structure involved in the computation of interaural time differences, a cue to the azimuth of a sound source [31]. These neurons detect
coincidences between inputs from the contralateral side and from the ipsilateral side. It was found that PSPs from the contralateral side peak about 500 µs later than those from the ipsilateral side,
and are also shallower, which makes coincidence detection problematic (the required precision is about a few tens of microseconds). But threshold adaptation reduces the peak time of the shallower
contralateral PSP, so that PSPs from both sides have similar latency. Another interesting consequence of the compression of peak times by threshold adaptation is that it also minimizes the impact of
dendritic propagation on the effective latency of PSPs.
A, Normalized biexponential PSPs obtained with non-instantaneous synaptic currents (i.e., postsynaptic currents are exponentially decaying with time constant τ[s] between 1 ms and 20 ms). B, As for
exponential PSPs (Figure 8), effective PSPs (ePSPs) are narrower and change sign (only the positive part is shown). The time to peak is also shorter. Threshold adaptation parameters were and dθ/dV=
1. C, The peak time increases with the synaptic filtering time constant τ[s], but less rapidly for ePSPs than for PSPs. D, ePSP peak time vs. PSP peak time. Threshold adaptation makes peak times
shorter and compressed.
As is illustrated in Figure 10A, the reduction of PSP width makes the neuron more sensitive to coincidences at the timescale of threshold dynamics, i.e., of Na inactivation. This property only arises
when the neuron is sufficiently depolarized, i.e., when V>V[i] (Figure 10B). In high-conductance states that are typical of in vivo activity [32], [33], the mean membrane potential is depolarized,
typically around −60 mV, which is slightly higher than the average V[i] in the dataset of Na channels in central neurons in situ (; Figure 3C). Thus, neurons in vivo should be more sensitive to
coincidences at the timescale of Na inactivation. This comes in addition to the fact that the membrane time constant is about 5 times shorter in vivo than in vitro because of increased total
conductance [34], [35]. More precisely, the shape of effective PSPs depends on depolarization: as the neuron is more depolarized, the fast component of the effective PSP (which decays with time
constant ) becomes more dominant, so that the neuron becomes more sensitive to fine correlations (Figure 8C).
A, The iLIF model was simulated with random inputs (exponentially decaying PSPs), temporally distributed according to a Poisson process. Top: Spikes are produced when the membrane potential V (black)
exceeds the threshold θ (red). Bottom: This is equivalent to a model with fixed zero threshold (red) and potential V-θ (black), which is the sum of effective PSPs. Effective PSPs are sharper than
PSPs. B, Top: The threshold is more adaptive when the neuron is depolarized (right) than near resting potential (left). Bottom: When the mean input is increased (4 different levels shown), effective
PSPs become sharper and their negative part cancels the input mean (see Figure 8). C, Random inhibitory PSPs are added to a depolarizing current ramp. Without inhibitory inputs (dashed), the
threshold adapts and the neuron does not spike. With inhibitory inputs (solid), the sign change in effective PSPs (see Figure 8) acts as a rebound and triggers spikes. This phenomenon is often called
postinhibitory facilitation [36]. D, When the voltage dependence of the Na inactivation time constant is taken into account (see Methods), effective PSPs become sharper as the neuron is more
depolarized, which implies an adaptive coincidence detection property [2].
For inhibitory PSPs (IPSPs), threshold adaptation is equivalent to simultaneous excitation with a slower time constant. Thus, in some cases, the later part of the effective PSP can be positive (
Figure 8D), and therefore an IPSP can trigger a spike (Figure 10C). This phenomenon is generally called postinhibitory facilitation. It has been previously observed in different systems, and can be
mediated by other mechanisms than Na inactivation [36], [37]. Figure 10C shows an example of postinhibitory facilitation due to Na inactivation, where a slow depolarization fails to trigger a
postsynaptic spike but additional IPSPs do.
Finally, while we have previously ignored the voltage dependence of the time constant of Na inactivation, we show in Figure 10D how it affects synaptic integration. The time constant decreases when
the neuron is depolarized above V[i] (see Methods), which reduces the half-width of effective PSPs (Figure 8C,D). This property was termed adaptive coincidence detection in previous experimental
studies [2].
Based on voltage clamp measurements of Na channel properties, we have found that Na inactivation can produce by itself large threshold variability, as observed in experiments in vivo [1]−[4]. Our
analysis led us to a simple theoretical criterion on Na channel properties ( for moderate variability and for unbounded variability). Threshold dynamics are then inherited from the dynamics of Na
inactivation, which implies that the threshold adapts to the membrane potential. As a consequence, the threshold is correlated with the preceding membrane potential and inversely correlated with the
depolarization rate. Both properties were observed in experiments and the quantitative relationships are close to what we predict from the properties of Na inactivation. Our analysis also provides a
simple adaptive equation which describes threshold dynamics.
The criterion for large threshold variability () depends on the precise values of the half-activation (k[a]) and half-inactivation voltages (k[i]), obtained from Boltzmann fits. However, the relevant
voltage range for these fits is the spike initiation range, and reported experimental values generally correspond to fits over the entire voltage range. This could contribute a significant
measurement error in these values, as we previously showed [23]. Another potential source of error is the overlap between activation and inactivation. If the inactivation time constant is very short
(comparable to the activation time constant), then voltage-clamp measurements tend to overestimate k[a] [23]. Thus, there is some uncertainty about the precise value of k[a]/k[i] in Na channels.
One consequence of threshold adaptation is to reduce the sensitivity of neurons to their mean input, and to make them more sensitive to fluctuations. In vitro, Arsiero et al. [38] indeed observed
that pyramidal cells of the prefrontal cortex were very sensitive to the variance of their inputs, even when the mean was high. In vivo, Ringach and Malone [39] described the responses of neurons of
the primary visual cortex as linear filtering of the visual input followed by (stochastic) spiking when a threshold was exceeded. They found that the threshold (defined on an abstract variable)
adapted to the input statistics, so that neurons responded only to positive fluctuations above the mean.
Threshold adaptation implies that a presynaptic spike has an effect on both the membrane potential (the classical PSP) and the spike threshold. We defined an effective PSP by subtracting the
threshold effect from the PSP. Thus, a neuron model with adaptive threshold where the membrane potential is a sum of PSPs is equivalent to a model with fixed threshold where the potential is a sum of
effective PSPs. We found that effective PSPs were briefer than PSPs, which makes neurons more sensitive to input correlations at the timescale of Na inactivation. The effect of threshold adaptation
can be understood as simultaneous inhibition for EPSPs and simultaneous excitation for IPSPs. These effective PSPs become briefer as the neuron is more depolarized, which can be seen as a form of
adaptive coincidence detection: as the neuron is more depolarized, it requires more precisely coincident inputs to fire. This suggests that the effective integration time constant of neurons might be
even shorter in vivo than expected from conductance measurements [34] because neurons are significantly depolarized in high conductance states [33]. A similar sharpening effect was recently found
with Kv1 channels in neurons of the medial superior olive (MSO) [40]; a linear treatment of temporal sharpening by active conductances along dendrites was also recently done [41] (although
independently of threshold properties).
Although Na channel inactivation can account for all the properties that have been experimentally observed, other mechanisms could potentially contribute to threshold variability: somatic measurement
when spikes are initiated in the axon, channel noise and other ionic mechanisms. We discuss below these alternative mechanisms and evaluate whether they may account for threshold adaptation.
Remote spike initiation
A recent debate about the validity of the Hodgkin-Huxley model for cortical neurons has highlighted the fact that, for central neurons, spikes are initiated in the axon while in vivo measurements of
the spike threshold were done at the soma, which could be an artifactual cause of threshold variability [5]-[7]. However, it is unclear whether distal initiation could account for the inverse
correlation between the threshold and the preceding slope of depolarization.
To address this question, we consider a simplified situation where spikes are initiated in the axon hillock when the potential is above a fixed threshold V[T] (Figure 11A). Suppose the membrane
potential increases linearly in the soma (blue line) and spreads to the spike initiation site with a delay (black line). A spike is initiated when the propagated potential reaches threshold (dashed
red line), and backpropagated to the soma with a delay . As a result, the spike “threshold” (in fact, spike onset) is higher when measured at the soma, by an amount of , where s is the slope of
depolarization. This has two consequences: 1) threshold variability is increased for fluctuating inputs, 2) the threshold is positively correlated with the slope of depolarization. Based on passive
cable properties, the forward delay can be estimated as and the backward delay as , where C is the specific membrane capacitance, (resp. ) is the membrane surface of the spike initiation site (resp.
soma) and is the coupling conductance between the two sites [23]. Considering active conductances would reduce these values, but these estimations are already close to experimental measurements [42].
Thus, the total delay (forward + backward) is smaller than 1 ms.
A, Illustration of the effect of depolarization slope s on somatic spike onset. In cortical neurons, spikes are initiated in the axon initial segment (AIS, black), then backpropagated to the soma
(blue). Somatic depolarization is propagated forward to the spike initiation site in the axon with delay . A spike is initiated in the axon when the threshold V[T] is reached (dashed red line). The
spike is backpropagated to the soma with delay . During time , the somatic voltage has increased by and the spike onset is seen higher (red dot). B, Slope-threshold relationship in the
multicompartmental model of Yu et al. (2008) [7] with fluctuating inputs (mean 0.7 nA, standard deviation 0.2 nA, time constant 10 ms), measured at the AIS (top) and at the soma (bottom). As
expected, the slope-threshold relationship is less pronounced at the soma than at the AIS. C, The effect of channel noise is modeled by a stochastic threshold (red; , , and ) and the neuron is
linearly depolarized. With slow depolarization (left), the threshold (at spike time) is lower than the average instantaneous threshold. With fast depolarization (right), the threshold distribution
(at spike time) follows the distribution of θ. D, As a result, the threshold is positively correlated with the depolarization slope (blue dots: threshold vs. slope for all spikes in the simulations;
black dots: average threshold for each slope).
We confirmed this reasoning by simulating the response of the multicompartmental model of Yu et al. (2008) [7] to fluctuating inputs and measuring the slope-threshold relationship both at the soma
and at the axon initial segment (AIS) (Figure 11B). As we expected, we found that this relationship was more pronounced at the AIS than at the soma, meaning that the net effect of backpropagation is
a positive correlation between slope and threshold. More precisely, the net effect corresponds to a total delay of (difference between the two slopes of the linear regressions), in accordance with
the estimation above. Thus, since distal spike initiation predicts the opposite relationship between depolarization rate and threshold than experimentally observed, it cannot be the dominant cause of
threshold variability and cannot account for the properties of threshold dynamics.
Channel noise
The Hodgkin-Huxley formalism describes the dynamics of the macroscopic average of many sodium channels, but individual channels have stochastic dynamics [43], [44]. It results in threshold
variability which is not significantly correlated with input properties [45], [46], [43], [47], [48]. As previously, we examine whether this mechanism may account for the slope-threshold relationship
in a simplified model. We consider an integrate-and-fire model with a threshold that fluctuates randomly, according to an Ornstein-Uhlenbeck process:where is the mean voltage threshold, is the
standard deviation of the threshold distribution, is a gaussian white noise and is the time constant of fluctuations (related to the time constant of Na activation).
When depolarization is very slow, spikes will be initiated lower than on average, because the stochastic threshold has time for many excursions below its mean, i.e., the threshold reaches the
membrane potential rather than the converse (Figure 11C, left). In fact if the membrane is not depolarized (zero slope), a spike will be initiated at resting potential (although after a potentially
very long time) because there is a positive probability that reaches that potential. On the contrary, if depolarization is very fast, spike initiation occurs at , where t is near the time of
depolarization, and therefore the distribution of the threshold at spike times is the same as the distribution of (at all times), with mean (Figure 11C, right). Therefore, the threshold is positively
correlated with the slope of depolarization. We confirmed this reasoning with a numerical simulation of the model for different depolarization slopes (Figure 11D). Thus, as for distal spike
initiation, channel noise produces threshold variability but induces a (weak) positive slope-threshold relationship, which is contrary to experimental findings.
Synaptic conductances
The spike threshold increases with the total non-sodium conductance, because spike initiation requires more Na channels to be open in order to counteract a larger total conductance. Thus, fluctuating
synaptic conductances could be a source of threshold variability. We previously estimated the effect of total conductance on spike threshold through the following formula [23]:where is the total
conductance, including excitatory (g[e]) and inhibitory (g[i]) conductances, and we ignored the effects of Na inactivation. Threshold variability is determined by the variability of total conductance
at spike time. In low-conductance states (in vitro or down states in vivo), spikes are preferentially triggered by increases in excitatory conductance g[e] [49]. In this case, the depolarization rate
is positively correlated with g[e], and therefore with the threshold. Besides threshold variability can only be mild because the total conductance is low (relative to the leak conductance). In
high-conductance states (up states in vivo), spikes are preferentially triggered by decreases in inhibitory conductance g[i] [49]. In this case, the depolarization rate is negatively correlated with
g[i], and therefore with the threshold. Therefore, in high-conductance states but not in low-conductance states, the slope-threshold relationship induced by synaptic conductances is qualitatively
consistent with experimental observations in vivo. However, with the same reasoning, the membrane potential increases when inhibition decreases and therefore, if inhibition is the main source of
variability, the threshold should be negatively correlated with the preceding membrane potential, which contradicts experimental observations in vivo. Therefore, synaptic conductances cannot
simultaneously account for the slope-threshold relationship and for the dependence on membrane potential observed in vivo.
Sodium channel activation
In our analysis, we assumed that Na activation is instantaneous. Voltage clamp measurements indeed show that its time constant is only a fraction of millisecond [50], [29], [51], [52]. However, with
this approximation, we might have neglected a source of threshold variability. As previously, let us examine the potential contribution of this cause of threshold variability to the slope-threshold
relationship. If depolarization is slow (compared to the activation time constant), then the proportion of open channels is given by the steady-state activation curve and our analysis applies. If
depolarization is very fast, fewer channels are opened than at steady state and therefore the threshold is higher. Thus, non-instantaneous activation of Na channels contributes a positive correlation
between depolarization rate and threshold, contrary to experimental findings.
Other voltage-gated channels
In the same way as synaptic conductances, voltage-gated channels may also modulate the spike threshold [23]. In particular, the delayed-rectifier potassium channel (e.g. Kv1) has been previously
proposed by several authors as the source of threshold variability [2], [10], [11], [14]–[16], [21]. Indeed, a similar model to our iLIF model was previously introduced in the context of threshold
accommodation by potassium channels [36]. To account for the positive correlation between membrane potential and threshold, the conductance must increase with depolarization, i.e., the activation
curve must be an increasing function of the voltage. We only consider this case in this discussion. The threshold depends on the voltage-gated conductance g[K] through the following formula:where we
ignored the effect of Na inactivation. To account for significant threshold variability, two conditions must be met: 1) the maximal conductance must be large (compared to the leak) and 2) the
half-activation voltage must be low enough. In this case, the spike threshold adapts to the membrane potential, which implies a positive correlation between membrane potential and threshold and a
negative correlation between depolarization rate and threshold, as experimentally observed. It is also possible to differentiate the threshold equation and obtain a differential equation that
describes the threshold dynamics as for Na inactivation, although it takes a different form [23]. However, there are several differences with threshold modulation induced by Na inactivation. Firstly,
the threshold is always bounded by the value obtained with the maximal conductance. Secondly, the relationship between membrane potential and threshold is in general sigmoidal and can only be linear
in a limited range, where the voltage is below half-activation but the conductance is still very large (the slope of this relationship is then k[a]^Na/k[a]^K). The impact on synaptic integration is
also different, because the conductance impacts not only the threshold but also the PSPs and effective membrane time constant.
Finally, we discuss below the possible interactions of several Na channel subtypes and of slow and fast Na inactivation.
Inactivation with several sodium channel subtypes
We assumed that a single Na channel type (e.g. Nav1.6) was present. It is possible to extend our analysis to the case of multiple subtypes. Suppose the Na current is made of two components
corresponding to two channel types:
To simplify, we assumed that the two channels have the same activation Boltzmann factor k[a], which is not unreasonable. Then the Na current can be equivalently expressed as: where:
In other words, when several subtypes are present, inactivation in the threshold equation is replaced by a linear combination of inactivation variables of all subtypes. For example, Nav1.2 and Nav1.6
are both found in the axon initial segment [8], and Nav1.2 channels activate and inactivate at more depolarized potentials than Nav1.6 [53]. According to the threshold equation above, at
hyperpolarized voltages, threshold modulation should be mainly determined by Nav1.6 (the inactivation variable h[2] for Nav1.2 is less voltage-dependent and its threshold is higher); at more
depolarized voltages (assuming the threshold has not been reached), Nav1.6 channels inactivate (h[1]≈0) and threshold modulation is then determined by Nav1.2 channels. Note however that with several
channel subtypes, it is not possible to express threshold dynamics as a single kinetic equation for anymore (without the use of the hidden variables h[1] and h[2]).
Slow sodium channel inactivation
In the present study, we focused on fast Na inactivation. We have briefly mentioned that the threshold equation applies when Na inactivation is slow, and implies that the threshold increases after
each spike, which induces a negative correlation between threshold and preceding inter-spike interval. This effect is expected, but it gets more interesting when the interaction between slow and fast
components is considered. One way to model this interaction is to consider two Na currents, as in the previous section. But since inactivation in the same channel can show slow and fast components,
it might be more relevant to include this interaction in the gating variables. The simplest way is to consider these components as independent gating processes, that is:where the gating variables h
[slow] and h[fast] have slow and fast dynamics, respectively [54], [55]. Since the interaction is multiplicative for the Na current, it is additive for the threshold:
In this case, it is possible to write a kinetic equation for each component of the threshold ( and ), in the same way as before (note that increases after each spike, whereas this effect can be
neglected for since its impact on subsequent spikes is negligible). Here, the effect of slow inactivation can be thought of as a slow change of an effective minimal threshold with firing activity.
Interesting interactions appear because, as we have seen, threshold variability depends on the value of that minimal threshold (relative to V[i]). Suppose for example that V[T]<V[i]. At low firing
rates (when interspike intervals are larger than the slow inactivation time constant), and the threshold is not variable. If the firing rate is high enough, then and the threshold becomes variable
with fast inactivation. In the same way, the time constant of synaptic integration should be larger at low rates than at high rates. Thus, slow inactivation controls threshold modulation by fast
In summary, many mechanisms may contribute to the variability of the spike threshold, but only two can account for its observed adaptive properties: Na inactivation and adaptive conductances (most
likely K channels). Although threshold dynamics is qualitatively similar for both mechanisms, they can be distinguished by the fact that Na inactivation has no subthreshold effect on the membrane
potential. Specifically, if the threshold is mainly modulated by adaptive conductances, then we can make two predictions:
1. The relationship between membrane potential and threshold should be determined by the I-V curve in the region where Na channels are closed: , where is a constant, and the I-V curve should be
highly nonlinear (this derives from the threshold equation above and the fact the total conductance is dI/dV).
2. The effective membrane time constant (as measured e.g. by the response to current pulses) should be inversely correlated with the threshold, through a similar formula: , because is inversely
proportional to the total conductance.
In a few experimental studies, the application of α-dendrotoxin, a pharmacological blocker of low-voltage-activated potassium channels, greatly reduces threshold variability [16], which suggests a
strong role for these channels in threshold adaptation. Our results suggest an alternative interpretation of these observations. The application of a blocker reduces the total conductance, which also
reduces the minimum threshold V[T] (see the threshold equation with voltage-gated channels), possibly below half-inactivation voltage V[i], where there is no threshold adaptation due to Na
inactivation. Thus, it could be that threshold adaptation was due to Na inactivation, but that suppressing K conductances shifted the minimum threshold out of the operating range of this mechanism.
This hypothesis could be tested by simultaneously injecting a fixed conductance in dynamic clamp, to compensate for the reduction in total conductance of the cell.
Although we cannot draw a universal conclusion at this point, and while it is possible that either or both mechanisms are present in different cells, we observe that Na inactivation is a
metabolically efficient way for neurons to shorten and regulate the time constant of synaptic integration. Indeed, Na inactivation implies no charge movement across the membrane while K+ conductances
modulate the threshold by counteracting the Na current, which implies a large transfer of charges across the membrane (Na+ inward and K+ outward) in the entire region where the threshold is variable.
Recently, it was found in hippocampal mossy fibers that K+ channels open only after spike initiation, in a way that minimizes charge movements [56]. Since energy consumption in the brain is a strong
evolutionary pressure [57]-[59], we suggest that Na inactivation may be the main source of threshold variability when this variability has functional benefits.
All numerical simulations were implemented with the Brian simulator [60] on a standard PC.
Inactivating exponential model (iEIF)
Near spike initiation, the Na current can be approximated by an exponential function of the voltage [18], [24]. If the inactivation variable h is not discarded (see Text S1 A), we obtain the
following model (membrane equation and inactivation dynamics):(1)(2)where V is the membrane potential, h is the Na inactivation variable, I is the input current, C is the membrane capacitance, (resp.
) is the leak conductance (resp. the reversal potential), is the Na activation slope factor, V[T] is the threshold when Na channels are not inactivated, is the Na steady-state inactivation function,
and is the Na inactivation time constant, which we consider constant for simplification (except in Figure 10D). Since the model does not include K+ channels and the exponential approximation is not
valid beyond spike initiation, action potentials are not realistically reproduced, but we only focus on spike initiation. We call this model iEIF (inactivating exponential integrate-and-fire model,
equations (1–2)). The membrane potential is reset to when it crosses 0 mV (h is unchanged). In Figure 2, we used , (typical membrane time constant in vivo [34]), V[T]=−58 mV, k[a]=5 mV, , and the
inactivation function was a Boltzmann function with parameters V[i]=−63 mV and k[i]=6 mV.
Adaptive threshold model and iLIF model
A very good approximation of the Na current is an exponential function of V [18], [24], [61]. The spike threshold can then be expressed with the threshold equation [23]:(3)where(4)is the minimum
threshold, i.e., when Na channels are not inactivated (h=1). By differentiating the threshold equation and substituting the differential equation for h, we obtain a differential equation for as
function of the membrane potential (see Text S1), which can be approximated by:(5)with , where is the steady-state threshold, which can be approximated by a piecewise linear function (see Text S1):
We refer to the differential equation of together with the expression of above as the adaptive threshold model. In simulations, we used this model with a passive membrane equation:(8)where R is the
membrane resistance and I is the input current, and a spike is produced when . The membrane potential is then reset to E[L]. Refractoriness is implemented either by maintaining V at resting potential
for 5 ms (Figure 10) or by increasing the threshold by 3.6 mV (Figures 4, 6– 8), corresponding to a spike duration of 3 ms and k[a]=6 mV (see Text S1 A, effect of output spikes on threshold). We
call this model iLIF (inactivating leaky integrate-and-fire model, equations (5–8)). In Figure 10 we used and Na parameters from a recent study of the role of Na inactivation in the temporal
precision of auditory neurons [17]: ; ; ; . For Figures 4– 8, we used V[T]=−55 mV, V[i]=−63 mV (average value in the in situ dataset), . Unless otherwise specified, we chose k[a]/k[i]=1
(average in the dataset: 1.05).
In Figure 10D, the time constant of Na inactivation is voltage-dependent, as in [17]:
Fluctuating inputs
Fluctuating inputs (Figures 2C–E, 6– 10) were generated according to Ornstein-Uhlenbeck processes: where is the mean, is the standard deviation, is the autocorrelation time constant, and is a
gaussian white noise of zero mean and unitary variance. We chose in Figure 2 and in other figures.
Empirical threshold measurement
To measure spike onset in models with no explicit threshold (Figures 2, 10, 11), we used the first derivative method [62], which consists in measuring the membrane potential V when its derivative dV/
dt crosses an empirical criterion . Since the input is not controlled, it measures spike onset and is an overestimate of the spike threshold. These two quantities can be related in simple models [23]
Slope-threshold relationship
To calculate the relationship between the slope of depolarization and the threshold, we consider a linear depolarization with slope s: V(t)=st, and we calculate the intersection with the threshold
(Figure 5A), described by the adaptive threshold model. By integrating the dynamic threshold equation, we find that when (), the threshold is implicitly determined by the following equation:
For low values of s, this equation may have no solution (i.e., the neuron does not spike). Using the piecewise linear approximation of the steady-state threshold, we obtain:which simplifies to:
This is also an implicit equation for , but it can be easily (numerically) calculated with a nonlinear solver. A closed formula can be obtained in the case when :
Supporting Information
Slope-threshold relationship in the multicompartmental model of Hu et al. (2009), measured with linear regression over 5 ms (black dots), superimposed on the calculated relationship (red dashed
line), using the Na channel properties of the model (as in Platkiewicz and Brette, 2010, Fig. 8H).
(0.17 MB PDF)
Author Contributions
Conceived and designed the experiments: JP RB. Performed the experiments: JP RB. Analyzed the data: JP RB. Wrote the paper: JP RB. | {"url":"https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1001129","timestamp":"2024-11-13T15:06:05Z","content_type":"text/html","content_length":"315989","record_id":"<urn:uuid:2d90e6ad-f36e-417d-819f-8516fa4ffc15>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00535.warc.gz"} |
Exploring Optimizers in Machine Learning - Fritz ai
Exploring Optimizers in Machine Learning
A guide to the widely used optimizer functions and a breakdown of their benefits and limitations
In this post we’re going to embark on a journey to explore and dive deep into the world of optimizers for machine learning models. We will also try and understand the foundational mathematics behind
these functions and discuss their use cases, merits, and demerits.
So, what are we waiting for? Let’s get started!
What is an Optimizer?
Don’t we all love when neural networks work their magic and provide us with tremendous accuracy? Let’s get to the core of this magic trick by understanding how our networks find the most optimal
parameters for our model.
We know that loss functions are used to understand how good/bad our model performs on the data provided to it. Loss functions are essentially the summation of the difference between the predicted and
calculated values for given training samples. For training a neural network to minimize its losses so as to perform better, we need to tweak the weights and parameters associated with the model and
the loss function. This is where optimizers play a crucial role.
Optimizers associate loss function and model parameters together by updating the model, i.e. the weights and biases of each node based on the output of the loss function.
An example:
Let us imagine a person hiking down the hill with no sense of direction. He doesn’t know the right way to reach the valley, but, he can understand whether he is moving closer (going downhill) or
further away (uphill) from his destination. If he keeps taking steps in the correct direction, he will reach the valley.
This is the intuition behind optimizers — to reach a global minima with respect to loss function.
Types of Optimizers
Let’s discuss the most frequently used and appreciated optimizers in machine learning:
Gradient Descent
Well, of course we need to start off with the biggest star of our post — gradient descent. Gradient descent is an iterative optimization algorithm. It is dependent on the derivatives of the loss
function for finding minima. Running the algorithm for numerous iterations and epochs helps to reach the global minima (or closest to it).
Role of Gradient:
Gradient refers to the slope of the equation in general. Gradients are partial derivatives and can be considered as the small change reflected in the loss function with respect to the small change in
weights or parameters of the function. Now, this slight change can tell us what to do next to reduce the output of loss function — reduce this weight by 0.02 or increase this parameter by 0.005.
Learning Rate:
Learning rate is the size of the steps our algorithm takes to reach the global minima. Taking very large steps may skip the global minima, and the model will never reach the optimal value for loss
function. On the other hand, taking very small steps will take forever to converge. Thus, size of the step is also dependent on the gradient value.
In the above formula, α is the learning rate, J is the cost function, and ϴ is the parameter to be updated. As you can see, partial derivative of J with respect to ϴj gives us the gradient. Note
that, as we get closer to the global minima, the slope/gradient of the curve becomes less and less steep, which gives us a smaller value of derivative, which in turn reduces the step size
Instances of Gradient Descent
• Vanilla/Batch Gradient Descent- gradient calculation with all training samples (slow and heavy computation)
• Stochastic Gradient Descent- gradient calculation with each training sample (unstable and may shoot even after getting global minima, but is memory efficient)
• Mini Batch Gradient Descent- dividing the training data into mini batches (32 , 64… usually in the powers of 2) and gradient calculation for the same (fast and memory efficient with lesser
Advantages of Gradient Descent
It is quite fast and is easy and intuitive to understand and compute.
Disadvantages of Gradient Descent
It may get stuck at a local minima and running gradient descent on a complete dataset takes too long if the dataset is very large (always use batches).
Momentum Based Optimizers
These optimizers help models converge by focusing on movement in the correct direction (downhill) and reduces the variance in every other insignificant direction. Thus, adding another parameter ‘Y’
to our optimization algorithm to move steeply in the relevant direction helps us move faster towards convergence. The momentum parameter ‘Y’ typically has a value around 0.9. But, at the same time,
one needs to hit and try multiple values and choose manually for the most optimal results.
Nesterov Accelerated Gradient (NAG)
If the value of the momentum is too high, our model can even cross the minima and again start going up. Thus, just like a ball, if it has too much energy, it would not stop on the flat surface after
rolling down, but will continue to move up.
But, if our algorithm knows when to slow down, it will reach the minima. Thus, in NAG algorithms, the gradient for time step t also takes into account the estimated gradient for time step t+1, which
gives a better idea whether to slow down or not. It works better than the conventional momentum algorithm.
In the above equations, (ϴ -ϒvt-1) is the next approximated values of the parameters and ∇J(ϴ -ϒvt-1) is the gradient with respect to the future values.
Advantages of NAG
It has an understanding of motion or future gradients. Thus, it can decrease momentum when the gradient value is lesser or the slope is low and can increase momentum when the slope is steep.
Disadvantages of NAG
NAG is not adaptive with respect to the parameter importance. Thus, all parameters are updated in a similar manner.
AdaGrad — Adaptive Gradient Algorithm
How cool would it be if our algorithm adapts different learning rates for different parameters? This is the intuition behind AdaGrad — some weights will have different learning rates based on how
frequently or by how much their values are updated so as to perform optimally. Larger learning rate updates are made for less frequent parameters and smaller ones for the frequent parameters. The
learning rate however, always decreases.
Here, as you can see, AdaGrad uses the sum of the square of past gradients to calculate the learning rate for each parameter (∇L(ϴ) is the gradient or partial derivative of cost function with respect
to ϴ).
Advantages of AdaGrad
Works great for datasets which have missing samples or are sparse in nature.
Disadvantages of AdaGrad
The learning might be very slow, since, according to the formula above, division by bigger numbers (sum of past gradients become bigger and bigger with time) means that the learning rate is
decreasing over time — therefore the pace of learning would also decrease.
RMS-Prop — Root Mean Square Propagation
RMS-Prop is almost similar to AdaGrad, with just a minute difference: It uses exponentially decaying average instead of the sum of gradients. Thus, RMS-Prop basically combines momentum with AdaGrad.
Also, instead of using all the gradients for momentum, it takes into account only the most recent gradients for momentum calculation. This gives the model an understanding of whether it should
increase or decrease its learning rate, given the current scenario.
Advantages of RMS-Prop
AdaGrad decreases the learning rate with each time step, but RMS-Prop can adapt to an increase or a decrease in the learning rate with each epoch.
Disadvantages of RMS-Prop
The learning can be pretty slow, same reason as in AdaGrad.
Adam — Adaptive Moment Estimation
Adam optimizer also uses adaptive learning rate technique to calculate current gradient from the first and second moments of the past gradients. Adam optimizer can be viewed as a combination of
momentum and RMS-prop and is the most widely used optimizer for a wide range of problems.
Rather than influencing the learning rate with respect to first moments as in RMSProp, Adam also uses the average of the second moments of the gradients. Also, Adam calculates an exponential moving
average of the gradient and the squared gradient. Thus, Adam can be considered as a combination of AdaGrad and RMS-Prop.
In the above formula, α is the learning rate, β1(usually ~0.9) is the exponential decay rate with respect to the first moments, β2 is the exponential decay rate with respect to the second moments
(usually ~0.999). ∈ is just a small value to avoid division by zero.
Advantages of Adam
Adam optimizer is well suited for large datasets and is computationally efficient.
Disadvantages of Adam
There are few disadvantages as the Adam optimizer tends to converge faster, but other algorithms like the Stochastic gradient descent focus on the datapoints and generalize in a better manner. Thus,
the performance depends on the type of data being provided and the speed/generalization trade-off.
Race to the Global Minima
Here is an animation of how different optimizers help the model reach the global minima for a particular data space:
In this post we discussed about various optimizers like gradient descent and its variations, Nesterov accelerated gradient, AdaGrad, RMS-Prop, and Adam along with their primary use cases. These are
the most widely-used optimizer functions and are essential for developing efficient neural networks. Choosing an optimizer is influential to building efficient and fast neural networks.
I hope this article has helped you learn and understand more about these fundamental ML concepts. All feedback is welcome. Please help me improve!
Until next time!
Comments 0 Responses | {"url":"https://fritz.ai/optimizers-in-machine-learning/","timestamp":"2024-11-14T05:15:31Z","content_type":"text/html","content_length":"148824","record_id":"<urn:uuid:cc2e2f71-cd12-49c6-b27e-6368b164f236>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00466.warc.gz"} |
Legacy Portal: A Guide to the Statistics
You can evaluate the performance of assessments, questions, and distractors by using the available ExamSoft reports. You can see how your questions are performing and make improvements in your
questions and/or your curriculum.
Psychometrics are the mathematical statistics that can be interpreted very differently based on the purpose of the question. No single statistic can give you the entire picture of how an item should
be interpreted. Likewise, there are no ideal psychometrics that are accurate for every item. The best practice is always to evaluate all pieces of information available about an item while accounting
for the intention of the question.
A question that was created by the faculty member as an “easy” question may serve that specific purpose, but it has very different psychometric results than that of a question that was created to be
discriminatory. Additionally, take into account any outside factors that could be influencing the statistics (including content delivery method, conflicting information given to the exam-takers,
testing environment, etc.)
Lastly, always keep in mind how many exam-takers took the exam. If the number is very low, then the statistics are significantly less reliable than if it is a large group of exam-takers.
Exam Statistics
Exam Statistics are statistics that are determined based on the performance of all exam-takers on all questions of the exam. This data can be found on a number of reports (including the Item Analysis
and Summary Reports).
• Mean: The mean is the average score of all exam-takers who took the exam. It is found by dividing the sum of the scores by the total number of exam-takers who took the exam.
• Median: The median is the score that marks the midpoint of all exam-takers’ scores. It Is the score that is halfway between the highest and lowest scores.
• Standard Deviation: The standard deviation indicates the variation of exam scores. A low standard deviation indicates that exam-taker’s score were all close to the average, while a high standard
deviation indicates that there was a large variation in scores.
• Reliability KR-20 (Kuder-Richardson Formula) (0.00-1.00): The KR-20 measures internal consistency reliability. It takes into account all dichotomous questions and how many exam takers answered
each question correctly. A high KR-20 indicates that if the same exam-takers took the same assessment there is a higher chance that the results would be the same. A low KR-20 means that the
results would be more likely to be different.
Question Statistics
Question Statistics are statistics that assess a single question. These can be found on the item analysis as well as in the question history for each question. These can be calculated based on
question performance from a single assessment or across the life of the question.
• Difficulty Index (p) (0.00-1.00): The difficulty index measures the proportion of exam-takers who answered an item correctly. A higher value indicates a greater proportion of exam-takers
responded to an item correctly. A lower value indicates that fewer exam-takers got the question correct.
In addition to being provided the overall difficulty index, there is an Upper Difficulty Index and Lower Difficulty Index. These follow the same format as above but only take into account the top 27%
of the class and the lower 27% of the class respectively. Thus the Upper Difficulty Index/Lower Difficulty index reflects what percentage of the top 27%/lower 27% of scorers on an exam answered the
question correctly. The Upper and Lower Groups of exam-takers are based on the top 27% and bottom 27% of performers respectively. 27% is an industry standard in item analyses.
• Discrimination Index (-1.00-1.00): The discrimination index of a question shows the difference in performance between the upper 27% and the lower 27%. It is determined by subtracting the
difficulty index of the lower 27% from the difficulty index of the upper 27%. A score close to 0 indicates that the upper exam-takers and the lower exam-takers performed similarly on this
question. As a discrimination index becomes negative, this indicates that more of the lower performers got this question correct than the upper performers. As it becomes more positive, more of
the upper performers got this question correct.
Determining an acceptable item discrimination score depends on the intention of the item. For example, if it is intended to be a mastery-level item, then a score as low as 0 to .2 is acceptable. If
it is intended to be a highly discriminating item, target a score of .25 to .5.
• Point Bi-Serial (-1.00-1.00): The point bi-serial measures the correlation between an exam-taker's response on a given item and how the exam-taker performed on the overall exam.
• Greater than 0: Indicates a positive correlation between the performance on the question and the performance on the exam. Exam takers who succeeded on the exam also succeeded on this question,
while exam-takers who had trouble on the exam also had trouble on this question. A point bi-serial closer to 1 indicates a very strong correlation; success/failure on this question is a strong
predictor of success/failure on the exam as a whole.
• Near 0: There was little correlation between the performance of this item and performance on the test as a whole. Possibly this question covered on material outside of the other learning
outcomes, so that all or most exam-takers struggled with this question. Possibly this question was a review item, so that all or most exam-takers were able to answer it correctly.
• Less than 0: Indicates a negative correlation between the performance on the question and the performance on the exam. Exam takers who succeeded on this question had trouble with the exam as a
whole, while exam-takers who had trouble with this question did well on the exam as a whole.
• Response Frequencies: This details the percentage of exam-takers who selected each answer choice. If there is an incorrect distractor that is receiving a very large portion of the answers, you
may need to assess if that was the intention for this question or if something in that answer choice is causing confusion. Additionally, an answer choice with very low proportions of responses
may need to be reviewed as well.
When reviewing response frequencies, you may wish to also examine the distribution of responses from your top 27% and lower 27%. If a large portion of your top 27% picked the same incorrect
answer choice, it could indicate the need for further review. | {"url":"https://support.examsoft.com/hc/en-us/articles/11166781396237-Legacy-Portal-A-Guide-to-the-Statistics","timestamp":"2024-11-06T05:07:58Z","content_type":"text/html","content_length":"42660","record_id":"<urn:uuid:b0bff9e2-0ba4-4c3d-9630-3ff3c4b259f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00712.warc.gz"} |
TARDIS Models
Had a bit of free time (what with the whole lockdown) so I thought I would have a play with my TARDISes and see if I could get some decent foreground model photo's.
Granted, most of them look the same; however there's only so many ways you can photograph something that looks more or less the same from each side | {"url":"https://tardisbuilders.com/index.php?topic=10416.0","timestamp":"2024-11-06T07:23:15Z","content_type":"text/html","content_length":"68758","record_id":"<urn:uuid:0d7894ad-9a8d-4fed-84fc-1dfb2b9ecbeb>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00172.warc.gz"} |
JDK 17 java.base.jmod - Base Module
⏎ java/lang/StrictMath.java
* Copyright (c) 1999, 2021, Oracle and/or its affiliates. All rights reserved.
* ORACLE PROPRIETARY/CONFIDENTIAL. Use is subject to license terms.
package java.lang;
import java.util.Random;
import jdk.internal.math.DoubleConsts;
import jdk.internal.vm.annotation.IntrinsicCandidate;
* The class {@code StrictMath} contains methods for performing basic
* numeric operations such as the elementary exponential, logarithm,
* square root, and trigonometric functions.
* <p>To help ensure portability of Java programs, the definitions of
* some of the numeric functions in this package require that they
* produce the same results as certain published algorithms. These
* algorithms are available from the well-known network library
* {@code netlib} as the package "Freely Distributable Math
* Library," <a
* href="https://www.netlib.org/fdlibm/">{@code fdlibm}</a>. These
* algorithms, which are written in the C programming language, are
* then to be understood as executed with all floating-point
* operations following the rules of Java floating-point arithmetic.
