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Using the Lagrange Error Bound to Approximate the Value of a Logarithmic Function at a Point
Question Video: Using the Lagrange Error Bound to Approximate the Value of a Logarithmic Function at a Point Mathematics • Higher Education
Find the error bound when using the third Taylor polynomial for the function π (π ₯) = ln 3π ₯ at π ₯ = 1/3 to approximate the value of π (1/2). Give your answer in scientific form to three
significant figures.
Video Transcript
Find the error bound when using the third Taylor polynomial for the function π of π ₯ is equal to the natural logarithm of three π ₯ at π ₯ is equal to one-third to approximate the value of π
evaluated at one-half. Give your answer in scientific form to three significant figures.
The question gives us a function π of π ₯, and it wants us to determine the error bound if we were to approximate this function by using the third Taylor polynomial. We want to center our Taylor
polynomial at one-third. And weβ re going to use this to approximate the value of π evaluated at one-half. Finally, we need to give our answer in scientific form to three significant figures.
To start, recall if we can approximate π of π ₯ by using an π -term Taylor polynomial, we can also add on a remainder term to help us see the difference between our function π of π ₯ and
our approximation π π of π ₯. And we can actually find a bound on this remainder term. We need to recall the following. For the π th Taylor polynomial centered at a value of π where we
want to approximate at a value of π ₯, we have the absolute value of π π of π ₯ is less than or equal to the absolute value of π times π ₯ minus π all raised to the power of π plus
one divided by π plus one factorial. And this value of π will be an upper bound on the absolute value of the π plus oneth derivative of π of π ₯ on an interval containing both π and π
And itβ s worth pointing out there are a few pieces of information weβ re missing about this, for example, how many times can we differentiate π of π ₯ and are these derivatives continuous on
the interval containing π ₯ and π . These are definitely worth thinking about. However, usually problems of this will arise when we try and find our value of π or we try and construct our
Taylor polynomial. So weβ ll ignore this and move on to find our bound. This is a very complicated-looking expression. However, it can be made easier by substituting in our values of π and π ₯.
First, weβ re told to center our Taylor polynomial at π ₯ is equal to one-third. So weβ ll set our value of π equal to one-third and update all of our bounds. Next, we see weβ re using this to
estimate the value of π at one-half. So we can set our value of π ₯ equal to one-half. And once again we can update our bound to include the value of π ₯ at one-half. We can also update our
equation for π of π ₯ involving the Taylor polynomial. However, this is not necessary. Finally, the question tells us to use the third Taylor polynomial. So weβ ll set our value of π equal to
three. And once again, we can update all of our bounds with the value of π set to be three.
The question is asking us to find a bound on our error, which is the following inequality. In other words, all we need to do is calculate the absolute value of π times one-half minus one-third all
raised to the power of three plus one divided by three plus one factorial. And the only part of this we donβ t know is the value of π . So all we need to do is find our value of π . And we know
that π is an upper bound on the π plus oneth derivative of π ₯ on an interval.
In this case, since π is three, this will be the fourth derivative of π of π ₯ with respect to π ₯. We know this needs to be an interval containing both π and π ₯. So we need an interval
containing one-third and one-half. When weβ re only using this on individual values β for example, in this case, weβ re only using this to approximate π evaluated at one-half β we can just
choose the closed interval between π and π ₯. In other words, we can just choose the closed interval from one-third to one-half.
Weβ re now almost ready to find our value of π . We just need to find an expression for the fourth derivative of π of π ₯ with respect to π ₯. To do this, we just need to differentiate π
of π ₯ four times. First, we need to find π prime of π ₯. Thatβ s the derivative of the natural logarithm of three π ₯ with respect to π ₯.
Thereβ s a few different ways of doing this. One way is to use the product rule for logarithms to rewrite π of π ₯ as the natural algorithm of three plus the natural logarithm of π ₯. Then, the
natural logarithm of three is a constant. So its derivative is just going to be equal to zero. And we know the derivative of the natural algorithm of π ₯ with respect to π ₯ is the reciprocal
function, one over π ₯. So π prime of π ₯ is one over π ₯. Because weβ re going to differentiate this again, weβ ll write this as π ₯ to the power of negative one.
We can now find all of our remaining derivatives by using the power rule for differentiation. We want to multiply by our exponent of π ₯ and then reduce this exponent by one. We get π double
prime of π ₯ will be equal to negative one times π ₯ to the power of negative two. Doing the same again, we get π triple prime of π ₯ will be equal to two times π ₯ to the power of negative
three. And doing this one more time, we get the fourth derivative of π of π ₯ with respect to π ₯ is equal to negative six times π ₯ to the power of negative four. And weβ ll use our laws of
exponents to rewrite this as negative six divided by π ₯ to the fourth power. The reason we do this is because weβ re going to want to find an upper bound for this expression. And itβ s easier to
do this in this form.
First, itβ s important to realize we donβ t want an upper bound on the fourth derivative of π of π ₯. We want an upper bound on the sides of π of π ₯. So we need to take the absolute value
of this expression, so weβ ll just take the absolute value of this. This gives us the absolute value of the fourth derivative of π of π ₯ with respect to π ₯ is equal to the absolute value of
negative six divided by π ₯ to the fourth power.
And we can simplify this. First, in our denominator, π ₯ to the fourth power is equal to π ₯ squared all squared. And we know this is greater than or equal to zero for all values of π ₯. So taking
the absolute value of our denominator is not going to change its value. This means all we need to do is take the absolute value of our numerator. And of course, we know the absolute value of negative
six is just equal to six. So we can just simplify this expression to get six divided by π ₯ to the fourth power.
Now weβ re ready to find our value of π . We want to find an upper bound for this expression on the closed interval from one-third to one-half. We can see on this interval, we have a positive
number divided by another positive number. We want to make this as big as possible. So to make this number as big as possible, we want to divide by the smallest number we can. The smallest number on
our interval is when π ₯ is equal to one-third. Therefore, on the closed interval from one-third to one-half, the absolute value of the fourth derivative of π of π ₯ with respect to π ₯ will be
less than or equal to six divided by one-third raised to the fourth power.
All weβ re really saying here is weβ re dividing by the smallest positive number we can on our interval, and this will be our value of π . And we can just calculate this expression. Six divided
by one-third raised to the fourth power is equal to 486. Now, all we need to do is substitute this into our error bound. Substituting π is equal to 486, π is equal to one-third, π ₯ is equal
to one-half, and π is equal to three into our error bound formula, we get the absolute value of π three will be less than or equal to the absolute value of 486 times one-half minus one-third
all raised to the power of three plus one divided by three plus one factorial. And itβ s worth pointing out since weβ re only interested in the error at our point, we will simplify our remainder
polynomial to just be π three.
Now, all thatβ s left to do is evaluate this expression. First, one-half minus one-third is one-sixth, three plus one is equal to four, and we know that four factorial is equal to 24. Then, if we
just calculate this expression, we get one divided by 64. But remember, the question doesnβ t want us to give this as an exact number. It wants us to give this in scientific form to three
significant figures. To do this, weβ ll start by writing this out as a decimal expansion. Itβ s equal to 0.015625. Next, we want this to three significant figures. We need to remember what this
means. First, the initial zeros will never count. So in this case, our three significant figures will be the one, five, and six.
Next, we need to check the first emitted digit to see if we need to round up. In this case, itβ s two, so we donβ t need to round up. Itβ s less than five. So to three significant figures, the
absolute value of π three is approximately 0.0156. Finally, we need to write this in scientific notation. We start with a number between negative 10 and 10 and multiply this by 10 to the power of
some integer. In this case, by looking at our three significant figures, we can see we need to start with 1.56. Then, we can see that 0.0156 is just equal to 1.56 times 10 to the power of negative
two, which is our final answer.
Therefore, we were able to find the error bound when using the third Taylor polynomial for the function π of π ₯ is equal to the natural logarithm of three π ₯ at π ₯ is equal to one-third to
approximate the value of π of one-half. In scientific form to three significant figures, we got 1.56 times 10 to the power of negative two. | {"url":"https://www.nagwa.com/en/videos/732103694062/","timestamp":"2024-11-10T06:37:47Z","content_type":"text/html","content_length":"263108","record_id":"<urn:uuid:fd392961-0100-4519-be8b-c94a70e235c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00212.warc.gz"} |
What Is Stable Diffusion and How Does It Work?
Knowledge base
What Is Stable Diffusion and How Does It Work?
Aleksa Nikolić Categories: Knowledge Base Date 16-Jan-2023 3 minute to read
For the past few years, revolutionary models in the field of AI image generators have appeared. Stable diffusion is a text-to-image model of Deep Learning published in 2022. It is possible to create
images which are conditioned by textual descriptions. Simply put, the text we write in the prompt will be converted into an image! How is this possible?
Stable diffusion is a version of the latent diffusion model. Latent spaces are used to get the benefits of the low-dimensional representation of the data. After that, diffusion models and methods of
adding and removing the noise are used to generate the image based on the text. In the following chapters, I will describe latent spaces in more detail as well as the way diffusion models function
and I will provide an interesting example of the image which the model can generate based on the given text.
Latent space
Disguised, latent space is, simply put, the representation of compressed data. The compression of data is defined as a process of encoding information by using smaller bits than in the original
representation. Let’s imagine that we have to present a 20-dimensional vector using a 10-dimensional vector. By reducing dimensionality we are losing data. However, in this case, this is not a bad
thing. Reducing dimensionality allows us to filter less important information and keep only the most important information.
In short, let’s say that we want to train the model that classifies images by using fully connected convolutional neural networks. When we say that the model is learning, we mean that it is learning
specific attributes on each layer of the neural network. These are for instance edges, specific angles, shapes, etc. Each time when the model has to learn by using data (an already existing image),
the dimensions of the image are reduced before they go back to their original size. In the end, the model reconstructs the image from compressed data by using a decoder, while learning all the
relevant information beforehand. Therefore, the space becomes smaller so that the most important attributes are extricated and kept. This is why latent space is suitable for diffusion models. It is
very useful that there is a way to single out the most important attributes from a training set of large number of images where there are many details, and that these attributes can be used to
classify two arbitrary objects in the same or different category.
Extricating the most important attributes by using convolutional neural networks (Source: Hackernoon)
Extricating the most important attributes by using convolutional neural networks (Source: Pytorch)
Diffusion models
Diffusion models are generative models. They are used to generate data that are similar to the data on which they have been trained. Fundamentally, diffusion models function in a way they “destroy”
trained data by iteratively adding Gaussian noise and then they learn how to bring back the data by eliminating the noise.
Gaussian noise (Source: Hasty.ai)
Forward diffusion process is the process where more and more noise is added to the picture. Therefore, the image is taken and the noise is added in t different temporal steps where in the point T,
the whole image is just the noise. Backward diffusion is a reversed process when compared to forward diffusion process where the noise from the temporal step t is iteratively removed in temporal step
t-1. This process is repeated until the entire noise has been removed from the image using U-Net convolutional neural network which is, besides all of its applications in machine and deep learning,
also trained to estimate the amount of noise on the image.
(Source: NVidia Ddeveloper)
From the left to the right, the picture demonstrates iterative adding of noise, and from the right to the left, it showcases iterative removing of the noise.
For estimation and removing of the noise, U-Net is used most often. It is interesting that that neural network has an architecture which reminds us of the letter U, which is how it got its name.
U-Net is a fully connected convolutional neural network which makes it very useful for processing on the images. U-Net is distinguished by its ability to take the image as the entrance and find
low-dimensional representation of that image by reducing the sampling, which makes it more suitable for processing and finding the important attributes, and then it reverts the image to the first
dimension by increasing the sampling.
(Source: https://lmb.informatik.uni-freiburg.de/people/ronneber/u-net/)
In more details, removing the noise, that is transitioning from arbitrary temporal step t in the temporal step t-1, where t number is the number between T0 (the image without the noise) and the final
number TMAX (total noise) happens in the following way: input is the image in the temporal step t, and in that temporal step there is a specific noise on the image. Using the U-Net neural network, a
total amount of noise can be predicted, and then a “part” of the total noise is removed from the image in the temporal step t. This is how you get the image in the temporal step t-1 where there is
less noise.
(Source: NVidia Ddeveloper)
Mathematically, there is much more sense to conduct this method T number of times than try to remove the entire noise. By repeating this method, the noise will be gradually removed and we will get a
much “cleaner” image. A simplified process is as follows: there is an image with the noise and we try to predict the image without the noise by adding complete noise on the initial and removing it
(Source: Prog.World)
Inserting the text in this model is done by “embedding” words using language transformers, which means that numbers (tokens) are added to the words, and then this representation of the text is added
to the input (to image) in U-Net, it goes through each layer of U-Net neural network and transforms together with the image. This is done from the first temporal iteration and the same text is added
to each following iteration after the first estimation of the noise. We could say that the text “serves as a guideline” for generating the image starting from the first iteration where there is a
complete noise and then further down to the entire iterative method.
Stable Diffusion
Why stable diffusion? The biggest problem diffusion models have is the fact that they are extremely “expensive” when it comes to time and calculations. Taking into account the way U-Net functions,
stable diffusion overcomes the above-mentioned problems. If we wanted to generate the image, with 1024x1024 dimensions, U-Net should use the noise of 1024x1024 size and then make the image out of it.
This can be an expensive approach for only one diffusion step, especially if that method would be repeated at times where t can be even hundred. In the following example, the number of temporal steps
is 45. Stable diffusion model generated two images for the given number of steps in around 6 seconds while some other diffusion models needed even between 5 and 20 minutes depending on GPU
specification and the size of the image. The most interesting part is that a stable diffusion model can be set in motion successfully locally with even 8 GB VRAM memory. So far, this problem has been
solved by training on a smaller image with the size of 256x256, and then an additional neural network would be used which would produce an image in a bigger resolution (super-resolution diffusion).
Latent diffusion model has a different approach. Concept latent diffusion models do not operate directly on the image but in the latent space! The initial image is actually encoded in much smaller
space, so that the noise is added and removed on a low-dimensional representation of our image by using U-Net.
The final architecture of the stable diffusion model
Considering the complex architecture of the mode, stable diffusion can generate the image "from scratch" by using a text prompt where it can be described which elements should be shown or left out
from the output. Also, it is possible to add new elements on the already existing image also by using the text prompt. Finally, I’d like to show you one example — on the website huggingface.co, there
is a space where you can try different things with a stable diffusion model. Here are the results when a text “A pikachu fine dining with a view to the Eiffel Tower” is written in the prompt:
A direct link to the space: https://huggingface.co/spaces/stabilityai/stable-diffusion
Stable diffusion gained a huge publicity over the last couple of months. The reasons for that are the fact that the source code is available to everybody and that stable diffusion does not hold any
rights to generated images, and it also allows people to be creative which can be seen from a large number of incredible published images which this model created. Generative models are developed and
improved, we can freely say, on a weekly basis and it will be very interesting to keep track of their future progress. | {"url":"https://www.vegait.co.uk/media-center/knowledge-base/what-is-stable-diffusion-and-how-does-it-work","timestamp":"2024-11-05T22:06:42Z","content_type":"text/html","content_length":"88564","record_id":"<urn:uuid:883d02de-7326-42ad-8b5e-da869d1bc1c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00414.warc.gz"} |
Concept of Fire Triangle: Understanding the Basics
Concept Of Fire Triangle: Understanding the Basics
The concept of the fire triangle is a fundamental principle in fire safety and prevention. It illustrates the three essential elements required for a fire to ignite and sustain: heat, fuel, and
By understanding how these components interact, you can better grasp how fires start, how they can be prevented, and what measures can be taken to extinguish them.
Whether you’re dealing with fire safety in the workplace, at home, or in an industrial setting, knowing the basics of the fire triangle is key to minimizing fire risks and ensuring safety.
Concept of Fire Triangle
The three sides of the fire triangle represent:
1. Heat
2. Fuel
3. Oxygen
Each of these elements plays a crucial role in the fire process, and removing just one can prevent or extinguish a fire.
The Fire Triangle: How Is Fire Created?
1. Heat
Heat is the first essential component of the fire triangle. It refers to the energy required to raise the temperature of a material to its ignition point. Heat sources can include anything from
matches and lighters to electrical sparks or friction. Once heat is applied, it causes the fuel to release gases that can ignite and sustain the fire.
2. Fuel
Fuel refers to any material that can combust, serving as the substance that burns and feeds the fire. It can be solid, liquid, or gas. Common types of fuel include wood, paper, gasoline, and natural
gas. Without fuel, a fire will have nothing to burn, making it one of the critical components of the triangle.
3. Oxygen
Oxygen supports the chemical reactions that fuel combustion. Fire typically requires oxygen concentrations of at least 16% to burn effectively. In most cases, this oxygen is drawn from the air around
us. Cutting off the oxygen supply, such as by smothering a fire with a blanket or using a fire extinguisher, will put out the fire.
Breaking the Fire Triangle
The fire triangle teaches us that by removing one of the three elements, we can prevent or extinguish a fire. Here’s how each element can be addressed:
• Heat: Cooling the fire, typically with water, can lower the temperature below the ignition point, stopping combustion.
• Fuel: Removing or isolating the fuel source will starve the fire, causing it to die out.
• Oxygen: Blocking oxygen, such as using foam or CO2 fire extinguishers, can suffocate the fire.
Process of Fire Dissemination
Fire dissemination, or the spread of fire, refers to how a fire propagates from its point of origin to other areas. This process occurs through the transfer of heat and the ignition of surrounding
materials. Understanding fire dissemination is critical for preventing and controlling fires. The key mechanisms by which fire spreads are:
1. Conduction
2. Convection
3. Radiation
4. Direct Flame Contact
1. Conduction
Conduction is the process by which heat is transferred through solid materials. When fire heats one part of a material, the heat travels through it, causing other parts to increase in temperature. If
the heat reaches flammable materials, it can ignite them. Metals, for instance, are good conductors of heat and can spread fire over distances by conducting heat to other areas.
• Example: Fire spreading through steel beams or walls made of metal.
2. Convection
Convection is the transfer of heat by the movement of fluids (gases or liquids). In fires, hot air or gases rise and carry heat upwards. As these heated gases accumulate, they can ignite combustible
materials in their path. Convection is a primary way fires spread vertically, such as in multi-story buildings.
• Example: A fire in a lower floor spreading to upper floors due to rising hot air and gases.
3. Radiation
Radiation is the transfer of heat through electromagnetic waves. Heat radiates from the flames and hot surfaces to surrounding objects, raising their temperature without direct contact. If the
radiated heat is intense enough, it can ignite nearby materials, even across distances.
• Example: A fire radiating heat from one building to another, causing the adjacent building to catch fire.
4. Direct Flame Contact
Direct flame contact occurs when flames physically touch and ignite other materials. This is the most obvious form of fire dissemination, where anything in direct contact with flames will ignite if
it’s flammable.
• Example: A fire spreading across a room by flames moving from one combustible object to another.
Phases of Fire Dissemination
1. Incipient Stage: The fire is in its early stage, with flames localized at the point of ignition. At this stage, it spreads slowly.
2. Growth Stage: The fire begins to spread as more materials catch fire, with heat transfer mechanisms like convection and radiation coming into play.
3. Fully Developed Stage: The fire has reached its peak, consuming all available fuel and spreading rapidly to adjacent areas.
4. Decay Stage: As the fuel is consumed or firefighting efforts succeed, the fire diminishes and eventually goes out.
Fire dissemination is a complex process that can occur through various mechanisms, including conduction, convection, radiation, and direct flame contact. Understanding these processes helps in the
development of fire safety protocols and firefighting techniques to control and prevent the rapid spread of fires.
What Are Flammable Materials?
Flammable materials are substances that can easily ignite and burn when exposed to heat, sparks, or flames. They have low ignition points, meaning they require relatively low temperatures to catch
fire, and they often release large amounts of energy in the form of heat and light when they burn.
These materials are dangerous because they can fuel fires or cause explosions if not handled properly. Flammable materials are categorized into different types based on their physical state, such as
gases, liquids, and solids.
Types of Flammable Materials:
1. Flammable Gases: These gases can easily mix with air and ignite when exposed to a heat source. They are particularly dangerous because they can spread quickly and are often invisible.
□ Examples: Propane, methane, hydrogen, acetylene.
2. Flammable Liquids: Flammable liquids are liquids with low flash points, meaning they produce enough vapor to ignite at relatively low temperatures. The vapor is what burns, not the liquid itself.
□ Examples: Gasoline, ethanol, acetone, paint thinner, kerosene.
3. Flammable Solids: These materials catch fire quickly when exposed to open flames or sparks. They can either burn themselves or release flammable gases when they decompose.
□ Examples: Wood, paper, magnesium, charcoal, plastic materials, sulfur.
4. Flammable Chemicals: Certain chemicals, whether solid, liquid, or gas, are highly reactive and ignite easily.
□ Examples: Sodium, potassium, phosphorus, organic peroxides.
Characteristics of Flammable Materials:
1. Low Flash Point: Flammable materials typically have a low flash point, which is the lowest temperature at which they can vaporize to form an ignitable mixture in the air.
2. High Vapor Pressure: They tend to produce a significant amount of vapor that can mix with air, increasing the risk of ignition.
3. Rapid Combustion: Once ignited, these materials burn quickly, spreading fire rapidly.
4. Explosion Risk: Certain flammable materials, especially gases, can cause explosions if they ignite in a confined space.
Examples of Common Flammable Materials:
• Everyday Items: Nail polish remover (acetone), lighter fluid, alcohol, hairspray, cooking oils, and gasoline.
• Construction and Industrial Materials: Solvents, paint thinners, and adhesives.
• Natural Materials: Dry leaves, sawdust, coal, and straw.
Safety Precautions:
• Storage: Flammable materials should be stored in well-ventilated areas, away from sources of heat and ignition.
• Handling: Always use proper protective equipment when handling flammable substances, such as gloves and goggles.
• Fire Extinguishers: Ensure fire extinguishers suitable for flammable materials (Class B for liquids, Class C for gases) are available nearby.
• Labeling: Flammable materials should be clearly labeled to alert users of potential fire hazards.
Understanding what flammable materials are and how to handle them safely is crucial in preventing fires and accidents, particularly in environments like kitchens, laboratories, and industrial
Frequently Asked Questions
What is the concept of fire?
The concept of fire involves a chemical reaction called combustion, where heat, fuel, and oxygen combine to produce heat, light, and often smoke. This process is essential for understanding fire
behavior and safety.
What is the meaning of a triangle fire?
A “triangle fire” refers to the fire triangle concept, which illustrates the three essential elements required for a fire: heat, fuel, and oxygen. Removing any one of these elements will extinguish
the fire.
What is the fire triangle in the workplace?
The fire triangle in the workplace represents the three components needed for a fire: heat, fuel, and oxygen. Managing these elements through safety practices helps prevent and control workplace
The concept of the fire triangle is fundamental in fire safety and prevention. By understanding how heat, fuel, and oxygen interact, we can take the necessary steps to control or prevent fires. This
concept serves as the foundation for firefighting strategies and fire prevention protocols used in homes, workplaces, and industries. | {"url":"https://safetysection.com/concept-of-fire-triangle/","timestamp":"2024-11-07T07:26:21Z","content_type":"text/html","content_length":"310044","record_id":"<urn:uuid:f627084a-f1fb-4c8e-9455-244c6a951f68>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00288.warc.gz"} |
Chapter 23: Including variants on randomized trials
Julian PT Higgins, Sandra Eldridge, Tianjing Li
Key Points:
• Non-standard designs, such as cluster-randomized trials and crossover trials, should be analysed using methods appropriate to the design.
• If the authors of studies included in the review fail to account for correlations among outcome data that arise because of the design, approximate methods can often be applied by review authors.
• A variant of the risk-of-bias assessment tool is available for cluster-randomized trials. Special attention should be paid to the potential for bias arising from how individual participants were
identified and recruited within clusters.
• A variant of the risk-of-bias assessment tool is available for crossover trials. Special attention should be paid to the potential for bias arising from carry-over of effects from one period to
the subsequent period of the trial, and to the possibility of ‘period effects’.
• To include a study with more than two intervention groups in a meta-analysis, a recommended approach is (i) to omit groups that are not relevant to the comparison being made, and (ii) to combine
multiple groups that are eligible as the experimental or comparator intervention to create a single pair-wise comparison. Alternatively, multi-arm studies are dealt with appropriately by network
This chapter should be cited as: Higgins JPT, Eldridge S, Li T (editors). Chapter 23: Including variants on randomized trials. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch
VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.1 (updated September 2020). Cochrane, 2020. Available from www.training.cochrane.org/handbook.
23.1 Cluster-randomized trials
In cluster-randomized trials, groups of individuals rather than individuals are randomized to different interventions. We say the ‘unit of allocation’ is the cluster, or the group. The groups may be,
for example, schools, villages, medical practices or families. Cluster-randomized trials may be done for one of several reasons. It may be to evaluate the group effect of an intervention, for example
herd-immunity of a vaccine. It may be to avoid ‘contamination’ across interventions when trial participants are managed within the same setting, for example in a trial evaluating training of
clinicians in a clinic. A cluster-randomized design may be used simply for convenience.
One of the main consequences of a cluster design is that participants within any one cluster often tend to respond in a similar manner, and thus their data can no longer be assumed to be independent.
It is important that the analysis of a cluster-randomized trial takes this issue into account. Unfortunately, many studies have in the past been incorrectly analysed as though the unit of allocation
had been the individual participants (Eldridge et al 2008). This is often referred to as a ‘unit-of-analysis error’ (Whiting-O’Keefe et al 1984) because the unit of analysis is different from the
unit of allocation. If the clustering is ignored and cluster-randomized trials are analysed as if individuals had been randomized, resulting confidence intervals will be artificially narrow and P
values will be artificially small. This can result in false-positive conclusions that the intervention had an effect. In the context of a meta-analysis, studies in which clustering has been ignored
will receive more weight than is appropriate.
In some trials, individual people are allocated to interventions that are then applied to multiple parts of those individuals (e.g. to both eyes or to several teeth), or repeated observations are
made on a participant. These body parts or observations are then clustered within individuals in the same way that individuals can be clustered within, for example, medical practices. If the analysis
is by the individual units (e.g. each tooth or each observation) without taking into account that the data are clustered within participants, then a unit-of-analysis error can occur.
There are several useful sources of information on cluster-randomized trials (Murray and Short 1995, Donner and Klar 2000, Eldridge and Kerry 2012, Campbell and Walters 2014, Hayes and Moulton 2017).
A detailed discussion of incorporating cluster-randomized trials in a meta-analysis is available (Donner and Klar 2002), as is a more technical treatment of the problem (Donner et al 2001). Evidence
suggests that many cluster-randomized trials have not been analysed appropriately when included in Cochrane Reviews (Richardson et al 2016).
23.1.2 Assessing risk of bias in cluster-randomized trials
A detailed discussion of risk-of-bias issues is provided in Chapter 7, and for the most part the Cochrane risk-of-bias tool for randomized trials, as outlined in Chapter 8, applies to
cluster-randomized trials.
A key difference between cluster-randomized trials and individually randomized trials is that the individuals of interest (those within the clusters) may not be directly allocated to one intervention
or another. In particular, sometimes the individuals are recruited into the study (or otherwise selected for inclusion in the analysis) after the interventions have been allocated to clusters,
creating the potential for knowledge of the allocation to influence whether individuals are recruited or selected into the analysis (Puffer et al 2003, Eldridge et al 2008). The bias that arises when
this occurs is referred to in various ways, but we use the term identification/recruitment bias, which distinguishes it from other types of bias. Careful trial design can protect against this bias
(Hahn et al 2005, Eldridge et al 2009a).
A second key difference between cluster-randomized trials and individually randomized trials is that identifying who the ‘participants’ are is not always straightforward in cluster-randomized trials.
The reasons for this are that in some trials:
1. there may be no formal recruitment of participants;
2. there may be two or more different groups of participants on whom different outcomes are measured (e.g. outcomes measured on clinicians and on patients); or
3. data are collected at two or more time points on different individuals (e.g. measuring physical activity in a community using a survey, which reaches different individuals at baseline and after
the intervention).
For the purposes of an assessment of risk of bias using the RoB 2 tool (see Chapter 8) we define participants in cluster-randomized trials as those on whom investigators seek to measure the outcome
of interest.
The RoB 2 tool has a variant specifically for cluster-randomized trials. To avoid very general language, it focuses mainly on cluster-randomized trials in which groups of individuals form the
clusters (rather than body parts or time points). Because most cluster-randomized trials are pragmatic in nature and aim to support high-level decisions about health care, the tool currently
considers only the effect of assignment to intervention (and not the effect of adhering to the interventions as they were intended). Special issues in assessing risk of bias in cluster-randomized
trials using RoB 2 are provided in Table 23.1.a.
Table 23.1.a Issues addressed in the Cochrane risk-of-bias tool for cluster-randomized trials
Bias domain Additional or different issues compared with individually randomized trials
Bias arising from the randomization • Processes for randomizing clusters vary: clusters may be randomized sequentially, in batches or all at once. Minimization is quite common and should be
process treated as equivalent to randomization. Cluster randomization is often performed at a single point in time by a methodologist, who may have less
motivation or knowledge to subvert randomization.
• The number of clusters can be relatively small, so chance imbalances are more common than in individually randomized trials. Such chance imbalances should
not be interpreted as evidence of risk of bias.
Bias arising from the timing of • This bias domain is specific to cluster-randomized trials.
identification and recruitment of • It is important to consider when individual participants were identified and recruited in relation to the timing of randomization.
participants • If identification or recruitment of any participants in the trial happened after randomization of the cluster, then their recruitment could have been
affected by knowledge of the intervention, introducing bias.
• Baseline imbalances in characteristics of participants (rather than of clusters) can suggest a problem with identification/recruitment bias.
Bias due to deviations from intended When the review authors’ interest is in the effect of assignment to intervention (see Chapter 8, Section 8.4):
• If participants are not aware that they are in a trial, then there will not be deviations from the intended intervention that arise because of the trial
context. It is these deviations that we are concerned about in this domain.
• If participants, carers or people delivering interventions are aware of the assigned intervention, then the issues are the same as for individually
randomized trials.
Bias due to missing outcome data • Data may be missing for clusters or for individuals within clusters.
• Considerations when addressing either type of missing data are the same as for individually randomized trials, but review authors should ensure that they
cover both.
Bias in measurement of the outcome • If outcome assessors are not aware that a trial is taking place, then their assessments should not be affected by intervention assignment.
• If outcome assessors are aware of the assigned intervention, then the issues are the same as for individually randomized trials.
Bias in selection of the reported result • The issues are the same as for individually randomized trials.
* For the precise wording of signalling questions and guidance for answering each one, see the full risk-of-bias tool at www.riskofbias.info.
23.1.3 Methods of analysis for cluster-randomized trials
One way to avoid a unit-of-analysis error in a cluster-randomized trial is to conduct the analysis at the same level as the allocation. That is, the data could be analysed as if each cluster was a
single individual, using a summary measurement from each cluster. Then the sample size for the analysis is the number of clusters. However, this strategy might unnecessarily reduce the precision of
the effect estimate if the clusters vary in their size.
Alternatively, statistical analysis at the level of the individual can lead to an inappropriately high level of precision in the analysis, unless methods are used to account for the clustering in the
data. The ideal information to extract from a cluster-randomized trial is a direct estimate of the required effect measure (e.g. an odds ratio with its confidence interval) from an analysis that
properly accounts for the cluster design. Such an analysis might be based on a multilevel model or may use generalized estimating equations, among other techniques. Statistical advice is recommended
to determine whether the method used is appropriate. When the study authors have not conducted such an analysis, there are two approximate approaches that can be used by review authors to adjust the
results (see Sections 23.1.4 and 23.1.5).
Effect estimates and their standard errors from correct analyses of cluster-randomized trials may be meta-analysed using the generic inverse-variance approach (e.g. in RevMan).
Unfortunately, many cluster-randomized trials have in the past failed to report appropriate analyses. They are commonly analysed as if the randomization was performed on the individuals rather than
the clusters. If this is the situation, approximately correct analyses may be performed if the following information can be extracted:
• the number of clusters (or groups) randomized to each intervention group and the total number of participants in the study; or the average (mean) size of each cluster;
• the outcome data ignoring the cluster design for the total number of individuals (e.g. the number or proportion of individuals with events, or means and standard deviations for continuous data);
• an estimate of the intracluster (or intraclass) correlation coefficient (ICC).
The ICC is an estimate of the relative variability within and between clusters (Eldridge and Kerry 2012). Alternatively it describes the ‘similarity’ of individuals within the same cluster (Eldridge
et al 2009b). In spite of recommendations to report the ICC in all trial reports (Campbell et al 2012), ICC estimates are often not available in published reports.
A common approach for review authors is to use external estimates obtained from similar studies, and several resources are available that provide examples of ICCs (Ukoumunne et al 1999, Campbell et
al 2000, Health Services Research Unit 2004), or use an estimate based on known patterns in ICCs for particular types of cluster or outcome. ICCs may appear small compared with other types of
correlations: values lower than 0.05 are typical. However, even small values can have a substantial impact on confidence interval widths (and hence weights in a meta-analysis), particularly if
cluster sizes are large. Empirical research has observed that clusters that tend to be naturally larger have smaller ICCs (Ukoumunne et al 1999). For example, for the same outcome, regions are likely
to have smaller ICCs than towns, which are likely to have smaller ICCs than families.
An approximately correct analysis proceeds as follows. The idea is to reduce the size of each trial to its ‘effective sample size’ (Rao and Scott 1992). The effective sample size of a single
intervention group in a cluster-randomized trial is its original sample size divided by a quantity called the ‘design effect’. The design effect is approximately
where M is the average cluster size and ICC is the intracluster correlation coefficient. When cluster sizes vary, M can be estimated more appropriately in other ways (Eldridge et al 2006). A common
design effect is usually assumed across intervention groups. For dichotomous data, both the number of participants and the number experiencing the event should be divided by the same design effect.
Since the resulting data must be rounded to whole numbers for entry into meta-analysis software such as RevMan, this approach may be unsuitable for small trials. For continuous data, only the sample
size need be reduced; means and standard deviations should remain unchanged. Special considerations for analysis of standardized mean differences from cluster-randomized trials are discussed by White
and Thomas (White and Thomas 2005).
As an example, consider a cluster-randomized trial that randomized 10 school classrooms with 295 children into a treatment group and 11 classrooms with 330 children into a control group. Suppose the
numbers of successes among the children, ignoring the clustering, are:
Treatment: 63/295
Control: 84/330.
Imagine an intracluster correlation coefficient of 0.02 has been obtained from a reliable external source or is expected to be a good estimate, based on experience in the area. The average cluster
size in the trial is
(295 + 330) ÷ (10 + 11) = 29.8.
The design effect for the trial as a whole is then
1 + (M – 1) ICC = 1 + (29.8 – 1) × 0.02 = 1.576.
The effective sample size in the treatment group is
295 ÷ 1.576 = 187.2
and for the control group is
330 ÷ 1.576 = 209.4.
Applying the design effects also to the numbers of events (in this case, successes) produces the following modified results:
Treatment: 40.0/187.2
Control: 53.3/209.4.
Once trials have been reduced to their effective sample size, the data may be entered into statistical software such as RevMan as, for example, dichotomous outcomes or continuous outcomes. Rounding
the results to whole numbers, the results from the example trial may be entered as:
Treatment: 40/187
Control: 53/209.
A clear disadvantage of the method described in Section 23.1.4 is the need to round the effective sample sizes to whole numbers. A slightly more flexible approach, which is equivalent to calculating
effective sample sizes, is to multiply the standard error of the effect estimate (from an analysis ignoring clustering) by the square root of the design effect. The standard error may be calculated
from the confidence interval of any effect estimate derived from an analysis ignoring clustering (see Chapter 6, Sections 6.3.1 and 6.3.2). Standard analyses of dichotomous or continuous outcomes may
be used to obtain these confidence intervals using standard meta-analysis software (e.g. RevMan). The meta-analysis using the inflated variances may be performed using the generic inverse-variance
As an example, the odds ratio (OR) from a study with the results
Treatment: 63/295
Control: 84/330
is OR=0.795 (95% CI 0.548 to 1.154). Using methods described in Chapter 6 (Section 6.3.2), we can determine from these results that the log odds ratio is lnOR=–0.23 with standard error 0.19. Using
the same design effect of 1.576 as in Section 23.1.4.1, an inflated standard error that accounts for clustering is given by 0.19×√1.576=0.24. The log odds ratio (–0.23) and this inflated standard
error (0.24) may be used as the basis for a meta-analysis using a generic inverse-variance approach.
23.1.6 Issues in the incorporation of cluster-randomized trials
Cluster-randomized trials may, in principle, be combined with individually randomized trials in the same meta-analysis. Consideration should be given to the possibility of important differences in
the effects being evaluated between the different types of trial. There are often good reasons for performing cluster-randomized trials and these should be examined. For example, in the treatment of
infectious diseases an intervention applied to all individuals in a community may be more effective than treatment applied to select (randomized) individuals within the community, since it may reduce
the possibility of re-infection (Eldridge and Kerry 2012).
Authors should always identify any cluster-randomized trials in a review and explicitly state how they have dealt with the data. They should conduct sensitivity analyses to investigate the robustness
of their conclusions, especially when ICCs have been borrowed from external sources (see Chapter 10, Section 10.14). Statistical support is recommended.
23.1.7 Stepped-wedge trials
In a stepped-wedge trial, randomization is by cluster. However, rather than assign a predefined proportion of the clusters to the experimental intervention and the rest to a comparator intervention,
a stepped-wedge design starts with all clusters allocated to the comparator intervention and sequentially randomizes individual clusters (or groups of clusters) to switch to the experimental
intervention. By the end of the trial, all clusters are implementing the experimental intervention (Hemming et al 2015). Stepped-wedge trials are increasingly used to evaluate health service and
policy interventions, and are often attractive to policy makers because all clusters can expect to receive (or implement) the experimental intervention.
The analysis of a stepped-wedge trial must take into account the possibility of time trends. A naïve comparison of experimental intervention periods with comparator intervention periods will be
confounded by any variables that change over time, since more clusters are receiving the experimental intervention during the later stages of the trial.
The RoB 2 tool for cluster-randomized trials can be used to assess risk of bias in a stepped-wedge trial. However, the tool does not address the need to adjust for time trends in the analysis, which
is an important additional source of potential bias in a stepped-wedge trial.
23.1.8 Individually randomized trials with clustering
Issues related to clustering can also occur in individually randomized trials. This can happen when the same health professional (e.g. doctor, surgeon, nurse or therapist) delivers the intervention
to a number of participants in the intervention group. This type of clustering raises issues similar to those in cluster-randomized trials in relation to the analysis (Lee and Thompson 2005, Walwyn
and Roberts 2015, Walwyn and Roberts 2017), and review authors should consider inflating the variance of the intervention effect estimate using a design effect, as for cluster-randomized trials.
23.2 Crossover trials
Parallel-group trials allocate each participant to a single intervention for comparison with one or more alternative interventions. In contrast, crossover trials allocate each participant to a
sequence of interventions. A simple randomized crossover design is an ‘AB/BA’ design in which participants are randomized initially to intervention A or intervention B, and then ‘cross over’ to
intervention B or intervention A, respectively. It can be seen that data from the first period of a crossover trial represent a parallel-group trial, a feature referred to in Section 23.2.6. In
keeping with the rest of the Handbook, we will use E and C to refer to interventions, rather than A and B.
Crossover designs offer a number of possible advantages over parallel-group trials. Among these are that:
1. each participant acts as his or her own control, significantly reducing between-participant variation;
2. consequently, fewer participants are usually required to obtain the same precision in estimation of intervention effects; and
3. every participant receives every intervention, which allows the determination of the best intervention or preference for an individual participant.
In some trials, randomization of interventions takes place within individuals, with different interventions being applied to different body parts (e.g. to the two eyes or to teeth in the two sides of
the mouth). If body parts are randomized and the analysis is by the multiple parts within an individual (e.g. each eye or each side of the mouth) then the analysis should account for the pairing (or
matching) of parts within individuals in the same way that pairing of intervention periods is recognized in the analysis of a crossover trial.
A readable introduction to crossover trials is given by Senn (Senn 2002). More detailed discussion of meta-analyses involving crossover trials is provided by Elbourne and colleagues (Elbourne et al
2002), and some empirical evidence on their inclusion in systematic reviews by Lathyris and colleagues (Lathyris et al 2007). Evidence suggests that many crossover trials have not been analysed
appropriately when included in Cochrane Reviews (Nolan et al 2016).
Crossover trials are suitable for evaluating interventions with a temporary effect in the treatment of stable, chronic conditions (at least over the time period under study). They are employed, for
example, in the study of interventions to relieve asthma, rheumatoid arthritis and epilepsy. There are many situations in which a crossover trial is not appropriate. These include:
1. if the medical condition evolves over time, such as a degenerative disorder, a temporary condition that will resolve within the time frame of the trial, or a cyclic disorder;
2. when an intervention (or its cessation) can lead to permanent or long-term modification (e.g. a vaccine). In this situation, either a participant will be unable (or ineligible) to enter a
subsequent period of the trial; or a ‘carry-over’ effect is likely (see Section 23.2.3);
3. if the elimination half-life of a drug is very long so that a ‘carry-over’ effect is likely (see Section 23.2.3); and
4. if wash-out itself induces a withdrawal or rebound effect in the second period.
In considering the inclusion of crossover trials in meta-analysis, authors should first address the question of whether a crossover trial is a suitable method for the condition and intervention in
question. For example, one group of authors decided that crossover trials were inappropriate for studies in Alzheimer’s disease (although they are frequently employed in the field) due to the
degenerative nature of the condition, and included only data from the first period of crossover trials in their systematic review (Qizilbash et al 1998). The second question to be addressed is
whether there is a likelihood of serious carry-over, which relies largely on judgement since the statistical techniques to demonstrate carry-over are far from satisfactory. The nature of the
interventions and the length of any wash-out period are important considerations.
It is only justifiable to exclude crossover trials from a systematic review if the design is inappropriate to the clinical context. Very often, however, even where the design has been appropriate, it
is difficult or impossible to extract suitable data from a crossover trial. In Section 23.2.6 we outline some considerations and suggestions for including crossover trials in a meta-analysis.
The principal problem associated with crossover trials is that of carry-over (a type of period-by-intervention interaction). Carry-over is the situation in which the effects of an intervention given
in one period persist into a subsequent period, thus interfering with the effects of the second intervention. These effects may be because the first intervention itself persists (such as a drug with
a long elimination half-life), or because the effects of the intervention persist. An extreme example of carry-over is when a key outcome of interest is irreversible or of long duration, such as
mortality, or pregnancy in a subfertility study. In this case, a crossover study is generally considered to be inappropriate. A carry-over effect means that the observed difference between the
treatments depends upon the order in which they were received; hence the estimated overall treatment effect will be affected (usually under-estimated, leading to a bias towards the null). Many
crossover trials include a period between interventions known as a wash-out period as a means of reducing carry-over.
A second problem that may occur in crossover trials is period effects. Period effects are systematic differences between responses in the second period compared with responses in the first period
that are not due to different interventions. They may occur, for example, when the condition changes systematically over time, or if there are changes in background factors such as underlying
healthcare strategies. For an AB/BA design, period effects can be overcome by ensuring the same number of participants is randomized to the two sequences of interventions or by including period
effects in the statistical model.
A third problem for crossover trials is that the trial might report only analyses based on the first period. Although the first period of a crossover trial is in effect a parallel group comparison,
use of data from only the first period will be biased if, as is likely, the decision to use first period data is based on a test for carry-over. Such a ‘two-stage analysis’ has been discredited but
is still used (Freeman 1989). This is because the test for carry-over is affected by baseline differences in the randomized groups at the start of the crossover trial, so a statistically significant
result might reflect such baseline differences. Reporting only the first period data in this situation is particularly problematic. Crossover trials for which only first period data are available
should be considered to be at risk of bias, especially when the investigators explicitly report using a two-stage analysis strategy.
Another potential problem with crossover trials is the risk of dropout due to their longer duration compared with comparable parallel-group trials. The analysis techniques for crossover trials with
missing observations are limited.
The Cochrane risk-of-bias tool for randomized trials (RoB 2, see Chapter 8) has a variant specifically for crossover trials. It focuses on crossover trials with two intervention periods rather than
with two body parts. Carry-over effects are addressed specifically. Period effects are addressed through examination of the allocation ratio and the approach to analysis. The tool also addresses the
possibility of selective reporting of first period results in the domain 'Bias in selection of the reported result'. Special issues in assessing risk of bias in a crossover trials using RoB 2 are
provided in Table 23.2.a.
Table 23.2.a Issues addressed in version 2 of the Cochrane risk-of-bias tool for randomized crossover trials
Bias domain Additional or different issues addressed compared with parallel-group trials
Bias arising from the • The issues surrounding methods of randomization are the same as for parallel-group trials.
randomization process • If an equal proportion of participants is randomized to each intervention sequence, then any period effects will cancel out in the analysis (providing there is not
differential missing data).
• If unequal proportions of participants are randomized to the different intervention sequences, then period effects should be included in the analysis to avoid bias.
• When using baseline differences to infer a problem with the randomization process, this should be based on differences at the start of the first period only.
Bias due to deviations • Carry-over is the key concern when assessing risk of bias in a crossover trial. Carry-over effects should not affect outcomes measured in the second period. A long
from intended period of wash-out between periods can avoid this but is not essential. The important consideration is whether sufficient time passes before outcome measurement in the
interventions second period, such that any carry-over effects have disappeared.
• All other issues are the same as for parallel-group trials.
Bias due to missing • The issues are the same as for parallel-group trials. Use of last observation carried forward imputation may be particularly problematic if the observations being
outcome data carried forward were made before carry-over effects had disappeared. Some analyses of crossover trials will automatically exclude (for an AB/BA design) all patients with
missing data in either period.
Bias in measurement of • The issues are the same as for parallel-group trials.
the outcome
Bias in selection of the • An additional concern is the selective reporting of first period data on the basis of a test for carry-over.
reported result
* For the precise wording of signalling questions and guidance for answering each one, see the full risk-of-bias tool at www.riskofbias.info.
23.2.4 Using only the first period of a crossover trial
One option when crossover trials are anticipated in a review is to plan from the outset that only data from the first periods will be used. Including only the first intervention period of a crossover
trial discards more than half of the information in the study, and often substantially more than half. A sound rationale is therefore needed for this approach, based on the inappropriateness of a
crossover design (see Section 23.2.2), and not based on lack of methodological expertise.
If the review intends (from the outset) to look only at the first period of any crossover trial, then review authors should use the standard version of the RoB 2 tool for parallel group randomized
trials. Review authors must, however, be alert to the potential impact of selective reporting if first-period data are reported only when carry-over is detected by the trialists. Omission of trials
reporting only paired analyses (i.e. not reporting data for the first period separately) may lead to bias at the meta-analysis level. The bias will not be picked up using study-level assessments of
risk of bias.
If neither carry-over nor period effects are thought to be a problem, then an appropriate analysis of continuous data from a two-period, two-intervention crossover trial is a paired t-test. This
evaluates the value of ‘measurement on experimental intervention (E)’ minus ‘measurement on control intervention (C)’ separately for each participant. The mean and standard error of these difference
measures are the building blocks of an effect estimate and a statistical test. The effect estimate may be included in a meta-analysis using a generic inverse-variance approach (e.g. in RevMan).
A paired analysis is possible if the data in any one of the following bullet points is available:
• individual participant data from the paper or by correspondence with the trialist;
• the mean and standard deviation (or standard error) of the participant-level differences between experimental intervention (E) and comparator intervention (C) measurements;
• the mean difference and one of the following: (i) a t-statistic from a paired t-test; (ii) a P value from a paired t-test; (iii) a confidence interval from a paired analysis;
• a graph of measurements on experimental intervention (E) and comparator intervention (C) from which individual data values can be extracted, as long as matched measurements for each individual
can be identified as such.
For details see Elbourne and colleagues (Elbourne et al 2002).
Crossover trials with dichotomous outcomes require more complicated methods and consultation with a statistician is recommended (Elbourne et al 2002).
If results are available broken into subgroups by the particular sequence each participant received, then analyses that adjust for period effects are straightforward (e.g. as outlined in Chapter 3 of
Senn (Senn 2002)).
Unfortunately, the reporting of crossover trials has been very variable, and the data required to include a paired analysis in a meta-analysis are often not published (Li et al 2015). A common
situation is that means and standard deviations (or standard errors) are available only for measurements on E and C separately. A simple approach to incorporating crossover trials in a meta-analysis
is thus to take all measurements from intervention E periods and all measurements from intervention C periods and analyse these as if the trial were a parallel-group trial of E versus C. This
approach gives rise to a unit-of-analysis error (see Chapter 6, Section 6.2) and should be avoided. The reason for this is that confidence intervals are likely to be too wide, and the trial will
receive too little weight, with the possible consequence of disguising clinically important heterogeneity. Nevertheless, this incorrect analysis is conservative, in that studies are under-weighted
rather than over-weighted. While some argue against the inclusion of crossover trials in this way, the unit-of-analysis error might be regarded as less serious than some other types of
unit-of-analysis error.
A second approach to incorporating crossover trials is to include only data from the first period. This might be appropriate if carry-over is thought to be a problem, or if a crossover design is
considered inappropriate for other reasons. However, it is possible that available data from first periods constitute a biased subset of all first period data. This is because reporting of first
period data may be dependent on the trialists having found statistically significant carry-over.
A third approach to incorporating inappropriately reported crossover trials is to attempt to approximate a paired analysis, by imputing missing standard deviations. We address this approach in detail
in Section 23.2.7.
Table 23.2.b presents some results that might be available from a report of a crossover trial, and presents the notation we will use in the subsequent sections. We review straightforward methods for
approximating appropriate analyses of crossover trials to obtain mean differences or standardized mean differences for use in meta-analysis. Review authors should consider whether imputing missing
data is preferable to excluding crossover trials completely from a meta-analysis. The trade-off will depend on the confidence that can be placed on the imputed numbers, and on the robustness of the
meta-analysis result to a range of plausible imputed results.
Table 23.2.b Some possible data available from the report of a crossover trial
Data relate to Core statistics Related, commonly reported statistics
Intervention E N, M[E], SD[E] Standard error of M[E].
Intervention C N, M[C], SD[C] Standard error of M[C].
Difference between E and C N, MD, SD[diff] Standard error of MD;
Confidence interval for MD;
Paired t-statistic;
P value from paired t-test.
The point estimate of mean difference for a paired analysis is usually available, since it is the same as for a parallel-group analysis (the mean of the differences is equal to the difference in
The standard error of the mean difference is obtained as
where N is the number of participants in the trial, and SD[diff] is the standard deviation of within-participant differences between E and C measurements. As indicated in Section 23.2.5, the standard
error can also be obtained directly from a confidence interval for MD, from a paired t-statistic, or from the P value from a paired t-test. The quantities MD and SE(MD) may be entered into a
meta-analysis under the generic inverse-variance outcome type (e.g. in RevMan).
When the standard error is not available directly and the standard deviation of the differences is not presented, a simple approach is to impute the standard deviation, as is commonly done for other
missing standard deviations (see Chapter 6, Section 6.5.2.7). Other studies in the meta-analysis may present standard deviations of differences, and as long as the studies use the same measurement
scale, it may be reasonable to borrow these from one study to another. As with all imputations, sensitivity analyses should be undertaken to assess the impact of the imputed data on the findings of
the meta-analysis (see Chapter 10, Section 10.14).
If no information is available from any study on the standard deviations of the within-participant differences, imputation of standard deviations can be achieved by assuming a particular correlation
coefficient. The correlation coefficient describes how similar the measurements on interventions E and C are within a participant, and is a number between –1 and 1. It may be expected to lie between
0 and 1 in the context of a crossover trial, since a higher than average outcome for a participant while on E will tend to be associated with a higher than average outcome while on C. If the
correlation coefficient is zero or negative, then there is no statistical benefit of using a crossover design over using a parallel-group design.
A common way of presenting results of a crossover trial is as if the trial had been a parallel-group trial, with standard deviations for each intervention separately (SD[E] and SD[C]; see Table
23.2.b). The desired standard deviation of the differences can be estimated using these intervention-specific standard deviations and an imputed correlation coefficient (Corr):
The most appropriate standardized mean difference (SMD) from a crossover trial divides the mean difference by the standard deviation of measurements (and not by the standard deviation of the
differences). A SMD can be calculated by pooled intervention-specific standard deviations as follows:
A correlation coefficient is required for the standard error of the SMD:
Alternatively, the SMD can be calculated from the MD and its standard error, using an imputed correlation:
In this case, the imputed correlation impacts on the magnitude of the SMD effect estimate itself (rather than just on the standard error, as is the case for MD analyses in Section 23.2.7.1). Imputed
correlations should therefore be used with great caution for estimation of SMDs.
The value for a correlation coefficient might be imputed from another study in the meta-analysis (see below), it might be imputed from a source outside of the meta-analysis, or it might be
hypothesized based on reasoned argument. In all of these situations, a sensitivity analysis should be undertaken, trying different plausible values of Corr, to determine whether the overall result of
the analysis is robust to the use of imputed correlation coefficients.
Estimation of a correlation coefficient is possible from another study in the meta-analysis if that study presents all three standard deviations in Table 23.2.b. The calculation assumes that the mean
and standard deviation of measurements for intervention E is the same when it is given in the first period as when it is given in the second period (and similarly for intervention C).
Before imputation is undertaken it is recommended that correlation coefficients are computed for as many studies as possible and compared. If these correlations vary substantially then sensitivity
analyses are particularly important.
As an example, suppose a crossover trial reports the following data:
Intervention E M[E] = 7.0,
(sample size 10) SD[E] = 2.38
Intervention C M[C] = 6.5,
(sample size 10) SD[C] = 2.21
Mean difference, imputing SD of differences (SD[diff])
The estimate of the mean difference is MD=7.0–6.5=0.5. Suppose that a typical standard deviation of differences had been observed from other trials to be 2. Then we can estimate the standard error of
MD as
The numbers 0.5 and 0.632 may be entered into RevMan as the estimate and standard error of a mean difference, under a generic inverse-variance outcome.
Mean difference, imputing correlation coefficient (Corr)
The estimate of the mean difference is again MD=0.5. Suppose that a correlation coefficient of 0.68 has been imputed. Then we can impute the standard deviation of the differences as:
The standard error of MD is then
The numbers 0.5 and 0.583 may be entered into a meta-analysis as the estimate and standard error of a mean difference, under a generic inverse-variance outcome. Correlation coefficients other than
0.68 should be used as part of a sensitivity analysis.
Standardized mean difference, imputing correlation coefficient (Corr)
The standardized mean difference can be estimated directly from the data:
The standard error is obtained thus:
The numbers 0.218 and 0.256 may be entered into a meta-analysis as the estimate and standard error of a standardized mean difference, under a generic inverse-variance outcome.
We could also have obtained the SMD from the MD and its standard error:
The minor discrepancy arises due to the slightly different ways in which the two formulae calculate a pooled standard deviation for the standardizing.
23.2.8 Issues in the incorporation of crossover trials
Crossover trials may, in principle, be combined with parallel-group trials in the same meta-analysis. Consideration should be given to the possibility of important differences in other
characteristics between the different types of trial. For example, crossover trials may have shorter intervention periods or may include participants with less severe illness. It is generally
advisable to meta-analyse parallel-group and crossover trials in separate subgroups, irrespective of whether they are also combined.
Review authors should explicitly state how they have dealt with data from crossover trials and should conduct sensitivity analyses to investigate the robustness of their conclusions, especially when
correlation coefficients have been borrowed from external sources (see Chapter 10, Section 10.14). Statistical support is recommended.
23.2.9 Cluster crossover trials
A cluster crossover trial combines aspects of a cluster-randomized trial (Section 23.1.1) and a crossover trial (Section 23.2.1). In a two-period, two-intervention cluster crossover trial, clusters
are randomized to either the experimental intervention or the comparator intervention. At the end of the first period, clusters on the experimental intervention cross over to the comparator
intervention for the second period, and clusters on the comparator intervention cross over to the experimental intervention for the second period (Rietbergen and Moerbeek 2011, Arnup et al 2017). The
clusters may involve the same individuals in both periods, or different individuals in the two periods. The design introduces the advantages of a crossover design into situations in which
interventions are most appropriately implemented or evaluated at the cluster level.
The analysis of a cluster crossover trial should consider both the pairing of intervention periods within clusters and the similarity of individuals within clusters. Unfortunately, many trials have
not performed appropriate analyses (Arnup et al 2016), so review authors are encouraged to seek statistical advice.
The RoB 2 tool does not currently have a variant for cluster crossover trials.
23.3 Studies with more than two intervention groups
23.3.1 Introduction
It is not uncommon for clinical trials to randomize participants to one of several intervention groups. A review of randomized trials published in December 2000 found that a quarter had more than two
intervention groups (Chan and Altman 2005). For example, there may be two or more experimental intervention groups with a common comparator group, or two comparator intervention groups such as a
placebo group and a standard treatment group. We refer to these studies as ‘multi-arm’ studies. A special case is a factorial trial, which addresses two or more simultaneous intervention comparisons
using four or more intervention groups (see Section 23.3.6).
Although a systematic review may include several intervention comparisons (and hence several meta-analyses), almost all meta-analyses address pair-wise comparisons. There are three separate issues to
consider when faced with a study with more than two intervention groups.
1. Determine which intervention groups are relevant to the systematic review.
2. Determine which intervention groups are relevant to a particular meta-analysis.
3. Determine how the study will be included in the meta-analysis if more than two groups are relevant.
23.3.2 Determining which intervention groups are relevant
For a particular multi-arm study, the intervention groups of relevance to a systematic review are all those that could be included in a pair-wise comparison of intervention groups that would meet the
criteria for including studies in the review. For example, a review addressing only a comparison of nicotine replacement therapy versus placebo for smoking cessation might identify a study comparing
nicotine gum versus behavioural therapy versus placebo gum. Of the three possible pair-wise comparisons of interventions in this study, only one (nicotine gum versus placebo gum) addresses the review
objective, and no comparison involving behavioural therapy does. Thus, the behavioural therapy group is not relevant to the review, and can be safely left out of any syntheses. However, if the study
had compared nicotine gum plus behavioural therapy versus behavioural therapy plus placebo gum versus placebo gum alone, then a comparison of the first two interventions might be considered relevant
(with behavioural therapy provided as a consistent co-intervention to both groups of interest), and the placebo gum alone group might not.
As an example of multiple comparator groups, a review addressing the comparison ‘acupuncture versus no acupuncture’ might identify a study comparing ‘acupuncture versus sham acupuncture versus no
intervention’. The review authors would ask whether, on the one hand, a study of ‘acupuncture versus sham acupuncture’ would be included in the review and, on the other hand, a study of ‘acupuncture
versus no intervention’ would be included. If both of them would, then all three intervention groups of the study are relevant to the review.
As a general rule, and to avoid any confusion for the reader over the identity and nature of each study, it is recommended that all intervention groups of a multi-intervention study be mentioned in
the table of ‘Characteristics of included studies’. However, it is necessary to provide detailed descriptions of only the intervention groups relevant to the review, and only these groups should be
used in analyses.
The same considerations of relevance apply when determining which intervention groups of a study should be included in a particular meta-analysis. Each meta-analysis addresses only a single pair-wise
comparison, so review authors should consider whether a study of each possible pair-wise comparison of interventions in the study would be eligible for the meta-analysis. To draw the distinction
between the review-level decision and the meta-analysis-level decision, consider a review of ‘nicotine therapy versus placebo or other comparators’. All intervention groups of a study of ‘nicotine
gum versus behavioural therapy versus placebo gum’ might be relevant to the review. However, the presence of multiple interventions may not pose any problem for meta-analyses, since it is likely that
‘nicotine gum versus placebo gum’, and ‘nicotine gum versus behavioural therapy’ would be addressed in different meta-analyses. Conversely, all groups of the study of ‘acupuncture versus sham
acupuncture versus no intervention’ might be considered eligible for the same meta-analysis. This would be the case if the meta-analysis would otherwise include both studies of ‘acupuncture versus
sham acupuncture’ and studies of ‘acupuncture versus no intervention’, treating sham acupuncture and no intervention both as relevant comparators. We describe methods for dealing with the latter
situation in Section 23.3.4.
Bias may be introduced in a multiple-intervention study if the decisions regarding data analysis are made after seeing the data. For example, groups receiving different doses of the same intervention
may be combined only after looking at the results. Also, decisions about the selection of outcomes to report may be made after comparing different pairs of intervention groups and examining the
findings. These issues would be addressed in the domain ‘Bias due to selection of the reported result’ in the Cochrane risk-of-bias tool for randomized trials (RoB 2, see Chapter 8).
Juszczak and colleagues reviewed 60 multiple-intervention randomized trials, of which over a third had at least four intervention arms (Juszczak et al 2003). They found that only 64% reported the
same comparisons of groups for all outcomes, suggesting selective reporting analogous to selective outcome reporting in a two-arm trial. Also, 20% reported combining groups in an analysis. However,
if the summary data are provided for each intervention group, it does not matter how the groups had been combined in reported analyses; review authors do not need to analyse the data in the same way
as the study authors.
There are several possible approaches to including a study with multiple intervention groups in a particular meta-analysis. One approach that must be avoided is simply to enter several comparisons
into the meta-analysis so that the same comparator intervention group is included more than once. This ‘double-counts’ the participants in the intervention group(s) shared across more than one
comparison, and creates a unit-of-analysis error due to the unaddressed correlation between the estimated intervention effects from multiple comparisons (see Chapter 6, Section 6.2). An important
distinction is between situations in which a study can contribute several independent comparisons (i.e. with no intervention group in common) and when several comparisons are correlated because they
have intervention groups, and hence participants, in common. For example, consider a study that randomized participants to four groups: ‘nicotine gum’ versus ‘placebo gum’ versus ‘nicotine patch’
versus ‘placebo patch’. A meta-analysis that addresses the broad question of whether nicotine replacement therapy is effective might include the comparison ‘nicotine gum versus placebo gum’ as well
as the independent comparison ‘nicotine patch versus placebo patch’, with no unit of analysis error or double-counting. It is usually reasonable to include independent comparisons in a meta-analysis
as if they were from different studies, although there are subtle complications with regard to random-effects analyses (see Section 23.3.5).
Approaches to overcoming a unit-of-analysis error for a study that could contribute multiple, correlated, comparisons include the following.
• Combine groups to create a single pair-wise comparison (recommended).
• Select one pair of interventions and exclude the others.
• Split the ‘shared’ group into two or more groups with smaller sample size, and include two or more (reasonably independent) comparisons.
• Include two or more correlated comparisons and account for the correlation.
• Undertake a network meta-analysis (see Chapter 11).
The recommended method in most situations is to combine all relevant experimental intervention groups of the study into a single group, and to combine all relevant comparator intervention groups into
a single comparator group. As an example, suppose that a meta-analysis of ‘acupuncture versus no acupuncture’ would consider studies of either ‘acupuncture versus sham acupuncture’ or studies of
‘acupuncture versus no intervention’ to be eligible for inclusion. Then a study with three intervention groups (acupuncture, sham acupuncture and no intervention) would be included in the
meta-analysis by combining the participants in the ‘sham acupuncture’ group with participants in the ‘no intervention’ group. This combined comparator group would be compared with the ‘acupuncture’
group in the usual way. For dichotomous outcomes, both the sample sizes and the numbers of people with events can be summed across groups. For continuous outcomes, means and standard deviations can
be combined using methods described in Chapter 6 (Section 6.5.2.10).
The alternative strategy of selecting a single pair of interventions (e.g. choosing either ‘sham acupuncture’ or ‘no intervention’ as the comparator) results in a loss of information and is open to
results-related choices, so is not generally recommended.
A further possibility is to include each pair-wise comparison separately, but with shared intervention groups divided out approximately evenly among the comparisons. For example, if a trial compares
121 patients receiving acupuncture with 124 patients receiving sham acupuncture and 117 patients receiving no acupuncture, then two comparisons (of, say, 61 ‘acupuncture’ against 124 ‘sham
acupuncture’, and of 60 ‘acupuncture’ against 117 ‘no intervention’) might be entered into the meta-analysis. For dichotomous outcomes, both the number of events and the total number of patients
would be divided up. For continuous outcomes, only the total number of participants would be divided up and the means and standard deviations left unchanged. This method only partially overcomes the
unit-of-analysis error (because the resulting comparisons remain correlated) so is not generally recommended. A potential advantage of this approach, however, would be that approximate investigations
of heterogeneity across intervention arms are possible (e.g. in the case of the example here, the difference between using sham acupuncture and no intervention as a comparator group).
Two final options are to account for the correlation between correlated comparisons from the same study in the analysis, and to perform a network meta-analysis. The former involves calculating an
average (or weighted average) of the relevant pair-wise comparisons from the study, and calculating a variance (and hence a weight) for the study, taking into account the correlation between the
comparisons (Borenstein et al 2008). It will typically yield a similar result to the recommended method of combining across experimental and comparator intervention groups. Network meta-analysis
allows for the simultaneous analysis of multiple interventions, and so naturally allows for multi-arm studies. Network meta-analysis is discussed in more detail in Chapter 11.
Two possibilities for addressing heterogeneity between studies are to allow for it in a random-effects meta-analysis, and to investigate it through subgroup analyses or meta-regression (Chapter 10,
Section 10.11). Some complications arise when including multiple-intervention studies in such analyses. First, it will not be possible to investigate certain intervention-related sources of
heterogeneity if intervention groups are combined as in the recommended approach in Section 23.3.4. For example, subgrouping according to ‘sham acupuncture’ or ‘no intervention’ as a comparator group
is not possible if these two groups are combined prior to the meta-analysis. The simplest method for allowing an investigation of this difference, across studies, is to create two or more comparisons
from the study (e.g. ‘acupuncture versus sham acupuncture’ and ‘acupuncture versus no intervention’). However, if these contain a common intervention group (here, acupuncture), then they are not
independent and a unit-of-analysis error will occur, even if the sample size is reduced for the shared intervention group(s). Nevertheless, splitting up the sample size for the shared intervention
group remains a practical means of performing approximate investigations of heterogeneity.
A more subtle problem occurs in random-effects meta-analyses if multiple comparisons are included from the same study. A random-effects meta-analysis allows for variation by assuming that the effects
underlying the studies in the meta-analysis follow a distribution across studies. The intention is to allow for study-to-study variation. However, if two or more estimates come from the same study
then the same variation is assumed across comparisons within the study and across studies. This is true whether the comparisons are independent or correlated (see Section 23.3.4). One way to overcome
this is to perform a fixed-effect meta-analysis across comparisons within a study, and a random-effects meta-analysis across studies. Statistical support is recommended; in practice the difference
between different analyses is likely to be trivial.
In a factorial trial, two (or more) intervention comparisons are carried out simultaneously. Thus, for example, participants may be randomized to receive aspirin or placebo, and also randomized to
receive a behavioural intervention or standard care. Most factorial trials have two ‘factors’ in this way, each of which has two levels; these are called 2x2 factorial trials. Occasionally 3x2 trials
may be encountered, or trials that investigate three, four, or more interventions simultaneously. Often only one of the comparisons will be of relevance to any particular review. The following
remarks focus on the 2x2 case but the principles extend to more complex designs.
In most factorial trials the intention is to achieve ‘two trials for the price of one’, and the assumption is made that the effects of the different active interventions are independent, that is,
there is no interaction (synergy). Occasionally a trial may be carried out specifically to investigate whether there is an interaction between two treatments. That aspect may more often be explored
in a trial comparing each of two active treatments on its own with both combined, without a placebo group. Such three intervention group trials are not factorial trials.
The 2x2 factorial design can be displayed as a 2x2 table, with the rows indicating one comparison (e.g. aspirin versus placebo) and the columns the other (e.g. behavioural intervention versus
standard care):
Randomization of B
Behavioural intervention (B) Standard care
(not B)
Aspirin (A) A and B A, not B
Randomization of A
Placebo (not A) B, not A Not A, not B
A 2x2 factorial trial can be seen as two trials addressing different questions. It is important that both parts of the trial are reported as if they were just a two-arm parallel-group trial. Thus, we
expect to see the results for aspirin versus placebo, including all participants regardless of whether they had behavioural intervention or standard care, and likewise for the behavioural
intervention. These results may be seen as relating to the margins of the 2x2 table. We would also wish to evaluate whether there may have been some interaction between the treatments (i.e. effect of
A depends on whether B or ‘not B’ was received), for which we need to see the four cells within the table (McAlister et al 2003). It follows that the practice of publishing two separate reports,
possibly in different journals, does not allow the full results to be seen.
McAlister and colleagues reviewed 44 published reports of factorial trials (McAlister et al 2003). They found that only 34% reported results for each cell of the factorial structure. However, it will
usually be possible to derive the marginal results from the results for the four cells in the 2x2 structure. In the same review, 59% of the trial reports included the results of a test of
interaction. On re-analysis, 2/44 trials (6%) had P <0.05, which is close to expectation by chance (McAlister et al 2003). Thus, despite concerns about unrecognized interactions, it seems that
investigators are appropriately restricting the use of the factorial design to those situations in which two (or more) treatments do not have the potential for substantive interaction. Unfortunately,
many review authors do not take advantage of this fact and include only half of the available data in their meta-analysis (e.g. including only aspirin versus placebo among those that were not
receiving behavioural intervention, and excluding the valid investigation of aspirin among those that were receiving behavioural intervention).
When faced with factorial trials, review authors should consider whether both intervention comparisons are relevant to a meta-analysis. If only one of the comparisons is relevant, then the full
comparison of all participants for that comparison should be used. If both comparisons are relevant, then both full comparisons can be included in a meta-analysis without a need to account for the
double counting of participants. Additional considerations may apply if important interaction has been found between the interventions.
23.4 Chapter information
Editors: Julian PT Higgins, Sandra Eldridge, Tianjing Li
Acknowledgements: We are grateful to Doug Altman, Marion Campbell, Michael Campbell, François Curtin, Amy Drahota, Bruno Giraudeau, Barnaby Reeves, Stephen Senn and Nandi Siegfried for contributions
to the material in this chapter.
23.5 References
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Whiting-O’Keefe QE, Henke C, Simborg DW. Choosing the correct unit of analysis in medical care experiments. Medical Care 1984; 22: 1101–1114. | {"url":"https://training.cochrane.org/handbook/archive/v6.1/chapter-23","timestamp":"2024-11-13T13:22:05Z","content_type":"text/html","content_length":"115772","record_id":"<urn:uuid:eaa02ed8-ceb2-4713-8d54-92259d421880>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00879.warc.gz"} |
Radu Purice
Radu Purice
Institute of Mathematics of the Romanian Academy,
Calea Grivitei 21, 010702 Bucharest, Romania
Mail Address: P.O. Box 1-764, Bucuresti, ROMANIA
Tel.: +4021-319.65.09
fax: +4021-319.65.05
E-mail: Radu dot Purice at imar dot ro
Academic Degree:
Habilitation in Mathematics; IMAR (2014)
Ph.D. in Physics; Bucharest University (1990)
Current Position:
Senior Researcher, Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Bucuresti
Evolution Equations and Control Theory, Partial Differential Equations and Mathematical Physics
"Gheorghe Titeica" Prize of the Romanian Academy 2002
Field of Activity:
Mathematical Physics - quantum mechanics.
of the CNRS European Associated Laboratory "LEA Mathématique et Modélisation Franco-Roumain"
in the IUPAP Commission C18 (on Mathematical Physics)
of the IAMP
in the Romanian Team of the RTN "Network in Noncommutative Geometry"
Coordinator of the Romanian Team on the International Research Network ECO-Math organised by CNRS, the Romanian Academy and "Simion Stoilow" Institute of Mathematics of the Romanian Academy.
Coordinator of the European Programme EURROMMAT
Coorganizer of
XIII-eme Colloque Franco-Roumain de Mathematiques Appliques, Iasi, 2016
International Conference: Mathematical aspects of solid state physics, quantum transport and spectral analysis, Bucuresti, 2014
XII-eme Colloque Franco-Roumain de Mathematiques Appliques, Lyon, 2014
XI-eme Colloque Franco-Roumain de Mathematiques Appliques, Bucuresti, 2012
The LEA Math-Mode Workshop , Bucuresti, 2011
7-th Congress of Romanian Mathematicians , Brasov, 2011
4-th annual meeting of the EU-NCG Network , Bucuresti, 2011
Workshop in Nonlinear Analysis and Mathematical Physics - Romanian - German Symposium on Mathematics and its Applications , Sibiu, May 2009
Qmath10 International Conference , Moeciu, 2007
Workshop on Aspects mathématiques du transport dans les systèmes mésoscopiques, Marseille, December 2006
Former Positions:
Deputy Director, Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Bucuresti 2012 - 2014
Scientific Secretary, Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Bucuresti 1999 - 2012
Assistant doctorand, Ecole de Physique, Université de Genève, Genève 1984 - 1985
Researcher, National Institute of Physics and Nuclear engineering - Department of Fundamental Physics, Bucuresti 1979 - 1990
Research activity:
study of some models in field theory: Markoff property for the Free Euclidean Electromagnetic Field; solutions to the Ghinzburg- Landau equations in a two-dimensional bounded domain
study of the Dirac Hamiltonian: nonrelativistic limit; time - dependent perturbations; singular perturbations; scattering theory
the Dirac Hamiltonian with time dependent perturbations
Current interest in:
spectral analysis and propagation estimations for quantum Hamiltonians
the conjugate operator method and its applications in spectral theory and evolution properties
weighted estimations of Hardy type for quantum Hamiltonians
operator algebra methods in spectral theory and dynamics of quantum systems
non-equilibrium steady states and transport theory
functional calculus for quantum observables in a magnetic field
Recent Papers:
Graduate Lecture Series:
• Floquet theory - a quantum mechanical point of view. Seminar talk at University of Illinois at Chicago; April 3, 2019 (lecture notes)
• Non-equilibrium steady states and currents. Graduate lectures given in the frame of a Common Seminar organized by the University of Aalborg and the Unicersity of Århus; Denmark, May 3-15, 2013.
(lecture notes)
• Non-equilibrium steady states and currents. (Quantum Transport and Related Problems in Mathematical Physics - Summer School, Hammamet, Tunisia 2-7th Sepember 2012
Some Recent Talks:
• Analyse spectrale des familles de Bloch isolés dans un champ magnétique regulier. Talk given at the 16th French Romanian Colloquium, Bucharest, 26-30th August, 2024.
• Some results in the spectral analysis of quantum Hamiltonians with magnetic fields. Talk given at the Trans-Carpatian Seminar on Geometry and Physics, on-line, March 13, 2024.
• Spectral regularity for a class of pseudo-differential operators with “dilation” perturbation. Talk given at the 10-th Congress of Romanian Mathematicians (CRM10), Piteşti, June 30 - July 5,
• Magnetic pseudo-differential operators and Gabor frames. Talk given at the 28-th International Conference on Operator Theory (OT28), Timisoara, June 27 - July 1, 2022.
• Spectral analysis near a conic point of a 2 dimensional periodic Hamiltonian in a weak magnetic field. Talk given at the International Conference: Solid-Mathematics 2021, Marne la Vallée, August
25-27, 2021, (hybrid mode).
• Spectral analysis of the bottom of the spectrum of a 2 dimensional periodic magnetic Hamiltonian. Talk given at the International Conference: Mathematkical challenge of Quantum Transport in
Nanosystems - Pierre Duclos Workshop, Saint Petersburg, September 14 – 16, 2020.
• Peirl's substitution at the bottom of the spectrum in the absence of Wannier functions. Talk given at the University of Illinois at Chicago, on April 8, 2019. presentation.
• Les hamiltoniens effectifs de Peierls-Onsager en tant que OPD magnétiques. Workshop - Transitions de phase et équations non locales, Centre Francophone de Mathématiques, IMAR Bucarest. 25 - 27
April 2018. presentation
• Low lying spectral gaps induced by slowly varying magnetic fields. Workshop: Spectral Theory and Mathematical Physics. Université de Lorraine - Metz, 16 - 18 Mai 2017. presentation
• Spectral gaps for periodic Hamiltonians in slowly varying magnetic fields. In the frame of the Spectral Analysis Seminar at the Pontificia Universidad Catolica de Chile, Santiago de Chile.
December 1-st, 2016.
• The Peierls-Onsager substitution in the framework of magnetic ΨDO. Workshop - Magnetic fields and semi-classical analysis, Centre &ldquot;Henri Lebesgue&rdquot;, Rennes France, May 19 - 22, 2015.
• The Magnetic Coherent States. XXXII Workshop on Geometric Methods in Physics. Biaƚowieża, Poland, June 30 - July 6, 2013. pesentation
• The Magnetic Moyal algebras. Seminar of Harmonic Analysis. Bucharest, September 21-22, 2012. pesentation
• Decay of Eigenfunctions of Magnetic Hamiltonians. (7-th Workshop Mathematical Challenges of Quantum Transport in Nano-Optoelectronic Systems, WIAS Berlin, February 04-05, 2011 ).pdf
• A Non Equilibrium Steady State as an Adiabatic Limit. (10-ème Colloque Franco - Roumain de Mathématiques Appliquées, Poitiers, 26 - 31 Aoû 2010).pdf
• The algebra of quantum observables in a magnetic field: Spectral continuity with respect to the magnetic field. (Workshop Spectral Problems for Quantum Hamiltonians, Centre Interfacultaire
Bernoulli, EPF Lausanne, February 2010).pdf | {"url":"http://www.imar.ro/~purice/","timestamp":"2024-11-08T11:52:40Z","content_type":"text/html","content_length":"25850","record_id":"<urn:uuid:d44c436f-38c5-49da-acef-7b9ebe4228fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00640.warc.gz"} |
Calculating Compound Interest
Compound interest means that the interest will include interest calculated on interest.
• At the end of the first year the interest would be ($5,000 * 0.10) or $500
• In the second year the interest rate of 10% will applied not only to the $5,000 but also to the $500 interest of the first year. Thus, in the second year the interest would be (0.10 * $5,500) or
Unless simple interest is stated one assumes interest is compounded.
When compound interest is used we must always know how often the interest rate is calculated each year. Generally the interest rate is quoted annually. e.g. 10% per annum.
Compound interest may involve calculations for more than once a year, each using a new principal (interest + principal). The first term we must understand in dealing with compound interest is
conversion period. Conversion period refers to how often the interest is calculated over the term of the loan or investment. It must be determined for each year or fraction of a year.
e.g.: If the interest rate is compounded semiannually, then the number of conversion periods per year would be two. If the loan or deposit was for five years, then the number of conversion periods
would be ten.
Compound Interest Formula:
S = P(1+i)^n
S = amount
P = principal
i = Interest rate per conversion period
n = total number of conversion periods
Alan invested $10,000 for five years at an interest rate of 7.5% compounded quarterly
P = $10,000
i = 0.075 / 4, or 0.01875
n = 4 * 5, or 20, conversion periods over the five years
Therefore, the amount, S, is:
S = $10,000(1 + 0.01875)^20
= $ 10,000 x 1.449948
= $14,499.48
So at the end of five years Alan would earn $ 4,499.48 ($14,499.48 – $10,000) as interest.
Note: How to calculate 1.449948,
(1 + 0.01875)^20 = multiply 1.01875 twenty (20) times
1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875
(you will find the number is used 20 times)
If he had invested this amount for five years at the same interest rate offering the simple interest option, then the interest that he would earn is calculated by applying the following formula:
S = P(1 + rt),
P = 10,000
r = 0.075
t = 5
Thus, S = $10,000[1+0.075(5)]
= $ 13,750
Here, the interest that he would have earned is $3,750.
A comparison of the interest amounts calculated under both the method indicates that Alan would have earned $749.48($4,499.48 – $3,750) more under the compound interest method than under the simple
interest method.
Categories Finance | {"url":"https://content.moneyinstructor.com/937/compound-interest.html","timestamp":"2024-11-03T19:32:22Z","content_type":"text/html","content_length":"41521","record_id":"<urn:uuid:3c394b76-8d9a-4a07-aa17-434cf31a8eb8>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00615.warc.gz"} |
What Do You Know About Hydrostatics?
Questions and Answers
Hydrostatics is rather an enjoyable topic, fluid at rest, its conditions, and balance state contrary to fluid dynamic which is not stable. Find a chair, sit down and enjoy hydrostatics, it’ll be fun.
• 1.
Hydrostatics is fundamental to hydraulic, the engineering of for?
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. All of the above
Hydrostatics is the study of fluids at rest and their behavior under various conditions. It is fundamental to hydraulic engineering, which involves the use of fluids for various purposes such as
transporting, using, and storing. Therefore, all of the given options - transporting, using fluids, and storing - are correct explanations of how hydrostatics is fundamental to hydraulic
• 2.
Hydrostatics is relevant to the field of geophysics and?
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. AstopHysics
Hydrostatics is relevant to the field of astrophysics because it deals with the study of fluids at rest and the forces exerted on them. In astrophysics, the behavior of fluids such as gases and
plasma in celestial bodies like stars and planets is of great importance. Understanding the principles of hydrostatics helps in analyzing the structure, composition, and dynamics of these
celestial bodies, as well as in explaining phenomena such as stellar evolution, planetary formation, and the behavior of interstellar medium.
• 3.
Hydrostatics offers physical explanations for many phenomena of?
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Everyday life
Hydrostatics is a branch of physics that deals with the study of fluids at rest and the forces exerted by these fluids. It provides physical explanations for various phenomena that occur in
everyday life, such as the behavior of fluids in containers, the buoyancy of objects in water, and the pressure exerted by fluids on surfaces. Therefore, the correct answer is "Everyday life."
• 4.
Why wood and oil float on water, why the surface of the water is always flat and horizontal, these studies are related to the subject of?
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Hydrostatics
Hydrostatics is the branch of fluid mechanics that deals with the study of fluids at rest. It includes the study of buoyancy, which explains why wood and oil float on water. The surface of the
water is always flat and horizontal due to the principle of hydrostatic equilibrium, which states that the pressure at any point in a fluid at rest is the same in all directions. Therefore, the
correct answer is Hydrostatics.
• 5.
Some principles of hydrostatics have been known in an empirical and?
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Intuitive sense
The given question is asking for an explanation for the correct answer "Intuitive sense." In the context of hydrostatics, the term "intuitive sense" refers to an understanding or knowledge that
is gained through intuition or instinct rather than through formal study or empirical evidence. This suggests that some principles of hydrostatics were discovered or understood by individuals
through their natural ability to perceive or sense the behavior of fluids, rather than through systematic scientific investigation. This explanation aligns with the idea that certain aspects of
hydrostatics were known before they were formally studied or explained.
• 6.
Archimedes is credited with the discovery of Archimedes’ principle, which relates the?
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. Buoyancy force
Archimedes is credited with the discovery of Archimedes' principle, which relates to the buoyancy force. Buoyancy force is the upward force exerted on an object immersed in a fluid, such as
water, due to the difference in pressure between the top and bottom of the object. This principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the
object. This discovery by Archimedes has significant implications in understanding the behavior of objects in fluids and is a fundamental principle in hydrostatics.
• 7.
Archimedes principle state a force on an object is submerged in a fluid to the weight of the fluid displaced by the?
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. Object
Archimedes' principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. This means that the upward force exerted
on the object (buoyant force) is equal to the downward force exerted on the fluid (weight of the fluid displaced). Therefore, the correct answer is "Object" as the force on the object is equal to
the weight of the fluid displaced by it.
• 8.
The concept of pressure and the way it is transmitted by fluids were formulated by?
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. Blaise Pascal
Blaise Pascal formulated the concept of pressure and the way it is transmitted by fluids. He conducted experiments and developed the principle known as Pascal's law, which states that when a
pressure is applied to a fluid in a confined space, it is transmitted equally in all directions. This concept is fundamental in various fields such as engineering, physics, and hydraulics.
• 9.
The ‘fair cup’ or Pythagorean cup, dates from about the 6^th century?
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. BC
The correct answer is BC. The fair cup or Pythagorean cup is believed to have originated around the 6th century BC. This cup has a unique mechanism that prevents excessive drinking. When the cup
is filled beyond a certain level, it automatically drains all the liquid, ensuring moderation in drinking. This invention is attributed to the ancient Greek philosopher and mathematician
Pythagoras, hence the name Pythagorean cup.
• 10.
Pascal made contributions to developments in both hydrostatics and?
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. Hydrodynamics
Pascal made contributions to developments in hydrostatics, which is the study of fluids at rest, and hydrodynamics, which is the study of fluids in motion. Hydrodynamics involves understanding
the behavior of fluid flow and the forces that act on it. Pascal's contributions in this field helped advance our understanding of fluid mechanics and laid the foundation for further developments
in areas such as fluid dynamics and aerodynamics. | {"url":"https://www.proprofs.com/quiz-school/story.php?title=3dq-what-do-you-know-about-hydrostatics","timestamp":"2024-11-05T15:36:08Z","content_type":"text/html","content_length":"441828","record_id":"<urn:uuid:4cdd89b0-6b65-429d-bd42-4ef58eb43968>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00431.warc.gz"} |
[SNU Number Theory Seminar 2021.12.03] Algebraization theorems in complex and non-archimedean geometry
• Date : 12월 3일 (금) 10:30 AM
• Place : Zoom 896 5654 6548 / 157067
• Speaker : Abhishek Oswal (Caltech)
• Title : Algebraization theorems in complex and non-archimedean geometry
• Abstract : Algebraization theorems originating from o-minimality have found striking applications in recent years to Hodge theory and Diophantine geometry. The utility of o-minimality originates
from the 'tame' topological properties that sets definable in such structures satisfy. O-minimal geometry thus provides a way to interpolate between the algebraic and analytic worlds. One such
algebraization theorem that has been particularly useful is the definable Chow theorem of Peterzil and Starchenko which states that a closed analytic subset of a complex algebraic variety that is
simultaneously definable in an o-minimal structure is an algebraic subset. In this talk, I shall discuss a non-archimedean version of this result and some recent applications of these
algebraization theorems.
• Website: https://sites.google.com/view/snunt/seminars | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&document_srl=2024&listStyle=viewer&page=6","timestamp":"2024-11-07T04:04:57Z","content_type":"text/html","content_length":"22101","record_id":"<urn:uuid:3e5a8a9b-2981-4624-b0a2-6a995caec01d>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00319.warc.gz"} |
Numbers to Ten
Read aloud the Headline Story to the class and encourage children to describe what the domino might look like.
Once children have described the domino, you might want to draw three blank dominoes on the board and ask children to complete them to make three different dominoes that have a total of seven dots
each. Then encourage children to compare the dominoes.
Headline Story
I am holding a domino that has seven dots in all.
Possible student responses
• The domino could have 1 dot on one part and 6 dots on the other part.
• The domino could have 2 dots on one part and 5 dots on the other part.
• The domino could have 3 dots on one part and 4 dots on the other part.
Possible student responses:
• All of the dominoes have dots on both parts.
• None of the dominoes have the same number of dots on both parts.
• There are seven dots in all on each of these dominoes.
• As the number of dots on one part goes up, the number of dots on the other part goes down.
• There are no dominoes with seven dots on one side. | {"url":"https://elementarymath.edc.org/mindset/numbers-to-ten-lesson-9/","timestamp":"2024-11-09T09:23:38Z","content_type":"text/html","content_length":"124197","record_id":"<urn:uuid:69c42a01-68d0-4897-8126-a8bd78a47e98>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00022.warc.gz"} |
Perch² to µm²
Units of measurement use the International System of Units, better known as SI units, which provide a standard for measuring the physical properties of matter. Measurement like area finds its use in
a number of places right from education to industrial usage. Be it buying grocery or cooking, units play a vital role in our daily life; and hence their conversions. unitsconverters.com helps in the
conversion of different units of measurement like Perch² to µm² through multiplicative conversion factors. When you are converting area, you need a Square Perch to Square Micrometer converter that is
elaborate and still easy to use. Converting Perch² to Square Micrometer is easy, for you only have to select the units first and the value you want to convert. If you encounter any issues to convert
Square Perch to µm², this tool is the answer that gives you the exact conversion of units. You can also get the formula used in Perch² to µm² conversion along with a table representing the entire | {"url":"https://www.unitsconverters.com/en/Perch2-To-Microm2/Utu-330-306","timestamp":"2024-11-14T15:06:13Z","content_type":"application/xhtml+xml","content_length":"131106","record_id":"<urn:uuid:f5930ca1-4c23-4f5f-b371-30d1fa9a1a28>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00373.warc.gz"} |
DOI Number
The convergence of normal S-iterative method to solution of a nonlinear
Fredholm integral equation with modied argument is established. The corresponding
data dependence result has also been proved. An example in support of the established results is included in our analysis.
Fredholm equation; data dependency; Fixed-point theorem
Y. Atalan and V. Karakaya: Iterative solution of functional Volterra-Fredholm integral equation with deviating argument. J. Nonlinear Convex Anal. 18 (2017), 675-684.
V. Berinde: Existence and approximation of solutions of some first order iterative differential equations. Miskolc Math. Notes. 11 (2010), 13-26.
V. Berinde: Picard iteration converges faster than the Mann iteration in the class of quasi-contractive operators. Fixed Point Theory A. 2004 (2004), 97-105.
M. Dobritoiu: System of integral equations with modified argument. Carpathian J. Math. 24 (2008), 26-36.
M. Dobritoiu and A. M. Dobritoiu: An approximating algorithm for the solution of an integral equation from epidemics. Ann. Univ. Ferrara, 56 (2010), 237-248.
M. Dobritoiu: A class of nonlinear integral equations. Transylvanian Journal of Mathematics and Mechanics, 4 (2012), 117-123.
F. Gursoy: Applications of normal S-iterative method to a nonlinear integral equation. The Scientific World Journal, 2014 (2014), 1-5.
E. Hacioglu, F. Gursoy, S. Maldar, Y. Atalan, and G. V. Milovanovic: Iterative approximation of fixed points and applications to two-point second-order boundary value problems and to machine
learning. Appl. Numer. Math., 167, (2021), 143-172.
N. Hussain, V. Kumar and M. A. Kutbi: On rate of convergence of Jungck-Type
iterative schemes. Abstr. Appl. Anal. 2013 (2013), 1-15.
S. Ishikawa: Fixed points by a new iteration method. Proc. Amer. Math. Soc. 44
(1974), 147-150.
S. M. Kang, A. Rafiq and S. Lee: Strong convergence of an implicit S-iterative process for lipschitzian hemicontractive mappings. Abstr. Appl. Anal. 2012 (2012), 1-7.
S. H. Khan: A Picard-Mann hybrid iterative process. Fixed Point Theory A., 2013 (2013),1-10.
S. H. Khan: Fixed points of contractive-like operators by a faster iterative process. WASET International Journal of Mathematical, Computational Science and Engineering. 7 (2013), 57-59.
A. R. Khan, V. Kumar and N. Hussain: Analytical and numerical treatment of Jungck-type iterative schemes. Appl. Math. Comput. 231 (2014), 521-535.
A. R. Khan, F. Gursoy and V. Karakaya: Jungck-Khan iterative scheme and higher convergence rate. Int. J. Comput. Math. 93 (2016), 2092-2105.
S. A. Khuri and A. Sayfy: Variational iteration method: Green's functions and fixed point iterations perspective. Appl. Math. Lett., 32 (2014), 24-34.
S. A. Khuri and A. Sayfy: A novel fixed point scheme: Proper setting of variational
iteration method for BVPs. Appl. Math. Lett., 48 (2015), 75-84.
V. Kumar, A. Latif, A. Rafiq and N. Hussain: S-iteration process for quasi-contractive mappings. J. Inequal. Appl. 2013 (2013), 1-15.
M. Lauran: Existence results for some differential equations with deviating argument. Filomat, 25 (2011), 21-31.
S. Maldar, F. Gursoy, Y. Atalan and M. Abbas: On a three-step iteration process for multivalued Reich-Suzuki type $alpha$-nonexpansive and contractive mappings. J.
Appl. Math. Comput., (2021), 1-21.
W. R. Mann: Mean value methods in iteration. Proc. Amer.Math. Soc. 4 (1953), 506-510.
E. Picard: Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives. J. Math. Pure. Appl. 6 (1890), 145-210.
D. R. Sahu: Applications of S iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory, 12 (2011), 187-204.
D. R. Sahu and A. Petrusel: Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces. Nonlinear Anal. Theor. 74 (2011), 6012-6023.
S. M. Soltuz and T. Grosan: Data dependence for Ishikawa iteration when dealing with contractive like operators. Fixed Point Theory Appl. 2008 (2008), 1-7.
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jMonkeyEngine Docs
Physically Based Rendering – Part Two
In Part one, I explained what you had to know about Physically Based Rendering if you were an artist. If you’re a developer, and reading this article, you may have tried, or are planning to implement
your own PBR system. If you started to read some of the available literature, you’ve probably been struck by the math complexity of it, and by the lack of explanation of the big picture. You usually
see articles that focus on specifics parts of the process, and don’t talk much about other parts as they are assumed easier. At some point you have to assemble all these parts, and I had a hard time
figuring out how to do it in my readings. I guess it’s considered basic stuff for other authors, but I think it deserves its proper explanation.
I don’t pretend these articles will enlighten you to the point you are ready to implement your own system, but I hope they will give you solid basis and understanding to start reading the literature
without saying “WTF??” on every line as I did.
You can find a lexical, at the end, with all the strange words you’ll come across and their explanations.
So here is what I figured out about using PBR and lighting in general in a 3D rendering pipeline.
So first, lets talk about lighting in games. It all boils down to 2 things :
• Computing Diffuse reflection: This represent the light that reflects off a surface in all directions
• Computing Specular reflection : This represent the light that reflects off a surface directly to your eye.
This image from wikipedia is the most simple and yet the most helpful to understand this
To compute each of these factors, we’re going to use a function. This function answers to the delicate name of Bidirectional Reflectance Distribution Function or BRDF.
Don’t be scared by this, it’s a big name for just a function really. Usually, it will be a shader function.
Of course there are different BRDFs depending on what you want to compute, and on the lighting model you use. The BRDFs are usually called by the name of the guys that discovered/elaborated them.
Also, most of the time, in implementations for real time rendering, those BRDFs are approximated for the sake of performance. And incidentally, those approximations also have names, that can be
people names or technique names…
Lighting in PBR
Computing lighting for PBR is exactly the same as with the current rendering ( the system we use today with ambient, diffuse, specular, sometimes called ad-hoc system in the literature) :
For each light source, compute the diffuse factor and the specular factor. The main difference is that the BRDFs used are different, more physically accurate, and works predictably under different
light sources with few parameter entries.
So what is a light source?
Direct light source
Something that emits light. In games the most common light sources are Directional lights (think of the sun), Spot lights (think of a torch light), Point lights (think of a light bulb).
That’s what is commonly used in the ad-hoc system, and PBR also handle those types of lights.
Indirect light source
Something that reflects light and indirectly lights its surroundings. Think for example of a red wall next to a car at daytime, the sunlight hits the wall and the wall reflects red light that, in
turn, lights up the car.
This is not handled by the ad-hoc system, or very poorly faked with ambient lighting.
This part is optional for PBR, but that’s actually the part you really want. because that’s what make things pretty!
In games, indirect lighting is done by using an environment map as a light source. This technique is called Image Based Lighting (IBL).
So let’s say we’re looking for the full package. we need to compute diffuse and specular contribution for each light source be it direct or indirect.
To do so we need a BRDF for diffuse and a BRDF for specular, and stick to them for each light source for consistency. Also those BRDF should accept as entry the parameters we want to expose for the
artists (base color, metalness, roughness), or derived parameters with minimal transformation.
So the pseudo code for a complete lighting is this :
//direct lighting
for each directLightSource {
directDiffuseFactor = DiffuseBRDF(directlightSource)
directSpecularFactor = SpecularBRDF(directLightSource)
directLighting += Albedo * directDiffuseFactor + SpecColor * directSpecularFactor
//Indirect Lighting, done through Image Based Rendering with an environment map
indirectDiffuseFactor = DiffuseBRDF(EnvMap)
indirectSpecularFactor = SpecularBRDF(EnvMap)
indirectLighting = Albedo * indirectDiffuseFactor + SpecColor * indirectSpecularFactor
Lighting = directLighting + indirectLighting
I’ll go into more details, in the posts series, on how to compute each factor, but that’s pretty much it.
Choosing your BRDFs
There is a vast choice of BRDF, and I’m not going to talk about all of them but focus on the ones that I use in my implementation. I’ll just guide you to alternatives and provide links to relevant
articles for more details.
I chose to use the same BRDF as the ones used in Unreal Engine 4 from this article by Brian Karis, as I completely trust his judgement. The provided code helped a great deal, but it was far from
straight forward to integrate. In the end I had to fully research, and understand all the whereabouts of BRDFs.
Diffuse BRDF : Lambert
The most used diffuse BRDF in games. It’s very popular because it’s very cheap to compute and gives good results. This is the most simple way of computing diffuse. here are the details
Diffuse Lambert factor for a direct light source (directional light) with a yellow surface color.
Oren-Nayar : Gives better visual results than classic Lambert, and has the advantage of using an entry parameter called roughness…rings a bell? Unfortunately, the additional computation cost is not
really worth it,IMO. Details here
Harahan-Krueger : Takes into consideration sub-surface scattering for diffuse lighting (every material surface has layers and light scatters into those different layers before going out of the
material in a random direction). A lot of computations compared to Lambert, but may be important if you want to have a good sub surface scattering look for skin for example. more details in this
Specular BRDF : Cook-Torrance
This is a bit more complicated for specular. We need a physically plausible BRDF. We use what is called a Microfacet BRDF. So what is it?
It states that at a micro level a surface is not plane, but formed of a multitude of little randomly aligned surfaces, the microfacets. Those surfaces acts as small mirrors that reflects incoming
light. The idea behind this BRDF is that only some of those facets may be oriented so that the incoming light reflects to your eye. The smoother the surface, the more all facets are aligned, and the
most neat is the light reflection. In the contrary, if a surface is rough, the facets are more randomly oriented so the light reflection is scattered on the surface, and the reflection looks more
Microfacet specular factor for a direct light source. On the left a smooth surface, on the right a rough one. Note how the reflection is scattered on the surface when it’s rough.
The microfacet BRDF we use is called Cook-Torrance. From my readings, I couldn’t find any implementation that use another specular BRDF. It seems like this is the global form of any microfacet BRDF.
f = D * F * G / (4 * (N.L) * (N.V));
N.L is the dot product between the normal of the shaded surface and the light direction.
N.V is the dot product between the normal of the shaded surface and the view direction.
• Normal Distribution Function called D (for distribution). You may also find some references to it as NDF. It computes the distribution of the microfacets for the shaded surface
• Fresnel factor called F. Discovered by Augustin Fresnel, it describes how light reflects and refracts at the intersection of two different media (most often in computer graphics : Air and the
shaded surface)
• Geometry shadowing term G. Defines the shadowing from the microfacets
That’s where it gets complicated. For each of these terms, there are several models or approximations to computed them.
I’ve settled to use those models and approximations :
• D : Trowbridge-Reitz/GGX normal Distribution function.
• F : Fresnel term Schlick’s approximation
• G : Schlick-GGX approximation
I won’t go into the details of all the alternatives I just want to expose an overview of the whole process first. But I’ll dive into more technical details on the terms I use, in following posts. To
have a neat overview of all alternatives you can see this post on Brian Karis’s blog.
That sums up the whole process, but there is still much to explain. In next post I’ll make a focus on indirect lighting, as it’s the part that gave me the hardest time to wrap my head around. I’ll
explain the Image Based Lighting technique used, and how you can compute diffuse and specular from an Environment Map.
Lexical :
Diffuse reflection : light that reflects from a surface in every direction.
Specular reflection : light that reflects from a surface toward the viewer.
Bidirectional Reflectance Distribution Function or BRDF : a function to compute Diffuse or Specular reflection.
Image Based Rendering or IBL : a technique that uses an image as a light source
Microfacet Specular BRDF : A specular BRDF that assumes a surface is made of a multitude of very small randomly aligned surfaces: the microfacets. it depends on 3 factors called D, F and G.
Normal Distribution Function called D (for distribution). You may also find some references to it as NDF. It computes the distribution of the microfacets for the shaded surface
Fresnel factor called F. Discovered by Augustin Fresnel, it describes how light reflects and refracts at the intersection of two different media (most often in computer graphics : Air and the shaded
Geometry shadowing term G. Defines the shadowing from the micro facets | {"url":"https://wiki.jmonkeyengine.org/docs/3.3/tutorials/how-to/articles/pbr/pbr_part2.html","timestamp":"2024-11-05T07:29:03Z","content_type":"text/html","content_length":"58382","record_id":"<urn:uuid:2bfc68d5-7f78-419f-8ee1-2a01ab89dfdc>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00540.warc.gz"} |
Closure property is satisfied in whole numbers with respect to which of the following?
Hint: First define the closure property and then check for the given operations for satisfying the closure property in whole numbers.
Complete step-by-step answer:
A set is closed under an operation if performance of that operation on members of the set always produces a member of that set only.
Given in the problem, we are given whole numbers.
Whole numbers are the set of positive integers that include zero.
Closure property of an operation in whole numbers means that if $x$ and $y$ are two whole numbers, then the operation $*$ satisfies the closure property if the result of $x*y$ is also a whole number.
We need to test the closure property of whole numbers with respect to addition, multiplication, subtraction and division.
Performing the operations one by one.
In case of addition , if we add two whole numbers say $a$ and $b$ such that $a + b = c$, their sum $c$ is always a whole number.
For example: $3 + 4 = 7,0 + 0 = 0$,etc.
Hence closure property for addition in whole numbers is always true.
In case of subtraction , if we subtract two whole numbers say $a$ and $b$ such that $a - b = c$, their difference $c$ is need not to be always a whole number.
For example: $3 - 4 = - 1$ , which is not a whole number.
Hence closure property for subtraction in whole numbers is not always satisfied.
In case of division , if we divide two whole numbers say $a$ and $b$ such that $a \div b = c$, their quotient $c$ is need not to be always a whole number.
For example: $3 \div 4 = 0.75$ , which is not a whole number.
Hence closure property for division in whole numbers is not always satisfied.
Lastly in case of multiplication , if we multiply two whole numbers say $a$ and $b$ such that $a \times b = c$, their product $c$ is always a whole number.
For example: $3 \times 4 = 12,0 \times 0 = 0$,etc.
Hence closure property for multiplication in whole numbers is always true.
Hence closure property is satisfied in whole numbers with respect to addition and multiplication.
Therefore, option (C). Addition and multiplication are the correct answer.
Note: The definition of whole numbers and closure property should be kept in mind in problems like above. One should try to look up examples in order to contradict an operation for given conditions
in problems like above. | {"url":"https://www.vedantu.com/question-answer/closure-property-is-satisfied-in-whole-numbers-class-11-maths-cbse-5ee9c8dd8cab265287ddbb7c","timestamp":"2024-11-05T20:31:07Z","content_type":"text/html","content_length":"164226","record_id":"<urn:uuid:293d8188-2c70-4ade-93eb-81b8c304d176>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00322.warc.gz"} |
Chess News
What is Chess calculation? Create a virtual board What is Chess calculation? Chess calculation is a critical aspect of the game of chess. It refers to the mental process of analyzing the position on
the chess board, considering various possible moves and their consequences, and ultimately determining the best move to play. Chess calculation is a skill that separates top-level chess players from
amateur players. It is the foundation of good chess strategy, as it enables players to look ahead, plan ahead, and predict their opponent’s moves. | {"url":"https://getchess.com/chess-news/","timestamp":"2024-11-12T22:26:20Z","content_type":"text/html","content_length":"35991","record_id":"<urn:uuid:042efb38-cfb1-478b-9a49-a64f5047f111>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00773.warc.gz"} |
2.5 CPC Collectors - Concentration of Diffuse Radiation
2.5 CPC Collectors - Concentration of Diffuse Radiation
The compound parabolic concentrators (CPC) are typical representatives of non-imaging concentrators, which are capable of collecting all available radiation - both beam and diffuse - and directing it
to the receiver. These concentrators do not have such strict requirements for the incidence angle as the parabolic troughs have, which makes them attractive from the point of view of system
simplicity and flexibility. Like parabolic and other shapes, CPC concentrators can be applied in both linear (troughs) and three-dimensional (parabolocylinder) versions. The same as in "pure"
parabola case, troughs are most widespread and useful for this type of concentrator.
The geometry of a CPC collector is demonstrated in Figure 2.12. If we consider a CPC trough, this diagram represents its cross-section. Each side of the shape is a parabola, and each of the parabolas
has its focus at the lower edge of the other parabola (e.g., F is the focus of the right-hand parabola in Figure 2.12). Each parabola axis is tilted relative to the axis of the CPC shape. One of its
key parameters is acceptance half-angle (${\theta }_{c}$), which is the angle between the axis of the collector and the line connecting the focus of one of the parabolas with the opposite edge of the
aperture. The collector is designed in such a way that each ray coming into the CPC aperture at an angle smaller that $\theta$ reaches the receiver; if this angle is greater than ${\theta }_{c}$ ,
the ray will return (Figure 2.13). The relationship between the size of the aperture (2a), the size of the receiver (2a') and the acceptance half-angle is expressed through the following equation:
$2a"=2a\mathrm{sin}{\theta }_{c}$ (2.11)
Knowing that the geometric concentration ratio is the quotient of the aperture area to the receiver area (see Section 2.3), for a linear CPC concentrator, we can obtain the relationship between the
concentration ratio and the acceptance angle:
${C}_{geo}=\frac{2a}{2a"}=\frac{1}{\mathrm{sin}{\theta }_{c}}$ (2.12)
Click for a text description of the image
One large parabolic mirror with a second mirror sitting tangent to the parabolic axis with an end at mirror #1’s focus. The distance between the two upper ends of the parabolas is labeled aperture
(2a) and the bottom two ends is labeled receiver. Dashed lines connect one top end to the opposite bottom end. The angle between their y intercept, y-axis and upper tip represents the acceptance
Credit: Mark Fedkin after Duffie and Beckman, 2013
There are some other useful expressions that describe the design of CPC concentrators. The following equations relate the focal distance of the side parabola (f) to the acceptance angle, receiver
size, and height of the collector (Duffie and Beckman, 2013):
$f=a"\left(1+\mathrm{sin}{\theta }_{c}\right)$ (2.13)
$h=\frac{f\mathrm{cos}{\theta }_{c}}{{\mathrm{sin}}^{2}{\theta }_{c}}$ (2.14)
Please complete the following reading to further explore the work principle of CPC concentrators.
Reading Assignment
Book chapter: Duffie, J.A. Beckman, W., Solar Engineering of Thermal Processes, Chapter 7: Sections 7.6 and 7.7 - pp. 337-349. This book is available online through the PSU Library system and can
also be accessed through e-reserves (via the Library Resources tab).
Section 7.6. of this book covers the fundamental optical principles of CPC collectors and also considers particular cases of truncated collector. Some practical examples are also presented. Section
7.7. talks about the orientation of CPC collectors. While CPC technology does not require continuous tracking, proper orientation with respect to the sun position is crucial to maximize absorbed
radiation. The theoretical material in this section is also supported by practical examples.
The following self-check questions will help you to test check your learning of the principles of CPC collectors:
Check Your Understanding - Questions 1-4
Check Your Understanding - Question 5
Can you calculate what would be the acceptance angle for a CPC collector with side parabola focal distance f=20 cm and width of the receiver 2a'=30 cm? | {"url":"https://www.e-education.psu.edu/eme812/node/558","timestamp":"2024-11-06T16:52:10Z","content_type":"text/html","content_length":"73895","record_id":"<urn:uuid:931fbd34-15af-41a1-bcdb-d3f9bf538b1b>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00288.warc.gz"} |
Slump Test - Civil Engineering X
Slump Test
The slump test is the most well-known and widely used test method to characterize the
workability of fresh concrete. The inexpensive test, which measures consistency, is used on job sites to determine rapidly whether a concrete batch should be accepted or rejected. The test method is
widely standardized throughout the world, including in ASTM C143 in the United States and EN 12350-2 in Europe.
The apparatus consists of a mold in the shape of a frustum of a cone with a base diameter of 8 inches, a top diameter of 4 inches, and a height of 12 inches. The mold is filled with concrete in three
layers of equal volume. Each layer is compacted with 25 strokes of a tamping rod. The slump cone mold is lifted vertically upward and the change in height of the concrete is measured.
Four types of slumps are commonly encountered, as shown in Figure 3. The only type of slump permissible under ASTM C143 is frequently referred to as the true slump, where the concrete remains intact
and retains a symmetric shape. A zero slump and a collapsed slump are both outside the range of workability that can be measured with the slump test. Specifically, ASTM C143 advises caution in
interpreting test results less than ½ inch and greater than 9 inches. If part of the concrete shears from the mass, the test must be repeated with a different sample of concrete. A concrete that
exhibits a shear slump in a second test is not sufficiently cohesive and should be rejected.The slump test is not considered applicable for concretes with a maximum coarse aggregate size
greater than 1.5 inches. For concrete with aggregate greater than 1.5 inches in size, such larger particles can be removed by wet sieving.
Additional qualitative information on the mobility of fresh concrete can be obtained after reading the slump measurement. Concretes with the same slump can exhibit different behavior when tapped with
a tamping rod. A harsh concrete with few fines will tend to fall apart when tapped and be appropriate only for applications such as pavements or mass concrete. Alternatively, the concrete may be very
cohesive when tapped, and thus be suitable for difficult placement conditions.
Slump is influenced by both yield stress and plastic viscosity; however, for most cases the effect of plastic viscosity on slump is negligible. Equations have been developed for calculating yield
stress in terms of slump, based on either analytical or experimental analyses. Since different rheometers measure different absolute values for the yield stress of identical samples of concrete, the
experimental equations are largely depended on the specific device used to measure yield stress.
Based on a finite element model of a slump test, Hu et al. (1996) developed an expression for yield stress in terms of slump and density, as shown in Equation [1]. The finite element calculations
were performed for concretes with slumps ranging from zero to 25 cm. The
equation is not appropriate for concretes with a plastic viscosity greater than 300 Pa.s, above which viscosity sufficiently slows flow and causes thixotropy, resulting in a reduction of the actual
slump value. An experimental study to verify the results of the finite element model showed satisfactory agreement between Equation [1] and yield stress measurements from the BTRHEOM rheometer. It
should be noted that the finite element calculations were preformed for concrete with slumps as low as zero, while the BTRHEOM rheometer can only measure
concretes with slumps greater than approximately 10 cm.
Using a viscoplastic finite element model, Tanigawa and Mori (1989) developed threedimensional graphs relating slump, yield stress, and plastic viscosity for concretes with slumps ranging from 1 to
26 cm. Schowalter and Christensen (1998) developed a simple analytical equation to relate slump to yield stress and the height of the unyielded region of the slump cone, defined as the region where
the weight of concrete above a given point is insufficient to overcome the yield stress. Other, more complex analytical analyses have been developed.
Additionally, Tattersall and Banfill (1983) have presented experimental data showing a
relationship between slump and yield stress.
The slump test is the most widely used device worldwide. In fact, the test is so well known that often the terms workability and slump are used interchangeably, even though
they have different meanings.
Specifications are typically written in terms of slump.
The slump test is simple, rugged, and inexpensive to perform. Results are obtained
The results of the slump test can be converted to yield stress in fundamental units based on various analytical treatments and experimental studies of the slump test.
Compared to other commonly used concrete tests, such as for air content and
compressive strength, the slump test provides acceptable precision.
The slump test does not give an indication of plastic viscosity.
The slump test is a static, not dynamic, test; therefore, results are influenced by concrete thixotropy. The test does not provide an indication of the ease with which concrete can be moved under
dynamic placing conditions, such as vibration.
The slump test is less relevant for newer advanced concrete mixes than for more
conventional mixes. | {"url":"https://civilengineeringx.com/concrete-workability/slump-test/","timestamp":"2024-11-14T00:00:23Z","content_type":"text/html","content_length":"281690","record_id":"<urn:uuid:953ac324-d948-455e-bdb0-09c4883d0300>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00541.warc.gz"} |
TS ICET 2015 Question Paper
Pipes A and canfill a tank in 9 minutes and 12 minutes respectively. A is opened and after sometime B is opened; and the tank is full in 4 minutes. The time difference (in minutes) between the
opening of A and is
Pipes A, B, C canfill a tank individually in 2 hours, 3 hours and 4 hours respectively. All the three are opened for 15 minutes and pipe C is closed. The time (in minutes) required furtherto fill the
tank is
In a joint business, A invests ₹ 20,000 for six months while B invested certain amount for the whole year. In the year—end profit of ₹ 9,000 the share of A is ₹ 6,000. The amount(in rupees)
invested by is
Three trains A, B and C moving at speeds $$s_1, s_2$$ and $$s_3$$ respectively take times $$t_1, t_2$$ and $$t_3$$ respectively to cover a distance of x kms. If $$t_1 : t_2 : t_3 = 20 : 15 : 12$$,
then $$s_1 : s_2 : s_3 = ...........$$
The time (in seconds) taken by a train of 240 metres long travelling at 70 kmphto cross another train of length 110 metres standing on a parallel track is
10 men and 15 women can complete a work in 6 days; and 12 men and 27 women can complete the same work in 4 days. The number of days required for 2 men and 12 women to complete the same work is
A and B can doa work in 8 hours; B and C can complete the same work in 6 hours while C and A require 12 hours to complete that work. The time required by C alone to complete that work is
The area of a square S is equal to the area of the rectangle of sides 56 metres and 14 metres. The length (in metres) of diagonal of S is
The area of a rhombus is 144 square units. If one of its diagonals is 18 units then the length of the other diagonal (in units) is
A brick measures 20 cms x 10 cms x 7.5 ems. The number ofbricks required to build a wall of dimensions 25 m x 2 m x 0.75 m is | {"url":"https://cracku.in/ts-icet-2015-question-paper-solved?page=9","timestamp":"2024-11-04T20:26:31Z","content_type":"text/html","content_length":"163422","record_id":"<urn:uuid:a5b2c831-3c4d-4e1b-aea4-aeb147ab16a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00687.warc.gz"} |
Influence of wavy surfaces on coherent structures in a turbulent... Simon Kuhn , Carsten Wagner , Philipp Rudolf von Rohr
13th Int. Symp on Appl. Laser Techniques to Fluid Mechanics, Lisbon, Portugal, June 26 – 29, 2006
Influence of wavy surfaces on coherent structures in a turbulent flow
Simon Kuhn1, Carsten Wagner2, Philipp Rudolf von Rohr3
1: Institute of Process Engineering, ETH Zurich, Switzerland, kuhn@ipe.mavt.ethz.ch
2: Institute of Process Engineering, ETH Zurich, Switzerland, wagner@ ipe.mavt.ethz.ch
3: Institute of Process Engineering, ETH Zurich, Switzerland, vonrohr@ ipe.mavt.ethz.ch
Keywords: PIV, Wall bounded flows, Surface roughness effects, Coherent structures
We describe how outer flow turbulence phenomena
depend on the interaction with the wall. We investigate
coherent structures in turbulent flows over different wavy
surfaces and specify the influence of the different surface
geometries on the coherent structures. The most important
contribution to the turbulent momentum transport is
attributed to these structures, therefore this flow
configuration is of large engineering interest. In order to
achieve a homogeneous and inhomogeneous reference flow
situation two different types of surface geometries are
considered: (i) three sinusoidal bottom wall profiles with
different amplitude-to-wavelength ratios of =2a/ =0.2
( =30mm), =0.2 ( =15mm), and =0.1 ( =30mm); and
(ii) a profile consisting of two superimposed sinusoidal
waves with =0.1 ( =30mm). Measurements are carried out
in a wide water channel facility (aspect ratio 12:1). Digital
particle image velocimetry (PIV) is performed to examine
the spatial variation of the streamwise, spanwise and
wall-normal velocity components in three measurement
planes. Measurements are performed at a Reynolds number
of 11200, defined with the half channel height h and the bulk
velocity UB. We apply the method of snapshots and perform
a proper orthogonal decomposition (POD) of the streamwise,
spanwise, and wall-normal velocity components to extract
the most dominant flow structures. From this orthogonal
decomposition of velocity fields we find similar large-scale
structures in the vicinity of the complex surface. These
large-scale structures are identified as streamwise-oriented,
counter-rotating vortices exhibiting a characteristic scale in
the spanwise coordinate direction. We quantitatively
describe this spanwise scale by applying the proper
orthogonal decomposition onto the streamwise velocity
component in the (x,z)-plane. We found similar spanwise
scales in the order of z=1.5H for the basic wavy surfaces of
different amplitude-to-wavelength ratios in the first two
eigenmodes. This scale is observed to be slightly smaller for
=0.2 ( =15mm) and slightly larger for =0.2 ( =30mm).
In addition, the oscillation and the location of the first two
streamwise-averaged eigenfunction extrema are nearly
identical. This scaling indicates that the size of the largest
structures neither depends solely on the solid wave
amplitude, nor on the wavelength. For the profile described
by the superposition of two sinusoidal waves a spanwise
scale of z=0.85H was identified for the first eigenmode.
This scaling is not confirmed by the second eigenfunction
where a spanwise distance of z=1.3H is observed. The
location of the extrema of the second eigenmode is shifted
by a spanwise distance of z/H=0.4. The eigenvalue spectra
from the orthogonal decomposition in the (x,y)- and
(x,z)-plane become increasingly broader for the profile with
doubled amplitude ( =0.2 ( =30mm)), half the wavelength
( =0.2 ( =15mm)), and the superimposed waves. Thus by
increasing the surface complexity more modes contribute to
the energy containing range. We conclude that by increasing
the complexity of the bottom surface and thus altering the
flow homogeneity from homogeneous in spanwise
coordinate direction as for the basic wave profiles to
completely inhomogeneous as for the superimposed waves
the energy spectrum of the turbulent flow and the number of
significant modes is increased. The flow over the
superimposed waves can be described as the superposition
of dominant eigenmodes with different spanwise scales.
(h) 2,u; 2/E=6.9%
(g) 2,u; 2/E=13.0%
(f) 2,u; 2/E=8.4%
(e) 2,u; 2/E=7.5%
Fig. 1 Comparison of the first two eigenfunctions for a decomposition of u/UB(x,y/H=0.26,z,t) for =0.2
( =15mm) (first column), =0.2 ( =30mm) (second column), =0.1 ( =30mm) (third column), and the
superimposed waves (fourth column), Reh=11200. | {"url":"https://studylib.net/doc/10549578/influence-of-wavy-surfaces-on-coherent-structures-in-a-tu...","timestamp":"2024-11-04T22:05:49Z","content_type":"text/html","content_length":"63886","record_id":"<urn:uuid:b656e55e-cbbf-4144-bcc1-43f30c9e3b90>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00281.warc.gz"} |
Methods of and apparatus for encoding and decoding data in data processing systems
Methods of and apparatus for encoding and decoding data in data processing systems
To encode and compress a data array 30, the data array 30 is first divided into a plurality of blocks 31. A quadtree representation is then generated for each block 31 by initializing each leaf node
of the quadtree to the value of the data element of the block 31 of the data array 30 that the leaf node corresponds to, and initializing each non-leaf node to the minimum value of its child nodes,
and then subtracting from each node except the root node the value of its parent node. A set of data indicating the differences between respective parent and child node values in the quadtree
representing the block of the data array is then generated and stored, together with a set of data representing a quadtree indicating the number of bits that have been used to signal the respective
difference values.
Latest Arm Limited Patents:
This application is a continuation-in-part of U.S. patent application Ser. No. 13/198,462, “METHODS OF AND APPARATUS FOR ENCODING AND DECODING DATA IN DATA PROCESSING SYSTEMS,” filed on Aug. 4, 2011,
now U.S. Pat. No. 8,542,939 which is incorporated herein by reference in its entirety.
The technology described herein relates to a method of and apparatus for encoding data, e.g. for storage, in data processing systems, and in particular to such a method and apparatus for use to
compress and store texture data and frame buffer data in computer graphics processing systems. It also relates to the corresponding decoding processes and apparatus.
It is common in computer graphics systems to generate colours for sampling positions in the image to be displayed by applying so-called textures or texture data to the surfaces to be drawn. For
example, surface detail on objects may be generated by applying a predefined “texture” to a set of polygons representing the object, to give the rendered image of the object the appearance of the
“texture”. Such textures are typically applied by storing an array of texture elements or “texels”, each representing given texture data (such as colour, luminance, and/or light/shadow, etc. values),
and then mapping the texels onto the corresponding elements, such as (and, indeed, typically) a set of sampling positions, for the image to be displayed. The stored arrays of texture elements (data)
are typically referred to as “texture maps”.
Such arrangements can provide high image quality, but have a number of drawbacks. In particular, the storage of the texture data and accessing it in use can place, e.g., high storage and bandwidth
requirements on a graphics processing device (or conversely lead to a loss in performance where such requirements are not met). This is particularly significant for mobile and handheld devices that
perform graphics processing, as such devices are inherently limited in their, e.g., storage, bandwidth and power resources and capabilities.
It is known therefore to try to store such texture data in a compressed form so as to try to reduce, e.g., the storage and bandwidth burden that may be imposed on a device.
A further consideration when storing texture data (whether compressed or not) for use in graphics processing is that typically the graphics processing system will need to be able to access the stored
texture data in a random access fashion (as it will not be known in advance which part or parts of the texture map will be required at any particular time). This places a further constraint on the
storage of the texture data, as it is accordingly desirable to be able to store the texture data in a manner that is suitable for (and efficient for) random access to the stored data.
The Applicants believe that there remains scope for improved arrangements for compressing data for use in data processing systems, such as texture data for use in graphics processing.