* <p>The Java math library is defined with respect to
* {@code fdlibm} version 5.3. Where {@code fdlibm} provides
* more than one definition for a function (such as
* {@code acos}), use the "IEEE 754 core function" version
* (residing in a file whose name begins with the letter
* {@code e}). The methods which require {@code fdlibm}
* semantics are {@code sin}, {@code cos}, {@code tan},
* {@code asin}, {@code acos}, {@code atan},
* {@code exp}, {@code log}, {@code log10},
* {@code cbrt}, {@code atan2}, {@code pow},
* {@code sinh}, {@code cosh}, {@code tanh},
* {@code hypot}, {@code expm1}, and {@code log1p}.
* <p>
* The platform uses signed two's complement integer arithmetic with
* int and long primitive types. The developer should choose
* the primitive type to ensure that arithmetic operations consistently
* produce correct results, which in some cases means the operations
* will not overflow the range of values of the computation.
* The best practice is to choose the primitive type and algorithm to avoid
* overflow. In cases where the size is {@code int} or {@code long} and
* overflow errors need to be detected, the methods {@code addExact},
* {@code subtractExact}, {@code multiplyExact}, {@code toIntExact},
* {@code incrementExact}, {@code decrementExact} and {@code negateExact}
* throw an {@code ArithmeticException} when the results overflow.
* For the arithmetic operations divide and absolute value, overflow
* occurs only with a specific minimum or maximum value and
* should be checked against the minimum or maximum as appropriate.
* <h2><a id=Ieee754RecommendedOps>IEEE 754 Recommended
* Operations</a></h2>
* The {@link java.lang.Math Math} class discusses how the shared
* quality of implementation criteria for selected {@code Math} and
* {@code StrictMath} methods <a
* href="Math.html#Ieee754RecommendedOps">relate to the IEEE 754
* recommended operations</a>.
* @author Joseph D. Darcy
* @since 1.3
public final class StrictMath {
* Don't let anyone instantiate this class.
private StrictMath() {}
* The {@code double} value that is closer than any other to
* <i>e</i>, the base of the natural logarithms.
public static final double E = 2.7182818284590452354;
* The {@code double} value that is closer than any other to
* <i>pi</i>, the ratio of the circumference of a circle to its
* diameter.
public static final double PI = 3.14159265358979323846;
* Constant by which to multiply an angular value in degrees to obtain an
* angular value in radians.
private static final double DEGREES_TO_RADIANS = 0.017453292519943295;
* Constant by which to multiply an angular value in radians to obtain an
* angular value in degrees.
private static final double RADIANS_TO_DEGREES = 57.29577951308232;
* Returns the trigonometric sine of an angle. Special cases:
* <ul><li>If the argument is NaN or an infinity, then the
* result is NaN.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.</ul>
* @param a an angle, in radians.
* @return the sine of the argument.
public static native double sin(double a);
* Returns the trigonometric cosine of an angle. Special cases:
* <ul><li>If the argument is NaN or an infinity, then the
* result is NaN.
* <li>If the argument is zero, then the result is {@code 1.0}.
* </ul>
* @param a an angle, in radians.
* @return the cosine of the argument.
public static native double cos(double a);
* Returns the trigonometric tangent of an angle. Special cases:
* <ul><li>If the argument is NaN or an infinity, then the result
* is NaN.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.</ul>
* @param a an angle, in radians.
* @return the tangent of the argument.
public static native double tan(double a);
* Returns the arc sine of a value; the returned angle is in the
* range -<i>pi</i>/2 through <i>pi</i>/2. Special cases:
* <ul><li>If the argument is NaN or its absolute value is greater
* than 1, then the result is NaN.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.</ul>
* @param a the value whose arc sine is to be returned.
* @return the arc sine of the argument.
public static native double asin(double a);
* Returns the arc cosine of a value; the returned angle is in the
* range 0.0 through <i>pi</i>. Special case:
* <ul><li>If the argument is NaN or its absolute value is greater
* than 1, then the result is NaN.
* <li>If the argument is {@code 1.0}, the result is positive zero.
* </ul>
* @param a the value whose arc cosine is to be returned.
* @return the arc cosine of the argument.
public static native double acos(double a);
* Returns the arc tangent of a value; the returned angle is in the
* range -<i>pi</i>/2 through <i>pi</i>/2. Special cases:
* <ul><li>If the argument is NaN, then the result is NaN.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
* <li>If the argument is {@linkplain Double#isInfinite infinite},
* then the result is the closest value to <i>pi</i>/2 with the
* same sign as the input.
* </ul>
* @param a the value whose arc tangent is to be returned.
* @return the arc tangent of the argument.
public static native double atan(double a);
* Converts an angle measured in degrees to an approximately
* equivalent angle measured in radians. The conversion from
* degrees to radians is generally inexact.
* @param angdeg an angle, in degrees
* @return the measurement of the angle {@code angdeg}
* in radians.
public static double toRadians(double angdeg) {
return Math.toRadians(angdeg);
* Converts an angle measured in radians to an approximately
* equivalent angle measured in degrees. The conversion from
* radians to degrees is generally inexact; users should
* <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
* equal {@code 0.0}.
* @param angrad an angle, in radians
* @return the measurement of the angle {@code angrad}
* in degrees.
public static double toDegrees(double angrad) {
return Math.toDegrees(angrad);
* Returns Euler's number <i>e</i> raised to the power of a
* {@code double} value. Special cases:
* <ul><li>If the argument is NaN, the result is NaN.
* <li>If the argument is positive infinity, then the result is
* positive infinity.
* <li>If the argument is negative infinity, then the result is
* positive zero.
* <li>If the argument is zero, then the result is {@code 1.0}.
* </ul>
* @param a the exponent to raise <i>e</i> to.
* @return the value <i>e</i><sup>{@code a}</sup>,
* where <i>e</i> is the base of the natural logarithms.
public static double exp(double a) {
return FdLibm.Exp.compute(a);
* Returns the natural logarithm (base <i>e</i>) of a {@code double}
* value. Special cases:
* <ul><li>If the argument is NaN or less than zero, then the result
* is NaN.
* <li>If the argument is positive infinity, then the result is
* positive infinity.
* <li>If the argument is positive zero or negative zero, then the
* result is negative infinity.
* <li>If the argument is {@code 1.0}, then the result is positive
* zero.
* </ul>
* @param a a value
* @return the value ln {@code a}, the natural logarithm of
* {@code a}.
public static native double log(double a);
* Returns the base 10 logarithm of a {@code double} value.
* Special cases:
* <ul><li>If the argument is NaN or less than zero, then the result
* is NaN.
* <li>If the argument is positive infinity, then the result is
* positive infinity.
* <li>If the argument is positive zero or negative zero, then the
* result is negative infinity.
* <li>If the argument is equal to 10<sup><i>n</i></sup> for
* integer <i>n</i>, then the result is <i>n</i>. In particular,
* if the argument is {@code 1.0} (10<sup>0</sup>), then the
* result is positive zero.
* </ul>
* @param a a value
* @return the base 10 logarithm of {@code a}.
* @since 1.5
public static native double log10(double a);
* Returns the correctly rounded positive square root of a
* {@code double} value.
* Special cases:
* <ul><li>If the argument is NaN or less than zero, then the result
* is NaN.
* <li>If the argument is positive infinity, then the result is positive
* infinity.
* <li>If the argument is positive zero or negative zero, then the
* result is the same as the argument.</ul>
* Otherwise, the result is the {@code double} value closest to
* the true mathematical square root of the argument value.
* @param a a value.
* @return the positive square root of {@code a}.
public static native double sqrt(double a);
* Returns the cube root of a {@code double} value. For
* positive finite {@code x}, {@code cbrt(-x) ==
* -cbrt(x)}; that is, the cube root of a negative value is
* the negative of the cube root of that value's magnitude.
* Special cases:
* <ul>
* <li>If the argument is NaN, then the result is NaN.
* <li>If the argument is infinite, then the result is an infinity
* with the same sign as the argument.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
* </ul>
* @param a a value.
* @return the cube root of {@code a}.
* @since 1.5
public static double cbrt(double a) {
return FdLibm.Cbrt.compute(a);
* Computes the remainder operation on two arguments as prescribed
* by the IEEE 754 standard.
* The remainder value is mathematically equal to
* <code>f1 - f2</code> × <i>n</i>,
* where <i>n</i> is the mathematical integer closest to the exact
* mathematical value of the quotient {@code f1/f2}, and if two
* mathematical integers are equally close to {@code f1/f2},
* then <i>n</i> is the integer that is even. If the remainder is
* zero, its sign is the same as the sign of the first argument.
* Special cases:
* <ul><li>If either argument is NaN, or the first argument is infinite,
* or the second argument is positive zero or negative zero, then the
* result is NaN.
* <li>If the first argument is finite and the second argument is
* infinite, then the result is the same as the first argument.</ul>
* @param f1 the dividend.
* @param f2 the divisor.
* @return the remainder when {@code f1} is divided by
* {@code f2}.
public static native double IEEEremainder(double f1, double f2);
* Returns the smallest (closest to negative infinity)
* {@code double} value that is greater than or equal to the
* argument and is equal to a mathematical integer. Special cases:
* <ul><li>If the argument value is already equal to a
* mathematical integer, then the result is the same as the
* argument. <li>If the argument is NaN or an infinity or
* positive zero or negative zero, then the result is the same as
* the argument. <li>If the argument value is less than zero but
* greater than -1.0, then the result is negative zero.</ul> Note
* that the value of {@code StrictMath.ceil(x)} is exactly the
* value of {@code -StrictMath.floor(-x)}.
* @param a a value.
* @return the smallest (closest to negative infinity)
* floating-point value that is greater than or equal to
* the argument and is equal to a mathematical integer.
public static double ceil(double a) {
return floorOrCeil(a, -0.0, 1.0, 1.0);
* Returns the largest (closest to positive infinity)
* {@code double} value that is less than or equal to the
* argument and is equal to a mathematical integer. Special cases:
* <ul><li>If the argument value is already equal to a
* mathematical integer, then the result is the same as the
* argument. <li>If the argument is NaN or an infinity or
* positive zero or negative zero, then the result is the same as
* the argument.</ul>
* @param a a value.
* @return the largest (closest to positive infinity)
* floating-point value that less than or equal to the argument
* and is equal to a mathematical integer.
public static double floor(double a) {
return floorOrCeil(a, -1.0, 0.0, -1.0);
* Internal method to share logic between floor and ceil.
* @param a the value to be floored or ceiled
* @param negativeBoundary result for values in (-1, 0)
* @param positiveBoundary result for values in (0, 1)
* @param increment value to add when the argument is non-integral
private static double floorOrCeil(double a,
double negativeBoundary,
double positiveBoundary,
double sign) {
int exponent = Math.getExponent(a);
if (exponent < 0) {
* Absolute value of argument is less than 1.
* floorOrceil(-0.0) => -0.0
* floorOrceil(+0.0) => +0.0
return ((a == 0.0) ? a :
( (a < 0.0) ? negativeBoundary : positiveBoundary) );
} else if (exponent >= 52) {
* Infinity, NaN, or a value so large it must be integral.
return a;
// Else the argument is either an integral value already XOR it
// has to be rounded to one.
assert exponent >= 0 && exponent <= 51;
long doppel = Double.doubleToRawLongBits(a);
long mask = DoubleConsts.SIGNIF_BIT_MASK >> exponent;
if ( (mask & doppel) == 0L )
return a; // integral value
else {
double result = Double.longBitsToDouble(doppel & (~mask));
if (sign*a > 0.0)
result = result + sign;
return result;
* Returns the {@code double} value that is closest in value
* to the argument and is equal to a mathematical integer. If two
* {@code double} values that are mathematical integers are
* equally close to the value of the argument, the result is the
* integer value that is even. Special cases:
* <ul><li>If the argument value is already equal to a mathematical
* integer, then the result is the same as the argument.
* <li>If the argument is NaN or an infinity or positive zero or negative
* zero, then the result is the same as the argument.</ul>
* @param a a value.
* @return the closest floating-point value to {@code a} that is
* equal to a mathematical integer.
* @author Joseph D. Darcy
public static double rint(double a) {
* If the absolute value of a is not less than 2^52, it
* is either a finite integer (the double format does not have
* enough significand bits for a number that large to have any
* fractional portion), an infinity, or a NaN. In any of
* these cases, rint of the argument is the argument.
* Otherwise, the sum (twoToThe52 + a ) will properly round
* away any fractional portion of a since ulp(twoToThe52) ==
* 1.0; subtracting out twoToThe52 from this sum will then be
* exact and leave the rounded integer portion of a.
double twoToThe52 = (double)(1L << 52); // 2^52
double sign = Math.copySign(1.0, a); // preserve sign info
a = Math.abs(a);
if (a < twoToThe52) { // E_min <= ilogb(a) <= 51
a = ((twoToThe52 + a ) - twoToThe52);
return sign * a; // restore original sign
* Returns the angle <i>theta</i> from the conversion of rectangular
* coordinates ({@code x}, {@code y}) to polar
* coordinates (r, <i>theta</i>).
* This method computes the phase <i>theta</i> by computing an arc tangent
* of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
* cases:
* <ul><li>If either argument is NaN, then the result is NaN.
* <li>If the first argument is positive zero and the second argument
* is positive, or the first argument is positive and finite and the
* second argument is positive infinity, then the result is positive
* zero.
* <li>If the first argument is negative zero and the second argument
* is positive, or the first argument is negative and finite and the
* second argument is positive infinity, then the result is negative zero.
* <li>If the first argument is positive zero and the second argument
* is negative, or the first argument is positive and finite and the
* second argument is negative infinity, then the result is the
* {@code double} value closest to <i>pi</i>.
* <li>If the first argument is negative zero and the second argument
* is negative, or the first argument is negative and finite and the
* second argument is negative infinity, then the result is the
* {@code double} value closest to -<i>pi</i>.
* <li>If the first argument is positive and the second argument is
* positive zero or negative zero, or the first argument is positive
* infinity and the second argument is finite, then the result is the
* {@code double} value closest to <i>pi</i>/2.
* <li>If the first argument is negative and the second argument is
* positive zero or negative zero, or the first argument is negative
* infinity and the second argument is finite, then the result is the
* {@code double} value closest to -<i>pi</i>/2.
* <li>If both arguments are positive infinity, then the result is the
* {@code double} value closest to <i>pi</i>/4.
* <li>If the first argument is positive infinity and the second argument
* is negative infinity, then the result is the {@code double}
* value closest to 3*<i>pi</i>/4.
* <li>If the first argument is negative infinity and the second argument
* is positive infinity, then the result is the {@code double} value
* closest to -<i>pi</i>/4.
* <li>If both arguments are negative infinity, then the result is the
* {@code double} value closest to -3*<i>pi</i>/4.</ul>
* @apiNote
* For <i>y</i> with a positive sign and finite nonzero
* <i>x</i>, the exact mathematical value of {@code atan2} is
* equal to:
* <ul>
* <li>If <i>x</i> {@literal >} 0, atan(abs(<i>y</i>/<i>x</i>))
* <li>If <i>x</i> {@literal <} 0, π - atan(abs(<i>y</i>/<i>x</i>))
* </ul>
* @param y the ordinate coordinate
* @param x the abscissa coordinate
* @return the <i>theta</i> component of the point
* (<i>r</i>, <i>theta</i>)
* in polar coordinates that corresponds to the point
* (<i>x</i>, <i>y</i>) in Cartesian coordinates.
public static native double atan2(double y, double x);
* Returns the value of the first argument raised to the power of the
* second argument. Special cases:
* <ul><li>If the second argument is positive or negative zero, then the
* result is 1.0.
* <li>If the second argument is 1.0, then the result is the same as the
* first argument.
* <li>If the second argument is NaN, then the result is NaN.
* <li>If the first argument is NaN and the second argument is nonzero,
* then the result is NaN.
* <li>If
* <ul>
* <li>the absolute value of the first argument is greater than 1
* and the second argument is positive infinity, or
* <li>the absolute value of the first argument is less than 1 and
* the second argument is negative infinity,
* </ul>
* then the result is positive infinity.
* <li>If
* <ul>
* <li>the absolute value of the first argument is greater than 1 and
* the second argument is negative infinity, or
* <li>the absolute value of the
* first argument is less than 1 and the second argument is positive
* infinity,
* </ul>
* then the result is positive zero.
* <li>If the absolute value of the first argument equals 1 and the
* second argument is infinite, then the result is NaN.
* <li>If
* <ul>
* <li>the first argument is positive zero and the second argument
* is greater than zero, or
* <li>the first argument is positive infinity and the second
* argument is less than zero,
* </ul>
* then the result is positive zero.
* <li>If
* <ul>
* <li>the first argument is positive zero and the second argument
* is less than zero, or
* <li>the first argument is positive infinity and the second
* argument is greater than zero,
* </ul>
* then the result is positive infinity.
* <li>If
* <ul>
* <li>the first argument is negative zero and the second argument
* is greater than zero but not a finite odd integer, or
* <li>the first argument is negative infinity and the second
* argument is less than zero but not a finite odd integer,
* </ul>
* then the result is positive zero.
* <li>If
* <ul>
* <li>the first argument is negative zero and the second argument
* is a positive finite odd integer, or
* <li>the first argument is negative infinity and the second
* argument is a negative finite odd integer,
* </ul>
* then the result is negative zero.
* <li>If
* <ul>
* <li>the first argument is negative zero and the second argument
* is less than zero but not a finite odd integer, or
* <li>the first argument is negative infinity and the second
* argument is greater than zero but not a finite odd integer,
* </ul>
* then the result is positive infinity.
* <li>If
* <ul>
* <li>the first argument is negative zero and the second argument
* is a negative finite odd integer, or
* <li>the first argument is negative infinity and the second
* argument is a positive finite odd integer,
* </ul>
* then the result is negative infinity.
* <li>If the first argument is finite and less than zero
* <ul>
* <li> if the second argument is a finite even integer, the
* result is equal to the result of raising the absolute value of
* the first argument to the power of the second argument
* <li>if the second argument is a finite odd integer, the result
* is equal to the negative of the result of raising the absolute
* value of the first argument to the power of the second
* argument
* <li>if the second argument is finite and not an integer, then
* the result is NaN.
* </ul>
* <li>If both arguments are integers, then the result is exactly equal
* to the mathematical result of raising the first argument to the power
* of the second argument if that result can in fact be represented
* exactly as a {@code double} value.</ul>
* <p>(In the foregoing descriptions, a floating-point value is
* considered to be an integer if and only if it is finite and a
* fixed point of the method {@link #ceil ceil} or,
* equivalently, a fixed point of the method {@link #floor
* floor}. A value is a fixed point of a one-argument
* method if and only if the result of applying the method to the
* value is equal to the value.)
* @apiNote
* The special cases definitions of this method differ from the
* special case definitions of the IEEE 754 recommended {@code
* pow} operation for ±{@code 1.0} raised to an infinite
* power. This method treats such cases as indeterminate and
* specifies a NaN is returned. The IEEE 754 specification treats
* the infinite power as a large integer (large-magnitude
* floating-point numbers are numerically integers, specifically
* even integers) and therefore specifies {@code 1.0} be returned.
* @param a base.
* @param b the exponent.
* @return the value {@code a}<sup>{@code b}</sup>.
public static double pow(double a, double b) {
return FdLibm.Pow.compute(a, b);
* Returns the closest {@code int} to the argument, with ties
* rounding to positive infinity.
* <p>Special cases:
* <ul><li>If the argument is NaN, the result is 0.
* <li>If the argument is negative infinity or any value less than or
* equal to the value of {@code Integer.MIN_VALUE}, the result is
* equal to the value of {@code Integer.MIN_VALUE}.
* <li>If the argument is positive infinity or any value greater than or
* equal to the value of {@code Integer.MAX_VALUE}, the result is
* equal to the value of {@code Integer.MAX_VALUE}.</ul>
* @param a a floating-point value to be rounded to an integer.
* @return the value of the argument rounded to the nearest
* {@code int} value.
* @see java.lang.Integer#MAX_VALUE
* @see java.lang.Integer#MIN_VALUE
public static int round(float a) {
return Math.round(a);
* Returns the closest {@code long} to the argument, with ties
* rounding to positive infinity.
* <p>Special cases:
* <ul><li>If the argument is NaN, the result is 0.
* <li>If the argument is negative infinity or any value less than or
* equal to the value of {@code Long.MIN_VALUE}, the result is
* equal to the value of {@code Long.MIN_VALUE}.
* <li>If the argument is positive infinity or any value greater than or
* equal to the value of {@code Long.MAX_VALUE}, the result is
* equal to the value of {@code Long.MAX_VALUE}.</ul>
* @param a a floating-point value to be rounded to a
* {@code long}.
* @return the value of the argument rounded to the nearest
* {@code long} value.
* @see java.lang.Long#MAX_VALUE
* @see java.lang.Long#MIN_VALUE
public static long round(double a) {
return Math.round(a);
private static final class RandomNumberGeneratorHolder {
static final Random randomNumberGenerator = new Random();
* Returns a {@code double} value with a positive sign, greater
* than or equal to {@code 0.0} and less than {@code 1.0}.
* Returned values are chosen pseudorandomly with (approximately)
* uniform distribution from that range.
* <p>When this method is first called, it creates a single new
* pseudorandom-number generator, exactly as if by the expression
* <blockquote>{@code new java.util.Random()}</blockquote>
* This new pseudorandom-number generator is used thereafter for
* all calls to this method and is used nowhere else.
* <p>This method is properly synchronized to allow correct use by
* more than one thread. However, if many threads need to generate
* pseudorandom numbers at a great rate, it may reduce contention
* for each thread to have its own pseudorandom-number generator.
* @return a pseudorandom {@code double} greater than or equal
* to {@code 0.0} and less than {@code 1.0}.
* @see Random#nextDouble()
public static double random() {
return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
* Returns the sum of its arguments,
* throwing an exception if the result overflows an {@code int}.
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows an int
* @see Math#addExact(int,int)
* @since 1.8
public static int addExact(int x, int y) {
return Math.addExact(x, y);
* Returns the sum of its arguments,
* throwing an exception if the result overflows a {@code long}.
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows a long
* @see Math#addExact(long,long)
* @since 1.8
public static long addExact(long x, long y) {
return Math.addExact(x, y);
* Returns the difference of the arguments,
* throwing an exception if the result overflows an {@code int}.
* @param x the first value
* @param y the second value to subtract from the first
* @return the result
* @throws ArithmeticException if the result overflows an int
* @see Math#subtractExact(int,int)
* @since 1.8
public static int subtractExact(int x, int y) {
return Math.subtractExact(x, y);
* Returns the difference of the arguments,
* throwing an exception if the result overflows a {@code long}.
* @param x the first value
* @param y the second value to subtract from the first
* @return the result
* @throws ArithmeticException if the result overflows a long
* @see Math#subtractExact(long,long)
* @since 1.8
public static long subtractExact(long x, long y) {
return Math.subtractExact(x, y);
* Returns the product of the arguments,
* throwing an exception if the result overflows an {@code int}.
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows an int
* @see Math#multiplyExact(int,int)
* @since 1.8
public static int multiplyExact(int x, int y) {
return Math.multiplyExact(x, y);
* Returns the product of the arguments, throwing an exception if the result
* overflows a {@code long}.
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows a long
* @see Math#multiplyExact(long,int)
* @since 9
public static long multiplyExact(long x, int y) {
return Math.multiplyExact(x, y);
* Returns the product of the arguments,
* throwing an exception if the result overflows a {@code long}.
* @param x the first value
* @param y the second value
* @return the result
* @throws ArithmeticException if the result overflows a long
* @see Math#multiplyExact(long,long)
* @since 1.8
public static long multiplyExact(long x, long y) {
return Math.multiplyExact(x, y);
* Returns the argument incremented by one,
* throwing an exception if the result overflows an {@code int}.
* The overflow only occurs for {@linkplain Integer#MAX_VALUE the maximum value}.
* @param a the value to increment
* @return the result
* @throws ArithmeticException if the result overflows an int
* @see Math#incrementExact(int)
* @since 14
public static int incrementExact(int a) {
return Math.incrementExact(a);
* Returns the argument incremented by one,
* throwing an exception if the result overflows a {@code long}.
* The overflow only occurs for {@linkplain Long#MAX_VALUE the maximum value}.
* @param a the value to increment
* @return the result
* @throws ArithmeticException if the result overflows a long
* @see Math#incrementExact(long)
* @since 14
public static long incrementExact(long a) {
return Math.incrementExact(a);
* Returns the argument decremented by one,
* throwing an exception if the result overflows an {@code int}.
* The overflow only occurs for {@linkplain Integer#MIN_VALUE the minimum value}.
* @param a the value to decrement
* @return the result
* @throws ArithmeticException if the result overflows an int
* @see Math#decrementExact(int)
* @since 14
public static int decrementExact(int a) {
return Math.decrementExact(a);
* Returns the argument decremented by one,
* throwing an exception if the result overflows a {@code long}.
* The overflow only occurs for {@linkplain Long#MIN_VALUE the minimum value}.
* @param a the value to decrement
* @return the result
* @throws ArithmeticException if the result overflows a long
* @see Math#decrementExact(long)
* @since 14
public static long decrementExact(long a) {
return Math.decrementExact(a);
* Returns the negation of the argument,
* throwing an exception if the result overflows an {@code int}.
* The overflow only occurs for {@linkplain Integer#MIN_VALUE the minimum value}.
* @param a the value to negate
* @return the result
* @throws ArithmeticException if the result overflows an int
* @see Math#negateExact(int)
* @since 14
public static int negateExact(int a) {
return Math.negateExact(a);
* Returns the negation of the argument,
* throwing an exception if the result overflows a {@code long}.
* The overflow only occurs for {@linkplain Long#MIN_VALUE the minimum value}.
* @param a the value to negate
* @return the result
* @throws ArithmeticException if the result overflows a long
* @see Math#negateExact(long)
* @since 14
public static long negateExact(long a) {
return Math.negateExact(a);
* Returns the value of the {@code long} argument, throwing an exception
* if the value overflows an {@code int}.
* @param value the long value
* @return the argument as an int
* @throws ArithmeticException if the {@code argument} overflows an int
* @see Math#toIntExact(long)
* @since 1.8
public static int toIntExact(long value) {
return Math.toIntExact(value);
* Returns the exact mathematical product of the arguments.
* @param x the first value
* @param y the second value
* @return the result
* @see Math#multiplyFull(int,int)
* @since 9
public static long multiplyFull(int x, int y) {
return Math.multiplyFull(x, y);
* Returns as a {@code long} the most significant 64 bits of the 128-bit
* product of two 64-bit factors.
* @param x the first value
* @param y the second value
* @return the result
* @see Math#multiplyHigh(long,long)
* @since 9
public static long multiplyHigh(long x, long y) {
return Math.multiplyHigh(x, y);
* Returns the largest (closest to positive infinity)
* {@code int} value that is less than or equal to the algebraic quotient.
* There is one special case, if the dividend is the
* {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
* then integer overflow occurs and
* the result is equal to the {@code Integer.MIN_VALUE}.
* <p>
* See {@link Math#floorDiv(int, int) Math.floorDiv} for examples and
* a comparison to the integer division {@code /} operator.
* @param x the dividend
* @param y the divisor
* @return the largest (closest to positive infinity)
* {@code int} value that is less than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero
* @see Math#floorDiv(int, int)
* @see Math#floor(double)
* @since 1.8
public static int floorDiv(int x, int y) {
return Math.floorDiv(x, y);
* Returns the largest (closest to positive infinity)
* {@code long} value that is less than or equal to the algebraic quotient.
* There is one special case, if the dividend is the
* {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
* then integer overflow occurs and
* the result is equal to {@code Long.MIN_VALUE}.
* <p>
* See {@link Math#floorDiv(int, int) Math.floorDiv} for examples and
* a comparison to the integer division {@code /} operator.
* @param x the dividend
* @param y the divisor
* @return the largest (closest to positive infinity)
* {@code int} value that is less than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero
* @see Math#floorDiv(long, int)
* @see Math#floor(double)
* @since 9
public static long floorDiv(long x, int y) {
return Math.floorDiv(x, y);
* Returns the largest (closest to positive infinity)
* {@code long} value that is less than or equal to the algebraic quotient.
* There is one special case, if the dividend is the
* {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
* then integer overflow occurs and
* the result is equal to the {@code Long.MIN_VALUE}.
* <p>
* See {@link Math#floorDiv(int, int) Math.floorDiv} for examples and
* a comparison to the integer division {@code /} operator.
* @param x the dividend
* @param y the divisor
* @return the largest (closest to positive infinity)
* {@code long} value that is less than or equal to the algebraic quotient.
* @throws ArithmeticException if the divisor {@code y} is zero
* @see Math#floorDiv(long, long)
* @see Math#floor(double)
* @since 1.8
public static long floorDiv(long x, long y) {
return Math.floorDiv(x, y);
* Returns the floor modulus of the {@code int} arguments.
* <p>
* The floor modulus is {@code x - (floorDiv(x, y) * y)},
* has the same sign as the divisor {@code y}, and
* is in the range of {@code -abs(y) < r < +abs(y)}.
* <p>
* The relationship between {@code floorDiv} and {@code floorMod} is such that:
* <ul>
* <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
* </ul>
* <p>
* See {@link Math#floorMod(int, int) Math.floorMod} for examples and
* a comparison to the {@code %} operator.
* @param x the dividend
* @param y the divisor
* @return the floor modulus {@code x - (floorDiv(x, y) * y)}
* @throws ArithmeticException if the divisor {@code y} is zero
* @see Math#floorMod(int, int)
* @see StrictMath#floorDiv(int, int)
* @since 1.8
public static int floorMod(int x, int y) {
return Math.floorMod(x , y);
* Returns the floor modulus of the {@code long} and {@code int} arguments.
* <p>
* The floor modulus is {@code x - (floorDiv(x, y) * y)},
* has the same sign as the divisor {@code y}, and
* is in the range of {@code -abs(y) < r < +abs(y)}.
* <p>
* The relationship between {@code floorDiv} and {@code floorMod} is such that:
* <ul>
* <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
* </ul>
* <p>
* See {@link Math#floorMod(int, int) Math.floorMod} for examples and
* a comparison to the {@code %} operator.
* @param x the dividend
* @param y the divisor
* @return the floor modulus {@code x - (floorDiv(x, y) * y)}
* @throws ArithmeticException if the divisor {@code y} is zero
* @see Math#floorMod(long, int)
* @see StrictMath#floorDiv(long, int)
* @since 9
public static int floorMod(long x, int y) {
return Math.floorMod(x , y);
* Returns the floor modulus of the {@code long} arguments.
* <p>
* The floor modulus is {@code x - (floorDiv(x, y) * y)},
* has the same sign as the divisor {@code y}, and
* is in the range of {@code -abs(y) < r < +abs(y)}.
* <p>
* The relationship between {@code floorDiv} and {@code floorMod} is such that:
* <ul>
* <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
* </ul>
* <p>
* See {@link Math#floorMod(int, int) Math.floorMod} for examples and
* a comparison to the {@code %} operator.
* @param x the dividend
* @param y the divisor
* @return the floor modulus {@code x - (floorDiv(x, y) * y)}
* @throws ArithmeticException if the divisor {@code y} is zero
* @see Math#floorMod(long, long)
* @see StrictMath#floorDiv(long, long)
* @since 1.8
public static long floorMod(long x, long y) {
return Math.floorMod(x, y);
* Returns the absolute value of an {@code int} value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
* <p>Note that if the argument is equal to the value of {@link
* Integer#MIN_VALUE}, the most negative representable {@code int}
* value, the result is that same value, which is negative. In
* contrast, the {@link StrictMath#absExact(int)} method throws an
* {@code ArithmeticException} for this value.
* @param a the argument whose absolute value is to be determined.
* @return the absolute value of the argument.
* @see Math#absExact(int)
public static int abs(int a) {
return Math.abs(a);
* Returns the mathematical absolute value of an {@code int} value
* if it is exactly representable as an {@code int}, throwing
* {@code ArithmeticException} if the result overflows the
* positive {@code int} range.
* <p>Since the range of two's complement integers is asymmetric
* with one additional negative value (JLS {@jls 4.2.1}), the
* mathematical absolute value of {@link Integer#MIN_VALUE}
* overflows the positive {@code int} range, so an exception is
* thrown for that argument.
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument, unless overflow occurs
* @throws ArithmeticException if the argument is {@link Integer#MIN_VALUE}
* @see Math#abs(int)
* @see Math#absExact(int)
* @since 15
public static int absExact(int a) {
return Math.absExact(a);
* Returns the absolute value of a {@code long} value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
* <p>Note that if the argument is equal to the value of {@link
* Long#MIN_VALUE}, the most negative representable {@code long}
* value, the result is that same value, which is negative. In
* contrast, the {@link StrictMath#absExact(long)} method throws
* an {@code ArithmeticException} for this value.
* @param a the argument whose absolute value is to be determined.
* @return the absolute value of the argument.
* @see Math#absExact(long)
public static long abs(long a) {
return Math.abs(a);
* Returns the mathematical absolute value of an {@code long} value
* if it is exactly representable as an {@code long}, throwing
* {@code ArithmeticException} if the result overflows the
* positive {@code long} range.
* <p>Since the range of two's complement integers is asymmetric
* with one additional negative value (JLS {@jls 4.2.1}), the
* mathematical absolute value of {@link Long#MIN_VALUE} overflows
* the positive {@code long} range, so an exception is thrown for
* that argument.
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument, unless overflow occurs
* @throws ArithmeticException if the argument is {@link Long#MIN_VALUE}
* @see Math#abs(long)
* @see Math#absExact(long)
* @since 15
public static long absExact(long a) {
return Math.absExact(a);
* Returns the absolute value of a {@code float} value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
* Special cases:
* <ul><li>If the argument is positive zero or negative zero, the
* result is positive zero.
* <li>If the argument is infinite, the result is positive infinity.
* <li>If the argument is NaN, the result is NaN.</ul>
* @apiNote As implied by the above, one valid implementation of
* this method is given by the expression below which computes a
* {@code float} with the same exponent and significand as the
* argument but with a guaranteed zero sign bit indicating a
* positive value: <br>
* {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))}
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument.
public static float abs(float a) {
return Math.abs(a);
* Returns the absolute value of a {@code double} value.
* If the argument is not negative, the argument is returned.
* If the argument is negative, the negation of the argument is returned.
* Special cases:
* <ul><li>If the argument is positive zero or negative zero, the result
* is positive zero.
* <li>If the argument is infinite, the result is positive infinity.
* <li>If the argument is NaN, the result is NaN.</ul>
* @apiNote As implied by the above, one valid implementation of
* this method is given by the expression below which computes a
* {@code double} with the same exponent and significand as the
* argument but with a guaranteed zero sign bit indicating a
* positive value: <br>
* {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)}
* @param a the argument whose absolute value is to be determined
* @return the absolute value of the argument.
public static double abs(double a) {
return Math.abs(a);
* Returns the greater of two {@code int} values. That is, the
* result is the argument closer to the value of
* {@link Integer#MAX_VALUE}. If the arguments have the same value,
* the result is that same value.
* @param a an argument.
* @param b another argument.
* @return the larger of {@code a} and {@code b}.
public static int max(int a, int b) {
return Math.max(a, b);
* Returns the greater of two {@code long} values. That is, the
* result is the argument closer to the value of
* {@link Long#MAX_VALUE}. If the arguments have the same value,
* the result is that same value.
* @param a an argument.
* @param b another argument.
* @return the larger of {@code a} and {@code b}.
public static long max(long a, long b) {
return Math.max(a, b);
* Returns the greater of two {@code float} values. That is,
* the result is the argument closer to positive infinity. If the
* arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If one
* argument is positive zero and the other negative zero, the
* result is positive zero.
* @param a an argument.
* @param b another argument.
* @return the larger of {@code a} and {@code b}.
public static float max(float a, float b) {
return Math.max(a, b);
* Returns the greater of two {@code double} values. That
* is, the result is the argument closer to positive infinity. If
* the arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If one
* argument is positive zero and the other negative zero, the
* result is positive zero.
* @param a an argument.
* @param b another argument.
* @return the larger of {@code a} and {@code b}.
public static double max(double a, double b) {
return Math.max(a, b);
* Returns the smaller of two {@code int} values. That is,
* the result the argument closer to the value of
* {@link Integer#MIN_VALUE}. If the arguments have the same
* value, the result is that same value.
* @param a an argument.
* @param b another argument.
* @return the smaller of {@code a} and {@code b}.
public static int min(int a, int b) {
return Math.min(a, b);
* Returns the smaller of two {@code long} values. That is,
* the result is the argument closer to the value of
* {@link Long#MIN_VALUE}. If the arguments have the same
* value, the result is that same value.
* @param a an argument.
* @param b another argument.
* @return the smaller of {@code a} and {@code b}.
public static long min(long a, long b) {
return Math.min(a, b);
* Returns the smaller of two {@code float} values. That is,
* the result is the value closer to negative infinity. If the
* arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If
* one argument is positive zero and the other is negative zero,
* the result is negative zero.
* @param a an argument.
* @param b another argument.
* @return the smaller of {@code a} and {@code b.}
public static float min(float a, float b) {
return Math.min(a, b);
* Returns the smaller of two {@code double} values. That
* is, the result is the value closer to negative infinity. If the
* arguments have the same value, the result is that same
* value. If either value is NaN, then the result is NaN. Unlike
* the numerical comparison operators, this method considers
* negative zero to be strictly smaller than positive zero. If one
* argument is positive zero and the other is negative zero, the
* result is negative zero.
* @param a an argument.
* @param b another argument.
* @return the smaller of {@code a} and {@code b}.
public static double min(double a, double b) {
return Math.min(a, b);
* Returns the fused multiply add of the three arguments; that is,
* returns the exact product of the first two arguments summed
* with the third argument and then rounded once to the nearest
* {@code double}.
* The rounding is done using the {@linkplain
* java.math.RoundingMode#HALF_EVEN round to nearest even
* rounding mode}.
* In contrast, if {@code a * b + c} is evaluated as a regular
* floating-point expression, two rounding errors are involved,
* the first for the multiply operation, the second for the
* addition operation.
* <p>Special cases:
* <ul>
* <li> If any argument is NaN, the result is NaN.
* <li> If one of the first two arguments is infinite and the
* other is zero, the result is NaN.
* <li> If the exact product of the first two arguments is infinite
* (in other words, at least one of the arguments is infinite and
* the other is neither zero nor NaN) and the third argument is an
* infinity of the opposite sign, the result is NaN.
* </ul>
* <p>Note that {@code fusedMac(a, 1.0, c)} returns the same
* result as ({@code a + c}). However,
* {@code fusedMac(a, b, +0.0)} does <em>not</em> always return the
* same result as ({@code a * b}) since
* {@code fusedMac(-0.0, +0.0, +0.0)} is {@code +0.0} while
* ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fusedMac(a, b, -0.0)} is
* equivalent to ({@code a * b}) however.
* @apiNote This method corresponds to the fusedMultiplyAdd
* operation defined in IEEE 754-2008.
* @param a a value
* @param b a value
* @param c a value
* @return (<i>a</i> × <i>b</i> + <i>c</i>)
* computed, as if with unlimited range and precision, and rounded
* once to the nearest {@code double} value
* @since 9
public static double fma(double a, double b, double c) {
return Math.fma(a, b, c);
* Returns the fused multiply add of the three arguments; that is,
* returns the exact product of the first two arguments summed
* with the third argument and then rounded once to the nearest
* {@code float}.
* The rounding is done using the {@linkplain
* java.math.RoundingMode#HALF_EVEN round to nearest even
* rounding mode}.
* In contrast, if {@code a * b + c} is evaluated as a regular
* floating-point expression, two rounding errors are involved,
* the first for the multiply operation, the second for the
* addition operation.
* <p>Special cases:
* <ul>
* <li> If any argument is NaN, the result is NaN.