A number of embodiments of the technology described herein will now be described by way of example only and with reference to the accompanying drawings, in which:
FIG. 1 shows schematically an array of data that may be encoded in accordance with an embodiment of the technology described herein;
FIG. 2 shows schematically the generation of a quadtree representing a block of data elements in accordance with an embodiment of the technology described herein;
FIG. 3 shows schematically the use of difference values and a bit count quadtree to represent a quadtree representing a block of data elements in an embodiment of the technology described herein;
FIG. 4 shows a set of tree node value storage patterns that are used in an embodiment of the technology described herein;
FIG. 5 shows the layout of the bits for quadtree node values in an embodiment of the technology described herein;
FIG. 6 shows schematically the storing of an array of data in accordance with embodiments of the technology described herein;
FIGS. 7 and 9 show schematically the arrangement for a block of a data array of data in a header data block and a body buffer in memory in a second embodiment of the technology described herein;
FIG. 8 shows schematically the order of the stored sub-block data in the embodiment of FIG. 7;
FIG. 10 shows schematically the order of the stored sub-block data in the embodiment of FIG. 9;
FIG. 11 shows schematically an encoding arrangement for YUV 420 data in an embodiment of the technology described herein;
FIG. 12 shows schematically the arrangement for a block of a data array of data in a header data block and a body buffer in memory for YUV 420 data in an embodiment of the technology described
FIG. 13 shows schematically the order of the stored sub-block data in the embodiment of FIG. 12;
FIG. 14 shows schematically an encoding arrangement for YUV 422 data in an embodiment of the technology described herein;
FIG. 15 shows schematically the arrangement for a block of a data array of data in a header data block and a body buffer in memory for YUV 422 data in an embodiment of the technology described
FIG. 16 shows schematically the order of the stored sub-block data in the embodiment of FIG. 15; and
FIG. 17 shows schematically a graphics processing system that may use data arrays encoded in accordance with the technology described herein.
A first embodiment of the technology described herein comprises a method of encoding an array of data elements for storage in a data processing system, the method comprising:
□ generating at least one tree representation for representing the array of data elements, the tree being configured such that each leaf node of the tree represents a respective data element of
the data array, and the data values for the nodes of the tree being set such that the data value that the tree indicates for the data element of the data array that a leaf node of the tree
represents is given by the sum of the data values in the tree for the leaf node and each preceding parent node in the branch of the tree that the leaf node belongs to; and
□ generating and storing data representing the at least one tree representing the data array as an encoded version of the array of data elements.
A second embodiment of the technology described herein comprises an apparatus for encoding an array of data elements for storage in a data processing system, the apparatus comprising:
□ processing circuitry configured to:
□ generate at least one tree representation for representing the array of data elements, the tree being configured such that each leaf node of the tree represents a respective data element of
the data array, and the data values for the nodes of the tree being set such that the data value that the tree indicates for the data element of the data array that a leaf node of the tree
represents is given by the sum of the data values in the tree for the leaf node and each preceding parent node in the branch of the tree that the leaf node belongs to; and
□ generate and store data representing the at least one tree representing the data array as an encoded version of the array of data elements.
In some embodiments, the processing circuitry may be in communication with one or more memory devices that store the array of data and/or store the data described herein and/or store software for
performing the processes described herein. The processing circuitry may also be in communication with a display for displaying images based on the data described above, or a graphics processor for
processing the data described above.
A third embodiment of the technology described herein comprises a data structure (and/or data format) representing an encoded version of an array of data elements for use in a data processing system,
□ data representing at least one tree representation representing the array of data elements; wherein:
□ the at least one tree representation that the data represents is configured such that each leaf node of the tree represents a respective data element of the data array, and the data values
for the nodes of the tree are set such that the data value that the tree indicates for the data element of the data array that a leaf node of the tree represents is given by the sum of the
data values in the tree for the leaf node and each preceding parent node in the branch of the tree that the leaf node belongs to.
In one example implementation, the stored set of data is stored on a computer-readable storage medium in the data format described above.
In the technology described herein, an array of data elements (which may, as discussed above, be an array of texture data for use in graphics processing) is represented using a tree representation in
which each leaf node of the tree represents a respective data element of the data array. Furthermore, the data values for the nodes of the tree are set such that the data value for the data element
of the data array that a leaf node of the tree represents is given by the sum of the data values in the tree for the leaf node and each preceding parent node in the branch of the tree that the leaf
node belongs to (in other words, to reproduce the data value for a data element that a leaf node of the tree represents, the value for the leaf node in the tree and the values for all the parent
nodes in the tree along the branch in the tree that the leaf node resides on must be added together).
As will be discussed further below, representing the data array using a tree of this form facilitates using less data to represent the data array (and thereby compressing the data array relative to
its original form). For example, it transforms the data array into a format that can facilitate efficient entropy coding.
In particular, because the values of the nodes in the tree are such that the data value that a leaf node represents (corresponds to) is indicated by the sum of the tree values for the leaf node and
the values of all the preceding parent nodes on the branch of the tree that the leaf node resides on, each node of the tree can effectively be set to a “minimum” data value that can still allow the
correct leaf node values to be determined. The tree can thus be thought of as a “minimum value” tree representation of the data array, and this “minimum value” tree representation can facilitate a
compressed version of the original data array, because rather than storing, for example, the actual, original data array values for each data element in the array, a set of “minimum” values from
which the actual, original data array values can be determined is stored instead.
Moreover, the compression can be lossless, because the tree can be configured such that the original value of the data element that each leaf node corresponds can be determined.
Also, as each leaf node in the tree indicates the value of a respective data element of the data array, that can facilitate random access to data elements in the compressed data.
Embodiments of the technology described herein can accordingly provide a method and apparatus for efficiently and losslessly compressing data arrays and in a way that facilitates parallel
decompression with hardware-based decoders and that can fit well with a bus friendly memory layout. This can then help, e.g., to reduce the bandwidth required to communicate, e.g. image, data between
different image data producing and consuming nodes in a graphics processing system, for example. This can all help to reduce issues with bus congestion and power requirements, for example.
The data array that is to be compressed in the technology described herein can be any suitable data array. It should comprise a plurality of data elements (entries), each occupying different
positions in the array. The data array is in an embodiment an image (in an embodiment represents an image). As discussed above, in an embodiment the data array is a graphics texture.
However, the technology described herein is not exclusively applicable to graphics textures, and may equally be used for other forms of data array, e.g. image. For example, the Applicants believe the
technology described herein may be equally useful for comprising frame buffer data (for use as a frame buffer format), e.g. in graphics processing systems and for use with display controllers. Thus,
in an embodiment the data array is a frame of data to be stored in a frame buffer, e.g. for display.
In an embodiment, the technology described herein is used both to compress texture data and as the frame buffer format in a graphics processing system. Thus, the technology described herein also
extends to a graphics processing system that uses the arrangement of the technology described herein both for compressing texture data and as its frame buffer format.
The technology described herein may also be used, for example, in image signal processing (image signal processors) and video encoding and decoding (video encoders and decoders).
While it would be possible to generate a single tree representation for a given data array to be compressed, in an embodiment, the data array is divided into plural separate regions or blocks, and a
respective tree representation is generated for each different block (region) that the data array is divided into.
In an embodiment, plural tree representations are generated for each different block (region) that the data array is divided into, with each tree representation representing a respective sub-block
(sub-region) of the block in question.
The blocks (regions) that the data array (e.g. texture) to be compressed is divided into for the purposes of the tree representations can take any suitable and desired form. Each block should
comprise a sub-set of the data elements (positions) in the array, i.e. correspond to a particular region of the array. In an embodiment the array is divided into non-overlapping and regularly sized
and shaped blocks. The blocks are in an embodiment square, but other arrangements could be used if desired. The blocks in an embodiment correspond to a block size that will otherwise be used in the
data processing system in question. Thus, in the case of a tile-based graphics processing system, the blocks in an embodiment correspond to (have the same size and configuration as) the tiles that
the rendering process of the graphics processing system operates on.
(As is known in the art, in tile-based rendering, the two dimensional output array of the rendering process (the “render target”) (e.g., and typically, that will be displayed to display the scene
being rendered) is sub-divided or partitioned into a plurality of smaller regions, usually referred to as “tiles”, for the rendering process. The tiles (sub-regions) are each rendered separately
(typically one after another). The rendered tiles (sub-regions) are then recombined to provide the complete output array (frame) (render target), e.g. for display.
Other terms that are commonly used for “tiling” and “tile based” rendering include “chunking” (the sub-regions are referred to as “chunks”) and “bucket” rendering. The terms “tile” and “tiling” will
be used herein for convenience, but it should be understood that these terms are intended to encompass all alternative and equivalent terms and techniques.)
In an embodiment, the data array is divided into 16×16 blocks (i.e. blocks of 16×16 array positions (entries)). In one such arrangement, a single tree representation is generated for the 16×16 block.
Thus, in the case of a texture map, for example, a separate tree representation would be generated for each (non-overlapping) 16×16 texel region of the texture map, and in the case of a frame for the
frame buffer, a tree representation would be generated for each 16×16 pixel or sampling position region of the frame.
In another embodiment, plural tree representations are generated for each 16×16 block. In an embodiment, a tree representation is generated for each 4×4 sub-block of a 16×16 block. Thus, in an
embodiment, each tree representation represents a 4×4 block or region of the data array (i.e. the data array is divided into 4×4 blocks for the purposes of the tree representations).
Thus, in the case of a texture map, for example, a separate tree representation would be generated for each (non-overlapping) 4×4 texel region of the texture map, and in the case of a frame for the
frame buffer, a tree representation would be generated for each 4×4 pixel or sampling position region of the frame.
Generating respective tree representations for smaller regions of the data array can be advantageous, as it means that less data needs to be stored for the individual tree representations.
Other arrangements would, of course, be possible. For example, four trees, each representing an 8×8 or a 16×4 block within a 16×16 block could be generated (in effect therefore, the data array would
be divided for the purposes of the tree representations into 8×8 or 16×4 blocks).
The data array element data values that the tree represents and indicates, i.e. that can be derived from the tree, can take any suitable and desired form, and will depend, as will be appreciated by
those skilled in the art, upon the nature of the data array being compressed, e.g. whether it is a texture, a frame for the frame buffer, etc.
In the case of a texture, for example, the data array element data values that the tree represents and indicates should be data values to allow appropriate texture data (texel values) for the data
array elements that the quadtree leaf nodes represent to be determined. Such texture data could comprise, e.g., a set of colour values (Red, Green, Blue (RGB), a set of colour and transparency values
(Red, Green, Blue, Alpha (RGBa)), a set of luminance and chrominance values, a set of shadow (light)-map values, a set of a normal-map (bump-map) values, z values (depth values), stencil values,
luminance values (luminance textures), luminance-alpha-textures, and/or gloss-maps, etc.
In the case of a frame for display, to be stored in a frame buffer, the data array element data values that the tree represents and indicates should be data values to allow appropriate pixel and/or
sampling position data values for the data array elements that the tree leaf nodes represent to be determined. Such pixel data could comprise, e.g., appropriate colour (RGB) values, or luminance and
chrominance values.
Where the data elements of the data array have plural components (channels) associated with them, such as would be the case, for example, for RGBa textures (in which each data element will have four
values associated with it), then in an embodiment a separate tree is generated (and stored) in respect of each different data component (channel). Thus, for example, in an embodiment a separate tree
is constructed for each different component of the data elements in the data array, e.g. for each of the three or four colour components that are present in the original data array. (In other words,
a given tree in an embodiment represents the values of only one component (channel) of the data elements of the data array.)
The tree or trees representing the data array can be generated in any suitable and desired manner. Each node of the tree will in an embodiment have plural child nodes, each representing a respective
non-overlapping and equal-sized region of the region of the data array that the parent node represents, save for the end leaf nodes that represent the individual data elements themselves.
In an embodiment, the tree representations are in the form of quadtrees, i.e. one or more quadtrees are generated to represent the data array. However, other tree structures, i.e. having greater or
fewer than four children per node could be used if desired. It would also be possible to use a hybrid tree structure, for example that is a quadtree but which only has two children per node for the
next to lowest layer (i.e. the leaf node parent nodes only have two child leaf nodes each).
Where the data array is represented using a quadtree or quadtrees, then each quadtree will accordingly have a root node representing the whole of the block or region of the data array that the
quadtree is encoding (or the whole of the data array where a single quadtree is to be generated for the entire data array). The root node will then have four child nodes (as it is a quadtree), each
representing a respective non-overlapping and equal-sized region (and in an embodiment a quadrant) of the block of the data array that the root node (that the quadtree) represents. Each child node of
the root node will then have four child nodes, each representing a respective non-overlapping and equal-sized region (and in an embodiment a quadrant) of the region of the data block that the child
node of the root node (that the quadtree) represents, and so on (i.e. with each node having four child nodes, each representing a respective non-overlapping and equal-sized region of the block of the
data array that the parent node represents), down to the leaf nodes that represent individual data elements (e.g. texels or pixels) of the data array.
Thus, for example, in the case of a quadtree for a 16×16 data element block, there will be a root node representing the whole 16×16 block. The root node will have four child nodes, each representing
an 8×8 block of data elements within the 16×16 block. Each such child node will have four child nodes, each representing a 4×4 block of data elements with the 8×8 block in question, and so on, down
to the leaf nodes that represent the individual data elements.
Similarly, in the case of a quadtree for a 4×4 data element block, there will be a root node representing the whole 4×4 block. The root node will have four child nodes, each representing an 2×2 block
of data elements within the 4×4 block. Each such child node will have four leaf nodes, each representing one of the individual data elements within the 2×2 block of data elements in question.
As discussed above, the (or each) tree, e.g. quadtree, should be generated such that each leaf node of the tree corresponds to a given data element (e.g. texel or pixel) of the data array. Thus there
will be one leaf node in the tree for each data element in the block (region) of the data array that the tree is to represent.
Each node of the tree (e.g. quadtree) should have a respective data value associated with it. As discussed above, the data values for the nodes of the tree are set such that the data value for the
data element of the data array that a leaf node of the tree represents is given by the sum of the data values in the tree for the leaf node and of each preceding parent node in the branch of the tree
that the leaf node belongs to. Thus, the data-value that is associated with and stored for each node of the tree will be (and is in an embodiment) the value that is needed to reproduce the desired
leaf node values when all the node values along the branches of the tree are summed together in the appropriate fashion.
The data values that are set for and allocated to the nodes of a tree (e.g. quadtree) could be such that the sum of the node values gives an approximation to the value of the data array element that
a tree leaf node corresponds to. This would then mean that the tree is a lossy representation of the original data array. However, in an embodiment, the tree or trees (e.g. quadtree or quadtrees) is
generated to be a lossless representation of the original data array. Thus, in an embodiment, the data values that are generated for and allocated to the nodes of the tree(s) are such that the sums
of the tree node values gives the correct, original values of the data array elements that the tree leaf nodes correspond to.
The data values to be associated with (set for) each node of the tree can be determined in any suitable and desired manner. In an embodiment, they are determined by performing two processing (data)
In the first processing pass, each leaf node in the tree is initialised with (set to) the value that the tree is to indicate for the data element in the data array to be encoded that the leaf node
represents (corresponds to), and each non-leaf node is initialised with (set to) a selected value that is, e.g., based on the values of the leaf nodes in the tree. This calculation is in an
embodiment performed from the bottom of the tree upwards. (As discussed above, the value that the tree is to indicate for a data element in the data array, i.e. that the corresponding leaf node will
be set to in this data pass, could be the actual value of that data element in the data array (for a lossless representation of the original data array), or an approximation to that actual value
(where the tree is to be a lossy representation of the data array), as desired.)
The value that each non-leaf node is set to in this first processing (data) pass is in an embodiment based and/or related to, the value of its child nodes. It is in an embodiment determined in a
predetermined manner from the values of its child nodes. In an embodiment, each non-leaf node is initialised with (set to) the value of one of its child nodes. In one such embodiment, each non-leaf
node is initialised with (set to) the minimum value of its child nodes (to the value of its lowest value child node) in this processing pass. In another embodiment, each non-leaf node is initialised
with (set to) the maximum value of its child nodes (to the value of its highest value child node) in this processing pass.
The second (subsequent) processing pass in an embodiment then subtracts from each node the value of its parent node. This is done for all nodes except the root node (which has no parent) and is again
in an embodiment done in a bottom-up fashion.
The resulting node values following the second pass are then the values that are associated with (set for) each node in the tree.
The data values for the data elements that are encoded in the quadtree could be transformed (converted) to a different form prior to generating the tree representing the data array, if desired.
For example, in an embodiment, where the data array to be compressed uses an RGB format, the RGB data is first transformed to a YUV format, and it is the so-transformed data in the YUV format that is
compressed and encoded in the manner of the technology described herein. In an embodiment a lossless YUV transform is used. In an embodiment the YUV transform is such that the U and V components in
the transformed data are expanded by 1 bit compared to their RGB counterparts.
The YUV transform is in an embodiment of the form:
F is a constant to avoid negative values. It is calculated from the size of G as:
F=1<<(bit width of the G component)
With this transform, the Y, U and V components will be expanded by 0-2 bits compared to their RGB counterparts.
(The inverse to this transform is:
Transforming RGB data to YUV data can simplify the coding of the chroma channels, as there may then be only one channel that will depict detail in the image (data array) instead of three. This can
then mean that even if the amount of uncompressed data per data element is expanded (as discussed above), there will still be a gain once the data has been compressed in the manner of the technology
described herein.
Once the tree, such as the quadtree, representing the data array has been generated, it is then necessary to generate data representing the tree, which data is then stored for the encoded version of
the original data array. The generated and stored data representing the tree can take any suitable and desired form. However, in an embodiment it represents the tree in an encoded form. In an
embodiment the data representing the tree (that is generated and stored to represent the tree) is a compressed representation of the tree that represents the data array (or part of the data array).
The stored data representing the tree, e.g. quadtree, in an embodiment includes a set of data representing or indicating the node values for the tree. This tree node value indicating data could, e.g.
simply comprise the actual data values of (set for) each node in the tree, which are then stored, e.g. in some particular, e.g. predetermined, order. However, in an embodiment, the data representing
the tree node values represents the tree node values in an encoded form.
In an embodiment, the tree, e.g. quadtree, representing (all or part of) the data array is encoded for storage by storing for each tree node, the difference between the value for the node in the tree
and the value of the parent node for the node in question. Thus, in an embodiment for some or all of the nodes in the tree, the difference between the value for the node in the tree, and the value of
the parent node in the tree for the node in question is determined, and that difference value is then used as the value that should be stored for the tree node in question in the stored, encoded
version of tree (in the stored data representing the tree). The value for a given node in the tree will then correspondingly be determined from the stored representation of the tree by determining
the value for the parent node of the node in question in the tree and adding the difference value indicated for the node of tree to the determined value for the parent node.
This arrangement, i.e. representing the tree, e.g. quadtree, node values by indicating the difference between the values of parent and child nodes in the tree, can allow the tree to be represented in
a compressed form. In particular, the Applicants have recognised that storing difference values allows the “minimum value” tree data values to be represented in a more compact form, particularly
where for at least one child node of each parent node, the difference between the child node's value and the parent node's value in the tree will be zero (such that that child node's value can be
indicated in a very efficient manner, as will be discussed further below). Also, for most images (such as might be used in graphics textures, for example), the amount of bits required to describe the
difference values is predictable, which can again facilitate representing and storing the tree(s) representing the data array in a compressed form.
Thus, in an embodiment, once a tree representing a block (or all) of the original data array to be encoded has been determined, a representation of that tree to be stored as the representation of the
tree from which the values of the data elements in the original data array that the tree represents are to be determined is generated by determining the differences between the values of respective
parent and child nodes in the tree, and then data representative of those difference values (and from which the difference values can derived) is stored as the data representing the tree.
It is also believed that using data representing difference values to represent a tree, such as a quadtree, representing a data array is new and advantageous in its own right, and irrespective, for
example, of whether the tree is a “minimum value” tree or not.
Thus, a fourth embodiment of the technology described herein comprises a method of encoding an array of data elements for storage in a data processing system, the method comprising:
□ generating at least one tree representation for representing the array of data elements, the data values for the nodes of the tree being set such that data values for the data elements of the
data array that the tree represents can be derived from the data values for the nodes of the tree;
□ generating data representing the at least one tree representing the data array by determining the differences between the values of respective parent and child nodes in the tree; and
□ storing data representative of the determined difference values as a set of data representing an encoded version of the array of data elements.
A fifth embodiment of the technology described herein comprises an apparatus for encoding an array of data elements for storage in a data processing system, the apparatus comprising:
□ processing circuitry configured to:
□ generate at least one tree representation for representing the array of data elements, the data values for the nodes of the tree being set such that data values for the data elements of the
data array that the tree represents can be derived from the data values for the nodes of the tree;
□ generate data representing the at least one tree representing the data array by determining the differences between the values of respective parent and child nodes in the tree; and
□ store data representative of the determined difference values as a set of data representing an encoded version of the array of data elements.
In some embodiments, the processing circuitry may be in communication with one or more memory devices that store the array of data and/or store the data described herein and/or store software for
performing the processes described herein. The processing circuitry may also be in communication with a display for displaying images based on the data described above, or a graphics processor for
processing the data described above.
A sixth embodiment of the technology described herein comprises a data structure (and/or data format) representing an encoded version of an array of data elements for use in a data processing system,
□ data representing at least one tree representation representing the array of data elements, the data values for the nodes of the tree being set such that data values for the data elements of
the data array that the tree represents can be derived from the data values for the nodes of the tree; wherein:
□ the data representing the at least one tree representation representing the array of data elements comprises data representative of the differences between the values of respective parent and
child nodes in the tree.
In one example implementation, the stored set of data is stored on a computer-readable storage medium in the data format described above.
As will be appreciated by those skilled in the art, these embodiments of the technology described herein can and in some embodiments do include any one or more or all of the optional features of the
technology described herein, as appropriate. Thus, for example, the tree (or trees) representing the data array is in an embodiment a quadtree (or quadtrees).
The data values (e.g. difference values) that are stored for each node of the tree in the stored representation of the tree can be arranged and configured in any suitable and desired manner in the
stored representation of the tree.
In an embodiment, the arrangement is such that individual node values or sets of individual node values can be extracted from the stored data representing the tree, without, e.g. having to decode the
entire set of the data. This then facilitates random access to original data element values or to sets of original data element values that the stored tree represents.
In an embodiment, where each tree is a quadtree and represents a 16×16, 16×4, 8×8, or 4×4 block of data elements, the arrangement is such that individual 4×4 blocks can be decoded without having to
decode any other 4×4 blocks, and such that 16×16, 16×4 or 8×8 blocks can be decoded independently of other 16×16, 16×4, or 8×8 blocks, respectively. In this arrangement, the minimum granularity will
be decoding a single 4×4 block. This is acceptable and efficient, as this will typically correspond to the minimum amount of data that can be fetched from memory in one operation in typical memory
Such arrangements could be achieved, for example, by allocating, a fixed size, known position, field within the set of data representing the tree for each node value or set of node values. However,
this may not be the most efficient arrangement from a data compression point of view.
Thus, in an embodiment, the data values (e.g. the difference values) for the nodes of a tree representing all or part of the data array can be (and are) stored in variable-sized fields within the set
of data representing the tree node values, in an embodiment in a contiguous fashion. This can allow the data representing the tree to be stored in a more compressed form.
Where the data values for the tree nodes are stored in variable sized fields in the set of data representing those values, then the position within the set of data representing the tree node values
of the field (the bits) representing the value (e.g. difference value) for a given node of the tree will not be at a fixed or predetermined position. A decoder decoding the stored data representing
the tree may therefore need to be able to identify (e.g. determine the position of) the data for a given tree node in use. This could be done in any desired and suitable manner.
However, in an embodiment, the generated and stored data representing the tree, as well as including a set of data indicating the tree node values (e.g., by indicating difference values to allow the
tree node values to be determined), as discussed above, also includes a set of data to be used when (and for) identifying the data for each respective tree node in the set of data indicating the tree
node values. This set of node value identifying data can then be used by a decoder for identifying the data for a given tree node within the stored set of data representing the tree node values.
The node value identifying data can take any suitable and desired form. In an embodiment, it indicates the number of bits that have been used for the respective node values in the set of data
representing the tree node values. Thus, in an embodiment, the data representing the tree representing the data array includes an indication of the number of bits (a bit count) that has been used to
represent (signal) the value of each node in the tree in the tree representing data. Accordingly, when the tree node values are stored in the form of a set of difference values, as discussed above,
the node value identifying data will comprise an indication of the number of bits (a bit count) used in the stored representation of the tree to signal the difference value for each node of the tree.
Thus, the data that is generated and stored to represent the tree (or each tree) representing the data array in an embodiment comprises a set of data representing the tree node values (and from which
the tree node values can be derived), together with a set of data indicating the number of bits (a bit count) that has been used to represent (signal) the value of each node in the tree in the set of
data representing the tree node values.
The data indicating the number of bits (the bit count) used in the stored representation of the tree to signal the, e.g. difference, values for each node of the tree can be arranged and configured in
the data representing the tree in any desired and suitable manner. However, in an embodiment, it too is arranged as a tree representation, with the nodes of the “bit count” tree indicating the number
of bits used in the set of data indicating the tree node values for corresponding nodes in the tree representing the data array.
Thus, in an embodiment, the data representing the tree representing the data array that is generated and stored in the technology described herein comprises data indicating node values for the tree
representing the data array, and data representing a tree that indicates the number of bits (the bit count) used to indicate the respective node values in the data indicating the node values for the
tree representing the data array. In other words, in an embodiment a “bit count” tree indicating the amount of bits used to signal the tree node values (e.g. difference values, as discussed above) in
the stored representation of the tree is generated and then data representing that “bit” count tree is stored and maintained in parallel with the set of data representing (indicating) the tree node
values (and from which the values of the data array elements that the tree represents will be determined).
The Applicants have found that using such a “bit count” tree together with data representing the node values of the tree representing the data array can allow the tree representation of the original
data array to be compressed in an efficient manner, particularly where the stored tree node values indicate difference values, as discussed above.
Indeed, it is believed that such an arrangement that uses parallel data value and “bit count” tree representations could be new and advantageous in its own right.
Thus, a seventh embodiment of the technology described herein comprises a method of generating an encoded version of an array of data elements for storage in a data processing system, the method
□ generating at least one tree representation for representing the array of data elements, the data values for the nodes of the tree being set such that data values for the data elements of the
data array that the tree represents can be derived from the data values for the nodes of the tree;
□ generating data representing the node values of the at least one tree representing the data array;
□ generating at least one further tree representation in which the node values for the at least one further tree indicate the number of bits used to indicate respective node values in the data
generated to represent the node values of the at least one tree representing the data array;
□ generating data representing the at least one further tree indicating the number of bits used to indicate the respective node values in the data generated to represent the node values of the
at least one tree representing the data array; and
□ storing the generated data representing the node values of the at least one tree representing the data array and the generated data representing the at least one further tree indicating the
number of bits used to indicate the respective node values in the data generated to represent the node values of the at least one tree representing the data array, as an encoded version of
the array of data elements.
An eighth embodiment of the technology described herein comprises an apparatus for generating an encoded version of an array of data elements for storage in a data processing system, the apparatus
□ processing circuitry configured to:
□ generate at least one tree representation for representing the array of data elements, the data values for the nodes of the tree being set such that data values for the data elements of the
data array that the tree represents can be derived from the data values for the nodes of the tree;
□ generate data representing the node values of the at least one tree representing the data array;
□ generate at least one further tree representation in which the node values for the at least one further tree indicate the number of bits used to indicate respective node values in the data
generated to represent the node values of the at least one tree representing the data array;
□ generate data representing the at least one further tree indicating the number of bits used to indicate the respective node values in the data generated to represent the node values of the at
least one tree representing the data array; and
□ store the generated data representing the node values of the at least one tree representing the data array and the generated data representing the at least one further tree indicating the
number of bits used to indicate the respective node values in the data generated to represent the node values of the at least one tree representing the data array, as an encoded version of
the array of data elements.
In some embodiments, the processing circuitry may be in communication with one or more memory devices that store the array of data and/or store the data described herein and/or store software for
performing the processes described herein. The processing circuitry may also be in communication with a display for displaying images based on the data described above, or a graphics processor for
processing the data described above.
A ninth embodiment of the technology described herein comprises a data structure (and/or data format) representing an encoded version of an array of data elements for use in a data processing system,
□ data representing at least one tree representation representing the array of data elements, the data values for the nodes of the tree being set such that data values for the data elements of
the data array that the tree represents can be derived from the data values for the nodes of the tree; wherein:
□ the data representing at least one tree representation representing the array of data elements, comprises:
□ data representing the node values of the at least one tree representing the data array; and
□ data representing at least one further tree representation in which the node values for the at least one further tree indicate the number of bits used to indicate respective node values in
the data representing the node values of the at least one tree representing the data array.
In one example implementation, the stored set of data is stored on a computer-readable storage medium in the data format described above.
As will be appreciated by those skilled in the art, these embodiments of the technology described herein can, and in some embodiments do, include any one or more or all of the optional features of
the technology described herein, as appropriate. Thus, for example, the respective tree representations are in an embodiment in the form of quadtrees.
In these embodiments, the “bit count” tree or trees can be configured in any suitable and desired manner. The bit count tree or trees in an embodiment have the same or a similar configuration to the
tree (or trees) representing the data array (but in an embodiment with one level of nodes less than the corresponding tree representing the data array). Thus, where one or more quadtrees are used to
represent the data array, the bit count trees are in an embodiment also in the form of quadtrees. Thus, in an embodiment, the bit count tree(s) are in the form of quadtrees (in which case each node
of a bit count quadtree will have four child nodes, save for the end leaf nodes of the quadtree). It would also be possible, e.g., to have bit count trees that are more or less sparse than the tree
(or trees) representing the data array.
As discussed above, the data values for the nodes of a bit count tree (e.g. quadtree) are in an embodiment set such that the number of bits used to signal the value of a corresponding (associated)
node or nodes of the tree (e.g. quadtree) representing the data array in the set of data representing those tree node values can be determined. Thus, the data-value that is associated with and stored
for a node of the bit count tree is in an embodiment a value that can be used either on its own, or in combination with other node values in the bit count tree, to determine the number of bits used
to signal the value of a corresponding node or nodes in the tree representing the data array in the set of data representing the node values of the tree representing the data array.
In an embodiment, the bit count tree (or trees) is configured such that each node of the bit count tree indicates the number of bits used to signal each child node value of a corresponding node of
the tree representing the data array that is indicated in the data representing the tree representing the data array. Thus, for example, a bit count tree node value of 3 will indicate that three bits
have been used for each child node value (have been used to signal each child node value) that is (explicitly) indicated in the data representing the tree representing the data array of the node of
the tree representing the data array that the bit count tree node corresponds to.
Accordingly, in an embodiment, the same number of bits is used for each child node (is used to signal the value of each child node) of a given node of the tree representing the data array whose value
is indicated in the set of data representing the node values of the tree representing the data array, and it is that same number of bits that a given (and the appropriate) node in the bit count tree
Thus, each “bit count” tree in an embodiment has a root node whose value will indicate or allow to be derived the number of bits used for the value of each child node of the root node that is
indicated in the set of data representing the node values for the tree representing the data array. The bit count tree root node will then have four child nodes, each child node indicating or
allowing to be derived, the number of bits used for the value of each child node of a respective (and in an embodiment corresponding) child node of the root node in the tree representing the data
array that is included in the set of data representing the node values for the tree representing the data array. Each child node of the bit count tree root node will then have four child nodes, each
such child node indicating or allowing to be derived, the number of bits used for the value of each child node of a respective child node of the child node of the root node in the tree representing
the data array that is included in the set of data representing the node values for the tree representing the data array, and so on (i.e. with each bit count tree node having four child nodes, each
child node indicating or allowing to be derived, the number of bits used for the value of each child node of a respective corresponding node in the tree representing the data array that is included
in the set of data representing the node values for the tree representing the data array), down to the leaf nodes in the bit count tree that indicate or allow to be derived, the number of bits used
for the value of each leaf node of a given (and respective) leaf node parent node in the tree representing the data array that is included in the set of data representing the node values for the tree
representing the data array.
Thus, for example, in the case of a quadtree that represents a 16×16 data element block (array), the corresponding “bit count” tree is in an embodiment a quadtree and in an embodiment has a root node
whose value will indicate or allow to be derived, the number of bits used for the value of each child node of the root node representing the whole 16×16 block that is included in the set of data
representing the node values for the quadtree representing the data array. The bit count quadtree root node will then have four child nodes, each such child node indicating or allowing to be derived,
the number of bits used for the value of each child node of a respective and corresponding node representing an 8×8 block of data elements that is included in the set of data representing the node
values for the quadtree representing the data array. Each child node of the root node in the bit count quadtree will then have four child nodes, each such child node indicating or allowing to be
derived, the number of bits used for the value of each child node of a respective and corresponding node representing a 4×4 block of data elements that is included in the set of data representing the
node values for the quadtree representing the data array, and so on, down to the leaf nodes of the bit count quadtree that will each indicate or allow to be derived the number of bits used for the
value of each leaf node of a respective and corresponding set of four leaf nodes (as it is a quadtree) that represent the individual data elements that is included in the set of data representing the
node values for the quadtree representing the data array.
Similarly, in the case of a quadtree that represents a 4×4 data element block (array), the corresponding “bit count” tree is in an embodiment a quadtree and in an embodiment has a root node whose
value will indicate or allow to be derived, the number of bits used for the value of each child node of the root node representing the whole 4×4 block that is included in the set of data representing
the node values for the quadtree representing the data array. The bit count quadtree root node will then have four leaf nodes, each such leaf node indicating or allowing to be derived, the number of
bits used for the value of each child node (representing an individual data element) of a respective and corresponding node representing a 2×2 block of data elements that is included in the set of
data representing the node values for the quadtree representing the data array.
It will be appreciated here that in these arrangements, the bit count tree will not signal the number of bits used for the root node value of the tree indicating the data values of the data array.
That root node value is in an embodiment signalled using the same number of bits as the input data element format uses. This number of bits can, if necessary, be communicated to the decoder in some
other way, for example by setting it in a data register that the decoder can read, or by including it in a file header associated with the encoded data array, etc. (All the blocks for a given data
array in an embodiment have the same size of root node.)
The data that is generated and stored to represent the bit count quadtree can indicate and represent the node values for the bit count quadtree in any suitable and desired manner.
For example, the data representing the bit count tree could store for each node in the bit count tree, the “true” bit count (the number of bits) used for representing the data values for the
corresponding nodes in the data representing the node values of the tree representing the data array. However, in an embodiment, the data representing the bit count tree represents the bit count tree
in an encoded form. In an embodiment, the data values for the nodes of the bit count tree are stored in an encoded form. In an embodiment the data representing the bit count tree (that is stored to
represent the bit count tree) is an encoded representation of the bit count tree.
In an embodiment, the bit count tree is encoded for storage by storing for the bit count tree nodes other than the root node, the difference between the bit count value (i.e. the number of bits the
node is to indicate) for the node in the bit count tree and the bit count value of its parent node in the bit count tree.
Thus, in an embodiment, for some or all of the nodes in the bit count tree, the difference between the bit count value that the node in the bit count tree is to indicate, and the bit count value that
the node's parent node in the bit count tree is to indicate is determined, and that difference value is then used as the value that should be stored for the bit count tree node in question in the
stored, encoded version of the bit count tree. The bit count value indicated by a given node in the bit count tree will then correspondingly be determined from the stored representation of the bit
count tree by determining the bit count value for the parent node of the node in question in the bit count tree, and then adding the difference value indicated for the node of the bit count tree to
the determined bit count value for its parent node.
This arrangement (i.e. representing the bit count tree node values by indicating the difference between the bit count values of parent and child nodes in the bit count tree) can allow the bit count
tree to be represented in a compressed form.
Thus, in an embodiment, the data that is generated and stored to represent the bit count tree comprises data representative of the differences between the bit count values to be indicated by
respective parent and child nodes in the bit count tree (and from which the difference values can derived) for every node of the bit count tree except the root node (which should have its actual bit
count value (i.e. the actual number of bits that has been used to signal the value of each child node of the root node in the data representing the node values for the tree representing the data
array) stored for it in the data representing the bit count tree).
Thus, in an embodiment, the bit count value to be indicated for the root node of the “bit count” tree (i.e. the node of the bit count tree that indicates how many bits have been used to signal the
value of each child node of the root node of the tree representing the data array in the set of data representing those node values) is given as the actual bit count (number of bits) that has been
used for those values (for each of those values) in the set of data representing the node values of the tree representing the data array (i.e. the root node bit count is given in an uncompressed form
in the data representing the bit count tree).
However, the data values for the remaining nodes in the bit count tree are given in the data representing the bit count tree as difference values, indicating the difference in the bit count for the
node in question, compared to its parent node.
The values (e.g. difference values) that are stored for the nodes of the bit count tree in the stored (e.g. compressed) representation of the bit count tree can be arranged and configured in any
suitable and desired manner in the stored representation of the bit count tree.
In an embodiment, the bit count tree root node is signalled using a sufficient number of bits to provide some “spare” bit count tree root node values (i.e. bit count tree root node values that will
never be needed to signal actual bit count values), such as the amount of bits that would be needed to signal the largest possible bit count for that specific bit count tree component +2. This is to
provide “spare” bit count tree root node values that can then be used to signal “special cases” to the decoder (as will be discussed below).
In an embodiment, the arrangement is such that individual node values or sets of individual node values can be extracted from the stored data representing the tree, without, e.g., having to decode
the entire set of the data. This then facilitates random access to bit count values or to sets of bit count values that the stored bit count tree represents.
As discussed above, in an embodiment, where each tree is a quadtree and represents a 16×16, 16×4, 8×8 or 4×4 block of data elements, the arrangement is such that individual 4×4 blocks can be decoded
without having to decode any other 4×4 blocks, and such that 16×16, 16×4 or 8×8 blocks can be decoded independently of other 16×16, 16×4 or 8×8 blocks, respectively. In this arrangement, the minimum
granularity will be decoding a single 4×4 block. This is acceptable and efficient, as this will typically correspond to the minimum amount of data that can be fetched from memory in one operation in
typical memory subsystems.
Such arrangements can be achieved in any suitable and desired manner, but in an embodiment are achieved by allocating fixed-size, known position, fields within the set of data representing the bit
count tree to each node value other than the root node.
The Applicants have found that storing the data representing the bit count tree in this form, particularly when in combination with storing the data representing the node values for the tree
representing the data array in the form discussed above, can provide a compressed, but still efficient to access, representation of the original data array.
In particular, this arrangement can facilitate determining where the individual data value syntax elements are in the stored data representing the tree representing the array of data elements with
relatively less serial processing compared to other entropy encoding schemes, such as Huffman coding or Arithmetic coding, whilst still being able to provide relatively high (and lossless, if
desired) compression of the original data array.
In an embodiment, the data values for the non-root nodes in the bit count tree are indicated in the data representing the bit count tree using fixed, in an embodiment predetermined, and in an
embodiment identical, size fields in the data representing the bit count tree. Thus, in an embodiment the value for each non-root node in the bit count tree whose value is indicated in the data
representing the bit count tree is indicated by using the same, fixed number of bits for each such node.
This then allows a decoder to readily determine where the relevant bit count tree node values are in the data representing the bit count tree.
In an embodiment, the value for each non-root node bit count is indicated using a 2-bit value in the data representing the bit count tree, in an embodiment using a 2-bit signed value. Accordingly, in
an embodiment the value for each non-root node bit count difference is indicated using a 2-bit value in the data representing the bit count tree, in an embodiment using a 2-bit signed value.
In an embodiment, certain bit count values that can be indicated by the bit count tree indicate predetermined, e.g. special, cases and/or sets of node values in the tree representing the data array.
This can then be used to further compress the data representing the node values in the tree representing the data array and/or the data representing the bit count tree (and thus to further compress
the stored data that represents the tree representing the data array).
For example, in an embodiment one bit count tree node value is set aside for indicating (and predefined as indicating) that the root node of the tree representing the data array should be set to a
predefined, default value, and that the values of all the child nodes of the tree representing the data array below that root node, have the same, predefined, default value as the root node.
In an embodiment the bit count value −2 is set aside for this purpose, particularly where the bit count difference for each non-root node is indicated using a 2-bit signed value.
This then allows the data representing the bit count tree to efficiently indicate the situation where all the leaf node values of the tree representing the data array will be the same as a given
default value indicated for the root node of the tree representing the data array.
In an embodiment, this bit count tree node value indicating that the root node of the tree representing the data array, and that all the child nodes of the tree representing the data array below that
root node, have the same default value, also indicates (and is also taken as indicating) that no bit count values are stored in the data representing the bit count tree for the nodes of the bit count
tree that are below the node in question (and, accordingly, in an embodiment, no bit count values are stored in the data representing the bit count tree for the nodes of the bit count tree that are
below the node in question). This is possible because as the lower level child node values will all be the same as the higher level root node, any information relating to those child nodes is
redundant and can therefore be omitted.
In effect therefore, this arrangement will trigger the decoder to assume that each lower level child node of the bit count tree also has its bit count node value set to the predetermined value, such
as −2, that indicates that the root node, and all the child nodes below that node, in the tree representing the data array have the same predefined default value.
Similarly, in an embodiment, this bit count tree node value indicating that the root node of the tree representing the data array, and that all the child nodes of the tree representing the data array
below that root node, have the same default value, correspondingly also or instead indicates (and is taken as indicating) that no data representing the node values of the tree that represents the
data array is stored for the block of the data array in question (and, accordingly, in an embodiment, no data representing the node values of the tree that represents the data array for the block of
the data array in question is stored). Again, this is possible because as all the lower level child node values will all be the same as the default value, any information relating to those child
nodes is redundant and can therefore be omitted.
In effect therefore, this arrangement will trigger the decoder to assume that each leaf node of the tree representing the data array will have the default value in question.
These arrangements can be used to efficiently indicate that data elements that the tree represents should all have the same, default value. This may be the case, for example, for a texture or frame
where all the data elements have fully opaque alpha values. In this case, the bit count value for the root node of the bit count tree for the alpha component (for example) could be set to the bit
count value that indicates that a default value, such as alpha=1, should be used for the entire block of data elements that the tree representation represents, thereby allowing blocks of data
elements having these properties to be encoded more efficiently. The default data value (e.g. colour) to use in these cases should be predefined, and, e.g., known to both the encoder and decoder.
In an embodiment one bit count tree node value is set aside for indicating (and predefined as indicating) that the value of that node of the tree representing the data array, and that the value of
all the child nodes of the tree representing the data array below that node, have the same value as the parent node of the tree representing the data array for the node in question.
In an embodiment the bit count value −1 is set aside for this purpose, particularly where the bit count difference for each non-root node is indicated using a 2-bit signed value.
This then allows the data representing the bit count tree to efficiently indicate the situation where all the leaf node values along a branch of the tree representing the data array will be the same
as a given value indicated for a node of the tree that is closer to the root node of the tree representing the data array.
In an embodiment, this bit count tree node value indicating that a node of the tree representing the data array, and that all the child nodes of the tree representing the data array below that node,
have the same value as the parent node of the node of the tree representing the data array in question, also indicates (and is also taken as indicating) that no bit count values are stored in the
data representing the bit count tree for the nodes of the bit count tree that are below the node in question (and, accordingly, in an embodiment, no bit count values are stored in the data
representing the bit count tree for the nodes of the bit count tree that are below the node in question). This is possible because as the lower level child node values will all be the same as the
higher level parent node, any information relating to those child nodes is redundant and can therefore be omitted.
In effect therefore, this arrangement will trigger the decoder to assume that each lower level child node of the bit count tree also has its bit count node value set to the predetermined value, such
as −1, that indicates that that node, and all the child nodes below that node, in the tree representing the data array have the same value as the parent node for the node in question.
Similarly, in an embodiment, this bit count tree node value indicating that a node of the tree representing the data array, and that all the child nodes of the tree representing the data array below
that node, have the same value as the parent node of the node of the tree representing the data array in question, correspondingly also or instead indicates (and is taken as indicating) that no node
values are stored in the data representing the node values of the tree that represents the data array for the nodes of the tree representing the data array that are below the node in question (and,
accordingly, in an embodiment, no node values are stored in the data representing the node values of the tree that represents the data array for the nodes of the tree representing the data array that
are below the node in question). Again, this is possible because as the lower level child node values will all be the same as the higher level parent node, any information relating to those child
nodes is redundant and can therefore be omitted.
In effect therefore, this arrangement will trigger the decoder to assume that the corresponding node of the tree representing the data array, and all the child nodes below that node in the tree
representing the data array, have the same value as the parent node in the tree representing the data array for the node in question.
Similarly, in an embodiment one bit count tree node value is set aside for indicating (and predefined as indicating) that all the child nodes of a node of the tree representing the data array have
the same value as the node in question.
In an embodiment the bit count value 0 is set aside for this purpose, particularly where the bit count difference for each non-root node is indicated using a 2-bit signed value.
This then allows the data representing the bit count tree to efficiently indicate the situation where all the child nodes of a node of the tree representing the data array will have the same value as
their parent node in the tree representing the data array.
In an embodiment, this bit count tree node value indicating that all the child nodes of a node of the tree representing the data array have the same value as their parent node also indicates (and is
also taken as indicating) that no node values are stored in the data representing the node values of the tree that represents the data array for the child nodes of the node in question (and,
accordingly, in an embodiment, no node values are stored in the data representing the node values of the tree that represents the data array for the child nodes of the node in question). Again, this
is possible because as the child node values will all be the same as their parent node, any information relating to those child nodes is redundant and can therefore be omitted.
In effect therefore, this arrangement will trigger the decoder to assume that all the child nodes of the node in question in the tree representing the data array have the same value as their parent
In an embodiment, one bit count tree node value is set aside for indicating (and predefined as indicating) that the values of all the child nodes of a node of the tree representing the data array
differ from the value of that node of the tree representing the data array by zero or one.
In an embodiment the bit count value 1 is set aside for this purpose, particularly where the bit count difference for each non-root node is indicated using a 2-bit signed value.
This then allows the data representing the bit count tree to efficiently indicate the situation where all the child nodes of a node of the tree representing the data array will differ from the value
of their parent node in the tree representing the data array by zero or one.
In an embodiment, this bit count tree node value indicating that the values of all the child nodes of a node of the tree representing the data array differ from the value of their parent node by zero
or one also indicates (and is also taken as indicating) that the node values for the child nodes of the node in question will be represented in the data representing the node values of the tree that
represents the data array using a bit map which has one bit for each child node (and with the value of each child node being determined as the value of the parent node plus the value of the bit in
the bit map for that node).
Accordingly, in this situation, the child node values for the node of the tree representing the data array in question are in an embodiment represented (and indicated) in the data representing the
node values of the tree representing the data array with a bit map having one bit for each child node, and with the value of each child node being determined as the value of the parent node plus the
value of the bit in the bit map for that node (i.e. by including such a bit map in the data representing the node values of the tree representing the data array).
This allows the child node values in this situation to be indicated in an efficient manner.
The above describes (the encoding of) special cases where all the child nodes of a node of the tree representing the data array have the same value as, or only differ in value by one from, their
parent node. However, there may also be (and typically will also be) situations in which one or more child nodes of the tree representing the data array differ in value from their parent node by more
than one.
Accordingly, in an embodiment other (at least one) bit count tree node values are set aside for this situation, i.e. for indicating (and predefined as indicating) that the value of at least one child
node of a node of the tree representing the data array differs from the value of that node of the tree representing the data array by more than one.
In an embodiment bit count values >=2 are set aside for this purpose, particularly where the bit count difference for each non-root node is indicated using a 2-bit signed value.
This then allows the data representing the bit count tree to efficiently indicate the situation where at least one child node of a node of the tree representing the data array differs from the value
of its parent node in the tree representing the data array by more than one.
As discussed above, even in this situation where at least one child node of a node of the tree representing the data array differs from the value of its parent node by more than one, in some
embodiments of the technology described herein at least (i.e. where each parent node of the tree representing the data array is initialised to a value of one of its child nodes (such as the value of
its minimum or maximum value child node) there will be at least one other child node of that parent node (of the node in question) that has the same value as the parent node (as the node in
question), because of the way the tree representing the data array is configured.
The Applicants have further recognised that this means that it is not necessary to store any data value for that “zero difference” child node in the stored data representing the tree node values
(since its value will be the same as its parent node).
Thus, if a node of the tree representing the data array has a child node whose value differs from it by more than one, the encoding process in an embodiment identifies a child node of that parent
node (of the node of the tree representing the data array in question) that has the same value as that parent node (i.e. a “zero difference” child node), and then stores in the data representing the
node values of the tree representing the data array, data representing the values of the other child nodes of the node in question, but does not store any data explicitly indicating the value of the
identified “zero difference” child node. Thus, in the case where a quadtree is used to represent the data array, data representing the values of three child nodes will be stored using for each of
these child node values the number of bits indicated by the bit count tree, but no data value will be stored for the identified “zero difference” child node. This can help to further compress the
data array.
To facilitate decoding of the data representing the tree node values in this situation, in an embodiment the values (e.g. difference values) for the non-zero difference child nodes of a node of the
tree representing the data array are stored in a predetermined sequence (order) relative to the child node with zero difference to the parent node (i.e. for which no data value is stored) in the data
representing the node values of the tree representing the data array. The predetermined sequence (order) in an embodiment depends upon which of the four child nodes is the “zero difference” child
node. This then means that a decoder can straightforwardly determine the order in which the data representing the node values is stored in the stored data representing the tree node values, once it
knows which child node of a tree node is the “zero difference” child node for which no value has been stored.
The designated “zero difference” child node can be indicated to the decoder as desired. However, in an embodiment, this information is included in the set of data representing (indicating) the node
values of the tree representing the data array. For example, in the case of quadtree, a 2-bit “zero child” designator could be, and is in an embodiment, included in the set of data representing
(indicating) the node values of tree representing the data array (and, e.g., then followed by bit fields indicating the values of each of the three remaining child nodes of the node in question, in
the predetermined order that is appropriate to the “zero child” node in question (and using for each of those bit fields, the number of bits indicated by the corresponding bit count tree)).
Thus, the stored data representing the node values of the tree representing the data array accordingly in an embodiment includes, for one or more nodes of the tree, an indication of which child node
of the respective node of the tree has been designated as a “zero difference” child node for which no value has been stored in the data representing the node values of the tree representing the data
The values of the (e.g. three) non-zero child nodes could be stored separately, one after another. However, in an embodiment, the bits representing the values of the non-zero child nodes are bit
interleaved (i.e. the values are stored with their bits interleaved, so bit 0 (e.g.) for each non-zero child node is stored first, followed by bit 1 (e.g.) for each non-zero child node, and so on).
This can help to simplify the hardware that is used to implement the encoding and decoding process.
The above describes the layout and configuration of the bit count tree and some arrangements of the data that is stored for representing the bit count tree. The bit count tree itself can be generated
in any suitable and desired manner from the tree representing the data array. For example, once the value to be stored for each node of the tree representing the data array has been determined, the
actual bit count needed to indicate the value of each node of the tree representing the data array could be determined, and then used as the node values to be stored for (to populate) the bit count
However, the Applicants have recognised that in arrangements, as discussed above, where a constrained, and fixed, number of bits, such as a 2-bit signed difference value, is used to indicate the bit
counts for the respective nodes of the bit count tree, then the actual bit count (number of bits) required for indicating the value of a given node of the tree representing the data array may in fact
be smaller or larger than what it is possible to signal with the fixed size bit count field to be used in the representation of the bit count tree.
To take account of this, in an embodiment, the data representing the node values of the tree representing the data array tree is constrained such that for each node, the number of bits that is used
to signal the value for that node in the data representing the node values of the tree representing the data array is a number of bits (a bit count) that can be indicated using the fixed size bit
count node fields to be used for the bit count tree.
Thus, for example, where the bit counts are indicated using a signed 2-bit difference value, the data representing the node values of the tree representing the data array is in an embodiment
constrained to make sure that the number of bits used for the value of each node except for the root node, in the data representing the node values of the tree representing the data array, is no
smaller than the bit count of its largest child node minus one, and no smaller than the bit count of its parent node minus two.
Such constraint on the number of bits used to signal the data values for the tree representing the data array can be achieved in any suitable and desired manner. In an embodiment, it is done by
performing two data passes.
First, a bottom-up pass is performed, in which for each node of the bit count tree, except the root node, the bit count for the node is constrained to be no smaller relative to the bit count of its
largest child node than the value that the bit count tree configuration can indicate. This is in an embodiment done by increasing the node's bit count by whatever amount is necessary to satisfy the
Following this first pass, a second pass is then performed from the top-down. In this second pass, the bit count for each node is constrained to be no smaller relative to the bit count of its parent
node than the value that the bit count tree configuration can indicate. This is again in an embodiment achieved by increasing the node's bit count by whatever amount is necessary to satisfy the
Thus, where the bit count tree uses signed 2-bit values to indicate bit count differences for all nodes except the root node, in the first bottom-up pass, for each node of the bit count tree, except
the root node, the bit count for the node is in an embodiment constrained to be no smaller than the bit count of its largest child node minus one (in an embodiment by increasing the node's bit count
by whatever amount is necessary to satisfy the constraint), and in the second, top-down pass, the bit count for each node is in an embodiment constrained to be no smaller than the bit count of its
parent node minus two (again in an embodiment by increasing the node's bit count by whatever amount is necessary to satisfy the constraint).
The resulting node bit count values following the second pass are then the values that are associated with (set for) each node in the bit count tree. A set of data representing (encoding) the bit
count tree can then be generated and stored, e.g., in the manner discussed above.
Similarly, once the bit count tree has been derived, the data representing the node values for the tree representing the data array can be generated and configured, using the configuration (such as
node value field sizes) corresponding to, and indicated by, the bit count tree.
In this regard, where, as discussed above, constraints on the number of bits used to indicate the node values of the tree representing the data array are imposed, then the set of data representing
the node values of the tree representing the data array should correspondingly use the determined number of bits for each respective node value. If necessary, the stored data representing the node
values can be padded with zeros (or some other recognisable “dummy” bits) to achieve this. (It will be appreciated in this regard that this arrangement accordingly may mean that the set of data
representing the node values of the tree representing the data array may for certain node values include more bits than the minimum necessary to convey the node value. However, the Applicants believe
that not withstanding this, the technology described herein can still provide a compressed version of the original data array, and still yield better compression than when using, for example, other
node value and bit count tree arrangements.)
It will be appreciated from the above, that the technology described herein, in some embodiments at least, in effect provides an entropy coding system for storing and representing the tree, e.g.,
quadtree, representing the data array, i.e. an encoding system in which the syntax elements (namely the nodes of the tree representing the data array and of the bit count tree) are mapped to
sequences of bits with a fixed length and may be placed in a continuous fashion in memory. Thus, the technology described herein, in some embodiments at least, finds an efficient way of representing
the data array with bit values. Thus, in an embodiment of the technology described herein, the data representing at least one tree representing the data array is generated using an entropy coding
The data for the data array can be processed to generate the tree representing the data array in any suitable and desired manner. For example, a suitable processor or processing circuitry may read
the original data array to be compressed from memory, and/or receive a stream of data corresponding to the original data array to be compressed, and then process the stream of data accordingly, e.g.
to divide it into blocks, generate the necessary tree or trees, and then generate data representing the quadtree(s) (and store that tree-representing data in memory), etc.
As discussed above, in an embodiment, this process will accordingly first comprise generating for the data array, or for each block that the data array has been divided into, a “minimum value” tree
(e.g. quadtree) having the form described above, in which the leaf nodes of the tree correspond to respective data elements of the data array (or block of the data array). In an embodiment a bit
count tree (e.g. quadtree) of the form described above is then derived, based on the number of bits that are needed to indicate, and/or that will be used to indicate, the value of each node of the
minimum value tree in a set of data representing the node values of that minimum value tree. A set of data representing that bit count tree is in an embodiment then generated, together with a set of
data representing the node values of the minimum value tree, which set of node value indicating data is configured according to, and has the configuration indicated by, the corresponding bit count
In an embodiment, this process further comprises identifying special cases of minimum value tree node values, for example, as discussed above, and configuring the bit count tree to indicate those
special cases, and the set of data representing the minimum value tree node values accordingly.
The data representing the tree representing the data array, such as the set of data representing the tree node values and the set of data representing the corresponding bit count tree, is in an
embodiment then stored appropriately for use as the compressed set of data representing the data array.
In an embodiment the system of the technology described herein employs arithmetic wrapping, i.e. the result of all additions when calculating the value of a node in the tree representing the data
array is in an embodiment calculated modulo (1<<bit depth of the (potentially yuv transformed) component).
Furthermore, the Applicants have recognised that when using such wrapping arithmetic, the rate of compression may be improved by exploiting the use of the wrapping arithmetic, in particular, for
example, for high contrast images, such as text and schematics.
For example, the Applicants have recognised that in some circumstances by initialising each parent node in the tree representing the data array to a value that is higher than the minimum value of its
child nodes, and then storing differences relative to that higher value for the child nodes, and using arithmetic wrapping of the results when the stored node values are summed to derive the actual
child node values, the compression may be improved.
For example, in an image that consists of mostly black or white data elements (such as could be the case for high contrast images (data arrays) like text and schematics), it may be better to
initialise each non-leaf node value (if a node has a “white value” child) to the higher “white” value and signal the dark data elements using the arithmetic wrap-around, as the counting distance for
example between the “white” and “dark” values when using wrapping arithmetic may be less than the counting distance between the “dark” and “white” values.
For example, if the data array contains only the numbers 0 (dark) and 255 (white), then if using 8-bit arithmetic with arithmetic wrapping, it would be more efficient to initialise any non-leaf nodes
that have a child node having the value 255 to 255, and to rely on arithmetic wrapping (i.e. 255+1=0 when utilising 8-bit arithmetic) to generate the desired node values when decoding the stored
data. In this particular example, each parent node in the tree representing the data array will accordingly be initialised to the maximum value of its child nodes (to the value of its highest child
node), rather than to the minimum value of its child nodes.
This will then allow smaller difference values (1 instead of 255) to be used to represent the node values in the data representing the node values of the tree representing the data array, and thus
allow smaller numbers of bits to be used to describe (signal) the differences (the difference values) (i.e. 1 bit instead of 8 bits). Such arrangements can accordingly use significantly less bits for
images that have this kind of characteristic.
This can thereby decrease the number of bits that are needed to convey the difference values in the set of data representing the node values of the tree representing the data array.
Thus, in an embodiment, the values that the non-leaf nodes in the tree representing the data array are initialised to are set so as to minimise the differences between those values (and thus the
differences that will need to be stored for the nodes) when taking account of any wrapping arithmetic that is used in the system in question (and the system will rely on arithmetic wrapping to
determine the actual node values when decoding the stored data).
In these arrangements each non-leaf node is in an embodiment still initially set to the value of one of its child nodes, but rather than, e.g., always initially setting it to the minimum value of its
child nodes, it is in an embodiment initially set to the lowest value of its child nodes that is equal to or greater than a selected threshold minimum value (i.e. to the value of its lowest value
child node whose value is equal to or above a selected threshold minimum value). Thus, in these arrangements, each non-leaf node will be initially set (e.g. in the first processing pass discussed
above) to the minimum value of its child nodes that is equal to or greater than a selected threshold minimum value (i.e. to the value of the lowest value child node whose value is equal to or above a
selected threshold minimum value).
Where the threshold minimum value is greater than zero (as may typically be the case in these arrangements), each non-leaf node may accordingly typically be initially set to the value of one of its
non-minimum value child nodes (unless all its child nodes have the same value, in which case it will be initially set to that value).
In this case, a non-leaf node will be initially set (e.g. in the first processing pass discussed above) to a value that is higher than the minimum value of its child nodes (i.e. to the value of one
of its higher value child nodes). Thus, in an embodiment, at least one non-leaf node in the tree representing the data array is initialised to a value that is higher than the minimum value of its
child nodes, such as to the maximum value of its child nodes.
In these arrangements, the threshold minimum value to be used when initialising the non-leaf node values to use to exploit the arithmetic wrapping efficiently can be determined in any desired and
suitable manner. For example, some or all of the leaf node values for the tree could be binned into a fixed amount of bins (such as four bins), and then the threshold minimum value to be used when
initialising the non-leaf node values set as being the value of a child node (and in an embodiment of the lowest value child node) in the bin that has the lowest total distance (in normal counting
order) to the other filled bins (to the values of the other child nodes). This will then generate a threshold minimum value that should have the minimum differences to all the other possible node
Thus, in an embodiment, a threshold minimum value for the tree representing the data array is determined based on the leaf node values to be used in the tree, and then the value each non-leaf node is
to be initialised to is determined using (based on) that determined threshold minimum value (in an embodiment by initialising the non-leaf nodes to the lowest value of their child nodes that is equal
to or greater than the determined threshold minimum value (i.e. to the value of the lowest value child node whose value is equal to or above the selected threshold minimum value).
The determination of the lowest value child node whose value is equal to or above the threshold minimum value should, and in an embodiment does, take account of and use arithmetic wrapping, such that
if there is no child node value between the threshold value and the upper limit of the arithmetic range, the determination should “wrap” back to zero and find (and select) the next child node value
starting at zero. (The effect of this then will be that when none of the child node values for a node is above (higher than) the threshold minimum value, the node should and will be initialised to
the minimum value of its child nodes.)
Essentially therefore, the process selects the child node value that is equal to or above (in the counting order, with wrapping), and closest to, the selected (determined) threshold value as the
value to initialise the parent node to. Thus, each non-leaf node is in an embodiment initially set (e.g. in the first processing pass discussed above) to its child node value that is equal to or
above (in the counting order, with wrapping) and closest to the selected (determined) threshold value. Thus, in an embodiment, the non-leaf nodes in the tree representing the data array are each
initialised to the value of their child node whose value is equal to or above (in the counting order, with wrapping) and closest to the selected (determined) threshold value.
In these arrangements, the root node of the tree representing the data array will accordingly be, and is in an embodiment, set to the selected (determined) threshold minimum value.
These arrangements will be particularly useful where the values that the data elements in the data array can take tend to be grouped, e.g., as high and low values, rather than being spread more
evenly across the possible data value range, such as could be the case for text, schematics, purely black and white images, etc.
It should also be noted here that in arrangements where each parent node is set to the minimum value of its child nodes, that is equivalent to setting the threshold minimum value to be used when
determining the node values to zero (where all the node values (data element values) are zero or greater). Thus, in one embodiment, the threshold minimum value is set to zero, and in another
embodiment, the threshold minimum value is set to a value that is not zero, and in an embodiment to a value that is greater than zero.
Although the technology described herein has been described above with particular reference to the storing of the data for the data array, as will be appreciated by those skilled in the art, the
technology described herein also extends to the corresponding process of reading (and decoding) data that has been stored in the manner of the technology described herein.
Thus, another embodiment of the technology described herein comprises a method of determining the value of a data element of a data array for use in a data processing system, the method comprising:
□ using stored data representing a tree representing some or all of the data elements of the data array to determine the value of each node of a branch of the tree representing some or all of
the data elements of the data array; and
□ determining the value to be used for a data element of the data array by summing the determined values for the leaf node of the branch of the tree and for each preceding parent node in the
branch of the tree that the leaf node belongs to.
Another embodiment of the technology described herein comprises an apparatus for determining the value of a data element of a data array for use in a data processing system, the apparatus comprising:
□ processing circuitry configured to:
□ use stored data representing a tree representing some or all of the data elements of the data array to determine the value of each node of a branch of the tree representing some or all of the
data elements of the data array; and
□ determine the value to be used for a data element of the data array by summing the determined values for the leaf node of the branch of the tree and for each preceding parent node in the
branch of the tree that the leaf node belongs to.
In some embodiments, the processing circuitry may be in communication with one or more memory devices that store the determined value to be used for a data element and/or store the data described
herein and/or store software for performing the processes described herein. The processing circuitry may also be in communication with a display for displaying images based on the data described
above, or a graphics processor for processing the data described above.
As will be appreciated by those skilled in the art, these embodiments of the technology described herein can and in some embodiments do include any one or more or all of the optional features of the
technology described herein, as appropriate.
Thus, for example, the tree representing some or all of the data elements of the data array is in an embodiment a quadtree, is in an embodiment a tree having one of the forms discussed above, and in
an embodiment represents either the entire data array, or a block of a respective data array. Similarly, the data processing system is in an embodiment a graphics processing system or a display
controller, and the data array in an embodiment comprises a texture map or a frame for display, and the data elements of the data array accordingly in an embodiment comprise texture elements and/or
pixel and/or sampling position data values, respectively.
In an embodiment, the stored data representing the tree representing some or all of the data elements of the data array accordingly in an embodiment comprises data representing the differences
between the values of respective parent and child nodes in the tree, and the value of a node of a branch of the tree accordingly is in an embodiment determined by determining the value of its parent
node and then adding the difference value indicated for the node of the tree in the stored data representing the tree to the determined value for the parent node.
Again, it is believed that such an arrangement may be new and advantageous in its own right.
Thus, another embodiment of the technology described herein comprises a method of determining the value of a data element of a data array for use in a data processing system, the method comprising:
□ using stored data representing the differences between the values of respective parent and child nodes in a tree representing some or all of the data elements of the data array to determine
the value of a node or nodes of the tree representing some or all of the data elements of the data array; and
□ using the determined node values to determine the value to be used for a data element of the data array.
Another embodiment of the technology described herein comprises an apparatus for determining the value of a data element of a data array for use in a data processing system, the apparatus comprising:
□ processing circuitry configured to:
□ use stored data representing the differences between the values of respective parent and child nodes in a tree representing some or all of the data elements of the data array to determine the
value of a node or nodes of the tree representing some or all of the data elements of the data array; and
□ use the determined node values to determine the value to be used for a data element of the data array.
In some embodiments, the processing circuitry may be in communication with one or more memory devices that store the determined value to be used for a data element and/or store the data described
herein and/or store software for performing the processes described herein. The processing circuitry may also be in communication with a display for displaying images based on the data described
above, or a graphics processor for processing the data described above.
Similarly, in all these decoding embodiments of the technology described herein, there may be, in addition to data representing the tree node values (e.g. difference values), a set of data to be used
for identifying the data for each respective node in the set of data indicating the tree node values. These decoding embodiments of the technology described herein accordingly in an embodiment
further comprise using stored data to be used for identifying the data for each respective tree node in the set of data indicating the tree node values to identify the data for a given tree node
within the stored set of data representing the tree node values. In an embodiment this comprises using stored data to determine the number of bits that have been used to represent (signal) the value
of a node or nodes in the stored data representing the tree node values, and this stored “bit count” data is in an embodiment in the form of a “bit count” tree, as discussed above.
Again, it is believed that such decoding arrangements may be new and advantageous in their own right.
Thus, another embodiment of the technology described herein comprises a method of determining the value of a data element of a data array for use in a data processing system, the method comprising:
□ using data representing a tree indicating the number of bits used to indicate respective node values in a set of stored data that represents the node values of a tree representing some or all
of the data elements of the data array to determine the number of bits used to indicate one or more node values in the stored data that represents the node values of the tree representing
some or all of the data elements of the data array;
□ using the determined number of bits to identify the stored data representing the data values of a node or nodes of the tree representing some or all the data elements of the data array;
□ using the identified stored data representing the data values of a node or nodes of the tree representing some or all the data elements of the data array to determine the data values of a
node or nodes of the tree representing some or all the data elements of the data array; and
□ using the determined node values to determine the value to be used for a data element of the data array.
Another embodiment of the technology described herein comprises an apparatus for determining the value of a data element of a data array for use in a data processing system, the apparatus comprising:
□ processing circuitry configured to:
□ use data representing a tree indicating the number of bits used to indicate respective node values in a set of stored data that represents the node values of a tree representing some or all
of the data elements of the data array to determine the number of bits used to indicate one or more node values in the stored data that represents the node values of the tree representing
some or all of the data elements of the data array;
□ use the determined number of bits to identify the stored data representing the data values of a node or nodes of the tree representing some or all the data elements of the data array;
□ use the identified stored data representing the data values of a node or nodes of the tree representing some or all the data elements of the data array to determine the data values of a node
or nodes of the tree representing some or all the data elements of the data array; and
□ use the determined node values to determine the value to be used for a data element of the data array.
In some embodiments, the processing circuitry may be in communication with one or more memory devices that store the determined value to be used for a data element and/or store the data described
herein and/or store software for performing the processes described herein. The processing circuitry may also be in communication with a display for displaying images based on the data described
above, or a graphics processor for processing the data described above.
In the decoding embodiments of the technology described herein that use a bit count tree, then as discussed above, the bit count tree node values may be stored in the form of difference values
between respective parent and child nodes in the bit count tree, and the bit count value indicated by a given node in the bit count tree is accordingly in an embodiment determined from the stored
representation of the bit count tree by determining the bit count value for the parent node of the node in question in the bit count tree, and then adding the difference value indicated for the node
of the bit count tree to the determined bit count value for a parent node.
The decoding method and apparatus of these embodiments of the technology described herein similarly may further comprise steps of identifying, or processing circuitry configured to identify, certain
bit count values that indicate special cases, and to then process those special cases accordingly, as discussed above.
Thus, for example, in an embodiment, the decoding process comprises identifying whether a bit count value indicates that the value of the root node of the tree representing the data array, and that
the value of all the child nodes of the tree representing the data array below that root node, have the same, default value, and if so, determining the tree node values accordingly (i.e. assuming
that the corresponding root node of the tree representing the data array, and all the child nodes below that node in the tree representing the data array (i.e. that the leaf nodes of the tree
representing the data array), have the default value).
Similarly, in an embodiment, the decoding process comprises identifying whether a bit count value indicates that the value of the node of the tree representing the data array, and that the value of
all the child nodes of the tree representing the data array below that node, have the same value as the parent node of the tree representing the data array for the node in question, and if so,
determining the tree node values accordingly (i.e. assuming that the corresponding node of the tree representing the data array, and all the child nodes below that node in the tree representing the
data array, have the same value as the parent node in the tree representing the data array for the node in question).
Similarly, there is in an embodiment a bit count tree value that indicates that all the child nodes of a node of the tree representing the data array have the same value as the node in question, and
if the decoder identifies this bit count node value, in an embodiment it determines the child node values accordingly, i.e. assumes that all the child nodes of the node in question in the tree
representing the data array have the same value as their parent node.
Similarly, in response to identifying a particular bit count value, the decoding process in an embodiment then determines from the stored data representing the tree representing the data array, a
child node in the tree representing the data array that will have the same value as its parent node, and the sequence that the data values for the other child nodes of the parent node in question
have been stored in in the data representing the tree representing the data array.
Equally in an embodiment of the technology described herein, the decoding process employs arithmetic wrapping to determine the data element values.
It will be appreciated that in these decoding arrangements, the stored data representing the data array may be provided, e.g., via a storage medium or over the Internet, etc., to the processor, such
as the graphics processor or display controller, that needs to use that data (and the decoding processor will then load the stored data and process it in the manner discussed above).
The apparatus for determining the value of a data element of the data array in a data processing system is in an embodiment incorporated in a graphics processor or a display controller.
The technology described herein also extends to a method and system that both stores and then decodes the data for the data array in the manners discussed above.
The methods and apparatus of the technology described herein can be implemented in any appropriate manner, e.g. in dedicated hardware or programmable hardware, and in (and be included in) any
appropriate device or component.
The actual device or component which is used to store the data in the manner of the technology described herein will, for example, depend upon the nature of the data array that is being stored. Thus,
for example, in the case of a graphics texture, an appropriate processor, such as a personal computer, may be used to generate and store the textures in the manner of the technology described herein,
e.g. by an application developer, and the so-stored textures then provided as part of the content of a game, for example. In the case of the stored data array being a frame for display, then it may
accordingly be a graphics processor that generates and stores the data in the manner required.
Similarly, on the data reading (decoding) side of the operation, in the case of texture data, for example, it could be a graphics processor that reads (decodes) the stored data array, and in the case
of a frame for display, it could be a display controller for a display that reads (decodes) the stored data array.
In an embodiment the technology described herein is implemented in a graphics processor, a display controller, an image signal processor, a video decoder or a video encoder, and thus the technology
described herein also extends to a graphics processor, a display controller, an image signal processor, a video decoder or a video encoder configured to use the methods of the technology described
herein, or that includes the apparatus of the technology described herein, or that is operated in accordance with the method of any one or more of the embodiments of the technology described herein.
Subject to any hardware necessary to carry out the specific functions discussed above, such a graphics processor, display controller, image signal processor, video decoder or video encoder can
otherwise include any one or more or all of the usual functional units, etc., that graphics processors, display controllers, image signal processors, video decoders or video encoders include. In an
embodiment, the methods and apparatus of the technology described herein are implemented in hardware, in an embodiment on a single semi-conductor platform.
The technology described herein is particularly, but not exclusively, suitable for use in low power and portable devices. Thus, in an embodiment, the technology described herein is implemented in a
portable device, such as a mobile telephone or PDA.
Similarly, the memory where the data representing the tree representing the data array is stored may comprise any suitable such memory and may be configured in any suitable and desired manner. For
example, it may be an on-chip buffer or it may be an external memory (and, indeed, may be more likely to be an external memory). Similarly, it may be dedicated memory for this purpose or it may be
part of a memory that is used for other data as well. In an embodiment, this data is stored in main memory of the system that incorporates the graphics processor.
In the case of a texture data array, the memory is in an embodiment a texture buffer of the graphics processing system (which buffer may, e.g., be on-chip, or in external memory, as desired).
Similarly, in the case of a frame for the display, the memory is in an embodiment a frame buffer for the graphics processing system and/or for the display that the graphics processing system's output
is to be provided to.
All the data representing the tree representing the data array is in an embodiment stored in the same physical memory, although this is not essential.
Other memory arrangements would, of course, be possible.
The technology described herein can be implemented in any suitable system, such as a suitably configured micro-processor based system. In an embodiment, the technology described herein is implemented
in computer and/or micro-processor based system.
The various functions of the technology described herein can be carried out in any desired and suitable manner. For example, the functions of the technology described herein can be implemented in
hardware or software, as desired. Thus, for example, the various functional elements of the technology described herein may comprise a suitable processor or processors, controller or controllers,
functional units, circuitry, processing logic, microprocessor arrangements, etc., that are operable to perform the various functions, etc., such as appropriately dedicated hardware elements and/or
programmable hardware elements that can be programmed to operate in the desired manner.
It should also be noted here that, as will be appreciated by those skilled in the art, the various functions, etc., of the technology described herein may be duplicated and/or carried out in parallel
on a given processor and/or share processing circuitry.
The technology described herein is applicable to any suitable form or configuration of graphics processor and renderer, such as tile-based graphics processors, immediate mode renderers, processors
having a “pipelined” rendering arrangement, etc.
It will also be appreciated by those skilled in the art that all of the described embodiments of the technology described herein can include, as appropriate, any one or more or all of the optional
features of the technology described herein.
The methods in accordance with the technology described herein may be implemented at least partially using software e.g. computer programs. It will thus be seen that embodiments of the technology
described herein comprise computer software specifically adapted to carry out the methods herein described when installed on data processing means, a computer program element comprising computer
software code portions for performing the methods herein described when the program element is run on data processing means, and a computer program comprising code means adapted to perform all the
steps of a method or of the methods herein described when the program is run on a data processing system. The data processing system may be a microprocessor, a programmable FPGA (Field Programmable
Gate Array), etc.
The technology described herein also extends to a computer software carrier (or medium) comprising such software which when used to operate a graphics processor, renderer or other system comprising
data processing means causes in conjunction with said data processing means said processor, renderer or system to carry out the steps of the methods of the technology described herein. Such a
computer software carrier could be a physical storage medium such as a ROM chip, RAM, flash memory, CD ROM or disk.
It will further be appreciated that not all steps of the methods of the technology described herein need be carried out by computer software and thus further broad embodiments of the technology
described herein comprise computer software and such software installed on a computer software carrier for carrying out at least one of the steps of the methods set out herein.
The technology described herein may accordingly suitably be embodied as a computer program product for use with a computer system. Such an implementation may comprise a series of computer readable
instructions fixed on a tangible, non-transitory medium, such as a computer readable medium, for example, diskette, CD ROM, ROM, RAM, flash memory or hard disk. The series of computer readable
instructions embodies all or part of the functionality previously described herein.
Those skilled in the art will appreciate that such computer readable instructions can be written in a number of programming languages for use with many computer architectures or operating systems.
Further, such instructions may be stored using any memory technology, present or future, including but not limited to, semiconductor, magnetic, or optical, or transmitted using any communications
technology, present or future, including but not limited to optical, infrared, or microwave. It is contemplated that such a computer program product may be distributed as a removable medium with
accompanying printed or electronic documentation, for example, shrink wrapped software, pre loaded with a computer system, for example, on a system ROM or fixed disk, or distributed from a server or
electronic bulletin board over a network, for example, the Internet or World Wide Web.
A number of embodiments of the technology described herein will now be described.
As discussed above, the technology described herein relates to the encoding and compression of data arrays for use in data processing systems, such as for use in graphics processing systems. A number
of embodiments of the encoding and compression techniques used in the technology described herein will now be described.
FIG. 1 shows schematically an exemplary original data array 30 that may be encoded in the manner of the technology described herein. The array of data 30 is a two-dimensional data array containing a
plurality of data elements (i.e. containing data array entries at a plurality of particular positions within the array). The data array 30 could be any suitable and desired array of data, such as
data representing an image.
In a graphics processing context, the data array could, for example, be a texture map (i.e. an array of texture elements (texels)), or an array of data representing a frame to be displayed (in which
case the data array may be an array of pixels to be displayed). In the case of a texture map, each data entry (position) in the data array will represent an appropriate texel value (e.g. a set of
colour values, such as RGBA, or luminance and chrominance, values for the texel). In the case of a frame for display, each data entry (position) in the array will indicate a set of colour values
(e.g. RGB values) to be used for displaying the frame on a display.
In the technology described herein, the data array 30 is encoded and compressed to provide a set of data representing the data array 30 that can then be stored in memory, and from which set of data,
the data values of individual data elements in the data array 30 can be derived by decoding the data representing the data array 30.
An embodiment of the process for encoding and compressing the data array 30 will now be described.
In this embodiment, as shown in FIG. 1, to encode and compress the data array 30, the data array 30 is first divided into a plurality of non-overlapping, equal size and uniform blocks 31, each block
corresponding to a particular region of the data array 30. In the present embodiment, each block 31 of the data array corresponds to a block of 4×4 elements (positions) within the data array 30 (i.e.
a block of 4×4 texels in the case of a texture map). (Other arrangements would, of course, be possible.)
Each such block 31 of the data array 30 is then encoded to provide a compressed representation of the block 31 of the data array 30.
To do this, a particular form of quadtree representation representing the block 31 of the data array 30 is first generated. This is done as follows.
The quadtree is constructed to have a root node that represents the whole of the data block 31 (thus the whole 4×4 block in the present embodiment). That root node then has four child nodes, each
representing a respective non-overlapping, uniform and equal-size 2×2 sub-block 32 of the 4×4 block 31 of the data array 30. Each such 2×2 sub-block representing child node then itself has four leaf
nodes, that each represent respective individual data elements 33 of the 4×4 block 31 of the data array 30.
The data value for each node of the quadtree is determined by performing two processing (data) passes.
In the first processing pass in this embodiment, each leaf node of the quadtree is initialised with (set to) the value of the data element of the block 31 of the data array 30 that the leaf node
corresponds to. Each non-leaf node is then initialised with (set to) the minimum value of its child nodes (to the value of its lowest value child node). This calculation is performed from the bottom
A second processing pass is then performed, in which each node except the root node (which has no parent) has the value of its parent node subtracted from it. This is again done from the bottom-up.
The node values following this second processing pass are then the node values to be used for each node of the quadtree representing the block 31 of the data array 30.
The effect of this process is that the value of the data element in the data array that a leaf node represents will be given by the sum of the value for the leaf node and of the values all the
preceding nodes along the branch of the quadtree that the leaf node resides on (in other words, to determine the value of a data element that a leaf node of the quadtree represents from the quadtree
representation of the data array, the value for the leaf node and of the values all the preceding nodes along the branch of the quadtree that the leaf node resides on must be summed (added
FIG. 2 illustrates the construction of such a quadtree for a representative 4×4 block 40 of data elements.
As shown in FIG. 2, the quadtree 45 representing the 4×4 array of data elements 40 has a root node 41, which has four child nodes 42, each corresponding to a respective 2×2 block 48 of the 4×4 block
40 of data elements. Each such child node 42 of the quadtree 45 then has 4 child nodes which form the leaf nodes 43 of the quadtree 45. The leaf nodes 43 of the quadtree 45 each correspond to a
respective individual data element 49 of the 4×4 block 40 of data elements.
As shown in FIG. 2, and as discussed above, in a first processing pass 44 each of the leaf nodes 43 in the quadtree 45 is set to the value of its respective data element 49 in the block of data
elements 40 (i.e. to the value of the data element 49 that the leaf node 43 corresponds to), and each non-leaf node 41, 42 in the tree 45 representing the data array 40 is set to the minimum value of
its child nodes. This calculation is performed from the bottom-up.
Then, in a second processing pass 46, each node, except the root node 41 (which has no parent) has the value of its parent node subtracted from it. This is again done from the bottom-up.
The resulting node values 47 following this second processing pass are the node values for the tree representing the data array 40 that are stored to represent the data array 40.
In the present embodiment, a separate such quadtree representation is constructed for each different component (data channel) of the data elements of the data array that is being encoded. Thus, for
example, in the case where each data element has four components, such as RGBA colour components, a separate quadtree representation of the form described above is constructed for each colour
component, i.e. such that there will be a “red” component quadtree, a “green” component quadtree, a “blue” component quadtree, and an “alpha” component quadtree. One such set of quadtrees will be
generated for each block of the data array that quadtree representations are being generated for.
Once the value to be stored for each node of the quadtree representing the block of the data array has been determined in the above manner, a set of data representing those node values (and from
which the node values can be determined) is then generated (which data can then be stored as an encoded and compressed version of the original data array block 31).
In the present embodiment, this is done by generating and storing data indicating the difference between respective parent and child node values in the quadtree representing the block of the data
array. In other words, the data value that is actually stored for a node of the quadtree representing the block of the data array in the data representing the quadtree indicates the difference
between the value of the node in question and its parent node. Thus, for some or all of the nodes in the tree, data indicating the difference between the value for the node in the tree, and the value
of its parent node is generated (and then stored in the stored data representing the tree).
The value for a given node in the tree will then correspondingly be determined from the stored representation of the tree by determining the value for the parent node of the node in question and
adding the difference value indicated for the node of tree to the determined value for the parent node.
Thus, in the present embodiment, once a “minimum value” quadtree representing a block 31 of the original data array 30 to be encoded has been determined as discussed above, a set of data representing
that quadtree to be stored as the representation of the quadtree from which the values of the data elements in the block 31 of the original data array 30 that the tree represents are to be determined
is generated by determining the differences between the values of respective parent and child nodes in the quadtree, and then generating data representative of those difference values (and from which
the difference values can derived), which data is then stored as the data representing the quadtree.
The left-hand quadtree representation 51 in FIG. 3 shows, by way of illustration, the difference values that would be stored for some exemplary nodes of a quadtree representing an exemplary block of
a data array. (FIG. 3 shows on the left-hand side respective exemplary difference values 51 to be stored for nodes of a quadtree representing a block of a data array (as discussed above), and on the
right-hand side, a “bit count” quadtree 55 that indicates the amount of bits that have been used to signal the difference values in the quadtree to the left (this will be discussed in more detail
below). The dotted line in FIG. 3 indicates the relationship between a node of the bit count tree 55 and the differential tree 51.)
As shown in FIG. 3, if it is assumed that the quadtree representing the data array has a root node value of 5 (following the generation of the node values using two processing passes as discussed
above), the value to be stored for the root node 52 of the quadtree 51 that represents the 4×4 block of data elements will be set to 5.
If it is then assumed that the four child nodes 60 of that root node 52 have values 12, 5, 8 and 9, respectively, from left-to-right in FIG. 3, then the difference values that are accordingly stored
for these child nodes 60 will be 7, 0, 3, and 4, respectively, as shown in FIG. 3. (For example, as the value for the root node of the quadtree representing the data elements is 5, and the first
child node 53 of that root node 52 has a value 12, the difference to be stored is 7.)
Similarly, if the child nodes 58 of the first child node 53 of the root node 52 have values 13, 13, 12 and 12, respectively, from left-to-right in FIG. 3, the difference values that are accordingly
stored for these child nodes 58 will be 1, 1, 0, and 0, respectively, as shown in FIG. 3. (These child nodes 58 are leaf nodes of the quadtree 51.)
Once the difference values to be stored for each node of the quadtree representing the block 30 of the data array have been determined (i.e. a quadtree of the form of the quadtree 51 shown on the
left-hand side of FIG. 3 has been generated), a set of data representing that “difference value” quadtree is then generated using an entropy coding scheme.
The entropy coding process used in the present embodiment basically determines how many bits are to be used to signal the respective difference values for the quadtree representing the data array.
Once this has been done, a set of data representing the quadtree representing the block of the data array that uses the determined number of bits to signal the respective difference values is
generated and stored to represent the node values of the quadtree representing the block of the data array.
Furthermore, to facilitate decoding of that stored data representing the node values of the quadtree, a set of data indicating the number of bits that have been used to signal the respective
difference values, which is also in the form of a quadtree, is also generated and stored.
Thus the coding and storage of the data representing the difference values for the nodes of the quadtree representing the block of the data array is based on a quadtree that is maintained in parallel
with the tree representing the values of the data array, which quadtree represents (and allows to be derived) the number of bits that have been used to signal the differences for the children of a
node of the quadtree representing the values of the data elements of the relevant block of the data array (and thus this parallel tree can accordingly be considered as a “bit count” tree).
This process will now be described with reference to FIG. 3, which shows in parallel to the quadtree 51 representing the node values of the quadtree representing the data elements of the block of the
data array, a bit count quadtree 55 which indicates how many bits have been used to signal the difference values for the respective node values in the quadtree 51.
As shown in FIG. 3, the bit count quadtree 55 is constructed to have a root node 56 that indicates how many bits have been used to signal the difference between the value of the root node of the
quadtree 51 representing the data array and each of its child nodes. Thus, in the example shown in FIG. 3, the root node 56 of the bit count quadtree 55 has a value 3, indicating that three bits have
been used to signal the difference values for the child nodes 60 of the root node 52 of the quadtree 51 representing the data array (i.e. that three bits have been used to signal each such difference
value that is indicated explicitly in the data representing the quadtree node values) (as shown by the dashed line in FIG. 3). This bit count quadtree root node 56 is set to indicate that three bits
are used, because the highest difference between the root node 52 and one of its child nodes 60 in the quadtree representing the values of the data elements is 7 (for the child node 53), thereby
requiring three bits to indicate that difference value.
It should be noted here that each difference value for the child nodes 60 of the root node 52 that is included explicitly in the data representing the node values of the quadtree representing the
block of the data array will be signalled using three bits (as that is the bit count indicated by the root node 56 of the bit count quadtree 55), even if the actual difference value does not require
as many as three bits to signal. Any redundant bits will be padded using zeros. This is the case for each respective bit count value indicated by a node of the bit count tree 55.
The next level down 57 in the bit count quadtree 55 has set as its node values the respective number of bits that have been used to signal the difference between each respective child node 60 of the
root node of the quadtree indicating the data values and its child nodes. Thus, for example, because one bit is sufficient to signal all the differences between the child nodes 58 of the node 53 in
the quadtree 51, the corresponding node 59 of the bit count quadtree 55 is set to 1.
It should be noted here that there is no need to include in the bit count tree a value indicating the number of bits that are used for the root node value of the tree representing the block of the
data array, because the value for that root node is signalled using the number of bits that is required to write an uncompressed value for the component of the data array in question (i.e. if the
data array uses 8-bit component, such as colour, values, then the root node value for the root node of the tree representing the values of the data array will be signalled using 8 bits). As the size
(i.e. number of bits) used for the uncompressed components in the data array will be known, the number of bits used for the root node will accordingly be known.
(Although FIG. 3 only shows the values for certain nodes in the bit count quadtree 55 and the quadtree 51 representing the data values of the data array, as will be appreciated, in practice each bit
count quadtree node and each node of the quadtree representing the values of the data array will have a value that will be set and determined in the manner of the present embodiment.)
In the present embodiment, the bit count quadtree 55 is encoded for storage by again storing the differences between node values in the bit count tree, rather than their actual values, as follows.
Firstly, the actual value of the root node for the bit count tree is stored for that node (i.e. the root node 56 of the bit count tree 55 is uncompressed). The number of bits used to signal the value
of this node in the data representing the bit count quadtree is set to the number of bits that will be needed for the largest possible bit count that the root node of the bit count quadtree could be
required to indicate +2. (For example, the bit count tree root node for a data array that uses 8-bit components will be sent as a 4-bit value, so that that root node of the bit count tree can
indicate any value (any number of bits) between 0 and 8, since the largest possible difference between the root node and its child nodes in the data value indicating tree where 8-bit component values
are being used will be 255, which would accordingly require 8 bits to signal (encode).) The +2 is added to facilitate the encoding of special cases by the root node of the bit count tree by setting
all the bits to 1 or all the bits but the last to 1 (which are interpreted as bit count values of −1 or −2, respectively).
Thus, for example, in the case of encoding a data array that uses 8-bit component values, the root node for the tree representing the data array values will be sent as an 8-bit value, indicating any
number between 0 and 255. If the largest difference then to the 8×8 level is 23, the root node of the bit count tree will be set to the value 5, to indicate that 5 bits are needed (and used) in the
data representing the tree representing the node values to represent that difference value of 23. The root bit count node value of 5 will be represented in the data representing the bit count tree
using 4 bits (as the largest possible difference value when using 8-bit component values between respective levels in the tree representing the node values will be 255, which would require 8 bits to
indicate, and so the root node of the bit count tree must contain enough bits to signal a value up to 8, i.e. 4 bits).
Then, the difference between the amount of bits required between the root 4×4 level and the 2×2 child node first level in the quadtree representing the data values and the amount of bits required
between each 2×2 level node in the quadtree representing the data values and its child leaf nodes (which can be thought of as a level 2 bit count) is sent as a 2-bit signed value [−2, 1] (i.e. the
value for the bit count quadtree for the child nodes 57 of the root node 56 of the bit count quadtree shown in FIG. 3 is indicated by using a signed 2-bit value to indicate the difference between the
bit count of the respective child node 57 and the parent root node 56).
The above describes the layout and configuration of the bit count tree 55 and of the data that is stored for representing the bit count tree 55 in the present embodiment. The actual bit count tree
values to use are determined as follows.
Firstly, once the (difference) value to be stored for each node of the tree representing the data array has been determined, as discussed above, the number of bits (the bit count) needed to indicate
the respective (difference) value of each node of the tree representing the data array is determined.
However, in the present embodiment, these bit count values are not simply then used as the node values to be stored for (to populate) the bit count tree, because the Applicants have recognised that
as the present embodiment uses a constrained, and fixed, number of bits (namely a 2-bit signed value) to indicate the bit counts for the respective nodes of the bit count tree, then the actual bit
count (number of bits) required for indicating the value of a given node of the tree representing the data array may in fact be smaller or larger than what it is possible to signal with the fixed
size bit count field to be used in the representation of the bit count tree.
To take account of this, the bit count quadtree 55 is constrained to make sure that the bit count to be signalled for each respective node in the bit count quadtree other than the root node is no
smaller than the bit count of the node's largest child node minus one, and no smaller than the bit count of the node's parent node minus two.
This is achieved in the following manner. First, each node in the bit count quadtree is initialised to its “true” bit count value. Then, a first bottom-up pass is performed in which the bit count for
each node of the bit count quadtree, except the root node, is constrained to be no smaller than the bit count of its largest child node minus 1. This is done by increasing the node's “true” bit count
by whatever amount is necessary to satisfy the constraint.
Following this first pass, a top-down pass is then performed in which the current bit count for each node is constrained to be no smaller than the bit count of its parent node minus 2. This is again
done by increasing the node's current bit count by whatever amount is necessary to satisfy the constraint.
The resulting node bit count values following the second pass are then the values that are associated with (set for) each node in the bit count tree. A set of data representing (encoding) the bit
count tree in the manner discussed above is then generated and stored.
Once the bit count tree has been derived in this manner, the data representing the node values for the tree representing the data array can be generated and stored, using the configuration (and in
particular the node value bit counts (field sizes)) corresponding to, and indicated by, the bit count tree. Thus the data representing the node values of the tree representing the data array will be
configured such that for each node, the number of bits that is used to signal the difference value for that node in the data representing the node values of the tree representing the data array is
the number of bits that the bit count tree indicates.
The effect of this then is that the data representing the node values of the tree representing the data array will be configured such that for each node, the number of bits that is used to signal the
difference value for that node in the data representing the node values of the tree representing the data array is a number of bits (a bit count) that can be indicated using the fixed configuration
of the bit count tree.
As in this arrangement the set of data representing the node values of the tree representing the data array must use the number of bits indicated by the bit count tree for respective node values, if
necessary the stored data representing the node values is padded with zeros (or some other recognisable “dummy” bits) to achieve this.
The root node value in the set of data representing the node values of the tree representing the data array is signalled in this embodiment using the same amount (number) of bits as are used in the
input data element format for the data array. As this number of bits is not signalled by the bit count tree, if necessary this number of bits used in the input data element format may be communicated
to the decoder in some other way. For example, in a chip where different processing blocks communicate, this number of bits could be set to a data register in the processing block that has to
decompress the data. Alternatively, for example for use in software encoding and decoding, a file header that contains the pixel format and information on how many bits are used could be associated
with the data representing the data array. All the blocks for a given data array in the present embodiment for which tree representations are generated use the same size of root node.
In the present embodiment, certain bit count values that can be indicated by the bit count tree are used to indicate predetermined, special, cases of node values in the quadtree representing the
block 31 of the data array. This is used to further compress the data representing the node values for the tree representing the block 31 of data array and the data representing the bit count tree
(and thus to further compress the stored data that represents the tree representing the data array).
Firstly, the bit count value −2 is set aside for indicating (and predefined as indicating) that the value of the root node of the tree representing the data array that that bit count node value
corresponds to and that the value of all the child nodes of the tree representing the data array below that root node, have the same, predefined default value (such as 1 or 0).
In effect therefore, a bit count value of −2 will trigger the decoder to assume that the root node of the tree representing the data array, and all the child nodes below that root node in the tree
representing the data array, have the same, predefined value.
This then allows the data representing the bit count tree to efficiently indicate the situation where all the leaf node values along a branch of the tree representing the block 31 of the data array
will be the same as a given default value.
In this embodiment, when a node of the bit count tree is set to this bit count value −2 indicating that the root node of the tree representing the data array, and that all the child nodes of the tree
representing the data array below that root node, have the same default value, then no bit count values are stored in the data representing the bit count tree for the nodes of the bit count tree that
are below the root node, and no node values are stored in the data representing the node values of the tree that represents the data array for the nodes of the tree representing the data array that
are below the node in question. This is possible because as the lower level child node values will all be the same as the higher level parent node, any information relating to those child nodes is
redundant and can therefore be omitted.
In effect therefore, this arrangement will trigger the decoder to assume that each node of the bit count tree has its bit count node value set to the predetermined value, such as −2, that indicates
that the root node, and all the child nodes below that root node, in the tree representing the data array have the predefined default value.
The bit count value −1 is set aside for indicating (and predefined as indicating) that the value of the node of the tree representing the data array that that bit count node value corresponds to and
that the value of all the child nodes of the tree representing the data array below that node, have the same value as the parent node of the tree representing the data array for the node in question.
In effect therefore, a bit count value of −1 will trigger the decoder to assume that the corresponding node of the tree representing the data array, and all the child nodes below that node in the
tree representing the data array, have the same value as the parent node in the tree representing the data array for the node in question.
This then allows the data representing the bit count tree to efficiently indicate the situation where all the leaf node values along a branch of the tree representing the block 31 of the data array
will be the same as a given value indicated for a node of the tree that is closer to the root node of the tree representing the block of the data array.
In this embodiment, when a node of the bit count tree is set to this bit count value −1 indicating that a node of the tree representing the data array, and that all the child nodes of the tree
representing the data array below that node, have the same value as the parent node of the node of the tree representing the data array in question, then no bit count values are stored in the data
representing the bit count tree for the nodes of the bit count tree that are below the node in question, and no node values are stored in the data representing the node values of the tree that
represents the data array for the nodes of the tree representing the data array that are below the node in question. This is possible because as the lower level child node values will all be the same
as the higher level parent node, any information relating to those child nodes is redundant and can therefore be omitted.
In effect therefore, this arrangement will trigger the decoder to assume that each lower level child node of the bit count tree also has its bit count node value set to the predetermined value, such
as −1, that indicates that that node, and all the child nodes below that node, in the tree representing the data array have the same value as the parent node for the node in question.
Secondly, the bit count value 0 is set aside for indicating (and predefined as indicating) that all the child nodes of the node of the tree representing the data array that that bit count value
corresponds to have the same value as the node in question. In effect therefore, a bit count value of 0 will trigger the decoder to assume that all the child nodes of the node in question in the tree
representing the data array have the same value as their parent node.
This then allows the data representing the bit count tree to efficiently indicate the situation where all the child nodes of a node of the tree representing the data array will have the same value as
their parent node in the tree representing the data array.
In this embodiment, when a node of the bit count tree is set to this bit count value 0 indicating that all the child nodes of a node of the tree representing the data array have the same value as
their parent node, then no node values are stored in the data representing the node values of the tree that represents the data array for the child nodes of the node in question. Again, this is
possible because as the child node values will all be the same as their parent node, any information relating to those child nodes is redundant and can therefore be omitted.
Thirdly, the bit count value 1 is set aside for indicating (and predefined as indicating) that the values of all the child nodes of the node of the tree representing the data array that that bit
count value corresponds to differ from the value of that node of the tree representing the data array by zero or one.
This then allows the data representing the bit count tree to efficiently indicate the situation where all the child nodes of a node of the tree representing the data array will differ from the value
of their parent node in the tree representing the data array by zero or one.
In this embodiment, when a node of the bit count tree is set to this bit count value 1, indicating that the values of all the child nodes of a node of the tree representing the data array differ from
the value of their parent node by zero or one, then the node values for the child nodes of the node in question are represented in the data representing the node values of the tree that represents
the data array using a bit map which has one bit for each child node. The value of each child node is then determined as the value of the parent node plus the value of the bit in the bit map for that
This allows the child node values to be indicated in an efficient manner in this situation.
The above describes (the encoding of) special cases where all the child nodes of a node of the tree representing the data array have the same value as, or only differ in value by one from, their
parent node.
In the event that a child node of the tree representing the data array differs in value from its parent node by more than one, the following arrangement is used.
Firstly, bit count values >=2 are set aside for this situation, i.e. for indicating (and predefined as indicating) that the value of at least one child node of the node of the tree representing the
data array that that bit count value corresponds to differs from the value of that node of the tree representing the data array by more than one.
As discussed above (and as can be seen from FIG. 2, for example), even in this situation where at least one child node of a node of the tree representing the data array differs from the value of its
parent node by more than one, in the present embodiment there will still be at least one other child node of that parent node (of the node in question) that has the same value as the parent node (as
the node in question), because of the way that the tree representing the (block of the) data array is configured.
Thus, if a node of the tree representing the data array has a child node whose value differs from it by more than one, the encoding process first identifies a child node of that parent node (of the
node of the tree representing the data array in question) that has the same value as that parent node (i.e. a “zero difference” child node), and then stores in the data representing the node values
of the tree representing the data array, data representing the values of the other child nodes of the node in question, but does not store any data indicating the value of the identified “zero
difference” child node.
Moreover, the difference values for the non-zero difference child nodes are stored in a predetermined sequence (order) in the data representing the node values of the tree representing the data
array, depending upon which of the four child nodes is the “zero difference” child node. FIG. 4 illustrates this, and shows four predetermined node value storage sequences, depending upon which node
is the designated “zero difference” child node Z. The values for the other nodes are stored in the sequence A, B, C.
This then means that a decoder can straightforwardly determine the order in which the data representing the node values is stored in the stored data representing the tree node values, once it knows
which child node of a tree node is the “zero difference” child node for which no data value has been stored.
The designated “zero difference” child node is indicated (to the decoder) by including a 2-bit “zero child” designator (zerocd_1* in FIG. 4) in the set of data indicating the node values of the tree
representing the (block of the) data array. This “zero child” designator is then followed by bit fields indicating the values of each of the three remaining child nodes of the node in question, in
the predetermined order that is appropriate to the “zero child” node in question.
In the present embodiment, the bit fields indicating the values of each of the three “non-zero” child nodes of the node in question are stored in a bit interleaved fashion, rather than keeping them
as three separate bit fields, as this can simplify the hardware implementation when decoding the bit fields. FIG. 5 illustrates this and shows that where the bit fields for the three child nodes A,
B, C are bit-interleaved (as shown in the right-hand side of FIG. 5), the existing bits for each child node will stay in the same positions when the widths of the bit fields change, whereas if three
separate fields, one for each child node, are stored one after the other without interleaving (as shown in the left-hand side of FIG. 5), the existing bits for the bit fields will need to move when
the widths of the child node difference value bit fields change.
Thus, for example, where the bit count indicated by the bit count tree in this situation indicates that three bits are used for the difference values, the set of data indicating the values of the
tree representing the block of the data array will contain a 2-bit “zero child” designator followed by three bit interleaved 3-bit fields signalling the difference values for the other three children
of the node.
In operation to encode a data array 30 in the manner of the present embodiment, the data for the data array can be processed in any suitable and desired manner. For example, a suitable processor or
processing circuitry may read the original data array to be compressed from memory, and/or receive a stream of data corresponding to the original data array to be compressed, and then process the
stream of data accordingly, e.g. divide it into blocks, generate the necessary quadtrees, and then generate data representing the quadtree(s) and store that tree-representing data in memory, e.g. in
memory and/or on a removable storage medium, etc.
As discussed above, in the present embodiment, this process will accordingly first comprise generating for the data array, or for each block that the data array has been divided into, a “minimum
value” quadtree having the form described above, in which the leaf nodes of the tree correspond to respective data elements of the data array (or block of the data array). Then a bit count quadtree
of the form described above is derived, based on the number of bits that are needed to indicate, and that will be used to indicate, the value of each node of the minimum value quadtree in a set of
data representing the node values of that minimum value quadtree. A set of data representing that bit count quadtree of the form discussed above will then be generated, together with a set of data
representing the node values of the minimum value quadtree (in the form of difference values), which set of node value indicating data is configured according to, and has the configuration indicated
by, the corresponding bit count quadtree.
The process will further comprise identifying special cases of minimum value quadtree node values, as discussed above, and configuring the bit count quadtree to indicate those special cases, and the
set of data representing the minimum value quadtree node values accordingly.
The so-generated set of data representing the tree node values and the set of data representing the corresponding bit count tree will then be stored appropriately for use as a compressed set of data
representing the data array.
FIG. 6 shows schematically an embodiment for storing the data that is generated to represent the data array in embodiments of the technology described herein in memory.
FIG. 6 again shows schematically an array of original data 20 that is a two-dimensional data array containing a plurality of data elements (containing data entries at a plurality of particular
positions within the array) and that is to be encoded and compressed and stored. As discussed above, the data array 20 could be any suitable and desired array of data, but in a graphics processing
context, it could, for example, be a texture map (i.e. an array of texture elements (texels)), or an array of data representing a frame to be displayed (in which case the data array may be an array
of pixels to be displayed). In the case of a texture map, each data entry (position) in the data array will represent an appropriate texel value (e.g. a set of colour values, such as RGBa, or
luminance and chrominance, values for the texel). In the case of a frame for display, each data entry (position) in the array will indicate a set of colour values (e.g. RBG values) to be used for
displaying the frame on a display.
As shown in FIG. 6, for the purposes of storing the data array 20 in memory, the data array 20 is first divided into a plurality of non-overlapping, equal-size and uniform blocks 21, each block
corresponding to a particular region of the data array 20. In the present embodiments, each block 21 that the data array is divided into for storing purposes corresponds to a block of 16×16 elements
(positions) within the data array 20 (i.e. a block of 16×16 texels in the case of a texture map). (Other arrangements would, of course, be possible.)
Each block 21 that the data array 20 is divided into for storing purposes is further sub-divided into a set of non-overlapping, uniform and equal-size sub-blocks 22. In the present embodiment each
sub-block 22 corresponds to a 4×4 data element region within the block 21 (e.g. 4×4 texels in the case of a texture map). (FIG. 6 only shows the division of a few of the blocks 21 of the data array
20 into sub-blocks for simplicity. However, each and every block 21 that the original data array 20 is divided into is correspondingly sub-divided into a set of plural sub-blocks 22.) A quad-tree
representation (or set of quadtree representations where there are plural data components (channels)) is generated in the manner discussed above for each such 4×4 sub-block 22.
To store the data array 20 in memory, firstly a header data block 23 is stored for each block 21 that the data array 20 has been divided into. These header data blocks are stored in a header buffer
24 in memory. The header buffer 24 starts at a start address A in memory, and the header data blocks 23 are each stored at a predictable memory address within the header buffer 24.
FIG. 6 shows the positions of the header data blocks 23 in the header buffer 24 for some of the blocks 21 that the data array 20 is divided into. Each block 21 that the data array 20 is divided into
has a corresponding header data block 23 in the header buffer 24.
The position that each header data block 23 is stored at within the header buffer 24 is determined from (predicted from) the position within the data array of the block 21 that the header data block
23 relates to. In particular, the address of the header data block 21 in the header buffer 24 for a data array element (e.g. texel or pixel) at a position x, y within the data array 20 is given by:
header data block address=A+16*(x/16+(y*xsize/16)
where A is the start address of the header buffer, xsize and ysize are the vertical and horizontal dimensions, respectively, of the data array 20, and it is assumed that the data array 20 is divided
into 16×16 blocks and each header data block occupies 16 bytes, the divisions are done as integer divisions (rounding down), and the array width (xsize) is an exact multiple of the block size. (If
necessary, padding data can be added to the input data array to ensure that it has a width (and height) evenly divisable by the block size (in this case 16) (such padding data can then be
appropriately cropped by the decoder when decoding the data array if required).)
In the present embodiment, each header data block 23 in the header buffer 24 has the same, fixed size, of 16-bytes. This means that the header data blocks 23 are of a size that can be fetched using a
system-friendly burst size.
As well as storing a respective header data block 23 in the header buffer 24 for each block 21 that the original data 20 is divided into, the data storage arrangement of the present embodiment also
stores for each block 21 that the original data 20 is divided into the appropriate quad-tree and bit-count tree representing data (i.e. the “payload” data) for each sub-block 22 that the data block
has been divided into. This sub-block data is stored in a continuous fashion in a body data buffer 30 in memory. In the present embodiment, the body buffer 30 is stored directly after the header
buffer 24 (but may appear in random order there). This allows the pointer data in the header data blocks to be in the form of offsets from the end of the header buffer 24. (This is not essential, and
the body buffer 30 may reside anywhere in memory, if desired.)
The sets of quad-tree and bit count tree data for each respective set of sub-blocks (i.e. for each respective block that the data array has been divided into for storage purposes) are stored in the
body buffer 30 one after another.
The header buffer 24 and body buffer 30 may be stored in any suitable memory of the data processing system in question. Thus, for example, they may be stored in an on-chip memory or in an external
memory, such as a main memory of the data processing system. They are in an embodiment stored in the same physical memory, but that is not essential.
Some or all of the header buffer and body buffer data could also be copied to a local memory (e.g. cached), in use, if desired.
The data that is stored for each sub-block in the body buffer 30 comprises respective data from the encoded representation of the original data array that has been generated in the manner of the
embodiments of the technology discussed above (i.e. data representing node values of a quadtree representing the respective 4×4 sub-block 22 of the data array 20, together with data representing a
corresponding bit count quadtree).
Each header data block 23 contains pointer data indicating the position within the body buffer 30 where the data for the sub-blocks for the block 21 that that header data block 23 relates to is
In these embodiments, the pointer data in the header data blocks 23 indicates the start position in the body buffer 30 of the stored data for the respective sub-blocks that the header data block
relates to, in the form of an offset from the start location A of the header buffer 24, together with a “size” indication value (in bytes) for each sub-block that the block that the header data block
relates to has been divided into (encoded as). The size indication value for a sub-block indicates the amount of memory (in bytes) that has been used to represent (signal) the data for the sub-block
in the body buffer 30.
To locate the data for an individual sub-block in the body buffer 30, the decoder then accordingly uses the pointer in the header data block 23 to determine the start position in the body buffer 30
of the data for the set of sub-blocks that the header data block 23 relates to, and then uses the size information in the header data block 23 to sum the sizes of the stored data for the sub-blocks
that are stored prior to the sub-block of interest, to determine the start position for the data in the body buffer 30 for the sub-block of interest. The end position of the data for the sub-block of
interest is correspondingly determined using the indicated stored data size for the sub-block in question.
In these embodiments, the header data blocks 23 are configured to contain only (to store only) the indication of the start position of the body buffer 30 of the stored data for the respective
sub-blocks that the header data block relates to and the respective size indication values for each sub-block that the header data block relates to. Thus, the header data blocks do not store any
“payload” sub-block data (i.e. any data that is to be used when decoding a given sub-block). This has the advantage that only the sub-block data and not any data that is in the header data block
needs to be decoded when decoding a given sub-block.
When encoding with multiple different encoders, the encoding processes in these embodiments are configured to make sure that each encoder before it starts encoding has access to a continuous set of
memory space that is the size of an uncompressed block of the data array 20. This is achieved by dividing the body buffer 30 into as many different parts as there are encoders available, and
allocating each encoder one of the respective parts of the body buffer 30. Then when one encoder has filled its own buffer part to the point where there is not enough memory space available to
guarantee that an encoded block of the data array will fit in the allocated space, the encoder is configured to take half of the space allocated for another encoder (rounded to the size of the
uncompressed body buffer) and to then carry on its compression. As long as the granularity of the allocated body buffer parts is as big as the maximum size that an encoded block of the data array can
occupy, this should not require any extra data.
In these embodiments, certain sub-block size indication values that can be indicated by the size fields in the header data blocks 23 are predefined as (set aside for) indicating special cases of
sub-block, so as to facilitate signalling such special cases to the decoder in an efficient manner. In the present embodiments, the size indication values that are used for this purpose are for sizes
which will not in fact occur in use, namely size indications of 0 or 1.
A size indication value of 1 is used to indicate that the data for the sub-block to which the size indication value relates has been stored in an uncompressed form in the body buffer.
A size indication value of 0 is used to indicate that the same data as was used for the preceding sub-block should be used when decoding the sub-block to which the size indication value of 0 relates
(i.e. the same data as was used for sub-block n−1 should be used when decoding the sub-block n to which the size indication value of 0 relates). This effectively can be used to indicate the situation
where a given sub-block can be considered to be a copy of (and thus can be copied from when decoding) another sub-block.
Where such a copy sub-block is identified in the encoding process, then no data for the copy sub-block is stored in the body buffer 30. This can allow the sub-block data to be stored in a more
compressed form in the body buffer 30, by avoiding the storage of duplicated sub-block data. For example, for a constant colour block, a set of sub-block data for the first sub-block of the block can
be stored, but then all the remaining blocks can simply be allocated a size indication of 0 in the header data block, with no data being stored for those sub-blocks in the body buffer 30.
To further enhance the potential benefit of the use of a “copy” sub-block size value indication, in these embodiments the sub-blocks for a given block of the data array are encoded and stored in an
order that follows a space filling curve, such as a Peano curve, U-order, Z-order, etc., so as to ensure that each sub-block is always encoded (stored) next to a spatially near neighbour sub-block.
This can also help to enhance the benefits of caching of any sub-block data.
In the present embodiments, as discussed above, 16-byte header blocks which contain a pointer to the start of the sub-block data in the body buffer 30 together with a size indication value for each
sub-block, are used. The basic layout of each header data block is for the pointer to the start of the set of sub-blocks' data for the block in question to come first, followed by size indication
values for each respective sub-block (in the order that the sub-blocks are encoded and stored in the body buffer 30). The actual sizes of the pointer and size indication values and the layout of the
sub-blocks in the body buffer 30 are, however, configured differently, depending upon the form of the data that is being encoded. In particular, different storage arrangements are used for RGB and
YUV data. This will be discussed and illustrated below, with reference to FIGS. 4-13.
FIGS. 7-10 show the arrangement of a header data block and the sub-blocks' data for a respective block of the data array 20 when encoding and storing RGB or RGBA data. FIGS. 7 and 8 show the
arrangement for a block of a data array that does not contain any predefined “special case” sub-blocks, and FIGS. 9 and 10 show the arrangement for a block of the data array that does include some
predefined special case sub-blocks (namely a “copy” sub-block and an uncompressed sub-block).
In this case, where each block that the data array is divided into represents 16×16 data entries, then for RGB or RGBA data each block of the data array is divided into and encoded as 16 4×4
sub-blocks. Each sub-block will accordingly be stored as a three or four component texture, with each component encoded as a respective quadtree in the manner discussed above. Thus, in this
embodiment sixteen quadtree representations will be generated per data component for each 16×16 block that the data array has been divided into for storage purposes.
As shown in FIGS. 7 and 8, the header data block 23 includes a 32-bit pointer 70 in the form of an offset to the start of the sub-block data for the data block in the body buffer 30, and then 16
6-bit size indication values 71, one for each sub-block that the block of the data array has been divided into for storing purposes (as discussed above), thereby providing in total a 16-byte header
data block. As shown in FIGS. 7 and 8, each size indication value in the header data block 23 indicates the memory size that has been used to store the data to be used to decode the respective
sub-block in the body buffer 30 that the size indication value relates to.
FIG. 8 shows the order that the 16 4×4 sub-blocks of the block of each data array are encoded and stored in. As can be seen from FIG. 8, the encoding order follows a space filling curve, in this case
a Peano curve.
FIGS. 9 and 10 show the data storage arrangement for RGB or RGBA data for a block where it has been determined that the third sub-block (sub-block number 2) can be considered to be a copy of the
second sub-block (sub-block number 1). In this case, the size indication value S2 for the third sub-block is set to the value “0” which has been predefined as indicating a copy sub-block, and as
shown in FIGS. 9 and 10, no sub-block data is stored for the third sub-block (sub-block number 2) in the body buffer 30, but rather the data stored for the second sub-block (sub-block number 1) will
be reused by the decoder to decode the third sub-block (sub-block number 2).
FIG. 9 also shows an exemplary uncompressed sub-block (sub-block 5), which accordingly has its size indication value S5 set to the value 1 that has been predefined as indicating an uncompressed
These arrangements will be used for other formats of data where it is desired to store all the data components together in the same sub-block.
FIGS. 11-16 illustrate the storage arrangement that is used in the present embodiments for YUV data. FIGS. 11-13 show the arrangement for YUV 420 data and FIGS. 14-16 show the arrangement for YUV 422
data. As can be seen from these Figures, in these arrangements the Y-plane data and the UV-plane data is stored in separate sub-blocks. This facilitates reading and decoding that data separately, and
also can be used to allow for the fact that the two different types of data are stored at different resolutions.
FIG. 11 shows the principle of the storage arrangement for YUV 420 format data. In this format, four Y samples are used for every set of UV values. Thus, as illustrated in FIG. 11, the Y-plane values
are encoded and stored at a higher resolution than the UV-plane values. The effect of this then is that one 4×4 block of UV (chrominance) values will be required for each set of four 4×4 sub-blocks
of Y (luminance) values.
As shown in FIGS. 12 and 13, to account for this, for each 16×16 block of YUV data, the Y-plane and the UV-plane are encoded and stored separately, with 16 4×4 sub-blocks being used and stored for
the Y-plane, and 4 4×4 sub-blocks being used and stored for the UV-plane data. Thus, for each block that the data array has been divided into, twenty sub-blocks will be stored, sixteen representing
the Y (luminance) data and four representing the UV (chrominance) data. The Y (luminance) data sub-blocks will be stored as one component textures (encoded as quadtrees in the manner discussed
above), and the UV (chrominance) data sub-blocks will be stored as 2-component textures (with each component encoded as a respective quadtree in the manner discussed above).
Each header data block will accordingly, as shown in FIG. 12, contain 20 sub-block size indication values 71. To allow each header data block still to occupy 16-bytes, in these arrangements the
offset data 70 to the start of the sub-blocks' data is configured to occupy 28 bits, and each sub-block size indication value 71 occupies 5 bits.
Again, the sub-blocks are encoded and stored in an order that essentially follows a space-filling curve, but as shown in FIG. 13, the arrangement is such that the UV data sub-blocks are interleaved
in the order of the sub-blocks with the Y data sub-blocks, such that the corresponding UV data sub-blocks are stored locally to the Y data sub-blocks that they are to be used with when decoding the
FIGS. 14-16 show the corresponding arrangement for YUV 422 data. In this case, as is known in the art, two Y samples will share a set of UV samples (as shown in FIG. 14). Thus in this case, as shown
in FIG. 16, the Y data will again be encoded as 16 4×4 sub-blocks, but there will be 8 4×4 UV sub-blocks. The effect of this then is that twenty-four sub-blocks will be stored for each block that the
data array has been divided into, sixteen Y sub-blocks and eight UV data sub-blocks.
In this case, as shown in FIG. 15, each header data block is configured to have a 32-bit offset pointer 70, and then 24 4-bit sub-block size indication values 71. As shown in FIG. 16, the order that
the sub-blocks are encoded and stored in again essentially follows a space filling curve, but is configured to appropriately interleave the UV data sub-blocks with their corresponding Y data
In these arrangements where, for example, a copy sub-block is indicated, then a “copy” chrominance sub-block should be copied from the previous chrominance sub-block, and a “copy” luminance sub-block
should be copied from the previous luminance sub-block, by the decoder.
These arrangements for storing YUV data can correspondingly be used for other forms of data where it is desired to store different components of the data separately, in different sub-blocks.
In operation to encode a data array 20 in the manner of the above embodiments, suitably configured and/or programmed processing circuitry will receive and/or fetch from memory a stream of data
representing the original data array 20, and operate to divide the data array 20 into blocks and sub-blocks as discussed above, generate and store appropriate header data blocks, generate encoded
tree representations of the sub-blocks of the data array (in the manner discussed above), and store the data for the encoded representations of the sub-blocks of the data array in memory. If
necessary, padding data can be added to the input data array to ensure that it has a width and height evenly dividable by 16 (such padding data can then be appropriately cropped by the decoder when
decoding the data array if required).
The above primarily describes the way in the present embodiments that the encoded version of the data array is generated and stored in memory for use. When the so-stored data array comes to be used,
for example to apply to fragments to be rendered (where the stored data array is a texture map for use in graphics processing), then the reading and decoding processes for the stored data array will
essentially comprise the reverse of the above storing and encoding processes.
Thus, the decoding device, such as a graphics processor (e.g. where the stored data array is texture map) or a display controller (e.g., where the stored data array is a frame to be displayed), will
first identify the position(s) of the particular element or elements in the data array that are of interest (i.e., whose values are to be determined). It will then determine the start address A of
the header buffer for the stored data array (if necessary, this can be communicated to the decoder by the, e.g., software that controls the encoder and decoder setting a control register with a
pointer to the header buffer), and then use that start address together with the position of the data array element or elements that are of interest to determine the location of the header data block
for the block of the data array that the data element(s) falls within in the manner discussed above.
The decoder will then read the header data block from the identified memory location and determine therefrom the pointer data and sub-block size data indicating the memory location in the body buffer
30 of the relevant sub-block data to be used to reproduce the sub-block of the block of the data array that the data element or elements falls within. The decoder will then read the relevant
sub-block data from the determined location in the body buffer 30.
Once the decoder has the necessary sub-block data, it can then decode that data to determine the value of the data element or elements of interest.
This decoding process will essentially be the reverse of the above-described encoding process. Thus, once the decoder has loaded the necessary data relating to the sub-block, it will first determine
the required bit count tree node values from the stored data representing the bit count tree, and then use those determined bit count tree node values to identify the data for the relevant nodes of
the quadtree representing the values of the data elements of the block of the data array of interest, and use those node values to determine the value of the data element or elements of interest.
As part of this process, the decoder will, accordingly, look at the bit count node values and identify the special cases discussed above (if present), for example, and interpret the stored data
representing the quadtree representing the values of the data elements of the data block, and determine the values for the nodes of that quadtree, accordingly.
This sub-block data reading process should also take account of any predefined “special case” sub-block size indication values, as discussed above. Thus, if the decoder identifies in the header data
block a sub-block size value indicating a “copy” sub-block, the decoder should accordingly use the sub-block data that was (or that would be) used for the preceding block to determine the value of
the data element or elements in question.
In the case of decoding YUV data where the luminance and chrominance values are stored as separate sub-blocks, for example, the decoding process should accordingly, where required, identify both the
relevant luminance sub-block data and the relevant chrominance sub-block data and decode both sets of sub-block data to determine the luminance and chrominance values needed for the data element or
elements in question.
This process can then be repeated for each data element of interest (whose value is required).
FIG. 17 shows schematically an arrangement of a graphics processing system 1 that can store and use data arrays that have been stored in the manner of the present embodiments.
FIG. 17 shows a tile-based graphics processing system. However, as will be appreciated, and as discussed above, the technology described herein can be implemented in other arrangements of graphics
processing system as well (and, indeed, in other data processing systems).
The system includes, as shown in FIG. 17, a tile-based graphics processor (GPU) 1. This graphics processor 1 generates output data arrays, such as output frames intended for display on a display
device, such as a screen or printer, in response to instructions to render graphics objects, etc. that it receives.
As shown in FIG. 17, the graphics processor 1 includes a vertex shader 2, a binning unit 3, a state management unit 4, a rasterising stage 5, and a rendering stage 6 in the form of a rendering
The vertex shader 2 receives descriptions of graphics objects to be drawn, vertices, etc., e.g. from a driver (not shown) for the graphics processor 1, and performs appropriate vertex shading
operations on those objects and vertices, etc., so as to, for example, perform appropriate transform and lighting operations on the objects and vertices.
The binning unit 3 sorts (bins) the various primitives, objects, etc., required for an output to be generated by the graphics processor 1 (such as a frame to be displayed) into the appropriate bins
(tile lists) for the tiles that the output to be generated is divided into (since, as discussed above, this exemplary graphics processing system is a tile-based graphics processing system).
The state management unit 4 stores and controls state data and the state of the graphics processing units to control the graphics processing operation.
The rasteriser 5 takes as its input primitives to be displayed, and rasterises those primitives to sampling positions and fragments to be rendered.
The rendering pipeline 6 takes fragments from the rasteriser 5 and renders those fragments to generate the output data (the data for the output (e.g. frame to be displayed) of the graphics processor
As is known in the art, the rendering pipeline will include a number of different processing units, such as fragment shaders, blenders, texture mappers, etc.