* <li> If one of the first two arguments is infinite and the
* other is zero, the result is NaN.
* <li> If the exact product of the first two arguments is infinite
* (in other words, at least one of the arguments is infinite and
* the other is neither zero nor NaN) and the third argument is an
* infinity of the opposite sign, the result is NaN.
* </ul>
* <p>Note that {@code fma(a, 1.0f, c)} returns the same
* result as ({@code a + c}). However,
* {@code fma(a, b, +0.0f)} does <em>not</em> always return the
* same result as ({@code a * b}) since
* {@code fma(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while
* ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fma(a, b, -0.0f)} is
* equivalent to ({@code a * b}) however.
* @apiNote This method corresponds to the fusedMultiplyAdd
* operation defined in IEEE 754-2008.
* @param a a value
* @param b a value
* @param c a value
* @return (<i>a</i> × <i>b</i> + <i>c</i>)
* computed, as if with unlimited range and precision, and rounded
* once to the nearest {@code float} value
* @since 9
public static float fma(float a, float b, float c) {
return Math.fma(a, b, c);
* Returns the size of an ulp of the argument. An ulp, unit in
* the last place, of a {@code double} value is the positive
* distance between this floating-point value and the {@code
* double} value next larger in magnitude. Note that for non-NaN
* <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive or negative infinity, then the
* result is positive infinity.
* <li> If the argument is positive or negative zero, then the result is
* {@code Double.MIN_VALUE}.
* <li> If the argument is ±{@code Double.MAX_VALUE}, then
* the result is equal to 2<sup>971</sup>.
* </ul>
* @param d the floating-point value whose ulp is to be returned
* @return the size of an ulp of the argument
* @author Joseph D. Darcy
* @since 1.5
public static double ulp(double d) {
return Math.ulp(d);
* Returns the size of an ulp of the argument. An ulp, unit in
* the last place, of a {@code float} value is the positive
* distance between this floating-point value and the {@code
* float} value next larger in magnitude. Note that for non-NaN
* <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive or negative infinity, then the
* result is positive infinity.
* <li> If the argument is positive or negative zero, then the result is
* {@code Float.MIN_VALUE}.
* <li> If the argument is ±{@code Float.MAX_VALUE}, then
* the result is equal to 2<sup>104</sup>.
* </ul>
* @param f the floating-point value whose ulp is to be returned
* @return the size of an ulp of the argument
* @author Joseph D. Darcy
* @since 1.5
public static float ulp(float f) {
return Math.ulp(f);
* Returns the signum function of the argument; zero if the argument
* is zero, 1.0 if the argument is greater than zero, -1.0 if the
* argument is less than zero.
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive zero or negative zero, then the
* result is the same as the argument.
* </ul>
* @param d the floating-point value whose signum is to be returned
* @return the signum function of the argument
* @author Joseph D. Darcy
* @since 1.5
public static double signum(double d) {
return Math.signum(d);
* Returns the signum function of the argument; zero if the argument
* is zero, 1.0f if the argument is greater than zero, -1.0f if the
* argument is less than zero.
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, then the result is NaN.
* <li> If the argument is positive zero or negative zero, then the
* result is the same as the argument.
* </ul>
* @param f the floating-point value whose signum is to be returned
* @return the signum function of the argument
* @author Joseph D. Darcy
* @since 1.5
public static float signum(float f) {
return Math.signum(f);
* Returns the hyperbolic sine of a {@code double} value.
* The hyperbolic sine of <i>x</i> is defined to be
* (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2
* where <i>e</i> is {@linkplain Math#E Euler's number}.
* <p>Special cases:
* <ul>
* <li>If the argument is NaN, then the result is NaN.
* <li>If the argument is infinite, then the result is an infinity
* with the same sign as the argument.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
* </ul>
* @param x The number whose hyperbolic sine is to be returned.
* @return The hyperbolic sine of {@code x}.
* @since 1.5
public static native double sinh(double x);
* Returns the hyperbolic cosine of a {@code double} value.
* The hyperbolic cosine of <i>x</i> is defined to be
* (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2
* where <i>e</i> is {@linkplain Math#E Euler's number}.
* <p>Special cases:
* <ul>
* <li>If the argument is NaN, then the result is NaN.
* <li>If the argument is infinite, then the result is positive
* infinity.
* <li>If the argument is zero, then the result is {@code 1.0}.
* </ul>
* @param x The number whose hyperbolic cosine is to be returned.
* @return The hyperbolic cosine of {@code x}.
* @since 1.5
public static native double cosh(double x);
* Returns the hyperbolic tangent of a {@code double} value.
* The hyperbolic tangent of <i>x</i> is defined to be
* (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>),
* in other words, {@linkplain Math#sinh
* sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note
* that the absolute value of the exact tanh is always less than
* 1.
* <p>Special cases:
* <ul>
* <li>If the argument is NaN, then the result is NaN.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
* <li>If the argument is positive infinity, then the result is
* {@code +1.0}.
* <li>If the argument is negative infinity, then the result is
* {@code -1.0}.
* </ul>
* @param x The number whose hyperbolic tangent is to be returned.
* @return The hyperbolic tangent of {@code x}.
* @since 1.5
public static native double tanh(double x);
* Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
* without intermediate overflow or underflow.
* <p>Special cases:
* <ul>
* <li> If either argument is infinite, then the result
* is positive infinity.
* <li> If either argument is NaN and neither argument is infinite,
* then the result is NaN.
* <li> If both arguments are zero, the result is positive zero.
* </ul>
* @param x a value
* @param y a value
* @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>)
* without intermediate overflow or underflow
* @since 1.5
public static double hypot(double x, double y) {
return FdLibm.Hypot.compute(x, y);
* Returns <i>e</i><sup>x</sup> -1. Note that for values of
* <i>x</i> near 0, the exact sum of
* {@code expm1(x)} + 1 is much closer to the true
* result of <i>e</i><sup>x</sup> than {@code exp(x)}.
* <p>Special cases:
* <ul>
* <li>If the argument is NaN, the result is NaN.
* <li>If the argument is positive infinity, then the result is
* positive infinity.
* <li>If the argument is negative infinity, then the result is
* -1.0.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
* </ul>
* @param x the exponent to raise <i>e</i> to in the computation of
* <i>e</i><sup>{@code x}</sup> -1.
* @return the value <i>e</i><sup>{@code x}</sup> - 1.
* @since 1.5
public static native double expm1(double x);
* Returns the natural logarithm of the sum of the argument and 1.
* Note that for small values {@code x}, the result of
* {@code log1p(x)} is much closer to the true result of ln(1
* + {@code x}) than the floating-point evaluation of
* {@code log(1.0+x)}.
* <p>Special cases:
* <ul>
* <li>If the argument is NaN or less than -1, then the result is
* NaN.
* <li>If the argument is positive infinity, then the result is
* positive infinity.
* <li>If the argument is negative one, then the result is
* negative infinity.
* <li>If the argument is zero, then the result is a zero with the
* same sign as the argument.
* </ul>
* @param x a value
* @return the value ln({@code x} + 1), the natural
* log of {@code x} + 1
* @since 1.5
public static native double log1p(double x);
* Returns the first floating-point argument with the sign of the
* second floating-point argument. For this method, a NaN
* {@code sign} argument is always treated as if it were
* positive.
* @param magnitude the parameter providing the magnitude of the result
* @param sign the parameter providing the sign of the result
* @return a value with the magnitude of {@code magnitude}
* and the sign of {@code sign}.
* @since 1.6
public static double copySign(double magnitude, double sign) {
return Math.copySign(magnitude, (Double.isNaN(sign)?1.0d:sign));
* Returns the first floating-point argument with the sign of the
* second floating-point argument. For this method, a NaN
* {@code sign} argument is always treated as if it were
* positive.
* @param magnitude the parameter providing the magnitude of the result
* @param sign the parameter providing the sign of the result
* @return a value with the magnitude of {@code magnitude}
* and the sign of {@code sign}.
* @since 1.6
public static float copySign(float magnitude, float sign) {
return Math.copySign(magnitude, (Float.isNaN(sign)?1.0f:sign));
* Returns the unbiased exponent used in the representation of a
* {@code float}. Special cases:
* <ul>
* <li>If the argument is NaN or infinite, then the result is
* {@link Float#MAX_EXPONENT} + 1.
* <li>If the argument is zero or subnormal, then the result is
* {@link Float#MIN_EXPONENT} -1.
* </ul>
* @param f a {@code float} value
* @return the unbiased exponent of the argument
* @since 1.6
public static int getExponent(float f) {
return Math.getExponent(f);
* Returns the unbiased exponent used in the representation of a
* {@code double}. Special cases:
* <ul>
* <li>If the argument is NaN or infinite, then the result is
* {@link Double#MAX_EXPONENT} + 1.
* <li>If the argument is zero or subnormal, then the result is
* {@link Double#MIN_EXPONENT} -1.
* </ul>
* @param d a {@code double} value
* @return the unbiased exponent of the argument
* @since 1.6
public static int getExponent(double d) {
return Math.getExponent(d);
* Returns the floating-point number adjacent to the first
* argument in the direction of the second argument. If both
* arguments compare as equal the second argument is returned.
* <p>Special cases:
* <ul>
* <li> If either argument is a NaN, then NaN is returned.
* <li> If both arguments are signed zeros, {@code direction}
* is returned unchanged (as implied by the requirement of
* returning the second argument if the arguments compare as
* equal).
* <li> If {@code start} is
* ±{@link Double#MIN_VALUE} and {@code direction}
* has a value such that the result should have a smaller
* magnitude, then a zero with the same sign as {@code start}
* is returned.
* <li> If {@code start} is infinite and
* {@code direction} has a value such that the result should
* have a smaller magnitude, {@link Double#MAX_VALUE} with the
* same sign as {@code start} is returned.
* <li> If {@code start} is equal to ±
* {@link Double#MAX_VALUE} and {@code direction} has a
* value such that the result should have a larger magnitude, an
* infinity with same sign as {@code start} is returned.
* </ul>
* @param start starting floating-point value
* @param direction value indicating which of
* {@code start}'s neighbors or {@code start} should
* be returned
* @return The floating-point number adjacent to {@code start} in the
* direction of {@code direction}.
* @since 1.6
public static double nextAfter(double start, double direction) {
return Math.nextAfter(start, direction);
* Returns the floating-point number adjacent to the first
* argument in the direction of the second argument. If both
* arguments compare as equal a value equivalent to the second argument
* is returned.
* <p>Special cases:
* <ul>
* <li> If either argument is a NaN, then NaN is returned.
* <li> If both arguments are signed zeros, a value equivalent
* to {@code direction} is returned.
* <li> If {@code start} is
* ±{@link Float#MIN_VALUE} and {@code direction}
* has a value such that the result should have a smaller
* magnitude, then a zero with the same sign as {@code start}
* is returned.
* <li> If {@code start} is infinite and
* {@code direction} has a value such that the result should
* have a smaller magnitude, {@link Float#MAX_VALUE} with the
* same sign as {@code start} is returned.
* <li> If {@code start} is equal to ±
* {@link Float#MAX_VALUE} and {@code direction} has a
* value such that the result should have a larger magnitude, an
* infinity with same sign as {@code start} is returned.
* </ul>
* @param start starting floating-point value
* @param direction value indicating which of
* {@code start}'s neighbors or {@code start} should
* be returned
* @return The floating-point number adjacent to {@code start} in the
* direction of {@code direction}.
* @since 1.6
public static float nextAfter(float start, double direction) {
return Math.nextAfter(start, direction);
* Returns the floating-point value adjacent to {@code d} in
* the direction of positive infinity. This method is
* semantically equivalent to {@code nextAfter(d,
* Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
* implementation may run faster than its equivalent
* {@code nextAfter} call.
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
* <li> If the argument is positive infinity, the result is
* positive infinity.
* <li> If the argument is zero, the result is
* {@link Double#MIN_VALUE}
* </ul>
* @param d starting floating-point value
* @return The adjacent floating-point value closer to positive
* infinity.
* @since 1.6
public static double nextUp(double d) {
return Math.nextUp(d);
* Returns the floating-point value adjacent to {@code f} in
* the direction of positive infinity. This method is
* semantically equivalent to {@code nextAfter(f,
* Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
* implementation may run faster than its equivalent
* {@code nextAfter} call.
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
* <li> If the argument is positive infinity, the result is
* positive infinity.
* <li> If the argument is zero, the result is
* {@link Float#MIN_VALUE}
* </ul>
* @param f starting floating-point value
* @return The adjacent floating-point value closer to positive
* infinity.
* @since 1.6
public static float nextUp(float f) {
return Math.nextUp(f);
* Returns the floating-point value adjacent to {@code d} in
* the direction of negative infinity. This method is
* semantically equivalent to {@code nextAfter(d,
* Double.NEGATIVE_INFINITY)}; however, a
* {@code nextDown} implementation may run faster than its
* equivalent {@code nextAfter} call.
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
* <li> If the argument is negative infinity, the result is
* negative infinity.
* <li> If the argument is zero, the result is
* {@code -Double.MIN_VALUE}
* </ul>
* @param d starting floating-point value
* @return The adjacent floating-point value closer to negative
* infinity.
* @since 1.8
public static double nextDown(double d) {
return Math.nextDown(d);
* Returns the floating-point value adjacent to {@code f} in
* the direction of negative infinity. This method is
* semantically equivalent to {@code nextAfter(f,
* Float.NEGATIVE_INFINITY)}; however, a
* {@code nextDown} implementation may run faster than its
* equivalent {@code nextAfter} call.
* <p>Special Cases:
* <ul>
* <li> If the argument is NaN, the result is NaN.
* <li> If the argument is negative infinity, the result is
* negative infinity.
* <li> If the argument is zero, the result is
* {@code -Float.MIN_VALUE}
* </ul>
* @param f starting floating-point value
* @return The adjacent floating-point value closer to negative
* infinity.
* @since 1.8
public static float nextDown(float f) {
return Math.nextDown(f);
* Returns {@code d} × 2<sup>{@code scaleFactor}</sup>
* rounded as if performed by a single correctly rounded
* floating-point multiply. If the exponent of the result is
* between {@link Double#MIN_EXPONENT} and {@link
* Double#MAX_EXPONENT}, the answer is calculated exactly. If the
* exponent of the result would be larger than {@code
* Double.MAX_EXPONENT}, an infinity is returned. Note that if
* the result is subnormal, precision may be lost; that is, when
* {@code scalb(x, n)} is subnormal, {@code scalb(scalb(x, n),
* -n)} may not equal <i>x</i>. When the result is non-NaN, the
* result has the same sign as {@code d}.
* <p>Special cases:
* <ul>
* <li> If the first argument is NaN, NaN is returned.
* <li> If the first argument is infinite, then an infinity of the
* same sign is returned.
* <li> If the first argument is zero, then a zero of the same
* sign is returned.
* </ul>
* @param d number to be scaled by a power of two.
* @param scaleFactor power of 2 used to scale {@code d}
* @return {@code d} × 2<sup>{@code scaleFactor}</sup>
* @since 1.6
public static double scalb(double d, int scaleFactor) {
return Math.scalb(d, scaleFactor);
* Returns {@code f} × 2<sup>{@code scaleFactor}</sup>
* rounded as if performed by a single correctly rounded
* floating-point multiply. If the exponent of the result is
* between {@link Float#MIN_EXPONENT} and {@link
* Float#MAX_EXPONENT}, the answer is calculated exactly. If the
* exponent of the result would be larger than {@code
* Float.MAX_EXPONENT}, an infinity is returned. Note that if the
* result is subnormal, precision may be lost; that is, when
* {@code scalb(x, n)} is subnormal, {@code scalb(scalb(x, n),
* -n)} may not equal <i>x</i>. When the result is non-NaN, the
* result has the same sign as {@code f}.
* <p>Special cases:
* <ul>
* <li> If the first argument is NaN, NaN is returned.
* <li> If the first argument is infinite, then an infinity of the
* same sign is returned.
* <li> If the first argument is zero, then a zero of the same
* sign is returned.
* </ul>
* @param f number to be scaled by a power of two.
* @param scaleFactor power of 2 used to scale {@code f}
* @return {@code f} × 2<sup>{@code scaleFactor}</sup>
* @since 1.6
public static float scalb(float f, int scaleFactor) {
return Math.scalb(f, scaleFactor);
⏎ java/lang/StrictMath.java
Or download all of them as a single archive file:
File name: java.base-17.0.5-src.zip
File size: 8883851 bytes
Release date: 2022-09-13
⇒ JDK 17 java.compiler.jmod - Compiler Module | {"url":"http://jar.fyicenter.com/3606_JDK_17_java_base_jmod-Base_Module.html?C=java.lang.StrictMath","timestamp":"2024-11-03T20:56:18Z","content_type":"text/html","content_length":"109793","record_id":"<urn:uuid:ff6a1759-12ab-4f4f-a60f-2b30c57c4930>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00189.warc.gz"} |
I’m not a string theorist and who am I to doubt Urs’ word, but this paragraph doesn’t gel with what I know of F-theory
More precisely, this process relates 11-dimensional supergravity on 11-dimensional torus-bundles / elliptic fibrations to type IIA supergravity. What is called F-theory is the explicit
description of type IIB supergravity vacua in terms of such elliptic fibrations.
In particular, I thought F-theory was about 12-dimensional elliptic fibrations. (to forestall any comments, I would have edited if I could be sure I had my facts straight)
Hi, I added a reference for phenomenology (first edit here at nLab). I also have some other good ones in my collection that I'm considering adding too, depending on which ones seem the most
started an entry F-theory (the string-theoretic notion)
Hi Cliff,
thanks for helping add references and content!
Hi David,
so far I chose to mention the M-theory dual picture only. There it is indeed an 11d elliptic fibration. From this perspective the 12d fibration that you are looking for appears from first shrinking
the fiber to vanishing volume and then applying T-duality in one fiber direction to make another large dimension appear again:
$\array{ M-theory in 11 d && F-theory in 12d \\ {}^{\mathllap{ellitpic fibration}}\downarrow && \downarrow^{\mathrlap{ellitpic fibration}} \\ type IIA in 9d &\stackrel{T-duality}{\leftrightarrow}&
type IIB in 10 d } \,.$
In actual fact, what is mostly discussed in the literature are real 8d elliptic fibrations $Y_8 \to B_6$ over a real 6d base. In F-theory phenomenology then one considers simply the 12 d product $\
mathbb{R}^{1,3} \times Y_8$ which gets eventually compactified to Minkowski spacetime $\mathbb{R}^{3,1}$, while in the dual M-theory picture one considers the 11d product $\mathbb{R}^{1,2} \times
I’ll try to find the time now to put more comments along these lines also into the entry…
Okay, I have now expanded F-theory as indicated above.
Have expanded a bit the Idea-section
Someone unhelpfully and ungrammatically added “It also have relation to multiverse” in the first paragraph, so I rolled it back.
Thanks David. That sort of speculation is not well-founded.
added pointer to Weigand 18
diff, v49, current
added some references on the F-theory realization of the exact SM gauge group (which is what the recent article with “quadrillion” in the title is based on…):
Discussion of the exact gauge group of the standard model of particle physics, $G = \big( SU(3) \times SU(2) \times U(1)\big)/\mathbb{Z}_6$ including its $\mathbb{Z}_6$-quotient (see there) and the
exact fermion field content, realized in F-theory is in
• Denis Klevers, Damian Kaloni Mayorga Pena, Paul-Konstantin Oehlmann, Hernan Piragua, Jonas Reuter, Denis Klevers, Damian Kaloni Mayorga Pena, Paul-Konstantin Oehlmann, Hernan Piragua, Jonas
Reuter, JHEP01(2015)142 (arXiv:1408.4808)
• Mirjam Cvetic, Ling Lin, section 3.3 of The global gauge group structure of F-theory compactifications with $U(1)$s (arXiv:1706.08521)
diff, v54, current
That’s an interesting title for the Klevers et al paper….
Fixed that ’title’ :-)
diff, v55, current
With a full description of M-theory available also F-theory should be a full non-perturbative description of type IIB string theory, but absent that it is some kind of approximation.
Can you see implications for F-theory already from the cohomotopic picture of M-theory you’re devising?
Yeah, there is a reason for a bunch of new $n$Lab entries, such as M-theory on 8-manifolds. More later. Busy typing up proofs… :-)
added pointer to today’s
• Washington Taylor, Andrew P. Turner, Generic construction of the Standard Model gauge group and matter representations in F-theory (arXiv:1906.11092)
diff, v59, current | {"url":"https://nforum.ncatlab.org/discussion/3591/ftheory/?Focus=29538","timestamp":"2024-11-03T19:37:03Z","content_type":"application/xhtml+xml","content_length":"63538","record_id":"<urn:uuid:1006927f-ab74-4291-92b4-cccd0ce1aba8>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00028.warc.gz"} |
Paper Review: Shape Classification Using Zernike Moments
A: What is a moment?
A moment is defined as
$m_{p,q}(x,y) = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} x^p y^q f(x,y) dxdy$
In other words, it's the summation of the figure w.r.t function f for both axes
A: What are Zernike moments?
Zernike moments are complex polynomial functions that we use to sum the elements of a shape. It is was first introduced in 1930s. The higher the order of it, the more complex shape appears. Order 1
ZMs are ellipsoid planes of which one side is higher than the other.
A: What's the difference between Hu moments and Zernike moments?
The importance of Zernike moments are their rotationally invariant features. However Hu moments are said to have these properties as well. So it looks Hu Moments are simpler alternatives to this.
A: What are their properties?
It uses polar coordinates and hence easy to describe the rotational invariance. | {"url":"https://emresahin.net/shape-classification-using-zernike-moments/","timestamp":"2024-11-05T00:12:39Z","content_type":"text/html","content_length":"4614","record_id":"<urn:uuid:7428f06e-c6a8-474d-b49b-d775a07ddcc1>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00597.warc.gz"} |
Addition of Similar Digit Numbers Shortcut tricks - Math Shortcut Tricks
Addition of Similar Digit Numbers Shortcut tricks:
Traditionally this is very time consuming to calculate the Addition of Similar Digit Numbers, And it create confusion at that moment in speed, So don’t worry here we given Quick shortcut tricks to do
quick calculation by using of Quick shortcut tricks. Sometime we face this type of Addition in Exam hall.
27 comments
1. sunil gyani says:
My son is preparing for competition exams,kindly let me know the supportive mat With you & how to make order
□ Rabi Mandal says:
At first memorized 1 to 30 square and 1 to 30 cube and table of odd number from 1 to 30, which help you fast calculation. Onward you have to perform multiplication addition and division
shortcut trick for that follow www.math-shortcut-tricks.com
□ VEGATEELA SAI PHANINDRA says:
pls these teqnics are add to my gmail
□ rajvirroy says:
plis send me on my gmail adress plisssssss
□ ASHISH JAWANJAL says:
plz send me on my gmail address…..plz
□ nikk says:
pls send mi all kind of math tricks
□ Amit kumar says:
Sir plz tell me how to add 3+333+33333+3333333+333333 (short trick)
☆ monu says:
there is no short method my son
☆ R.I.Sivaprasath says:
it is too easy , you try 11334455×3=34003365
○ sreekanth says:
its 1233445
○ sachin porwal says:
please send me all topics aptitude tricks on my gmail porwalsachin26@gmail.com
○ Nishanth g says:
If suppose like this series 333+3+333+33+33
How to do short cut
○ Mitul Jambukiya says:
please provide me shortcut of 7+7+77+77+777+777+….
like vise
○ sagar jadhav says:
please sir forward all maths short ricks on my mail jsagar22@yahoo.com
○ Shakthi says:
Can you please send me the shortcuts tips and tricks for clearing out the aptitude round …I am tired of it ..Please do help me ,My mail shakthijayabal@gmail.com
○ nagu nagarajan says:
Please send me all kind of tricks
○ vikash sah says:
Sir send all shorcut techniques of maths in my email
○ Shabnam S Ram says:
How shall we calculate the addition of the similar digits when they are not increasing consecutively.Foe example:
According to the above trick,it should be 6×1224=7344.
however the actual answer is 6804.
Please revert.
■ Ashish Kumar says:
it is 6×1134=6840
Plz understand the logic clearly
Here after taking common we add things like this
■ Akhtar says:
whatis siries of start with 2 digits like :-
■ sunny gambhir says:
what will be short cut trick for below
in above the difference between digit is not in continuation and end result when used ur trick is different.
Also please share tricks on sunnylive16@gmail.com
■ anamika says:
sir/madam i find difficult in clearing aptitude paper pls help to solve easily and interestingly kindly guide me for my bank exams
■ tina says:
how can we solve easily trignometery ratios sum
■ Krishna says:
i need a mathematical steps for 12345 = 555
(1+4)(3+2)(5) like that 95381 = 888 (9-1)(5+3)(8) like that if any number given then the answer is similar value
■ Kashyap says:
This shortcut is good but only if number series are into the ascending order for example
if have a case of
6+66+666+6666 = 6(1234) = 7404
but what if we have
6+666+66+6666+6 = 6(13241) = 79446 which is wrong correct answer is 7410
Is there is any short cut for the same?
■ Jatinder Kumar Ahooja says:
As at the start of the page, you gave the question – add:
——————I need not write the answer to this question. As five eights followed by 4, by 3, by 2, by 1 are seen above. Who doesn’t know the table of 8? As we see five eights on the
right most row, we know that 8 x 5 = 40. Here after, we see 4 eights, and we know that 8 x 4 = 32, write 32 in a complete manner. Likewise, we see that only 3 eights are to be
added, write down 24, likewise 2 eights = 16 and finally a single 8, write down that as well. You may be wondering why I told you to do so! You’d understand the method in no time.
As, I’d written 40, followed by 32, likewise 24, 16 and 8, write these numbers in a particular way,i.e., 40
———————– All you’ve to do is to note down 0 of 40, add 4 + 2 of 40 and 32 respectively
Same way add 3 of 32 to 4 of 24; add 2 of 24 to 6 of 16 and finally note down the single 8 left. Do the addition from right to left, so to get the final total of the above in
hardly any time.
■ Ashwini /Chashma says:
Hi Guys,
Could you please send me the required aptitude materials and also basic shortcut tricks for the same to my below mentioned email address at the earliest?
Leave a Reply
If you have any questions or suggestions then please feel free to ask us. You can either comment on the section above or mail us in our e-mail id. One of our expert team member
will respond to your question with in 24 hours. You can also Like our Facebook page to get updated on new topics and can participate in online quiz.
All contents of this website is fully owned by Math-Shortcut-Tricks.com. Any means of republish its content is strictly prohibited. We at Math-Shortcut-Tricks.com are always try
to put accurate data, but few pages of the site may contain some incorrect data in it. We do not assume any liability or responsibility for any errors or mistakes in those pages.
Visitors are requested to check correctness of a page by their own.
© 2024 Math-Shortcut-Tricks.com | All Rights Reserved. | {"url":"https://www.math-shortcut-tricks.com/addition-of-similar-digit-numbers-shortcut-tricks/","timestamp":"2024-11-06T17:41:53Z","content_type":"text/html","content_length":"259035","record_id":"<urn:uuid:40b73d2b-ccf4-4b88-9197-bcef16ef99e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00805.warc.gz"} |
[Solved] A conductor 2 m long carrying 3A is place | SolutionInn
A conductor 2 m long carrying 3A is placed parallel to the z-axis at distance ?0 =
A conductor 2 m long carrying 3A is placed parallel to the z-axis at distance ?0 = 10 cm as shown in figure. If the field in the region is cos (?/3) a? Wb/m2, how much work is required to rotate the
conductor one revolution about the z-axis?
Transcribed Image Text:
3A Po o
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Answer rating: 57% (14 reviews)
Answered By
Aysha Ali
my name is ayesha ali. i have done my matriculation in science topics with a+ . then i got admission in the field of computer science and technology in punjab college, lahore. i have passed my final
examination of college with a+ also. after that, i got admission in the biggest university of pakistan which is university of the punjab. i am studying business and information technology in my
university. i always stand first in my class. i am very brilliant client. my experts always appreciate my work. my projects are very popular in my university because i always complete my work with
extreme devotion. i have a great knowledge about all major science topics. science topics always remain my favorite topics. i am also a home expert. i teach many clients at my home ranging from
pre-school level to university level. my clients always show excellent result. i am expert in writing essays, reports, speeches, researches and all type of projects. i also have a vast knowledge
about business, marketing, cost accounting and finance. i am also expert in making presentations on powerpoint and microsoft word. if you need any sort of help in any topic, please dont hesitate to
consult with me. i will provide you the best work at a very reasonable price. i am quality oriented and i have 5 year experience in the following field. matriculation in science topics; inter in
computer science; bachelors in business and information technology _embed src=http://www.clocklink.com/clocks/0018-orange.swf?timezone=usa_albany& width=200 height=200 wmode=transparent type=
4.40+ 11+ Reviews 14+ Question Solved
Students also viewed these Electrodynamics questions
Study smarter with the SolutionInn App | {"url":"https://www.solutioninn.com/conductor-2-m-long-carrying-3a-is-placed-parallel","timestamp":"2024-11-08T21:58:44Z","content_type":"text/html","content_length":"81714","record_id":"<urn:uuid:0c1803aa-ee06-4b2e-bfba-e902baa45acf>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00682.warc.gz"} |
Nonlinear Optics, Back-of-Envelope Numbers
If you take a flashlight and shine it at a wall or through a prism or do pretty much anything with the light, you'll find that the effects of the light change in direct proportion to the amount of
light the flashlight's putting out. Twice the light, twice the reflection from the wall. Twice the light, twice the brightness in each color component of the prism spectrum. Nothing really changes
qualitatively as you change the light intensity, the only difference is quantitative.
This property of light is called linear optics. Until the last half-century or so it was just called "optics". The reason for this is the fact that in fact there are physical situations where the
response of a material to light qualitatively changes as a function of the intensity of the light. For instance, we now know that the refractive index of a material changes slightly when exposed to
intense light. This can result in all kinds of weird phenomena such as self-focusing and filamentation. Other effects such as frequency doubling and Raman scattering crop up, and in fact have found
uses in practical technology. Many green laser pointers use a nonlinear effect where an infrared laser is focused into a nonlinear crystal which doubles the frequency of the light into the green
range. But this doesn't happen for the very same crystal in infrared light of everyday intensity.
So why is it that we don't see nonlinear optical effects when we stand in sunlight or turn on our headlights? Well, we might correctly surmise that (at least classically) light interacts with
materials mostly by virtue of the electromagnetic field causing the electrons orbiting the atoms to wiggle around. This wiggle itself is an accelerating electric charge, which combines with the
original electromagnetic wave and thereby slightly alters it. We might expect that the linear nature of this process would start to fail when the strength of the electric field of the incoming light
starts to be comparable in magnitude to the electric field that holds the atomic electrons to the nucleus in the first place.
Fig. 1: Me, illuminated by 536nm frequency-doubled laser light
We want to know how much electric field is being felt by the electron, holding it in its orbital around the nucleus. The electric field produced by a point charge - in this case the nucleus - at a
distance r is given by:
For a hydrogen atom, the classical separation between the nucleus and the electron is the Bohr radius, which is about 5.29 x 10^-11 meters. We can look up the electron charge and electric constant
pretty easily, and plugging into the equation the characteristic electric field inside an atom is about 5.14 x 10^11 volts per meter. This is a pretty stout electric field - air breaks down with a
lightning flash at a mere three million or so volts per meter.
So how intense does a light beam need to be before you get field strengths that high? We have an equation for that as well - the derivation is too long for this post but you can find it in any E&M
textbook if you're interested. The intensity of light as a function of its peak field strength E is:
Where c is the speed of light. Plugging in our value for E, and we get that the intensity of the light is about 3.5 x 10^20 watts per square meter. For reference, this is about a hundred thousand
trillion times more intense than direct sunlight. No wonder we don't see nonlinear optics much.
But on the other hand if you take a relatively ordinary laboratory laser that emits 1mJ pulses with a duration of 35 femtoseconds and focus it down to about a square micron, suddenly you have light
of about 100 times the threshold intensity without breaking a sweat.
Now this is just a back-of-the-envelope calculation. It's easily possible to see nonlinear optical effects at vastly lower powers than our hydrogen-field calculation. But this kind of argument does
give an idea as to why they're uncommon with everyday incoherent light.
More like this
Shortly after the invention of the laser, a torrent of discoveries began pouring in thanks to the previously unreachable intensities that became available. Many of these discoveries fall under the
general category of "nonlinear optics", which you could more or less say is the study of the behavior…
Whew! Back from a very successful wedding and honeymoon, moving into a new apartment, writing thank-you notes, and all the fun jazz that comes with being newly married. But hey, we've got to get this
blog cranking again at some point, and now's as good a time as any. We'll kick things back off…
Most of the time, when we talk about seeing quantum effects from light, we talk about extremely weak beams-- looking at intensities where one photon more or less represents a significant change in
the intensity of the light. Last week, though, Physics Buzz wrote up a paper that goes in the other…
Weekend posting here is usually pretty light, but it's only the second day here so I think a little extra is a nice way to kick things off. How about a little bit of solar sailing, since it fits
pretty well with what I'm teaching in my intro class? We all know light carries energy. Go outside on…
For two equal point charges separated by a distance r, the electric field between them is given by:
With the equation you've written, it would be more accurate to say that's the field due to one point charge +e, which would be the field felt by the other charge. But your quote makes it seem like
you're looking at the total field due to both charges, in which case you'd have to use the dipole field.
Good point. I've reworded it a bit for clarity.
dear matt springer , i have a silly question .
what is the source of electric field , only charges ?
electric charges can be the source of an electric field.
also, a changing magnetic field can be the source of an electric field. look up Faraday's law of induction to read about it.
what is the percentage carbon composition in liquidfy natural gas | {"url":"https://scienceblogs.com/builtonfacts/2010/08/27/if-you-take-a-flashlight","timestamp":"2024-11-05T10:57:23Z","content_type":"text/html","content_length":"47732","record_id":"<urn:uuid:b9532a78-f8a5-4470-92ca-2e1283d13566>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00601.warc.gz"} |
Math Worksheets Coins
Money worksheets counting math grade 2nd worksheet coins count dollar pdf printable salamanders sheet coin 1st kids maths sheets second 2nd grade money worksheets up to $2 Count the coins and tell
the amount worksheets
How many coins will be needed to make the given rupees – Worksheets
Counting money worksheets up to $1 Coins many money grade worksheets rupees currency worksheet math make given makes maths needed will mathsdiary comment leave Money – grade 1 math worksheets
Count the coins and tell the amount worksheets
Many coins worksheets make rupees given needed will worksheet mathsdiary makes grade moneyCoins worksheets grade money counting currency worksheet mathsdiary comparing math pare comment leave
Worksheets grade 2nd money math coins pdf count dollars counting sheet versionUs coins worksheets kindergarten.
Worksheets grade money 1st math printable first kindergarten learning coins counting worksheet salamanders kids chart pdf know school different writingHow many coins will be needed to make the given
rupees – worksheets How many coins will be needed to make the given rupees – worksheetsCounting worksheets money coins 2p 1p math sheet pdf salamanders.
How many coins needed
Coins preschool counting identify penny education recognition printables bukaninfoWorksheets kindergarten math coins money printable identify grade fun first activities teaching salamanders 1st
counting sheet pdf currency preschool sheets How many coins will be needed to make the given rupees – worksheetsCoins many worksheets grade rupees make given math needed will money makes worksheet
currency mathsdiary maths.
How many coins will be needed to make the given rupees – worksheetsWorksheets coins grade counting math first printable worksheet money kids printables problem visit Free counting money worksheets uk
coinsKindergarten money worksheets 1st grade.
Coins worksheets count amount worksheet tell mathsdiary math
Coins many needed grade worksheet maths money mathsdiaryCount sheet salamanders Money worksheets for first gradeCounting money worksheets up to $1.
Many coins worksheet grade worksheets rupees given needed make will math currency mathsdiary money mathsWorksheets coins money count amount rupees worksheet grade maths tell math counting mathsdiary
coin choose board .
How many coins will be needed to make the given rupees – Worksheets
Money – Grade 1 Math Worksheets
Kindergarten Money Worksheets 1st Grade
counting-coins-worksheets - Tim's Printables
Money Worksheets for First Grade
How many coins will be needed to make the given rupees – Worksheets
How many coins needed - Math Worksheets - MathsDiary.com
Counting Money Worksheets up to $1
How many coins will be needed to make the given rupees – Worksheets | {"url":"https://studyschoolwhipworm.z14.web.core.windows.net/math-worksheets-coins.html","timestamp":"2024-11-13T16:16:37Z","content_type":"text/html","content_length":"40579","record_id":"<urn:uuid:640d50ae-34e6-410b-830f-6e9ded4876ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00149.warc.gz"} |
Module 2 – Mathlets in Group Work - MIT Mathlets
Module 2 – Mathlets in Group Work
The first segment of this module contains an example of the use of a Mathlet to organize group work by students in a lecture setting. This second segment provides samples of what you can ask groups
of students to do using Mathlets, inside or outside a classroom. Each segment is followed by questions, my remarks about the lecture fragments and the features of Mathlets, and examples that
demonstrate their use. At the end there are a couple of exercises.
1. In this module…
You will first try out the Mathlets in a separate window. Next, you will watch video lectures where you will be instructed to pause and engage in a variety of activities, as well as think about the
questions posed.
Download Module slides [ PDF, 1.9MB ]
Download Complete Module 2 video [ ZIP, 72MB, 540p ]
2. Learning Objectives
After completing this module, the participant will be able to use Mathlets to:
• Encourage group work in the classroom.
• Encourage genuine mathematical reasoning by students.
• Provide a focus for small group projects.
3. Segment 1: Mathlets in Group Work
Module 1 Mathlets in Lecture discussed the use of Mathlets during in-class lectures. One advantage to using Mathlets is the ability to allow students to explore and learn collaboratively as they
complete problems and projects. This module will introduce the use of Mathlets in group work by providing a recorded example from a teachers workshop.
3.1 Activity
Think About: Isoclines
1. Did you find my image of the slope field as a field of grass knocked down by the wind helpful? And how about the small animal running through it, as an image of an integral curve?
2. Suggest one other way in which I could have gotten the students working with each other using Isoclines.
3. What are your comments about the Isoclines Mathlet. Could some part of this be useful in one of the classes you teach?
Segment 1 Review
Below, you will find:
• Lecture Fragment Topics $-$ A listing of the topics covered by the lecture fragment
• Lecture Components $-$ A listing of the components of the lecture fragment
• Discussion and Questions $-$ Dr. Miller’s reflection on this lecture fragment
• Further use of the Isoclines Mathlet $-$ Further ideas of things one can do with the Isoclines Mathlet in a classroom group work setting.
Lecture Fragment Topics
1. Direction fields and integral curves
2. Isoclines, nullcline
3. Fences, entrapment of integral curves
Lecture Components
1. Slope field hay field
2. Integral curves deer tracks
3. Where do extreme points occur?
4. Talking with students
5. Gathering students for an answer
6. Showing isoclines
7. Group discussion about maxima vs minimum
Was my concept about a hay field with deer running though it successful?
My casual cursor drag, generating a whole family of solutions, was carefully rehearsed. If I had dragged carelessly the integral curves would have jumped abruptly, confounding the impression I wished
to give that the solutions formed a smoothly changing family of functions.
Did you notice my gesture of where the nullcline was? It was reversed from how it looked to the audience! I still have not learned that skill.
Clearly students wanted to keep working on things themselves in this segment!
Then I asked the question about why the extrema are maxima. In retrospect, I should have asked the participants to work on this on their own.
Note that I systematically talked about what I was doing with the Mathlet. Staying inside the parabola: I asked them to work on it, but then switched to an open discussion. This worked in the
context, and got quite a few different participants chipping in, but it might have been better to have them explain it to each other for a while.