In particular, as shown in FIG. 17, the rendering unit 6 will, inter alia, access texture maps 10 stored in a memory 9 that is accessible to the graphics processor 1, so as to be able to apply the
relevant textures to fragments that it is rendering (as is known in the art). The memory 9 where the texture maps 10 are stored may be an on-chip buffer or external memory (e.g. main system memory)
that is accessible to the graphics processor 1.
The graphics processor 1 generates its output data arrays, such as output frames, by generating tiles representing different regions of a respective output data array (as it is a tile-based graphics
processor). Thus, the output from the rendering pipeline 6 (the rendered fragments) is output to tile buffers 7.
The tile buffers' outputs are then written to a frame buffer 8, e.g. for display. The frame buffer 8 may reside, e.g. in main memory (which memory may be DDR-SDRAM) of the system (not shown). The
data from the tile buffers may be downsampled before it is written to the frame buffer, if desired.
The texture maps 10 and the frame buffer 8 may be stored in the same physical memory, or they may be stored in different memories, as desired.
Sometime later, the data array in the frame buffer 3 will be read by a display controller and output to a display device for display (not shown).
The graphics processing system shown in FIG. 17 uses the data array encoding and decoding and storing arrangement of the embodiments described above in respect of both the stored texture maps 10 in
the memory 9, and when storing its output data in the frame buffer 8.
Thus, each texture map 10 that is stored in the memory 9 for use by the rendering unit 6 is stored in one of the forms described above. Accordingly, when the rendering unit 6 needs to access a
texture map, it will read and decode the texture map data in the manners described above.
Similarly, when the generated output data from the graphics processor 1 is written to the frame buffer 8 from the tile buffer 7, that data is processed in the manner described above, to take the data
from the tile buffers 7 and store it in the format of one of the embodiments of the technology described herein in the frame buffer 8. This data can then be read and decoded from the frame buffer 8
in the manners described above by, e.g., the display controller (not shown) of the display on which the frame is to be displayed.
It will be appreciated that each of the stages, elements, and units, etc., of the graphics processor as shown in FIG. 17 may be implemented as desired and will accordingly comprise, e.g., appropriate
circuitry, and/or processing logic, programmable logic, etc., for performing the necessary operations and functions, and will provide the appropriate control and processing circuitry, etc., for
performing the technology described herein.
It will also be appreciated here that FIG. 17 simply shows the arrangements schematically, and thus, for example, the data flow in operation of the technology described herein need not and may not be
as shown in FIG. 17, but may, for example, involve the looping back of data as between the various units and stages shown in FIG. 17 as appropriate.
A number of modifications and variations to the above described embodiments of the technology described herein would be possible.
For example, the data values for the data elements that are encoded in the quadtree could be transformed (converted) to a different form prior to generating the tree representing the data array, if
For example, where the data array to be compressed uses an RGB format, the RGB data may be first transformed to a YUV format, and the so-transformed data in the YUV format then compressed and encoded
in the manner of the present embodiments.
In such an arrangement, the YUV transform is in an embodiment of the form:
where F is a constant to avoid negative values, and is calculated from the size of G as:
F=1<<(bit width of the G component)
With this transform, the Y, U and V components will be expanded by 0-2 bits compared to their RGB counterparts.
(The inverse to this transform (to be used when decoding the encoded and compressed data representing the data array) is:
Transforming RGB data to YUV data can simplify the coding of the chroma channels, as there may then be only one channel that will depict detail in the image (data array) instead of three. This can
then mean that even if the amount of uncompressed data per data element is expanded, there will still be a gain once the data has been compressed in the manner of the present embodiments.
In an embodiment, the system of the technology described herein employs arithmetic wrapping, i.e. the result of all additions when calculating the value of a node in the tree representing the data
array are calculated modulo (1<<bit depth of the (potentially yuv transformed) component).
Furthermore, in an embodiment when using such wrapping arithmetic, each parent node in the tree representing the data array is initialised to a value that is higher than the minimum value of its
child nodes, and then differences relative to that higher value are stored for the child nodes, and arithmetic wrapping of the results used when the stored node values are summed to derive the actual
child node values.
This can help to improve the rate of compression for certain forms of data array, for example, for arrays where the values that the data elements can take tend to fall into distinct groups, such as
being either very bright or dark in images as could be the case, for example, for high contrast images, such as text and schematics.
For example, if the data array contains only the numbers 0 (dark) and 255 (white), then if using 8-bit arithmetic with arithmetic wrapping, it would be more efficient to initialise each parent node
in the tree representing the data array to the maximum value of its child nodes (i.e. to 255, where a parent node has a child node having the value 255), rather than to the minimum value of its child
nodes, and to rely on arithmetic wrapping (i.e. 255+1=0 when utilising 8-bit arithmetic) to generate the desired node values when decoding the stored data.
This will then allow smaller difference values (1 instead of 255) to be used to represent the node values in the data representing the node values of the tree representing the data array, and thus
allow smaller numbers of bits to be used to describe (signal) the differences (the difference values) (i.e. 1 bit instead of 8 bits).
Thus, in an embodiment, the values that the non-leaf nodes in the tree representing the data array are initialised to are set so as to minimise the differences between those values (and thus the
differences that will need to be stored for the nodes) when taking account of any wrapping arithmetic that is used in the system in question (and the system will rely on arithmetic wrapping to
determine the actual node values when decoding the stored data).
It should be noted in these arrangements it doesn't necessarily have to be the maximum child node value that is used. For example, for a parent node having child nodes with the values 2, 250, 253 and
251, if using 8-bit arithmetic (i.e. such that the largest value is 256), setting the parent node to the child node value 2 would mean that the differences between the parent node and its child nodes
would be 0, 248, 251 and 249, respectively. If the parent node is instead set to the child node value 250 and wrapping arithmetic is relied on, the differences to be stored will be 10, 0, 3 and 1,
respectively. Since these difference values are smaller, they will require less bits to store. In this case, the parent node should be set to the value of the smallest value child node in the set of
higher value child nodes.
In general, in these arrangements of the technology described herein, the parent node may be set to the value of its lowest value child node whose value is equal to or above a selected threshold
minimum value. This can be seen from the above example, where the threshold minimum value would be 250, and so the parent node is set to the lowest value child node whose value equals or exceeds that
selected threshold minimum value. This should take the wrapping arithmetic into account, such that the parent node will be set to the minimum value of its child nodes where there is no child node
that has a value higher than the selected minimum threshold value.
In these arrangements, the threshold minimum value to use to exploit the arithmetic wrapping efficiently can be determined, for example, by binning some or all the leaf node values into a fixed
amount of bins (such as four bins), and then setting the threshold minimum value set as being a value, and in an embodiment the minimum value, in the bin that has the lowest total distance (in normal
counting order) to the other filled bins. In an embodiment the threshold minimum value is selected by binning (sorting) all the leaf node values (i.e. the data element values) in the data array or
block of the data array for which the tree is being generated into groups (bins) of values, and then determining an appropriate minimum value on that basis.
For example, for a given set of leaf node values, the values of all the leaf nodes could be placed in four bins having, in the case of 8-bit arithmetic, values from 0-63, 64-127, 128-191 and 192-255.
It could then be considered which of these bins have node values in them (in the above example there will be one node value in the first bin and three in the last bin), and the distance for each
respective node value to the bins with data in them determined.
For example, with the above example leaf node values (i.e. data element values) of 2, 250, 253 and 251, if the value 2 which is in bin zero was selected as the threshold minimum node value to use,
the distance between that “minimum value” bin and the other bins with leaf node values in them would be 0 (0→0) and 3 (0→3). If instead, the threshold minimum node value was set to a node value in
the last bin (e.g. 250 as discussed above), the distances will be 0 (3→3) and 1(3→0, using wrapping). Similarly, if the leaf node values were such that there were leaf node values in the first and
last two bins, the minimum distance from the first bin would be 0+2+3, while the total distance from the third bin would be 0+1+2 and from the fourth bin 0+1+3, i.e. the value to use for the
threshold minimum node value should be the value in the third bin.
Once the selected minimum threshold value has been selected, then the value for each parent node in the tree representing the data array can be initialised to the value of its lowest value child node
whose value is equal to or greater than the selected threshold minimum value. Once this has been done, the differences between the node values can be determined and data representing those
differences stored, as discussed above. If the threshold minimum value has been selected appropriately, the difference values should be lower than would otherwise be the case if the wrapping
arithmetic was not being exploited in this manner. The root node value will be set to the threshold minimum value (as it has no parent node from which to subtract its value from).
To determine the node values to use in these arrangement, if the selected minimum value is denoted as M, then working with 8-bit arithmetic, for each leaf node having a value p, one could initialise
the leaf node with the value p′=(p+256−m) modulo 256. This essentially sets the leaf nodes to the difference between their value p and the threshold M, taking account of the arithmetic wrapping. Each
non-leaf node can then be set to the smallest value of its child nodes to build the tree representing the data array, and the root node set to the threshold minimum value M. Thus, an example with
leaf node values of 2, 250, 253 and 251, and setting the threshold minimum value M to 250, would generate transformed leaf node values of 8, 0, 3, 1 (i.e. the respective difference values when
compared to M) and the lowest of these difference values, 0, would be chosen as the value to set the parent node to (but the root node of the tree would be set to M=250 so that the tree will
decompress correctly to its original form).
If desired, the use of the “zero child” arrangement discussed above can also be modified to provide a further compression gain. In particular, if two of the “non-zero” child nodes B and C that are
geometrically closest to the zero value are typically smaller than the node A which is further away, then that can be used to map the concatenation of the upper two bits of A, B and C to a table with
64 entries. Twenty four of these entries (which represent the most common combinations) can then be mapped to thirty-two different 5-bit values (i.e. saving 1-bit), and the other forty entries will
map to 8-bit values. The difference between the 8- and 5-bit values can be determined by looking at the first 2 bits of the value. For example if the first 2-bits are “11”, then an 8-bit entry should
be read for the look-up, but otherwise a 5-bit entry. When using appropriate look-up table entries for appropriate images, any penalty for the forty entries mapping to 8-bit values should be smaller
than the gain that is derived from having 24 entries mapping to thirty-two different 5-bit values, thereby giving some reduction in the overall number of bits required.
Although the present embodiment has been described above as generating a tree representation for a 4×4 block of data elements, other arrangements could be used. For example, a separate tree
representation could be generated for each 16×16 data element block of the data array, or for each 8×8 or 16×4 data element block of the data array (i.e. such that the data array will be divided into
16×16, 8×8 or 16×4 blocks for the purposes of the tree representation encoding, respectively).
Also, although the present embodiments have been described above with particular reference to the use of the techniques of the present embodiment with graphics processors and display controllers, the
techniques of the technology described herein can be used for other data array processing and in particular for other image processing arrangements.
For example, the techniques of the technology described herein may be used in image signal processors and video decoders and encoders (MPEG/h.264, etc.). In these cases, the techniques of the
technology described herein could be used, for example, to encode an image generated by an image signal processor which is processing data received from an image sensor to make a watchable image out
of it. A video encoder/decoder could, for example, decode images (video frames) encoded in the form of the technology described herein to then compress the image using some other standard like h.264,
and correspondingly encode frames of video data using the technique of the technology described herein for provision, for example, to a graphics processor or a display controller.
As can be seen from the above, the technology described herein, in some embodiments at least, provides a method and apparatus for encoding data arrays that can allow the encoded data to take less
memory space (to be stored more efficiently), reduce the amount of memory traffic for reading the encoded data, and/or make more efficient the memory traffic for reading the encoded data. It can
accordingly, thereby reduce power consumption.
This is achieved, in some embodiments at least by representing the data array using a particular form of tree representation, together with a separate bit count tree indicating the number of bits
that have been used to signal the value of the nodes in the tree representing the data elements of the data array. For example, by using a quadtree of the form described above to represent data
elements of the data array, together with a bit count tree communicated by 2-bit differences for each node, and the node value elimination arrangements described above, the tree representing the data
elements of the data array can be compressed in an efficient manner.
The data encoding arrangement of the technology described herein is particularly suited to use for textures and frame buffers, and can decrease external bandwidth as well as facilitating random
access to the encoded data and being decodable at line speed for the texture cache. The arrangement of the technology described herein, in some embodiments at least, allows the efficient decoding of
the data for a given block within an overall data array, and with little overhead.
The technology described herein, in some embodiments at least, in effect provides an entropy coding system for storing and representing a tree, such as a quadtree, representing all or part of a data
array, i.e. an encoding system in which the syntax elements (such as the nodes of the tree representing the data array and of the bit count tree) are mapped to sequences of bits with a fixed length
and may be placed in a continuous fashion in memory. This provides an efficient way of representing the data array with bit values.
Thus the technology described herein, in some embodiments at least, provides a way to efficiently and losslessly compress a data array in a way that allows fast decompression with a hardware-based
decoder and fits well with a bus friendly memory layout. It can accordingly help to minimise the bandwidth required to communicate image data between different graphics producing and consuming nodes
in a system on-chip, for example. This helps reduce issues with bus congestion and power requirements.
The technology described herein is advantageous over other possible entropy encoding schemes such as Huffman or arithmetic encoding, etc., because those schemes require, for example, more serial
processing to determine where individual syntax elements can be located.
The foregoing detailed description has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed.
Many modifications and variations are possible in the light of the above teaching. The described embodiments were chosen in order to best explain the principles of the technology and its practical
application, to thereby enable others skilled in the art to best utilise the technology in various embodiments and with various modifications as are suited to the particular use contemplated. It is
intended that the scope be defined by the claims appended hereto.
1. A method of encoding an array of data elements for storage in a data processing system, the method comprising:
generating at least one tree representation for representing the array of data elements, the tree being configured such that each leaf node of the tree represents a respective data element of the
data array, and the data values for the nodes of the tree being set such that the data value that the tree indicates for the data element of the data array that a leaf node of the tree represents
is given by the sum of the data values in the tree for the leaf node and each preceding parent node in the branch of the tree that the leaf node belongs to;
generating and storing data representing the at least one tree representing the data array as an encoded version of the array of data elements; and
for at least one parent node of the tree, interleaving bits representing the data values of child nodes of the parent node of the tree with each other in the stored data representing the at least
one tree representing the data array.
2. The method of claim 1, further comprising:
dividing the data array into plural separate blocks, and generating a respective tree representation for each different block that the data array is divided into.
3. The method of claim 1, comprising determining the data values to be associated with each node of the at least one tree representation by:
setting, in a first processing pass, each leaf node in the tree to the value that the tree is to indicate for the data element in the data array to be encoded that the leaf node represents, and
each non-leaf node in the tree to the value of one of its child nodes; and
then, in a second processing pass, subtracting from each node the value of its parent node.
4. The method of claim 3, wherein:
each non-leaf node is set to the value of its lowest value child node in the first processing pass.
5. The method of claim 3, further comprising:
selecting the values that the non-leaf nodes in the tree representing the data array are set to in the first processing pass so as to minimise the differences between those values when taking
account of any wrapping arithmetic that is to be used to determine the node values when decoding the stored data.
6. The method of claim 1, wherein the step of generating and storing data representing the at least one tree representing the data array comprises:
generating data representing the at least one tree representing the data array by determining the differences between the values of respective parent and child nodes in the tree; and
storing data representative of the determined difference values as the set of data representing an encoded version of the array of data elements.
7. The method of claim 1, wherein:
the data that is generated and stored to represent the at least one tree representing the data array comprises a set of data representing the tree node values, together with a set of data
indicating the number of bits that has been used for signalling the value for each node in the tree in the set of data representing the tree node values.
8. The method of claim 7, wherein:
at least one of the values that can be included in the set of data indicating the number of bits that have been used for signalling the value for each node in the tree in the set of data
representing the tree node values is predefined as indicating that the value of a node of the tree representing the data array, and that the values of all the child nodes of the tree representing
the data array below that node, are the same as the value of that node's parent node in the tree representing the data array, and no node values are stored in the data representing the node
values of the tree that represents the data array for the nodes of the tree representing the data array that are below the node in question.
9. A method of generating an encoded version of an array of data elements for storage in a data processing system, the method comprising:
generating at least one tree representation for representing the array of data elements, the data values for the nodes of the tree being set such that data values for the data elements of the
data array that the tree represents can be derived from the data values for the nodes of the tree; and
responsive to the root node of the tree and to all of the child nodes of the tree representing the data array below the root node not being the same predefined value: generating data representing
the node values of the at least one tree representing the data array; generating at least one further tree representation in which the node values for the at least one further tree indicate the
number of bits used to indicate respective node values in the data generated to represent the node values of the at least one tree representing the data array; generating data representing the at
least one further tree indicating the number of bits used to indicate the respective node values in the data generated to represent the node values of the at least one tree representing the data
array; and storing the generated data representing the node values of the at least one tree representing the data array and the generated data representing the at least one further tree
indicating the number of bits used to indicate the respective node values in the data generated to represent the node values of the at least one tree representing the data array, as an encoded
version of the array of data elements; and
responsive to the root node of the tree and to all of the child nodes of the tree representing the data array below the root node being the same predefined value: storing as a set of data for
indicating the number of bits that have been used for signalling the value of each node in the tree representing the data array, a data value that is predefined as indicating that the root node
of the tree representing the data array, and that all of the child nodes of the tree representing the data array below that root node, have the same predefined value; and omitting storing data
representing the node values of the at least one tree representing the data array.
10. The method of claim 9, wherein:
the data that is generated and stored to represent the at least one further tree indicating the number of bits used to indicate the respective node values in the data generated to represent the
node values of the at least one tree representing the data array comprises data representative of the differences between the values to be indicated by respective parent and child nodes in the at
least one further tree indicating the number of bits used to indicate the respective node values in the data generated to represent the node values of the at least one tree representing the data
11. A method of determining the value of a data element of a data array for use in a data processing system, the method comprising:
using stored data representing a tree representing some or all of the data elements of the data array to determine the value of each node of a branch of the tree representing some or all of the
data elements of the data array; and
determining the value to be used for a data element of the data array by summing the determined values for the leaf node of the branch of the tree and for each preceding parent node in the branch
of the tree that the leaf node belongs to; and
wherein, for at least one parent node of the tree, bits representing the data values of child nodes of the parent node of the tree in the stored data representing the at least one tree
representing the data array are interleaved with each other in the stored data representing the at least one tree representing the data array.
12. The method of claim 11, wherein:
the stored data representing the tree representing some or all of the data elements of the data array comprises data representing the differences between the values of respective parent and child
nodes in the tree, and the value of a node of a branch of the tree is determined by determining the value of the node's parent node and then adding the difference value indicated for the node of
the tree in the stored data representing the tree to the determined value for the parent node.
13. A method of determining a value of a data element of a data array for use in a data processing system, the method comprising:
during traversal of a tree indicating a number of bits used to indicate one or more node values of a tree representing some or all of the data elements of the data array:
responsive to identifying a data value associated with a node of the tree indicating the number of bits used to indicate one or more node values of a tree representing some or all of the data
elements of the data array indicating that the root node and all of the child nodes below the root node in the tree representing some or all of the data elements of the data array have a
predefined node value, determining the value of the data element of the data array based on the predefined node value; and
responsive to not identifying the data value associated with a node of the tree indicating the number of bits used to indicate one or more node values in a tree representing some or all of the
data elements of the data array the data value indicating that the root node and all of the child nodes below the root node in the tree representing some or all of the data elements of the data
array have a predefined node value: using data representing the tree indicating the number of bits used to indicate one or more node values of a tree representing some or all of the data elements
of the data array, to determine the number of bits used to indicate one or more node values in a set of stored data that represents the node values of the tree representing some or all of the
data elements of the data array; using the determined number of bits to identify the stored data representing the data values of a node or nodes of the tree representing some or all the data
elements of the data array,
using the identified stored data representing the data values of a node or nodes of the tree representing some or all the data elements of the data array to determine the data values of a node or
nodes of the tree representing some or all the data elements of the data array; and using the determined node values to determine the value to be used for the data element of the data array.
14. The method of claim 13, wherein:
data representing the tree indicating the number of bits used to indicate respective node values in a set of stored data that represents the node values of the tree representing some or all of
the data elements of the data array, is stored in the form of difference values between respective parent and child nodes in a bit count tree, and the bit count value indicated by a node in the
bit count tree is determined from the stored data by determining the bit count value for the parent node of the node in the bit count tree, and then adding the difference value indicated for the
node of the bit count tree to the determined bit count value for its parent node.
15. An apparatus for encoding an array of data elements for storage in a data processing system, the apparatus comprising:
processing circuitry that generates at least one tree representation for representing the array of data elements, the tree being configured such that each leaf node of the tree represents a
respective data element of the data array, and the data values for the nodes of the tree being set such that the data value that the tree indicates for the data element of the data array that a
leaf node of the tree represents is given by the sum of the data values in the tree for the leaf node and each preceding parent node in the branch of the tree that the leaf node belongs to, the
processing circuitry generates and stores data representing the at least one tree representing the data array as an encoded version of the array of data elements; and
wherein the processing circuitry, for at least one parent node of the tree, interleaves bits representing the data values of child nodes of the parent node of the tree with each other in the
stored data representing the at least one tree representing the data array.
16. The apparatus of claim 15, wherein:
the processing circuitry divides the data array into plural separate blocks, and generates a respective tree representation for each different block that the data array is divided into.
17. The apparatus of claim 15, wherein:
the processing circuitry determines the data values to be associated with each node of the at least one tree representation by setting, in a first processing pass, each leaf node in the tree to
the value that the tree is to indicate for the data element in the data array to be encoded that the leaf node represents, and each non-leaf node in the tree to the value of one of its child
nodes, and then, in a second processing pass, subtracting from each node the value of its parent node.
18. The apparatus of claim 17, wherein:
each non-leaf node is set to the value of its lowest value child node in the first processing pass.
19. The apparatus of claim 17, wherein:
the processing circuitry selects the values that the non-leaf nodes in the tree representing the data array are set to in the first processing pass so as to minimise the differences between those
values when taking account of any wrapping arithmetic that is to be used to determine the node values when decoding the stored data.
20. The apparatus of claim 15, wherein:
the processing circuitry generates and stores data representing the at least one tree representing the data array by generating data representing the at least one tree representing the data array
by determining the differences between the values of respective parent and child nodes in the tree, and storing data representative of the determined difference values as the set of data
representing an encoded version of the array of data elements.
21. The apparatus of claim 15, wherein:
the data that is generated and stored to represent the at least one tree representing the data array comprises a set of data representing the tree node values, together with a set of data
indicating the number of bits that has been used for signalling the value for each node in the tree in the set of data representing the tree node values.
22. The apparatus of claim 21, wherein:
one of the values that can be included in the set of data indicating the number of bits that have been used for signalling the value for each node in the tree in the set of data representing the
tree node values is predefined as indicating that the value of a node of the tree representing the data array, and that the values of all the child nodes of the tree representing the data array
below that node, are the same as the value of that node's parent node in the tree representing the data array, and no node values are stored in the data representing the node values of the tree
that represents the data array for the nodes of the tree representing the data array that are below the node in question.
23. An apparatus for generating an encoded version of an array of data elements for storage in a data processing system, the apparatus comprising:
a memory;
processing circuitry having access to the memory that generates at least one tree representation for representing the array of data elements, the data values for the nodes of the tree being set
such that data values for the data elements of the data array that the tree represents can be derived from the data values for the nodes of the tree;
wherein the processing circuitry is configured to:
responsive to the root node of the tree and to all of the child nodes of the tree representing the data array below the root node not being the same predefined value: generate data representing
the node values of the at least one tree representing the data array, generate at least one further tree representation in which the node values for the at least one further tree indicate the
number of bits used to indicate respective node values in the data generated to represent the node values of the at least one tree representing the data array, generate data representing the at
least one further tree indicating the number of bits used to indicate the respective node values in the data generated to represent the node values of the at least one tree representing the data
array, and store the generated data representing the node values of the at least one tree representing the data array and the generated data representing the at least one further tree indicating
the number of bits used to indicate the respective node values in the data generated to represent the node values of the at least one tree representing the data array, as an encoded version of
the array of data elements; and
responsive to the root node of the tree and to all of the child nodes of the tree representing the data array below the root node being the same predefined value: storing as a set of data for
indicating the number of bits that have been used for signalling the value of each node in the tree representing the data array, a data value that is predefined as indicating that the root node
of the tree representing the data array, and that all of the child nodes of the tree representing the data array below that root node, have the same predefined value; and omitting storing data
representing the node values of the at least one tree representing the data array.
24. The apparatus of claim 23, wherein:
the data that is generated and stored to represent the at least one further tree indicating the number of bits used to indicate the respective node values in the data generated to represent the
node values of the at least one tree representing the data array comprises data representative of the differences between the values to be indicated by respective parent and child nodes in the at
least one further tree indicating the number of bits used to indicate the respective node values in the data generated to represent the node values of the at least one tree representing the data
25. An apparatus for determining the value of a data element of a data array for use in a data processing system, the apparatus comprising:
processing circuitry that uses stored data representing a tree representing some or all of the data elements of the data array to determine the value of each node of a branch of the tree
representing some or all of the data elements of the data array, and determines the value to be used for a data element of the data array by summing the determined values for the leaf node of the
branch of the tree and for each preceding parent node in the branch of the tree that the leaf node belongs to;
wherein, for at least one parent node of the tree, bits representing the data values of child nodes of the parent node of the tree in the stored data representing the at least one tree
representing the data array are interleaved with each other in the stored data representing the at least one tree representing the data array.
26. The apparatus of claim 25, wherein:
the stored data representing the tree representing some or all of the data elements of the data array comprises data representing the differences between the values of respective parent and child
nodes in the tree, and the value of a node of a branch of the tree is determined by determining the value of the node's parent node and then adding the difference value indicated for the node of
the tree in the stored data representing the tree to the determined value for the parent node.
27. An apparatus for determining a value of a data element of a data array for use in a data processing system, the apparatus comprising:
a memory for storing data representing a tree representing some or all of the elements of the data array and a tree indicating a number of bits used to indicate one or more node values of the
tree representing some or all of the data elements of the data array;
processing circuitry having access to the memory for traversing the tree indicating the number of bits used to indicate the one or more node values of the tree representing some or all of the
elements of the data array;
the processing circuitry being configured to during traversal of the tree indicating the number of bits used to indicate one or more node values of a tree representing some or all of the data
elements of the data array:
responsive to identifying a data value associated with a node of the tree indicating the number of bits used to indicate one or more node values of a tree representing some or all of the data
elements of the data array indicating that the root node and all of the child nodes below the root node in the tree representing some or all of the data elements of the data array have a
predefined node value, determine the value of the data element of the data array based on the predefined node value; and
responsive to not identifying the data value associated with a node of the tree indicating the number of bits used to indicate one or more node values in a tree representing some or all of the
data elements of the data array the data value indicating that the root node and all of the child nodes below the root node in the tree representing some or all of the data elements of the data
array have a predefined node value: use data representing the tree indicating the number of bits used to indicate one or more node values of a tree representing some or all of the data elements
of the data array to determine the number of bits used to indicate one or more node values in a set of stored data that represents the node values of the tree representing some or all of the data
elements of the data array, use the determined number of bits to identify the stored data representing the data values of a node or nodes of the tree representing some or all the data elements of
the data array, use the identified stored data representing the data values of a node or nodes of the tree representing some or all the data elements of the data array to determine the data
values of a node or nodes of the tree representing some or all the data elements of the data array, and use the determined node values to determine the value to be used for the data element of
the data array.
28. The apparatus of claim 27, wherein:
data representing the tree indicating the number of bits used to indicate respective node values in a set of stored data that represents the node values of the tree representing some or all of
the data elements of the data array, is stored in the form of difference values between respective parent and child nodes in a bit count tree, and the bit count value indicated by a node in the
bit count tree is determined from the stored data by determining the bit count value for the parent node of the node in the bit count tree, and then adding the difference value indicated for the
node of the bit count tree to the determined bit count value for its parent node.
29. A non-transitory computer readable storage medium storing computer software code which when executing on a processor performs a method of encoding an array of data elements for storage in a data
processing system, the method comprising:
generating at least one tree representation for representing the array of data elements, the tree being configured such that each leaf node of the tree represents a respective data element of the
data array, and the data values for the nodes of the tree being set such that the data value that the tree indicates for the data element of the data array that a leaf node of the tree represents
is given by the sum of the data values in the tree for the leaf node and each preceding parent node in the branch of the tree that the leaf node belongs to;
generating and storing data representing the at least one tree representing the data array as an encoded version of the array of data elements; and
for at least one parent node of the tree, interleaving bits representing the data values of child nodes of the parent node of the tree with each other in the stored data representing the at least
one tree representing the data array.
30. A non-transitory computer readable storage medium storing computer software code which when executing on a processor performs a method of generating an encoded version of an array of data
elements for storage in a data processing system, the method comprising:
generating at least one tree representation for representing the array of data elements, the data values for the nodes of the tree being set such that data values for the data elements of the
data array that the tree represents can be derived from the data values for the nodes of the tree; and
responsive to the root node of the tree and to all of the child nodes of the tree representing the data array below the root node not being the same predefined value: generating data representing
the node values of the at least one tree representing the data array; generating at least one further tree representation in which the node values for the at least one further tree indicate the
number of bits used to indicate respective node values in the data generated to represent the node values of the at least one tree representing the data array; generating data representing the at
least one further tree indicating the number of bits used to indicate the respective node values in the data generated to represent the node values of the at least one tree representing the data
array; and storing the generated data representing the node values of the at least one tree representing the data array and the generated data representing the at least one further tree
indicating the number of bits used to indicate the respective node values in the data generated to represent the node values of the at least one tree representing the data array, as an encoded
version of the array of data elements; and responsive to the root node of the tree and to all of the child nodes of the tree representing the data array below the root node being the same
predefined value: storing as a set of data for indicating the number of bits that have been used for signalling the value of each node in the tree representing the data array, a data value that
is predefined as indicating that the root node of the tree representing the data array, and that all of the child nodes of the tree representing the data array below that root node, have the same
predefined value; and omitting storing data representing the node values of the at least one tree representing the data array.
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7885911 February 8, 2011 Cormode et al.
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Other references
• Office Action dated Feb. 13, 2013 in U.S. Appl. No. 13/198,462, 11 pages.
• Response to Office Action filed May 13, 2013 in U.S. Appl. No. 13/198,462, 14 pages.
• Notice of Allowance and Fee(s) Due dated May 29, 2013 in U.S. Appl. No. 13/198,462, 10 pages.
Patent History
Patent number
: 9014496
: Aug 3, 2012
Date of Patent
: Apr 21, 2015
Patent Publication Number
20130195352 Assignee
Arm Limited
Jorn Nystad
Oskar Flordal
Jeremy Davies
Ola Hugosson
Primary Examiner
Amir Alavi Application Number
: 13/566,894
Current U.S. Class
Pyramid, Hierarchy, Or Tree Structure (382/240); Compression Of Color Images (382/166) International Classification
: G06K 9/46 (20060101); G06K 9/00 (20060101); G06T 9/40 (20060101); H04N 19/196 (20140101); H04N 19/96 (20140101); G06K 9/34 (20060101); H04N 19/176 (20140101); H04N 19/186 (20140101); H04N 19/42 | {"url":"https://patents.justia.com/patent/9014496","timestamp":"2024-11-04T06:28:06Z","content_type":"text/html","content_length":"321705","record_id":"<urn:uuid:8d80e5dc-d83d-4cb8-91d0-064a55907eaa>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00397.warc.gz"} |
There are \(36\) warrior tomcats standing in a \(6 \times 6\) square formation. Each cat has several daggers strapped to his belt. Is it possible that the total number of daggers in each row is more
than \(50\) and the total number of daggers in each column is less than \(50\)? | {"url":"https://problems.org.uk/problems/443/?return=/problems/&category_id__in=105&from_difficulty=3.0&to_difficulty=3.0","timestamp":"2024-11-14T17:23:35Z","content_type":"text/html","content_length":"8071","record_id":"<urn:uuid:83790b8f-db81-4fe5-90ee-332267478249>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00039.warc.gz"} |
Free Online 1st Grade Math Test/Quiz | Your Homework Help
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How to Use A Tensor to Initialize A Variable In Tensorflow?
To use a tensor to initialize a variable in TensorFlow, you first need to create a tensor object with the desired values using the TensorFlow library. Once you have the tensor object, you can pass it
as the initial value when defining a TensorFlow variable. This can be done by using the tf.Variable function and providing the tensor object as the initial_value parameter. When the variable is
initialized, its value will be set to the values of the tensor object. This allows you to customize the initial values of variables using tensors in TensorFlow.
How to reshape a tensor in TensorFlow?
In TensorFlow, you can reshape a tensor using the tf.reshape function. This function takes in the tensor you want to reshape and the new shape you want to reshape it into. Here is an example of how
to reshape a tensor in TensorFlow:
1 import tensorflow as tf
3 # Create a tensor
4 tensor = tf.constant([[1, 2, 3],
5 [4, 5, 6]])
7 # Reshape the tensor
8 reshaped_tensor = tf.reshape(tensor, [3, 2])
10 # Print the reshaped tensor
11 print(reshaped_tensor)
In this example, we first create a tensor with shape (2, 3) and then use the tf.reshape function to reshape it into a tensor with shape (3, 2). The resulting reshaped tensor will have the same data
as the original tensor but in a different shape.
How to utilize a tensor to store user-defined constants in TensorFlow?
To utilize a tensor to store user-defined constants in TensorFlow, you can simply create a tensor with the desired constant value and store it in a variable.
Here is an example code snippet that demonstrates how to create a tensor to store a user-defined constant value in TensorFlow:
1 import tensorflow as tf
3 # Define the user-defined constant value
4 user_constant = 10
6 # Create a tensor to store the user-defined constant
7 user_constant_tensor = tf.constant(user_constant, dtype=tf.float32)
9 # Print the tensor value
10 with tf.Session() as sess:
11 print(sess.run(user_constant_tensor))
In this example, we first define the user-defined constant value (10 in this case) and then create a tensor using the tf.constant function. We specify the data type of the constant as tf.float32.
Finally, we run a TensorFlow session to evaluate and print the value of the tensor.
You can store multiple user-defined constants in tensors by creating additional tensors with different constant values. These tensors can then be used in TensorFlow operations and computations.
How to perform tensor reshaping in TensorFlow?
In TensorFlow, you can reshape a tensor using the tf.reshape() function. Here's how you can do it:
1. Import the TensorFlow library:
1 import tensorflow as tf
1. Define a tensor:
1 # create a tensor with shape [2, 3]
2 tensor = tf.constant([[1, 2, 3], [4, 5, 6]])
1. Reshape the tensor to a new shape:
1 # reshape the tensor to shape [3, 2]
2 reshaped_tensor = tf.reshape(tensor, [3, 2])
1. Run the TensorFlow session to evaluate the reshaped tensor:
1 with tf.Session() as sess:
2 result = sess.run(reshaped_tensor)
3 print(result)
This will output:
1 [[1 2]
2 [3 4]
3 [5 6]]
You can reshape a tensor to any shape as long as the total number of elements remains the same. | {"url":"https://article-blog.kdits.ca/blog/how-to-use-a-tensor-to-initialize-a-variable-in","timestamp":"2024-11-01T19:00:39Z","content_type":"text/html","content_length":"157552","record_id":"<urn:uuid:36dfa31b-f37e-493a-b82d-a6108fd78c0f>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00826.warc.gz"} |
SubQuery – SQLServerCentral Forums
Points: 17191
Points: 6879
SQLRNNR (9/19/2011)
Kenneth Wymore (9/19/2011)
Any idea as to why they would allow this behavior in a join? Seems to me like it would actually introduce more confusion than convenience.
I think they have to allow it due to the requirement that a subquery that is based on values instead of a query requires the same syntax.
Here's an article on that. http://jasonbrimhall.info/2011/08/31/bitwise-and-derived-table-revisited/
I reviewed your other post about this and I see what you mean about the subquery using values instead of a table select. I have never seen values used in that way exactly but I am sure there are
times when it is necessary. When there is a list of static values to reference, I have usually seen it coded as follows.
SELECT *
FROM (
SELECT 1 as a, 2 as b
SELECT 3 as a, 4 as b
SELECT 5 as a, 6 as b
SELECT 7 as a, 8 as b
SELECT 9 as a, 10 as b
) as MyTable;
--OR using a temp table
IF OBJECT_ID(N'TempDB..#MyTable') IS NOT NULL
DROP TABLE #MyTable
CREATE TABLE #MyTable
(a INT, b INT)
INSERT INTO #MyTable
SELECT 1 as a, 2 as b
SELECT 3 as a, 4 as b
SELECT 5 as a, 6 as b
SELECT 7 as a, 8 as b
SELECT 9 as a, 10 as b
SELECT * FROM #MyTable;
Using the union all statements is a bit tedious but that is what I have normally seen. If the same set needs to be used differently for multiple queries then it is typically dropped into a temp table
or a regular table. I have seen this option used before just to keep the main query from looking overly complicated too.
I am guessing that using a set of values like you showed on your post would be more common when dealing with applications? For example, where you don't want to insert user supplied values into a
table but instead are just using them temporarily in the subquery?
SSC Guru
Points: 281334
Points: 5847
Points: 11130 | {"url":"https://www.sqlservercentral.com/forums/topic/subquery-7/page/3","timestamp":"2024-11-04T14:28:03Z","content_type":"text/html","content_length":"81522","record_id":"<urn:uuid:b6738c5d-693e-4fbb-acce-db0997b2acd6>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00520.warc.gz"} |
Math Mind
To solve questions quickly you need to use the right strategy. Sometimes to work out a tough puzzle you might need a combination of different strategies too. By combining them we can create a
‘Library of Hows’
How to Enrich Your Library of Hows
Remembering how you solve a math or logic problem will help you when you come across another difficult question. By building a library of ‘Hows’ inside your brain, you give yourself a toolbox that
will help you take on a wide range of puzzles & problems. By tying together different Math Minds, your Library of ‘Hows’ becomes richer. And with a richer library, the faster ideas will come to your | {"url":"https://mmind.sonyged.com/mmind/fthinking/","timestamp":"2024-11-12T10:48:50Z","content_type":"text/html","content_length":"349370","record_id":"<urn:uuid:645029c2-23bf-4292-9339-a3b325b267aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00145.warc.gz"} |
Function ASpsider transforms vector x as follows \[ ASpsi(u)= \left\{ \begin{array}{cc} (1/c) \cos(u/c) & |u/c| \leq \pi \\ 0 & |u/c|> \pi \\ \end{array} \right. \]
Remark: Andrew's sine functionon is almost linear around $u = 0$ in accordance with Winsor's principle that all distributions are normal in the middle.
This means that $\psi (u)/u$ is approximately constant over the linear region of $\psi$, so the points in that region tend to get equal weight. | {"url":"http://rosa.unipr.it/fsda/ASpsider.html","timestamp":"2024-11-05T16:42:04Z","content_type":"application/xhtml+xml","content_length":"17155","record_id":"<urn:uuid:e8939e5e-9008-473f-8db2-64d5cb1b1990>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00109.warc.gz"} |
(PDF) The Body Mass Index recalculated
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All content in this area was uploaded by Koen Van de Moortel on Feb 20, 2022
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The Body Mass Index recalculated
Koen Van de moortel, MSc experimental physics, independent math tutor
Jules de Saint-Genoisstraat 98, 9050 Gentbrugge, Belgium, info@lerenisplezant.be
15 june 2021 - small addition 20 february 2022
Abstract: The so-called ‘least squares regression’ for mathematical modeling is a widely used
technique. It’s socommon that one might think nothing could be improved to the algorithm
anymore. Butit can. By minimizing the squares of the differences between measured and
predicted values not only in the vertical, but also in the horizontal direction. I call this
‘multidirectional regression’. The difference is very significant, especially for power function
models, often used in biomedical sciences. And it makes the regression invariant if the
dependent and independent variables are switched.This was a neglected problem with the
traditional method. The Body Mass Index and the Corpulence Index and their correlation with
body fat percentage are studied here as an example showing that this regression technique
produces better results.
Keywords: nonlinear regression, mathematical modeling, multidirectional regression, curve
fitting, software, algorithm, body mass index, BMI, corpulence index, ponderal index, body fat
percentage, anthropometry, scaling law, power function.
Introduction: the BMI mystery
In the process of writing a book about measuring methodology and regression analysis, I
thought the so called “Body Mass Index” (BMI) might be a good example of quantization, how to
put a number on ‘overweight’. As you probably know, it is calculated by taking a person’s mass
m (in kg) and divide it by the square of the height h (in meters). Now, this is quite awkward,
since the masses of objects with the same shape and similar density distributions are
proportional with the
power of the height (or any longitudinal dimension).
So I started digging... Why did the inventor, Adolphe Quêtelet, who happens to have lived in the
same city as me (Ghent, Belgium), define this index with h² in 1832? I wanted to find the
original data that he analyzed, the ‘reference people’ to calibrate it. Strangely, there seems to
be no trace ofthem on theinternet, and also no other dataset could be found! Thousands of
sites offer ‘ideal mass’ tables or calculators, and many use obscure disclosed formulas,
clearly not using h², some even using a linear relationship!
For me as a physicist, it’s hard to believe, but apparently it took almost a century before
someone (Fritz Rohrer, CH) came up with the idea to calculate the index with h³ anyway. This
number: m/h³ is thencalled ‘Corpulence Index’ (CI) or ‘Ponderal Index’ (PI) [Rohrer 1921]. It
took another century until someone (Sultan Babar, SA) found what was to be expected: “It has
the advantagethat it doesn’t need to be adjusted for age after adolescence.” [Babar 2015] In
spite of that, the general public still only knows the BMI. Today, june 2021, a search on
Academia.edu on BMI produces more than 795000 results, while only 1735 articles mention the
CI, and 6428 the PI. It took until 2013 before someone like Nick Trefethen (numerical analyst at
the University of Oxford, GB) raised his eyebrows and dared to make this remark in The
Body Mass Index recalculated
Economist: “The body-mass index that you (and the National Health Service) count on to
assess obesity is a bizarre measure. We live in a three-dimensional world, yet the BMI is
defined as weight divided by height squared. It was invented in the 1840s, before calculators,
when a formula had to be very simple to be usable. As a consequence of this ill-founded
definition, millions of short people think they are thinner than they are, and millions of tall
people think they are fatter.” And then he said: “You might think that the exponent should
simply be 3, but
that doesn’t match the data at all
. It has been known for a long time that
people don’t scale in a perfectly linear fashion as they grow. I propose that a better
approximation to the actual sizes and shapes of healthy bodies might be given by an exponent
of 2.5. So here is the formula I think is worth considering as an alternative to the standard
BMI: ‘new BMI’=1.3m/h2.5. [Trefethen, 2013, my emphasis]
Now, how could it “not match the data”? I was curious now to inspect some data myself. After
a long search, I came in touch with Nir Krakauer (The City College of New York), who was also
doing BMI-related modeling, and he was so kind to refer me to his data:
From this large collection, I extracted the masses and heights of 90 adult men who had a more
or less ‘ideal’ body fat percentage: between 11.6 and 13.8%. I’m not a medical doctor, but
according to different sources these percentages seem to be considered good for young
adults. The most important point for this selection was to have a more or less homogeneous
group with a range of sizes, but with similar densities. Of course I know other factors like
bone density and body type play a role as well, but this is the best I could do.
First, I put the data in the popular math program ‘GeoGebra’ (version 5 Classic):
This calculated the following relationship as ‘best fitting’: m = 18.8541 h2.1419.
Aha, that must have been the reason why Quêtelet decided to round the exponent of h to 2,
because the empirical value seems to be 2.1419! Then I realized that this program takes the
logarithms of the variables, in order to reduce the regression problem to a linear one. This
causes errors, as I illustrated elsewhere [Van de moortel 2021].
So I decided to put the data in my own softwareprogram, called ‘FittingKVdm’, which uses an
iterative algorithm to estimate the parameters. This produced: m = 19.331 h2.1084. Now the
exponent was even closer to 2! Strange!
GraphPad Prism 9.0.2, a program that seems well designed to me, and also uses iteration,
produced an identical result. Their writers also condemn the logarithm habit, by the way.
Still not being happy, I wanted to see the difference between a fit with a fixed exponent of 2
and one with exponent 3. The results: m = 20.5918 h2 and m = 11.4640 h3.
The value of 20.5918 is indeed a good BMI, and 11.4640 is close to the ‘good’ value of 12 for the
CI according to Sultan Babar.
Now, a picture is worth a thousand words, so I would like you to take a look on the mass
versus height graphs of both fitted curves (don’t mind if you can’t read the small letters,just
look at the dots and the lines):
Body Mass Index recalculated
Which line visually fits the best through the cloud of points? Everyone I asked this question,
answered: the one on the right, obviously!
So nowthe question came up: is there something wrong with the regression method itself?
Well, there is definitely an asymmetry: the classical algorithm that everybody uses, minimizes
the sum of the (weighted)
distances betweenthe measured (yi) and the predicted y
values f(xi). The weights are inversely proportional to the square of the measuring errors σy,i.
The parameters in the model function are adjusted in order to minimize this sum:
Sy f(x )
i i
i 1
Would it make any difference if we would usethe sum of the
distances? Why is it
not done? Well, in the case of non-invertible functions, especially periodical functions, there
are many such distances for every y value, but for a bijective function like the one above, it’s
perfectly possible. The simplest way totry it, isby switching theso called ‘independent’ and
the ‘dependent’ variable.
If the ‘best fit’ for our data, with free moving exponent, m = 19.331 h2.1084, was indeed the best fit,
it shouldn’t make any differenceif we switched the h and m columns and fitted again, should
it? The expected outcome of this procedure would be:
19.331 0.24534 m
2.1084 0.47429
Now, what was the actual outcome? h = 0.72877 m0.21394 Or, inverted: m = 4.38814 h4.67421
I double-checked it using GraphPad... same result. GeoGebra gave almost the same: h =
0.7225 m0.2159
This is not just a small difference, like a ‘rounding error’ or so. This is obviously shocking and
Is it possible that nobody ever noticed this? Or that nobody cared?Well, after a long search, I
found some people who made the same observation, but I couldn’t find any textbooks or
software manuals mentioning this problem.
I experimented with other data and other invertible functions. The same happens every time.
Body Mass Index recalculated
Solution: multidirectional regression
The classical regression, minimizing the squares of the vertical distances (or residues ry,i, see
the graph below), seems to pull the line through a cloud of points too much horizontally.
Minimizing the horizontal distances rx,i (i.e. by switching x and y) pulls the line too much
Because of symmetry reasons, there is no reason to favor one of both if f is a bijection.
Therefore it seems only logical to give the two ‘pulling forces’ equal rights, and minimize this
Sy f(x ) x f (y )
i i
i 1
I implemented this in a Windows program called ‘FittingKVdm, version 1.0’, and I wouldcall it
‘multidirectional regression’, or shorter ‘xy-fitting’, abbreviated ‘MDLS’ (as opposed to ‘OLS’=
‘Ordinary Least Squares’). It can be expanded in multiple directions of course, if there are
more variables.
Fitting the same data now, yielded:
m=ahb, with a=10.482±0.058*, b=3.1581±0.0092*.
That exponent b is a lot closer to 3, as we physicists always expected! And 3.1581 is ap-
proximately equal to the geometric mean of 2.1084 and 4.67421, which makes sense.
Now again switching the variables, we obtain an exponent of 1/3.1581 as expected by the
(*The confidence intervals are estimated by doing 100 fittings with the data+random noise with
the same magnitude as the probable error on the measurements, i.e. the xi values are
replaced by xi+g(σx,i) and yi+g(σy,i), g being a Gauss distributed random number function.)
Body Mass Index recalculated
Confirmation: relationship with body fat percentage
The same analysis was done for men and women (aged 16 and more) with different fat
percentages, first with parameters a and b floating, thenwith a fixed value of b=2, so in that
case ‘a’ represents the classic ‘BMI’, and then with bfixed to 3, so in this case ‘a’ represents
the ‘BMI with h³’ aka ‘CI’.
Men a and b floating b=2 fixed b=3 fixed
fat % (f) n a b a=‘BMI’ a= ‘BMI3’=‘CI’
[10, 12[
[12, 14[
[14, 16[
[16, 18[
[18, 20[
[20, 22[
[22, 24[
[24, 26[
[26, 28[
[28, 30[
[30, 32[
[32, 34[
[34, 36[
[36, 38[
[38, 40[
[40, 42[
[42, 44[
[44, 46[
[46, 48[
[48, 50[
19.318 ± 0.017
20.5621 ± 0.0051
21.2382 ± 0.0043
21.4028 ± 0.0033
22.3881 ± 0.0038
23.2707 ± 0.0035
24.5169 ± 0.0030
25.8724 ± 0.0029
26.6912 ± 0.0022
28.0654 ± 0.0027
29.3692 ± 0.0035
30.8169 ± 0.0031
32.9015 ± 0.0041
34.6159 ± 0.0046
38.3387 ± 0.0063
39.8371 ± 0.0084
44.184 ± 0.014
43.407 ± 0.024
46.445 ± 0.032
43.711 ± 0.054
10.921 ± 0.012
11.4527 ± 0.0036
11.9307 ± 0.0029
12.0430 ± 0.0022
12.5053 ± 0.0022
13.0892 ± 0.0022
13.9356 ± 0.0022
14.5751 ± 0.0022
15.0336 ± 0.0019
15.8937 ± 0.0019
16.6075 ± 0.0021
17.4054 ± 0.0024
18.4909 ± 0.0030
19.5014 ± 0.0035
21.5662 ± 0.0041
22.4331 ± 0.0067
24.8267 ± 0.0099
25.308 ± 0.021
27.236 ± 0.024
24.836 ± 0.049
When a and b are left free to be adjusted, the fitting is not very stable. We can presume that
this is because many other factors besides the body fat percentage play a role, like the body
type, muscle weight etc. The measurement points form clouds rather than precise lines.
Anyhow, the exponents are almost always more than 3. The average is even 3.668, and the
standard deviation 0.579. With fixed exponents, we see a very nice correlation of a versus the
fat percentage. The ‘goodness of fit’ (indicated by the χ² per degree of freedom, but also by the
estimated errors on the fitted parameters) was always better with b=3 than with b=2.
Body Mass Index recalculated
Women a and b floating b=2 fixed b=3 fixed
fat % (f) n a b a=‘BMI’ a= ‘BMI3’=‘CI’
[16, 18[
[18, 20[
[20, 22[
[22, 24[
[24, 26[
[26, 28[
[28, 30[
[30, 32[
[32, 34[
[34, 36[
[36, 38[
[38, 40[
[40, 42[
[42, 44[
[44, 46[
[46, 48[
[48, 50[
[50, 52[
[52, 54[
[54, 56[
[56, 58[
17.209 ± 0.031
18.000 ± 0.021
18.834 ± 0.011
18.869 ± 0.092
19.8013 ± 0.0077
20.4095 ± 0.0055
21.1685 ± 0.0054
21.6030 ± 0.0039
23.1021 ± 0.0046
24.5112 ± 0.0042
25.5553 ± 0.0031
26.9707 ± 0.0030
28.7631 ± 0.0030
30.6793 ± 0.0030
32.4786 ± 0.0032
34.6952 ± 0.0039
38.0436 ± 0.0049
41.1029 ± 0.0074
46.046 ± 0.011
47.842 ± 0.027
52.369 ± 0.048
10.927 ± 0.023
11.302 ± 0.015
11.5395 ± 0.0076
11.5032 ± 0.0072
12.0853 ± 0.0054
12.3724 ± 0.0037
12.9417 ± 0.0040
13.2474 ± 0.0027
13.9777 ± 0.0030
14.8582 ± 0.0028
15.6460 ± 0.0028
16.5106 ± 0.0025
17.6573 ± 0.0022
18.8671 ± 0.0023
19.8652 ± 0.0029
21.3284 ± 0.0034
23.2991 ± 0.0038
25.2133 ± 0.0057
28.013 ± 0.011
29.441 ± 0.020
32.142 ± 0.036
We can make the same remarks for the women. The average b is here even more: 3.859, and
the standard deviation 0.556.
If the body mass is better correlated with h³ than with h², as it was found, we would also
expect that the body fat percentage (f) is better correlated with the CI (=BMI3) than with the
classic BMI.
The graphs below show CI vs f for men (from the table above). The relationship is clearly not
linear, but it seems to follow an exponential pattern. The dotted lines are ‘worst case
scenarios', with parameters at the limits of their confidence intervals.
Body Mass Index recalculated
The curve through the data points is the best fitting exponential function B=b af+c, with a, b and
c fitted parameters and B=‘classic BMI’ or ‘CI’ (CI in the above graph). The graphs are similar
for men and women, for BMI and CI, but with different parameters ofcourse. They are listed
Corpulence Index vs Body Fat % (men 16 and older)
Corpulence Index vs Body Fat % (women 16 and older)
Body Mass Index recalculated
men aged 16 or more (n=8039) women aged 16 or more (n=7475)
B=classic BMI B=BMI3 = CI B=classic BMI B=BMI3 = CI
a 1.0424±0.0011 1.0433±0.0011 1.0671±0.0019 1.0687±0.0019
b 5.08±0.26 2.77±0.14 0.97±0.081 0.564±0.050
c 11.42±0.37 6.48±0.21 14.7±0.31 9.15±0.16
χ²x per degree of
1.79432 1.5959 0.581528 0.444813
χ²y per degree of
11953.5 6344.76 3654.17 2086.1
By comparing the χ² values (in both x and y directions), we see that the body fat percentage (f)
is better correlated with the CI than with the BMI.
We see that the Corpulence Index can be estimated from the body fat % (f), using:
‘CI’ = 2.77 1.0433f + 6.43 (men)
‘CI’ = 0.564 1.0687f + 9.15 (women)
I hope I have awakened your interest and you will be curious to test multidirectional
regression with your own data. Use it whenever the dependent and independent variables can
be switched. The necessary software ‘FittingKVdm’ can be downloaded from:
www.lerenisplezant.be/fitting.htm and a 25 day free trial is possible. More examples of the
method and some practical thoughts can be found in another text [Van de moortel, April 2021].
I also hope it has become clear now that there is no reason anymore to use the classic BMI.
The ‘BMI’ with h³, aka Corpulence Index should logically be a better estimator for all kinds of
health issues. Maybe the theoretical exponent of h should even be bigger than 3, as suggested
by the empiric evidence. Also the reason why the relationship betweenbody fat and the CI
seems to be exponential, is a puzzle to be solved by biologists. Maybe they already did, but my
main point here is to remark that the correlation is better with m/h³ than with m/h².
All your remarks and suggestions are most welcome, of course.
Body Mass Index recalculated
Babar, Sultan (March 2015): “Evaluating the Performance of 4 Indices in Determining
Adiposity”. Clinical Journal of Sport Medicine, Lippincott Williams & Wilkins, 25 (2): 183.
Rohrer, Fritz (1921): “Der Index der Körperfülle als Maß des Ernährungszustandes”. Münchner
Med. WSCHR. 68: 580–582.
Trefethen, Nick (2013) on his own website:people.maths.ox.ac.uk/trefethen/bmi.html
Van de moortel, Koen (February2021): “Non-linear regression - Why you shouldn’t take the
logarithms of your variables”, DOI: 10.13140/RG.2.2.18442.80324
Van de moortel, Koen (April 2021): “Multidirectional regression analysis”, DOI:
Also published here:
Conflict of interest:
The author declares no conflict of interest.
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What Gauge Wire for 30 AMP 220v - The Engineering KnowledgeWhat Gauge Wire for 30 AMP 220v - The Engineering Knowledge
Here we will discuss What Gauge Wire for 30 AMP 220v. The 8 AWG copper or 8 AWG aluminum wire is best to use for 30 amp service. The ampacity value for 8 AWg wire is 50 amps at 75 centigrade and 55
at 90 degrees. the aluminum 8 AWG wire has an ampacity of about 75 degrees and 45 amps at 90 degrees. Many resources suggested that 10 AWG copper wire is best to use for a 30 amp circuit breaker.
However it is not correct since it follows the 80 percent NEC rule. So best to minimize the effect of overheating use use 8 AWG copper and 8 AWG aluminum wire for a 30 Amp breaker wire size. Let’s
get started with What Gauge Wire for 30 AMP 220v.
What Size Wire For 30 Amps?
• To get the accurate wire size for 30 we must follow the NEC parameter. We cannot use any wire size without getting proper knowledge of it as the highest loading for any circuit is 80 percent of
the rating of the circuit for wire ampacity according to the NEC 80 percent rule.
• According to 80 % percent 30 amps have to show 80 percent of the ampacity of wire here, The process explains
30 Amp Wire = 30A *100% / 80% = 37.5A Ampacity
• So for a 30 amp break use wire that can easily handle about 37.5 amps. 8AWG wire has 50A ampacity so best to use it for 30 amp service.
30 amp wire size chart
The factor for Installing a 30 Amp Circuit Breaker
• Follow these points for a 30-amp circuit.
• The maximum length of 150 feet must be used when you are using 10 gauge wire size.
• It is best to use 8 gauge wire size if lengths larger than 10 gauge wire to fulfill the current need of load.
• So use 80 percent of the circuit breaker capacity to avoid overheating or any other problem. The circuit becomes overloaded when the load connected is higher than 80 percent of the breaker rating
when devices are needed for continuous loads.
What Happens if Wrong Length and Wire Size for 30 Amp Circuit?
• If the wire size is less than the required size that can affect the circuit.
• If there is overloading in the circuit, the circuit breaker connected will not easily detect the overloading and casues the damaging of wires. Damaged wire can get fired and burn the circuit.
• But if thick wire is used it will not affect the circuit. for this wire, the circuit breaker can detect overloading conditions with short circuits and any other faults in the circuit. So thick
wire is suggested for use
• The main disadvantage of 8-gauge wire for 30 Amp 220VÂ is its high prices for required wire sizes.
• For 10 gauge wire, about twenty percent extra cost is needed with the costs based on wire type and brand.
What is the maximum distance of 10 gauge wire for 30 amps?
• 10 gauge wire comes with a length of 128 feet for using the 30 amp breaker. it also operates well for the 100-foot range. This wire will not handle the same load if the distance is increased. If
there is a need to pass larger curent use a big wire. That fan easily handles larger curent but its prices will be higher.
• Small gauge wire faces the strain if they are connected with high loads.
• Diameter is also the main factor in running curent from one point to another. Dia must be according to wire requirements. 4 AGW wire is best to use for 30 Amps running to 200 feet. Copper wire is
best to use for longer distances than aluminum.
Can 12/2 wire handle 30 amps?
Normally 12 gauge wire easily handles about 20 amps. But if there is a need to run appliances for a distance of more than 150 feet, use high gauge wire. Such as 12-2 wire is best for 20 amps load.
And can be used for a 20-amp circuit breaker. The gauge of any wire is based on the diameter of the conductor. 12 gauge wire is thicker than 14 gauge wire with a large dia. 12-2 wire is best to use
for water heaters.
What Happens If You use the wrong size wire and length for a 30-Amp Circuit?
• It can get overheated and melt wire if dia of write is small for the current that passes through it.
• Due to a small sire voltage drop occurring, less voltage will received at the load end and connected devices will not operate well.
• If the wire size is tiny, the breaker will also get a trip so the wire is small for connected load.
• Connected load can also damaged if the wire size is small such as electrical components
• Using the wrong wire size is also a violation of codes that can cause penalties and legal repercussions.
What size wire is needed for 30 amp?
• The wire size is based on the current passing through the circuit. Such as there is a need for a circuit that uses 30 amps curent but has a single outlet. 10 AWg wire can be used for 30-amp
circuits. Thick wire can be used for good curent supply. the 10 gauge wire is best to use for a short distance of 30 amps and for a larger distance use thick wire.
• In-home wiring must use thick conductors to save from any damage to the circuit.
• Normally 8 gauge wires will be best to use for 30 amps bu best to use 10 gauge. Less wire size can damage the circuit
What size wire is needed for 30 amp 240 volts Circuit?
The American wire gauge system defines the wire size and distance in reverse relation to the diameter of the wire. For this, we can use wire of thin size and small number to show larger wire.
Normally used wire gauges 6, 8, 12, and 14 are based on curent that can wire carry.
The use of copper or aluminum wire is best in home wiring. Copper wire is thick so preferred to aluminum. the accurate wire size for a 30 Amp 220V circuit is 10 gauge. This wire comes with a diameter
of 2.6mm but it is not best to use for longer distances. Before using any wire make sure that it is according to the requirements of the circuit.
What Wire is needed for 100 Feet of a 30-Amp Breaker?
The wire size to run the 30 amp circuit for a distance of more than 100 feet is based on wire type, circuit voltage, and temperature. Normally wire size for a 30 amp circuit is copper wire in the 10
AWG size. For distances 100 feet or longer use a larger dia since voltage losses can occur. Normmlay8 AWg wire or larger gauge ire used to handle the voltage loss for distances of more than 100 feet.
What size wire for a 30 amp dryer?
The design of the dryer wiring is such that it can handle a minimum of 30 amps. New dryers can be wired in older wiring with 3 wire outlets. In new models of dryers, there is a need to use a copper
grounding strap that is connected to a green grounding screw on the dryer chassis. Dryer cors has three wires red, green, and neutral. The wires in the dryer plug connect with external power terminal
blocks. If there are four four-wire outlets use a dryer with 4 wires.
│Dryer Amperage │Wire Gauge │
│30-amp │10 AWG copper wire │
│40-amp │8 AWG copper wire │
│50-amp │6 AWG copper wire │
How many outlets can be on a 30 amp circuit?
• It is based on circuit demand, normally thirty receptacles are used on 30 amps. But in some circuits value can be higher. For more than 30 outlets use double pole receptacles that come in single
and duplex configurations.
• The standard 30A breaker can support 20 receptacles in a similar circuit. For more than 20 outlets on similar circuit receptacles can be overheated.
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What gauge wire for 50 amp 220V
For a 50 amp circuit, 6 AWG (American Wire Gauge) wire size is typically recommended.
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10 AWG wire is rated for 30A peak / 24A and can run for 50 ft.
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For 30 amps, you’ll need a 10 gauge wire
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8 AWG. is the recommended wire size for 40 amp load
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25 amp breaker with #10 AWG wire
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10AWG is best for 30 AMP Wire Size
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│4 AWG Copper │85 │100 │
│2 AWG Aluminum │90 │100 │
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Amperage Gauge
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6 AWG wire best for 50 AMP
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│AWG 3 │85 Amps │100 Amps │
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For a 60-amp circuit, a 4 AWG wire is best to use
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12-gauge or 10-gauge wire best to use for 20 amps
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The 2.5mm2 electric wire can carry 20 Amp current, for  20℃, and lay without any conduit. For high-temperature values, 2.5mm2 wire can handle 18 Amp electric current.
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For measuring the wire size first find the of wire using i vernier caliper or a micrometer, and then find the section. The wire with a diameter of 1.76 is 1.76/2 × 3.14 = 2.76 square == 2.5 square,
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6mm wire is best to use for 32 amps
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Diffusion-Weighted MRI for Treatment Response Assessment in Osteoblastic Metastases—A Repeatability Study
Institute of Biostatistics and Clinical Research, University of Münster, 48149 Münster, Germany
Department of Nuclear Medicine, University Hospital Münster, 48149 Münster, Germany
Clinic for Radiology, University Hospital Münster, 48149 Münster, Germany
Authors to whom correspondence should be addressed.
These authors contributed equally to this work.
Submission received: 19 June 2023 / Revised: 19 July 2023 / Accepted: 21 July 2023 / Published: 25 July 2023
Simple Summary
Patients with many advanced cancers develop osteoblastic bone metastases that cannot be assessed by conventional imaging. There is an unmet need for a quantitative imaging technique that can assess
the treatment response of osteoblastic metastases to further improve treatment of these patients. This article examines the difference in apparent diffusion coefficient (ADC) values found between
viable and nonviable metastases in relation to the variability of repeated measurements as a basis for the potential use of diffusion-weighted MRI (DWI) for treatment response assessment. DWI is
based on observing the movement of water molecules, which is often restricted in tumor tissue and is quantified using the ADC. It is shown that viable and nonviable metastases differ significantly in
ADC value and that these differences are considerably higher than the variability of repeated measurements. This shows that DWI meets the basic technical requirements for reliable treatment response
assessment of osteoblastic metastases.
The apparent diffusion coefficient (ADC) is a candidate marker of treatment response in osteoblastic metastases that are not evaluable by morphologic imaging. However, it is unclear whether the ADC
meets the basic requirement for reliable treatment response evaluation, namely a low variance of repeated measurements in relation to the differences found between viable and nonviable metastases.
The present study addresses this question by analyzing repeated in vivo ADC[median] measurements of 65 osteoblastic metastases in nine patients, as well as phantom measurements. PSMA-PET served as a
surrogate for bone metastasis viability. Measures quantifying repeatability were calculated and differences in mean ADC values according to PSMA-PET status were examined. The relative repeatability
coefficient %RC of ADC[median] measurements was 5.8% and 12.9% for phantom and in vivo measurements, respectively. ADC[median] values of bone metastases ranged from $595 × 10 − 6 mm 2 / s$ to $2090 ×
10 − 6 mm 2 / s$ with an average of 63% higher values in nonviable metastases compared with viable metastases (p < 0.001). ADC shows a small repeatability coefficient in relation to the difference in
ADC values between viable and nonviable metastases. Therefore, ADC measurements fulfill the technical prerequisite for reliable treatment response evaluation in osteoblastic metastases.
1. Introduction
The Response Evaluation Criteria In Solid Tumors (RECIST) guidelines are widely accepted for measuring tumor response in solid tumor patients and are used by regulatory agencies for drug approval [
]. The guidelines are based on morphologic imaging, measuring the size of target lesions and assessing changes in size over time. However, they have limitations in evaluating bone metastases, which
can be osteolytic, osteoblastic, or radio-occult [
]. Osteolytic metastases are characterized by bone destruction and are measurable if they have a soft tissue component of at least 1 cm, while osteoblastic metastases, characterized by increased bone
formation, are generally defined as not measurable [
]. Bone scintigraphy, a molecular imaging technique, also has limitations in distinguishing treatment response from progression [
]. One of these limitations is that progression can only be ascertained with delay [
]. This is particularly problematic in prostate cancer and breast cancer, with prostate cancer being the second most diagnosed cancer in men and breast cancer being the most common cancer in women [
]. Up to 70% of patients with breast and prostate cancer have evidence of metastatic bone disease in advanced stages. Bone metastases of prostate cancer are almost exclusively osteoblastic and
therefore virtually always not measurable within the RECIST framework, while breast cancer often presents with mixed osteolytic and osteoblastic metastases, often rendering these metastases
unmeasurable as well [
]. The inability of current imaging techniques to recognize treatment response or progression in a timely manner can potentially delay treatment switching when disease burden is still relatively low.
Therefore, new imaging techniques are needed to provide a reliable assessment of treatment response in bone metastases, to improve the drug approval process and better serve the patient population
suffering from these common forms of cancer.
An imaging technique which has shown promise in treatment response assessment in osteoblastic metastases is diffusion-weighted MRI (DWI), which is based on the Brownian motion of water and can be
quantified by the apparent diffusion coefficient (ADC). The ADC is inversely correlated with cellularity [
]. The treatment response of osteoblastic metastases, resulting in the loss of cell membrane integrity and cellular density, has been shown to result in an increase in the ADC in preclinical mouse
models, suggesting its potential clinical utility [
]. The promising results obtained in preclinical models could be replicated by first studies on humans [
], albeit not by all [
It is crucial to be aware that ADC measurements are subject to random measurement variability, as with all quantitative biomarkers. A change in ADC value in longitudinal studies may not necessarily
indicate a real change but rather be a result of this random variability. Therefore, before using a quantitative biomarker such as ADC in longitudinal studies, it is essential to assess its
measurement repeatability through test–retest studies. Measurement repeatability can be quantified by the repeatability coefficient (RC) or the within-subject coefficient of variation (wCV), which
are used in longitudinal studies to determine if a measured difference represents real change or is within the range of expected random measurement variability.
Furthermore, it is important to consider the degree of repeatability in relation to the changes that occur with treatment response or progression. A quantitative marker like the ADC has diagnostic
potential only when the differences between the measurements of viable and nonviable metastases are significantly greater than the measurement uncertainty.
The range of ADC values that can be expected in the spectrum from highly viable to nonviable metastases is difficult to determine, since conventional morphologic imaging cannot determine viability of
metastases and routine biopsies are not indicated. For our study, we intend to use longitudinal PSMA-PET as a surrogate for viability of bone metastases. PSMA-PET uses radiotracers specifically
targeting the prostate-specific membrane antigen (PSMA), a surface protein highly overexpressed by most prostate cancer cells, allowing for highly sensitive and specific detection of prostate cancer
manifestations. Recent studies indicate that PSMA-PET is not only highly sensitive in the detection of prostate cancer lesions but that change in PSMA uptake under therapy can serve as a marker for
treatment response [
So far, there has been only one study investigating the repeatability of ADC measurements of osteoblastic metastases, which has not been corroborated so far. Also, currently, it is unclear what range
of ADC values can be expected in viable and nonviable metastases. Therefore, our study aims to further the understanding of ADC measurement repeatability of osteoblastic metastases and to determine
how it relates to the range of ADC values seen in viable and nonviable metastases, as determined by PSMA-PET.
2. Materials and Methods
2.1. Study Design, Patients and Imaging Protocol
Nine men with prostate cancer and known bone metastases, presenting for clinically indicated PSMA-PET between June 2022 and January 2023 at the University Hospital Münster, were included. Patient
characteristics are shown in
Table 1
The study was approved by the local ethics committee (2021-825-f-S) and performed in accordance with the 1964 Declaration of Helsinki and its amendments. Written informed consent was obtained from
all patients.
First, patients were injected with 3 MBq/kg body weight [
F]PSMA-1007 for clinically indicated PSMA-PET. One hour after PET tracer injection, two repeated T1w and DWI MRI measurements were conducted on a Biograph mMR PET-MR-System (Siemens, Erlangen,
Germany) to determine in vivo repeatability of DWI measurements. There was no concurrent PET scan at this time. Diffusion acquisition was performed using a Spin-Echo-EPI Sequence (FOV 380 × 275 mm²,
21 axial slices, voxel size 2 × 2 × 5 mm³, TE 86 ms, TR 6400 ms, fat suppression SPAIR, 1 × b = 50 s/mm², 3 × 400 s/mm², 3 × 800 s/mm²). ADC-maps were automatically calculated with the standard
settings provided by the vendor. For T1, an axial volumetric interpolated breath hold examination (VIBE) sequence was used (FOV: 420 × 342, acquisition matrix: 320 × 224, slice thickness 5 mm, TR
1.96 ms, TR 4.07, no fat suppression). Between the two MRI measurements, the patients were moved out of the MRI, repositioned and moved back in again. The area covered by repeated DWI measurement in
each patient can be found in
Table 1
Subsequently, all patients underwent clinically indicated PSMA-PET/CT or PET/MRI scans from skull to tibia two hours after tracer injection. The patients were asked to void their bladder before the
PET scan. PET image reconstruction was performed with onboard software using OSEM with 21 subsets and 3 iterations.
Five patients were scanned on the integrated 3-Tesla Biograph PET-MRI, also used for repeated DWI measurements; the other four patients were scanned on a PET/CT System (Biograph mCT, Siemens,
Erlangen, Germany). Four of the nine patients had previously undergone PSMA-PET.
2.2. Phantom Measurements
MRI-phantom measurements were performed on a commercially available DWI phantom (
, accessed on 12 June 2023) with an MR-readable thermometer to adjust target diffusion values for temperature. Six measurement runs with five repetitions each were performed under identical
conditions, using the same MRI sequences and flex body coil as for the patient study. Phantom measurements were evaluated using MIM
Version 7.2.4 (MIM Software Inc., Cleveland, OH, USA).
2.3. Image Analysis
To identify bone lesions, b400 DWI images were reviewed for areas of bone marrow hyperintensity relative to background marrow with corresponding hypointensity on T1w [
]. Additionally, MR images were correlated to PSMA-PET. Same-day and previous PSMA-PET served as a surrogate for viability of bone metastases, with metastases showing strong uptake considered viable
and those currently showing no uptake but having had strong uptake in pretreatment PET considered nonviable. The osteoblastic nature of metastases was visually confirmed on CT scans of same-day PET/
CT in four patients, previous CT in four patients and on radiographs of the spine and pelvis in one patient. Volumes of interest (VOIs) were delineated by a board-certified nuclear medicine
specialist on the ADC maps using MINT-Lesion (Mint Medical, Dossenheim, Germany). b50, b400, b800, T1w and PSMA-PET images were used to check for plausibility of the contours. In cases of diffuse
disease, the VOI was drawn around the entire vertebral body or pelvic bone [
]. The mean and median ADC values (ADC
and ADC
) within the VOI were measured. The maximum standardized uptake value (
$S U V m a x$
) was measured using a VOI on PET images. For lesions showing no tracer accumulation above background, a VOI was drawn in correlation to DWI signal alterations and alterations on T1w images or CT.
2.4. Statistical Analysis
Normally distributed data are described using mean ± standard deviation (SD) and non-normally distributed data using median and interquartile range (IQR, 25th–75th percentile). Data were checked for
normality using histograms. Association of quantitative variables was analyzed using Spearman’s correlation coefficient ($r S p e a r m a n$) or linear regression.
In the phantom measurements, no correlation within the measurement runs was observed. Therefore, all measurements are pooled by concentration. To enable comparability with the in vivo measurements,
results are reported as estimates of standard deviation (SD) and repeatability coefficient (
$R C = 2.77 × S D$
), as well as coefficient of variation (CV) [
] and relative repeatability coefficient %RC, defined as
$2.77 × C V$
For analysis of the repeatability of the in vivo measurements, the lesions are treated as independent subjects, as differences between repeated measurements of lesions showed no relevant correlation
within patients. The within-subject standard deviation (wSD) and the resulting repeatability coefficient, as well as the within-subject coefficient of variation (wCV) and the corresponding %RC, are
estimated [
The agreement of repeated measurements is visualized using the methods of Bland and Altman [
]. Agreement is further quantified using an intraclass correlation coefficient (ICC) based on a one-way random effects model [
]. The repeatability of different regions or classes of lesion size is compared using the Kruskal–Wallis test and the Wilcoxon rank-sum test, respectively. The repeatability of ADC
and ADC
is compared with the Wilcoxon signed-rank test. Two-sided
-values are reported for all tests.
In contrast to the differences between repeated measurements, the magnitude of the ADC measurements shows relevant intraindividual correlation within patients. Therefore, the comparison of ADC by
PSMA status was performed using a linear mixed model, including a random intercept for both patient and lesion [
]. Since the ADC values were log-transformed for this analysis, the multiplicative factor by which the groups differ regarding their geometric mean is estimated. Model assumptions were assessed using
residual plots.
A classical power analysis was not performed. The sample size of 9 patients resulted in 65 lesions that could be included in the analyses. The approach of Danzer et al. [
] allows to determine the effective specificity resulting from a repeatability study as a function of the sample size. The calculation is based on the RC commonly used in the literature for a target
specificity of 95% (i.e.,
$R C = 2.77 × w S D$
). For 65 lesions that are measured twice, the mean effective specificity is 94.6%. An effective specificity of at least 90% is achieved with a probability of 96.6%.
R version 4.3.0 was used for the analyses [
3. Results
3.1. Phantom Measurements
All phantom measurements were conducted at 22 °C, as indicated by the built-in MR-readable thermometer. For ADC[mean], the mean deviation from the respective target values over all vials was +$23.6 ×
10 − 6 mm 2 / s$ and the mean absolute deviation $27.5 × 10 − 6 mm 2 / s$, with a standard deviation of $17.8 × 10 − 6 mm 2 / s$. The repeatability coefficient averaged over all vials for ADC[mean]
was $54.3 × 10 − 6 mm 2 / s$ and the %RC was 5.83%.
For ADC[median], the mean deviation from the respective target values over all vials was +$23.9 × 10 − 6 mm 2 / s$ and the mean absolute deviation $27.8 × 10 − 6 mm 2 / s$, with a standard deviation
of $17.8 × 10 − 6 mm 2 / s$. The repeatability coefficient over all vials for ADC[median] was $54 × 10 − 6 mm 2 / s$ and the %RC was 5.81%.
As shown in
Figure 1
, a positive correlation between the ADC target value and the RC was found (
-value of regression slope for target value
$p = 0.001$
$p = 0.0003$
for ADC
and ADC
, respectively).
It has been suggested in the literature that the coefficient of variation (CV) and the deducted %RC should be used in the case of a positive correlation between the magnitude of the measurement and
the measurement error [
]. Unlike RC, which uses absolute values, %RC expresses the repeatability in proportion to the magnitude of the measurement. However, using %RC in the case of the phantom measurements overcompensated
for the positive correlation found between the ADC target value and RC, resulting in a negative correlation between the ADC target value and %RC (
Figure 1
-value of regression slope for target value
$p = 0.01$
$p = 0.01$
$A D C m e a n$
$A D C m e d i a n$
, respectively).
Measurement deviations from the respective ADC target values and further results on repeatability stratified by ADC target value can be found in
Table A1
Table A2
, and
Table A3
3.2. Bone Metastases Characteristics
Overall, 87 bone metastases were identified in nine patients. Of those, 14 could be identified in PSMA-PET but not unequivocally on DWI images or ADC maps. Four metastases that could be identified on
both PSMA-PET and DWI images had to be excluded from ADC measurements due to movement artifacts. Another 4 metastases had to be excluded because the metastatic nature could not be proven, leaving 65
metastases suitable for ADC measurements.
Six metastases that could be clearly identified on DWI showed only faint uptake on PSMA-PET. Nine metastases could be identified on DWI but showed no uptake on PSMA-PET above background. Of the
metastases without discernible tracer accumulation, all showed uptake on previous PSMA-PET examinations (
Figure 2
). There were 9 metastases in the thoracic spine, 15 in the lumbar spine and 41 in the pelvic region. The detailed location of single metastases can be found in
Table A4
. The median short axis of the evaluable metastases was 17 mm [IQR: 12–23], and the median long axis was 32 mm [IQR: 19–52]. The median volume of the metastases, as measured on the ADC maps, was 3.1
[IQR: 1.1–12.8]. The median
$S U V m a x$
was 12.1 [IQR: 6.6–17.7].
3.3. Repeatability of ADC Measurements in Bone Metastases
The within-subject standard deviation (wSD) for ADC
and ADC
$60 × 10 − 6 mm 2 / s$
$51 × 10 − 6 mm 2 / s$
, respectively (
Table 2
). Consequently, the repeatability coefficients (RC) for ADC
and ADC
were determined to be
$166 × 10 − 6 mm 2 / s$
$141 × 10 − 6 mm 2 / s$
, respectively. Furthermore, the within-subject coefficient of variation (wCV) for ADC
and ADC
was 5.5% and 4.7%, respectively.
With the lower wSD and wCV, the repeatability of the in vivo measurements was better for ADC[median] compared with ADC[mean] (p = 0.04 for comparison of the SD of repeated measurements of ADC[median]
and ADC[mean]). For this reason, the following analyses were conducted using the ADC[median].
The ICC of
(95% CI [0.972–0.990]) indicated an excellent agreement of the two ADC
measurements. In the Bland–Altman analysis, the mean difference between the two repeated measurements was
$3.09 × 10 − 6 mm 2 / s$
, with limits of agreement of
$± 141.52 × 10 − 6 mm 2 / s$
. The Bland-Altman analysis showed that the difference between repeated measurements increased with the height of the measured value (correlation between mean and absolute difference
$r S p e a r m a n = 0.31 , p = 0.01$
Figure A1
). Expressing the differences as a percentage of their average [
] removed the relationship between repeatability and size of the measurement (
$r S p e a r m a n = 0.008 , p = 0.95$
Figure A1
). The distance of the limits of agreement to the mean difference of
$− 0.32 %$
$± 12.60 %$
(95% CI [9.86–15.33], shown in
Figure 3
), which is very close to the %RC reported in
Table 2
. Based on these results, wCV and %RC seem suitable to quantify the repeatability of the in vivo measurements.
To investigate whether the anatomical region had an impact on the repeatability of ADC measurements, we compared the standard deviations of repeated measurements in the thoracic spine, lumbar spine
and pelvic region. Among the anatomical regions investigated, the standard deviation of repeated ADC
measurements was the highest in the thoracic spine, followed by the lumbar spine, and it was the lowest in the pelvic region (
Figure 3
). However, the differences were not statistically significant (
$p = 0.18$
) with corresponding RCs of
, and
$116.3 × 10 − 6 mm 2 / s$
To investigate whether metastasis size had an impact on ADC measurement repeatability, we categorized the metastases into those with a volume of less or equal
$5 mm 3$
(n = 40) and those with a volume of more than
$5 mm 3$
(n = 25). Our analysis showed a difference (
$p = 0.007$
) in the standard deviation of repeated measurements in the dependence of lesion volume (
Figure 3
). The RC for lesions
$≤ 5 mm 3$
$170.4 × 10 − 6 mm 2 / s$
, and the RC for lesions
$> 5 mm 3$
$69.9 × 10 − 6 mm 2 / s$
3.4. ADC Range in Bone Metastases and Association with PSMA-PET Uptake
The mean value of repeated ADC
measurements ranged from
$595 × 10 − 6 mm 2 / s$
$2090 × 10 − 6 mm 2 / s$
Figure 4
). The lowest mean ADC
was observed in lesions with strong PSMA uptake with
$930 × 10 − 6 mm 2 / s$
. A markedly higher mean ADC
was found in lesions with only faint tracer uptake (
$1529 × 10 − 6 mm 2 / s$
) and with PET signal on background level (
$1683 × 10 − 6 mm 2 / s$
). According to the linear mixed-model analysis, the ADC
was on average
$64.1 %$
(95% CI [41.6–90.4],
$p < 0.001$
) and
$63.2 %$
(95% CI [44.6–84.8],
$p < 0.001$
) higher in lesions with faint tracer uptake and PET signal on background level, respectively.
The median
$S U V m a x$
of the metastases was
(IQR: 6.6–17.7). A marked decrease in
$S U V m a x$
with an increase in ADC
can be seen (
Figure 3
$r S p e a r m a n = − 0.72$
, 95% CI [−0.82–−0.58]).
4. Discussion
A crucial requirement for a quantitative treatment response biomarker is low variability between repeated measures, indicated by a small RC or %RC, relative to the differences observed between
pathological states and treatment response. This study assessed the repeatability of ADC measurements in osteoblastic bone metastases compared with the ADC values obtained from viable and nonviable
metastases using phantom and in vivo studies. The RC and %RC of ADC measurements were found to be small when compared with the difference observed between viable and nonviable metastases. Therefore,
ADC measurements meet the essential technical prerequisite for reliable treatment response evaluation in osteoblastic metastases.
Measurement repeatability of quantitative imaging biomarkers can be quantified using different metrics. It has been suggested that the within-subject standard deviation (wSD) and its derived
repeatability coefficient (RC) should be utilized when repeatability is independent of biomarker magnitude. Conversely, for biomarkers where measurement variability increases with larger values, the
within-subject coefficient of variation (wCV) and the relative repeatability coefficient (%RC) have been proposed as measures of repeatability [
]. To our knowledge, it has not been investigated so far whether the repeatability of ADC measurements depends on the magnitude of the measured values. We could demonstrate in our phantom and in vivo
study that the measurement error does indeed increase with the magnitude of the target value. Using measures of relative change to quantify repeatability allowed to handle this association for in
vivo measures and resulted in stable limits over the range of observed values (
Figure 3
Also, the quantification of ADC in a region or volume of interest can be performed using different measures. In the in vivo study, we observed a better repeatability for ADC[median] compared with ADC
[mean] ($p = 0.04$). In contrast, both measures were equally precise in the phantom study. The absence of a difference in the phantom study can be easily explained. Since the vials in the phantom
contain a homogenous fluid, every voxel should have the same ADC value under ideal conditions. Image noise and other artifacts should add a random, symmetrically distributed measurement error. Hence,
mathematically, ADC[mean] and ADC[median] should be identical in the phantom study under ideal conditions. In contrast, the situation is fundamentally different in vivo. Focal bone lesions are
surrounded by fatty bone marrow in older individuals, who represent the majority of patients with osteoblastic metastases. Due to the fat suppression techniques used in diffusion-weighted MRI, fatty
regions have a very low ADC, often zero. Hence, inclusion of surrounding fatty bone marrow into the volume of interest, be it due to imperfect segmentation or partial volume effect, will result in
significant outliers in the array of ADC values obtained for the measurement. We believe that the better measurement repeatability observed for ADC[median] in vivo is explained by the robustness of
the median, as a measure of the average, to outliers, unlike the arithmetic mean.
To date, only a limited number of studies have explored the repeatability of ADC measurements in bone, particularly in focal lesions exhibiting osteoblastic characteristics. The only other study that
we are aware of exploring the repeatability of ADC measurements in osteoblastic metastases is by Reischauer et al., investigating 34 pelvic bone metastases of prostate cancer in twelve men. Employing
a monoexponential fitting for ADC calculation, a wCV of 4.4% was reported for ADC
, closely aligning with the 4.7% found in our study [
]. Messiou et al. investigated the repeatability of ADC measurements in the pelvic bone marrow of nine healthy volunteers, with a repeated scan performed within a 7-day interval. Their estimated %RC
of 14.8% is well in line with our results. Their Bland–Altman limits of agreement of mean ADC of bone marrow in normal subjects, however, are much narrower for absolute measurements (
$2.0 ± 86 × 10 − 6 mm 2 / s$
), possibly due to a much smaller range of measured values [
]. Donners et al. assessed the value of multiparametric MRI to identify viable bone metastases for biopsies. In their sample of 43 lesions, they observed lower, though not statistically significant,
mean/median ADC in tumor-positive biopsies [
]. In contrast to the studies by Reischauer et al. [
], Messiou et al. [
] and Donners et al. [
], which employed regions of interest (ROI), we used volume of interests (VOIs). Going beyond previous studies, the validity of ADC measurements were ensured in our study by the use of a phantom
containing the complete range of measured values found in vivo.
As previously emphasized, it is crucial not only to ascertain the absolute values of repeatability but also to assess the relative magnitude of repeatability in comparison with the differences
observed in measurements between pathological states and treatment responses. In our study, we used PSMA-PET as a surrogate marker of bone lesion viability. Lesions with strong, focal tracer
accumulation above background were considered viable. Lesions that were detectable on DWI-MRI but did not show tracer uptake on concurrent PSMA-PET, but had shown focal tracer accumulation on
previous PSMA-PET prior to cancer treatment, were considered nonviable, i.e., showing treatment response. In our study, the lesions with focal tracer accumulation had a mean ADC
$930 × 10 − 6 mm 2 / s$
, and those with uptake on background level one of
$1683 × 10 − 6 mm 2 / s$
. In relative terms, the ADC
of metastases considered nonviable was
$63 %$
higher than in those considered viable. Accordingly, a negative correlation between the ADC
and SUV
of bone lesions could be shown. Our results corroborate the findings of Perez-Lopez et al., correlating ADC values to the detectability of cancer cells in 43 histologic samples of osteoblastic bone
metastases. Median ADC was significantly lower in biopsies with tumor cells versus nondetectable tumor cells (
$898 × 10 − 6 mm 2 / s$
$1617 × 10 − 6 mm 2 / s , p < 0.001 )$
. Tumor cellularity was inversely correlated with ADC (
< 0.001). In serial biopsies taken in three patients before and after treatment, changes in MRI parameters paralleled histological changes [
]. The same group also showed a correlation between change in median ADC of bone metastases and treatment response in metastasized prostate cancer [
We found a moderate inverse correlation between the ADC and tracer uptake of bone metastases in PSMA-PET, quantified by SUV[max]. To our knowledge, this is the first study investigating this
relationship. The imperfect correlation indicates that DWI and PSMA-PET could have a complementary value in treatment response assessment in prostate cancer metastases. Further studies should
investigate this possibility. Going beyond PSMA-PET, ADC quantification might allow for treatment response assessment in osteoblastic metastases of breast cancer and other cancers which do not
express PSMA.
Our study has limitations. For one, there were only a limited number number of metastases in the thoracic spine. Also, we used PSMA-PET uptake as a surrogate for viability, which is not a perfect
gold standard. Still, our results are in very close alignment to those of Perez-Lopez et al., who had histology available [
]. Moreover, interscanner variability was not examined in our study. However, our study establishes a groundwork for future investigations into interscanner variability, i.e., reproducibility. A
comprehensive analysis of repeatability serves as a fundamental step in assessing reproducibility, since repeatability limits the extent of agreement achievable when comparing different scanners [
5. Conclusions
In conclusion, ADC measurements demonstrate a favorable repeatability in relation to the differences found between viable and nonviable metastases. This fulfillment of essential metrological
prerequisites establishes a reliable foundation for assessing treatment response in osteoblastic metastases.
Author Contributions
Conceptualization, B.N., J.B. and D.G.; methodology, D.G., M.E., J.B. and B.N.; software, M.E., D.G. and B.N.; validation, M.E., P.R., A.R., Z.M., J.B., D.G. and B.N.; formal analysis, M.E., D.G. and
B.N.; investigation, P.R. and B.N.; resources, P.R. and B.N.; data curation, M.E., P.R. and B.N.; writing—original draft preparation, M.E. and B.N.; writing—review and editing, A.R., Z.M., J.B. and
D.G.; visualization, M.E. and B.N.; supervision, B.N. and D.G.; project administration, B.N. All authors have read and agreed to the published version of the manuscript.
There was no dedicated funding for this study. B.N. was supported as a Clinician Scientist by the Medical Faculty, University of Münster, Germany. We acknowledge support from the Open-Access
Publication Fund of the University of Münster.
Institutional Review Board Statement
The study was conducted in accordance with the Declaration of Helsinki and approved by the Ethics Committee of the Medical Association of Westphalia-Lippe and the University Münster (Protocol code:
2021-825-f-S, Date of approval: 23 February 2023).
Informed Consent Statement
Informed consent was obtained from all subjects involved in the study.
Data Availability Statement
The data are not publicly available because the informed consent signed by the patients did not provide for it.
The authors would like to thank Anne Exler and Stanislav Milachowski for their ongoing excellent technical support.
Conflicts of Interest
The authors declare no conflict of interest.
The following abbreviations are used in this manuscript:
ADC apparent diffusion coefficient
CI confidence interval
CT computed tomography
CV coefficient of variation
DWI diffusion-weighted (magnetic resonance) imaging
ICC intraclass correlation coefficient
IQR interquartile range
MRI magnetic resonance imaging
PET positron emission tomography
PSA prostate-specific antigen
PSMA prostate-specific membrane antigen
PVP polyvinylpyrrolidone
RC repeatability coefficient
RECIST Response Evaluation Criteria in Solid Tumors
SD standard deviation
SUV standardized uptake value
VOI volume of interest
wCV within-subject coefficient of variation
wSD within-subject standard deviation
Appendix A
Table A1. Deviation of ADC[mean] and ADC[median] measurements from the target value in the phantom stratified by ADC target values.
ADC[mean] ADC[median]
PVP Concentration Temperature-Adjusted Target Value Mean Deviation from Target Value Mean Absolute Deviation from Target Mean Deviation from Target Value Mean Absolute Deviation from Target
Value Value
% $10 − 6 mm 2 / s$ $10 − 6 mm 2 / s$ $10 − 6 mm 2 / s$ $10 − 6 mm 2 / s$ $10 − 6 mm 2 / s$
0 2106.00 33.81 36.28 (24.33) 33.58 36.07 (24.20)
10 1640.00 21.69 25.20 (17.88) 21.90 25.50 (17.78)
20 1258.00 3.50 18.54 (10.38) 3.45 18.38 (10.44)
30 886.00 16.80 19.09 (12.97) 17.08 19.52 (13.09)
40 545.00 31.46 32.17 (14.45) 32.38 32.68 (14.09)
50 293.00 29.13 29.13 (9.93) 30.27 30.27 (10.40)
All measurements were conducted at a temperature of 22 °C, as indicated by the built-in MR-readable thermometer. PVP: polyvinylpyrrolidone. Numbers in brackets are standard deviations.
Table A2. Deviation and repeatability of ADC[median] measurements in the phantom, stratified by ADC target values.
PVP Concentration Temperature-Adjusted Target Value Number of Measurements SD RC CV %RC
% $10 − 6 mm 2 / s$ $10 − 6 mm 2 / s$ $10 − 6 mm 2 / s$ % %
0 2106.00 6 × 5 × 3 27.6 76.5 1.3 3.6
(24.1–32.3) (66.7–89.6) (1.1–1.5) (3.1–4.2)
10 1640.00 6 × 5 × 2 22.1 61.4 1.3 3.7
(18.8–27.0) (52.0–74.8) (1.1–1.6) (3.1–4.5)
20 1258.00 6 × 5 × 2 21.0 58.2 1.7 4.6
(17.8–25.6) (49.3–71.0) (1.4–2.0) (3.9–5.7)
30 886.00 6 × 5 × 2 16.2 44.9 1.8 5.0
(13.7–19.7) (38.0–54.7) (1.5–2.2) (4.2–6.1)
40 545.00 6 × 5 × 2 14.8 41.0 2.6 7.1
(12.5–18.0) (34.7–50.0) (2.2–3.1) (6.0–8.7)
50 293.00 6 × 5 × 2 10.4 28.8 3.23 9.0
(8.8–12.7) (24.4–35.2) (2.7–4.0) (7.6–11.0)
All measurements were conducted at a temperature of 22 °C, as indicated by the built-in MR-readable thermometer. PVP: polyvinylpyrrolidone, SD: standard deviation, RC: repeatability coefficient, CV:
coefficient of variation. Number in brackets are 95% confidence intervals.
Table A3. Deviation and repeatability of ADC[mean] measurements in the phantom, stratified by ADC target values.
PVP Concentration Temperature-Adjusted Target Value Number of Measurements SD RC CV %RC
% $10 − 6 mm 2 / s$ $10 − 6 mm 2 / s$ $10 − 6 mm 2 / s$ % %
0 2106.00 6 × 5 × 3 27.7 76.8 1.3 3.6
(24.2–32.5) (66.9–90.0) (1.1–1.5) (3.1–4.2)
10 1640.00 6 × 5 × 2 22.1 61.1 1.3 3.7
(18.7–26.9) (51.9–74.6) (1.1–1.6) (3.1–4.5)
20 1258.00 6 × 5 × 2 21.1 58.5 1.7 4.7
(17.9–25.7) (49.6–71.3) (1.4–2.1) (3.9–5.7)
30 886.00 6 × 5 × 2 15.9 44.0 1.8 4.9
(13.5–19.4) (37.3–53.7) (1.5–2.2) (4.1–6.0)
40 545.00 6 × 5 × 2 16.0 44.3 2.8 7.7
(13.5–19.5) (37.5–54.0) (2.4–3.4) (6.5–9.4)
50 293.00 6 × 5 × 2 9.9 27.5 3.1 8.6
(8.4–12.2) (23.3–33.6) (2.6–3.8) (7.3–10.5)
All measurements were conducted at a temperature of 22 °C, as indicated by the built-in MR-readable thermometer. PVP: polyvinylpyrrolidone, SD: standard deviation, RC: repeatability coefficient, CV:
coefficient of variation. Number in brackets are 95% confidence intervals.
Thoracic Spine
(n = 9)
T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
Lumbar Spine
(n = 15)
L1 L2 L3 L4 L5
(n = 41)
right iliac bone left iliac bone sacral bone right pubic bone left pubic bone right ischium left ischium right femur left femur
Figure A1. (A) Mean of repeated ADC[median] measurements versus absolute difference of repeated ADC[median] measurements. The dashed line indicates the linear relationship between mean ADC[median]
and absolute deviation of the repeated measurements. Monotonic association between mean and absolute difference: $r S p e a r m a n = 0.31 , p = 0.01$. (B) Mean of repeated ADC[median] measurements
versus absolute difference of repeated ADC[median] measurements as a percentage of their mean. The dashed line indicates the linear relationship between mean ADC[median] and absolute deviation of the
repeated measurements. Monotonic association between mean and absolute difference: $r S p e a r m a n = 0.008 ,$$p = 0.95$.
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Figure 1. (A) Repeatability coefficient (RC) with corresponding 95% confidence intervals of ADC measurements as measured with the phantom (p-value of regression slope for target value $p = 0.001$ and
$p = 0.0003$ for $A D C m e a n$ and $A D C m e d i a n$, respectively). (B) %RC with corresponding 95% confidence intervals of ADC measurements as measured with the phantom (p-value of regression
slope for target value $p = 0.01$ and $p = 0.01$ for $A D C m e a n$ and $A D C m e d i a n$, respectively).
Figure 2. (A) PSMA-PET CT of the pelvis in a patient with widespread bone metastases of prostate cancer prior to initiation of PSMA therapy showing homogeneous PSMA uptake of the right iliac and
sacral bone. (B) The same patient one year later after three cycles of PSMA therapy. Only focal PSMA uptake in the sacral bone and multifocal uptake in the right iliac bone. Large areas of sclerotic
bone with perviously high PSMA uptake, now showing at most faint uptake, i.e., considered minimal/nonviable. (C) T1w MRI acquired immediately before PSMA-PET in B, showing widespread sclerosis in the
right iliac and scaral bone. Note that it is not possible to differentiate areas with high and low uptake in PSMA-PET. (D) ADC map of the same location. Note the excellent correlation with PSMA-PET.
Areas showing vivid uptake in PSMA-PET are depicted in dark gray, corresponding to ADC values around 1000. Areas with no or minimal uptake are depicted in light gray, corresponding to ADC values from
1300 to 1500.
Figure 3. (A) Standard deviation (SD) of repeated ADC[median] measurements by region. (B) SD of repeated ADC[median] measurements dependent on lesion volume. (C) Bland–Altman plot of repeated ADC
[median] measurements. The mean of the repeated measurements is plotted against the differences as a percentage of their mean. The distance of the limits of agreement to the mean difference of $−
0.32 %$ is $± 12.60 %$ (95% CI [9.86–15.33]). (D) Correlation of average ADC[median] measurement and SUV[max], $r S p e a r m a n = − 0.72$, 95% CI [−0.82–−0.58]).
Figure 4. Agreement of ADC[median] measurements and association with visually assessed uptake of corresponding metastases in PSMA-PET. The diagonal line indicates perfect agreement of the 1st and 2nd
measurement. ICC 0.983 (95% CI [0.972–0.990]). Note the small variation between repeated measurements compared with the range of values and the difference between metastases with strong PSMA uptake
and those with low tracer uptake. Compared with lesions with strong PSMA uptake, the ADC[median] is on average $64.1 %$ (95% CI [41.6–90.4], $p < 0.001$) and $63.2 %$ (95% CI [44.6–84.8], $p < 0.001$
) higher in lesions with faint tracer uptake and PET signal on background level, respectively, according to a linear mixed-model analysis.
ID Age PSA ^1 Pretreatments Area Covered by DWI
Years ng/mL Px RTx LHRH Abi Enza PSMA RTxB CTX Apa Thorax Lumbar Pelvis
1 77 0.34 x x x x x x x x x x x
2 79 110.00 x x x x x x x x x x
3 83 323.00 x x x x x x
4 73 3.87 x x x x x x
5 ^* 53 3.98 x x x x x
6 65 110.00 x x x x x x x
7 78 407.00 x x x x x x x x
8 67 3.02 x x x x x
9 74 1.49 x x x
^1 measured at time of scan. PSA = prostate-specific antigen, Px = prostatectomy, RTx = radiation therapy to prostate region, LHRH = treatment with LHRH agonists, Abi = abiraterone, Enza =
enzalutamide, PSMA = 177Lu-PSMA therapy, RTxB = radiation therapy of bone metastases, CTX = taxane-based chemotherapy, Apa = apalutamide, DWI = diffusion-weighted imaging. * Patient five was also
treated with Denusomab.
ADC[mean] ADC[median]
wSD ($10 − 6 mm 2 / s$) 59.95 (51.18–72.37) 50.71 (43.29–61.22)
RC ($10 − 6 mm 2 / s$) 166.18 (141.87–200.61) 140.56 (120.00–169.68)
wCV (%) 5.51 (4.57–6.69) 4.65 (3.85–5.66)
%RC (%) 15.27 (12.66–18.55) 12.90 (10.68–15.70)
wSD: within-subject standard deviation, RC: repeatability coefficient, wCV: within-subject coefficient of variation. Numbers in brackets are 95% confidence intervals.
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A simple theory for quantum quenches in
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A simple theory for quantum quenches in the ANNNI model
by Jacob H. Robertson, Riccardo Senese, Fabian H. L. Essler
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Authors (as registered SciPost users): Fabian Essler · Jacob Robertson · Riccardo Senese
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Preprint Link: scipost_202301_00025v1 (pdf)
Date submitted: 2023-01-18 15:08
Submitted by: Robertson, Jacob
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties: • Condensed Matter Physics - Theory
In a recent numerical study [1] it was shown that signatures of proximate quantum critical points can be observed at early and intermediate times after certain quantum quenches. Said work focused
mainly on the case of the axial next-nearest neighbour Ising (ANNNI) model. Here we construct a simple time-dependent mean-field theory that allows us to obtain a quantitatively accurate description
of these quenches at short times and a surprisingly good approximation to the thermalization dynamics at late times. Our approach provides a simple framework for understanding the reported numerical
results as well as fundamental limitations on detecting quantum critical points through quench dynamics. We moreover explain the origin of the peculiar oscillatory behaviour seen in various
observables as arising from the formation of a long-lived bound state. [1] doi:https://doi.org/10.1103/physrevx.11.031062.
Current status:
Has been resubmitted
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-3-15 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202301_00025v1, delivered 2023-03-15, doi: 10.21468/SciPost.Report.6905
This manuscript studies quantum quenches in the ANNNI model using a self-consistent time-dependent mean field theory. Specifically, the analysis presented here is consistent with a recent exact
numerical study [1] (citations [.] are from the manuscript) that show signatures of quantum critical points in certain observables at finite times. In addition, the analysis presented here points to
an explanation of persistent oscillatory behavior in certain observables.
The manuscript can be evaluated along two lines:
a) Insights regarding the accuracy of the self-consistent time-dependent mean field theory as introduced in Refs. [27-33] and applied here.
b) Insights regarding the finite time signatures in the quench dynamics first reported in [1].
1) In the abstract the authors state that the mean field approximation is quantitatively accurate at short times and even a good approximation for the thermalization dynamics at late time. Regarding
the late time thermalization dynamics this statement is missing an important point: A mean field approximation (time dependent or not) in a translation invariant model cannot lead to a thermal
distribution function of moment resolved observables like mode occupation numbers. The authors allude to this when they mention prethermal behavior above Eq. (25), but do not go into detail. A
non-expert reader will draw the wrong conclusions from the presentation in this manuscript and believe the mean field analysis is better than it really is: Its shortcomings are known and should be
addressed openly.
2) Another question is the accuracy of the mean field approximation when quenching well into the paramagnetic phase. How do time-dependent quantities compare to exact numerical results even for local
(non-momentum resolved) observables?
3) It should also be mentioned that the phase diagram of the ANNNO model is considerably more complex if one looks at larger values of kappa/J (see e.g. C. Karrasch and D. Schuricht, Phys. Rev. B 87,
195104). How accurate is the mean field approximation if one starts exploring these parts of the phase diagram? And what about the signatures of the quantum phase transition reported in [1]? In order
to make progress regarding this question one also needs to look at other transitions and not only at what was already analyzed using more reliable methods in [1].
In summary, this manuscript is well written, but does not contribute much to a critical analysis of a) and does not add much to b) from the point of view of what was already seen in [1]. Point 1)
needs to be addressed before acceptance, points 2) and 3) would be desirable but are not essential.
We thank the referee for assessing our manuscript and for providing helpful feedback with which to improve it. The report suggests that our work can be evaluated along two lines:
“a) Insights regarding the accuracy of the self-consistent time-dependent mean field theory as introduced in Refs. [27-33] and applied here.
b) Insights regarding the finite time signatures in the quench dynamics first reported in [1].”
Feature (b) is the motivation and purpose of our work so we will address the reviewer’s comments surrounding this first. Regarding this the referee states our work “does not add much to b) from the
point of view of what was already seen in [1].” We do not think that this is a fair assessment. Our work offers several new and important insights that go beyond the work [1] in terms of (i)
providing a simple understanding of the cause of the finite time signatures; (ii) clarifying the conditions for their emergence and (iii) establishing a key requirement for them to provide
quantitative information about quantum critical behaviour. More precisely:
We point out that a necessary condition for the programme proposed in [1] is that the quantum quench results in an energy density that is small compared to the energy cutoff associated with the
critical theory. Our simple theoretical framework readily gives us access to this information, which was not considered in [1]. We observe that for some of the cases considered in Ref [1] this
criterion is not well met and the observed signatures are therefore not suitable for extracting any information about the quantum critical point.
By considering two-point functions we establish that the correlation length grows after the quench (as one may have expected) and this provides an explanation for signatures associated with quantum
critical scaling. This observation is entirely absent from Ref [1].
By directly working in terms of fermionic variables we show finite time signatures occur also for topological transitions, which provides a significant simplification compared to the discussion in
We explain a peculiar feature visible in the numerical results of [1] as resulting from the formation of a two-particle bound state.
We now turn to criterion (a) mentioned in the report. SCTDMFT and its accuracy in weak-coupling regimes has been previously assessed in considerable detail, for example in past work involving the
senior author, cf. Refs [36,37]. The current work concerns a somewhat different situation in that the initial states are such that there is no dynamics without the quench of the interaction
parameter. This is closer to the situation considered by Moeckel and Kehrein in Ref.[39]. In this situation the known limitations of SCTDMFT are somewhat less visible and it can provide a
quantitatively decent approximation also at late times (as long as we perform a time-average over a short time window). This is all we meant by our comment on the application on SCTDMFT at late
times. In hindsight our original discussion was at best unclear and at worst misleading, and we are very grateful for the referee’s comments which enabled us to clarify the why SCTDMFT provides in
some sense a decent approximation even at late times. We have now added a discussion of how this is compatible with previous studies of the accuracy of SCTDMFT to replace our previous comments.
We were primarily interested in the short-time regime, where SCTDMFT is expected to be quite good, and this is fully confirmed by comparison with the numerical results of [1]. Another result of our
work that we find to be quite interesting is that the mean field theory can capture the persistent oscillations due to the bound state of the exact theory, despite no such bound state existing in the
mean field theory, and that moreover within the mean field approximations these oscillations are undamped. In summary we feel that whilst evaluation criterion (a) was not a focus of our work we do in
fact contribute something valuable here as well.
Finally, we address the other points raised by the referee. We are of course aware of the fact that the phase diagram of the ANNNI model is quite rich and that there are other phase transitions
present. However, we very deliberately restricted our analysis to the parameter regime covered in Ref. [1], because shedding additional light on the findings of that work was our key objective. The
mean field approach in terms of Jordan-Wigner fermions is expected to work well for describing the Ising transition for which [1] provided numerical studies to compare its accuracy against. We have
not attempted to explore the rest of the phase diagram with this method because there is no a priori reason for the mean-field treatment to be good. Instead, we would advocate using the numerical
techniques employed in [1] to study quantum quenches in these regions of the phase diagram.
Report #1 by Anonymous (Referee 1) on 2023-3-14 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202301_00025v1, delivered 2023-03-14, doi: 10.21468/SciPost.Report.6896
This work is timely, targeting an interesting open question in the field.
In this work the authors develop a mean-field theory for a paradigmatic quantum spin model in one dimension, the so-called ANNNI model. They apply their method to a dynamical problem, which has been
recently studied from a purely numerical point of view. Specifically, this concerns signatures of quantum critical points visible in the dynamics after quantum quenches in certain quantum spin
models. The authors convincingly and impressively show that their mean-field theory successfully captures the essential behaviors of the previously numerically obtained data. Most importantly, this
work provides physical explanations for the numerical observations, such as the presence of a special type of bound state causing oscillatory dynamical behavior in various time-dependent observables.
This work is timely , targeting an interesting open question in the field. The manuscript is very well written and well structured. I therefore recommend publication of the article in Scipost.
I could imagine that the manuscript might be further improvable by accounting for the following point:
In Sec. 3 it is said that "... this thermal states should be amenable to a description in terms of a simple self-consistent mean-field theory...". The results shown later by the authors indeed
demonstrate that. However, I wouldn't have guessed naturally that such a mean-field theory might be suitable for 1D systems. Is this because of the effect of temperature? In case the authors have
further arguments for the applicability of mean-field theory for the considered model and/or in a more general context this might further improve the manuscript.
Overall, I recommend this manuscript for publication in Scipost. | {"url":"https://www.scipost.org/submissions/scipost_202301_00025v1/","timestamp":"2024-11-06T15:47:15Z","content_type":"text/html","content_length":"45428","record_id":"<urn:uuid:f983f3ec-8606-4090-a172-fe00d11fd492>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00435.warc.gz"} |
Pressure difference between aorta and aneurysm
• Thread starter dcramps
• Start date
In summary, the conversation discusses an aortic aneurysm and how it affects blood pressure. Using Bernoulli's Principle, the difference in pressure between the aorta and the aneurysm can be
calculated by subtracting the initial and final velocities. However, since the density of blood is not given, the value of 1/2p cannot be calculated. The density of blood is 1060 kg/m^3 and cannot be
cancelled out in the equation.
Homework Statement
In an aortic aneurysm, a bulge forms where the walls of the aorta are weakened. If blood flowing through the aorta (radius 1.0cm) enters an aneurysm with a radius of 3.0cm, how much on average is the
blood pressure higher inside the aneurysm than the pressure in the unenlarged part of the aorta? The average flow rate through the aorta is 120cm³/s. Assume the blood is nonviscous and the patient is
lying down so there is no change in height.
Homework Equations
Bernoulli's Principle?
P[1] + 1/2pv[1]² = P[2] + 1/2pv[2]² where:
P[1] = pressure in the aorta?
P[2] = pressure in the aneurysm?
p = density of blood
v[1] = initial velocity
v[2] = final velocity
v = flowrate/pi*r²
The Attempt at a Solution
Calculated the initial velocity to be 38.19718634cm/s
So, using Bernoulli's Principle:
P[1] + 1/2pv[1]² = P[2] + 1/2pv[2]²
I changed it around a bit:
P1 - P2 = 1/2pv[2]² - 1/2pv[1]²
ΔP = 1/2pv[2]² - 1/2pv[1]²
and ended up with:
ΔP = 1/2p(v[2]² - v[1]²)
But, there is no density for blood given, and I'm unsure what to do about that. Did I make a mistake when I modified the formula?
I am quite confused now. Do I just make up a value for blood density?
actually, do the 1/2p cancel each other out?1/2pv[2]²-1/2pv[1]² => v[2]²-v[1]²
Science Advisor
Homework Helper
Density of blood is rho = 1060 kg/m^3. The rhos do not cancel out.
FAQ: Pressure difference between aorta and aneurysm
1. What is the normal pressure difference between the aorta and aneurysm?
The normal pressure difference between the aorta and aneurysm varies depending on the individual and their health. Generally, the pressure in the aorta is around 120/80 mmHg, while an aneurysm may
have a lower pressure due to the weakening of the artery walls. It is important to monitor and manage this pressure difference to avoid complications.
2. How does the pressure difference affect the risk of aneurysm rupture?
A higher pressure difference between the aorta and aneurysm can increase the risk of aneurysm rupture. This is because the aneurysm is already weakened and a higher pressure can put more stress on
the artery walls, causing it to burst. It is important to keep the pressure difference under control to prevent such complications.
3. Can the pressure difference between the aorta and aneurysm be measured?
Yes, the pressure difference between the aorta and aneurysm can be measured using various imaging techniques such as ultrasound, CT scans, or MRI. These tests can also help in monitoring the pressure
difference over time and detecting any changes or abnormalities.
4. What are the symptoms of a high pressure difference between the aorta and aneurysm?
Some common symptoms of a high pressure difference between the aorta and aneurysm include chest or back pain, shortness of breath, dizziness, and a pulsating feeling in the abdomen. However, in some
cases, there may be no symptoms at all, making regular check-ups and monitoring even more important.
5. How can the pressure difference between the aorta and aneurysm be managed?
The pressure difference between the aorta and aneurysm can be managed through lifestyle changes such as maintaining a healthy blood pressure, quitting smoking, and exercising regularly. In some
cases, medication or surgery may be necessary to control the pressure difference and prevent complications. | {"url":"https://www.physicsforums.com/threads/pressure-difference-between-aorta-and-aneurysm.288194/","timestamp":"2024-11-14T02:09:49Z","content_type":"text/html","content_length":"81578","record_id":"<urn:uuid:79df9f3b-62ac-4f77-9fce-803edc301368>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00332.warc.gz"} |
Keynesian cross and the multiplier : 네이버 참여번역
Keynesian cross and the multiplier
76문장 100% 한국어 번역 8명 참여 출처 : 칸아카데미
Keynesian cross and the multiplier
In the last video, we saw how the Keynesian Cross could help us visualize an increase in government spending which was a shift in our aggregate planned expenditure line right over here and we saw how
the actual change,
the actual increase in output if you take all the assumptions that we took in this, the actual change in output and aggregate income was larger than the change in government spending.
You might say okay, Keynesian thinking, this is very left wing, this is the government's growing larger right here.
I'm more conservative. I'm not a believer in Keynesian thinking.
The reality is you actually might be.
Whether you're on the right or the left, although Keynesian economics tends to be poo-pooed more by the right and embraced more by the left, most of the mainstream right policies, especially in the
US, have actually been very Keynesian.
They just haven't been by manipulating this variable right over here.
For example, when people talk about expanding the economy by lowering taxes, they are a Keynesian when they say that because if we were to rewind and we go back to our original function so if we
don't do this, if we go back to just having our G here,
we're now back on this orange line, our original planned expenditure, you could, based on this model right over here, also shift it up by lowering taxes.
If you change your taxes to be taxes minus some delta in taxes, the reason why this is going to shift the whole curve up is because you're multiplying this whole thing by a negative number, by
negative C1.
C1, your marginal propensity to consume, we're assuming is positive.
There's a negative out here. When you multiply it by a negative, when you multiply a decrease by a negative, this is a negative change in taxes, then this whole thing is going to shift up again.
You would actually shift up. You would actually shift up in this case and depending on what the actual magnitude of the change in taxes are, but you would actually shift up and the amount that you
would shift up -
I don't want to make my graph to messy so this is our new aggregate planned expenditures - but the amount you would move up is by this coefficient down here, C1, -C1 x -delta T.
You're change, the amount that you would move up, is -C1 x -delta T, if we assume delta T is positive and so you actually have a C1, delta T.
The negatives cancel out so that's actually how much it would actually move up.
It's also Keynesian when you say if we increase taxes that will lower aggregate output because if you increase taxes, now all of a sudden this is a positive, this is a positive and then you would
shift the curve by that much.
You would actually shift the curve down and then you would get to a lower equilibrium GDP.
This really isn't a difference between right leaning fiscal policy or left leaning fiscal policy and everything I've talked about so far at the end of the last video and this video really has been
fiscal policy.
This has been the spending lever of fiscal policy and this right over here has been the taxing lever of fiscal policy.
If you believe either of those can effect aggregate output, then you are essentially subscribing to the Keynesian model.
Now one thing that I did touch on a little bit in the last video is whatever our change is, however much we shift this aggregate planned expenditure curve, the change in our output actually was some
multiple of that.
What I want to do now is show you mathematically that it actually all works out that the multiple is actually the multiplier.
If we go back to our original and this will just get a little bit mathy right over here so I'm just going to rewrite it all.
We have our planned expenditure, just to redig our minds into the actual expression, the planned expenditure is equal to the marginal propensity to consume times aggregate income and then you're
going to have all of this business right over here.
We're just going to go with the original one, not what I changed.
All this business, let's just call this B.
That will just make it simple for us to manipulate this so let's just call of this business right over here B.
We could substitute that back in later.
We know that an economy is in equilibrium when planned expenditures is equal to output.
That is an economy in equilibrium so let's set this.
Let's set planned expenditures equal to aggregate output, which is the same thing as aggregate expenditures, the same thing as aggregate income.
We can just solve for our equilibrium income.
We can just solve for it.
You get Y=C1xY+B, this is going to look very familiar to you in a second.
Subtract C1xY from both sides. Y-C1Y, that's the left-hand side now.
On the right-hand side, obviously if we subtract C1Y, it's going to go away and that is equal to B.
Then we can factor out the aggregate income from this, so Yx1-C1=B and then we divide both sides by 1-C1 and we get, that cancels out.
I'll write it right over here.
We get, a little bit of a drum roll, aggregate income, our equilibrium, aggregate income, aggregate output.
GDP is going to be equal to 1/1-C1xB.
Remember B was all this business up here.
Now what is this? You might remember this or if you haven't seen the video, you might want to watch the video on the multiplier.
This C1 right over here is our marginal propensity to consume.
1 minus our marginal propensity to consume is actually - And I don't think I've actually referred to it before which let me rewrite it here just so that you know the term - so C1 is equal to our
marginal propensity to consume.
For example, if this is 30% or 0.3, that means for every incremental dollar of disposable income I get, I want to spend $.30 of it.
Now 1-C1, you could view this as your marginal propensity to save.
If I'm going to spend 30%, that means I'm going to save 70%.
This is just saying I'm going to save 1-C1.
If I'm spending 30% of that incremental disposable dollar, then I'm going to save 70% of it.
This whole thing, this is the marginal propensity to consume.
This entire denominator is the marginal propensity to save and then one over that, so 1/1-C1 which is the same thing as 1/marginal propensity to save, that is the multiplier.
We saw that a few videos ago.
If you take this infinite geometric series, if we just think through how money spends, if I spend some money on some good or service, the person who has that money as income is going to spend some
fraction of it based on their marginal propensity to consume,
and we're assuming that it's constant throughout the economy at all income levels for this model right over here.
Then they'll spend some of it and then the person that they spend it on, they're going to spend some fraction.
When you keep adding all that infinite series up, you actually get this multiplier right over here.
This is equal to our multiplier.
For example, if B gets shifted up by any amount, let's say B gets shifted up and it could get shifted up by changes in any of this stuff right over here.
Net exports can change, planned investments can change, could be shifted up or down.
The impact on GDP is going to be whatever that shift is times the multiplier.
We saw it before. If, for example, if C1=0.6, that means for every incremental disposable dollar, people will spend 60% of it.
That means that the marginal propensity to save is equal to 40%.
They're going to save 40% of any incremental disposable dollar and then the multiplier is going to be one over that, is going to be 1/0.4 which is the same thing as one over two-fifths, which is the
same thing as five-halves, which is the same thing as 2.5.
For example, in this situation, we just saw that Y, the equilibrium Y is going to be 2.5 times whatever all of this other business is.
If we change B by, let's say, $1 billion and maybe if we increase B by $1 billion.
We might increase B by $1 billion by increasing government spending by $1 billion or maybe having this whole term including this negative right over here become less negative by $1 billion.
Maybe we have planned investment increase by $1 billion and that could actually be done a little bit with tax policy too by letting companies maybe depreciate their assets faster.
If we could increase net exports by $1 billion.
Essentially any way that we increase B by $1 billion, that'll increase GDP by $2.5 billion, 2.5 times our change in B.
We can write this down this way.
Our change in Y is going to be 2.5 times our change in B.
Another way to think about it when you write the expression like this, if you said Y is a function of B, then you would say look the slope is 2.5, so change in Y over change in B is equal to 2.5, but
I just wanted to right this to show you that this isn't some magical voodoo that we're doing.
This is what we looked at visually when we looked at the Keynesian Cross.
This is really just describing the same multiplier effect that we saw in previous videos and where we actually derived the actual multiplier.