I say it is an argument by contradiction; but this is not quite right, is it?
3.2 Further uses of the Isoclines Mathlet
There are many other things you can ask your students to think about using the Isoclines Mathlet. Below, I will state three observations you can make using this Mathlet.
After stating each observation, I will offer hints you can share if your students need them. I will also provide some answers that students have come up with. Remember that students will come up with
various ways to phrase arguments for these facts. The following provides a few of my own descriptions along with some alternatives to help you be better prepared to offer some ideas about suggestions
to groups that get stuck.
1. First Observation: Once an integral curve enters the region to the right of the isocline, it is trapped in that region forever (for all larger $x$). Appearances can be deceiving, though. Is this
really true? To be sure, you have to come up with an argument for it.
□ Hints:
☆ Point out that there are curves that cross this parabola from the inside. Why can they not be integral curves?
☆ Invite students to consider the slope of the tangent line of the parabola at the point of crossing.
□ Answers:
☆ Curves can certainly cross this parabola moving from inside to the outside (as you move to the right). But these cannot be integral curves.
☆ When an integral curve crosses the nullcline, it must cross with slope zero, and horizontal lines always cross from outside to inside, never from inside to outside.
2. Second Observation: If an integral curve goes through some point directly below the lower branch of the nullcline (so for some $a > 0$, $y(a) < -\sqrt a)$, then it will cross the nullcline
somewhere further to the right. Is this really true? To be sure, you have to come up with an argument for it.
□ Hint:
☆ Draw a horizontal straight line through the chosen point.
□ Answers:
☆ Below the nullcline the slopes are positive; so, until it crosses the nullcline, the solution stays above a horizontal line; but every horizontal line crosses the nullcline eventually, so
the integral curve has to also.
3. Third Observation: Many solutions become asymptotic to the lower branch of the nullcline as $x\rightarrow\infty$. Is this really true? To be sure, you have to come up with an argument for it.
□ Hints:
☆ This is tricky. Invite students to consider the $m=-1$ isocline. If that does not help your students recognize a solution, ask them whether solutions can ever cross it from below.
□ Answers:
☆ One way to see this is to focus on an isocline for some negative slope; say $m = -1$. This is another parabola, nested inside the nullcline. The two are asymptotic as $x\rightarrow\infty$
(because $\sqrt{x+1}-\sqrt x\rightarrow0$ as $x\rightarrow\infty$).
☆ It looks as though an an integral curve cannot cross the lower branch of the $-1$ isocline from the outside of that parabola. That would trap the solution between the two parabolas, and
show what we want.
☆ This is almost, but not quite true. Some solutions do cross from the outside. What is the dividing line? When an integral curve crosses the $-1$ isocline, it does so with slope $-1$. If
it comes from the outside, its slope must be greater than the (negative) slope of the tangent line to the isocline at that point. There are places where the slope of the tangent line is
less than $-1$; and at those points integral curves do cross from the outside. But eventually the slope of the tangent line to the $-1$ isocline become greater than $-1$, and from then on
the integral curves cannot cross from the outside. (Using calculus you can see that this occurs at $\left(\frac{5}{4},-\frac{1}{2}\right)$.)
☆ So if a solution $y$ has $y\left(\frac{5}{4}\right)<-\frac{1}{2}$, then it never crosses the $(-1)$ isocline. It is trapped between these two isoclines, and so asymptotic to both.
3.3 Segment 1 Conclusion
There are many other questions stimulated by the Isoclines Mathlet. If you want students to find reasons for them, you had better have a reason in mind yourself! For example, it seems clear enough
from the Mathlet that every integral curve falls into one of the following three categories, according to its behavior for large $x$.
i. The integral curve is asymptotic to $y=-\sqrt x$ (from above) as $x\rightarrow\infty$.
ii. The integral curve is asymptotic (from the left) to a vertical straight line $-$ so the solution fails to exist for large $x$.
iii. The integral curve is asymptotic (from above) to $y=\sqrt x$ as $x\rightarrow\infty$.
The third option has just one member, the separatrix, which separates the other two behaviors from each other.
Also, all integral curves are asymptotic from the right to a vertical straight line (as $x$ becomes smalls).
Some parts of this picture are elementary, but some are harder to prove.
Curves which integral curves can cross from only one side are called fences. Portions of isoclines often form useful fences; but in principle any curve can form a fence. For example, escape of
solutions to infinity can be verified using appropriate fences that are asymptotic to vertical lines.
Pairs of fences that trap solutions are called funnels.
4. Mathlets in Group Work, Segment 2
Segment 2 introduces the concept of Mathlets in Group Work and provides ideas for setting up group work in your course.
Next I provide a couple examples of group work projects involving various Mathlets that could be incorporated into your courses.
4.1 First Project: Beats and the Complex Exponential
This exercise would be appropriate in a class in which the complex exponential plays a big role.
The various formulas on the screen indicate that the green curve in the top window is the graph of the sum of the two sinusoidal functions represented by the red and yellow curves in the bottom
window. The [zoom] button lets you get a closer look at these graphs. Here are some questions for you to think about.
1. What is the angular frequency of the red sinusoid? What is the significance of the parameter $\omega$? How about $\phi$? $A$?
2. Zoom out and set $\omega=1.1$ (or any other number near to 1). Describe in words what you see on the top screen.
3. Assume that $\phi=0$ and $A=1$. Use a trigonometric identity to express the function represented by the green curve in terms of trigonometric functions. Does the result explain what you see?
Invoke the [Envelope] key now. Describe what the word envelope signifies, and write down the functions whose graphs give the envelope in this case.
4. Still assuming that $\phi=0$, watch what happens when $A$ is decreased from $1$ to $0$. Pick a nonzero value of $A$ and watch what happens as $\omega$ is increased. Here is the challenge: Find a
formula for the functions whose graphs are the envelope, for general $A$.
To get you started, it is a good idea to write $\sin(\omega t)=$$\rm{Im}$$e^{i\omega t}$ and use properties of the complex exponential.
Solutions to the project can be found on the theory page of the Beats Mathlet. In setting up this project, I tried to begin by encouraging exploration of the Mathlet.
Think About
What are your comments about the Beats project? Identify the parts of this project that will be useful in one of the classes you teach.
4.2 Second Project: Riemann Sums
A menu selects the function whose graph is drawn. The function is approximated by a step-function, whose integral is read out as Estimate. Various checkboxes produce different approximations.
1. Accept the default function, $f(x)=x^2-2x$, and the default choice of left endpoint evaluation. Let the Mathlet compute the estimates for the integral with $n$ rectangles, where $n=
2. Based on your findings in (1), do you have a prediction for the actual value of the integral? Compute the integral explicitly using an anti-derivative. Compare results.
3. Now select [Min]. This constructs an approximation using the minimal value of $f(x)$ in each interval, as you can see by choosing fairly small values for $n$. What happens to the estimates as $n$
increases? Please explain your observation. Can you predict what the trend is of the estimates using the maximum values in each interval? Check it on the Mathlet.
4. Return to the [Evaluation point] selection, and use the slider which appears next to that checkbox to vary the point at which the function is evaluated. What is the trend as $n$ gets large, using
left endpoint evaluation? How about for right endpoint evaluation? Can you explain what you observe?
5. Now select [Simpson’s Rule]. What is the trend now? Please explain!
The goals here are to get students to think about the various different ways to approximate an integral, and to see that (for this function, at least!) they all lead to the same result. Of course
Simpson’s method gives the correct answer outright because it is approximating a parabola by a parabola. This project could be continued by asking about the rate of convergence. In all the menu
items, Simpson’s rule appears to give the exact answer for very small $n$. The read out [Estimate] shows that this is not true, though, and students could be asked to see that the errors are
quadratic for Simpson’s rule and linear for any of the other approximations. It may appear that the trapezoidal rule should be better than the yellow approximations, but the Mathlet shows that this
is not so, and a good project is to explain why it is really a first order scheme.
Think About
What are your comments regarding the Riemann Sums project? Include a few ideas for ways that this all or part of this project could be incorporated into a course you teach. Are there any changes
that you will make to meet your needs of your course?
4.3 Segment 2 Conclusion
I now leave you with my parting thoughts regarding the use of Mathlets during group work.
5. Post-Module Activity
Practice writing instructions for group work using Mathlets. These instructions might be verbal or the might be a written script. In any case, begin with a specification of the context (what point in
what course) and learning objectives; then give the group work instructions.
Pick two of these three options:
• Pick any one of the Further questions mentioned in the discussion following Module 2, Session 1, in connection with Isoclines, and design some group work around it. Alternatively: pick a
different menu item from Isoclines and write out a group work project.
• Trigonometric Identity is very simple, but still a lot of fun. Notice that green is blue plus yellow. You could play a game with this: Player Alice sets A and $\phi$, and Player Bob has to match
the red curve with the green one by adjusting the a and b sliders. The key of course is the trigonometric identity reflecting the fact that if $(A, \phi)$ are the polar coordinates of $(a,b)$
then $a \cos(\omega t)+b \sin(\omega t)=A \cos(\omega t- \phi)$. Design a group work project around this Mathlet.
• Pick another Mathlet and design a project around some part of it.
These materials are Copyright © 2013, Massachusetts Institute of Technology and unless otherwise specified are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported | {"url":"https://mathlets.org/training/module-2/","timestamp":"2024-11-11T21:14:00Z","content_type":"text/html","content_length":"67999","record_id":"<urn:uuid:81c75add-a47a-47dd-ac86-177dbb01c3b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00802.warc.gz"} |
DC Series Motor, Working, Construction, Working & Applications - The Engineering KnowledgeDC Series Motor, Working, Construction, Working & Applications - The Engineering Knowledge
Hi reader welcomes to another interesting post. in this post, we will have a detailed look at DC Series Motor Working and Applications. With respect to construction or physical dimension, there are
2 basic categories of dc motors first one is self-excited and the second one is a separately excited dc motor. There are further three subtypes of self-excited dc motor which are the Dc shunt motor,
dc compound motor, and dc series motor.
In this post, we will cover the detail of the dc series motor, its construction working, operation, application, and related parameters. So let’s get started with Introduction to DC Series Motor.
Introduction to DC Series Motor
• The DC series Motor is such a dc motor that is used in such applications where a high quantity of torque is needed.
• Its construction is such that armature windings and field windings are linked in a series combination. Such configuration helps to generate a large quantity of torque.
• The number of turns required in field windings is less than the turns of windings used in armature windings.
• In the below figure, the equivalent circuit of a series dc motor can be seen. You can see that in this circuit same current passing through the armature windings, and field windings.
• If we apply KVL to this circuit then we have.
Induced Torque in Series DC Motor
• The terminal characteristics of the series dc motor is like the parameters of the shunt dc motor.
• Its operation is based on that flux has a direct relation with the current flowing through the armature or IA till the point where we get saturation.
• With the increment in the load connected to the motor flux also rises. Due to the increment in flux speed of the motor reduces.
• In a result, the torque-speed curve of the motor is sharply dropping.
• The induced torque for this motor is mentioned here.
• As the flux for this motor has a direct relation with the IA for the saturation point of material used in the motor rotor. The equation for flux will be.
• Here C is constant. The equation for the induced torque of this motor will become.
• From this equation, we can see that torque has a direct relation to the square of the armature. So this motor gives high torque than any other type of motor.
• So it prefers such applications where high torque is required. Like stator motors in cars, elevators, and tractors.
Terminal Characteristics of Series DC Motor
• For discussing the terminal characteristic of this motor we will make assumptions related to the linear magnetization curve and study the effect of saturation in the graph.
• The supposition of the linear magnetization curve indicates that flux in the motor has value.
• This equation will help us to make the torque-speed characteristic curve of the series motor.
• According to KVL law we have.
IA= √Tind/KC
• Putting the value of E=Kw Φ in equation A we have.
VT= KwΦ+√Tind/KC (RA+RS)
• If flux is deleted from the equation so there will be a direct relation between torque and speed of motor.
• Deleting flux from the equation note that.
• We equation for induced torque become.
Tind=K/c Φ^2
Construction of Series DC Motor
• The main parts of this motor are armature windings commutator stator field windings and brushes.
• Its outer part is a stator that is created through the steel and provides cover to the internal parts of the motor. Here in the place of electromagnet poles are also used in some cases.
• The Rotor of this motor comprises of armature windings these windings are linked to the commutators..
• The external power to this motor is given by the carbon brushes than armature windings.
Why Series Motor is Started with No-Load
• The value of the armature current of this motor depends on the load linked to the motor. When there is no load small armature current passes in the motor.
• But if we connect the link large load to the motor when it is starting so motor operates tremendously which causes the damage of motor windings.
Speed control of DC series Motor
• Normally three methods are used for speed control of this motor that are described here.
□ Armature resistance control
□ Tapped Field control
□ Field control
Armature Resistance Control Method
• The circuit for this speed control method can be seen in the below figure.
• From this circuitry, you can observe that the change in resistance speed is changed. As there is a series combination between field and armature windings and current flow also the same.
• So current passing in the motor relies on the value of resistance. So if we increase resistance current will decrease.
• The expression for voltage and speed can be seen as.
N ∝ E[b]/ɸ
• Speed has an inverse relation to the field current.
Field Control Method
• The circuit for this method can be seen here.
• In this circuit field, the diverter is linked to the field windings that is in series combination to the armature.
• The usage of this diverter is to help bypass the quantity of armature current in the motor.
• The changes in IA speed of the motor can be varied similar to the above-mentioned method. With the difference that in this current of armature windings, I bypassed through the passing the field
• For this resistance, if linked to the field windings. If the value of resistance offered by the diverted is high current passing in the field windings.
How does a Series DC Motor Work?
The series dc motor transforms electric energy into mechanic energy on the base of the electromagnetic principle. This DC motor power supply terminal is connected to one of the armature and field
coils. Through applying voltage, power starts in terminals and passes through armature and field windings.
Conductors in this winding are larger, they come with less resistance. As a result, the motor gets high power from the terminals.
Through this high current in the armature and field coils, a strong field is generated that produces high torque in shafts. This starting torque spins the armature and generates high mechanic energy.
Series DC motor comes with their errors and it is their speed based on load, the heavy load has a lower speed of motor armature and load is less, the speed will increase. So through removing load
completely, the motor will work fastly the armature can be damaged.
Why Series Motor is Started with No-Load
The DC series motor must be started with a load since at no load it rotates at high speed.
When a motor is connected to the supply without load, it gets less curent from the supply mains.
That current passes through a series field and armature, speed will increase so back emf can reach to applied voltage value.
The increment in back emf reduces armature current and is called field current.
That causes an increase in speed and back emf. That field continuously decreases and speed continous to increase until the armature generates a centrifugal force that comes out from its shaft and
gets damaged.
Read also:
Which motor should be started with no load?
• Starting shunt motor for high load needed high starting current. To avoid a high starting current, shunt motors not start at high loads and must start without a load
What is the construction and working of DC series motor?
• The construction of dc motor comes with a stator, and rotor part that rotates carry winding. If dc voltage is connected with winding, current flows through it and produces emf
What are the applications of a DC series motor?
• The series motor comes with high starting torque and is used for starting high inertia loads, like trains, elevators, or hoists. Speed features are good for applications like dragline excavators,
where the digging tool moves fast when unloaded but slowly for heavy loads.
What is the working principle of a DC motor?
• The machine that transforms direct current in mechncal work is called a dc motor. DC motor operated on “Faraday’s law of electromagnetic induction“. Faraday’s law of electromagnetic induction
defines that ” if a current-carrying conductor is placed in a magnetic field, it bear force”
What is the main function of DC series motor?
• The AC motor uses AC to transform electrical energy into mechanical energy. There are different types of AC motors, DC motor uses dc currents to transform electrical energy into mechanical
energy. The main benefit of using this is that helps to control speed and has less coverage area.
What is the construction and working of motors?
• The motor uses the phenomena of the magnetic force on the current-carrying conductor. The direction of force is based on current flow and magnetic field. But we can fix the magnetic field and
produce torque on a loop based on current flow. So electric motor made.
What is an application for a DC motor?
• Automotive Industries.
• Hoists & Cranes operations.
• Industrial tools power supply.
• Lifts.
• Air compressor.
• Winching system.
• Elevators.
• Electric traction.
What application is a DC series motor best suited for?
Motor Application
DC series motor Traction system, Cranes, air compressors
DC shunt motor Lathe Machines, Conveyors, Weaving Machines, Centrifugal Pumps, Blowers, Spinning machines
DC compounded motor Presses, Rolling Mills Shears, Elevators,
Stepper motor 3D printing equipment,, CNC milling machines Small robotics
What are the advantages of DC series motors?
• The benefits of a dc series motor are a starting torque with a high speed of about 500 percent, and a maximum momentary operating torque is about 400 percent. and speed regulation is variable and
high for no-load
What is the starting method of a DC series motor?
• Different methods of starting dc series motors exist like two-point starters. The simple starter used for small series dc motor. it comes with two contact points, one for supply and the other
common to field and armature coils. The manual levels is engaged to start the motor directly.
What is the construction of DC series motor?
• The components of the motor are the rotor, commutator, stator axle, field winding, and brushes.
• The fixed component of the motor is the stator and made with two electromagnet pole parts.
What are the applications of DC motor?
• Main applications are, rolling mills elevators, steel mills, locomotives, and excavators. For other rotating machines, DC motors result of the interaction of two magnets.
What is the principle of DC series motor?
• DC motor is an electrical machine that transforms electrical energy into mechanical. The main working of a DC motor is that when the current carrying conductor put in the field bears mechanical
What is the difference between DC motor and DC generator?
• DC motors are devices that transform electrical energy into mechanical energy. DC generator converts mechanical energy into electrical energy.
How does a DC motor work step by step?
• The DC motor uses a static set of agents in the stator and coil of wire with curent operating through it produces a field aligned with the center of the coil. One or more windings of insulated
wire are wounded about the core of the motor for concentrating magnetic field.
So that is a detailed post about the series DC motor if you have any further queries ask in the comments. Thanks for reading. Have a nice day.
Leave a Reply Cancel reply
Author: Henry
I am a professional engineer and graduate from a reputed engineering university also have experience of working as an engineer in different famous industries. I am also a technical content writer my
hobby is to explore new things and share with the world. Through this platform, I am also sharing my professional and technical knowledge to engineering students. Follow me on: Twitter and Facebook.
Related Posts | {"url":"https://www.theengineeringknowledge.com/introduction-to-dc-series-motor/","timestamp":"2024-11-10T01:39:14Z","content_type":"text/html","content_length":"231176","record_id":"<urn:uuid:a7b3846d-a13d-44b0-9197-de34556ea65e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00436.warc.gz"} |
Step-by-step Evolution of Young’s Double-slit Interference Fringes Using Boundary Diffraction Wave Theory
Step-by-step Evolution of Young’s Double-slit Interference Fringes Using Boundary Diffraction Wave Theory
Raj Kumar^*
CSIR - Central Scientific Instruments Organisation, Chandigarh 160030, India
Article Information
Identifiers and Pagination:
First Page:
Last Page:
Publisher Id: PHY-3-122 DOI: 10.2174/1874843001603010122
Article History:
Received Date: 11/12/2015
Revision Received Date: 13/5/2016
Acceptance Date: 28/10/2016
Electronic publication date: 30/11/2016
Collection year: 2016
© Raj Kumar; Licensee Bentham Open open-access license:
This is an open access article licensed under the terms of the Creative Commons Attribution-Non-Commercial 4.0 International Public License (CC BY-NC 4.0)
), which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.
Present work reports formation of Young’s double-slit interference fringes using the Young's own theory of diffraction called the theory of boundary diffraction wave. Theory demands that double-slit
interference fringes are generated due to superposition of boundary diffraction waves originating from the edges of the slits due to their physical interaction with the incident light. The
theoretical development is further verified with the experimental observations.
Keywords: Boundary diffraction wave, Double-slit interference fringes, Interferometry, Young’s double slit experiment.
Diffraction and interference are building blocks of physical optics. These play important role in applied optics including optical instrumentation, imaging and the emerging areas of nanophotonics,
plasmonics etc. Thomas Young reported his famous double-slit experiment in 1802 to conclusively demonstrate the wave nature of light [1]. This experiment is widely used to study spatial coherence of
light sources [2]. Application of double-slit experiment into the realm of quantum mechanics has resulted into discovery of new phenomena like quantum erasure and micromaser which-path detector [3, 4
]. The experiment had remained a talking point in 19^th century among the leading physicists Einstein and Bohr regarding uncertainty and complementarity principles of quantum mechanics [5], where
Einstein proposed that the lateral kick imparted by a photon to an interference screen could be used to identify which slit the photon travelled through on its way to the screen implying that
generation of interference fringes is a measure of photon’s momentum. Today also the experiment is as relevant and exciting as it had been in earlier times and has found many applications in new
areas of research like superresolution [6], plasmonics and nanophotonics [7-9]. In view of vast applications of this basic interferometer in established as well as in emerging areas of basic and
applied nature it is necessary to explore the process of formation of interference fringes. It is well known that interference fringes are generated by superposition of two or more coherent beams of
light. In present case two beams are generated by division of wavefront of incident beam due to presence of two small apertures (slits). Control experiments, which are performed by having a constant
control over the system performance parameters, play important role in understanding the evolution of Young’s double-slit interference fringes. Many control experiments have been reported in the
literature for demonstrating various features of the Young’s double slit interference pattern [3, 4, 9]. In one such experiment [10] control over the individual slits to observe probability
distributions from both single and double-slits, and the build-up of a diffraction pattern at single electron detection rates to achieve the full realization of Feynman’s thought experiment is
reported. Here a physical mask was used on slits to observe the diffraction phenomenon. The final build-up of the pattern took about 2 h. Most of the control experiments are performed using single
particle sources (electrons, photons) and detectors and thus involve complex and costly systems. To our knowledge there is no any control experiment which can demonstrate this process in simple
manner using the conventional light source and detectors. Only simulation work is reported in one of the papers where Iterative Fresnel Integrals Method is used for demonstrating the combination of
interference effects with Fresnel diffraction in the simulation images [11].
Generally the interaction of incident light with the slits is explained by using Fresnel's theory of diffraction which demands that diffraction patterns/effects are generated due to superposition of
Huygens secondary wavelets starting from every point of space located in the aperture. Here it may be noted that Huygens proposed his theory based on the existence of aether (the material supposed to
fill the region of the universe above the terrestrial sphere) so that aether particles were responsible for generation of secondary wavelets. But existence of aether has been ruled out long ago
through Michelson-Morley experiment [12]. Thus, again interest is gaining towards Young's theory of diffraction [13-18]. According to Young's theory of boundary diffraction wave, diffraction patterns
are a result of interference of a direct wave unaffected by the diffracting aperture and another wave generated by interaction of incident light with the edges of the diffracting apertures [19].
Recently the theory has been successfully applied to explain various phenomena [20-23].
Present paper reports investigations to make it explicit that Young’s double-slit fringes are formed due to superposition of boundary diffraction waves originating at the edges of individual slit.
Beams from individual slits, which generate double-slit interference fringes on superimposition, are in fact formed by superposition of two boundary diffraction waves from two edges of the slit. This
process is demonstrated experimentally by using diffraction through single slit as well as through double-slit by focusing incident laser beam on the slits. In this situation the diffraction patterns
of individual slits in the double-slit setup are well separated in space. Further, as one move slits axially away from the focus, beams of light passing through individual slit start coming closer to
each other. When a portion of these beams is superimposed it generates interference fringes in overlap region and on complete superposition of beams through two slits one obtains very nice Young’s
double-slit interference fringes.
Let a diverging beam of light with complex amplitude distribution
is incident on the diffracting aperture. According to Fresnel-Kirchhoff theory, diffracted field at observation point P[0 ] in terms of incident wave field and its first derivatives at an arbitrary
closed surface surrounding P[0 ] is
Where ∂/∂n denotes differentiation along the outward normal, Q is a point situated in the diffracting aperture Σ and exp(iks/s) is Green’s free space function, r is distance between source of light
and a point Q on diffracting aperture and s is distance between aperture point Q and observation point P[ 0]. Maggi and Rubinowicz converted double integrals used in above formulation into a line
integral using Stoke’s theorem, giving [19]
represents Young’s boundary diffraction wave generated at the edge of the aperture by it's interaction with incident light. Recently a quantitative criterion has been developed for classifying
whether diffraction pattern is of Fresnel or Fraunhofer type [24] giving
Fraunhofer rigion γ ≤ 0.8
Fresnel rigion γ > 0.8
Using parameters of our experimental setup the slit-width (1[X]) = 20 micron, wavelength of laser (λ) = 632.8nm, r = 2mm, and s = 30mm value of γ = 0.0019. Thus, we receive Fraunhofer diffraction
pattern at the detector's surface.
Using the method of stationary phase and some approximations the expression of boundary diffraction wave becomes [25]
Here dl is an infinitesimal element situated on illuminated edge Γ of the diffracting aperture, λ is wavelength of light used and ϕ is polar coordinate. In case of single slit having width l[x] the
diffracted field can be written as sum of two boundary diffraction waves arising from each edge of the slit, giving
This equation represents amplitude distribution at observation screen resulting due to single slit diffraction. Here effect of geometrical beam U[G], which is small due to small size of slit, has
been neglected. Here two boundary diffraction waves staring at each edge of the slit interfere to generate the single slit diffraction pattern having different diffraction orders. Diffracted light
propagates along the direction of incident beam with an additional effect of diverging out symmetrically with respect to initial direction of propagation as given by Keller’s geometrical theory of
diffraction [26]. If slit width is small than boundary diffraction waves originating from two edges of the slit interfere to generate a single fringe, which forms the beam of light passing through
the slit. This is also evident from our earlier discussion in a previous paper where it was demonstrated by using a Lloyd mirror on boundary diffraction wave from a knife-edge [21]. When incident
beams at individual slits of the double-slit system enclose large angle (in our case when slits are placed in proximity of focus), light striking at individual slits generates their own single slit
diffraction pattern which propagates in different directions (i.e. beams from individual slits are spatially separated) and thus individual slit diffraction pattern can be observed. When angle
between beams incident on slits decreases (this can be experimentally realized by moving slits away from focus) separation of light striking on individual slit also decreases and consequently
diffracted patterns come closer to each other. This is schematically shown in Fig. (1).
│ │Fig. (1). Change in angle θ with change in position of double-slit with respect to laser focal point F. When angle becomes small diffraction patterns from the two slits superimpose to generate │
│ │Young’s double-slit interference fringes. │
Angle θ between incident beams on slits is related with separation of individual slit diffraction patterns ‘x’ as:
Here z is distance of double-slits (having width l[x] and separation d) from focal point of the laser. When θ becomes small, separation between diffracted patterns also become correspondingly small
and ultimately diffracted light from both the slits superimposes resulting in Young’s double-slit interference fringes with an amplitude distribution:
This equation of field distribution derived using boundary diffraction wave theory is in full agreement with that obtainable with Fourier optics treatment as explained in reference [25].
A schematic representation of experimental arrangement is shown in Fig. (2). A He-Ne laser at 632.8nm wavelength is expanded and spatially filtered using the spatial filtering assembly SF (pin hole =
5μm and microscope objective 45 x). Lens L is used to focus the expanding laser beam at point F. Light diverging out from focus F in the direction of beam propagation is used to illuminate a single
slit of opening 100μm and a Young’s double-slits (slit width = 20μm and spacing between slits = 250μm), sequentially. Slits were mounted on a linear translation stage with travel length of 25 mm and
resolution of 0.01mm. Position of slits and of translation stage is aligned such that at 25mm marking on the translation stage, slits are at focus F. Now slits are translated in the direction of
propagation of laser beam such that diverging light falling on individual slits gets diffracted from these and forms individual single slit diffraction patterns. In case of double-slit these
diffraction patterns of individual slits are well separated in space on the observation screen. As slits are further translated the angle between light beams, which strike on individual slits,
decreases consequently diffraction patterns of individual slits come closer to each other. Finally, individual slit diffraction patterns superimpose on each other, generating the Young’s double-slit
interference fringes. Relation between position of double-slit relative to laser focal point F and separation between individual slit diffracted light is shown in Fig. (1). The diffraction patterns
are recorded with a CMOS sensor (Lumenera: Lu120MB) placed at a distance 30 mm from the slits. Complete system including slits and sensor is mounted on single platform which infact is installed on
the main linear translation stage.
│ │Fig. (2). Schematic representation of experimental arrangement. │
Experiments are performed to study the step-by-step evolution of single slit diffraction pattern and generation of Young’s double-slit interference fringes due to interaction of light diffracted from
the individual slits of the double-slit system. Initially slits are positioned symmetrically with respect to laser focus F and near to it. This position is determined by observing that at this
position most of the laser beam is transmitted through the single slit producing uniform illumination on the observation screen and in case of double slits most of the laser beam is blocked by opaque
strip between the slits and thereby only weak diffracted light from individual slits passes to observation screen and is well separated in space. Now translation, in steps of 2mm, is given to the
slits in the direction of propagation of laser beam and diffraction patterns are recorded, a few of these are shown in Fig. (3a-c) for single slit and in Fig. (4a-c) in case of diffraction from
│ │Fig. (3). Photographs of single slit diffraction patterns taken step-by-step by moving the slit away from laser focal point. (a) slit at 5 mm from F (b) slit at 11 mm from F and (c) slit at 25 mm│
│ │from F. │
Fig. (3a) shows a photograph of diffraction pattern generated by interaction of incident beam with the edges of a single slit which is placed at z = 5 mm from the focused image of laser light F. Here
since slit is quite close to the focus of laser beam so most of the incident light passes through the slit resulting in the central bright fringe of large width in the slit diffraction pattern and a
small amount of light also strikes edges of the slit and thus gets diffracted from these. This interaction of light with slit edges generates the boundary diffraction waves at each edge which further
propagate and interfere with boundary diffraction wave generated from the other edge of the slit to generate the interference fringes on either side of the central maxima. These fringes are known as
higher diffraction orders of the slit. If distance between slit and the laser focus is increased, only a small portion of incident light passed through the slit and more light gets diffracted from
the edges of the slit. This results in decrease in width of central bright fringe and increase of light in higher diffraction orders as is seen in diffraction patterns generated by the slit at
distances 11mm and 25mm from the laser focus in Fig. (3b and c) respectively. If number of closely placed slits is increased than interaction between boundary diffraction waves generated from
individual edge of the slits take place. For two properly spaced slits this interaction results in Young’s double slit interference fringes as discussed next and if slits are further increased then
it generates the effect of a diffraction grating.
Fig. (4a) shows a photograph of diffraction pattern through double-slit placed at z = 5 mm from F. Here individual slit diffraction patterns are well separated in space. As slits are further
translated away from the focus, angular separation between incident beams (two points of same diverging beam) on individual slit reduces and hence according to Keller’s geometrical theory of
diffraction [24], separation between diffraction patterns of the slits also decreases as shown in Fig. (4b) where distance z is 9 mm. Finally central maxima of diffraction patterns of both the slits
superimpose. In this situation diffracted light from individual slits interfere, generating the Young’s double-slit interference fringes as shown in Fig. (4c) corresponding to z = 25 mm. In
interference pattern, position of constructive and destructive fringe depends on the phase of interacting photons at that point. If photons are arriving in phase they generate constructive (bright)
fringe and if they are out of phase destructive (dark) fringe is produced. At other positions where there is different phase angle between the photons, intensity varies according to their phase
relationship. Thus, phase variations between interacting photons result in formation of sinusoidal interference fringe pattern. Here it may be noted that in Fig. (4b) interference fringes are
generated only in the area where diffraction patterns from both the slits superimpose while in other areas only diffracted light is available but interference fringes are not observable. In Fig. (4c)
where both the diffraction patterns almost superimpose, interference fringes are formed along all the width of the slit diffraction pattern. Here only central maxima of the single slit diffraction
pattern is observable. Generation of diffraction pattern of individual slits and formation of interference fringes by superposition of individual slit diffraction patterns clearly demonstrate that
two diffracted lights generated from a common incident coherent beam of light could interact with each other. Generation of interference fringes by superposition of boundary diffraction waves
originating from individual slits is also in agreement with our earlier investigations on Young’s boundary diffraction wave [22]. As individual slit diffraction pattern is a result of superposition
of two boundary diffraction waves emanating from two edges of the slit the double-slit interference fringes are results of superposition of four boundary diffraction waves. But boundary diffraction
waves from two edges of a slit combine constructively resulting in a single beam of light and hence Young's double-slit fringes are in fact fringes due to interference of two beams of light.
│ │Fig. (4). Photographs of diffraction patterns from Young’s double- slits taken step-by-step by moving double-slits away from laser focal point. (a) slits are 5 mm from F and thus diffraction │
│ │patterns are well separated in space (b) slits at 9 mm from F shows interference fringes only in small central overlapped region and (c) slits at 25 mm from F results in superposition of │
│ │individual slit diffraction patterns to generate Young’s fringes. │
The theory of Young's boundary diffraction wave is used to explain the formation of double-slit interference fringes. According to this theory individual slit gives rise to two boundary diffraction
waves corresponding to interaction of incident light with two edges of the slit. Due to small separation between the edges these boundary diffraction waves superimpose to generate the single slit
diffraction pattern. Further superposition of light from individual slit generates the well known double-slit interference fringes. An experimental procedure is developed to observe step-by-step the
process of evolution of double-slit interference fringes due to superposition of light passing through individual slit. These investigations may be helpful in developing a better understanding about
the phenomenon of fringe formation in Young’s double-slit interferometer and hence may provide a clue in solving the paradox of complementarity and uncertainty principles in quantum mechanics.
The author confirms that this article content has no conflict of interest.
Author thanks Mr. DP Chhachhia for interesting discussion and Mr Omendra Singh and Mr Naresh Sharma for technical help during the work. I also thank Prof. B C Chaudhary of NITTTR, Chandigarh for
lending his double-slit used in this experiment. This work is financially supported by Council of Scientific & Industrial Research, New Delhi.
[1] Young T. Bakerian lecture on the mechanism of the theory of light and colour. Philos Trans R Soc Lond 1802; 92: 12-48.
[2] Singh N, Vora HS. On the coherence measurement of a narrow bandwidth dye laser. Appl Phys B 2013; 110: 483-9.
[3] Scully MO, Drühl K. Quantum eraser: A proposed photon correlation experiment concerning observation and “delayed choice” in quantum mechanics. Phys Rev A 1982; 25: 2208-13.
[4] Aharonov Y, Zubairy MS. Time and the quantum: erasing the past and impacting the future. Science 2005; 307(5711): 875-9.
[5] Wheeler JA, Zurek WH. Quantum Theory and Measurement. USA: Princeton University Press 1984; p. 945.
[6] Jun X, Feng H, De-Zhong C, Hong-Guo L, Xu-Juan S, Kai-Ge W. Super-resolution of interference pattern with independent laser beams. Chin Phys Lett 2005; 22: 2824.
[7] Gan CH, Gbur G, Visser TD. Surface plasmons modulate the spatial coherence of light in Youngs interference experiment. Phys Rev Lett 2007; 98(4): 043908.
[8] Barnes WL, Dereux A, Ebbesen TW. Surface plasmon subwavelength optics. Nature 2003; 424(6950): 824-30.
[9] Kocsis S, Braverman B, Ravets S, et al. Observing the average trajectories of single photons in a two-slit interferometer. Science 2011; 332(6034): 1170-3.
[10] Bach R, Pope D, Liou SH, Batelaan H. Controlled double-slit electron diffraction. New J Phys 2013; 15: 033018.
[11] Al-Saiari FH, Rahman SM, Abedin KM. Computer simulation of Fresnel diffraction from triple apertures by iterative Fresnel integrals method. Photonics Optoelectron 2012; 1: 33-42.
[12] Michelson AA, Morley EW. On the relative motion of the earth and the luminiferous ether. Am J Sci 1887; 34: 333-45.
[13] Miyamoto K, Wolf E. Generalization of the maggi-rubinowicz theory of the boundary diffraction wave—part I and part II. J Opt Soc Am 1962; 52: 615-36.
[14] Kumar R, Kaura SK, Chhachhia DP, Aggarwal AK. Direct visualization of Young’s boundary diffraction wave. Opt Commun 2007; 276: 54-7.
[15] Kumar R. Structure of boundary diffraction wave revisited. Appl Phys B 2008; 90: 379-82.
[16] Umul YZ. The theory of the boundary diffraction wave. In: Advances in Imaging and Electron Physics. USA: Academic Press 2011; 165: pp. 265-82.
[17] Piyadasa CK. Detection of a cylindrical boundary diffraction wave emanating from a straight edge by light interaction. Opt Commun 2012; 285: 4878-83.
[18] Kumar R, Chhachhia DP. Formation of circular fringes by interference of two boundary diffraction waves using holography. J Opt Soc Am A Opt Image Sci Vis 2013; 30(8): 1627-31.
[19] Born M, Wolf E. Principles of Optics. Cambridge, UK: Cambridge University Press 1999; p. 499.
[20] Kumar R, Chhachhia DP, Aggarwal AK. Folding mirror schlieren diffraction interferometer. Appl Opt 2006; 45(26): 6708-11.
[21] Kumar R. Extraordinary optical transmission by interference of diffracted wavelets. Opt Appl 2010; 40: 491-9.
[22] Kumar R. Diffraction Lloyd mirror interferometer. J Optics (India) 2010; 39: 90-101.
[23] Piksarv P, Bowlan P, Lõhmus M, Valtna-Lukner H, Trebino R, Saari P. Diffraction of ultrashort Gaussian pulses within the framework of boundary diffraction wave theory. J Opt 2012; 14: 015701.
[24] Rueda EA, Medina FF, Barrera JF. Diffraction criterion for a slit under spherical illumination. Opt Commun 2007; 274: 32-6.
[25] Ganci S. Boundary diffraction wave theory for rectilinear. Eur J Phys 1997; 18: 229-36.
[26] Keller JB. Geometrical theory of diffraction. J Opt Soc Am 1962; 52: 116-30. | {"url":"https://openphysicsjournal.com/VOLUME/3/PAGE/122/","timestamp":"2024-11-14T21:58:52Z","content_type":"text/html","content_length":"77483","record_id":"<urn:uuid:f83935ec-7e54-47b1-8981-a52bd5ea7851>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00023.warc.gz"} |
Data visualization for survey data – Jihong Z. - Play Harder and Learn Harder
Data visualization for survey data
Many tutorials online are about general data visualization. This post aims to showcase some tricks for survey data
1 Question
I am looking for R package than can analyze big data of survey.
2 R package - survey
The survey package in R is designed specifically for analysis of data from complex surveys. It provides functions for descriptive statistics and general regression models for survey data that
includes design features such as clustering, stratification, and weighting.
Here are some of the core features of the survey package:
1. Descriptive Statistics: The package provides functions for computing descriptive statistics on survey data, including mean, total, and quantiles.
2. Regression Models: The package provides a variety of model fitting functions for binary and multi-category response, count data, survival data, and continuous response.
3. Design Effects: It allows calculation of design effects for complex survey designs.
4. Post-stratification and Raking: The package allows for adjusting the sampling weights to match known population margins.
5. Subpopulation Analysis: It includes functions for correctly handling analyses that are limited to a subset of the population (a subpopulation).
6. Variance Estimation: The survey package supports multiple methods of variance estimation, including Taylor series linearization, replication weights, and subbootstrap.
Remember that before you can use these functions, you will need to define a survey design object that specifies the features of your survey’s design (like the sampling method, strata, clustering, and
Here’s an example of how you might use it to calculate the mean of a variable from a survey:
variables Formula or data frame specifying the variables measured in the survey. If NULL, the data argument is used.
ids Formula or data frame specifying cluster ids from largest level to smallest level, ~0 or ~1 is a formula for no clusters.
probs Formula or data frame specifying cluster sampling probabilities
Please replace mydata, weight, and variable with your actual data frame, weight column, and the variable you’re interested in, respectively.
Remember, working with survey data can be complex due to the design features of surveys. The survey package in R provides a robust set of tools for dealing with this complexity.
3 An empirical example
The example I used here is a tody data exacted from a real data about eating disorders. The sample size is 500.
The measurement data contains 12 items, each ranging from 0 to 3. The demographic data contains 6 variables: age, gender, race, birthplace, height, weight. The very first thing is to visualize the
characteristics of the samples to have a big picture of respondents.
knitr::opts_chunk$set(echo = TRUE, message=FALSE, warnings=FALSE, include = TRUE)
library(formattable) # format styles of table
options(knitr.kable.NA = '')
mycolors = c("#4682B4", "#B4464B", "#B4AF46",
"#1B9E77", "#D95F02", "#7570B3",
"#E7298A", "#66A61E", "#B4F60A")
softcolors = c("#B4464B", "#F3DCD4", "#ECC9C7",
"#D9E3DA", "#D1CFC0", "#C2C2B4")
mykbl <- function(x, ...){
kbl(x, digits = 2, ...) |> kable_styling(bootstrap_options = c("striped", "condensed")) }
3.1 Descriptive statistics
We can use multiple R tools for descriptive statistics. bruceR is one of them.
freqTblVars = c("gender", "race", "birthplace")
freqTable <- function(tbl, var) {
tbl |> as.data.frame() |>
tibble::rownames_to_column("Levels") |>
dplyr::mutate(Variable = var)
freqTableComb = NULL
for (var in freqTblVars) {
tbl = bruceR::Freq(dplyr::select(description, gender:birthplace), varname = var)
freqTableComb = rbind(freqTableComb, freqTable(tbl = tbl, var = var))
freqTableComb <- freqTableComb |>
Or we can use survey package for descriptive analysis
dexample = svydesign(ids = ~1,
data = datList$measurement)
## summay statistics for all measurement indicators
vars <- colnames(datList$measurement)
svymean(make.formula(vars), design = dexample, na.rm = TRUE)
svytotal(make.formula(vars), design = dexample, na.rm = TRUE)
3.2 Stacked barplot for survey data responses
survey = datList$measurement
survey <- survey |>
mutate(ID = 1:nrow(survey)) |>
mutate(across(starts_with("EDEQS"), \(x) factor(x, levels = 0:3))) |>
pivot_longer(starts_with("EDEQS"), names_to = "items", values_to = "values") |>
group_by(items) |>
dplyr::count(values) |>
dplyr::mutate(perc = n/sum(n) * 100)
p = ggplot(survey) +
geom_col(aes(y = factor(items, levels = paste0("EDEQS", 1:12)),
x = perc,
fill = values),
position = position_stack(reverse = TRUE)) +
labs(y = "", x = "Proportion (%)", title = "N and proportion of responses for items")
p = p + geom_text(aes(y = factor(items, levels = paste0("EDEQS", 1:12)),
x = perc, group = items,
label = ifelse(n >= 50, paste0(n, "(", round(perc, 1), "%)"), "")),
size = 3, color = "white",
position = position_stack(reverse = TRUE, vjust = 0.5))
p = p + scale_fill_manual(values = mycolors)
We can clearly identify item 7 has highest proportion of level 0, and needed to be theoretically justified.
Back to top | {"url":"https://jihongzhang.org/posts/2023-07-04-data-visualization-for-survey-data/","timestamp":"2024-11-02T13:40:08Z","content_type":"application/xhtml+xml","content_length":"106682","record_id":"<urn:uuid:7a485236-28e8-426a-825b-9eaba1a6a3fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00173.warc.gz"} |
seminars - Optimal Retirement with Long-Run Income Risk
We develop a general retirement framework in which optimal retirement decision can be adjusted by both short-run and long-run income risk. The generalized framework is found to alter both
quantitative and qualitative features of existing retirement models.
Having generalized the retirement framework with the long-run income risk, we then investigate its effect on the optimal decision to buy more or fewer risky assets. Taking on more risk in the stock
market by adjusting retirement timing is no longer applicable with the long-run income risk.
Rather, retirement flexibility makes the optimal portfolio invest less in the stock market than without the long-run income risk. Finally, we can explain the non-participation puzzle and empirically
plausible portfolio share exhibiting an increasing and concave trend in wealth. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&page=46&document_srl=803295","timestamp":"2024-11-04T08:01:11Z","content_type":"text/html","content_length":"45365","record_id":"<urn:uuid:6104f947-a645-43c9-8822-cf979936bbd5>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00389.warc.gz"} |
Mark Range || CodeTo.win
Asad studies B.Sc. in CSE program at BUBT. Today their programming teacher returns their mid term examination scripts. There are varieties types of numbers that students got in the examination. The
teacher wants to find the mark range of numbers (Maximum and Minimum) that students got in the examination. But it is very difficult to find the mark range manually. Teacher gives assignment to write
a program that can be used to find the mark range.
Input Format
First line contain number of test case 1<=T<=100.For each case the input contains two lines. The first line contains a number N (1<=N<=100) denotes the number of students in the class. The second
line contains N different positive numbers that denotes the marks students got in the examination.
Output Format
Output contains two lines. The first line contains Lower limit and the second line contains Upper limit of numbers. see sample input and output .
Lower Limit: 5 Upper Limit: 25 Lower Limit: 0 Upper Limit: 33
Language Time Memory
GNU C 11 1s 512MB
GNU C++ 14 1s 512MB
GNU C++ 11 1s 512MB
Login To Submit | {"url":"https://codeto.win/problem/1020","timestamp":"2024-11-06T01:27:33Z","content_type":"text/html","content_length":"9566","record_id":"<urn:uuid:5387945d-3e23-44af-8d27-0fc0cfd9bec0>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00242.warc.gz"} |
Fluids at Rest
A barotropic, compressible fluid at rest is governed by the statics equation,
where z is the height above an arbitrary datum, and g is the gravity acceleration constant (9.81 m/s^2; 32.2 ft/s^2). This equation describes the pressure profile of the atmosphere, for example.
For an incompressible fluid, the statics equation simplifies to,
This equation describes the pressure profile in a body of water, or in a manometer.
If the fluid is compressible but barotropic, then the density and the pressure can be integrated into the "pressure per density" function
Note that the equation at the top of the page can still be applied though, as it makes no assumption on the fluid's equation of state.
Derivation from Navier-Stokes
The Navier-Stokes equation for a fluid at rest reduce to,
Rearranging, and assuming that the body force b is due to gravity only, we can integrate over space to remove any vector derivatives,
For the barotropic fluid case, the derivation can be repeated in a fashion similar to that of Bernoulli, | {"url":"https://www.efunda.com/formulae/fluids/fluid_statics.cfm","timestamp":"2024-11-14T23:53:32Z","content_type":"text/html","content_length":"24542","record_id":"<urn:uuid:0a3ad69f-402d-40bc-af2b-d8beb69b1449>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00000.warc.gz"} |
Content Markup
4.2 Strict Content MathML
4.2.1 The structure of MathML3 Content Expressions
MathML content encoding is based on the concept of an expression tree built up from
As a general rule, the terminal nodes in the tree represent basic mathematical objects such as numbers, variables, arithmetic operations and so on. The internal nodes in the tree generally represent
some kind of function application or other mathematical construction that builds up a compound object. Function application provides the most important example; an internal node might represent the
application of a function to several arguments, which are themselves represented by the terminal nodes underneath the internal node.
This section provides the basic XML Encoding of content MathML expression trees. General usage and the mechanism used to associate mathematical meaning with symbols are provided here. [mathml3cds]
provides a complete listing of the specific Content MathML symbols defined by this specification along with full reference information including attributes, syntax, and examples. It also describes
the intended semantics of those symbols and suggests default renderings. The rules for using presentation markup within content markup are explained in Section 5.4.2 Presentation Markup in Content
4.2.2 Encoding OpenMath Objects
Strict Content MathML is designed to be and XML encoding of OpenMath Objects (see [OpenMath2004]), which constitute the semantics of strict content MathML expressions. The table below gives an
element-by-element correspondence between the OpenMath XML encoding of OpenMath objects and strict content MathML.
Note that with this correspondence, strict content MathML also gains the OpenMath binary encoding as a space-efficient way of encoding content MathML expressions.
4.2.3 Numbers
The cn element is the MathML token element used to represent numbers. The supported types of numbers include integers, real numbers, double precision floating point numbers, rational numbers and
complex numbers. Where it makes sense, the base in which the number is written can be specified. For most numeric values, the content of a cn element should be either PCDATA or other cn elements.
The permissible attributes on the cn are:
│Name│ Values │Default│
│type│"integer" | "real" | "double" | "e-notation," | "rational" | "complex-cartesian" | "complex-polar" │real │
│base│number │10 │
The attribute type is used to specify the kind of number being represented. The pre-defined values are given in the table above. Unless otherwise specified, the default "real" is used.
The attribute base is used to specify how the content is to be parsed. The attribute value is a base 10 positive integer giving the value of base in which the PCDATA is to be interpreted. The base
attribute should only be used on elements with type "integer" or "real". Its use on cn elements of other type is deprecated. The default value for base is "10".
Each data type implies that the content be of a certain form, as detailed below.
An integer is represented by an optional sign followed by a string of one or more "digits". How a "digit" is interpreted depends on the base attribute. If base is present, it specifies the base
for the digit encoding, and it specifies it base 10. Thus base='16' specifies a hexadecimal encoding.
When base > 10, letters are used in alphabetical order as digits. For example,
<cn base="16">7FE0</cn>
encodes the number written as 32736 in base ten.
When base > 36, some integers cannot be represented using numbers and letters alone and it is up to the application what additional characters (if any) may be used for digits. For example,
<cn base="1000">10F</cn>
represents the number written in base 10 as 1,000,015. However, the number written in base 10 as 1,000,037 cannot be represented using letters and numbers alone when base is 1000.
A real number is presented in radix notation. Radix notation consists of an optional sign ("+" or "-") followed by a string of digits possibly separated into an integer and a fractional part by a
"decimal point". Some examples are 0.3, 1, and -31.56. If a different base is specified, then the digits are interpreted as being digits computed to that base (in the same was as described for
type "integer").
This type is used to mark up those double-precision floating point numbers that can be represented in the IEEE 754 standard. This includes a subset of the (mathematical) real numbers, negative
zero, positive and negative real infinity and a set of "not a number" values.
The content of a cn element may be PCDATA (representing numeric values as described below), a infinity element (representing positive real infinity), a minfinity element (representing negative
real infinity) or a notanumber element.
If the content is PCDATA, it is interpreted as a real number in scientific notation. The number then has one or two parts, a significand and possibly an exponent. The significand has the format
of a base 10 real number, as described above. The exponent (if present) has the format of a base 10 integer as described above. If the exponent is not present, it is taken to have the value 0.
The value of the number is then that of the significand times ten to the power of the exponent.
A special case of PCDATA content is recognized. If a number of the above form has a negative sign and all digits of the signifcand are zero, then it is taken to be a negative zero in the sense of
the IEEE 754 standard.
This type is deprecated. It is recommended to use double or real instead.
A real number may be presented in scientific notation using this type. Such numbers have two parts (a significand and an exponent) separated by a <sep/> element. The first part is a real number,
while the second part is an integer exponent indicating a power of the base. For example, 12.3<sep/>5 represents 12.3 times 10^5. The default presentation of this example is 12.3e5.
A rational number is given as two integers giving the numerator and denominator of a quotient. These should themselves be given as nested cn elements.
For backward compatibility, deprecated usage allows the two integers to be given as PCDATA separated by <sep/>. If a base is present in this deprecated use, it specifies the base used for the
digit encoding of both integers.
A complex cartesian number is given as two numbers giving the real and imaginary parts. These should themselves be given as nested cn elements. As for rational numbers, the deprecated use of <sep
/> is also allowed.
A complex polar number is given as two numbers giving the magnitude and angle. These should themselves be given as nested cn elements. As for rational numbers, the deprecated use of <sep/> is
also allowed.
This type was deprecated in MathML 2.0 and is now no longer supported.
4.2.4 Symbols and Identifiers
The notion of constructing a general expression tree is essentially that of applying an operator to sub-objects. For example, the sum "x+y" can be thought of as an application of the addition
operator to two arguments x and y. And the expression "cos(π)" as the application of the cosine function to the number π.
In Content MathML, elements are used for operators and functions to capture the crucial semantic distinction between the function itself and the expression resulting from applying that function to
zero or more arguments. This is addressed by making the functions self-contained objects with their own properties and providing an explicit apply construct corresponding to function application. We
will consider the apply construct in the next section.
In a sum expression "x+y" above, x and y typically taken to be "variables", since they have properties, but no fixed value, whereas the addition function is a "constant" or "symbol" as it denotes a
specific function, which is defined somewhere externally. (Note that "symbol" is used here in the abstract sense and has no connection with any presentation of the construct on screen or paper).
4.2.4.1 Content Identifiers
Strict Content MathML3 uses the ci element (for "content identifier") to construct a variable, or an identifier that is not a symbol. Its PCDATA content is interpreted as a name that identifies it.
Two variables are considered equal, iff their names are in the respective scope (see Section 4.2.6 Bindings and Bound Variables for a discussion). A type attribute indicates the type of object the
symbol represents. Typically, ci represents a real scalar, but no default is specified.
│Name│values │ default │
│type│string │unspecified │
4.2.4.2 Content Symbols
Due to the nature of mathematics the meaning of the mathematical expressions must be extensible. The key to extensibility is the ability of the user to define new functions and other symbols to
expand the terrain of mathematical discourse. The csymbol element is used represent a "symbol" in much the same way that ci is used to construct a variable. The difference is that csymbol should
refer to some mathematically defined concept with an external definition referenced via the content dictionary attributes, whereas ci is used for identifiers that are essentially "local" to the
MathML expression.
In MathML3, external definitions are grouped in Content Dictionaries (structured documents for the definition of mathematical concepts; see [OpenMath2004] and [mathml3cds]).
We need three bits of information to fully identify a symbol: a symbol name, a Content Dictionary name, and (optionally) a Content Dictionary base URI, which we encode in the textual content (which
is the symbol name) and two attributes of the csymbol element: cd and cdbase. The Content Dictionary is the location of the declaration of the symbol, consisting of a name and, optionally, a unique
prefix called a cdbase which is used to disambiguate multiple Content Dictionaries of the same name. There are multiple encodings for content dictionaries, this referencing scheme does not
distinguish between them. If a symbol does not have an explicit cdbase attribute, then it inherits its cdbase from the first ancestor in the XML tree with one, should such an element exist. In this
document we have tended to omit the cdbase for brevity.
│ Name │values│ default │
│cdbase│URI │inherited │
│cd │URI │required │
│Editorial note: MiKo ││
│need to fix the default URI here │
│ Issue default_cd │wiki (member only) │
│ Current CD default for csymbol │
│We might make the cd attribute optional? Then that would refer to the current CD if we are in one, or we could make cd inherit like cdbase. That would save bandwidth│
│ Resolution │None recorded │
There are other properties of the symbol that are not explicit in these fields but whose values may be obtained by inspecting the Content Dictionary specified. These include the symbol definition,
formal properties and examples and, optionally, a Role which is a restriction on where the symbol may appear in a MathML expression tree. The possible roles are described in Section 8.5 Symbol Roles.
<csymbol cdbase="http://www.example.com" cd="VectorCalculus">Christoffel</csymbol>
For backwards compatibility with MathML2 and to facilitate the use of MathML within a URI-based framework (such as RDF [rdf] or OWL [owl]), the content of the name, cd, and cdbase can be combined in
the definitionURL attribute: we provide the following scheme for constructing a canonical URI for an MathML Symbol, which can be given in the definitionURL attribute.
URI = cdbase-value + '/' + cd-value + '#' + name-value
In the case of the Christoffel symbol above this would be the URL
For backwards compatibility with MathML2, we do not require that the definitionURL point to a content dictionary. But if the URL in this attribute is of the form above, it will be interpreted as the
canonical URL of a MathML3 symbol. So the representation above would be equivalent to the one below:
<csymbol definitionURL="http://www.example.com/VectorCalculus">Christoffel</csymbol>
│ Issue MathML_CDs_URI │wiki (member only) │
│ What is the official URI for MathMLCDs │
│We still have to fix this. Maybe it should correspond to the final resting place for CDs. │
│ Resolution │None recorded │
│ Issue definitionURL_encoding │wiki (member only) ISSUE-17 (member only) │
│ URI encoding of cdbase/cd/name triplet │
│The URI encoding of the triplet we propose here does not work (not yet for MathMLCDs and not at all for OpenMath2 CDs). The URI reference proposed uses a bare name pointer #Christoffel at the end, │
│which points to the element that has and ID-type attribute with value Christoffel, which is not present in either of these formats. Moreover, it does not scale well with extended CD formats like │
│the OMDoc 1.8 format currently under development │
│ Resolution │None recorded │
│ Issue cdbase-default │wiki (member only) ISSUE-13 (member only) │
│ cdbase default value │
│For the inheritance mechanism to be complete, it would make sense to define a default cdbase attribute value, e.g. at the math element. We'd support expressions ignorant of cdbase as they all are │
│thus far. Something such as http://www.w3.org/Math/CDs/official ? Moreover the MathML content dictionaries should contain such. │
│ Resolution │None recorded │
4.2.5 Function Application
The most fundamental way of building a compound object in mathematics is by applying a function or an operator to some arguments. MathML supplies an infrastructure to represent this in expression
trees, which we will present in this section.
An apply element is used to build an expression tree that represents the result of applying a function or operator to its arguments. The tree corresponds to a complete mathematical expression.
Roughly speaking, this means a piece of mathematics that could be surrounded by parentheses or "logical brackets" without changing its meaning.
│ Name │values│ default │
│cdbase│URI │inherited │
For example, (x + y) might be encoded as
<apply><csymbol cd="algebra-logic">plus</csymbol><ci>x</ci><ci>y</ci></apply>
The opening and closing tags of apply specify exactly the scope of any operator or function. The most typical way of using apply is simple and recursive. Symbolically, the content model can be
described as:
<apply> op a b </apply>
where the operands a and b are MathML expression trees themselves, and op is a MathML expression tree that represents an operator or function. Note that apply constructs can be nested to arbitrary
An apply may in principle have any number of operands:
<apply> op a b [c...] </apply>
For example, (x + y + z) can be encoded as
<csymbol cd="algebra-logic">plus</csymbol>
Mathematical expressions involving a mixture of operations result in nested occurrences of apply. For example, a x + b would be encoded as
<apply><csymbol cd="algebra-logic">plus</csymbol>
<apply><csymbol cd="algebra-logic">times</csymbol>
There is no need to introduce parentheses or to resort to operator precedence in order to parse the expression correctly. The apply tags provide the proper grouping for the re-use of the expressions
within other constructs. Any expression enclosed by an apply element is viewed as a single coherent object.
An expression such as (F+G)(x) might be a product, as in
<apply><csymbol cd="algebra-logic">times</csymbol>
<apply><csymbol cd="algebra-logic">plus</csymbol>
or it might indicate the application of the function F + G to the argument x. This is indicated by constructing the sum
<apply><csymbol cd="algebra-logic">plus</csymbol><ci>F</ci><ci>G</ci></apply>
and applying it to the argument x as in
<apply><csymbol cd="algebra-logic">plus</csymbol>
Both the function and the arguments may be simple identifiers or more complicated expressions.
The apply element is conceptually necessary in order to distinguish between a function or operator, and an instance of its use. The expression constructed by applying a function to 0 or more
arguments is always an element from the codomain of the function. Proper usage depends on the operator that is being applied. For example, the plus operator may have zero or more arguments, while the
minus operator requires one or two arguments to be properly formed.
If the object being applied as a function is not already one of the elements known to be a function (such as sin or plus) then it is treated as if it were a function.
4.2.6 Bindings and Bound Variables
Some complex mathematical objects are constructed by the use of bound variables. For instance the integration variables in an integral expression is one.
4.2.6.1 Bindings
Such expressions are represented as MathML expression trees using the bind element. Its first child is a MathML expression that represents a binding operator (the integral operator in our example).
This can be followed by a non-empty list of bvar elements for the bound variables, possibly augmented by the qualifier element condition (see Section 4.2.7 Qualifiers. The last child is the body of
the binding, it is another content MathML expression.
│ Name │values│ default │
│cdbase│URI │inherited │
4.2.6.2 Bound Variables
The bvar element is a special qualifier element that is used to denote the bound variable of a binding expression, e.g. in sums, products, and quantifiers or user defined functions.
│ Name │values│ default │
│cdbase│URI │inherited │
4.2.6.3 Examples
<csymbol cd="algebra-logic">forall</csymbol>
<apply><csymbol cd="relations">eq</csymbol>
<apply><csymbol cd="algebra-logic">minus</csymbol><ci>x</ci><ci>x</ci></apply>
<csymbol cd="calculus_veccalc">int</csymbol>
<bvar><ci xml:id="var-x">x</ci></bvar>
<apply><csymbol cd="algebra-logic">power</csymbol>
<ci definitionURL="#var-x"><mi>x</mi></ci>
│Editorial note: MiKo ││
│We need to say something about alpha-conversion here for OpenMath compatibility. │
4.2.7 Qualifiers
The integrals we have seen so far have all been indefinite, i.e. the range of the bound variables range is unspecified. In many situations, we also want to specify range of bound variables, e.g. in
definitive integrals. MathML3 provides the optional condition element as a general restriction mechanism for binding expressions.
4.2.7.1 Conditions
A condition element contains a single child that represents a truth condition. Compound conditions are indicated by applying operators such as and in the condition. Consider for instance the
following representation of a definite integral.
│ Name │values│ default │
│cdbase│URI │inherited │
4.2.7.2 Examples
<apply><csymbol cd="sets">in</csymbol>
Here the condition element restricts the bound variables to range over the non-negative integers. A number of common mathematical constructions involve such restrictions, either implicit in
conventional notation, such as a bound variable, or thought of as part of the operator rather than an argument, as is the case with the limits of a definite integral.
A typical use of the condition qualifier is to define sets by rule, rather than enumeration. The following markup, for instance, encodes the set {x | x < 1}:
In the context of quantifier operators, this corresponds to the "such that" construct used in mathematical expressions. The next example encodes "for all x in N there exist prime numbers p, q such
that p+q = 2x".
<bind><csymbol cd="algebra-logic">forall</csymbol>
<apply><csymbol cd="sets">in</csymbol>
<csymbol cd="contstants">naturalnumbers</csymbol>
<bind><csymbol cd="algebra-logic">exists</csymbol>
<apply><csymbol cd="algebra-logic">and</csymbol>
<apply><csymbol cd="sets">in</csymbol><ci>p</ci><primes/></apply>
<apply><csymbol cd="sets">in</csymbol><ci>q</ci><primes/></apply>
<apply><csymbol cd="relations">eq</csymbol>
<apply><csymbol cd="algebra-logic">plus</csymbol><ci>p</ci><ci>q</ci></apply>
<apply><csymbol cd="algebra-logic">times</csymbol><cn>2</cn><ci>x</ci></apply>
This use extends to multivariate domains by using extra bound variables and a domain corresponding to a cartesian product as in
<apply><csymbol cd="algebra-logic">and</csymbol>
<apply><csymbol cd="relations">leq</csymbol><cn>0</cn><ci>x</ci></apply>
<apply><csymbol cd="relations">leq</csymbol><ci>x</ci><cn>1</cn></apply>
<apply><csymbol cd="relations">leq</csymbol><cn>0</cn><ci>y</ci></apply>
<apply><csymbol cd="relations">leq</csymbol><ci>y</ci><cn>1</cn></apply>
<csymbol cd="algebra-logic">times</csymbol>
<apply><csymbol cd="algebra-logic">power</csymbol><ci>x</ci><cn>2</cn></apply>
<apply><csymbol cd="algebra-logic">power</csymbol><ci>y</ci><cn>3</cn></apply>
4.2.8 Structure Sharing
To conserve space, MathML3 expression trees can make use of structure sharing
4.2.8.1 The share element
This element has an href attribute whose value is the value of a URI referencing an xml:id attribute of a MathML expression tree. When building the MathML expression tree, the share element is
replaced by a copy of the MathML expression tree referenced by the href attribute. Note that this copy is structurally equal, but not identical to the element referenced. The values of the share will
often be relative URI references, in which case they are resolved using the base URI of the document containing the share element.
│Name│values │default │
│href│URI │ │
For instance, the mathematical object f(f(f(a,a),f(a,a)),f(a,a),f(a,a)) can be encoded as either one of the following representations (and some intermediate versions as well).
<math> <math>
<apply> <apply>
<ci>f</ci> <ci>f</ci>
<apply> <apply xml:id="t1">
<ci>f</ci> <ci>f</ci>
<apply> <apply xml:id="t11">
<ci>f</ci> <ci>f</ci>
<ci>a</ci> <ci>a</ci>
<ci>a</ci> <ci>a</ci>
</apply> </apply>
<apply> <share href="#t11"/>
</apply> </apply>
<apply> <share href="#t1"/>
</math> </math>
4.2.8.2 An Acyclicity Constraint
We say that an element dominates all its children and all elements they dominate. An share element dominates its target, i.e. the element that carries the xml:id attribute pointed to by the href
attribute. For instance in the representation above the apply element with xml:id="t1" and also the second share dominate the apply element with xml:id="t11".
The occurrences of the share element must obey the following global acyclicity constraint: An element may not dominate itself. For instance the following representation violates this constraint:
<apply xml:id="foo">
<csymbol cd="algebra-logic">plus</csymbol>
<csymbol cd="algebra-logic">plus</csymbol>
<share href="foo"/>
Here, the apply element with xml:id="foo" dominates its third child, which dominates the share element, which dominates its target: the element with xml:id="foo". So by transitivity, this element
dominates itself, and by the acyclicity constraint, it is not an MathML expression tree. Even though it could be given the interpretation of the continued fraction
Note that the acyclicity constraints is not restricted to such simple cases, as the following example shows:
<apply xml:id="bar"> <apply xml:id="baz">
<csymbol cd="algebra-logic">plus</csymbol> <csymbol cd="algebra-logic">plus</csymbol>
<cn>1</cn> <cn>1</cn>
<share href="baz"/> <share href="bar"/>
</apply> </apply>
Here, the apply with xml:id="bar" dominates its third child, the share with href="baz", which dominates its target apply with xml:id="baz", which in turn dominates its third child, the share with
href="bar", this finally dominates its target, the original apply element with xml:id="bar". So this pair of representations violates the acyclicity constraint.
4.2.8.3 Structure Sharing and Binding
Note that the share element is a syntactic referencing mechanism: an share element stands for the exact element it points to. In particular, referencing does not interact with binding in a
semantically intuitive way, since it allows for variable capture. Consider for instance
<bind xml:id="outer">
<bind xml:id="inner">
<share xml:id="copy" href="#orig"/>
<apply xml:id="orig"><ci>g</ci><ci>X</ci></apply>
it represents the term xml:id="orig" (the one explicitly represented) and one with xml:id="copy", represented by the share element. In the original, the variable x is bound by the outer bind element,
and in the copy, the variable x is bound by the inner bind element. We say that the inner bind has captured the variable X.
It is well-known that variable capture does not conserve semantics. For instance, we could use α-conversion to rename the inner occurrence of x into, say, y arriving at the (same) object
4.2.9 Attribution via semantics
Content elements can be adorned with additional information via the semantics element, see Section 5.3 Semantic Annotations beyond Alternate Representations for details. As such, the semantics
element should be considered part of both presentation MathML and content MathML. MathML3 considers a semantics element (strict) content MathML, if and only if its first child is (strict) content
MathML. All MathML processors should process the semantics element, even if they only process one of those subsets.
│Editorial note: MiKo ││
│Give an elaborated example from the types note here (or in the primer?), reference Section 8.4 Type Declarations │
4.2.10 In Situ Error Markup
A content error expression is made up of a symbol and a sequence of zero or more MathML expression trees. This object has no direct mathematical meaning. Errors occur as the result of some treatment
on an expression tree and are thus of real interest only when some sort of communication is taking place. Errors may occur inside other objects and also inside other errors.
│ Name │values│ default │
│cdbase│URI │inherited │
To encode an error caused by a division by zero, we would employ a aritherror Content Dictionary with a DivisionByZero symbol with role error we would use the following expression tree:
<csymbol cd="aritherror">DivisionByZero</csymbol>
Note that the error should cover the smallest erroneous subexpression so cerror can be a subexpression of a bigger one, e.g.
<apply><csymbol cd="relations">eq</csymbol>
<csymbol cd="aritherror">DivisionByZero</csymbol>
If an application wishes to signal that the content MathML expressions it has received is invalid or is not well-formed then the offending data must be encoded as a string. For example:
<csymbol cd="parser">invalid_XML</csymbol>
<mtext> <<!--LESS-THAN SIGN-->apply><!--GREATER-THAN SIGN--><<!--LESS-THAN SIGN-->cos><!--GREATER-THAN SIGN--> <<!--LESS-THAN SIGN-->ci><!--GREATER-THAN SIGN-->v<<!--LESS-THAN SIGN-->/ci><!--GREATER-THAN SIGN--> <<!--LESS-THAN SIGN-->/apply><!--GREATER-THAN SIGN--> </mtext>
Note that the < and > characters have been escaped as is usual in an XML document.
4.3 Pragmatic Content MathML
MathML3 content markup differs from earlier versions of MathML in that it has been regularized and based on the content dictionary model introduced by OpenMath [OpenMath2004].
MathML3 also supports MathML2 markup as a pragmatic representation that is easier to read and more intuitive for humans. We will discuss this representation in the following and indicate the
equivalent strict representations. Thus the "pragmatic content MathML" representations inherit the meaning from their strict counterparts.
4.3.1 Numbers with "constant" type
The cn element can be used with the value "constant" for the type attribute and the Unicode symbols for the content. This use of the cn is deprecated in favor of the number constants exponentiale,
imaginaryi, true, false, notanumber, pi, eulergamma, and infinity in the content dictionary constants CD, or the use of csymbol with an appropriate value for the definitionURL attribute. For example,
instead of using the pi element, an instance of <cn type="constant">π</cn> could be used.
4.3.2 csymbol Elements with Presentation MathML
│ Issue csymbol_pmathml_strict │wiki (member only) │
│ Strict equivalent for csymbol with pMathML content │
│What is the strict equivalent for the case of a csymbol with pMathML content, we do not have a good way of determining that either from the pMathML (we could take the element content stripped of │
│elements; I am assuming this in the example below for now) or from the definitionURL. But as David convinced me, this does not work, so we still need to discuss this. We also need to keep the use │
│of symbol names as fragment identifiers in mind. │
│ Resolution │None recorded │
In pragmatic MathML3 the csymbol element can contain presentation MathML instead of the symbol name. For example,
<csymbol definitionURL="http://www.example.com/ContDiffFuncs.htm">
encodes an atomic symbol that displays visually as C^2 and that, for purposes of content, is treated as a single symbol representing the space of twice-differentiable continuous functions. This
pragmatic representation is equivalent to
<csymbol definitionURL="http://www.example.com/ContDiffFuncs.htm">C2</csymbol>
<annotation-xml encoding="MathMLP">
Both can be used interchangeably.
4.3.3 Symbols and Identifiers With Presentation MathML
In Pragmatic Content MathML, the ci and csymbol elements can contain a general presentation construct (see Section 3.1.6 Summary of Presentation Elements), which is used for rendering (see Section
8.6 Rendering of Content Elements). In this case, the definitionURL attribute can be used to associate a name with with a ci element, which identifies it. See the discussion of bound variables (
Section 4.2.6 Bindings and Bound Variables) for a discussion of an important instance of this. For example,
<ci definitionURL="c1"><msub><mi>c</mi><mn>1</mn></msub></ci>
encodes an atomic symbol that displays visually as c[1] which, for purposes of content, is treated as a atomic concept representing a real number.
Instances of the bound variables are normally recognized by comparing the XML information sets of the relevant ci elements after first carrying out XML space normalization. Such identification can be
made explicit by placing an xml:id on the ci element in the bvar element and referring to it using the definitionURL attribute on all other instances. An example of this approach is This xml:id based
approach is especially helpful when constructions involving bound variables are nested.
It can be necessary to associate additional information with a bound variable one or more instances of it. The information might be something like a detailed mathematical type, an alternative
presentation or encoding or a domain of application. Such associations are accomplished in the standard way by replacing a ci element (even inside the bvar element) by a semantics element containing
both it and the additional information. Recognition of and instance of the bound variable is still based on the actual ci elements and not the semantics elements or anything else they may contain.
The xml:id based approach outlined above may still be used.
A ci element with Presentation MathML content is equivalent to a semantics construction where the first child is a ci whose content is the value of the definitionURL attribute and whose second child
is an annotation-xml element with the MathML Presentation. For example the Strict Content MathML equivalent to the example above would be
<annotation-xml encoding="PMathML">
4.3.4 Elementary MathML Types on Tokens
The ci element uses the type attribute to specify the basic type of object that it represents. While any CDATA string is a valid type, the predefined types include "integer", "rational", "real",
"complex", "complex-polar", "complex-cartesian", "constant", "function" and more generally, any of the names of the MathML container elements (e.g. vector) or their type values. For a more advanced
treatment of types, the type attribute is inappropriate. Advanced types require significant structure of their own (for example, vector(complex)) and are probably best constructed as mathematical
objects and then associated with a MathML expression through use of the semantics element.
│Editorial note: MiKo ││
│Give the Strict equivalent here by techniques from the Types Note │
4.3.5 Token Elements
For convenience and backwards compatibility MathML3 provides empty token elements for the operators and functions of the K-14 fragment of mathematics. The general rule is that for any symbol defined
in the MathML3 content dictionaries (see Chapter 8 MathML3 Content Dictionaries), there is an empty content element with the same name. For instance, the empty MathML element
is equivalent to the element
<csymbol cdbase="http://w3.org/Math/CD" cd="algebra-logic" name="plus"/>
both can be used interchangeably.
In MathML2, the definitionURL attribute could be used to modify the meaning of an element to allow essentially the same notation to be re-used for a discussion taking place in a different mathematic
domain. This use of the attribute is deprecated in MathML3, in favor of using a csymbol with the same definitionURL attribute.
4.3.6 Tokens with Attributes
In MathML2, the meaning of various token elements could be specialized via various attributes, usually the type attribute. Strict Content MathML does not have this possibility, therefore these
attributes are either passed to the symbols as extra arguments in the apply or bind elements, or MathML3 adds new symbols for the non-default case to the respective content dictionaries.
We will summarize the cases in the following table:
│pragmatic Content MathML│ strict Content MathML │
│<diff type="function"/> │<csymbol cd="calculus_veccalc">diff</csymbol> │
│<diff type="algebraic"/>│<csymbol cd="calculus_veccalc">aDiff</csymbol> │
│Editorial note: MiKo ││
│systematically consider all the cases here │
4.3.7 Container Markup
To retain compatibility with MathML2, MathML3 provides an alternative representation for applications of constructor elements. For instance for the set element, the following two representations are
considered equivalent
and following the discussion in section Section 4.2.4 Symbols and Identifiers they are equivalent to
<apply><csymbol cd="sets">set</csymbol><ci>a</ci><ci>b</ci><ci>c</ci></apply>
Other constructors are interval, list, matrix, matrixrow, vector, apply, lambda, piecewise, piece, otherwise
4.3.8 Domain of Application in Applications
The domainofapplication element was used in MathML2 an apply element which denotes the domain over which a given function is being applied. In contrast to its use as a qualifier in the bind element,
the usage in the apply element only marks the argument position for the range argument of the definite integral.
MathML3 supports this representation as a pragmatic form. For instance, the integral of a function f over an arbitrary domain C can be represented as
in the Pragmatic Content MathML representation, it is considered equivalent to
│Editorial note: MiKo ││
│be careful with Int and int here │
4.3.9 Domain of Application in Bindings
The domainofapplication was intended to be an alternative to specification of range of bound variables for condition. Generally, a domain of application D can be specified by a condition element
requesting that the bound variable is a member of D. For instance, we consider the Pragmatic Content MathML representation
<domainofapplication><ci type="set">D</ci></domainofapplication>
<apply><ci type="function">f</ci><ci>x</ci></apply>
as equivalent to the Strict Content MathML representation
<condition><apply><in/><ci>x</ci><ci type="set">D</ci></apply></condition>
<apply><ci type="function">f</ci><ci>x</ci></apply>
4.3.10 Integrals with Calling patterns
MathML2 used the int element for the definite or indefinite integral of a function or algebraic expression on some sort of domain of application. There are several forms of calling sequences
depending on the nature of the arguments, and whether or not it is a definite integral. Those forms using interval, condition, lowlimit, or uplimit, provide convenient shorthand notations for an
appropriate domainofapplication.
│Editorial note: Miko ││
│the following must be reworked │
MathML separates the functionality of the int element into three different symbols: int, defint, and defintset. The first two are integral operators that can be applied to functions and the latter is
binding operators for integrating an algebraic expression with respect to a bound variable.
The following two indefinite function integrals are equivalent.
The following two definite function integrals are equivalent (see also Section 4.3.8 Domain of Application in Applications).
<domainofapplication><ci type="set">D</ci></domainofapplication>
<apply><defintfun/><ci type="set">D</ci><sin/></apply>
The following two indefinite integrals over algebraic expressions are equivalent.
The following two definite function integrals are equivalent.
<domainofapplication><ci type="set">D</ci></domainofapplication>
<domainofapplication><ci type="set">D</ci></domainofapplication>
4.3.11 degree
The degree element is a qualifier used by some MathML containers to specify that, for example, a bound variable is repeated several times.
│Editorial note: MiKo ││
│specify a complete list of containers that allow degree elements, so far I see diff, partialdiff, root │
The degree element is the container element for the "degree" or "order" of an operation. There are a number of basic mathematical constructs that come in families, such as derivatives and moments.
Rather than introduce special elements for each of these families, MathML uses a single general construct, the degree element for this concept of "order".
<degree><ci> n </ci></degree>
A variable that is to be bound is placed in this container. In a derivative, it indicates which variable with respect to which a function is being differentiated. When the bvar element is used to
qualify a derivative, the bvar element may contain a child degree element that specifies the order of the derivative with respect to that variable.
it is equivalent to
│Editorial note: MiKo ││
│what do we want to use for degree? │
Note that the degree element is only allowed in the container representation. The strict representation takes the degree as a regular argument as the second child of the apply or bind element.
│Editorial note: MiKo ││
│Make sure that all MMLdefinitions of degree-carrying symbols get a paragraph like the one for root. │
The default rendering of the degree element and its contents depends on the context. In the example above, the degree elements would be rendered as the exponents in the differentiation symbols:
4.3.12 Upper and Lower Limits
The uplimit and lowlimit elements are Pragmatic Content MathML qualifiers that can be used to restrict the range of a bound variable to an interval, e.g. in some integrals and sums. uplimit/lowlimit
pairs can be expressed via the interval element from the CD Basic Content Elements. For instance, we consider the Pragmatic Content MathML representation
<bvar><ci> x </ci></bvar>
<apply><ci type="function">f</ci><ci>x</ci></apply>
as equivalent to the following strict representation
<apply><ci type="function">f</ci><ci>x</ci></apply>
If the lowlimit qualifier is missing, it is interpreted as negative infinity, similarly, if uplimit is then it is interpreted as positive infinity.