Keynesian cross and the multiplier발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
In the last video, we saw how the Keynesian Cross could help us visualize an increase in government spending which was a shift in our aggregate planned expenditure line right over here and we saw how
the actual change,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
the actual increase in output if you take all the assumptions that we took in this, the actual change in output and aggregate income was larger than the change in government spending.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
You might say okay, Keynesian thinking, this is very left wing, this is the government's growing larger right here.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
I'm more conservative. I'm not a believer in Keynesian thinking.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
The reality is you actually might be.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Whether you're on the right or the left, although Keynesian economics tends to be poo-pooed more by the right and embraced more by the left, most of the mainstream right policies, especially in the
US, have actually been very Keynesian.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
They just haven't been by manipulating this variable right over here.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
For example, when people talk about expanding the economy by lowering taxes, they are a Keynesian when they say that because if we were to rewind and we go back to our original function so if we
don't do this, if we go back to just having our G here,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
we're now back on this orange line, our original planned expenditure, you could, based on this model right over here, also shift it up by lowering taxes.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
If you change your taxes to be taxes minus some delta in taxes, the reason why this is going to shift the whole curve up is because you're multiplying this whole thing by a negative number, by
negative C1.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
C1, your marginal propensity to consume, we're assuming is positive.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
There's a negative out here. When you multiply it by a negative, when you multiply a decrease by a negative, this is a negative change in taxes, then this whole thing is going to shift up again.발음
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
You would actually shift up. You would actually shift up in this case and depending on what the actual magnitude of the change in taxes are, but you would actually shift up and the amount that you
would shift up -발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
I don't want to make my graph to messy so this is our new aggregate planned expenditures - but the amount you would move up is by this coefficient down here, C1, -C1 x -delta T.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
You're change, the amount that you would move up, is -C1 x -delta T, if we assume delta T is positive and so you actually have a C1, delta T.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
The negatives cancel out so that's actually how much it would actually move up.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
It's also Keynesian when you say if we increase taxes that will lower aggregate output because if you increase taxes, now all of a sudden this is a positive, this is a positive and then you would
shift the curve by that much.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
You would actually shift the curve down and then you would get to a lower equilibrium GDP.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
This really isn't a difference between right leaning fiscal policy or left leaning fiscal policy and everything I've talked about so far at the end of the last video and this video really has been
fiscal policy.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
This has been the spending lever of fiscal policy and this right over here has been the taxing lever of fiscal policy.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
If you believe either of those can effect aggregate output, then you are essentially subscribing to the Keynesian model.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Now one thing that I did touch on a little bit in the last video is whatever our change is, however much we shift this aggregate planned expenditure curve, the change in our output actually was some
multiple of that.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
What I want to do now is show you mathematically that it actually all works out that the multiple is actually the multiplier.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
If we go back to our original and this will just get a little bit mathy right over here so I'm just going to rewrite it all.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We have our planned expenditure, just to redig our minds into the actual expression, the planned expenditure is equal to the marginal propensity to consume times aggregate income and then you're
going to have all of this business right over here.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We're just going to go with the original one, not what I changed.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
All this business, let's just call this B.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
That will just make it simple for us to manipulate this so let's just call of this business right over here B.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We could substitute that back in later.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We know that an economy is in equilibrium when planned expenditures is equal to output.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
That is an economy in equilibrium so let's set this.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Let's set planned expenditures equal to aggregate output, which is the same thing as aggregate expenditures, the same thing as aggregate income.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We can just solve for our equilibrium income.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We can just solve for it.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
You get Y=C1xY+B, this is going to look very familiar to you in a second.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Subtract C1xY from both sides. Y-C1Y, that's the left-hand side now.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
On the right-hand side, obviously if we subtract C1Y, it's going to go away and that is equal to B.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Then we can factor out the aggregate income from this, so Yx1-C1=B and then we divide both sides by 1-C1 and we get, that cancels out.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
I'll write it right over here.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We get, a little bit of a drum roll, aggregate income, our equilibrium, aggregate income, aggregate output.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
GDP is going to be equal to 1/1-C1xB.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Remember B was all this business up here.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Now what is this? You might remember this or if you haven't seen the video, you might want to watch the video on the multiplier.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
This C1 right over here is our marginal propensity to consume.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
1 minus our marginal propensity to consume is actually - And I don't think I've actually referred to it before which let me rewrite it here just so that you know the term - so C1 is equal to our
marginal propensity to consume.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
For example, if this is 30% or 0.3, that means for every incremental dollar of disposable income I get, I want to spend $.30 of it.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Now 1-C1, you could view this as your marginal propensity to save.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
If I'm going to spend 30%, that means I'm going to save 70%.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
This is just saying I'm going to save 1-C1.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
If I'm spending 30% of that incremental disposable dollar, then I'm going to save 70% of it.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
This whole thing, this is the marginal propensity to consume.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
This entire denominator is the marginal propensity to save and then one over that, so 1/1-C1 which is the same thing as 1/marginal propensity to save, that is the multiplier.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We saw that a few videos ago.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
If you take this infinite geometric series, if we just think through how money spends, if I spend some money on some good or service, the person who has that money as income is going to spend some
fraction of it based on their marginal propensity to consume,발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
and we're assuming that it's constant throughout the economy at all income levels for this model right over here.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Then they'll spend some of it and then the person that they spend it on, they're going to spend some fraction.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
When you keep adding all that infinite series up, you actually get this multiplier right over here.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
This is equal to our multiplier.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
For example, if B gets shifted up by any amount, let's say B gets shifted up and it could get shifted up by changes in any of this stuff right over here.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Net exports can change, planned investments can change, could be shifted up or down.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
The impact on GDP is going to be whatever that shift is times the multiplier.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We saw it before. If, for example, if C1=0.6, that means for every incremental disposable dollar, people will spend 60% of it.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
That means that the marginal propensity to save is equal to 40%.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
They're going to save 40% of any incremental disposable dollar and then the multiplier is going to be one over that, is going to be 1/0.4 which is the same thing as one over two-fifths, which is the
same thing as five-halves, which is the same thing as 2.5.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
For example, in this situation, we just saw that Y, the equilibrium Y is going to be 2.5 times whatever all of this other business is.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
If we change B by, let's say, $1 billion and maybe if we increase B by $1 billion.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We might increase B by $1 billion by increasing government spending by $1 billion or maybe having this whole term including this negative right over here become less negative by $1 billion.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Maybe we have planned investment increase by $1 billion and that could actually be done a little bit with tax policy too by letting companies maybe depreciate their assets faster.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
If we could increase net exports by $1 billion.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Essentially any way that we increase B by $1 billion, that'll increase GDP by $2.5 billion, 2.5 times our change in B.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
We can write this down this way.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Our change in Y is going to be 2.5 times our change in B.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
Another way to think about it when you write the expression like this, if you said Y is a function of B, then you would say look the slope is 2.5, so change in Y over change in B is equal to 2.5, but
I just wanted to right this to show you that this isn't some magical voodoo that we're doing.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
This is what we looked at visually when we looked at the Keynesian Cross.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요.
This is really just describing the same multiplier effect that we saw in previous videos and where we actually derived the actual multiplier.발음듣기
• 더 많은 번역문은 보기 옵션을 이용해 보세요. | {"url":"https://dict.naver.com/userstranslate/khanInfo.dict?fullTextId=15601&langCode=ko&tab=1&selector=1","timestamp":"2024-11-06T14:25:08Z","content_type":"text/html","content_length":"351073","record_id":"<urn:uuid:47c28692-af2b-4959-846e-cdc857f98d6a>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00865.warc.gz"} |
How to read and create Pareto Charts
Pareto Principle
We have all heard something about the Pareto Principle. Actually it goes with different names, such as:
• Pareto Low
• 80-20 Rule
• The Principle of Imbalance
• All kind of modifications of these
It is very powerful tool, which can help us perform a root cause analysis and detect the most important issues in our project.
The most common phrase related to Pareto principle is:
You can encounter the following as well:
• 80% of the results come from 20% of the efforts
• 80% of the sales come from 20% of the customers
• 80% of the work is completed by 20% of the team
• 80% of the customers uses only 20% of the features
In this article we will have an overview of the Pareto Diagrams, how to read them and how to create them.
First let’s take a look at the following scenario.
The problem
Our management is giving us a task to improve the user experience in the company’s website. So what should we do first?
One option is to collect feedback from the current users for what is bothering them.
The result of online survey, sent to 200 of our regular visitors, is a total of 162 responses.
What should we do next?
Create a list
Divide the responses to different categories (it is considered a good practice to have less than 10 different categories).
At the end we come up with the following list, which now should be analyzed.
That’s the moment when the Pareto Diagram comes to our mind.
Take a look at the graphic below, it looks complicated, but actually it is not at all.
As you can notice the blue bars represent the number of users complaining about similar issues. The interesting part here is the orange line which is also called cumulative line or Pareto curve.
This line gives us the cumulative weight of every bar.
So far, so good. But where exactly lies the Pareto principle here?
Imagine you can draw a line across from the 80% mark to the cumulative line and then down into the bars, as in the diagram above (the dotted red lines).
This means that 80 percent of our complains come from the first 2 issues – the site is too complicated to navigate and it is too slow.
This sounds awesome, right? If we manage to eliminate the two problems we will get significantly higher feedback.
Create your own Pareto Chart
Now when you already bought the idea of using Pareto Diagrams, let’s learn how to create them.
Step 1
We need to get our initial statistics and sort them out from largest to smallest:
Step 2
We need to create another column for the cumulative values. For each cell in the Cumulative Count row as value we need to set the total number of complains from the rows above.
So for instance, in the cell C2 we put the same value as the one in B2, as this is the first row we count.
Step 3
After that in every cell we put the sum of the users’ complains (the current row included).
As you can see in the last 7th row we have the same value as the total number of complains. This means that our calculations are correct, so we can proceed with getting the cumulative percentage.
Step 4
The next thing is to create a Cumulative Count % Column which can count the percent of every row from the totals. You can do this by dividing the cumulative value by the total and multiply by 100:
After we do that for the whole table it is time to create the graphic.
Step 5
First you need to select Column A, B and D at the same time without selecting the Total row.
Then click on Insert tab of the Excel and select a graphic.
Right after you do this, you will receive the following diagram, which of course needs to be modified a bit.
Step 6
You have to select the orange bars and click right button of the mouse, then select Change series char type.
You will get the following screen from where you need to pick the Combo type and select the second chart.
That was the last step before our creation is done.
I hope the article was useful to you.
If you like this, follow me on twitter! | {"url":"http://thelillysblog.com/2016/05/19/how-to-read-and-create-pareto-charts/","timestamp":"2024-11-04T21:41:29Z","content_type":"text/html","content_length":"10059","record_id":"<urn:uuid:0fceb015-aede-4d93-b822-e08392122616>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00211.warc.gz"} |
A quantitative nuisance factor that varies along with the response variable but is not related to the other Factors under study. Its effect must be adjusted for using Analysis of Covariance,
otherwise it may inflate the error variance and obscure the true factor effects. For instance, while studying the effect of Training (factor) on Task Performance (response), Experience could be a
covariate. Age is used as a covariate in many clinical studies. | {"url":"https://sigmapedia.com/includes/term.cfm?&word_id=1328&lang=ENG","timestamp":"2024-11-12T17:07:48Z","content_type":"text/html","content_length":"17886","record_id":"<urn:uuid:cfab4aaa-3f09-49e8-a1cb-86f0265e7d1c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00607.warc.gz"} |
Lesson 2
Finding Area by Decomposing and Rearranging
Let’s create shapes and find their areas.
2.1: What is Area?
You may recall that the term area tells us something about the number of squares inside a two-dimensional shape.
1. Here are four drawings that each show squares inside a shape. Select all drawings whose squares could be used to find the area of the shape. Be prepared to explain your reasoning.
2. Write a definition of area that includes all the information that you think is important.
2.2: Composing Shapes
This applet has one square and some small, medium, and large right triangles. The area of the square is 1 square unit.
Click on a shape and drag to move it. Grab the point at the vertex and drag to turn it.
1. Notice that you can put together two small triangles to make a square. What is the area of the square composed of two small triangles? Be prepared to explain your reasoning.
2. Use your shapes to create a new shape with an area of 1 square unit that is not a square. Draw your shape on paper and label it with its area.
3. Use your shapes to create a new shape with an area of 2 square units. Draw your shape and label it with its area.
4. Use your shapes to create a different shape with an area of 2 square units. Draw your shape and label it with its area.
5. Use your shapes to create a new shape with an area of 4 square units. Draw your shape and label it with its area.
Find a way to use all of your pieces to compose a single large square. What is the area of this large square?
2.3: Tangram Triangles
Recall that the area of the square you saw earlier is 1 square unit. Complete each statement and explain your reasoning.
1. The area of the small triangle is ____________ square units. I know this because . . .
2. The area of the medium triangle is ____________ square units. I know this because . . .
3. The area of the large triangle is ____________ square units. I know this because . . .
Here are two important principles for finding area:
1. If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.
2. We can decompose a figure (break a figure into pieces) and rearrange the pieces (move the pieces around) to find its area.
Here are illustrations of the two principles.
• Each square on the left can be decomposed into 2 triangles. These triangles can be rearranged into a large triangle. So the large triangle has the same area as the 2 squares.
• Similarly, the large triangle on the right can be decomposed into 4 equal triangles. The triangles can be rearranged to form 2 squares. If each square has an area of 1 square unit, then the area
of the large triangle is 2 square units. We also can say that each small triangle has an area of \(\frac12\) square unit.
• area
Area is the number of square units that cover a two-dimensional region, without any gaps or overlaps.
For example, the area of region A is 8 square units. The area of the shaded region of B is \(\frac12\) square unit.
• compose
Compose means “put together.” We use the word compose to describe putting more than one figure together to make a new shape.
• decompose
Decompose means “take apart.” We use the word decompose to describe taking a figure apart to make more than one new shape.
• region
A region is the space inside of a shape. Some examples of two-dimensional regions are inside a circle or inside a polygon. Some examples of three-dimensional regions are the inside of a cube or
the inside of a sphere. | {"url":"https://im.kendallhunt.com/MS/students/1/1/2/index.html","timestamp":"2024-11-12T17:01:10Z","content_type":"text/html","content_length":"102128","record_id":"<urn:uuid:73a458b2-c11c-450c-8f7d-3e728ba4d881>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00704.warc.gz"} |
[{"has_accepted_license":"1","oa_version":"Published Version","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":
[{"last_name":"Chatterjee","id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87","first_name":"Krishnendu","full_name":"Chatterjee, Krishnendu","orcid":"0000-0002-4561-241X"},
{"first_name":"Rasmus","id":"3B699956-F248-11E8-B48F-1D18A9856A87","last_name":"Ibsen-Jensen","full_name":"Ibsen-Jensen, Rasmus","orcid":"0000-0003-4783-0389"},
Krishnendu, et al. Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs. IST Austria, 2015, doi:10.15479/AT:IST-2015-330-v2-1.","ama":"Chatterjee K, Ibsen-Jensen R,
Pavlogiannis A. Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs. IST Austria; 2015. doi:10.15479/AT:IST-2015-330-v2-1","ieee":"K. Chatterjee, R. Ibsen-Jensen, and A.
Pavlogiannis, Faster algorithms for quantitative verification in constant treewidth graphs. IST Austria, 2015.","short":"K. Chatterjee, R. Ibsen-Jensen, A. Pavlogiannis, Faster Algorithms for
Quantitative Verification in Constant Treewidth Graphs, IST Austria, 2015.","apa":"Chatterjee, K., Ibsen-Jensen, R., & Pavlogiannis, A. (2015). Faster algorithms for quantitative verification in
constant treewidth graphs. IST Austria. https://doi.org/10.15479/AT:IST-2015-330-v2-1","ista":"Chatterjee K, Ibsen-Jensen R, Pavlogiannis A. 2015. Faster algorithms for quantitative verification in
constant treewidth graphs, IST Austria, 27p.","chicago":"Chatterjee, Krishnendu, Rasmus Ibsen-Jensen, and Andreas Pavlogiannis. Faster Algorithms for Quantitative Verification in Constant Treewidth
Graphs. IST Austria, 2015. https://doi.org/10.15479/AT:IST-2015-330-v2-1."},"oa":1,"day":"27","ddc":["000"],"page":"27","year":"2015","doi":"10.15479/
AT:IST-2015-330-v2-1","file_date_updated":"2020-07-14T12:46:54Z","pubrep_id":"333","alternative_title":["IST Austria Technical Report"],"related_material":{"record":
[{"id":"5430","status":"public","relation":"earlier_version"},{"status":"public","id":"1607","relation":"later_version"}]},"abstract":[{"lang":"eng","text":"We consider the core algorithmic problems
related to verification of systems with respect to three classical quantitative properties, namely, the mean-payoff property, the ratio property, and the minimum initial credit for energy property. \
r\nThe algorithmic problem given a graph and a quantitative property asks to compute the optimal value (the infimum value over all traces) from every node of the graph. We consider graphs with
constant treewidth, and it is well-known that the control-flow graphs of most programs have constant treewidth. Let $n$ denote the number of nodes of a graph, $m$ the number of edges (for constant
treewidth graphs $m=O(n)$) and $W$ the largest absolute value of the weights.\r\nOur main theoretical results are as follows.\r\nFirst, for constant treewidth graphs we present an algorithm that
approximates the mean-payoff value within a multiplicative factor of $\\epsilon$ in time $O(n \\cdot \\log (n/\\epsilon))$ and linear space, as compared to the classical algorithms that require
quadratic time. Second, for the ratio property we present an algorithm that for constant treewidth graphs works in time $O(n \\cdot \\log (|a\\cdot b|))=O(n\\cdot\\log (n\\cdot W))$, when the output
is $\\frac{a}{b}$, as compared to the previously best known algorithm with running time $O(n^2 \\cdot \\log (n\\cdot W))$. Third, for the minimum initial credit problem we show that (i)~for general
graphs the problem can be solved in $O(n^2\\cdot m)$ time and the associated decision problem can be solved in $O(n\\cdot m)$ time, improving the previous known $O(n^3\\cdot m\\cdot \\log (n\\cdot
W))$ and $O(n^2 \\cdot m)$ bounds, respectively; and (ii)~for constant treewidth graphs we present an algorithm that requires $O(n\\cdot \\log n)$ time, improving the previous known $O(n^4 \\cdot \\
log (n \\cdot W))$ bound.\r\nWe have implemented some of our algorithms and show that they present a significant speedup on standard benchmarks. "}],"publisher":"IST
Austria","status":"public","title":"Faster algorithms for quantitative verification in constant treewidth graphs","publication_status":"published","department": | {"url":"https://research-explorer.ista.ac.at/record/5437.json","timestamp":"2024-11-02T08:34:46Z","content_type":"application/json","content_length":"5622","record_id":"<urn:uuid:67f82a2e-25ba-4f9c-ae70-9e6614290a4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00623.warc.gz"} |
Jess's mathematics
Creative Commons CC BY 4.0
LaTeX is the best way to write mathematics. It completely pisses all over Word. However, it does take some time to get used to so might not be worth your while if you won't write too much. The way I
use it is to first download and install a latex editor and then get writing, but I would recommend that you use this website instead since you can get going a lot quicker. The upshot of the whole
business is that you type in here and then a pdf is generated with all the equations looking ace. I'll give you some examples. | {"url":"https://cn.overleaf.com/latex/examples/jesss-mathematics/cwtkcdvbpjcx","timestamp":"2024-11-06T05:44:09Z","content_type":"text/html","content_length":"38968","record_id":"<urn:uuid:75d7048d-725a-4e3a-ac42-ae9845063db0>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00379.warc.gz"} |
Building Molar Mass
Doug Ragan | Sun, 05/10/2015 - 21:27
An advantage to teaching on the trimester schedule allows me the opportunity to teach the same course again roughly twelve weeks later. So after grading my 2^nd trimester students’ Chemistry B final
exams, I was able to evaluate certain topics that caused my students problems, reflect on my teaching, and then determine how I was going to better prepare my students in the 3^rd trimester chemistry
B class.
One topic in particular that caused some problems for my students was the dissolving of ionic salts into their appropriate ions and then representing them correctly regarding the appropriate numbers
of ions. Having students mention to me that table salt when dissolved in water produces sodium and chlorine and not their respected ions, has always bugged me. Let alone even after correcting
students of this common misconception, asking students what is produced when MgCl[2] is dissolved in water often students give me chlorine, Cl[2,] as a response due to seeing Cl[2] in the formula and
often times when I ask students the reason for their answer they tell me because chlorine is a diatomic molecule.
Another topic that caused my students some problems was that dreaded dot that represents the bond within the hydrate formula such as CuSO[4]^ . 5H[2]O. As much as I stressed that the dot was not a
multiplication sign, still a percentage of my students missed that topic in some way or another. Regardless of doing some worksheet exercises beforehand, when my students performed the hydrate lab
and then answered questions regarding molar mass of the hydrate or the percentage of water in the hydrate then I would get some really strange numbers.
So how did I change? Well, my original teaching method consisted of supplementing my notes with technology to show my students the dissolving of ions in solution. The dissolving of table salt for
example, has been represented in several YouTube videos.
I have also used the PhET Simulation Salts and Solubility to show what happens when salts dissolve in solution and to focus on the numbers of ions that can dissolve in solution that are more than
just a 1:1 ratio. Next, I used notes and practice exercises to calculate everything involved with hydrates. However this method proved to be ineffective.
So with the start of the 3^rd trimester, I decided that rather than use the technology to show students things such as molar mass calculations and the dissolving of ionic salts then I would ask them
to simply build the salts using colored unifix cubes. Just search amazon for unifix cubes (or snap cubes also look popular).I have a box of 1000 that I distribute amongst my lab tables. Well, I don’t
know if it was the start of a new trimester of just the capability to literally build, but my students attacked my handout building ionic salts and hydrates and explaining how same colors were same
atoms and subscripts outside of the parenthesis were shown by replicating a certain pattern of blocks a certain way. It was amazing to watch and I began to see understanding. I added teacher
checkpoints onto my worksheet to check for this. Overall the activity was a big success and I will definitely continue this activity again using the unifix cubes. I gave an assessment and was pleased
with the overall average of my classes seeing a bit of an improvement overall. I have included a link to the document. If you have any questions or comments, please don’t hesitate to contact me. If I
missed anything or if you can think of another way to improve this then please share.
Calculating molar mass
Dissociation of ionic salts
Calculating the molar mass of a hydrate
See document and Teachers Guide
Provide materials for each group.
Search amazon for unifix cubes or snap cubes
Students who demonstrate understanding can use mathematical representations to support the claim that atoms, and therefore mass, are conserved during a chemical reaction.
Assessment Boundary:
Assessment does not include complex chemical reactions.
Emphasis is on using mathematical ideas to communicate the proportional relationships between masses of atoms in the reactants and the products, and the translation of these relationships to the
macroscopic scale using the mole as the conversion from the atomic to the macroscopic scale. Emphasis is on assessing students’ use of mathematical thinking and not on memorization and rote
application of problem - solving techniques.
Students who demonstrate understanding can develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of
particles (objects) and energy associated with the relative position of particles (objects).
*More information about all DCI for HS-PS3 can be found at https://www.nextgenscience.org/topic-arrangement/hsenergy.
Students who demonstrate understanding can develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of
particles (objects) and energy associated with the relative position of particles (objects).
Assessment Boundary:
Examples of phenomena at the macroscopic scale could include the conversion of kinetic energy to thermal energy, the energy stored due to position of an object above the earth, and the energy stored
between two electrically-charged plates. Examples of models could include diagrams, drawings, descriptions, and computer simulations.
Students who demonstrate understanding can develop and use a model of two objects interacting through electric or magnetic fields to illustrate the forces between objects and the changes in energy of
the objects due to the interaction.
*More information about all DCI for HS-PS3 can be found at https://www.nextgenscience.org/topic-arrangement/hsenergy.
Students who demonstrate understanding can develop and use a model of two objects interacting through electric or magnetic fields to illustrate the forces between objects and the changes in energy of
the objects due to the interaction.
Assessment Boundary:
Assessment is limited to systems containing two objects.
Examples of models could include drawings, diagrams, and texts, such as drawings of what happens when two charges of opposite polarity are near each other. | {"url":"https://www.chemedx.org/activity/building-molar-mass","timestamp":"2024-11-04T01:31:29Z","content_type":"text/html","content_length":"48450","record_id":"<urn:uuid:f6970726-adf3-41b0-a584-8d1c3b6df70c>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00431.warc.gz"} |
Mathematics,Probability and Statistics,Applied Mathematics Download ( 260 Pages | Free )
“ At the end of your life, you will never regret not having passed one more test, not winning one more verdict or not closing one more deal. You will regret time not spent with a husband, a friend, a
child, or a parent. ” ― Barbara Bush
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Ask yourself:
Are my beliefs about life, religion, my kids, my family, my spouse, or politics the absolute truth? | {"url":"http://www.pdfdrive.com/mathematicsprobability-and-statisticsapplied-mathematics-e16657497.html","timestamp":"2024-11-14T00:24:42Z","content_type":"application/xhtml+xml","content_length":"55497","record_id":"<urn:uuid:80dc0d94-ded3-4597-9d3a-717496c087b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00375.warc.gz"} |
calculate belt grinding
Belt Grinders, Combination Belt & Disc Grinders, Knife ...
Belt Grinders & Sanders We offer one of the widest selections of metal grinding equipment to fit your needs, whether you're a knife maker, home hobbyist, or full-time fabricator. Abrasive belt
grinders and sanders come in a variety of different sizes for deburring, shaping, sanding, polishing, grinding, sharpening, cleaning, and de-scaling metal.
اقرأ أكثر | {"url":"https://www.transports-aio.fr/Sep/03-21644.html","timestamp":"2024-11-04T10:54:54Z","content_type":"text/html","content_length":"35967","record_id":"<urn:uuid:15461187-6525-4255-9d4e-040ef38024a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00787.warc.gz"} |
a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.
b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.) | {"url":"https://problems.org.uk/problems/?from_difficulty=4.0&to_difficulty=4.0&","timestamp":"2024-11-14T16:38:15Z","content_type":"text/html","content_length":"41863","record_id":"<urn:uuid:84073a5c-4e89-4a95-967c-d88b04a85b4b>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00300.warc.gz"} |
dask.array.log(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature]) = <ufunc 'log'>¶
This docstring was copied from numpy.log.
Some inconsistencies with the Dask version may exist.
Natural logarithm, element-wise.
The natural logarithm log is the inverse of the exponential function, so that log(exp(x)) = x. The natural logarithm is logarithm in base e.
Input value.
outndarray, None, or tuple of ndarray and None, optional
A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple
(possible only as a keyword argument) must have length equal to the number of outputs.
wherearray_like, optional
This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original
value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
For other keyword-only arguments, see the ufunc docs.
The natural logarithm of x, element-wise. This is a scalar if x is a scalar.
Logarithm is a multivalued function: for each x there is an infinite number of z such that exp(z) = x. The convention is to return the z whose imaginary part lies in (-pi, pi].
For real-valued input data types, log always returns real output. For each value that cannot be expressed as a real number or infinity, it yields nan and sets the invalid floating point error
For complex-valued input, log is a complex analytical function that has a branch cut [-inf, 0] and is continuous from above on it. log handles the floating-point negative zero as an infinitesimal
negative number, conforming to the C99 standard.
In the cases where the input has a negative real part and a very small negative complex part (approaching 0), the result is so close to -pi that it evaluates to exactly -pi.
M. Abramowitz and I.A. Stegun, “Handbook of Mathematical Functions”, 10th printing, 1964, pp. 67. https://personal.math.ubc.ca/~cbm/aands/page_67.htm
Wikipedia, “Logarithm”. https://en.wikipedia.org/wiki/Logarithm
>>> import numpy as np
>>> np.log([1, np.e, np.e**2, 0])
array([ 0., 1., 2., -inf]) | {"url":"https://docs.dask.org/en/stable/generated/dask.array.log.html","timestamp":"2024-11-08T12:26:26Z","content_type":"text/html","content_length":"34533","record_id":"<urn:uuid:13041a3f-3bf5-45dd-827c-acde82d4c81c>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00716.warc.gz"} |
gradients for state-averaged MCSCF and MCSCF state tracking
State-specific gradients for state-averaged MCSCF and MCSCF state tracking.
State-specific gradients for state-averaged MCSCF
The approach used by Firefly to compute state-specific (SS) gradients for state-averaged (SA) MCSCF is described in details here and is based on the differentiation of effective gradient vector
computed with state-averaged density matrices over the weight of the state of interest. The differentiation is performed using second-order finite differences. Therefore, due to their seminumeric
nature, the computed gradients are more sensitive to numerical errors in the computed solution of SA-MCSCF than the ordinary MCSCF gradients are. Hence, one needs to increase the overall precision of
computations as will be discussed below. Computationally, state-specific gradients for SA-MCSCF are approximately two to thee times more costly than gradients for MCSCF without state averaging. The
MCSCF procedure itself can be arbitrary i.e. it can be of any supported type, can use both GUGA and ALDET CI code, and can utilize any of available MCSCF convergers. However, the use of Firefly’s
unique SOSCF converger is strongly recommended. The relevant input groups are $mcscf and $mcaver
The $mcscf input group contains several relevant keywords, namely istate, ntrack, acurcy, and engtol.
istate the state number for which SS gradient will be computed. States are numbered starting from one. For example, $mcscf istate=2 $end will compute gradients for the first excited state of given
multiplicity and symmetry type as specified by other sections of input file. The only exception is the case of ALDET CI with pures option disabled, for which the numbering will include all states
regardless of their multiplicity. One should not use a nonzero istate option for MCSCF runs without state-averaging or when state-specific gradients are not required! The default value is istate=0
i.e. state-specific gradients are disabled.
ntrack if nonzero, activates MCSCF states tracking and remapping feature of Firefly. More precisely, ntrack defines the number of lowest roots (states) to be tracked and, if necessary, remapped to
other states. The present implementation of state tracking in Firefly is rather simple and is based on the analysis of overlap matrix of current states with the previously computed ones. See the
description of $track group below for more details and for additional keywords. The default for ntrack is ntrack=0 i.e. do not track states at all. State tracking is cheap and can be of great help
e.g. during geometry optimization of excited states as it can detect root flipping and remap states accordingly so it is generally a good idea to activate it.
acurcy the major convergence criterion for MCSCF, the maximum permissible asymmetry in the Lagrangian matrix. While its default value is 1.0E-05 and is suitable for single-state MCSCF, computation of
SS gradients for SA-MCSCF requires tighter convergence. The recommended values of acurcy are in the range 1.0E-07 - 1.0E-08.
engtol the secondary convergence criterion, the MCSCF is considered converged when the energy change is smaller than this value. The default is engtol=1.0E-10. Again, SS gradients require tighter
convergence; the recommended values of engtol are in the range 1.0E-12 - 1.0E-13.
The $mcaver input group defines the details on how exactly SS gradients are computed. The relevant keywords are deltaw, conic, and hpgrad. Besides these, this group contains additional keywords
controlling parts of Firefly’s code used for location of interstate crossings and conical intersections, namely: ssgrad, jstate, a, b, target, xhess, xgrad, multiw, shift, and penlty. These keywords
will be described elsewhere.
deltaw the step size (dimensionless) over state’s weight used in finite differencing. The recommended values are in the range from 0.0005 to 0.0025. The default value is deltaw=0.0015.
conic selects one of the three programmed approaches to finite differencing. Valid values are conic=0, 1, or 2. The default is conic=0
Conic=0 selects the use of central (i.e. symmetric) second order finite differences. The calculations are performed as follows. First, calculations on SA-MCSCF energies and effective gradient are
performed with the original weight of state # istate increased by deltaw. Second, calculations on SA-MCSCF energies and effective gradient are performed with the original weight of state # istate
decreased by deltaw. Third, the calculations on SA-MCSCF energies are performed using unmodified original weights, and finally, state-specific expectation value type density matrix is computed for
state # istate. This is the most economical way of computations. In addition, it provides the way to obtain the state-specific properties for the state of interest.
Conic=1 selects the alternative approach based on the use of forward finite differencing scheme. The latter is more stable in the case of nearly quasi-degenerated CI roots and hence it is more
suitable for location of conical intersections. The calculations are organized as follows. First, calculations on SA-MCSCF energies and effective gradient are performed with the original weight of
state # istate increased by deltaw. Second, calculations on SA-MCSCF energies and effective gradient are performed with the original weight of state # istate increased by deltaw/2. Finally, the
calculations on SA-MCSCF energies and effective gradient are performed using unmodified original weights but state-specific density matrix is not computed hence no state-specific properties are
Finally, conic=2 selects another approach which is similar to conic=1 but is even more robust in the vicinities of conical intersections. First, calculations on SA-MCSCF energies and effective
gradient are performed with the original weight of state # istate. Second, calculations on SA-MCSCF energies and effective gradient are performed with the original weight of state # istate increased
by deltaw/2. Finally, the calculations on SA-MCSCF energies and effective gradient are performed with the original weight of target state increased by deltaw. Similarly to conic=1 no state-specific
density matrix is computed thus the state-specific properties are not available.
hpgrad logical variable. If set, requests extra high precision during computations of the two-electron contributions to the effective gradients. This may significantly slow down computations and
usually does not considerably increase the precision of the computed state-specific gradients. This is why this option is disabled by default (hpgrad=.false.).
Hints on increasing the overall precision of SA-MCSCF calculations
Typically, the separate stages involved into MSCSF procedure are: evaluation of two-electron integrals, integral transformation, CI procedure, computation of one- and two-particle density matrices,
and the orbital improvement step. For SS gradients, one needs to increase the precision of each of these steps. Here we provide a synopsis of related input groups and keywords and give some
recommendations on the optimal values for the latter.
1. Precision of two-electron integrals is controlled by inttyp, icut, and itol keywords of the $contrl group as described in the manual. For SS gradients, the recommended values are inttyp=hondo,
icut= at least 11 (value of 12 or 13 is even better), and itol=20
2. Precision of integral transformation stage is controlled by the cuttrf keyword of $trans group as described in manual. For SS gradients, the recommended value of cuttrf is 1.0d-13 or tighter.
3. Precision of CI step is controlled by a single keyword for ALDET CI code and by two keywords for GUGA CI code. For ALDET code, it is cvgtol of $det group. The recommended value of cvgtol is 1.0d-7
to 1.0d-8 to 1.0d-9. For GUGA CI, these are cutoff of $gugem group and cvgtol of $gugdia group. The recommended values are cutoff=1.0d-20 and cvgtol=1.0d-7 to 1.0d-8 to 1.0d-9.
4. For GUGA CI, precision of two-particle density matrix computation step is controlled by cutoff keyword of $gugdm2 input group. The recommended value is cutoff=1.0d-15 or tighter. For ALDET CI,
there are no additional keywords required.
5. The precision of the orbital improvement step can be improved using several keywords of $moorth and $system groups. It is recommended to set:
$system kdiag=0 nojac=1 $end
nostf=.t. nozero=.t. syms=.t. symden=.t. symvec=.t. symvx=.t.
tole=0.0d0 tolz=0.0d0
An example
An archive with commented sample input and output files for both single-state MCSCF gradient and SS gradient for SA-MCSCF computations can be found here
MCSCF state tracking.
The present implementation of MCSCF state tracking in Firefly is based on the analysis of overlap matrix of the current MCSCF states with the previously computed and possibly already remapped states
which serve as the reference vectors. First, the overlap matrix is computed every MSCSF iteration. The diagonal elements of this matrix are then scaled to decrease the probability of false root
flipping detection in the regions where MCSCF states undergo rapid changes i.e. near avoided crossings or conical intersections. Finally, the modified overlap matrix is decomposed using a dedicated
procedure and the optimal new mapping for the new states is computed. This scheme is very cheap and simple and typically works well but cannot ideally handle all possible situations. It may be
modified or retuned in the future Firefly versions. Note, tracking will abort the job if the number of available CI vectors is less than the number of MCSCF states to track. The latter is defined by
the $mcscf ntrack= option as described above.
The $track input group is used to control the details of MCSCF state tracking. This group contains the following keywords:
tol The scaling factor for diagonal of the overlap matrix. The default value is 1.2
update Logical variable. If set, reference vectors will be updated and replaced by the remapped current CI vectors at the end of each MCSCF iteration. If not set, the CI vectors from the very first
MCSCF iteration at the initial geometry will be used as the reference vectors throughout all calculations. The default is not to update reference vectors i.e. update=.false.
freeze Logical variable. If set, the final remapping scheme of the first of three MCSCF calculations used to compute SS gradient for SA-MCSCF will be applied “as is” during the second and third stage
of gradient computations. Otherwise, the dynamic tracking will be active throughout all three MCSCF procedures. Normally, this flag must be set to get reliable results is the MCSCF states are
quasi-degenerated. The default is freeze=.true.
reset Logical variable. If set, state tracking will be reset at the beginning of each MCSCF computations and the reference vectors will be re-initialized by the vectors from the first CI step.
Default is reset=.false.
sticky Logical variable to be used together with reset option. If reset and sticky are both set, state tracking will be reset at the beginning of each MCSCF computations but the existing reference
vectors will be reused. Default is sticky=.false.
delciv Logical variable. At the end of each CI procedure, converged CI vectors are stored in the special file. If delciv is set to .true., the file with converged CI vectors of previous CI stage will
be deleted before each new CI step so that these old vectors will not be used as the initial guess for the new CI procedure. The default value is to reuse old CI vectors as the initial guess and thus
to keep the file intact i.e. delciv=.false. | {"url":"http://classic.chem.msu.su/gran/gamess/ss-gradients.html","timestamp":"2024-11-05T19:20:46Z","content_type":"text/html","content_length":"22545","record_id":"<urn:uuid:aa62e083-3c69-498c-98ae-5a4b92c8e254>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00727.warc.gz"} |
32.17 centimeters per square second to meters per square second
32 Centimeters per square second = 0.32 Meters per square second
Acceleration Converter - Centimeters per square second to meters per square second - 32.17 meters per square second to centimeters per square second
This conversion of 32.17 centimeters per square second to meters per square second has been calculated by multiplying 32.17 centimeters per square second by 0.01 and the result is 0.3217 meters per
square second. | {"url":"https://unitconverter.io/centimeters-per-square-second/meters-per-square-second/32.17","timestamp":"2024-11-14T22:14:57Z","content_type":"text/html","content_length":"26991","record_id":"<urn:uuid:0567b7a2-6b5b-493f-9ba9-ca8bfc0306c9>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00107.warc.gz"} |
My take on Eden and Plymouth.
People are certainly free to hop in if they want, although i'm going to definitely be writing a few chapters.
Most of it was just 'random' numbers. I didn't quite really think about it.
I think TLC did a show like that, 21 kids and counting right?
So... I haven't seen the surprise. Is it still coming or is this a Christmas/New Years type of deal? | {"url":"https://forum.outpost2.net/index.php?topic=5769.0","timestamp":"2024-11-05T23:43:35Z","content_type":"application/xhtml+xml","content_length":"64307","record_id":"<urn:uuid:a246eeff-8edc-4f07-9fa3-6e0022513382>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00107.warc.gz"} |
TimeMixer: Exploring the Latest Model in Time Series Forecasting
Jul 23, 2024
Photo by sutirta budiman on Unsplash
The field of time series forecasting keeps evolving at a rapid pace, with many models being proposed and claiming state-of-the-art performance.
Deep learning models are now common methods for time series forecasting, especially on large datasets with many features.
Although numerous models have been proposed in recent years, such as the iTransformer, SOFTS, and TimesNet, their performance often falls short in other benchmarks against models like NHITS, PatchTST
and TSMixer.
In May 2024, a new model was proposed: TimeMixer. According to the original paper TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting, this model uses mixing of features along with
series decomposition in an MLP-based architecture to produce forecasts.
In this article, we first explore the inner workings of TimeMixer before running our own little benchmark in both short and long horizon forecasting tasks.
As always, make sure to read the original research article for more details.
Learn the latest time series forecasting techniques with my free time series cheat sheet in Python! Get the implementation of statistical and deep learning techniques, all in Python and
Let’s get started!
Discover TimeMixer
The motivation behind TimeMixer comes from the realization that time series data hold different information at different scales.
Illustrating different patterns arising at different scales in time series data. Image by the author.
In the figure above, we can see that depending on the scale at which we sample the data, different patterns arise.
Of course, at a small sampling scale (top of the figure), we have fine variations, while at a large sampling scale (as shown in the bottom portion of the figure), we see more coarse changes to the
series over a longer period of time.
Thus, TimeMixer looks to disentangle the microscopic and macroscopic information and apply feature mixing, which is an idea that was fully explored in TSMixer.
Architecture of TimeMixer
Below, we can see the general architecture of TimeMixer.
Overall architecture of TimeMixer. Image by S. Wang et al., “TimeMixer: Decomposable Multisclae Mixing For Time Series Forecasting.” Accessed: Jul. 19, 2024. [Online]. Available: https://arxiv.org/
In the figure above, we see that the input series is first downsampled using average pooling. That way, we can the decouple the fine from the coarse variations in the series.
The decoupled series is then sent to the Past-Decomposable-Mixing block, or PDM block, where information at different scales is learned by the model. Notice also a further decomposition, where trend
and seasonality are treated separately.
Finally, the model is sent to the Future-Multipredictor-Mixing block, or FMM block. This ensembles the prediction at each scale to get the final forecast.
Of course, there is much more information to learn about each step, so let’s cover them in more detail.
Past decomposable mixing (PDM)
The first step of average pooling the series at different scales is straightforward enough to move directly to the Past-Decomposable-Mixing block immediately.
Here, the series is further decomposed into its trend and seasonal components. Note that the seasonal component carries short-term and cyclical changes, while the trend component carries slow changes
over longer periods of time.
Notice that even when downsampled, a series can still exhibit trend and seasonal properties, hence the need of separating both components. Image by S. Wang et al., “TimeMixer: Decomposable Multisclae
Mixing For Time Series Forecasting.” Accessed: Jul. 19, 2024. [Online]. Available: https://arxiv.org/pdf/2405.14616
From the figure above, we can see that the top series is downsampled, but it still exhibits trend and seasonal properties. Therefore, it is beneficial to separate both components as they carry
different information.
To carry out this decomposition, the researchers reuse the logic from Autoformer, which basically uses average pooling to smooth out cyclical variations, thus highlighting the trend.
This is then combined with a Fourier transform, which transforms the input series into a function of frequencies and amplitude.
Example of a Fourier transform on a time series. Image by the author.
In the image above, we can see that the Fourier transform results in a function of amplitude and frequency. Frequencies at which the amplitude are the largest are then the most important, thus
indicating strong seasonal effects.
Seasonal mixing
Once the trend and seasonality components are separated, they both undergo mixing.
In the case of seasonal mixing, we realize that larger seasonal periods can be seen as aggregation of smaller seasonal periods.
For example, a weekly seasonality can appear due to the daily seasonality observed in the last seven days.
Thus, TimeMixer employs a bottom-up approach to seasonal mixing.
Illustrating seasonal mixing in TimeMixer. Image by S. Wang et al., “TimeMixer: Decomposable Multisclae Mixing For Time Series Forecasting.” Accessed: Jul. 19, 2024. [Online]. Available: https://
In the figure above, we can see that in seasonal mixing, we incorporate information from the fine-scale series up to the downsampled series.
Trend mixing
On the other hand, for trend mixing, TimeMixer uses a top-down approach, as shown below.
Illustrating trend mixing. Image by S. Wang et al., “TimeMixer: Decomposable Multisclae Mixing For Time Series Forecasting.” Accessed: Jul. 19, 2024. [Online]. Available: https://arxiv.org/pdf/
A top-down approach is used for the trend component, because noise from a fine-scale series can be introduced when trying to capture the macroscopic trend of the downsampled series.
Therefore, macro trends are used to further inform micro trends in this case.
This is how TimeMixer achieves mixing on different scales for both the trend and seasonality components.
Once this step is done, the data flows to the Future-Multipredictor-Mixing block
Future multipredictor mixing (FMM)
Here, the future-multipredictor-mixing block receives information at different scales. Therefore, it is responsible to aggregate this information to output the final prediction.
Ilustrating the future-multipredictor-mixing block. Image by S. Wang et al., “TimeMixer: Decomposable Multisclae Mixing For Time Series Forecasting.” Accessed: Jul. 19, 2024. [Online]. Available:
In the figure above, we can see what the FMM block looks like. Basically, because each predictor runs on a different scale, they are not all used at the same time.
For example, a predictor for on a finer scale will influence the prediction at many steps. On the other hand, a predictor on a coarse will influence the prediction at fewer time steps, since it
treats downsampled data.
Now that we have a deep understanding of TimeMixer and how it works, let’s apply it in our own little benchmark using Python.
Forecasting with TimeMixer
Let’s now work with TimeMixer and evaluate in both short and long horizon forecasting tasks.
For the short horizon benchmark, we use the M3 dataset, released under the Creative Commons license. This compiles yearly, quarterly and monthly data from different domains like demography, finance,
and others.
For the long horizon benchmark, we use the Electricity Transformer dataset (ETT) released under the Creative Commons License. This tracks the oil temperature of an electricity transformer from two
regions in a province of China. For both regions, we have a dataset sampled at each hour and every 15 minutes, for a total of four datasets. In our case, we only use the two datasets sampled at every
15 minutes.
Again, I extended the neuralforecast library with an adapted implementation of the TimeMixer model from their official repository. That way, we have a streamlined experience for using and testing
different forecasting models.
Note that at the time of writing this article, TimeMixer is not in a stable release of neuralforecast just yet.
To reproduce the results, you may need to clone the repository and work in this branch.
If the branch is merged, then you can run:
pip install git+https://github.com/Nixtla/neuralforecast.git
As always, the code for this experiment is available on GitHub.
Let’s get started!
Forecasting on a short horizon
First, let’s import the required packages. They will be used for both benchmarks. Notice that we use the datasetsforecast library to load the datasets in the format expected by neuralforecast.
import time
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from datasetsforecast.m3 import M3
from datasetsforecast.long_horizon import LongHorizon
from neuralforecast.core import NeuralForecast
from neuralforecast.losses.pytorch import MAE, MSE
from neuralforecast.models import TimeMixer, PatchTST, iTransformer, NHITS, NBEATS
from utilsforecast.losses import mae, mse, smape
from utilsforecast.evaluation import evaluate
Then, let’s define a function that load the dataset and with the appropriate frequency and horizon.
def get_dataset(name):
if name == 'M3-yearly':
Y_df, *_ = M3.load("./data", "Yearly")
horizon = 6
freq = 'Y'
elif name == 'M3-quarterly':
Y_df, *_ = M3.load("./data", "Quarterly")
horizon = 8
freq = 'Q'
elif name == 'M3-monthly':
Y_df, *_ = M3.load("./data", "Monthly")
horizon = 18
freq = 'M'
return Y_df, horizon, freq
Now, we can initialize our models and run the benchmark.
Here, we compare TimeMixer to NHITS and NBEATS, which are notoriously fast and accurate models in this type of task.
To run this experiment, we first initialize an empty list to store our results and start a for loop over each dataset. Once the dataset is loaded, we split it into a training and a test set.
results = []
DATASETS = ['M3-yearly', 'M3-quarterly', 'M3-monthly']
for dataset in DATASETS:
Y_df, horizon, freq = get_dataset(dataset)
test_df = Y_df.groupby('unique_id').tail(horizon)
train_df = Y_df.drop(test_df.index).reset_index(drop=True)
Inside that same for loop, we initialize our models. Here, we keep the default parameters for all models. The updated code then becomes:
results = []
DATASETS = ['M3-yearly', 'M3-quarterly', 'M3-monthly']
for dataset in DATASETS:
Y_df, horizon, freq = get_dataset(dataset)
test_df = Y_df.groupby('unique_id').tail(horizon)
train_df = Y_df.drop(test_df.index).reset_index(drop=True)
timemixer_model = TimeMixer(input_size=2*horizon,
nbeats_model = NBEATS(input_size=2*horizon,
nhits_model = NHITS(input_size=2*horizon,
Notice that we train for a maximum of 1000 steps, but stop training if the loss does not improve over three training steps.
Once the models are initialized, we can fit them and make predictions. Here, we also track the time it takes to complete the training process.
The update code block is shown below.
results = []
DATASETS = ['M3-yearly', 'M3-quarterly', 'M3-monthly']
for dataset in DATASETS:
Y_df, horizon, freq = get_dataset(dataset)
test_df = Y_df.groupby('unique_id').tail(horizon)
train_df = Y_df.drop(test_df.index).reset_index(drop=True)
timemixer_model = TimeMixer(input_size=2*horizon,
nbeats_model = NBEATS(input_size=2*horizon,
nhits_model = NHITS(input_size=2*horizon,
MODELS = [timemixer_model, nbeats_model, nhits_model]
MODEL_NAMES = ['TimeMixer', 'NBEATS', 'NHITS']
for i, model in enumerate(MODELS):
nf = NeuralForecast(models=[model], freq=freq)
start = time.time()
nf.fit(train_df, val_size=horizon)
preds = nf.predict()
end = time.time()
elapsed_time = round(end - start,0)
preds = preds.reset_index()
test_df = pd.merge(test_df, preds, 'left', ['ds', 'unique_id'])
All that is left to do is the evaluation of the models. Here, we use the mean absolute error (MAE) and symmetric mean absolute percentage error (sMAPE), using the utilsforecast library.
The full code block thus becomes:
results = []
DATASETS = ['M3-yearly', 'M3-quarterly', 'M3-monthly']
for dataset in DATASETS:
Y_df, horizon, freq = get_dataset(dataset)
test_df = Y_df.groupby('unique_id').tail(horizon)
train_df = Y_df.drop(test_df.index).reset_index(drop=True)
timemixer_model = TimeMixer(input_size=2*horizon,
nbeats_model = NBEATS(input_size=2*horizon,
nhits_model = NHITS(input_size=2*horizon,
MODELS = [timemixer_model, nbeats_model, nhits_model]
MODEL_NAMES = ['TimeMixer', 'NBEATS', 'NHITS']
for i, model in enumerate(MODELS):
nf = NeuralForecast(models=[model], freq=freq)
start = time.time()
nf.fit(train_df, val_size=horizon)
preds = nf.predict()
end = time.time()
elapsed_time = round(end - start,0)
preds = preds.reset_index()
test_df = pd.merge(test_df, preds, 'left', ['ds', 'unique_id'])
evaluation = evaluate(
metrics=[mae, smape],
evaluation = evaluation.drop(['unique_id'], axis=1).groupby('metric').mean().reset_index()
model_mae = evaluation[f"{MODEL_NAMES[i]}"][0]
model_smape = evaluation[f"{MODEL_NAMES[i]}"][1]
results.append([dataset, MODEL_NAMES[i], round(model_mae, 0), round(model_smape*100,2), elapsed_time])
results_df = pd.DataFrame(data=results, columns=['dataset', 'model', 'mae', 'smape', 'time'])
results_df.to_csv('./M3_benchmark.csv', header=True, index=False)
Once this is done running, the following results were obtained.
MAE, sMAPE and speed of TimeMixer, N-BEATS and NHITS on the M3 dataset. Time is reported in seconds. Image by the author.
From the table above, we can see that NHITS achieves the top performance overall.
Also, TimeMixer definitely takes the longest to run, being about seven times slower than the fastest model. Furthermore, it is far from achieving competitive error metrics when compared to NHITS and
It seems that the performance of TimeMixer on for short horizon forecasting is underwhelming, so let’s test it for longer horizons.
Forecasting on a long horizon
The setup is similar for running the model on the ETT dataset.
Again, we define a function to load the dataset, its validation size, test size, and frequency.
def load_data(name):
if name == 'Ettm1':
Y_df, *_ = LongHorizon.load(directory='./', group='ETTm1')
Y_df['ds'] = pd.to_datetime(Y_df['ds'])
freq = '15T'
h = 96
val_size = 11520
test_size = 11520
elif name == 'Ettm2':
Y_df, *_ = LongHorizon.load(directory='./', group='ETTm2')
Y_df['ds'] = pd.to_datetime(Y_df['ds'])
freq = '15T'
h = 96
val_size = 11520
test_size = 11520
return Y_df, h, val_size, test_size, freq
Here, we test TimeMixer against PatchTST and iTransformer, as they usually perform best on long-horizon forecasting.
Specifically, we test on a horizon of 96 time steps.
For this experiment, we run cross-validation to have a more robust assessment of the each model’s performance. We also use the optimal parameters as reported by their respective papers.
DATASETS = ['Ettm1', 'Ettm2']
for dataset in DATASETS:
Y_df, horizon, val_size, test_size, freq = load_data(dataset)
timemixer_model = TimeMixer(input_size=horizon,
patchtst_model = PatchTST(input_size=horizon,
iTransformer_model = iTransformer(input_size=horizon,
models = [timemixer_model, patchtst_model, iTransformer_model]
nf = NeuralForecast(models=models, freq=freq)
nf_preds = nf.cross_validation(df=Y_df, val_size=val_size, test_size=test_size, n_windows=None)
nf_preds = nf_preds.reset_index()
evaluation = evaluate(df=nf_preds, metrics=[mae, mse], models=['TimeMixer', 'PatchTST', 'iTransformer'])
evaluation.to_csv(f'{dataset}_results.csv', index=False, header=True)
To evaluate the performance of these models, we use the mean absolute error (MAE) and mean squared error (MSE) as they are typically used for these benchmarks.
Note that the metrics are averaged for the prediction of all seven series in both ETTm1 and ETTm2 datasets.
The results of this experiment are shown below.
MAE and MSE for TimeMixer, PatchTST and iTransformer on the ETTm datasets. Results averaged for all seven series, on a forecast horizon of 96. Image by the author.
From the table above, we can see that PatchTST achieves the best result for ETTm1, while TimeMixer gets the top place for ETTm2.
Of course, this is not a comprehensive benchmark, but it is interesting to see TimeMixer perform much better in forecasting longer horizons than shorter horizons.
Therefore, if you are forecasting more than one seasonal period, TimeMixer should be considered in your tests.
TimeMixer is an MLP-based model that uses feature mixing at different scales to capture both micro and macro variations in time series and make predictions.
In our small benchmark, we noticed that TimeMixer is not suitable for short-horizon forecasting, as it does not perform better than NHITS or N-BEATS, and it is much slower than the latter.
However, for long-horizon forecasting, TimeMixer achieved very good results, and it was the top performing model for the ETTm2 dataset when compared to iTransformer and PatchTST.
Now, these benchmarks are far from being comprehensive, and they were meant to teach how to use the model in Python, and see it in action.
As always, I believe that each problem requires its own unique solution, so make sure to test TimeMixer and other models to find the optimal model for your specific scenario.
Thanks for reading! I hope that you enjoyed it and that you learned something new!
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S. Wang et al., “TimeMixer: Decomposable Multisclae Mixing For Time Series Forecasting.” Accessed: Jul. 19, 2024. [Online]. Available: https://arxiv.org/pdf/2405.14616
Original code repository of TimeMixer: GitHub
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The Foolproof What Is a Resultant in Physics Strategy
Hearsay, Deception and What Is a Resultant in Physics
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But any 2 vectors can be used as long as they are the exact vector quantity. However, in the event the scalar is negative then we have to alter the direction of the vector. A step-by-step way of
applying the head-to-tail method to fix the sum of a few vectors is given below. Here you may see very easily that you just can’t utilize algebra to bring the magnitudes of these 2 forces, as they
aren’t parallel to one another.
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triangle are both wonderful whole numbers. Pay a visit to the webpage on vector addition.
I like the process of locating vectors graphically. It’s one example of locating the elements of a vector. The technique isn’t applicable for adding two or more vectors or for adding vectors that
aren’t at 90-degrees to one another. Basically, you’d be using the head-to-tail method of vector addition.
Problems using these numerical techniques are located at the conclusion of several Chapters. The Fourth Edition contains a larger proportion of section-specific exercises in comparison with general
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scalar numbers since they also include things like information regarding direction.
Top Choices of What Is a Resultant in Physics
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The moment of a couple is known as the torque. Consequently, she is able to spin for quite a while. Please or to read the remainder of this content.
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Publications about 'discrete-time'
Publications about 'discrete-time'
1. E.D. Sontag. Polynomial Response Maps, volume 13 of Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin, 1979. [PDF] Keyword(s): realization theory, discrete-time, real
algebraic geometry.
│ │(This is a monograph based upon Eduardo Sontag's Ph.D. thesis. The contents are basically the same as the thesis, except for a very few revisions and extensions.) This work deals the │
│ │realization theory of discrete-time systems (with inputs and outputs, in the sense of control theory) defined by polynomial update equations. It is based upon the premise that the │
│ │natural tools for the study of the structural-algebraic properties (in particular, realization theory) of polynomial input/output maps are provided by algebraic geometry and │
│ │commutative algebra, perhaps as much as linear algebra provides the natural tools for studying linear systems. Basic ideas from algebraic geometry are used throughout in │
│ │system-theoretic applications (Hilbert's basis theorem to finite-time observability, dimension theory to minimal realizations, Zariski's Main Theorem to uniqueness of canonical │
│ │realizations, etc). In order to keep the level elementary (in particular, not utilizing sheaf-theoretic concepts), certain ideas like nonaffine varieties are used only implicitly │
│ │(eg., quasi-affine as open sets in affine varieties) or in technical parts of a few proofs, and the terminology is similarly simplified (e.g., "polynomial map" instead of "scheme │
│Abstract:│morphism restricted to k-points", or "k-space" instead of "k-points of an affine k-scheme"). │
Articles in journal or book chapters
1. B. DasGupta and E.D. Sontag. A polynomial-time algorithm for checking equivalence under certain semiring congruences motivated by the state-space isomorphism problem for hybrid systems. Theor.
Comput. Sci., 262(1-2):161-189, 2001. [PDF] [doi:http://dx.doi.org/10.1016/S0304-3975(00)00188-2] Keyword(s): hybrid systems, computational complexity.
│ │The area of hybrid systems concerns issues of modeling, computation, and control for systems which combine discrete and continuous components. The subclass of piecewise linear (PL) │
│ │systems provides one systematic approach to discrete-time hybrid systems, naturally blending switching mechanisms with classical linear components. PL systems model arbitrary │
│ │interconnections of finite automata and linear systems. Tools from automata theory, logic, and related areas of computer science and finite mathematics are used in the study of PL │
│ │systems, in conjunction with linear algebra techniques, all in the context of a "PL algebra" formalism. PL systems are of interest as controllers as well as identification models. │
│ │Basic questions for any class of systems are those of equivalence, and, in particular, if state spaces are equivalent under a change of variables. This paper studies this state-space │
│ │equivalence problem for PL systems. The problem was known to be decidable, but its computational complexity was potentially exponential; here it is shown to be solvable in │
│Abstract:│polynomial-time. │
2. X. Bao, Z. Lin, and E.D. Sontag. Finite gain stabilization of discrete-time linear systems subject to actuator saturation. Automatica, 36(2):269-277, 2000. [PDF] Keyword(s): discrete-time,
saturation, input-to-state stability, stabilization, ISS, bounded inputs.
│ │It is shown that, for neutrally stable discrete-time linear systems subject to actuator saturation, finite gain lp stabilization can be achieved by linear output feedback, for all p> │
│ │1. An explicit construction of the corresponding feedback laws is given. The feedback laws constructed also result in a closed-loop system that is globally asymptotically stable, and │
│Abstract:│in an input-to-state estimate. │
3. D. Nesic, A.R. Teel, and E.D. Sontag. Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems. Systems Control Lett., 38(1):49-60, 1999. [PDF] Keyword(s):
input to state stability, sampled-data systems, discrete-time systems, sampling, ISS.
│ │We provide an explicit KL stability or input-to-state stability (ISS) estimate for a sampled-data nonlinear system in terms of the KL estimate for the corresponding discrete-time │
│ │system and a K function describing inter-sample growth. It is quite obvious that a uniform inter-sample growth condition, plus an ISS property for the exact discrete-time model of a │
│ │closed-loop system, implies uniform ISS of the sampled-data nonlinear system; our results serve to quantify these facts by means of comparison functions. Our results can be used as an│
│ │alternative to prove and extend results of Aeyels et al and extend some results by Chen et al to a class of nonlinear systems. Finally, the formulas we establish can be used as a tool│
│Abstract:│for some other problems which we indicate. │
4. E.D. Sontag and F.R. Wirth. Remarks on universal nonsingular controls for discrete-time systems. Systems Control Lett., 33(2):81-88, 1998. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(97)
00117-5] Keyword(s): discrete time, controllability, real-analytic functions.
│ │For analytic discrete-time systems, it is shown that uniform forward accessibility implies the generic existence of universal nonsingular control sequences. A particular application │
│ │is given by considering forward accessible systems on compact manifolds. For general systems, it is proved that the complement of the set of universal sequences of infinite length is │
│ │of the first category. For classes of systems satisfying a descending chain condition, and in particular for systems defined by polynomial dynamics, forward accessibility implies │
│Abstract:│uniform forward accessibility. │
5. Y. Yang, E.D. Sontag, and H.J. Sussmann. Global stabilization of linear discrete-time systems with bounded feedback. Systems Control Lett., 30(5):273-281, 1997. [PDF] [doi:http://dx.doi.org/
10.1016/S0167-6911(97)00021-2] Keyword(s): discrete-time, saturation, bounded inputs.
│ │This paper deals with the problem of global stabilization of linear discrete time systems by means of bounded feedback laws. The main result proved is an analog of one proved for the │
│ │continuous time case by the authors, and shows that such stabilization is possible if and only if the system is stabilizable with arbitrary controls and the transition matrix has │
│ │spectral radius less or equal to one. The proof provides in principle an algorithm for the construction of such feedback laws, which can be implemented either as cascades or as │
│Abstract:│parallel connections (``single hidden layer neural networks'') of simple saturation functions. │
6. E.D. Sontag. Interconnected automata and linear systems: a theoretical framework in discrete-time. In R. Alur, T.A. Henzinger, and E.D. Sontag, editors, Proceedings of the DIMACS/SYCON workshop
on Hybrid systems III : verification and control, pages 436-448. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 1996. [PDF] Keyword(s): hybrid systems.
│ │This paper summarizes the definitions and several of the main results of an approach to hybrid systems, which combines finite automata and linear systems, developed by the author in │
│Abstract:│the early 1980s. Some related more recent results are briefly mentioned as well. │
7. Y. Wang and E.D. Sontag. Orders of input/output differential equations and state-space dimensions. SIAM J. Control Optim., 33(4):1102-1126, 1995. [PDF] [doi:http://dx.doi.org/10.1137/
S0363012993246828] Keyword(s): identifiability, observability, realization theory, real-analytic functions.
│ │This paper deals with the orders of input/output equations satisfied by nonlinear systems. Such equations represent differential (or difference, in the discrete-time case) relations │
│ │between high-order derivatives (or shifts, respectively) of input and output signals. It is shown that, under analyticity assumptions, there cannot exist equations of order less than │
│ │the minimal dimension of any observable realization; this generalizes the known situation in the classical linear case. The results depend on new facts, themselves of considerable │
│ │interest in control theory, regarding universal inputs for observability in the discrete case, and observation spaces in both the discrete and continuous cases. Included in the paper │
│Abstract:│is also a new and simple self-contained proof of Sussmann's universal input theorem for continuous-time analytic systems. │
8. F. Albertini and E.D. Sontag. Further results on controllability properties of discrete-time nonlinear systems. Dynam. Control, 4(3):235-253, 1994. [PDF] [doi:http://dx.doi.org/10.1007/BF01985073
] Keyword(s): discrete-time, nonlinear control.
│ │Controllability questions for discrete-time nonlinear systems are addressed in this paper. In particular, we continue the search for conditions under which the group-like notion of │
│ │transitivity implies the stronger and semigroup-like property of forward accessibility. We show that this implication holds, pointwise, for states which have a weak Poisson stability │
│Abstract:│property, and globally, if there exists a global "attractor" for the system. │
9. F. Albertini and E.D. Sontag. Discrete-time transitivity and accessibility: analytic systems. SIAM J. Control Optim., 31(6):1599-1622, 1993. [PDF] [doi:http://dx.doi.org/10.1137/0331075] Keyword
(s): controllability, discrete-time systems, accessibility, real-analytic functions.