4.3.13 Lifted Associative Commutative Operators
│ Issue lifted_operators │wiki (member only) ISSUE-8 (member only) │
│ New Symbols for Lifted Operators │
│MathML2 allowed the use of n-ary operators as binding operators with bound variables induced by them. For instance union could be used as the equivalent for the TeX \cup as well as \bigcup. While │
│the relation between the nary and the set-based operators is deterministic, i.e. the induced big operators are fully determined by them, the concepts are quite different in nature (different │
│notational conventions, different types, different occurrence schemata. I therefore propose to extend the MathML K-14 CDs with symbols big operators, much like we already have sum as the big │
│operator for for the n-ary plus symbol, and prod for times. For the new symbols, I propose the naming convention of capitalizing the big operators (as an alternative, we could follow TeX and │
│pre-pend a bib). For example we could have Union as a big operator for union │
│ Resolution │None recorded │
MathML2 allowed to use a associative operators to be "lifted" to "big operators", for instance the n-ary union operator to the union operator over sets, as the union of the U-complements over a
family F of sets in this construction
While the relation between the nary and the set-based operators is deterministic, i.e. the induced big operators are fully determined by them, the concepts are quite different in nature (different
notational conventions, different types, different occurrence schemata). Therefore the MathML3 content dictionaries provides explicit symbols for the "big operators", much like MathML2 did with sum
as the big operator for for the n-ary plus symbol, and prod for times. Concretely, these are big_union, big_intersect, big_max, big_min, big_gcd, big_lcm, big_or, big_and, and big_xor. With these, we
can express all Pragmatic Content MathML expressions. For instance, the union above can be represented strictly as
For the exact meaning of the new symbols, consult the content dictionaries.
│ Issue large_ops │wiki (member only) ISSUE-18 (member only) │
│ Large Operators │
│The large operators can be solved in two ways, in the way described here, by inventing large operators (and David does not like symbol names distinguished only by case; and I agree tend to agree │
│with him). Or by extending the role of roles to allow duplicate roles per symbol, then we could re-use the symbols like we did in MathML2, but then we would have to extend OpenMath for that │
│ Resolution │None recorded │
4.3.14 Declare (declare)
│Editorial note: MiKo ││
│This should maybe be moved into a general section about changes or deprecated elements. Also Stan thinks the text should be improved. │
MathML2 provided the declare element that allowed to bind properties like types to symbols and variables and to define abbreviations for structure sharing. This element is deprecated in MathML3.
Structure sharing can obtained via the share element (see Section 4.2.8 Structure Sharing for details). | {"url":"https://www.w3.org/TR/2007/WD-MathML3-20071214/chapter4.html","timestamp":"2024-11-14T14:47:14Z","content_type":"text/html","content_length":"181760","record_id":"<urn:uuid:1b816471-fde1-4a00-a0d5-05582bd457d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00142.warc.gz"} |
49 research outputs found
Complicated generative models often result in a situation where computing the likelihood of observed data is intractable, while simulating from the conditional density given a parameter value is
relatively easy. Approximate Bayesian Computation (ABC) is a paradigm that enables simulation-based posterior inference in such cases by measuring the similarity between simulated and observed data
in terms of a chosen set of summary statistics. However, there is no general rule to construct sufficient summary statistics for complex models. Insufficient summary statistics will "leak"
information, which leads to ABC algorithms yielding samples from an incorrect (partial) posterior. In this paper, we propose a fully nonparametric ABC paradigm which circumvents the need for manually
selecting summary statistics. Our approach, K2-ABC, uses maximum mean discrepancy (MMD) as a dissimilarity measure between the distributions over observed and simulated data. MMD is easily estimated
as the squared difference between their empirical kernel embeddings. Experiments on a simulated scenario and a real-world biological problem illustrate the effectiveness of the proposed algorithm
The kernel mean embedding is known to provide a data representation which preserves full information of the data distribution. While typically computationally costly, its nonparametric nature has an
advantage of requiring no explicit model specification of the data. At the other extreme are approaches which summarize data distributions into a finite-dimensional vector of hand-picked summary
statistics. This explicit finite-dimensional representation offers a computationally cheaper alternative. Clearly, there is a trade-off between cost and sufficiency of the representation, and it is
of interest to have a computationally efficient technique which can produce a data-driven representation, thus combining the advantages from both extremes. The main focus of this thesis is on the
development of linear-time mean-embedding-based methods to automatically extract informative features of data distributions, for statistical tests and Bayesian inference. In the first part on
statistical tests, several new linear-time techniques are developed. These include a new kernel-based distance measure for distributions, a new linear-time nonparametric dependence measure, and a
linear-time discrepancy measure between a probabilistic model and a sample, based on a Stein operator. These new measures give rise to linear-time and consistent tests of homogeneity, independence,
and goodness of fit, respectively. The key idea behind these new tests is to explicitly learn distribution-characterizing feature vectors, by maximizing a proxy for the probability of correctly
rejecting the null hypothesis. We theoretically show that these new tests are consistent for any finite number of features. In the second part, we explore the use of random Fourier features to
construct approximate kernel mean embeddings, for representing messages in expectation propagation (EP) algorithm. The goal is to learn a message operator which predicts EP outgoing messages from
incoming messages. We derive a novel two-layer random feature representation of the input messages, allowing online learning of the operator during EP inference
Two semimetrics on probability distributions are proposed, given as the sum of differences of expectations of analytic functions evaluated at spatial or frequency locations (i.e, features). The
features are chosen so as to maximize the distinguishability of the distributions, by optimizing a lower bound on test power for a statistical test using these features. The result is a parsimonious
and interpretable indication of how and where two distributions differ locally. An empirical estimate of the test power criterion converges with increasing sample size, ensuring the quality of the
returned features. In real-world benchmarks on high-dimensional text and image data, linear-time tests using the proposed semimetrics achieve comparable performance to the state-of-the-art
quadratic-time maximum mean discrepancy test, while returning human-interpretable features that explain the test results
We propose two nonparametric statistical tests of goodness of fit for conditional distributions: given a conditional probability density function $p(y|x)$ and a joint sample, decide whether the
sample is drawn from $p(y|x)r_x(x)$ for some density $r_x$. Our tests, formulated with a Stein operator, can be applied to any differentiable conditional density model, and require no knowledge of
the normalizing constant. We show that 1) our tests are consistent against any fixed alternative conditional model; 2) the statistics can be estimated easily, requiring no density estimation as an
intermediate step; and 3) our second test offers an interpretable test result providing insight on where the conditional model does not fit well in the domain of the covariate. We demonstrate the
interpretability of our test on a task of modeling the distribution of New York City's taxi drop-off location given a pick-up point. To our knowledge, our work is the first to propose such
conditional goodness-of-fit tests that simultaneously have all these desirable properties.Comment: In UAI 2020. http://auai.org/uai2020/accepted.ph
We propose a new family of specification tests called kernel conditional moment (KCM) tests. Our tests are built on a novel representation of conditional moment restrictions in a reproducing kernel
Hilbert space (RKHS) called conditional moment embedding (CMME). After transforming the conditional moment restrictions into a continuum of unconditional counterparts, the test statistic is defined
as the maximum moment restriction (MMR) within the unit ball of the RKHS. We show that the MMR not only fully characterizes the original conditional moment restrictions, leading to consistency in
both hypothesis testing and parameter estimation, but also has an analytic expression that is easy to compute as well as closed-form asymptotic distributions. Our empirical studies show that the KCM
test has a promising finite-sample performance compared to existing tests.Comment: In Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI2020
We propose a novel adaptive test of goodness-of-fit, with computational cost linear in the number of samples. We learn the test features that best indicate the differences between observed samples
and a reference model, by minimizing the false negative rate. These features are constructed via Stein's method, meaning that it is not necessary to compute the normalising constant of the model. We
analyse the asymptotic Bahadur efficiency of the new test, and prove that under a mean-shift alternative, our test always has greater relative efficiency than a previous linear-time kernel test,
regardless of the choice of parameters for that test. In experiments, the performance of our method exceeds that of the earlier linear-time test, and matches or exceeds the power of a quadratic-time
kernel test. In high dimensions and where model structure may be exploited, our goodness of fit test performs far better than a quadratic-time two-sample test based on the Maximum Mean Discrepancy,
with samples drawn from the model.Comment: Accepted to NIPS 201 | {"url":"https://core.ac.uk/search/?q=author%3A(Jitkrittum%2C%20Wittawat)","timestamp":"2024-11-11T06:57:15Z","content_type":"text/html","content_length":"115697","record_id":"<urn:uuid:dcafc9b8-9dc2-4b49-975b-9c7620a5a162>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00488.warc.gz"} |
What is yard in physics?
A yard is a unit of length. The symbol of yard is “yd”. It is equal to 3 feet or 36 inches. If converted into meters, 1 yard is equal to 0.9144 meters.
What does yard mean in science?
yard, Unit of length equal to 36 inches, or 3 feet (see foot), in the U.S. Customary System or 0.9144 metre in the International System of Units.
What is a yard a measure of?
A yard is a unit of length in both US Customary and British Imperial Systems of Measurement. It is equivalent to 3 feet or 36 inches. Its symbol is yd. It is often used to measure the length of
medium-sized objects.
What do you mean by yards?
a unit of measurement equal to 3 feet or 91.4 centimetres. an area of land next to buildings used for a particular purpose such as building, storing, or selling goods: a builders' yard.
What is the difference between a yard and a meter?
The difference between meter and yard is that the meter is a SI unit of length and a yard is a unit of length. Also, 1 meter is about 1.09 yards.
How to Yard Stick (Physics)
Is 1 meter equal to 1 yard?
Even though meters and yards are used for measuring the same unit (distance/length), their values differ. The conversion factor that is used to convert meters to yards is 1.09361 because 1 meter (m)
= 1.09361 yards (yd).
Which is bigger 1 yard or 1 meter?
A meter is longer than a yard. A meter is the standard metric unit of measurement and is equal to 3.2 feet. A yard is equal to 3 feet.
What is a yard in math?
Yard Definition: A yard is a unit of length. The symbol of yard is “yd”. It is equal to 3 feet or 36 inches. If converted into meters, 1 yard is equal to 0.9144 meters.
How do you find yards?
3 feet equals 1 yard, so 9 feet equals 3 total yards in length. The width of 3 feet equals 1 yard. The height/depth is 12 inches (1 foot), which equals one-third of a yard.
What is the old meaning of yard?
Summary. A word inherited from Germanic. Old English geard strong masculine fence, dwelling, house, region = Old Saxon gard enclosure, field, dwelling, Middle Dutch, Dutch gaard garden, Old High
German gart circle, ring, Old Norse garðr… Show more.
Is a yard a metric unit?
The decimal based measuring system, with 'metre', 'litre', and 'gram' as units of length, capacity, and weight or mass respectively, is called metric system. On the other hand, a 'yard' is used to
measure length and is equal to 3 feet, which is not a metric unit of measure.
Is a yard blank than a meter?
A yard is roughly equivalent to a meter: one yard = 0.9144 meters.
Is a yard a measure of distance?
Yard: Yard is the fundamental unit of length in both the US Customary System and the British Imperial System of measurement. Yard is the distance between the end of the outstretched arm and the chin.
One yard = 3 feet. In terms of inches, one yard is equal to 36 inches and equivalent to 0.9144 meter.
What makes a yard of material?
Measuring a Yard of Fabric
Fabrics come in various widths, so a yard of the fabric refers to the length of material only. The material is unrolled from the bolt, and you should measure 36 inches or 3 feet. That's precisely how
much a yard of fabric is.
What is a yard in stone?
27 cubic feet equal 1 cubic yard (3'L x 3'W x 3'H). Soil weighs about 2,200 lbs per cubic yard. Stone weighs about 2,700 lbs per cubic yard.
How do you convert distance to yards?
All you need to do is start with a number of meters. Multiply the number of meters by 1.0936 to get the number of yards. There are 1.0936 yards in every meter. So if 1 meter is 1.0963 yards, then 2
meters is 2.1872 yards, and so on.
How do you measure 5 yards?
5 yards equals 15 feet because 5x3=15 or 180 inches because 15x12=180.
What is one yard divided into?
A yard is further divided into feet and inches for the measurement of smaller lengths. 1 yard = 3 feet and 1 foot = 12 inches.
Which is longer 2 meters or 3 yards?
A meter is slightly longer than a yard. How much longer is a meter than a yard? One yard equals 36 inches. One meter is approximately equivalent to 39.4 inches.
What is the kids definition of yard?
The grassy area right outside a house is a yard. A yard is often surrounded by a fence or marked by shrubs or other plants. As a unit of measurement, a yard is equal to three feet.
What is a yard of concrete?
Most concrete purchases will be made in cubic yards, which equates to 27 cubic feet. For example, a project measuring 10 ft in length by 10 ft in width with a depth of 3.5 in will be just over 1
cubic yard.
Is 1m equal to 3 feet?
One meter is approximately equal to 3.28084 feet. To convert meters to feet, multiply the given meter value by 3.28084 feet.
What is 10 meters called?
Deka- means 10; a dekameter is 10 meters. Hecto- means 100; a hectometer is 100 meters.
Is a yard longer than a mile?
How much longer is a mile than a yard? 1 mile = 1760 yards ⇒ 1 mile is 1759 yards longer than 1 yard or 1 mile is 1760 times as long as 1 yard. | {"url":"https://eboots.co.uk/wiki/what-is-yard-in-physics","timestamp":"2024-11-08T17:57:42Z","content_type":"text/html","content_length":"125181","record_id":"<urn:uuid:bac32d28-0350-47ba-8d7e-30355eb95c3d>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00070.warc.gz"} |
Measuring Water in a Tank
Over the years, we have received a huge number of questions asking about how to find the amount of liquid (water, oil, …) in a tank, usually a horizontal cylindrical tank. The simplest case involves
a rather complicated formula; from there, we can reverse the formula (finding the depth for a given volume), or we can add features to the tank. Here I will look at just a couple of these, with links
to others, most of which include derivations.
The basic horizontal cylindrical tank
Here is a typical, well-written question, from 1999:
Horizontal Gas Tank Content Formula
I'm trying to find a chart that can be used by our drivers to check the content of a fuel tank by inserting a sort of yardstick into the tank and checking the inches of liquid in the fuel tank. The tanks are 97-gallon horizontal tanks. I have a horizontal tank content indicator chart but it is for larger tanks. I need something that a driver, given the length and height of the tank, can use to calculate the remaining fuel by measuring the height of the remaining fuel. Our objective is to reduce the amount of fuel purchased on the road and more accurately calculate the amount of fuel left on board. The smallest tank size on the chart I have is 300 gal. Thanks in advance for your help.
The calculation is sufficiently complicated that most people would find it easiest to use a chart. Today, I suppose they would be more likely to use a spreadsheet to do the calculations; the formula
is available in many places. Doctor Anthony straightforwardly derived it:
The problem is not difficult. We simply need to calculate the area of a segment of a circle, representing the cross-section of the tank that contains the fuel. If this cross-sectional area is multiplied by L, the horizontal length of the tank, we have the volume.
Suppose r = the radius of the tank and h = the height of the fuel measured from the lowest point to the surface of the fuel (the dipstick reading):
If theta is the angle, in radians, between the vertical through the centre of the circle and a radius drawn from the centre to the point of contact of the fuel and the side of the tank, then
cos(theta) = (r-h)/r
So theta = cos^(-1)[(r-h)/r)]
Note that:
** area of sector of circle with angle 2 theta = r^2 theta
** area of triangle to be subtracted = (1/2)r^2 sin(2 theta)
Cross-section of fuel = r^2 theta - (1/2)r^2 sin(2 theta)
= r^2[theta - (1/2)sin(2 theta)]
= r^2[theta - sin(theta) cos(theta)]
And so:
Volume of fuel = r^2 L[theta - sin(theta) cos(theta)]
It is probably better to leave the formula in this form and use the other formula
theta = cos^(-1)[(r-h)/r] radians
to complete the calculation. It becomes rather untidy if you express
the whole thing in one big formula in terms of h. The sin(theta) term is the problem.
sqrt[r^2 - (r-h)^2] sqrt(2rh - h^2)
sin(theta) = ------------------- = ---------------
r r
and you end up with
Volume = r^2 L[cos^(-1)[(r-h)/r] - sqrt(2rh-h^2)/r (r-h)/r]
= r^2 L[cos^(-1)[(r-h)/r] - (r-h) sqrt(2rh-h^2)/r^2]
We will usually see this in a slightly different form,
\displaystyle V = L\left[r^2 \cos^{-1}\frac{r-h}{r} – (r-h) \sqrt{2rh-h^2}\right]\).
This (without the L) can also be found in the Formulas FAQ under Segment of a Circle.
We can factor out \(R^2\) and define \(f = \frac{h}{2r}\), then divide by the volume of the whole cylinder, yielding this nice formula for the percentage full in terms of the percentage depth:
\displaystyle V_{rel} = \frac{1}{\pi} \left(\cos^{-1}(1-2f) – 2(1-2f) \sqrt{(1-f)f}\right)\).
Let’s try an example. Suppose the radius r is 1 foot, and the length L is 4 feet. Then the volume of the whole tank is
\(V = \pi r^2 L = \pi\cdot12^2\cdot48 = 21,715 \text{ in}^3\),
which is \(21,715/231 = 94\text{ gal}\).
Close enough to 97! If the fuel is 9 inches deep, we have
\(V = 48\left[12^2 \cos^{-1}\frac{12-9}{12} – (12-9) \sqrt{2\cdot 12\cdot 9-9^2}\right] \) \(= 48\left[144 \cos^{-1} \frac{1}{4} – 3 \sqrt{135}\right] \) \(= 48\cdot 154.95 = 7438 \text{ in}^3 = 32 \
text{ gal}\).
So the tank is about 1/3 full. (Don’t forget to set your calculator to radians.)
[DEL::DEL]Using the second formula, the relative depth is 9/24 = 3/8, so the relative volume is
\displaystyle \frac{1}{\pi} \left( \cos^{-1}\left(1-2\frac{3}{8}\right) – 2
\left(1-2\frac{3}{8}\right) \sqrt{ \frac{5}{8} \frac{3}{8} }\right){\pi} = 0.3425\)
For an earlier, and longer, discussion of this problem, see
Finding the Volume of a Horizontal Tank
The inverse problem: Making a dipstick
In the next question, from 2000, the “patient” Mark has derived that formula himself, but has a much harder problem. He wants to reverse it:
Volume of a Partially Filled Cylindrical Tank
I am trying to find the inverse for the equation used to find the area of a circle's segment. I have managed to derive the area formula, but finding its inverse seems impossible.
One version of the formula follows:
Given a circle of radius 5, and the distance from the center of the circle to the center of the chord defining the segment, h,
a = 25arccos((5 - h) / 5) - (5 - h)sqrt(10h - h^2)
The formula itself is no trouble to find, being just the difference between the area of the sector the segment is part of and the triangle formed by the two endpoints of the chord and the circle's center. I need to transform it such that it is expressed in terms of h.
If the transformation is too difficult, it may help that this question is part of a bigger problem. Briefly, a tank, filled from the top and drained from the bottom, is made from a hemispherical prism, such that it lies with its axis parallel to the ground and its flat surface facing upwards (a D with the flat section upwards). The radius of the hemisphere is 5, and the tank is 10 deep. The problem is to design a dipstick such that it can be used to find the volume of liquid in the tank. I had planned to use the inverse to find h for convenient volumes of liquid, and use that value to specify markings on the dipstick. I have written a computer program that solves the problem using successive iterations to reduce error, but it is not really an acceptable solution.
I have done a little calculus (just differentiation and integration), but no solving of differential equations (which is what I think may be needed here, as the derivative of the area would be related to the derivative of h), so that is little help here unless I hit the books.
Any suggestions?
We first had to be sure what he means by “hemispherical prism”. It turns out that he meant something more like “semicircular prism”, or more simply, half a cylinder lying on its side, with the flat
surface up. So this is just a tank like the previous question, without its upper half. He got the formula right.
Doctor Jerry answered, first giving a different form of the same formula:
Consider a circle of radius a (the end of the tank) and imagine that the liquid has reached height h, measured from the lowest point on the circle. Note that 0 <= h <= 2a. The area A of the segment of the circle covered by the liquid is
A = pi*a^2/2 - a^2*arcsin(1-h/a) - (a-h)*sqrt(h(2a-h))
The volume of liquid is just A*L, where L is the length of the tank. From this, a dipstick can be calibrated.
This formula is related to the formula you gave, but it is expressed in terms of the depth of the liquid, which I think is useful.
It is not possible to solve either of our two formulas for h - at least not in "closed form."
After a discussion clarifying the shape, he pointed out that, in fact, an iterative approximation will be necessary; he showed how to do this more efficiently, using Newton’s Method as we recently
discussed here:
Using the formula:
V(h) = 10[pi*a^2/2 - a^2*arcsin(1-h/a) - (a-h)*sqrt(h(2a-h))]
for the volume of such a tank, where a = 5 and h, the depth of the liquid, lies between 0 and a = 5, our problem is to determine h when given various convenient values of V(h).
I find V(5) = 392.7 sq meters, approximately.
Suppose we want to determine h so that
V(h) = 350 sq meters.
I'll use Newton's method on the function
f(h) = V(h)-350.
Letting h_1 be our first guess, which we can take as, say, 4.5, the next guess comes from
h_2 = h_1-f(h_1)/f'(h_1).
All other guesses follow the same pattern.
I find
h_2 = 4.572538...
h_3 = 4.572487...
So, it looks as if a depth of 4.57 yields 350 sq meters. You could, of course, specify an amount in liters and, before using the above formulas, convert the liters to square meters.
By f'(h_1) I mean the derivative (quite messy) of f, evaluated at h_1.
Mark asked for a simpler method; Doctor Jerry added:
I don't see an easier way of solving the problem. However, I can offer a simplified version of the iteration formula, with the differentiation already done.
0.5*(5 + h_1 - (5*(-7 + 5*ArcCos[1 - h_1/5]))
h_2 = ---------------------------------------------
Sqrt[-(-10 + h_1)*h_1])
As before, h_2 means "h sub 2" and h_1 means "h sub 1". h_{n+1} is obtained similarly from h_n.
With this I obtained the results I mentioned earlier.
The same sort of problem was raised in 2002 on Car Talk, and Dr. Ian discussed it here, with a reference to our FAQ:
1/4 Tank Dipstick Problem (from Car Talk)
Many of the links at the end of that are dead, but you can find the original Car Talk episode and listener discussions by searching.
When the ends aren’t flat
We have had many questions about tanks with “dished” ends of various shapes. They may be hemispherical, as here, where the tank stands vertically:
Gas in a Cylindrical Tank with Hemispheric Caps
For a horizontal tank with ellipsoidal ends (a generalization of hemispheres), see:
Volume of a Horizontal Cylindrical Tank with Elliptic Heads
Volume of a Rounded Horizontal Tank
The first of these refers to the following for the volume of a partially filled ellipsoid:
Variable Volumes in an Oblate Spheroid
The formula turns out to be
\(\displaystyle V = L\left(R^2 \cos^{-1}\left(\frac{R-h}{R}\right) – (R-h)\sqrt{2Rh-h^2}\right)
+ \frac{a}{R}\frac{\pi}{3}\left(3hR – h^2\right)h\)
where L is the length of the cylindrical part, R is the radius of the cylinder, h is the depth of fluid and a is the thickness of the ellipsoidal ends.
In relative terms, using my \(f = \frac{h}{2R}\) from above, this becomes
\(\displaystyle V = L R^2 \left(\cos^{-1}\left(1-2f\right) – 2(1-2f)\sqrt{f(1-f)}\right)+ \frac{4}{3} \pi aR^2 f^2(3-2f)\)
If a = R, the ends are hemispheres.
In contrast to that, here is an elliptical cylinder (flat ends, elliptical cross-section):
Liquid in an Elliptical Tank
Segment of an Ellipse
That style of tank may be quite common; but it is also likely that what many people think of as an ellipse might really be some other sort of oval, such as a rectangle with semicircular ends, which I
don’t think we’ve ever covered.
Here is the formula from the first of these pages, with a = horizontal semiaxis and b = vertical semiaxis, and d = depth:
\(\displaystyle V = L\frac{a}{b}\left[\frac{\pi b^2}{2} + (d – b) \sqrt{b^2-(d-b)^2} + b^2 \sin^{-1}\left(\frac{d}{b}-1\right)\right]\)
For consistency with our other formulas, let’s replace d with h, simplify, then use my f:
\(\displaystyle V = Lab\left[\frac{\pi}{2} + \left(1-\frac{h}{b}\right) \sqrt{2\frac{h}{b} – \left(\frac{h}{b}\right)^2} – \sin^{-1}\left(1 – \frac{h}{b}\right)\right]\) \( = Lab\left[\cos^{-1}\left
(1-2f\right) +2(1-2f) \sqrt{f(1-f)}\right]\)
This becomes the same as for an ordinary cylinder when we let a = b = R.
But the most frustrating case over the years has been the case of spherical cap ends. I think there have been dozens of questions about this case, where each end is a portion of a sphere, and we have
found that the formula is too complicated to even try to post in the archives. (We have either given a table of values obtained from math software, or sent a PDF of the answer, which could not be
easily posted, and which appears to be lost.) I have occasionally searched for existing sources on this case, and failed.
Just recently, however, I ran across a question (on another site) with a picture, and the caps looked to me more parabolic than spherical. I realized that even if they really are spherical, this
might be easier to work with than the spherical cap, and I worked out an exact formula. It still isn’t pretty, but here it is:
\(\displaystyle V = LR^2\left(1+g\right)\cos^{-1}(1-2f) + 2(1-2f)
\sqrt{f(1-f)} \left(\frac{g(8f^2-8f-3)}{3}-1\right)\)
where L is the length of the cylinder, R is the radius, B is the “bulge” of the end, h is the depth of the liquid, \(f = \frac{h}{2R}\), and
\(g = \frac{B}{L}\) . I obtained this by integrating parabolic horizontal cross-sections, and again by integrating circular segment cross-sections (with a little help from WolframAlpha) for
After working that out, I found the following detailed article from Oil News of 1920 (Computation of Gauge Tables for Horizontal Tanks, Carl D. Miller), including a simplified version of the formula:
There, D is the diameter, h is the liquid depth, and \(A_T\) is the area of the segment of the circle expressed as a fraction of the area of the whole circle, which is \(\cos^{-1}\left(1-2\frac{h}{D}
\right) –
\left(1-2 \frac{h}{D} \right )-2\sqrt{\frac{h}{D} – \frac{h}{D}^2}\). I think there’s a small problem with this, as it gives close to 4 times what my formula gives, perhaps because they use the
diameter. I don’t know how they did their approximation.
This post has taken more time than usual, as I compared various formulas and put them in consistent formats. Please let me know if you find any errors!
1 thought on “Measuring Water in a Tank”
Leave a Comment
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://www.themathdoctors.org/measuring-water-in-a-tank/","timestamp":"2024-11-06T10:38:32Z","content_type":"text/html","content_length":"126527","record_id":"<urn:uuid:481c3f48-4dea-41a3-be6e-50218b1ce427>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00088.warc.gz"} |
Bayesian Model Selection - (Data, Inference, and Decisions) - Vocab, Definition, Explanations | Fiveable
Bayesian Model Selection
from class:
Data, Inference, and Decisions
Bayesian model selection is a statistical approach used to evaluate and compare different models based on their likelihood and prior distributions, allowing for the identification of the best-fitting
model given the data. This method incorporates prior beliefs about models and updates them with evidence from observed data, leading to a posterior probability for each model. It emphasizes
quantifying uncertainty in model choice and can account for model complexity through penalties for overfitting.
congrats on reading the definition of Bayesian Model Selection. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Bayesian model selection involves computing the Bayes factor, which compares the likelihood of two competing models given the same data.
2. One key advantage of Bayesian model selection is its ability to incorporate prior knowledge or expert opinion into the model evaluation process.
3. In Bayesian analysis, the complexity of a model can be taken into account by using techniques like Bayesian Information Criterion (BIC) or Deviance Information Criterion (DIC) to avoid
4. Unlike frequentist methods, which rely solely on p-values, Bayesian model selection provides a probabilistic framework that can yield more informative results about model uncertainty.
5. Bayesian model selection can be computationally intensive, especially for complex models, often requiring advanced techniques like Markov Chain Monte Carlo (MCMC) methods for estimation.
Review Questions
• How does Bayesian model selection use prior distributions in evaluating different models?
□ Bayesian model selection uses prior distributions to incorporate existing beliefs about a model's parameters before analyzing new data. By combining these priors with the likelihood of
observing the data under each model, it generates posterior probabilities that reflect both prior knowledge and empirical evidence. This allows for a more nuanced understanding of how well
each model fits the data, compared to methods that do not consider prior information.
• Discuss the significance of the Bayes factor in comparing models during Bayesian model selection.
□ The Bayes factor is crucial in Bayesian model selection as it quantifies the strength of evidence provided by the data in favor of one model over another. It is calculated as the ratio of the
marginal likelihoods of two competing models and allows researchers to determine which model is more plausible given the observed data. A higher Bayes factor indicates stronger evidence for
one model compared to another, aiding in making informed decisions about model choice.
• Evaluate how Bayesian model selection addresses issues of overfitting in complex models compared to traditional statistical methods.
□ Bayesian model selection tackles overfitting by incorporating penalties for model complexity directly into the evaluation process through methods like BIC or DIC. This contrasts with
traditional statistical methods that may focus solely on goodness-of-fit without accounting for how well a model generalizes to new data. By incorporating complexity into its framework,
Bayesian model selection provides a more balanced approach, ensuring that simpler models with good predictive performance can compete effectively against more complex ones, ultimately leading
to better overall decision-making regarding which model to use.
"Bayesian Model Selection" also found in:
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/data-inference-and-decisions/bayesian-model-selection","timestamp":"2024-11-07T19:01:56Z","content_type":"text/html","content_length":"163413","record_id":"<urn:uuid:18bbbd5d-8730-43cd-83a5-e6d713dc6ed1>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00840.warc.gz"} |
Dynamic Programming
Dynamic Programming refers to a very large class of algorithms. The idea is to break a large problem down (if possible) into incremental steps so that, at any given stage, optimal solutions are known
to sub-problems. When the technique is applicable, this condition can be extended incrementally without having to alter previously computed optimal solutions to subproblems. Eventually the condition
applies to all of the data and, if the formulation is correct, this together with the fact that nothing remains untreated gives the desired answer to the complete problem.
It is often the case in D.P. that the intermediate desirable condition implies more than seems to be strictly needed for the final answer. For example, the linear-time Fibonacci function, recursive
or iterative, could be said to be a DPA. It holds (at least) two values, fib(i-1) and fib(i). When `i' gets to `n', only one of these values is needed, fib(i)=fib(n). However it is possession of the
two values that allows the linear-time algorithm.
Examples of dynamic programming include | {"url":"https://allisons.org/ll/AlgDS/Dynamic/","timestamp":"2024-11-02T11:42:16Z","content_type":"text/html","content_length":"3699","record_id":"<urn:uuid:cfdf62e4-6938-45b5-94b5-64fea4b9e0b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00315.warc.gz"} |
Graphing Software
TeraPlot Graphing Software
TeraPlot scientific plotting software gives you everything you need for publication quality graph plotting in science and engineering. At its core is the ability to create 2D and 3D plots based on
tabular data and/or mathematical expressions. Single plot types can be used individually within a graph, or multiple plots can be combined to construct graphs ranging from basic line plots, through
3D scatter plots, to complex 3D visualizations comprising analytical expressions, tabular data, 3D objects, annotation, and image overlay. In addition to these primary features, a range of
capabilities such as flexible data import, and the ability to create graphs with just a few mouse clicks via graph wizards, make TeraPlot a highly effective scientific visualization tool. The program
also provides curve fitting and data analysis features. Follow the link to the overview page for a more detailed outline..
Evaluate TeraPlot
TeraPlot is a Microsoft Windows desktop application (Windows XP, Windows Vista, Windows 7, 8 and 10 ). Download a fully functional 30-day evaluation version..
Surface Plotting
Plot and combine surfaces based on mathematical expressions, regular gridded data, or arbitrary (x, y, z) points. Surfaces can be plotted in cartesian, polar, and cylindrical coordinate systems.
Display features include: colour mapping, texture mapping, overlay of secondary data, transparency, 3D contour display (lines and/or filled contours), general 3D transformation, wire frame display.
Surface plotting is a primary requirement of any 3D graphing software and TeraPlot caters for virtually any scenario.
Contour Plotting
Create contour plots based on functions, regular, or irregular data. Display contours and/or colourmapped filled levels. Add annotation labels with a wide range of label configuration options such as
font, orientation, foreground and background colour. Because contour plots can be based on mathematical expressions, they can be used to draw implicit functions. For example, plotting xy - 5 = 0,
would involve plotting z = xy - 5, and displaying the z = 0 contour value.
Windows Store App
If you don't need the full power of a desktop application, and/or you'd like the flexibility of graphing software that runs on both a PC and a tablet, try TeraPlot LT, a Windows Store graph plotting
app. TeraPlot LT allows you to create 2D line plots and 3D surface plots in various coordinate systems. As with the desktop version, TeraPlot LT plots can be based on mathematical expressions or
supplied data. TeraPlot LT uses the muParser scripting engine.
Scatter Plots
Teraplot graphing software allows you to create 2D scatter plots or 3D scatter plots in a range of coordinate systems. Use scatter point colour and size to visualise up to two further variables in
addition to z value. Distinguish between data sets using different symbol types (e.g. sphere, cube, cone). Combine scatter plots with planes and text to create multidimensional analysis diagrams.
Scatter plots can also be created with a few mouse clicks using Graph Wizards.
Isosurfaces are surfaces of constant w for functions of the form w = f(x, y, z), i.e. isosurfaces are the 3d equivalent of contour plots. The data from which the isosurfaces are generated can be
defined as a formula (analytical plot), or loaded from a file. For an analytical plot, the data is assumed to lie on a regular rectangular 3D grid. For data loaded from a file, the data can either
lie on a regular grid, or arbitrary individual cells can be specified. Isosurfaces can also be used to plot implicit functions.
Line/Series Plots
Create line plots based on arbitrary x-y values, or on Excel-like series/category data (series/category data can also be used to produce clustered bar plots, stacked bar plots, and area plots via
graph wizards). Enhance 2D plots using symbols, captions, legends, shading and colourmaps. Combine plots based on mathematical functions and tabular data in the same graph. Place text at specific x-y
positions using text plots. Copy graphs to the clipboard or export to image files.
Graph Wizards
Create fully annotated, publication quality graphs with just a few mouse clicks via Graph Wizards. With graph wizards, graph creation is broken down into a set of input stages: plot choice, data
input, plot parameter settings, and annotation (e.g. captions, colourmaps, legends) settings. Wizards are available for Excel-like series/category data, financial data, and XYZ point data. A graph
created via a graph wizard can be used as-is, or as a starting point for a more complex graph.
Function Plotting
TeraPlot brings the power of scripting to function graphing software. Using VBScript, function definitions can range from something as simple as the single line y = sin(x), to complex multi-line
function definitions containing constants, variables and conditional expressions. Graphs of implicit functions can be easily created, and mathematical concepts explored with ease in a range of
coordinate systems in both 2D and 3D.
Flexible Data Import
Take advantage of a flexible range of data import features. Data can be entered into a plot manually, pasted from Excel, or imported into the graph from a previously created internal spreadsheet.
These internal spreadsheets are created by importing data from character-delimited text files using an Excel-like data import feature. At each point in the plot creation, the spreadsheets are always
available, providing easy access to your data.
Data Analysis
Data analysis capabilities are provided in the form of statistical functions (Normal, Exponential, Lognormal, Weibull, Gamma, Binomial) and standard statistical analysis plots such as histograms
(reporting Mean, Median, Variance, Standard Deviation, Skewness and Kurtosis), box plots, probability plots and curve fitting via linear regression. The program also features a general
multidimensional nonlinear regression tool based on the Levenberg–Marquardt Method.
Program Automation
Automation provides the ability to remotely start TeraPlot via a driver program written in e.g. VB.Net or C#. Much of the menu-driven functionality which would normally be accessed manually can then
be accessed and manipulated programmatically. Controlling the program from a programming language provides a wide range of possibilities, from writing code which interactively controls the program
via your own dialogs, to creating movies utilising any combination of program features you wish.
Wide Range of Plots
Over 30 plot types, applicable over a wide range of disciplines, are available to meet your graphing software requirements. For 2D plotting there are basic line plots, series based plots such as
stacked/clustered bars, statistical plots such as histograms and box plots, and contour plots. 3D plot types include surface plots based on regular or irregular data, scatter plots, 3D lines, bars,
and object plots, which allow primitive objects to be laid out in 3D. | {"url":"https://www.teraplot.com/","timestamp":"2024-11-02T09:12:43Z","content_type":"application/xhtml+xml","content_length":"20115","record_id":"<urn:uuid:40a9e7d2-19be-477d-9a41-c7a92bece954>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00056.warc.gz"} |
Probability Shortcut Tricks Tips & Concept : How to Solve Probability Question Short Trick ? - GovernmentAdda
Home Short Tricks Quantitative Aptitude Tricks Probability Shortcut Tricks Tips & Concept : How to Solve Probability Question...
Probability Shortcut Tricks Tips & Concept : How to Solve Probability Question Short Trick ?
Probability Short Tricks & Tips
Probability Short-Cut Tricks & Tips : Probability Short-Cut Tricks & Tips Question Pdf for Banking, SSC, RRB, FCI, Railway, UPSC, State PCS, Insurance & other Competitive exams. Probability Short-Cut
Tricks & Tips shortcut Tricks Pdf, Probability Short-Cut Tricks & Tips MCQ, Probability Short-Cut Tricks & Tips Objective Question & Answer Pdf. “Probability Short-Cut Tricks & Tips Questions PDF” In
this post we are providing you the Probability Short-Cut Tricks & Tips pdf with detailed solution & Short Tricks. So that you can easily get the logic of question. This Probability Short-Cut Tricks &
Tips Pdf we are Providing is free to download. ” Most Important Probability Short-Cut Tricks & Tips Question PDF with Answers“
Probability Short-Cut Tricks & Tips Plays a vital role in Exam. In every exam you will get at least 5-10 questions from this topic. So candidates must focus on this topic and download this
Probability Short-Cut Tricks & Tips pdf to get important questions with best solution regarding Probability Short-Cut Tricks & Tips. We have put all Previous Year Questions of Probability Short-Cut
Tricks & Tips that are Asked in various Govt & Private Exam.
In everyday life, we come across the situations having either some certainty or uncertainty and we are generally interested to measure this certainty or uncertainty. We want to measure that up to
what extent this particular situation would occur. We generally achieve it qualitatively; not able to calculate it quantitatively. So what about if we want to measure it quantitatively? That is
achieved with the help of the Theory of Probability.
Probability is the Measure of Uncertainty.
Experiment:An operation which can produce some well-defined outcomes is called an experiment.
Random Experiment:
An experiment in which all possible outcomes are know and the exact output cannot be predicted in advance, is called a random experiment.
i. Rolling an unbiased dice.
ii. Tossing a fair coin.
iii. Drawing a card from a pack of well-shuffled cards.
iv. Picking up a ball of certain colour from a bag containing balls of different colours.
⇒When we throw a coin, then either a Head (H) or a Tail (T) appears.
⇒A dice is a solid cube, having 6 faces, marked 1, 2, 3, 4, 5, 6 respectively. When we throw a die, the outcome is the number that appears on its upper face.
⇒A pack of cards has 52 cards.
It has 13 cards of each suit, name Spades, Clubs, Hearts and Diamonds.
Cards of spades and clubs are black cards.
Cards of hearts and diamonds are red cards.
There are 4 honours of each unit.
There are Kings, Queens and Jacks. These are all called face cards.
Sample Space:
When we perform an experiment, then the set S of all possible outcomes is called the sample space
In tossing a coin, S = {H, T}
If two coins are tossed, the S = {HH, HT, TH, TT}.
In rolling a dice, we have, S = {1, 2, 3, 4, 5, 6}.
Probability of Occurrence of an Event:
Let S be the sample space and E be the event
1) ?(?) = ?
2) ≤ ??(?) ≤ ?
3) ?(?) = ? → ? ?? ?????? ? ???? ?????
4) ?(?) = ? → ? ?? ?????? ??????????
5) ?(?) + ?(?̅) = 1
EXAMPLE:-A fair coin is tossed at random. Find the probability of getting:
1) Head
2) Tail
EXAMPLE:- Two unbiased coins are tossed simultaneously at random.
Find the probability of getting:
1) Head on the first coin.
2) Head on the second coin.
3) Head on both the coins.
4) No heads.
5) At least one head.
6) At most one head.
1. Lets take an example..
Set your timer for 60 seconds, and see if you can get the correct answer. Then read below for the full explanation, and see if you’ve fallen into a trap or not!
Question – Sixty percent of the members of a study group are women, and 45 percent of those women are lawyers. If one member of the study group is to be selected at random, what is the probability
that the member selected is a woman lawyer?
(A) 0.10
(B) 0.15
(C) 0.27
(D) 0.33
(E) 0.45
Although the question uses the word “probability,” the concept it tests for and the trap laid within are percent-related—the piece of information you need from the wide field of probability to solve
this problem is the basic formula:
probability = number of wanted outcomes / total number of outcomes
In essence, probability, like a percentage, is a ratio between a part and a whole, expressed as a fraction.
So why did this relatively simple problem catch my eye? Precisely because it is deceptively easy, which is why a decent percent of GMAT test takers will get it wrong, at least when put under a time
crunch. For many test takers, the following (mistaken) thought pattern will ensue:
60% of the members are women. Imagine a pie chart, with a 60% chunk marked “women.”
Now, 45% are lawyers. Take a chunk of 45% (almost half of the pie), out of the original 60% chunk, and that’s your percent of women lawyers—45%, or 0.45 (answer choice E).
In extreme rush cases, a test taker may even forget what he’s looking for in the first place. Once you’re imagining a 45% chunk taken out of the 60% chunk, it’s deceptively easy to fall into the trap
of focusing on what remains: a 15% “slice,” which will lead the test-taker in a hurry into choosing B in the rush to move on to the next question.
Both of these thought processes and the resulting answer choices are wrong. The stumbling point that the GMAT test-writers are counting on is the failure to ask the simple questions whenever the word
“percent” appears: What is the whole? What number or quantity is the percent taken out of?
The first percent (60% women) is indeed taken out of the members of the study group. The next line has a crucial phrase: 45% of those women are lawyers. So the next percent is not taken out of the
entire pie chart, but out of the 65% chunk alone. We’re looking for 45% of the group titled women, which happens to be given as a percent of the whole, not just 45% of the entire group.
The actual calculation is therefore 45% of 60%, or {45/100}*{60/100} (think of any “of” in these cases as a multiplication sign).
One last note: instead of actually calculating the above expression, just ballpark it. The group you seek (women lawyers) is ‘slightly less than half’ of the women (as 45% is just under 50%). The
right answer will therefore need to be something slightly smaller than {1/2}*0.6 = 0.3. Only one answer choice out of the five answer choices presented fits that description, and that is C 0.27.
Answer choices A and B are too small, and D and E are already over half of 60%.
2. Consider the following question:
A canoe has two oars, left and right. Each oar either works or breaks. The failure or non-failure of each oar is independent of the failure or non-failure of the other. You can still row the canoe
with one oar. The probability that the left oar works is 3/5. The probability that the right oar works is also 3/5. What is the probability that you can still row the canoe?
A) 9/25
B) 10/25
C) 6/10
D) 2/3
E) 21/25
1. The wrong way
The temptation is to multiply the two probabilities given to reach the answer 9/25. Whenever you get to an answer choice very quickly, particularly when that answer is A, I would look at the question
again! Answer choice A is the first answer you see. If you are in a hurry and option A looks right, many test takers will go for A.
This calculation only gives you the probability that both oars work.
To get the right answer, you would also have to add the probability that the left oar works and the right fails.
Then you would also have to add the probability that the right works, but the left fails.
All this would be possible, but slow. Is there a better way? Yes!
2. The right way
Simply look at the question from the other side. What is the probability that you can’t row the canoe? This would be {2/5}*{2/5} = 4/25.
Using the idea that the probability of something happening is 1- the probability that it doesn’t happen, you can use the following equation to reach the right answer: 1 – 4/25 = 21/25. Answer choice
We call this the Forbidden Method: subtracting the ‘forbidden’ or unwanted probabilities from the total probability, which is 1.
When two dice are thrown The probability of sample points for the given sum can be known with below logic.
Total No. of sample points when two dice are thrown = (6*6) =36
Question 1: Find the probability to get sum 3 when two dice are thrown simultaneously?
Sol : P(S) = P(3) / P(Total) = 2/36 = 1/13.
Note: ( From the figure we know for sum 3 the probability of sample points are 2)
Question 2: Find the probability to get sum 10 when two dice are thrown simultaneously?
Sol: Probability to get sum 10 = P(10)/P(total )
= 3/36 ( From the figure we know for sum 10 the probability of sample points are 3)
When three dice are thrown The probability of sample points for the given sum can be known with below logic.
Example Question: Find the probability to get sum 5 when three dice are thrown simultaneously?
Sol :
Probability to get sum 5 = P(5)/P(total no. of sample points )
= 6/216.
Note: ( From the above figure we know for sum 5 the probability of sample points are 6)
Practice Set On Probability
1. A bag contains 6 red, 2 blue and 4 green balls. 3 balls are chosen at random. What is the probability that at least 2 balls chosen will be red?
A) 2/7
B) 1/2
C) 1/3
D) 2/5
E) 3/7
2. Tickets numbered 1 to 250 are in a bag. What is the probability that the ticket drawn has a number which is a multiple of 4 or 7?
A) 83/250
B) 89/250
C) 77/250
D) 93/250
E) 103/250
3. From a deck of 52 cards, 3 cards are chosen at random. What is the probability that all are face cards?
A) 14/1105
B) 19/1105
C) 23/1105
D) 11/1105
E) 26/1105
4. One 5 letter word is to be formed taking all letters – S, A, P, T and E. What is the probability that this the word formed will contain all vowels together?
A) 2/5
B) 3/10
C) 7/12
D) 3/5
E) 5/12
5. One 5-digit number is to be formed from numbers – 0, 1, 3, 5, and 6 (repetition not allowed). What is the probability that number formed will be even?
A) 8/15
B) 7/16
C) 7/15
D) 3/10
E) 13/21
Directions (6-8): There are 3 bags containing 3 colored balls – Red, Green and Yellow.
Bag 1 contains:
24 green balls. Red balls are 4 more than blue balls. Probability of selecting 1 red ball is 4/13
Bag 2 contains:
Total balls are 8 more than 7/13 of balls in bag 1. Probability of selecting 1 red ball is 1/3. The ratio of green balls to blue balls is 1 : 2
Bag 3 contains:
Red balls equal total number of green and blue balls in bag 2. Green balls equal total number of green and red balls in bag 2. Probability of selecting 1 blue ball is 3/14.
6. 1 ball each is chosen from bag 1 and bag 2, What is the probability that 1 is red and other blue?
A) 15/128
B) 21/115
C) 17/135
D) 25/117
E) 16/109
7. Some green balls are transferred from bag 1 to bag 3. Now probability of choosing a blue ball from bag 3 becomes 3/16. Find the number of remaining balls in bag 1.
A) 60
B) 58
C) 52
D) 48
E) 44
8. Green balls in ratio 4 : 1 from bags 1 and 3 respectively are transferred to bag 4. Also 4 and 8 red balls from bags 1 and 3 respectively . Now probability of choosing green ball from bag 4 is 5/
11. Find the number of green balls in bag 4?
A) 12
B) 15
C) 10
D) 9
E) 11
Directions (9-10): There are 3 people – A, B and C. Probability that A speaks truth is 3/10, probability that B speaks truth is 3/7 and probability that C speaks truth is 5/6. For a particular
question asked, at most 2 people speak truth. All people answer to a particular question asked.
9. What is the probability that B will speak truth for a particular question asked?
A) 7/18
B) 14/33
C) 4/15
D) 9/28
E) 10/33
10. A speaks truth only when B does not speak truth, then what is the probability that C does not speak truth on a question?
A) 11/140
B) 21/180
C) 22/170
D) 13/140
E) None of these
• There are 100 tickets in a box numbered 1 to 100. 3 tickets are drawn at one by one. Find the probability that the sum of number on the tickets is odd.
A) 2/7
B) 1/2
C) 1/3
D) 2/5
E) 3/7
• There are 4 green and 5 red balls in first bag. And 3 green and 5 red balls in second bag. One ball is drawn from each bag. What is the probability that one ball will be green and other red?
A) 85/216
B) 34/75
C) 95/216
D) 35/72
E) 13/36
• A bag contains 2 red, 4 blue, 2 white and 4 black balls. 4 balls are drawn at random, find the probability that at least one ball is black.
A) 85/99
B) 81/93
C) 83/99
D) 82/93
E) 84/99
• Four persons are chosen at random from a group of 3 men, 3 women and 4 children. What is the probability that exactly 2 of them will be men?
A) 1/9
B) 3/10
C) 4/15
D) 1/10
E) 5/12
• Tickets numbered 1 to 120 are in a bag. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?
A) 8/15
B) 5/16
C) 7/15
D) 3/10
E) 13/21
• There are 2 people who are going to take part in race. The probability that the first one will win is 2/7 and that of other winning is 3/5. What is the probability that one of them will win?
A) 14/35
B) 21/35
C) 17/35
D) 19/35
E) 16/35
• Two cards are drawn at random from a pack of 52 cards. What is the probability that both the cards drawn are face card (Jack, Queen and King)?
A) 11/221
B) 14/121
C) 18/221
D) 15/121
E) 14/221
• A committee of 5 people is to be formed from among 4 girls and 5 boys. What is the probability that the committee will have less number of boys than girls?
A) 7/12
B) 7/15
C) 6/13
D) 5/12
E) 7/13
• A bucket contains 2 red balls, 4 blue balls, and 6 white balls. Two balls are drawn at random. What is the probability that they are not of same color?
A) 5/11
B) 14/33
C) 2/5
D) 6/11
E) 2/3
• A bag contains 5 blue balls, 4 black balls and 3 red balls. Six balls are drawn at random. What is the probability that there are equal numbers of balls of each color?
A) 11/77
B) 21/77
C) 22/79
D) 13/57
E) 15/77
Directions (1-3): An urn contains some balls colored white, blue and green. The probability of choosing a white ball is 4/15 and the probability of choosing a green ball is 2/5. There are 10 blue
1. What is the probability of choosing one blue ball?
A) 2/7
B) 1/4
C) 1/3
D) 2/5
E) 3/7
2. What is the total number of balls in the urn?
A) 45
B) 34
C) 40
D) 30
E) 42
3. If the balls are numbered 1, 2, …. up to number of balls in the urn, what is the probability of choosing a ball containing a multiple of 2 or 3?
A) 3/4
B) 4/5
C) 1/4
D) 1/3
E) 2/3
4. There are 2 brothers A and B. Probability that A will pass in exam is 3/5 and that B will pass in exam is 5/8. What will be the probability that only one will pass in the exam?
A) 12/43
B) 19/40
C) 14/33
D) 21/40
E) 9/20
5. If three dices are thrown simultaneously, what is the probability of having a same number on all dices?
A) 1/36
B) 5/36
C) 23/216
D) 1/108
E) 17/216
6. There are 150 tickets in a box numbered 1 to 150. What is the probability of choosing a ticket which has a number a multiple of 3 or 7?
A) 52/125
B) 53/150
C) 17/50
D) 37/150
E) 32/75
7. There are 55 tickets in a box numbered 1 to 55. What is the probability of choosing a ticket which has a prime number on it?
A) 3/55
B) 5/58
C) 8/21
D) 16/55
E) 4/13
8. A bag contains 4 white and 5 blue balls. Another bag contains 5 white and 7 blue balls. What is the probability of choosing two balls such that one is white and the other is blue?
A) 61/110
B) 59/108
C) 45/134
D) 53/108
E) 57/110
9. The odds against an event are 2 : 3 and the odds in favor of another independent event are 3 : 4. Find the probability that at least one of the two events will occur.
A) 11/35
B) 27/35
C) 13/35
D) 22/35
E) 18/35
10. The odds against an event are 1 : 3 and the odds in favor of another independent event are 2 : 5. Find the probability that one of the event will occur.
A) 17/28
B) 5/14
C) 11/25
D) 9/14
E) 19/28
1. From a pack of 52 cards, 1 card is chosen at random. What is the probability of the card being diamond or queen?
A) 2/7
B) 6/15
C) 4/13
D) 1/8
E) 17/52
2. From a pack of 52 cards, 1 card is drawn at random. What is the probability of the card being red or ace?
A) 5/18
B) 7/13
C) 15/26
D) 9/13
E) 17/26
3. There are 250 tickets in an urn numbered 1 to 250. One ticket is chosen at random. What is the probability of it being a number containing a multiple of 3 or 8?
A) 52/125
B) 53/250
C) 67/125
D) 101/250
E) 13/25
4. There are 4 white balls, 5 blue balls and 3 green balls in a box. 2 balls are chosen at random. What is the probability of both balls being non-blue?
A) 23/66
B) 5/18
C) 8/21
D) 7/22
E) 1/3
5. There are 4 white balls, 3 blue balls and 5 green balls in a box. 2 balls are chosen at random. What is the probability that first ball is green and second ball is white or green in color?
A) 1/3
B) 5/18
C) 1/2
D) 4/21
E) 11/18
6. 2 dices are thrown. What is the probability that there is a total of 7 on the dices?
A) 1/3
B) 2/7
C) 1/6
D) 5/36
E) 7/36
7. 2 dices are thrown. What is the probability that sum of numbers on the two dices is a multiple of 5?
A) 5/6
B) 5/36
C) 1/9
D) 1/6
E) 7/36
8. There are 25 tickets in a box numbered 1 to 25. 2 tickets are drawn at random. What is the probability of the first ticket being a multiple of 5 and second ticket being a multiple of 3.
A) 5/11
B) 1/4
C) 2/11
D) 1/8
E) 3/14
9. What is the probability of selecting a two digit number at random such that it is a multiple of 2 but not a multiple of 14?
A) 17/60
B) 11/27
C) 13/30
D) 31/60
E) 17/30
10. There are 2 urns. 1st urn contains 6 white and 6 blue balls. 2nd urn contains 5 white and 7 black balls. One ball is taken at random from first urn and put to second urn without noticing its
color. Now a ball is chosen at random from 2nd urn. What is the probability of the second ball being a white colored ball?
A) 11/13
B) 6/13
C) 5/13
D) 5/12
E) 11/12
1. A bag contains 12 white and 18 black balls. Two balls are drawn in succession without replacement. What is the probability that first is white and second is black?
A) 36/135
B) 36/145
C) 18/ 91
D) 30/91
E) None of these
2. Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?
A) 3/16
B) 1/8
C) 3/4
D) 1/2
E) None of these
3. In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected is:
A) 21/46
B) 21/135
C) 42/135
D) Can’t be determined
E) None of these
4. A card is drawn from a pack of 52 cards. The probability of getting a queen of club or a king of heart is?
A) 3/26
B) 3/52
C) 1/26
D) 1/4
E) None of these
5. A bag contains 4 white, 5 red and 6 blue balls. Three balls are drawn at random from the bag. The probability that all of them are blue, is:
A) 1/91
B) 2/91
C) 3/91
D) 4/91
E) None of these.
6. A bag contains 2 yellow, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?
A) 5/7
B) 1/21
C) 10/21
D) 2/9
E) None of these
7. Three coins are tossed. What is the probability of getting at most two tails?
A) 1/8
B) 5/8
C) 3/8
D) 7/8
E) None of these
8. One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card (Jack, Queen and King only)?
A) 1/13
B) 2/13
C) 3/13
D) 3/52
E) None of these
9. P and Q sit in a ring arrangement with 10 persons. What is the probability that P and Q will sit together?
A) 2/11
B) 3//11
C) 4/11
D) 5/11
E) None of these
10. Two dice are thrown simultaneously. Find the probability of getting a multiple of 2 on one dice and multiple of 3 on the other dice.
A) 1/9
B) 11/36
C) 13/36
D) Data inadequate
E) None of these
2. C
3. A
4. C
5. D
6. C
7. D
8. C
9. A
10.BExplanation:1. The probability that first ball is white= 12c1/30c1= 2/5
Since, the ball is not replaced; hence the number of balls left in bag is 29.
Hence the probability the second ball is black= 18c1/29c1 =18/29
Required probability = 2/5*18/29 = 36/1452. In a simultaneous throw of two dice, we have n(S) = (6 x 6) = 36.
Then, E = {(1, 2), (1, 4), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (3, 4),
(3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 2), (5, 4), (5, 6), (6, 1),
(6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
n(E) = 27.
so probability = 27/36 = 3/43. Probability = 10c1*15c2/25c3
= 21/464. 2/52 = 1/265. 6c3/15c3 =4/916. 5c2/7c2 = 10/217. 7/88. 12/52 =3/139. n(S)= number of ways of sitting 12 persons at round table:
Since two persons will be always together, then number of persons:
So, 11 persons will be seated in (11-1)!=10! ways at round table and 2 particular persons will be seated in 2! ways.
n(A)= The number of ways in which two persons always sit together =10!×2
So probability = 10!*2!/11!= 2/1110. 11/36
LEAVE A REPLY Cancel reply | {"url":"https://governmentadda.com/probability-tricks-tips/","timestamp":"2024-11-07T16:57:24Z","content_type":"text/html","content_length":"302547","record_id":"<urn:uuid:cb4fe9b4-9446-453a-9256-667d41cc94d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00241.warc.gz"} |
What is the 300th Digit of 0.0588235294117647? An In-Depth Exploration - Inspire Zap
The world of numbers is filled with fascinating patterns and mysteries, particularly when it comes to decimals. One such curiosity involves the decimal expansion of fractions, particularly repeating
decimals. In this article, we explore the question: what is the 300th digit of 0.0588235294117647? Understanding this query not only unveils the significance of specific digits in decimals but also
sheds light on broader mathematical concepts. This decimal represents the fraction 117\frac{1}{17}171, which has a unique repeating pattern. By delving into the calculation, patterns, and
implications of this repeating decimal, we can uncover a deeper appreciation for the beauty and complexity of mathematics.
Understanding Decimal Expansions
What Are Decimal Expansions?
Decimal expansions are representations of numbers in the base-10 numeral system, where a number can be expressed as a whole number combined with a fractional part. For instance, the number 3.14
consists of a whole number (3) and a fractional part (0.14). Decimal expansions can be finite or infinite; finite decimals terminate, while infinite decimals can either repeat or be non-repeating.
The beauty of decimal expansions lies in their ability to convey not just quantities but also relationships between numbers, especially when we encounter fractions.
The Importance of Repeating Decimals
Repeating decimals are particularly interesting because they provide insight into the behavior of fractions in the decimal system. When a fraction results in a repeating decimal, it indicates a
relationship between its numerator and denominator that leads to a cycle in its decimal representation. This periodic nature is not only mathematically intriguing but also has practical implications
in fields such as computer science, engineering, and finance, where precise numerical representations are crucial. Understanding repeating decimals allows for better numerical calculations and can
enhance our comprehension of the underlying mathematical structures.
The Decimal Representation of 117\frac{1}{17}171
Calculating 117\frac{1}{17}171
To comprehend the decimal representation of 117\frac{1}{17}171, we can perform long division. Dividing 1 by 17, we start by determining how many times 17 fits into 1. Since 17 is larger than 1, we
move to 10, which 17 still does not fit into. Continuing this process, we extend our division into decimal places, yielding:
1.0000000… divided by 17 results in 0.0588235294117647…
This process reveals the repeating nature of the decimal, as it continues indefinitely, cycling through a set sequence. The method of long division is fundamental in understanding how decimal
representations arise from fractions.
Breaking Down 0.0588235294117647
The decimal 0.0588235294117647 is a representation of 117\frac{1}{17}171 and features a repeating block of digits. The sequence “0588235294117647” repeats infinitely, which means that if we were to
write out the decimal representation of 117\frac{1}{17}171 to any length, it would show these digits cycling. This periodicity indicates that any decimal representation of 117\frac{1}{17}171 will
always return to this sequence, making it predictable and allowing us to explore specific digits within that cycle.
Analyzing the Repeating Pattern
Identifying the Repeating Block
To identify the repeating block within 0.0588235294117647, we observe the digits after the decimal point. The entire repeating block is “0588235294117647,” consisting of 16 digits. This block repeats
indefinitely, and understanding this helps us answer specific questions about the decimal, such as the 300th digit or any other digit in its sequence.
The significance of recognizing repeating blocks lies in the ability to calculate any digit position in the decimal without writing out the entire sequence. For example, knowing the length of the
repeating block allows us to determine which digit corresponds to any given position, making complex calculations more manageable.
Length of the Repeating Sequence
The length of the repeating sequence for the decimal 0.0588235294117647 is 16 digits. This means that every 16 digits, the sequence restarts. Therefore, if we want to find a specific digit within the
decimal, we can use modular arithmetic to simplify our calculations. By dividing the desired digit position by the length of the repeating sequence, we can find the position within the repeating
block that corresponds to our query. This mathematical property allows us to answer questions like what is the 300th digit of 0.0588235294117647? with ease.
Finding the 300th Digit
Dividing to Find the Remainder
To find the 300th digit of 0.0588235294117647, we divide 300 by the length of the repeating sequence, which is 16. Performing this calculation, we get:
300÷16=18 R 12300 \div 16 = 18 \text{ R } 12 300÷16=18 R 12
This means that when we cycle through the repeating sequence 18 times, we are left with a remainder of 12. The remainder indicates the position within the repeating block that corresponds to the
300th digit. Understanding this process highlights the efficiency of using the properties of repeating decimals to pinpoint specific digits without needing to write out the entire sequence.
Locating the 12th Digit
Now that we have established that the 300th digit corresponds to the 12th digit in the repeating block “0588235294117647,” we can simply count to determine which digit this is.
1. 0 (1st digit)
2. 5 (2nd digit)
3. 8 (3rd digit)
4. 8 (4th digit)
5. 2 (5th digit)
6. 3 (6th digit)
7. 5 (7th digit)
8. 2 (8th digit)
9. 9 (9th digit)
10. 4 (10th digit)
11. 1 (11th digit)
12. 7 (12th digit)
Thus, the 12th digit, and consequently the 300th digit of 0.0588235294117647, is 7. This simple yet effective method allows us to easily find specific digits in repeating decimals without cumbersome
The Significance of the 300th Digit
Why the 300th Digit Matters
Knowing specific digits within a repeating decimal, such as the 300th digit, can provide insights into the properties of numbers and the relationships between fractions and their decimal equivalents.
In practical applications, this knowledge is particularly useful in fields like cryptography, computer science, and numerical analysis, where precise decimal representations are critical.
Understanding the placement of digits within these sequences can lead to greater efficiency in calculations and improved mathematical comprehension.
Moreover, the 300th digit exemplifies how seemingly simple mathematical concepts can lead to complex and rich inquiries. As mathematicians and researchers delve deeper into these questions, they
often uncover broader patterns and relationships that can impact various fields, from engineering to economics.
Applications of Decimal Expansions in Real Life
Decimal expansions and their properties play significant roles in numerous real-world applications. For example, in finance, accurate decimal representations are crucial for calculations involving
interest rates, loan payments, and investment returns. In computer science, algorithms often rely on precise numerical representations to ensure accurate data processing and computational efficiency.
Furthermore, fields like physics and engineering frequently use decimal expansions to model real-world phenomena and conduct simulations.
The understanding of repeating decimals also extends to education, where teaching students about these concepts can enhance their numerical literacy and critical thinking skills. By exploring
questions such as what is the 300th digit of 0.0588235294117647, students gain a deeper appreciation for the intricacies of mathematics and its applications.
Common Misconceptions About Decimal Expansions
Misunderstanding Repeating Decimals
One common misconception about repeating decimals is that they are simply approximations of fractions. In reality, repeating decimals are exact representations of certain fractions. For example, the
decimal 0.0588235294117647 is not an approximation of 117\frac{1}{17}171; it is precisely equal to it. This distinction is crucial for understanding the nature of fractions and their decimal
Additionally, some may believe that repeating decimals are less valuable than terminating decimals. However, both types serve essential roles in mathematics, and understanding repeating decimals can
lead to greater insights into the nature of numbers and their relationships.
Clarifying the Concept of Infinite Decimals
Infinite decimals, including repeating decimals, often lead to confusion among learners. Many people find it challenging to grasp the idea that a decimal can go on forever while still being a finite
representation of a fraction. Infinite decimals can be thought of as having a limit; they approach a certain value without ever reaching an endpoint.
For instance, while the decimal representation of 13\frac{1}{3}31 is 0.333…, it is understood to represent one-third exactly, despite its infinite appearance. Understanding this concept allows for a
more profound appreciation of mathematical notation and the various ways numbers can be represented.
In conclusion, the exploration of the question what is the 300th digit of 0.0588235294117647 reveals not just a single digit but a wealth of mathematical understanding. From decimal expansions and
repeating patterns to practical applications in everyday life, the study of decimals is a gateway to uncovering the beauty of mathematics. Through the process of calculating specific digits, we have
learned about the nature of fractions and the significance of their decimal representations.
As we continue to delve into the intricacies of mathematics, questions like the 300th digit of a decimal remind us that numbers are not just symbols but tools that unlock a deeper understanding of
the world around us. By embracing these concepts, we can enhance our numerical literacy and cultivate a lifelong curiosity for the subject.
What is the repeating pattern of 0.0588235294117647?
The repeating pattern of 0.0588235294117647 is “0588235294117647,” consisting of 16 digits that repeat indefinitely.
How can I calculate other digits in a repeating decimal?
To calculate other digits in a repeating decimal, divide the desired digit position by the length of the repeating sequence to find the corresponding position within that sequence.
Why is 117\frac{1}{17}171 significant in mathematics?
117\frac{1}{17}171 is significant because it produces a repeating decimal that illustrates the relationship between fractions and their decimal representations, providing insight into the properties
of numbers.
What are some other interesting repeating decimals?
Other interesting repeating decimals include 13=0.333…\frac{1}{3} = 0.333…31=0.333…, 16=0.1666…\frac{1}{6} = 0.1666…61=0.1666…, and 19=0.111…\frac{1}{9} = 0.111…91=0.111…, all of which have unique
repeating patterns that can be analyzed similarly.
You May Also Like | {"url":"https://inspirezap.com/what-is-the-300th-digit-of-0-0588235294117647-an-in-depth-exploration/","timestamp":"2024-11-09T22:58:49Z","content_type":"text/html","content_length":"87618","record_id":"<urn:uuid:16534ef8-54f6-4e8e-8286-fb2296381a0f>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00721.warc.gz"} |
NUM02-J. Ensure that division and remainder operations do not result in divide-by-zero errors
Division and remainder operations performed on integers are susceptible to divide-by-zero errors. Consequently, the divisor in a division or remainder operation on integer types must be checked for
zero prior to the operation. Division and remainder operations performed on floating-point numbers are not subject to this rule.
Noncompliant Code Example (Division)
The result of the / operator is the quotient from the division of the first arithmetic operand by the second arithmetic operand. Division operations are susceptible to divide-by-zero errors. Overflow
can also occur during two's-complement signed integer division when the dividend is equal to the minimum (negative) value for the signed integer type and the divisor is equal to −1 (see NUM00-J.
Detect or prevent integer overflow for more information). This noncompliant code example can result in a divide-by-zero error during the division of the signed operands num1 and num2:
long num1, num2, result;
/* Initialize num1 and num2 */
result = num1 / num2;
Compliant Solution (Division)
This compliant solution tests the divisor to guarantee there is no possibility of divide-by-zero errors:
long num1, num2, result;
/* Initialize num1 and num2 */
if (num2 == 0) {
// Handle error
} else {
result = num1 / num2;
Noncompliant Code Example (Remainder)
The % operator provides the remainder when two operands of integer type are divided. This noncompliant code example can result in a divide-by-zero error during the remainder operation on the signed
operands num1 and num2:
long num1, num2, result;
/* Initialize num1 and num2 */
result = num1 % num2;
Compliant Solution (Remainder)
This compliant solution tests the divisor to guarantee there is no possibility of a divide-by-zero error:
long num1, num2, result;
/* Initialize num1 and num2 */
if (num2 == 0) {
// Handle error
} else {
result = num1 % num2;
Risk Assessment
A division or remainder by zero can result in abnormal program termination and denial-of-service (DoS).
Rule Severity Likelihood Remediation Cost Priority Level
NUM02-J Low Likely Medium P6 L2
Automated Detection
Related Guidelines
[ISO/IEC 9899:1999] Subclause 6.5.5, "Multiplicative Operators"
[Seacord 05] Chapter 5, "Integers"
[Seacord 2015]
[Warren 02] Chapter 2, "Basics"
6 Comments
□ Code samples won't compile because of the signed keywords.
□ Handle error condition comments need to be replaced with proper exceptions
□ sl1 and sl2 are bad variable names as l can be confused with 1.
Fixed. For 2nd point, we have not been particularly consistent about how to handle errors...is there some normative text all code should rely on? My suggestion would be "// Handle Error"
(except in cases where more specific directions are called for.)
This seems to be covered in SonarQube too by S3518.
We could specify the difference between integers and floatting point numbers.
□ integer (int, long) : divide-by-zero result in a RuntimeException (witch result in abnormal program termination)
□ floatting point number (float, double) : result in an Infinite number (result in an unspecified behavior if unspecified, not necessarily abnormal termination)
I clarified that this rule only applies to integers. | {"url":"https://wiki.sei.cmu.edu/confluence/display/java/NUM02-J.+Ensure+that+division+and+remainder+operations+do+not+result+in+divide-by-zero+errors","timestamp":"2024-11-14T18:46:03Z","content_type":"text/html","content_length":"92598","record_id":"<urn:uuid:a606915c-5102-478c-b1a2-c942bf468671>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00014.warc.gz"} |
ECDSA: Elliptic Curve Signatures
The ECDSA (Elliptic Curve Digital Signature Algorithm) is a cryptographically secure digital signature scheme, based on the elliptic-curve cryptography (ECC). ECDSA relies on the math of the cyclic
groups of elliptic curves over finite fields and on the difficulty of the ECDLP problem (elliptic-curve discrete logarithm problem). The ECDSA sign / verify algorithm relies on EC point
multiplication and works as described below. ECDSA keys and signatures are shorter than in RSA for the same security level. A 256-bit ECDSA signature has the same security strength like 3072-bit RSA
signature. ECDSA uses cryptographic elliptic curves (EC) over finite fields in the classical Weierstrass form.
These curves are described by their EC domain parameters, specified by various cryptographic standards such as SECG: SEC 2. Elliptic curves, used in cryptography, define:
• Generator point G, used for scalar multiplication on the curve (multiply integer by ECpoint)
• Order n of the subgroup of EC points, generated by G, which defines the length of the private keys (e.g. 256 bits)
For example, the 256-bit elliptic curve secp256r1 has:
• Order n = 115792089210356248762697446949407573529996955224135760342422259061068512044369 (prime number)
• Generator point G {x = 48439561293906451759052585252797914202762949526041747995844080717082404635286, y =36134250956749795798585127919587881956611106672985015071877198253568414405109}
Key Generation
The ECDSA key-pair consists of:
• private key (integer): privKey
• public key (EC point): pubKey = privKey * G
The private key is generated as a random integer in the range [0...n-1]. The public key pubKey is a point on the elliptic curve, calculated by the EC point multiplication: pubKey = privKey * G (the
private key, multiplied by the generator point G).
The public key EC point {x, y} can be compressed to just one of the coordinates + 1 bit (parity).
For the secp256r1 curve, the private key is 256-bit integer (32 bytes) and the compressed public key is 257-bit integer (~ 33 bytes).
The ECDSA signing algorithm (RFC 6979) takes as input a message msg + a private key privKey and produces as output a signature, which consists of a pair of integers {r, s}. The ECDSA signing
algorithm is based on the ElGamal signature scheme and works as follows (with minor simplifications):
1. Calculate the message hash, using a cryptographic hash function like SHA-256: h = hash(msg)
2. Generate securely a random number k in the range [1..n-1]
In the case of deterministic-ECDSA, the value k is HMAC-derived from h + privKey (see RFC 6979)
1. Calculate the random point R = k * G and take its x-coordinate: r = R.x
2. Calculate the signature proof: s = k−1 ∗ (h+r ∗ privKey)(modn)
The modular inverse k−1(modn) is an integer, such that k ∗ k−1≡1(modn)
1. Return the signature {r, s}.
The calculated signature {r, s} is a pair of integers, each in the range [1...n-1]. It encodes the random point R = k * G, along with a proof s, confirming that the signer knows the message h and the
private key privKey. The proof s is by idea verifiable using the corresponding pubKey. ECDSA signatures are 2 times longer than the signer's private key for the curve used during the signing process.
For example, for 256-bit elliptic curves (like secp256r1) the ECDSA signature is 512 bits (64 bytes) and for 521-bit curves (like secp521r1) the signature is 1042 bits.
Verify Signature
The algorithm to verify a ECDSA signature takes as input the signed message msg + the signature {r, s} produced from the signing algorithm + the public key pubKey, corresponding to the signer's
private key. The output is boolean value: valid or invalid signature. The ECDSA signature verify algorithm works as follows (with minor simplifications):
1. Calculate the message hash, with the same cryptographic hash function used during the signing: h = hash(msg)
2. Calculate the modular inverse of the signature proof: s1 = s−1(modn)
3. Recover the random point used during the signing: R' = (h * s1) * G + (r * s1) * pubKey
4. Take from R' its x-coordinate: r' = R'.x
5. Calculate the signature validation result by comparing whether r' == r
The general idea of the signature verification is to recover the point R' using the public key and check whether it is the same point R, generated randomly during the signing process.
How Does it Work?
The ECDSA signature {r, s} has the following simple explanation:
• The signing signing encodes a random point R (represented by its x-coordinate only)
through elliptic-curve transformations using the private key privKey and the message hash h into a number s, which is the proof that the message signer knows the private key privKey.
• The signature {r, s} cannot reveal the private key due to the difficulty of the ECDLP
• The signature verification decodes the proof number s from the signature back to its original point R, using the public key pubKey and the message hash h and compares the x-coordinate of the
recovered R with the r value from the signature.
Public Key Recovery from Signature
It is important to know that the ECDSA signature scheme allows the public key to be recovered from the signed message together with the signature. The recovery process is based on some mathematical
computations (described in the SECG: SEC 1 standard) and returns 0, 1 or 2 possible EC points that are valid public keys, corresponding to the signature. To avoid this ambiguity, some ECDSA
implementations add one additional bit v to the signature during the signing process and it takes the form {r, s, v}. From this extended ECDSA signature {r, s, v} + the signed message, the signer's
public key can be restored with confidence.
The public key recovery from the ECDSA signature is very useful in bandwidth-constrained or storage-constrained environments (such as blockchain systems), when transmission or storage of the public
keys cannot be afforded. For example, the Ethereum blockchain uses extended signatures {r,s, v} for the signed transactions on the chain to save storage and bandwidth.
Public key recovery is possible for signatures, based on the ElGamal signature scheme (such as DSA and ECDSA).
Signature malleability
For any message hash there are two valid signatures. It is possible to calculate the second valid signature with the private key, just by knowing the first signature, which is public. One can take
any transaction, flip the s value from s to n - s, flip the v value ( 27 -> 28, 28 -> 27), and the resulting signature would still be valid.
Therefore allowing transactions with any s value with 0 < s < n, opens a transaction malleability concern. This is not a serious security flaw, especially since Ethereum uses addresses and not
transaction hashes as the input to an ether value transfer or other transaction, but all Ethereum nodes, including Besu, will only allow signatures where s <= n/2 . This makes sure that only one of
the two possible valid signatures will be accepted. As any ECDSA library, that is not focused on Blockchains (e.g. libcrypto from OpenSSL), will not take this constraint into consideration, any
created signature has to be normalized after the creation of the signature to fit the above mentioned criteria.
Implementation in Besu
The elliptic curve signature is made available in Besu via the interface org.hyperledger.besu.crypto.SignatureAlgorithm. It is injected in various places across the different modules. There are two
classes that implement the interface:
• org.hyperledger.besu.crypto.SECP256K1
• org.hyperledger.besu.crypto.SECP256R1
As the names of the classes suggest, they implement the SECP256K1 or SECP256R1 signature algorithms. SECP256K1 is the default signature algorithm, as it is used in Ethereum Mainnet and all public
testnets. SECP256R1 has been added as an alternative for private networks as it is NIST compliant while SECP256K1 is not.
The only difference between SECP256K1 and SECP256R1 is the curve parameters of their respective elliptic curves. Therefore all calculations are exactly the same, only the input parameters are
different. Because of this, the class org.hyperledger.besu.crypto.AbstractSECP256 contains all the logic of the ECDSA calculations and the classes SECP256K1 and SECP256R1 are extending it.
Native library
Because of performance reasons an implementation of the SECP256R signature algorithm has been created in C, as an addition to the Java implementation. The repository can be found at: https://
The library uses libcrypto from OpenSSL which contains all necessary functionality for the ECDSA signature algorithm. The functions for signing and verifying the signature are directly provided by
libcrypto. The public key recovery is implemented using the elliptic curve operations provided by libcrypto.
Debugging the library
For debugging the library gdb can be used. A basic tutorial for gdb can be found here: gdb tutorial
It needs an executable in order to work. The tests can be used for this purpose, as they are executables. They are in the directory build and have the file extension out.
To debug for example the tests for signing it would be started with:
gdb build/test_ec_sign.out
Testing for memory leaks
In C the memory management is a responsibility of the developer. Therefore it can happen easily that memory leaks are introduced. The pipeline in the repository is testing for it, to avoid that any
memory leaks are introduced in the code.
To test for them locally valgrind can be used. As gdb it needs an executable as well. Therefore the test can be used here as well.
For example to test the tests for signing for memory leaks the following command can be used:
valgrind --leak-check=full build/test_ec_sign.out
If a memory leak is detected an error similar to the following is displayed
1. ==586133==
2. ==586133== 26,780 (2,280 direct, 24,500 indirect) bytes in 15 blocks are definitely lost in loss record 64 of 64
3. ==586133== at 0x483B7F3: malloc (in /usr/lib/x86_64-linux-gnu/valgrind/vgpreload_memcheck-amd64-linux.so)
4. ==586133== by 0x4A086AD: CRYPTO_zalloc (in /home/daniel/projects/besu-native-ec/build/libs/libbesu_native_ec_crypto.so)
5. ==586133== by 0x49EE8EB: EVP_PKEY_new (in /home/daniel/projects/besu-native-ec/build/libs/libbesu_native_ec_crypto.so)
6. ==586133== by 0x49F3A3D: EVP_PKEY_fromdata (in /home/daniel/projects/besu-native-ec/build/libs/libbesu_native_ec_crypto.so)
7. ==586133== by 0x10F567: create_key (ec_key.c:89)
8. ==586133== by 0x10F73D: create_key_pair (ec_key.c:40)
9. ==586133== by 0x10B483: sign (ec_sign.c:65)
10. ==586133== by 0x10B6B9: p256_sign (ec_sign.c:36)
11. ==586133== by 0x10AB7B: p256_sign_should_create_valid_signatures (test_ec_sign.c:44)
12. ==586133== by 0x10ACF0: p256_sign_should_create_valid_signatures_from_sha224_hashes (test_ec_sign.c:76)
13. ==586133== by 0x10F105: UnityDefaultTestRun (in /home/daniel/projects/besu-native-ec/build/test_ec_sign.out)
14. ==586133== by 0x10A94B: main (test_ec_sign.c:118)
15. ==586133==
16. ==586133== LEAK SUMMARY:
17. ==586133== definitely lost: 9,120 bytes in 60 blocks
18. ==586133== indirectly lost: 95,600 bytes in 2,352 blocks
19. ==586133== possibly lost: 0 bytes in 0 blocks
20. ==586133== still reachable: 0 bytes in 0 blocks
21. ==586133== suppressed: 0 bytes in 0 blocks
22. ==586133==
The output has to be read starting at the bottom:
Lines 16 - 21 tell us that a memory leak has occurred and how many bytes have not been freed.
Line 14 is the starting point to the stack trace. It tells us that in line 7 that in the file ec_key.c in line 89 memory was allocated, by calling functions from the library besu_native_ec_crypto.so
(see lines 4 -6). Because this has caused an error, it means that this allocated memory was never freed.
Now that we know where the memory leak occurs, we need to free the memory at the appropriate place. In almost all cases this will be at the end of the function which has the memory allocated or if
the allocated memory is the return value at the end of the calling function.
Integration of the native library in Besu
The native library is integrated via another repository: https://github.com/hyperledger/besu-native
This repository uses JNA to create the bridge between the native library and Java. The corresponding code can be found here: https://github.com/hyperledger/besu-native/tree/main/secp256r1
The code calls the corresponding functions of the native library and converts the results to basic Java data types. Further it needs to convert the received parameters from the representation in Java
to one that the native library expects.
Especially important are the functions convertToNonNegativeRepresentation and convertToNativeRepresentation. They are responsible for converting the signatures into the correct representation. The
native library expects a big-endian encoded byte array which is always positive, while in Java byte arrays can be positive or negative. The leftmost bit of an byte array needs to be tested in order
to verify if a conversion is needed or not.
Testing local versions of besu-native
For development it can be necessary to test a new version of a besu-native library. This can be done locally. The following steps are necessary:
1. Create a new jar file by executing in besu-native:
cd secp256r1
../gradlew build
This creates a new jar file in secp256r1/build/libs
1. Create a directory for the jar file in besu:
mkdir libs
1. Copy the jar file from step 1 from besu-native to the newly created directory:
cp $BESU_NATIVE_DIR/secp256r1/build/libs/besu-native-secp256r1-$VERSION.jar $BESU_DIR/libs
1. Add the newly created directory as a Gradle repository in besu. Modify $BESU_DIR/build.gradle:
repositories {
// …
flatDir {
dirs 'libs'
1. Add the dependency to the crypto module. Modify $BESU_DIR/crypto/build.gradle:
dependencies {
// …
api 'org.hyperledger.besu:secp256r1:$VERSION_OF_JAR_FILE'
In past occasions the gradle task checkLicenses was failing when a local jar file was added like this. To avoid the error the task has to be excluded:
// for example
./gradlew build -x checkLicenses
Adding an additional signature algorithm
The native library currently only supports SECP256R1, but it is prepared to be extended to allow other ECDSA signature algorithms. To extend it the following steps have to be taken:
1. The 3 new functions to sign, verify and recover the public key have to be defined in /src/besu_native_ec.h. They can be directly copied from the existing ones for SECP256R1 (P256) and only have
to be renamed.
2. The public key recovery function, which was just defined, needs to be implemented in src/ec_key_recovery.c. The existing function p256_key_recovery can be copied. It calls the function
key_recovery. The new function needs to set the correct parameters for curve_nidand curve_byte_length. The correct value for curve_nid can be found in the file openssl/obj_mac.h
3. The signing function, which was just defined, needs to be implemented in src/ec_sign.c. The existing function p256_sign can be copied. It calls the function sign. The new function needs to set
the correct parameters for private_key_len, public_key_len, group_name and curve_nid. The parameter for curve_nid is the same as in the public key recovery function. The correct value for
group_name can be found as well in openssl/obj_mac.h
4. The verification function, which was just defined, needs to be implemented in src/ec_verify.c. The existing function p256_verify can be copied. It calls the function verify. The new function
needs to set the correct parameters for public_key_len,group_name and curve_nid. The parameters group_name and curve_nid are the same as in the signing function.
SECP256R1 support in 3rd party libraries
All Ethereum libraries support per default only SECP256K1 as this is the signature algorithm, which is used in the public Ethereum networks. The support for SECP256R1 is not existing and has to be
In Web3J the support can be added by using a custom transaction manager. The transaction manager is responsible for decoding, signing and sending a transaction. Therefore replacing the signing
functions to use SECP256R1 are sufficient to add basic support.
A working version has been already created here: https://github.com/daniel-iobuilders/besu-secp256r1-demo/tree/master/app/src/main/java/org/hyperledger/besu/secp256r1/demo/Web3jNist
To use the custom transaction manager it has to be provided as a parameter for the functions deploy and load.
In Truffle transactions are handled by so- called providers. The standard provider for Truffle is called hdwallet-provider. It’s implementation can be found here: https://github.com/trufflesuite/
To add functionality for SECP256R1, the hdwallet-provider could be used as a basis and the signing functions would need to be changed to use SECP256R1.
This newly created provider needs to be set in the network configuration in truffle-config.js | {"url":"https://lf-hyperledger.atlassian.net/wiki/spaces/BESU/pages/22154986/SECP256R1+Support?atl_f=content-tree","timestamp":"2024-11-13T23:55:56Z","content_type":"text/html","content_length":"969494","record_id":"<urn:uuid:4f483a2c-ddc7-4bef-a49b-148531306370>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00679.warc.gz"} |
(PDF) Localized Simultaneous Clustering and Classification
... And then, an improved multi-objective simultaneous learning framework for designing a classifier (IMSDC) [38] adopts a new initialization method in which the value of clusters membership degree
is calculated on the basis of randomly initialized cluster centers, rather than initially chosen at random. Next then, a model for simultaneous clustering and classification in a localized way (LSCC)
[39] is presented for focusing on small, distinct, local groups in big datasets. Inspired from them, our algorithm strengthens the SCC framework and adopts OBL strategy and AMI evaluation for
alleviating the imbalance of accuracy and efficiency. ... | {"url":"https://www.researchgate.net/publication/228749666_Localized_Simultaneous_Clustering_and_Classification","timestamp":"2024-11-12T10:27:49Z","content_type":"text/html","content_length":"537076","record_id":"<urn:uuid:0b80a929-099d-4128-93ee-2d61d649f032>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00080.warc.gz"} |
Preparing for Primary Maths Olympiad Competition
is another new book written by trainers from MSO Pte Ltd and published by Educational Publishing House Pte Ltd to provide pupils with ample practice for various Primary Mathematical Olympiad
It consists of 12 sets of 60-minute question papers, each comprising 15 questions that cover a wide range of topics and question styles that are typically set during Mathematical Olympiad
competitions. By exposing pupils to a diversity of mathematical problems, pupils will be able to hone their problem-solving skills and boost their confidence in tackling Mathematical Olympiad
questions. Step-by-step worked solutions are provided to facilitate thorough understanding and self-checking.
Preparing for Primary Maths Olympiad Competition is now available at Popular bookstores and all our centres nationwide. | {"url":"https://www.mathshub.com.sg/mathematics/publications/preparing-for-primary-maths-olympiad-competition/","timestamp":"2024-11-05T04:23:18Z","content_type":"text/html","content_length":"35950","record_id":"<urn:uuid:f83e3c40-0f7d-462c-844f-f413fda7f4d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00176.warc.gz"} |
ONA Chelsea Camera Bag (Giveaway) - Making it Lovely
'A' for Accessories• 'F' for Freebies
ONA Chelsea Camera Bag (Giveaway)
ONA recently wrote to me to ask if I was interested in reviewing and giving away a new camera bag they were coming out with. I did the same for their Brooklyn camera bag (which I still use and love),
so I agreed. They sent over their newest, the Chelsea camera bag in cognac leather.
(in case you’re curious: dress • cardigan • bracelet • shoes)
If I’m going out with my big camera in tow, I’m not going to carry a camera bag and a separate purse. With my other bags, I have to pare down a bit, but the Chelsea fits everything I usually carry,
plus my camera with a small lens attached and two additional large lenses. I also like the classic, feminine shape and gold hardware.
Would you like a Chelsea camera bag (in either cognac or black) of your own? ONA will give one away to a lucky Making it Lovely reader. Just leave a comment on this post to enter. Good luck!
One entry per person, open to U.S. and international entrants. The giveaway ends next Wednesday, January 30, 2013 at midnight CST, and a winner will be selected randomly.
p.s. You can keep up with ONA by liking their Facebook page.
* Congratulations to the winner, Molly!
January 24, 2013 at 4:44 pm
Beautiful bag! I would use it as a purse!
January 24, 2013 at 4:44 pm
What a beeeeautiful bag! Choose me! Choose me!
January 24, 2013 at 4:45 pm
Yes Please! in love with that leather
January 24, 2013 at 4:48 pm
I’d love one. It’s so stylish! And it would be perfect for my DSLR (a Christmas present.)
Thanks for the opportunity.
January 24, 2013 at 4:48 pm
Such a beautiful camera bag! Love the look and functionality of it. Definitely need one of these in my life! :)
January 24, 2013 at 4:49 pm
So pretty!
January 24, 2013 at 4:49 pm
So pretty for my camera!
January 24, 2013 at 4:53 pm
So pretty!
January 24, 2013 at 4:56 pm
This is such a nice alternative to the giant black camera bag, plus purse! Beautiful.
January 24, 2013 at 5:00 pm
So pretty!
January 24, 2013 at 5:17 pm
Lovely bag! my old camera bag is too small for my new camera, I’ll have to look into one of these.
January 24, 2013 at 5:23 pm
What a lovely bag (and skirt9)!
Thank you for all your hard work on your blog. It always brings something beautiful into my crazy mommy life.
January 24, 2013 at 5:24 pm
Soooo pretty!
Sarah N
January 24, 2013 at 5:24 pm
Beautiful bag – I love it!
January 24, 2013 at 5:25 pm
Love this bags shape it is so classic.
January 24, 2013 at 5:26 pm
Gorgeous bag! Love it, or I mean, I will love it!!
January 24, 2013 at 5:28 pm
This bag is amazing. I love that it fits two large lenses.
January 24, 2013 at 5:34 pm
Oh, I love it! My generous father just gave me a camera that’s much larger than the one I’m used to carrying – so I really need a new camera bag! I hope I win! :)
January 24, 2013 at 5:38 pm
What a lovely bag. Pick me! Pick me!
Nancy Reid
January 24, 2013 at 5:39 pm
Camera Bag? I would use it as my pocketbook! and so stunning in Cognac
Kathy Green
January 24, 2013 at 5:39 pm
This bag is beautiful and what I need for my DSLR. Since I carry my camera everywhere this bag would also go everywhere.
Sarah W
January 24, 2013 at 5:40 pm
I love this bag! Great giveaway!
January 24, 2013 at 5:42 pm
would i like his bag? why, yes, i would! thanks for offering! :)
January 24, 2013 at 5:43 pm
Beautiful! I really need a camera bag!
January 24, 2013 at 5:45 pm
Love this bag!
Tracey C
January 24, 2013 at 5:47 pm
Having an actual camera bag, rather than wrapping my camera in an old t-shirt and putting it in my backpack would be so nice!
January 24, 2013 at 5:52 pm
Nice! Will defiantly make me want to take the camera out more often!
January 24, 2013 at 5:56 pm
love it! love the color annnnd I just bought a nikon 5100 so I need a great bag to drag it around!
January 24, 2013 at 6:00 pm
It’s lovely! I would take it everywhere with me!
January 24, 2013 at 6:04 pm
Wow. So happy when products like this are so well executed.
January 24, 2013 at 6:04 pm
so gorgeous. so needed in my life.
pamela lance
January 24, 2013 at 6:07 pm
My daughter needs this! What a beautiful way to carry her camera and lenses!
Joy @ OSS
January 24, 2013 at 6:07 pm
This looks SO AMAZING. Please pick me :)
Tina C.
January 24, 2013 at 6:09 pm
What a fabulous giveaway!
January 24, 2013 at 6:12 pm
It’s simply lovely! Thank you.
Elizabeth @ HobbyLobbyist
January 24, 2013 at 6:17 pm
This is such a special bag, I love it!
Lisa F
January 24, 2013 at 6:23 pm
Nice looking bag!
January 24, 2013 at 6:28 pm
What a great bag! Hope I win!
Amy N
January 24, 2013 at 6:36 pm
Beautiful bag, love the cognac!
Nancy Haberman
January 24, 2013 at 6:39 pm
I’d love it for my new camera!
January 24, 2013 at 6:42 pm
Love that bag!
Rachel Clark
January 24, 2013 at 6:43 pm
Just got a Nikon D5100… The bag in cognac would be perfect to carry it in!!
January 24, 2013 at 6:52 pm
Love it! Want it!
Kim Kargas
January 24, 2013 at 7:07 pm
I I would love, Love, LOVE that bag!!
Jessica @ The Tiny Abode
January 24, 2013 at 7:15 pm
What a perfect bag for a camera! I LOVE consolidating as much as possible. A bag to not only fit my camera, but my all my lipsticks too would be fabulous!
January 24, 2013 at 7:15 pm
I love that bag…. my camera would look perfect in it….
January 24, 2013 at 7:16 pm
Oh thanks for the opportunity! I would love to have one!
Sadie Clifford
January 24, 2013 at 7:17 pm
Angie Sandy
January 24, 2013 at 7:20 pm
These bags are lovely, I hope I win :)
January 24, 2013 at 7:32 pm
Absolutely love the bag….but more importantly where did you get that skirt?
Christina W.
January 24, 2013 at 7:32 pm
1) Your dress is AH-MAZING. 2) What are the odds that we can beg/bully/make puppy dog eyes at this company until they agree to make diaper bags in these styles?
January 24, 2013 at 7:36 pm
As a first year design student with a brand new camera, I’d love a ladylike camera bag instead of a big black monster bag. That gorgeous bag is a gem!
Ashley Lee
January 24, 2013 at 7:41 pm
This is by far the loveliest camera bag I have ever seen. I so hate lugging around a camera bag and purse. I would have my camera with me even more with such a bag on my arm. Please pick me!!!
Kate Castelli
January 24, 2013 at 7:43 pm
What a delicious bag! I love the longer straps and the cognac colour is dreamy.
January 24, 2013 at 7:48 pm
What a stylish camera bag! And it won’t advertise that you’re carrying around all that expensive camera equipment. Nice.
January 24, 2013 at 7:56 pm
The Chelsea bag rocks!
January 24, 2013 at 7:57 pm
amazing! gorgeous, chic bag, makes me want to take the camera everywhere:)
January 24, 2013 at 7:58 pm
I desperately need a camera bag, so this would be amazing! I love the bags that look like purses. Enough with the grey sports fabric and glow in the dark trim! ha!
Thanks for the opportunity.
January 24, 2013 at 8:04 pm
Love these bags! And love both colors too!
January 24, 2013 at 8:08 pm
What a gorgeous camera bag!
January 24, 2013 at 8:08 pm
What a nice camera bag! Much more elegant than the black canvas bag I use today!
January 24, 2013 at 8:10 pm
Yes, please! That would be perfect for my new Nikon!
January 24, 2013 at 8:12 pm
A beautiful bag – the cognac would look so great with my camel coat!! And navy coat, and…
Linsey @ pretty preened
January 24, 2013 at 8:13 pm
cognac camera bag….something this girl can’t live without!!
January 24, 2013 at 8:21 pm
♥ this bag
Justine Semple
January 24, 2013 at 8:27 pm
Pure loveliness !!!!!
Paige Rutherford
January 24, 2013 at 8:32 pm
January 24, 2013 at 8:33 pm
Never had a REAL camera bag (not counting my ratty canvas one!).
January 24, 2013 at 8:39 pm
LOVE the cognac!!!
Jennifer Rodgers
January 24, 2013 at 8:39 pm
I would looooove to replace my clunky standard issue bag with something fashionable from ONA.
kristin p
January 24, 2013 at 8:42 pm
I’d love to win this bag….I’ve been eyeballing new bags for a while and would love something that can do double duty (purse/cam bag).
January 24, 2013 at 8:44 pm
What a fantastic bag! Thanks for the contest :)
January 24, 2013 at 8:46 pm
I just got my first DSLR and this camera would be the perfect little carrier for it!!!
January 24, 2013 at 8:51 pm
Oh what a lovely bag – you often inspire bag envy in me. The color and shape are both so lovely and classic.
Lauren P
January 24, 2013 at 8:51 pm
Wow! I love the bag. I would use it as a purse along with a camera bag, so perfect!
January 24, 2013 at 8:53 pm
I love everything about this post! Your dress is perfect and that bag is to die for! I wish I was in Alt Summit right now to hear you speak!
January 24, 2013 at 8:54 pm
Oh. I would love this. I don’t have a camera bag at all!
January 24, 2013 at 8:57 pm
It’s gorgeous!
Lisa W.
January 24, 2013 at 9:04 pm
Ahhh! That bag is amazing. I would absolutely love a camera bag as nice as that one!
January 24, 2013 at 9:05 pm
What a beautiful camera bag! Thank you and ONA for offering such a lovely giveaway!
January 24, 2013 at 9:08 pm
Such a great giveaway!
lyn reeves
January 24, 2013 at 9:10 pm
OH please, I am so in need of a camera bag. And this one is so beautiful!
January 24, 2013 at 9:11 pm
Lovely and timeless! I would go for the Chelsea in cognac.
January 24, 2013 at 9:12 pm
Would love a new camera bag, this one is lovely!
January 24, 2013 at 9:13 pm
I would love a Chelsea bag in cognac. So cute!!!
January 24, 2013 at 9:17 pm
A year after buying my camera and I’m still in need of a camera bag!
Jessica Mcconnel
January 24, 2013 at 9:17 pm
I’d love to trade up to the cognac Chelsea camera bag!
Holly B
January 24, 2013 at 9:20 pm
This bag is gorgeous! I need!
January 24, 2013 at 9:23 pm
I have been looking for a cute camera bag and this one is perfection. It’s definitely been added to my wish list
January 24, 2013 at 9:39 pm
Ahhhh so beautiful!
Whitney Kaye
January 24, 2013 at 9:41 pm
This is a beautiful bag! I would love the cognac.
Serenia Stegnet
January 24, 2013 at 9:50 pm
I.WOULD.LOVE. :)
Nicole Wiley
January 24, 2013 at 9:51 pm
This bag is gorgeous!!!!
Liz Hergert
January 24, 2013 at 9:58 pm
Crossing fingers and toes! Would love something slightly more girly than what I’m currently rocking. :)
January 24, 2013 at 10:00 pm
So pretty…
January 24, 2013 at 10:01 pm
I would adore this camera bag and treat it with so much love and adoration!
christine e-e
January 24, 2013 at 10:38 pm
what a gorgeous bag! I’d love to have a bag this love-ly… keeping my fingers crossed that my name is chosen. I would love it in either color – I won’t be picky!
January 24, 2013 at 10:40 pm
Oh oh! Me me! Love this bag sooo much! :)
amy johnson
January 24, 2013 at 10:45 pm
What a fun giveaway. Love it!
January 24, 2013 at 11:10 pm
I would LOVE this bag! I’m sick of my lenses being spread around the house.
« 1 … 12 13 14 15 16 … 18 »
Previous Post Fab Valentine's Day Gift Guide
Next Post Blue and Green Kitchen | {"url":"https://makingitlovely.com/2013/01/23/ona-chelsea-camera-bag-giveaway/comment-page-14/","timestamp":"2024-11-05T21:35:57Z","content_type":"text/html","content_length":"136294","record_id":"<urn:uuid:1a6cf28f-8f57-42d7-8253-d19e88f83ead>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00540.warc.gz"} |
H Model – Dividend Discount Model
The H model is another form of the Dividend Discount Model under Discounted Cash flow (DCF) method, which breaks down the cash flows (dividends) into two phases or stages. It is similar, or one can
say, a variation of a two-stage model; however, unlike the classical two-stage model, this model differs in how the growth rates are defined in the two stages.
The two-stage model assumes that the first stage goes through an extraordinary growth phase while the second stage goes through a constant growth phase. In the H model, the growth rate in the first
phase is not constant but reduces gradually to approach the constant growth rate in the second stage. The key point to note here is that the growth rate is assumed to reduce linearly in the initial
phase until it reaches a stable growth rate in the second stage. The model also assumes that dividend payout and cost of equity remain constant. Let us take an example illustrating firm value using
the H model dividend discount model.
Example of Valuation using H Model – Dividend Discount Model
Let us take an example of a company ABC Ltd. that has paid a dividend of $ 4 this year. Assuming a growth for the next 3 years at 13%, 10%, and 7%, respectively, in the first stage and stable growth
of 4% thereafter, let us calculate the firm value using the H model dividend discount model.
The dividend values will be as follows:
Current Dividend = $ 4.00
Dividend after 1^st year will be = $ 4.52 ($ 4.00 x 1.13 – growing at 13 %)
2^nd year will be = $ 4.972 ($ 4.52 x 1.10 – growing at 10%)
3^rd year will be = $ 5.32 ($ 4.972 x 1.07 – growing at 7%)
The dividend declared after the first stage will be $ 5.32 as calculated above.
Assuming a stable growth rate of 4% in the second stage; the dividend value after 4^th year will be
$ 5.32 x 1.04 = $ 5.5328.
Assuming this as the constant dividend for the rest of the life of the company, we arrive at the present values as follows.
P[0 ]= D/ (i – g)
Where, P[0] = Value of the stock/equity
D = per-share dividend paid by the company at the end of each year
i = discount rate, which is the required rate of return that an investor wants for the risk associated with the investment in equity as against investment in risk-free security.
g = growth rate
Now using the formula for calculating the value of the firm, we can arrive at the present value at the end of 3rd year for all future cash flows as follows:
Value = $ 5.5328 / (10% – 4%)
= $ 92.21
Assuming a constant discount rate of 10%, the firm’s value can now be calculated as the present value of future cash flows.
Tenor Cash Flow Discount Rate Present Value
1 4.52 10% 4.11
2 4.972 10% 4.11
3 5.32 10% 3.99
3 92.21 10% 69.28
Total Present Value 81.49
Present value calculations arrived as follows:
$ 4.11 = $ 4.52 / (1 + 10%) ^1
$ 4.11 = $ 4.972 / (1 + 10%) ^2
$ 5.32 = $ 5.32 / (1 + 10%) ^3
$ 69.28 = $ 92.21 / (1 + 10%) ^3
The sum of all the present values will be the value of the firm, which in our example comes to $81.49.
The H model tries to do away with some of the problems/shortcomings associated with the classical two-stage model; let us have a comparative look to better understand the H Model.
H Model V/s Two-Stage Model
• The classical two-stage model assumes an extraordinary growth rate (constant) in the initial stage. In contrast, the H model is free to use an increasing or declining rate in the initial phase
and then aligns itself with the constant second-stage growth rate.
• In the two-stage model, the growth rate drops suddenly from a very high rate to a stable rate as stages change. However, in the H model, the growth rate reduces linearly to reach a stable growth
rate, thereby avoiding any sudden jumps or falls.
• Like the two-stage model, the H model also assumes a constant dividend payout ratio and cost of equity which may not be a real-world scenario and may lead to estimation errors.
The main limitation of the H Model is that it assumes a linear fall in growth rates from an extraordinary growth rate period in stage 1 to a stable growth rate period in stage 2.
Some of the limitations can be handled by using the 3 stage model. A 3 stage model assumes 1st stage to have an extraordinarily high rate of growth, the second stage has a reducing growth rate, and
the third stage has a constant stable growth rate. One can say that it is a blend of the two-stage model and the H model.
In cases where the transition happens faster from the extraordinary phase to the stable growth rate phase, a three-stage model may behave the same as in the same way as the H model, and so would the
value of the firm using either model. As a result, many cases for the H model yield similar results as the three stages model.
Leave a Comment | {"url":"https://efinancemanagement.com/investment-decisions/h-model-dividend-discount-model","timestamp":"2024-11-03T09:37:38Z","content_type":"text/html","content_length":"253258","record_id":"<urn:uuid:cc2cdf38-a5ca-4f15-9a64-5641141e0d50>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00424.warc.gz"} |
What is the diameter of the wire in the fuse?
Determining the diameter of the wire in the fuse
The diameter of the fuse wire is approximately 2 mm. This is calculated using the formula for current density and the area of a circle to first determine the cross-sectional area and then derive the
The question asks for the diameter of the wire in a fuse that blows at a current of 1.0 A with a current density of 320 A/cm². The current density is defined as the current per unit area. Since the
current and current density are given, we can first calculate the cross-sectional area of the fuse wire and then determine its diameter.
First, we convert the current density to the SI unit by multiplying with 10² (since 1 cm² = 10² mm²), which gives us 32000 A/m². The formula for current density (J) is J = I/A, where I is the
current and A is the area. We can rearrange the formula to find the area (A = I/J).
For this fuse, the current (I) is 1 A and the current density (J) is 32000 A/m², so:
A = 1 A / 32000 A/m² = 3.125 à 10⠻⠵ m²
Now, using the area of a circle (A = Ï d²/4, where d is the diameter), we rearrange to solve for the diameter (d):
d = 2 â (A/Ï ) = 2 â ((3.125 à 10⠻⠵ m²)/Ï )
Using a calculator, we find the diameter is approximately 2 mm. | {"url":"https://airdocsolutions.com/physics/what-is-the-diameter-of-the-wire-in-the-fuse.html","timestamp":"2024-11-05T16:59:59Z","content_type":"text/html","content_length":"21131","record_id":"<urn:uuid:5a5f5ffb-c274-4299-a5d1-3b55411df405>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00384.warc.gz"} |
from soyabean:
m2's shiliang calculations
Hey, im eating dinner now.. nothing to do, translate the shiliang calculations for u guys ^^ First of all, this is an introduction to shiliang, because there are more styles to it, i will update it
when i have time. In gb, there are 2 methods of calculations known to me. One is the wind index method that most of us are familair with. The other, is shiliang. Shiliang deals with the horizontal
and vertical displacement due to the wind effect wheres wind index resolves these 2 displacement into a factor. The origins and details of shiliang can be found at knat's forums, in the original m2
Next, based on these 2 methods, a few styles of calculation has been derived. They are mainly these 3, fix power, fix angle, change angle change power. Some examples of fix power calculations include
the xiaopao, banpao, and man pao. And the change angle change power calculations are the Bian Li Suan Jiao formulas.
Below, im going to translate the introduction part for m2's guide, which deals with the second style of calculation, fix angle.
All of us know that it is impossible to make a complete wind chart based on wind index for fix angle, for 2 simple reasons. Firstly, effect varies with distance, secondly, effect varies with wind.
Therefore, shiliang becomes a more effective way of calculation.
Style: fix angle, calculate power
Wind chart:
Screen: 20 parts
x: horizontal displacement
y: vertical displacement
d: change in power
f: power at wind 0
a: angle
This formula uses armor as a reference. (all other bots can be used the same way except for mage and boomer, for boomer, just x2 the result)
Armor 60:
formula: 2x+y/100
Armor 40
formula: x+y/100
Complete formula: D=F*(Y+Xtga)/2(100-Y+Xtga)
Recommended formula: D=(Y+X)/(100-Y+X) for angle 40, D=(Y+2X)/(100-Y+2X) for angle 60
Explaination for simplification of the complete formula:
Taking angle to be 40 (tga = 1), therefore,
Taking power to be 2.0, therefore,
Usually, -Y+Xtga tends to be very small (there are exceptions, and they will be discussed below). Hence, we can further simplify it to become:
When angle becomes 60, tga=2. Thats how m2 derived 2x+y/100
I will use angle 60 as example, using simplified formula: 2x+y/100
opponent 14 parts away. power to use at wind 0=2.0
wind blowing at bearing of 045, 10
formula = 2x+y/100
power to use = 2.0 - 0.21 = 1.79 ~ 1.8
And now for the most important part, how to reduce errors. this formula has been simplified alot to allow easy calculations.
1)The change in power due to wind is only relevant for distance of 2.0 power, which is 14 parts.(We took F=2.0 previously to simplify the equation) If distance becomes further, like 1 screen(20
parts), the change in power increases. the reverse is true. u can either feel for the difference, or u can add some calculations, just take b/2*result. b being the power required to reach the
distance. for full screen, b will be 2.45.
2)Secondly, a few wind directions have weird effect. (when 100-Y+2Xtga becomes very big or very small). For example, wind blowing at a bearing of 315, 20, 100-Y+2Xtga for angle 60 = 100 - (0.7*20) +
(-0.7*20*2) = 58. You will need to multiply the result by some wind index to change the ans. Wind blowing around 000, 270, 315, *1.4 to result. Wind blowing around 090, 135, 180, *0.8 to result.
With that, the general idea of shiliang has been established. when i have time, i will translate the use of shiliang in xiaopao (2.4) banpao (2.8-2.95-3.05) for armor ^^
* If u are using the recommended formulas, u can ignore the second way of reducing errors.
Power at wind 0 for armor 40.
4 覧覧1.0
5 覧覧1.1
6 覧覧1.2
7 覧覧1.3
8 覧覧1.4
9 覧覧1.5
Armor 60 u can find it somewhere on this forums already. This one is provided by m2, so i suppose you will measure from the center of ur bot to where u want to hit.
Style: Fix power calculate angle
Firstly, just for your information, the calculations for this type of shooting style still contains some errors. It is not recommended to use it. m2 provided calculations for both boomer and armor,
but I will only translate armor as this method for boomer is very impractical.
Power: Full power
Screen: 11 parts
Angle: Power at wind 0 + Horizontal displacement + Vertical displacement
Horizontal displacement = X/10
Vertical displacment = Y/2
Wind blowing at a bearing of 045, 20.
Distance = 1 screen
Angle: 79 + (0.7*20/10) + (0.7*20/2) = 79 + 1.4 + 7 = 87.5
Xiaopao and Banpao:
Known to many of us, xiaopao and banpao can be used with 14 parts, 20 parts, and 30 parts. But since 14 parts is not very practical, unless u r shooting beyond 1 screen distance, I will only
translate for 20 parts and 30 parts.
Power: 2.4
Screen: 30 parts
Angle = Angle at wind 0 + Horizontal displacement + Vertical displacment
Horizontal displacement = X/2 (When X/2 is clsoe to 5, add 1 angle)
Vertical displacement = Y/8 (This is accurate for halfscreen - 1 screen distance, if distance is less than half, u will use Y/4, and as distance increases to 1-1.5 screen, use Y/16)
Wind blowing at the bearing of 079, 15
Distance: 24 parts
Angle = (90-24) + [(1*15/2)+1] + (0.5*15/8 ) = 75
Tired... later den translate banpao | {"url":"http://creedo.gbgl-hq.com/shiliang_formula.php","timestamp":"2024-11-11T14:30:10Z","content_type":"text/html","content_length":"7122","record_id":"<urn:uuid:19f4bc73-27bf-4c85-8189-3f89ab408d41>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00715.warc.gz"} |
Given the following boxplot where m is the median value, what
Show all work when appropriate. You may type your answers onto the test, or complete it by hand and submit a scanned copy.
1. Given the following boxplot where m is the median value, what statement could be made about the distribution of the data?
Explain your answer.
2. The Colorado State Legislature wants to estimate the length of time it takes a resident of Colorado to earn a bachelor’s degree from a state college or university. A random sample was taken of 265
recentin-state graduates.
a) Identify the variable.
b)Is the variable quantitative or qualitative?
c)What is the implied population?
3. For the information in parts (a) through (g) below, list the highest
level of measurement as ratio, interval, ordinal, or nominal, and
explain your choice.
A student advising file contains the following information:
(a) Name of student
(b) Student I.D. number
(c) Cumulative grade point average
(d) Dates of awards (scholarships, dean’s list, etc.)
(e) Declared major
(f) A number code representing class standing:
1 = freshman, 2 = sophomore, 3 = junior,
4 = senior, 5 = graduate student
(g) Entrance exam rating for competency in English:
excellent, satisfactory, unsatisfactory
4. Exam Scores For 108 randomly selected collegestudents, this exam score frequency distribution wasobtained.
Class Class frequency [*Frequency] [*Frequency]
Limits Boundaries
90–98 89.5 – 98.5 6 94 8836
99–107 98.5 – 107.5 22 103 10609
108–116 107.5-116.5 43 112 12544
117–125 116.5-125.5 28 121 14641
126–134 125.5-134.5 9 130 16900
Find by using the correct formulas:
Be sure to show all work.
a) Mean
b) Modal class
c) Variance
d) Standard deviation
e) Constructa histogram
f) Discuss the shape of the distribution
5 .Stories in the World’s Tallest Buildings Thenumber of stories in each of the world’s 30 tallestbuildings is listed below.
a) Construct a stem-and-leaf plot
b) Find the 5-number summary
c) Construct a box-and-whiskers-plot
d) Check for outliers
e) Discuss the shape of the distribution
1. Here clearly the data is not symmetrical as the very large dosage is expected to have a really high amount of drug dosage thus the data is somewhat positively skewed. And we know that for skewed
data the median is the best measure to describe the location. So here the median would be best measure of position.
2. Top quartile means top 25% of the data. Here the total number of mice are 40 so top quartile consists of (40*25%) = 10 mice. Now 40% of them survived so (10*40%) = 4 mice survived.
3. The percentile would give us what percent of the mice are in that category. So in this example it will pin point the information we are looking for.
4. The quartile is somewhat similar to percentile. The percentile divides the whole group into 100 equal parts and the quartiles into 4 equal parts. So quartiles would give the 25 percentiles
5. The standard score will give us the relative measures. It will give enough information to compare to different scores by standardizing them.
https://doneassignments.com/wp-content/uploads/2021/08/logo-300x75.png 0 0 admin https://doneassignments.com/wp-content/uploads/2021/08/logo-300x75.png admin2021-08-27 10:27:272021-08-27 10:27:27
Given the following boxplot where m is the median value, what | {"url":"https://doneassignments.com/2021/08/27/given-the-following-boxplot-where-m-is-the-median-value-what/","timestamp":"2024-11-02T02:30:56Z","content_type":"text/html","content_length":"64733","record_id":"<urn:uuid:3b23bc72-a367-4c96-b647-48541ea66b8b>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00531.warc.gz"} |
Using One Piece Twice
If we are allowed to use one hexabolo twice, we get 108 pieces with a total area of 316. This number factorises very well and a lot of constructions are possible.
Four isosceles rectangular triangles are versatile as they can be combined into a 18s x 18s square, two 9h x 9h squares, a large triangle and a parallelogram.
If we want the additional piece to be placed in the center of a 18s x 18s square a special construction is necessary. For the "house" as the additional piece a solution is shown on the Poly Pages .
There are only five symmetric pieces, which are balanced and have odd slope and therefore meet the the constraints for such a construction. The solutions for the four remaining pieces are shown here.
The cross is another nice shape to put the additional piece in the center of the construction.
Let's have a look at the rectangles. With the two 27s x 6s or the three 18s x 6s rectangles you can get the 54s x 6s and the 27s x 12s rectangles, too. The two 18s x 9s or the three 12s x 9s
rectangles can be combined into the 36s x 9s. The four 9s x 9s squares seem to be more difficult and I haven't got a solution yet.
Besides the two 9h x 9h squares, which can be derived form the four triangles, two sets of multiple h-rectangles are shown: three 18h x 3h and three 9h x 6h rectangles, which combine into a 54h x 3h
and a 27h x 6h respectively.
With 108=3*36 three 6-fold replicas of all pieces might be possible. It would be nice to use the replicated piece twice, but due to the constraints this can only be done for the balanced pieces with
odd slope. For the five symmetric pieces meeting this condition the solutions are shown.
I think a solution for the other pieces is also possible since I have found the replicas even for a piece with three acute corners
Among the pieces which need a different additional piece to get replicated there are some hard examples: the star with four acute corners and pieces with a boundary of 8 s-sides. For the "bar" three
18s x 6s rectangles are the solution as seen above.
index - previous - next | {"url":"http://polyforms.eu/polytans/hextanplus.html","timestamp":"2024-11-11T04:03:11Z","content_type":"text/html","content_length":"3153","record_id":"<urn:uuid:6314600b-b571-4164-a2e1-e03da1a5c49a>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00519.warc.gz"} |
GPA Calculator allows you to quickly calculate your average grade. This tool is suitable for anyone, regardless of where you study and in what major. With the GPA Calculator, you can keep track of
semester results and your entire academic progress for the entire college career.
This average calculator suits multiple user types. There is no doubt that this average calculator can be used by various user types. Some of them are listed below. Students working on mathematical
problems; There are very few students who like spending time on a mathematics assignment. Calculation of average is one of the many topics which
The most widely used method of calculating an average is the ‘mean’. 2020-06-24 GPA Calculator allows you to quickly calculate your average grade. This tool is suitable for anyone, regardless of
where you study and in what major. With the GPA Calculator, you can keep track of semester results and your entire academic progress for the entire college career. Weighted average is the average of
a set of numbers, each with different associated “weights” or values. To find a weighted average, multiply each number by its weight, then add the results. If the weights don’t add up to one, find
the sum of all the variables multiplied by their … Average of these individual results equals to 5.5 hours.
If the weights don’t add up to one, find the sum of all the variables multiplied by their weight, then divide by the sum of the weights. Average Calculator. The Average Calculator is the best option
for you to find the average of numbers within no time. You can also get assistance from this Mean calculator in the calculation of the mean of a massive set of numbers without any hassle. This
average calculator suits multiple user types. There is no doubt that this average calculator can be used by various user types. Some of them are listed below.
kundbetyg, se skärmavbilder och läs mer om Batting Average Calculator. Hämta och upplev Batting Average Calculator på din iPhone, iPad och iPod touch.
Real_inwCoin Mar 10, 2020. Ever wonder what is my average entry ? No need to use Calculate. After clicking Calculate, if the route is possible to establish*, the client will see: the estimated
distance, average price per km, the overall average price Trail Calculator.
The calculator not only shows the savings which will be made by optimising filling processes to reduce overfilling and unnecessary scrap it also calculates the ROI
1 Ratings.
Average calculator Weighted average calculation. The weighted average (x) is equal to the sum of the product of the weight (w i) times the data number (x i) divided by the sum of the weights:
Example. Find the weighted average of class grades (with equal weight) 70,70,80,80,80,90: Average Cost Basis Calculator. If you have Android device, you can find the average cost of your stock
purchases with the average cost basis calculator which you can install for free. Get stock average calculator for Play Store. Following is an average down stock formula that shows you how to
calculate average price.
Civilekonomprogrammet lund
The average value for North Carolina during that period was 0.58U.S. Research the E85 gasoline The geometric mean is the positive square root of the product of two numbers. Example.
In addition, learn about the definition of average or explore many other calculators. Calculator Use. Use this calculator to find the average or mean of a data set.
Triaden lörenskog
kommande kvartalsrapporter 2021kista engelska översättningtömma papperskorgen i mobilenstockksigrid bernson bröst
calculator lkbardiwpf dota 2 average matchmaking time calculator qkzpavnlwo dota 2 average matchmaking time calculator jczmuldfrg dota 2
Students working on mathematical problems; There are very few students who like spending time on a mathematics assignment. Calculation of average is one of the many topics which About Average
Deviation Calculator . The Average Deviation Calculator is used to calculate the average absolute deviation of a set of given numbers.
Arbetsgivarintyg foretagareblack diamond casino
Average Down Calculator. Simply enter your share purchase price above and the number of shares for each buy to get your average share price. Stock trading or investing is easy to get in, but it takes
a lot effort to make money from the stock market.
GPA är en förkortning på Grade Point Average. Detta är ett medelvärde av dina Calculators for health, exercise and weight loss. workload reached and how many seconds you managed on this workload. | {"url":"https://hurmanblirriksiar.web.app/61171/596.html","timestamp":"2024-11-07T02:23:56Z","content_type":"text/html","content_length":"10107","record_id":"<urn:uuid:8d1764b6-186c-4ff6-ab1e-07d599aa229e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00713.warc.gz"} |