│ │A basic open question for discrete-time nonlinear systems is that of determining when, in analogy with the classical continuous-time "positive form of Chow's Lemma", accessibility │
│ │follows from transitivity of a natural group action. This paper studies the problem, and establishes the desired implication for analytic systems in several cases: (i) compact state │
│ │space, (ii) under a Poisson stability condition, and (iii) in a generic sense. In addition, the paper studies accessibility properties of the "control sets" recently introduced in the│
│Abstract:│context of dynamical systems studies. Finally, various examples and counterexamples are provided relating the various Lie algebras introduced in past work. │
10. R. Koplon and E.D. Sontag. Linear systems with sign-observations. SIAM J. Control Optim., 31(5):1245-1266, 1993. [PDF] [doi:http://dx.doi.org/10.1137/0331059] Keyword(s): observability.
│ │This paper deals with systems that are obtained from linear time-invariant continuous- or discrete-time devices followed by a function that just provides the sign of each output. Such│
│ │systems appear naturally in the study of quantized observations as well as in signal processing and neural network theory. Results are given on observability, minimal realizations, │
│Abstract:│and other system-theoretic concepts. Certain major differences exist with the linear case, and other results generalize in a surprisingly straightforward manner. │
11. F. Albertini and E.D. Sontag. Transitivity and forward accessibility of discrete-time nonlinear systems. In Analysis of controlled dynamical systems (Lyon, 1990), volume 8 of Progr. Systems
Control Theory, pages 21-34. Birkhäuser Boston, Boston, MA, 1991.
12. B. Jakubczyk and E.D. Sontag. Controllability of nonlinear discrete-time systems: a Lie-algebraic approach. SIAM J. Control Optim., 28(1):1-33, 1990. [PDF] [doi:http://dx.doi.org/10.1137/0328001]
Keyword(s): discrete-time.
│ │This paper presents a geometric study of controllability for discrete-time nonlinear systems. Various accessibility properties are characterized in terms of Lie algebras of vector │
│Abstract:│fields. Some of the results obtained are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena appear. │
13. B. Jakubczyk and E.D. Sontag. Nonlinear discrete-time systems. Accessibility conditions. In Modern optimal control, volume 119 of Lecture Notes in Pure and Appl. Math., pages 173-185. Dekker, New
York, 1989. [PDF]
14. A. Arapostathis, B. Jakubczyk, H.-G. Lee, S. I. Marcus, and E.D. Sontag. The effect of sampling on linear equivalence and feedback linearization. Systems Control Lett., 13(5):373-381, 1989. [PDF]
[doi:http://dx.doi.org/10.1016/0167-6911(89)90103-5] Keyword(s): discrete-time, sampled-data systems, discrete-time systems, sampling.
│ │We investigate the effect of sampling on linearization for continuous time systems. It is shown that the discretized system is linearizable by state coordinate change for an open set │
│ │of sampling times if and only if the continuous time system is linearizable by state coordinate change. Also, it is shown that linearizability via digital feedback imposes highly │
│Abstract:│nongeneric constraints on the structure of the plant, even if this is known to be linearizable with continuous-time feedback. │
15. E.D. Sontag. A Chow property for sampled bilinear systems. In C.I. Byrnes, C.F. Martin, and R. Saeks, editors, Analysis and Control of Nonlinear Systems, pages 205-211. North Holland, Amsterdam,
1988. [PDF] Keyword(s): discrete-time, bilinear systems.
│ │This paper studies accessibility (weak controllability) of bilinear systems under constant sampling rates. It is shown that the property is preserved provided that the sampling period│
│ │satisfies a condition related to the eigenvalues of the autonomous dynamics matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the │
│Abstract:│linear case, and which, for observability, results in the classical Nyquist theorem. │
16. E.D. Sontag. Reachability, observability, and realization of a class of discrete-time nonlinear systems. In Encycl. of Systems and Control, pages 3288-3293. Pergamon Press, 1987. Keyword(s):
17. E.D. Sontag. An eigenvalue condition for sample weak controllability of bilinear systems. Systems Control Lett., 7(4):313-315, 1986. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(86)90045-9]
Keyword(s): discrete-time.
│ │Weak controllability of bilinear systems is preserved under sampling provided that the sampling period satisfies a condition related to the eigenvalues of the autonomous dynamics │
│Abstract:│matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the linear case. │
18. B.W. Dickinson and E.D. Sontag. Dynamic realizations of sufficient sequences. IEEE Trans. Inform. Theory, 31(5):670-676, 1985. [PDF] Keyword(s): realization theory, statistics, innovations,
sufficient statistics.
│ │Let Ul, U2, ... be a sequence of observed random variables and (T1(U1),T2(Ul,U2),...) be a corresponding sequence of sufficient statistics (a sufficient sequence). Under certain │
│ │regularity conditions, the sufficient sequence defines the input/output map of a time-varying, discrete-time nonlinear system. This system provides a recursive way of updating the │
│ │sufficient statistic as new observations are made. Conditions are provided assuring that such a system evolves in a state space of minimal dimension. Several examples are provided to │
│ │illustrate how this notion of dimensional minimality is related to other properties of sufficient sequences. The results can be used to verify the form of the minimum dimension │
│Abstract:│(discrete-time) nonlinear filter associated with the autoregressive parameter estimation problem. │
19. E.D. Sontag. On finitary linear systems. Kybernetika (Prague), 15(5):349-358, 1979. [PDF] Keyword(s): systems over rings.
│ │An abstract operator approach is introduced, permitting a unified study of discrete- and continuous-time linear control systems. As an application, an algorithm is given for deciding │
│ │if a linear system can be built from any fixed set of linear components. Finally, a criterion is given for reachability of the abstract systems introduced, giving thus a unified proof│
│Abstract:│of known reachability results for discrete-time, continuous-time, and delay-differential systems. │
20. E.D. Sontag. Realization theory of discrete-time nonlinear systems. I. The bounded case. IEEE Trans. Circuits and Systems, 26(5):342-356, 1979. [PDF] Keyword(s): discrete-time systems, nonlinear
systems, realization theory, bilinear systems, state-affine systems.
│ │A state-space realization theory is presented for a wide class of discrete time input/output behaviors. Although In many ways restricted, this class does include as particular cases │
│ │those treated in the literature (linear, multilinear, internally bilinear, homogeneous), as well as certain nonanalytic nonlinearities. The theory is conceptually simple, and │
│ │matrix-theoretic algorithms are straightforward. Finite-realizability of these behaviors by state-affine systems is shown to be equivalent both to the existence of high-order input/ │
│Abstract:│output equations and to realizability by more general types of systems. │
21. E.D. Sontag and Y. Rouchaleau. On discrete-time polynomial systems. Nonlinear Anal., 1(1):55-64, 1976. [PDF] Keyword(s): identifiability, observability, polynomial systems, realization theory,
│ │Considered here are a type of discrete-time systems which have algebraic constraints on their state set and for which the state transitions are given by (arbitrary) polynomial │
│ │functions of the inputs and state variables. The paper studies reachability in bounded time, the problem of deciding whether two systems have the same external behavior by applying │
│ │finitely many inputs, the fact that finitely many inputs (which can be chosen quite arbitrarily) are sufficient to separate those states of a system which are distinguishable, and │
│Abstract:│introduces the subject of realization theory for this class of systems. │
1. Z-P. Jiang, E.D. Sontag, and Y. Wang. Input-to-state stability for discrete-time nonlinear systems. In Proc. 14th IFAC World Congress, Vol E (Beijing), pages 277-282, 1999. [PDF] Keyword(s):
input to state stability, input to state stability, ISS, discrete-time.
│ │This paper studies the input-to-state stability (ISS) property for discrete-time nonlinear systems. We show that many standard ISS results may be extended to the discrete-time case. │
│ │More precisely, we provide a Lyapunov-like sufficient condition for ISS, and we show the equivalence between the ISS property and various other properties, as well as provide a small │
│Abstract:│gain theorem. │
2. D. Nesic, A.R. Teel, and E.D. Sontag. On stability and input-to-state stability ${\cal K}{\cal L}$ estimates of discrete-time and sampled-data nonlinear systems. In Proc. American Control Conf.,
San Diego, June 1999, pages 3990-3994, 1999. Keyword(s): input to state stability, sampled-data systems, discrete-time systems, sampling.
3. X. Bao, Z. Lin, and E.D. Sontag. Some new results on finite gain $l_p$ stabilization of discrete-time linear systems subject to actuator saturation. In Proc. IEEE Conf. Decision and Control,
Tampa, Dec. 1998, IEEE Publications, 1998, pages 4628-4629, 1998. Keyword(s): saturation, bounded inputs.
4. F. Albertini and E.D. Sontag. Controllability of discrete-time nonlinear systems. In Systems and Networks: Mathematical Theory and Applications, Proc. MTNS '93, Vol. 2, Akad. Verlag, Regensburg,
pages 35-38, 1993.
5. F. Albertini and E.D. Sontag. Identifiability of discrete-time neural networks. In Proc. European Control Conf., Groningen, June 1993, pages 460-465, 1993. Keyword(s): machine learning, neural
networks, recurrent neural networks.
6. F. Albertini and E.D. Sontag. Accessibility of discrete-time nonlinear systems, and some relations to chaotic dynamics. In Proc. Conf. Inform. Sci. and Systems, John Hopkins University, March
1991, pages 731-736, 1991.
7. E.D. Sontag and H.J. Sussmann. Accessibility under sampling. In Proc. IEEE Conf. Dec. and Control, Orlando, Dec. 1982, 1982. [PDF] Keyword(s): discrete-time.
│ │This note addresses the following problem: Find conditions under which a continuous-time (nonlinear) system gives rise, under constant rate sampling, to a discrete-time system which │
│Abstract:│satisfies the accessibility property. │
8. E.D. Sontag. Algebraic-geometric methods in the realization of discrete-time systems. In Proc. Conf. Inform. Sci. and Systems, John Hopkins Univ. (1978), pages 158-162, 1978.
1. E.D. Sontag and F.R. Wirth. Remarks on universal nonsingular controls for discrete-time systems. Technical report 381, Institute for Dynamical Systems, University of Bremen, 1996.
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Last modified: Wed Oct 30 12:09:15 2024
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Day 20: Calculus
Mathematica is the de facto standard for symbolic differentiation and integration. But many other languages also have great facilities for Calculus. For example, R has the deriv() function in the
base stats package as well as the numDeriv, Deriv and Ryacas packages. Python has NumPy and SymPy.
Let’s check out what Julia has to offer.
Numerical Differentiation
First load the Calculus package.
The derivative() function will evaluate the numerical derivative at a specific point.
derivative(x -> sin(x), pi)
derivative(sin, pi, :central) # Options: :forward, :central or :complex
There’s also a prime notation which will do the same thing (but neatly handle higher order derivatives).
f(x) = sin(x);
f'(0.0) # cos(x)
Functions exist for second derivatives, gradients (for multivariate functions) and Hessian matrices too. Related packages for derivatives are ForwardDiff and ReverseDiffSource.
Symbolic Differentiation
Symbolic differentiation works for univariate and multivariate functions expressed as strings.
differentiate("sin(x)", :x)
differentiate("sin(x) + exp(-y)", [:x, :y])
2-element Array{Any,1}:
It also works for expressions.
differentiate(:(x^2 \* y \* exp(-x)), :x)
:((2x) \* y \* exp(-x) + x^2 \* y \* -(exp(-x)))
differentiate(:(sin(x) / x), :x)
:((cos(x) * x - sin(x)) / x^2)
Have a look at the JuliaDiff project which is aggregating resources for differentiation in Julia.
Numerical Integration
Numerical integration is a snap.
integrate(x -> 1 / (1 - x), -1 , 0)
Compare that with the analytical result. Nice.
diff(map(x -> - log(1 - x), [-1, 0]))
1-element Array{Float64,1}:
By default the integral is evaluated using Simpson’s Rule. However, we can also use Monte Carlo integration.
integrate(x -> 1 / (1 - x), -1 , 0, :monte_carlo)
Symbolic Integration
The SymPy package supports the SymPy Python library. You might want to restart your Julia session before loading it.
Revisiting the same definite integral from above we find that we now have an analytical expression as the result.
integrate(1 / (1 - x), (x, -1, 0))
To perform symbolic integration we need to first define a symbolic object using Sym().
x = Sym("x"); # Creating a "symbolic object"
Sym (constructor with 6 methods)
sin(x) |> typeof # f(symbolic object) is also a symbolic object
Sym (constructor with 6 methods)
There’s more to be said about symbolic objects (they are the basis of pretty much everything in SymPy), but we are just going to jump ahead to constructing a function and integrating it.
f(x) = cos(x) - sin(x) * cos(x);
integrate(f(x), x)
sin (x)
- ─────── + sin(x)
What about an integral with constant parameters? No problem.
k = Sym("k");
integrate(1 / (x + k), x)
log(k + x)
We have really only grazed the surface of SymPy. The capabilities of this package are deep and broad. Seriously worthwhile checking out the documentation if you are interested in symbolic
I’m not ready to throw away my dated version of Mathematica just yet, but I’ll definitely be using this functionality often. Come back tomorrow when I’ll take a look at solving differential equations
with Julia. | {"url":"https://datawookie.dev/blog/2015/09/monthofjulia-day-20-calculus/","timestamp":"2024-11-04T15:22:16Z","content_type":"text/html","content_length":"23989","record_id":"<urn:uuid:8f69d17b-0713-4dd2-ac7a-bc958468bd4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00254.warc.gz"} |
What is an Integer?
Integers are positive whole numbers and their opposites, including zero.
You might be wondering, what is an opposite?
The opposite of 2 is -2. The opposite of 24 is -24. The opposite of -13 is 13. The opposite of 0... Well, it is just 0.
There is no opposite because zero is neither positive nor negative.
We can list the integers on a number line and then add arrows to show that integers continue on and on in both directions.
Integers are helpful when modeling real life situations. Here are some examples:
We can compare integers the same way that we compare positive whole numbers. Start by using a number line. The farther the number is to the right, the larger the value is.
Compare -9 and -2.
Here, -2 is farther to the right. Therefore, -2 is greater than -9. -2 > -9.
Compare -8 and 3.
You might be tempted to say that -8 is larger than 3 because 8 is larger than 3. Let's first look on the number line.
Here we can see that 3 is larger than -8. 3 > -8.
A positive number will always be greater than a negative.
Integers will become helpful when solving problems like those that include temperatures (above and below zero degrees), heights above and below sea level (sea level acts as 0), or even when we work
with money (where less than 0 shows we owe someone money).
Related Links: Math Fractions Factors | {"url":"https://softschools.com/math/topics/what_is_an_integer/","timestamp":"2024-11-03T20:04:14Z","content_type":"application/xhtml+xml","content_length":"16701","record_id":"<urn:uuid:cf79faa0-6009-4d47-a433-5f4b4959d1a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00666.warc.gz"} |
ils to micrometers
Conversion mils to micrometers, mil to μm.
The conversion factor is 25.4; so 1 mil = 25.4 micrometers. In other words, the value in mil multiply by 25.4 to get a value in μm. The calculator answers the questions:
110 mil is how many μm
? or
change mil to μm
. Convert mil to μm.
Conversion result:
1 mil = 25.4 μm
1 mil is 25.4 micrometers.
Choose other units (length)
Conversion of a length unit in word math problems and questions
more math problems » | {"url":"https://www.hackmath.net/en/calculator/conversion-of-length-units?unit1=mil&unit2=%CE%BCm&dir=1","timestamp":"2024-11-12T18:26:28Z","content_type":"text/html","content_length":"25690","record_id":"<urn:uuid:a0152e1a-6fc9-4dd3-9c07-b3838f2415fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00799.warc.gz"} |
Phase Portraits of Complex Functions with the R Package
Visualizing the complex plane
The package does not contain many functions, but provides a very versatile workhorse called phasePortrait. We will explore some of its key features now. Let us first consider a function that maps a
complex number \(z \in \mathbb{C}\) on itself, i.e. \(f(z)=z\). After attaching the package with library(viscomplexr), a phase portrait of this function is obtained very easily with:
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
xlab = "real", ylab = "imaginary", main = "f(z) = z",
nCores = 2) # Probably not required on your machine (see below)
# Note the argument 'nCores' which determines the number of parallel processes to
# be used. Setting nCores = 2 has been done here and in all subsequent
# examples as CRAN checks do not allow more parallel processes.
# For normal work, we recommend not to define nCores at all which will make
# phasePortrait use all available cores on your machine.
# The progress messages phasePortrait is writing to the console can be
# suppressed by including 'verbose = FALSE' in the call (see documentation).
Such a phase portrait is based on the polar representation of complex numbers. Any complex number \(z\) can be written as \(z=r\cdot\mathrm{e}^{\mathrm{i}\varphi}\) or equivalently \(z=r\cdot(\cos\
varphi+\mathrm{i}\cdot\sin\varphi)\), whereby \(r\) is the modulus and the angle \(\varphi\) is the argument. The argument, also called the phase angle, is the angle in the origin of the complex
number plane between the real axis and the position vector of the number in counter-clockwise orientation. The main feature of a phase portrait is to translate the argument into a color. In addition,
there are options for visualizing the modulus or, more precisely, its relative change.
The translation of the phase angle \(\varphi\) into a color follows the hsv color model, where radian values of \(\varphi=0+k\cdot2\pi\), \(\varphi=\frac{2\pi}{3}+k\cdot2\pi\), and \(\varphi=\frac{4\
pi}{3}+k\cdot2\pi\) with \(k\in\mathbb{Z}\) translate into the colors red, green, and blue, respectively, with a continuous transition of colors with values between. As all numbers with the same
argument \(\varphi\) obtain the same color, the numbers of the complex plane as visualized in the Figure above are colored along the chromatic cycle. In order to add visual structure, argument values
of \(\varphi=\frac{2\pi}{9}\), i.e. \(40°\) and their integer multiples are emphasized by black lines. Note that each of these lines follows exactly one color. Moreover, the zones between two
neighboring arguments \(\varphi_1=k\cdot\frac{2\pi}{9}\) and \(\varphi_2=(k+1)\cdot\frac{2\pi}{9}\) with \(k\in\mathbb{Z}\) are shaded in a way that the brightness of the colors inside one such zone
increases with increasing \(\varphi\), i.e. in counterclockwise sense of rotation.
The other lines visible in the figure above relate to the modulus \(r\). One such line follows the same value of \(r\); it is thus obvious that each of these iso-modulus lines must form a concentric
circle on the complex number plane (see the figure above). The distance between neighboring iso-modulus lines is chosen so that it always indicates the same relative change. For reasons to talk about
later (see also Wegert 2012), the default setting of the function phasePortrait is a relative change of \(b=\mathrm{e}^{2\pi/9}\) which is very close to \(2\). Thus, with a grain of salt, the modulus
of the complex numbers doubles or halves when moving from one iso-modulus line to the other. In the phase portrait, the zones between two adjacent iso-modulus lines are shaded in a way that the
colors inside such a zone become brighter in the direction of increasing modulus. The lines themselves are located at the moduli \(r=b^k\), with \(k\in\mathbb{Z}\). This is nicely visible in the
phase portrait above, where the outmost circular iso-modulus line indicates (approximately, as \(b\) is not exactly \(2\)) \(r=2^3=8\). Moving inwards, the following iso-modulus lines are at
(approximately) \(r=2^2=4\), \(r=2^1=2\), \(r=2^0=1\), \(r=2^{-1}=\frac{1}{2}\), \(r=2^{-2}=\frac{1}{4}\), etc. Obviously, as the modulus of the numbers on the complex plane is their distance from
the origin, the width of the concentric rings formed by adjacent iso-modulus lines approximately doubles from ring to ring when moving outwards.
Visual structuring - the argument pType
When working with the function phasePortrait, it might not always be desirable to display all of these reference lines and zonings. The argument pType allows for four different options as illustrated
in the next example:
# divide graphics device into four regions and adjust plot margins
op <- par(mfrow = c(2, 2),
mar = c(0.25, 0.55, 1.10, 0.25))
# plot four phase portraits with different choices of pType
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "p",
main = "pType = 'p'", axes = FALSE, nCores = 2)
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pa",
main = "pType = 'pa'", axes = FALSE, nCores = 2)
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pm",
main = "pType = 'pm'", axes = FALSE, nCores = 2)
phasePortrait("z", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pma",
main = "pType = 'pma'", axes = FALSE, nCores = 2)
par(op) # reset the graphics parameters to their previous values
As evident from the figure above, setting ptype to ‘p’ displays a phase portrait in the literal sense, i.e. only the phase of the complex numbers is displayed and nothing else. The option ‘pa’ adds
reference lines for the argument, the option ‘pm’ adds iso-modulus lines, and the (default) option ‘pma’ adds both. In addition to these options, the example shows phasePortrait in combination with R
’s base graphics. The first and the last line of the code chunk set and reset global graphics parameters, and inside the calls to phasePortrait, we use the arguments main (diagram title) and axes
which are generic plot arguments.
Visual structuring - the arguments pi2Div and logBase
For demonstrating options to adjust the density of the argument and modulus reference lines, consider the rational function \[ f(z)=\frac{(3+2\mathrm{i}+z)(-5+5\mathrm{i}+z)}{(-2-2\mathrm{i}+z)^2} \]
Evidently, this function has two zeroes, \(z_1=-3-2\mathrm{i}\), and \(z_2=5-5\mathrm{i}\). It also has a second order pole at \(z_3=2+2\mathrm{i}\). We make a phase portrait of this function over
the same cutout of the complex plane as we did in the figures above. When calling phasePortrait with such simple functions, it is most convenient to define them as as a quoted character string in R
syntax containing the variable \(z\). Run the code below for displaying the phase portrait (active 7” x 7” screen graphics device suggested, e.g. x11()).
op <- par(mar = c(5.1, 4.1, 2.1, 2.1), cex = 0.8) # adjust plot margins
# and general text size
xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
xlab = "real", ylab = "imaginary",
nCores = 2) # Increase or leave out for higher performance
par(op) # reset the graphics parameters to their previous values
The resulting figure nicely displays the function’s two zeroes and the pole. Note that all colors meet in zeroes and poles. Around zeroes, the colors cycle counterclockwise in the order red, green,
blue, while this order is reversed around poles. For \(n\)^th order (\(n\in\mathbb{N}\)) zeroes and poles, the cycle is passed through \(n\) times. I recommend to check this out with examples of your
Now, suppose we want to change the density of the reference lines for the phase angle \(\varphi\). This can be done by way of the argument pi2Div. For usual applications, pi2Div should be a natural
number \(n\:(n\in\mathbb{N})\). It defines the angle \(\Delta\varphi\) between two adjacent reference lines as a fraction of the round angle, i.e. \(\Delta\varphi=\frac{2\pi}{n}\). The default value
of pi2Div is 9, i.e. \(\Delta\varphi=\frac{2\pi}{9}=40°\). Let’s plot our function in three flavors of pi2Div, namely, 6, 9 (the default), and 18, resulting in \(\Delta\varphi\) values of \(\frac{\
pi}{3}=60°\), \(\frac{2\pi}{9}=40°\), and \(\frac{\pi}{9}=20°\). In order to suppress the iso-modulus lines and display the argument reference lines only, we are using pType = "pa". Visualize this by
running the code below (active 7” x 2.8” screen graphics device suggested, e.g. x11(width = 7, height = 2.8)).
# divide graphics device into three regions and adjust plot margins
op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
for(n in c(6, 9, 18)) {
phasePortrait("(3+2i+z)*(-5+5i+z)/(-2-2i+z)^2", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
pi2Div = n, pType = "pa", axes = FALSE, nCores = 2)
# separate title call (R base graphics) for nicer line adjustment, just cosmetics
title(paste("pi2Div =", n), line = -1.2)
par(op) # reset graphics parameters to previous values
So far, this is exactly, what had to be expected. But see what happens when we choose the default pType, "pma" which also adds modulus reference lines:
# divide graphics device into three regions and adjust plot margins
op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
for(n in c(6, 9, 18)) {
phasePortrait("(3+2i+z)*(-5+5i+z)/(-2-2i+z)^2", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
pi2Div = n, pType = "pma", axes = FALSE, nCores = 2)
# separate title call (R base graphics) for nicer line adjustment, just cosmetics
title(paste("pi2Div =", n), line = -1.2)
par(op) # reset graphics parameters to previous values
Evidently, the choice of pi2Div has influenced the density of the iso-modulus lines. This is because, by default, the parameter logBase, which controls how dense the iso-modulus lines are arranged,
is linked to pi2Div. As stated above, pi2Div is usually a natural number \(n\:(n \in\mathbb{N})\), and logBase is the real number \(b\:(b\in\mathbb{R})\) which defines the moduli \(r=b^k\:(k\in\
mathbb{Z})\) where the reference lines are drawn. When \(n\) is given, the default definition of \(b\) is \(b=\mathrm{e}^{2\pi/n}\). In the default case, \(n=9\), this results in \(b\approx2.009994\)
. Thus, by default, moving from one iso-modulus line to the adjacent one means almost exactly doubling or halving the modulus, depending on the direction. For the other two cases \(n=6\) and \(n=18\)
, the resulting values for \(b\) are \(b\approx2.85\) and \(b\approx1.42\), the latter obviously being the square root of \(\mathrm{e}^{2\pi/9}\). For \(n=9\), the modulus (approximately) doubles or
halves when traversing two adjacent iso-modulus lines.
Before we demonstrate the special property of this linkage between \(n\) and \(b\), i.e. between pi2Div and logBase, we shortly show, that they can be decoupled in phasePortrait without any
complication. In the following example, we want to define the density of the iso-modulus lines in a way that the modulus triples when traversing three zones in the direction of ascending moduli.
Clearly, this requires to define logBase as \(b=\sqrt[3]{3}\approx1.44\). Thus, when moving from one iso-modulus line to the next higher one, the modulus has increased by a factor of about \(1.4\).
One line further, it has about doubled (\({\sqrt[3]{3}}^{2}\approx2.08\)), and another line further it has exactly tripled. While varying pi2Div exactly as in the previous example, we now keep
logBase constant at \(\sqrt[3]{3}\). Run the code below for visualizing this (active 7” x 2.8” screen graphics device suggested, e.g. x11(width = 7, height = 2.8)).
# divide graphics device into three regions and adjust plot margins
op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
for(n in c(6, 9, 18)) {
phasePortrait("(3+2i+z)*(-5+5i+z)/(-2-2i+z)^2", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
pi2Div = n, logBase = sqrt(3), pType = "pma", axes = FALSE, nCores = 2)
# separate title call (R base graphics) for nicer line adjustment, just cosmetics
title(paste("pi2Div = ", n, ", logBase = 3^(1/3)", sep = ""), line = -1.2)
par(op) # reset graphics parameters to previous values
In order to understand why by default the parameters pi2Div and logBase are linked as described above, we consider the exponential function \(f(z)=\mathrm{e}^z\). We can write \(z=r\cdot(\cos\varphi+
\mathrm{i}\cdot\sin\varphi)\) and thus \(f(z)=\mathrm{e}^{r\cdot(\cos\varphi+\mathrm{i}\cdot\sin\varphi)}\) or \(w=f(z)=\mathrm{e}^{r\cdot\cos\varphi}\cdot\mathrm{e}^{\mathrm{i}\cdot r\cdot\sin\
varphi}\). The modulus of \(w\) is \(\mathrm{e}^{r\cdot\cos\varphi}\) and its argument is \(r\cdot\sin\varphi\) with \(\Re(z)=r\cdot\cos\varphi\) and \(\Im(z)=r\cdot \sin\varphi\). So, the modulus
and the argument of \(w=\mathrm{e}^z\) depend solely on the real and the imaginary part of \(z\), respectively. This can be easily verified with a phase portrait of \(f(z)=\mathrm{e}^z\). Run the
code below for displaying the phase portrait (active 7” x 7” screen graphics device suggested, e.g. x11()). Note that in the call to phasePortrait we hand over the exp function directly as an object.
Alternatively, the quoted strings "exp(z)" or "exp" can be used as well (see section ways to provide functions to phasePortrait below).
op <- par(mar = c(5.1, 4.1, 2.1, 2.1), cex = 0.8) # adjust plot margins
# and general text size
phasePortrait(exp, xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5), pType = "pm",
xlab = "real", ylab = "imaginary", nCores = 2)
par(op) # reset graphics parameters to previous values
If we now define the argument pi2Div as a number \(n\:(n\in\mathbb{N})\) and use it for determining the angular difference \(\Delta\varphi=\frac{2\pi}{n}\) between two subsequent phase angle
reference lines, our default link between pi2Div and logBase (which is the ratio \(b\) of the moduli at two subsequent iso-modulus lines) establishes \(b=\mathrm{e}^{\Delta\varphi}\). This means, if
we add \(\Delta\varphi\) to the argument of any \(w=\mathrm{e}^z\:(z\in\mathbb{C})\) or increase its modulus by the factor \(\mathrm{e}^{\Delta\varphi}\), both are equidistant reference line steps in
a plot of \(f(z)=\mathrm{e}^z\). You can visualize this with the following R code (active 7” x 2.8” screen graphics device suggested, e.g. x11(width = 7, height = 2.8)):
# divide graphics device into three regions and adjust plot margins
op <- par(mfrow = c(1, 3), mar = c(0.2, 0.2, 0.4, 0.2))
for(n in c(6, 9, 18)) {
phasePortrait("exp(z)", xlim = c(-8.5, 8.5), ylim = c(-8.5, 8.5),
pi2Div = n, pType = "pma", axes = FALSE, nCores = 2)
# separate title call (R base graphics) for nicer line adjustment, just cosmetics
title(paste("pi2Div = ", n, ", logBase = exp(2*pi/pi2Div)", sep = ""),
line = -1.2, cex.main = 0.9)
par(op) # reset graphics parameters to previous values
As expected, the default coupling of both arguments produces square patterns when applied to a phase portrait of the exponential function which can, insofar, serve as a visual reference. Recall, that
equidistant modulus reference lines (in ascending order) indicate an exponentially growing modulus. In the middle phase portrait one such steps means (approximately) doubling the modulus. From the
left to the right, the plot covers 24 of these steps, indicating a total increase of the modulus by factor \(2^{24}\) which amounts to almost 17 millions.
Fine tuning shading and contrast
For optimizing visualization in a technical sense, as well as for aesthetic purposes, it may be useful to adjust shading and contrast of the argument and modulus reference zones mentioned above. This
is done by modifying the parameters darkestShade (\(s\)) and lambda (\(\lambda\)) when calling phasePortrait. These two parameters can be used to steer the transition from the lower to the uper edge
of a reference zone. They address the v-value of the hsv color model, which can take values between 0 and 1, indicating maximum darkness (black), and no shading at all, respectively. Hereby, \(s\)
gives the v-value at the lower edge of a reference zone, and \(\lambda\) determines the interpolation from there to the upper edge, where no shading is applied. The intended use is \(\lambda > 0\)
where small values sharpen the contrast between shaded and non-shaded zones and vice versa. Exactly, the shading value \(v\) is calculated as: \[ v = s + (1-s)\cdot x^{1/\lambda} \] For modulus zone
shading at a point \(z\) in the complex plane when portraying a function \(f(z)\), \(x\) is the fractional part of \(\log_b{f(z)}\), with the base \(b\) being the parameter logBase that defines the
modulus reference zoning (see above). For shading argument reference zones, \(x\) is simply the difference between the upper and the lower angle of an argument reference zone, linearly mapped to the
range \([0, 1[\). The following code generates a \(3\times3\) matrix of phase portraits of \(f(z)=\tan{z^2}\) with \(\lambda\) and \(s\) changing along the rows and columns, respectively. Run the
code for visualizing these concepts (active 7” x 7” screen graphics device suggested, e.g. x11()).
op <- par(mfrow = c(3, 3), mar = c(0.2, 0.2, 0.2, 0.2))
for(lb in c(0.1, 1, 10)) {
for(dS in c(0, 0.2, 0.4)) {
phasePortrait("tan(z^2)", xlim = c(-1.7, 1.7), ylim = c(-1.7, 1.7),
pType = "pm", darkestShade = dS, lambda = lb,
axes = FALSE, xaxs = "i", yaxs = "i", nCores = 2)
Additional possibilities exist for tuning the interplay of modulus and argument reference zones when they are used in combination; this can be controlled with the parameter gamma when calling
phasePortrait). The maximum brightness of the colours in a phase portrait is adjustable with the parameter stdSaturation (see documentation of phasePortrait; we will also get back to these points in
the chapter aesthetic hints below).
Be aware of branch cuts
When exploring functions with phasePortrait, discontinuities of certain functions can become visible as abrupt color changes. Typical examples are integer root functions which, for a given point \(z,
z\in\mathbb{C}\setminus\lbrace0\rbrace\) in the complex plane, can attain \(n\) values with \(n\) being the root’s degree. It takes, so to speak, \(n\) full cycles around the origin of the complex
plane in order to cover all values obtained from a function \(f(z)=z^{1/n}, n\in\mathbb{N}\). The code below creates an illustration comprising three phase portraits with branch cuts (dashed lines),
illustrating the three values of \(f(z)=z^{1/3}\), \(z\in\mathbb{C}\setminus\lbrace0\rbrace\). The transitions between the phase portraits are indicated by same-coloured arrows pointing at the branch
cuts. For running the code, an open 7” x 2.7” graphics device is suggested, e.g. x11(width = 7, height = 2.8).
op <- par(mfrow = c(1, 3), mar = c(0.4, 0.2, 0.2, 0.2))
for(k in 0:2) {
FUNstring <- paste0("z^(1/3) * exp(1i * 2*pi/3 * ", k, ")")
phasePortrait(FUN = FUNstring,
xlim = c(-1.5, 1.5), ylim = c(-1.5, 1.5), pi2Div = 12,
axes = FALSE, nCores = 2)
title(sub = paste0("k = ", k), line = -1)
# emphasize branch cut with a dashed line segment
segments(-1.5, 0, 0, 0, lwd = 2, lty = "dashed")
# draw colored arrows
upperCol <- switch(as.character(k),
"0" = "black", "1" = "red", "2" = "green")
lowerCol <- switch(as.character(k),
"0" = "green", "1" = "black", "2" = "red")
arrows(x0 = c(-1.2), y0 = c(1, -1), y1 = c(0.2, -0.2),
lwd = 2, length = 0.1, col = c(upperCol, lowerCol))
After you have run the code, have a look at the leftmost diagram first. Note that the argument reference lines have been adjusted to represent angle distances of \(30°\), i.e. pi2Div = 12. Most
noticeable is the abrupt color change from yellow to magenta along the negative real axis (emphasized with a dashed line). This is what is called a branch cut, and it suggests that our picture of the
function \(f(z)=z^{1/3}\) is not complete. As the three third roots of any complex number \(z=r\cdot\mathrm{e}^{\mathrm{i}\varphi}, z\in\mathbb{C}\setminus\lbrace0\rbrace\) are \(r^{1/3}\cdot\mathrm
{e}^{\mathrm{i}\cdot(\varphi+k\cdot2\pi)/3}; k=0,1,2; \varphi\in\left[0,2\pi\right[\), we require three different phase portraits, one for each \(k\), as shown in the figure above. With the argument
reference line distance being \(30°\), it is easy to see that each phase portrait covers a total argument range of \(120°\), i.e. \(2\pi/3\).
Obviously, each of the three portraits has a branch cut along the negative real axis, and the colors at the branch cuts show, where the transitions between the phase portraits have to happen. In the
figure, we have illustrated this by arrows pointing to the branch cuts. Same-colored arrows in different phase portraits indicate the transitions. Thus, the first phase portrait (\(k = 0\)) links to
the second (\(k = 1\)) in their yellow zones (black arrows); the second links to the third (\(k = 2\)) in their blue zones (red arrows), and the third links back to the first in their magenta zones
(green arrows). Actually, one could imagine to stack the three face portraits in ascending order, cut them at the dashed line, and glue the branch cuts together according to the correct transitions.
The resulting object is a Riemann surface with each phase portrait being a ‘sheet.’ See more about this fascinating concept in Wegert (2012), Chapter7.
While the function \(f(z)=z^{1/3}\) could be fully covered with three phase portraits, \(f(z)=\log z\) has an infinite number of branches. As the (natural) logarithm of any complex number \(z=r\cdot\
mathrm{e}^{i\cdot\varphi}, r>0\) is \(\log z=\log r+\mathrm{i}\cdot\varphi\), it is evident that the imaginary part of \(\log z\) increases linearly with the argument of \(z\), \(\varphi\). In terms
of phase portraits, this means an infinite number of stacked ‘sheets’ in either direction, clockwise and counterclockwise. Neighboring sheets connect at a branch cut. Run the code below to illustrate
this with a phase portrait of \(\log z=\log r+\mathrm{i}\cdot(\varphi+k\cdot2\pi), r > 0, \varphi\in\left[0,2\pi\right[\) for \(k=-1, 0, 1\) (active 7” x 2.7” screen graphics device suggested,
e.g. x11(width = 7, height = 2.7)). In the resulting illustration, the branch cuts are marked with dashed white lines.
op <- par(mfrow = c(1, 3), mar = c(0.4, 0.2, 0.2, 0.2))
for(k in -1:1) {
FUNstring <- paste0("log(Mod(z)) + 1i * (Arg(z) + 2 * pi * ", k, ")")
phasePortrait(FUN = FUNstring, pi2Div = 36,
xlim = c(-2, 2), ylim = c(-2, 2), axes = FALSE, nCores = 2)
segments(-2, 0, 0, 0, col = "white", lwd = 1, lty = "dashed")
title(sub = paste0("k = ", k), line = -1)
Riemann sphere plots
A convenient way to visualize the whole complex number plane is based on a stereographic projection suggested by Bernhard Riemann (see Wegert (2012), p. 20 ff. and p. 39 ff.). The Riemann Sphere is a
sphere with radius 1, centered around the origin of the complex plane. It is cut into an upper (northern) and lower (southern) half by the complex plane. By connecting any point on the complex plane
to the north pole with a straight line, the line’s intersection with the sphere’s surface marks the location on the sphere where the point is projected onto. Thus, all points inside the unit disk on
the complex plane are projected onto the southern hemisphere, the origin being represented by the south pole. In contrast, all points outside the unit disk are projected onto the northern hemisphere,
the north pole representing the point at infinity. For visualizing both hemispheres as 2D phase portraits, they have to be projected onto a flat surface in turn.
If we perform a stereographic projection of the southern hemisphere from the north pole to the complex plane (and look at the plane’s upper - the northern - side), this obviously results in a phase
portrait on the untransformed complex plane as were all examples shown so far in this text. We can perform an analogue procedure for the northern hemisphere, projecting it from the south pole to the
complex plane. We now want to think of the northern hemisphere projection as layered on top of the southern hemisphere projection, for the northern hemisphere, which it depicts, is naturally also on
top of the southern hemisphere. If, in a ‘normal’ visualization of the complex plane (orthogonal real and imaginary axes), a point at any location represents a complex number \(z\), a point at the
same location in the northern hemisphere projection is mapped into \(1/z\). The origin is mapped into the point at infinity. Technically, this mapping can be easily achieved when calling the function
phasePortrait by setting the flag invertFlip = TRUE (default is FALSE). The resulting map is, in addition, rotated counter-clockwise around the point at infinity by an angle of \(\pi\). As Wegert
(2012) argues, this way of mapping has a convenient visual effect: Consider two phase portraits of the same function, one made with invertFlip = FALSE and the other one with invertFlip = TRUE. Both
are shown side by side (see the pairs of phase portraits in the next two figures below). This can be imagined as a view into a Riemann sphere that has been cut open along the equator and swung open
along a hinge in the line \(\Re(z)=1\) (if the southern hemisphere is at the left side) or \(\Re(z)=-1\) (if the northern hemisphere is at the left side). In order to highlight the Riemann sphere in
Phase Portraits if desired, we provide the function riemannMask. Let’s first demonstrate this for the function \(f(z)=z\).
op <- par(mfrow = c(1, 2), mar = rep(0.1, 4))
# Southern hemisphere
phasePortrait("z", xlim = c(-1.4, 1.4), ylim = c(-1.4, 1.4),
pi2Div = 12, axes = FALSE, nCores = 2)
riemannMask(annotSouth = TRUE)
# Northern hemisphere
phasePortrait("z", xlim = c(-1.4, 1.4), ylim = c(-1.4, 1.4),
pi2Div = 12, axes = FALSE, invertFlip = TRUE, nCores = 2)
riemannMask(annotNorth = TRUE)
The function riemannMask provides several options, among others adjusting the mask’s transparency or adding annotations to landmark points (see the function’s documentation). In the next example, we
will use it without any such features. Consider the following function: \[ f(z)=\frac{(z^{2}+\frac{1}{\sqrt{2}}+\frac{\mathrm{i}}{\sqrt{2}})\cdot(z+\frac{1}{2}+\frac{\mathrm{i}}{2})}{z-1} \] This
function has two zeroes exactly located on the unit circle, \(z_1=\mathrm{e}^{\mathrm{i}\frac{5\pi}{8}}\), and \(z_2=\mathrm{e}^{\mathrm{i}\frac{13\pi}{8}}\). Moreover, it has another zero inside the
unit circle, \(z_3=\frac{1}{\sqrt{2}}\cdot\mathrm{e}^{\mathrm{i}\frac{5\pi}{4}}\). Equally obvious, it has a pole exactly on the unit circle, \(z_4=1\). Less obvious, it has a double pole, \(z_5\),
at the point at infinity. The code required for producing the following figure looks somewhat bulky, but most lines are required for annotating the zeroes and poles. Note that the real axis
coordinates of the northern hemisphere’s annotation do not have to be multiplied with \(-1\) in order to take into account the rotation of the inverted complex plane. By calling phasePortrait with
invertFlip = TRUE the coordinate system of the plot is already set up correctly and will remain so for subsequent operations.
op <- par(mfrow = c(1, 2), mar = c(0.1, 0.1, 1.4, 0.1))
# Define function
FUNstring <- "(z^2 + 1/sqrt(2) * (1 + 1i)) * (z + 1/2*(1 + 1i)) / (z - 1)"
# Southern hemisphere
phasePortrait(FUNstring, xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
pi2Div = 12, axes = FALSE, nCores = 2)
title("Southern Hemisphere", line = 0)
# - annotate zeroes and poles
text(c(cos(5/8*pi), cos(13/8*pi), cos(5/4*pi)/sqrt(2), 1),
c(sin(5/8*pi), sin(13/8*pi), sin(5/4*pi)/sqrt(2), 0),
c(expression(z[1]), expression(z[2]), expression(z[3]), expression(z[4])),
pos = c(1, 2, 4, 2), offset = 1, col = "white")
# Northern hemisphere
phasePortrait(FUNstring, xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2),
pi2Div = 12, axes = FALSE, invertFlip = TRUE, nCores = 2)
title("Northern Hemisphere", line = 0)
# - annotate zeroes and poles
text(c(cos(5/8*pi), cos(13/8*pi), cos(5/4*pi)*sqrt(2), 1, 0),
c(sin(5/8*pi), sin(13/8*pi), sin(5/4*pi)*sqrt(2), 0, 0),
c(expression(z[1]), expression(z[2]), expression(z[3]),
expression(z[4]), expression(z[5])),
pos = c(1, 4, 3, 4, 4), offset = 1,
col = c("white", "white", "black", "white", "white")) | {"url":"https://cran-r.c3sl.ufpr.br/web/packages/viscomplexr/vignettes/viscomplexr-vignette.html","timestamp":"2024-11-15T03:14:31Z","content_type":"text/html","content_length":"1048850","record_id":"<urn:uuid:19423bb7-99aa-472f-a542-6ac20ac7e663>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00822.warc.gz"} |
Google Docs and LaTeX | Math ∞ Weblog
Google Docs is a good way to collaborate on paperwork and for lots of people can in all probability change massive, costly workplace suites with a free, on-line answer. After I first began utilizing
the service, the options had been fairly primary. I seen in the present day that there was a hyperlink to a listing of latest options. A few these had been significantly attention-grabbing. Since a
few of these options have been round for some time with out me noticing, I assumed it may be value a weblog submit.
The phrase processing program has a visible equation editor that works very similar to Microsoft Equation Editor. This isn’t a foul possibility for somebody who solely often wants to incorporate
equations of their writing. Nevertheless, professionals who write extensively on mathematical matters can be higher served by investing the trouble in studying to make use of LaTeX for typing
The brand new equation editor in Google Docs will acknowledge LaTeX symbols and can mechanically convert them to the suitable image. So, as an illustration, you possibly can kind alpha and get [tex]
This has the potential to be fairly useful for teams that have to collaborate on mathematical paperwork because it leverages the LaTeX shortcuts for customers who’re already acquainted in addition to
offering a graphical instrument for customers preferring that possibility.
If you happen to’re enthusiastic about utilizing these shortcuts, an inventory of widespread LaTeX symbols for arithmetic is obtainable right here.
Linked Drawings
The opposite massive function is that components in a drawing will be related in such a method that the connection will stretch when the weather are moved. Google offers an in depth clarification
right here together with this pattern picture.
One of these drawing is helpful in arithmetic (significantly graph principle), pc science and engineering. I’ve used the open supply GraphViz previously for drawing related graphs. Whereas this new
function received’t change a program like GraphViz, it’s a function that can definitely be helpful for scientific writers.
Though I wasn’t a giant fan of internet apps at first, Google Docs received me over by offering a minimal set of instruments and amazingly easy collaboration. I’m very glad to see that they’re
including options that can attraction to technical writers.
Tony McDaniel is a graduate scholar in computational engineering on the College of Tennessee at Chattanooga. His analysis pursuits embody computational arithmetic, algorithm design and evaluation,
and information visualization for numerical options of partial differential equations. Different pursuits embody pictures, mannequin rocketry and electronics.
Sponsor’s message: Try Math Higher Defined, an insightful e book and screencast collection that can assist you to see math in a brand new gentle and expertise extra of these superior “aha!” moments
when concepts all of a sudden click on. | {"url":"https://azmath.info/google-docs-and-latex-math-%E2%88%9E-weblog.html","timestamp":"2024-11-05T08:58:32Z","content_type":"text/html","content_length":"72748","record_id":"<urn:uuid:994e94c5-1a6d-4204-8864-291367a26c7a>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00776.warc.gz"} |
Re: How to read histograms and compute distances from them in VL53L3CX?
2022-12-07 10:05 AM
Despite many previous questions about how to read histograms from the VL53L3CX, I am still confused how the histogram data is formatted.
I currently read out the histogram as follows:
1. I call `VL53L3A2_RANGING_SENSOR_GetDistance`
2. Then right after getting the distance I use the following function to access the histogram data: VL53LX_GetAdditionalData
3. Then I print out the histogram to the console
As the data is output, I then have a python script parsing the data output on the /dev/ttyACM0 serial port.
Here is an example I get for 2 histograms that are printed out sequentially when imaging a wall 0.5m away (ignore bottom plot).
My questions are:
• Why does each histogram have 2 bumps even though I am only pointing the sensor at a flat wall?
• Why do the two consecutive histograms look different?
• At each frame, is the distance computed using both consecutive histograms or only one of them?
• How would I parse this histogram data to generate an actual histogram I can compute distances through simple peak finding?
2022-12-08 12:17 PM
2022-12-08 12:49 PM
2022-12-08 02:38 PM
2022-12-09 09:17 AM
2022-12-09 02:37 PM
2022-12-13 02:40 PM
2022-12-14 10:13 AM | {"url":"https://community.st.com/t5/imaging-sensors/how-to-read-histograms-and-compute-distances-from-them-in/m-p/128188","timestamp":"2024-11-09T22:04:20Z","content_type":"text/html","content_length":"309837","record_id":"<urn:uuid:42551a8e-5de5-4e9f-9db1-8a86f8cd78d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00533.warc.gz"} |
Stephan validate - on time travel
from http://www.orkut.com/CommMsgs.aspx?cmm=28433022&tid=2583773577075764617 (needs google login)
Unfortunately, laws of physics and special relativity has a limit speed of 'C' which is assigned with the nature for the maximum speed achievable for any particle.
That's relatively old Physics.
The statement "speed of light is a constant(C)" is a tautology given that standard units of distance and time are related by the speed of light.
Every observable object is given to be moving slower than light because it remains within the future/forward light cone propagating from its position at any instant.
Take a thought experiment instead.
Stand up in a clear space and spin round.
tgkprog It is not too difficult to turn at one revolution each two seconds.
12:13:28 Now, suppose the moon is on the horizon.
How fast is it spinning round your head?
It is about 385,000 km away so the answer is 1.21 million km/s, which is more than four times the speed of light!
It sounds ridiculous to say that the moon is going round your head when really it is you who is turning, but according to the General theory of relativity, all coordinate systems are
equally valid including revolving ones.
So isn't the moon going faster than the speed of light?
BTW, tachyons ( particles that travel faster than light ) haven't been experimentally validated, but they turn up quite often in modern physics of spontaneously broken symmetry of
scalar fields, and in some versions of string theory, and the likes.
The key, though, is that you have to rise above the concept of local causality, which requires a paradigm shift in human thinking.
--[but according to the General theory of relativity, all coordinate systems are equally valid including revolving ones.]--
Brooks Yes, but the equations of general relativity include the metric tensor g[μν] that defines distance in that coordinate system. The laws of physics in an arbitrary coordinate system also
2008-02-19 include the metric tensor in such a way that it cancels the effects from accelerating or rotating reference frames.
In reference frames moving at a constant velocity, the laws of physics require no corrections. But in an accelerating system (rotating or accelerating) you get correction terms
appearing as things like centrifugal force or being pushed to the back of the car as it accelerates.
2008-04-04 thanks | {"url":"https://stephenbrooks.org/forum/?thread=1325&bork=yohqnyvhoh","timestamp":"2024-11-05T00:26:12Z","content_type":"application/xhtml+xml","content_length":"9967","record_id":"<urn:uuid:879a9f5f-72a9-413e-af4f-25d064098957>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00161.warc.gz"} |
Extension Complexity
Last week I gave an introductory lecture on extension complexity for my communication complexity reading group. It is a fascinating topic, discussing if a polytope with exponentially many facets
might be described as a linear projection from a higher dimensional polytope with polynomially many facets. Specifically, the extension complexity is defined as xc(P) := min{# facets of Q : Q
projects to P}.
A classical example of this would be the permutohedron with dimension N; it has 2^N - 2 facets, but there is a polytope in dimension N^2 + N with O(N^2) facets which projects to the permutohedron. I
covered this example, as well as Yannakakis' theorem, which shows an equivalence between extension complexity and nonnegative rank of a certain matrix. I also highlight a connection with
communication complexity, which will be expanded upon in the upcoming semester.
I hope to give more in-depth lectures, with the intent of covering Rothvoss' 2017 result on exponential lower bounds for the matching polytope.
Here are the lecture notes if you are interested! Credit to Rothvoss' talk for several examples used. | {"url":"https://blog.catalangrenade.com/2020/12/extension-complexity.html","timestamp":"2024-11-02T21:58:08Z","content_type":"application/xhtml+xml","content_length":"88914","record_id":"<urn:uuid:4f772b2b-b882-49df-a938-68259a0861a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00590.warc.gz"} |
Bzoj 1233: [Usaco2009open] hay pile tower "idea question"
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• Alibaba Cloud offers highly flexible support services tailored to meet your exact needs. | {"url":"https://topic.alibabacloud.com/a/bzoj-1233-usaco2009open-hay-pile-tower-idea-question_8_8_31409383.html","timestamp":"2024-11-08T21:08:32Z","content_type":"text/html","content_length":"79248","record_id":"<urn:uuid:0c33bac3-bf8f-4e5e-8afa-558602dc7ccd>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00268.warc.gz"} |
Pattern Search Climbs Mount Washington
This example shows visually how pattern search optimizes a function. The function is the height of the terrain near Mount Washington, as a function of the x-y location. In order to find the top of
Mount Washington, we minimize the objective function that is the negative of the height. (The Mount Washington in this example is the highest peak in the northeastern United States.)
The US Geological Survey provides data on the height of the terrain as a function of the x-y location on a grid. In order to be able to evaluate the height at an arbitrary point, the objective
function interpolates the height from nearby grid points.
It would be faster, of course, to simply find the maximum value of the height as specified on the grid, using the max function. The point of this example is to show how the pattern search algorithm
operates; it works on functions defined over continuous regions, not just grid points. Also, if it is computationally expensive to evaluate the objective function, then performing this evaluation on
a complete grid, as required by the max function, will be much less efficient than using the pattern search algorithm, which samples a small subset of grid points.
How Pattern Search Works
Pattern search finds a local minimum of an objective function by the following method, called polling. In this description, words describing pattern search quantities are in bold. The search starts
at an initial point, which is taken as the current point in the first step:
1. Generate a pattern of points, typically plus and minus the coordinate directions, times a mesh size, and center this pattern on the current point.
2. Evaluate the objective function at every point in the pattern.
3. If the minimum objective in the pattern is lower than the value at the current point, then the poll is successful, and the following happens:
3a. The minimum point found becomes the current point.
3b. The mesh size is doubled.
3c. The algorithm proceeds to Step 1.
4. If the poll is not successful, then the following happens:
4a. The mesh size is halved.
4b. If the mesh size is below a threshold, the iterations stop.
4c. Otherwise, the current point is retained, and the algorithm proceeds at Step 1.
This simple algorithm, with some minor modifications, provides a robust and straightforward method for optimization. It requires no gradients of the objective function. It lends itself to
constraints, too, but this example and description deal only with unconstrained problems.
Preparing the Pattern Search
To prepare the pattern search, load the data in mtWashington.mat, which contains the USGS data on a 472-by-345 grid. The elevation, Z, is given in feet. The vectors x and y contain the base values of
the grid spacing in the east and north directions respectively. The data file also contains the starting point for the search, X0.
There are three MATLAB® files that perform the calculation of the objective function, and the plotting routines. They are:
1. terrainfun, which evaluates the negative of height at any x-y position. terrainfun uses the MATLAB function interp2 to perform two-dimensional linear interpolation. It takes the Z data and enables
evaluation of the negative of the height at all x-y points.
2. psoutputwashington, which draws a 3-d rendering of Mt. Washington. In addition, as the run progresses, it draws spheres around each point that is better (higher) than previously-visited points.
3. psplotwashington, which draws a contour map of Mt. Washington, and monitors a slider that controls the speed of the run. It shows where the pattern search algorithm looks for optima by drawing +
signs at those points. It also draws spheres around each point that is better than previously-visited points.
In the example, patternsearch uses terrainfun as its objective function, psoutputwashington as an output function, and psplotwashington as a plot function. We prepare the functions to be passed to
patternsearch in anonymous function syntax:
mtWashObjectiveFcn = @(xx) terrainfun(xx, x, y, Z);
mtWashOutputFcn = @(xx,arg1,arg2) psoutputwashington(xx,arg1,arg2, x, y, Z);
mtWashPlotFcn = @(xx,arg1) psplotwashington(xx,arg1, x, y, Z);
Pattern Search Options Settings
Next, we create options for pattern search. This set of options has the algorithm halt when the mesh size shrinks below 1, keeps the mesh unscaled (the same size in each direction), sets the initial
mesh size to 10, and sets the output function and plot function:
options = optimoptions(@patternsearch,'MeshTolerance',1,'ScaleMesh',false, ...
'InitialMeshSize',10,'UseCompletePoll',true,'PlotFcn',mtWashPlotFcn, ...
Observing the Progress of Pattern Search
When you run this example you see two windows. One shows the points the pattern search algorithm chooses on a two-dimensional contour map of Mount Washington. This window has a slider that controls
the delay between iterations of the algorithm (when it returns to Step 1 in the description of how pattern search works). Set the slider to a low position to speed the run, or to a high position to
slow the run.
The other window shows a three-dimensional plot of Mount Washington, along with the steps the pattern search algorithm makes. You can rotate this plot while the run progresses to get different views.
[xfinal ffinal] = patternsearch(mtWashObjectiveFcn,X0,[],[],[],[],[], ...
patternsearch stopped because the mesh size was less than options.MeshTolerance.
xfinal = 1×2
The final point, xfinal, shows where the pattern search algorithm finished; this is the x-y location of the top of Mount Washington. The final objective function, ffinal, is the negative of the
height of Mount Washington, 6280 feet. (This should be 6288 feet according to the Mount Washington Observatory).
Examine the files terrainfun.m, psoutputwashington.m, and psplotwashington.m to see how the interpolation and graphics work.
There are many options available for the pattern search algorithm. For example, the algorithm can take the first point it finds that is an improvement, rather than polling all the points in the
pattern. It can poll the points in various orders. And it can use different patterns for the poll, both deterministic and random. Consult the Global Optimization Toolbox User's Guide for details.
Related Topics | {"url":"https://au.mathworks.com/help/gads/patternsearch-climbs-mt-washington.html","timestamp":"2024-11-10T08:47:38Z","content_type":"text/html","content_length":"76182","record_id":"<urn:uuid:a3c6ed6e-8889-4b58-ab71-25568b7d005a>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00649.warc.gz"} |
Convert lm·min/m³ to lm·h/m³ (Luminous energy density)
lm·min/m³ into lm·h/m³
Direct link to this calculator:
Convert lm·min/m³ to lm·h/m³ (Luminous energy density)
1. Choose the right category from the selection list, in this case 'Luminous energy density'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), square root (√), brackets and
π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Lumen-Minutes per Cubic meter [lm·min/m³]'.
4. Finally choose the unit you want the value to be converted to, in this case 'Lumen-Hours per Cubic meter [lm·h/m³]'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
Utilize the full range of performance for this units calculator
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '489 Lumen-Minutes per Cubic meter'. In so doing, either the full
name of the unit or its abbreviation can be usedas an example, either 'Lumen-Minutes per Cubic meter' or 'lm·min/m3'. Then, the calculator determines the category of the measurement unit of measure
that is to be converted, in this case 'Luminous energy density'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also
to find the conversion you originally sought. Alternatively, the value to be converted can be entered as follows: '66 lm·min/m3 to lm·h/m3' or '80 lm·min/m3 into lm·h/m3' or '49 Lumen-Minutes per
Cubic meter -> Lumen-Hours per Cubic meter' or '32 lm·min/m3 = lm·h/m3' or '15 Lumen-Minutes per Cubic meter to lm·h/m3' or '97 lm·min/m3 to Lumen-Hours per Cubic meter' or '63 Lumen-Minutes per
Cubic meter into Lumen-Hours per Cubic meter'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which
of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over
for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(77 * 60) lm·min/m3'. But
different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '12 Lumen-Minutes per Cubic meter + 94 Lumen-Hours per Cubic
meter' or '43mm x 26cm x 9dm = ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
The mathematical functions sin, cos, tan and sqrt can also be used. Example: sin(π/2), cos(pi/2), tan(90°), sin(90) or sqrt(4).
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 2.496 148 125 433 2×1021. For this form of presentation, the number
will be segmented into an exponent, here 21, and the actual number, here 2.496 148 125 433 2. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket
calculators, one also finds the way of writing numbers as 2.496 148 125 433 2E+21. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at
this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 2 496 148 125 433 200 000 000. Independent of the presentation of the
results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications. | {"url":"https://www.convert-measurement-units.com/convert+lm+min+m3+to+lm+h+m3.php","timestamp":"2024-11-08T01:16:33Z","content_type":"text/html","content_length":"55016","record_id":"<urn:uuid:780c0e04-7da6-4cde-b2d1-f25e40d7f8d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00818.warc.gz"} |
Linear Supertypes
Type Hierarchy
1. final def !=(arg0: Any): Boolean
Definition Classes
AnyRef → Any
2. final def ##(): Int
Definition Classes
AnyRef → Any
3. def +(other: String): String
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to any2stringadd[Tuple9Codec[A, B, C, D, E, F, G, H, I]] performed by method any2stringadd in
Definition Classes
4. def ->[B](y: B): (Tuple9Codec[A, B, C, D, E, F, G, H, I], B)
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ArrowAssoc[Tuple9Codec[A, B, C, D, E, F, G, H, I]] performed by method ArrowAssoc in
Definition Classes
5. def :+:[B](left: Codec[B]): CoproductCodecBuilder[:+:[B, :+:[(A, B, C, D, E, F, G, H, I), CNil]], ::[Codec[B], ::[Codec[(A, B, C, D, E, F, G, H, I)], HNil]], :+:[B, :+:[(A, B, C, D, E, F, G, H, I
), CNil]]]
Supports creation of a coproduct codec.
6. def ::[B](codecB: Codec[B]): Codec[::[B, ::[(A, B, C, D, E, F, G, H, I), HNil]]]
When called on a Codec[A] where A is not a subytpe of HList, creates a new codec that encodes/decodes an HList of B :: A :: HNil.
When called on a Codec[A] where A is not a subytpe of HList, creates a new codec that encodes/decodes an HList of B :: A :: HNil. For example,
uint8 :: utf8
has type Codec[Int :: String :: HNil]. uint8 :: utf8 }}}
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ValueCodecEnrichedWithHListSupport[(A, B, C, D, E, F, G, H, I)] performed by method
ValueCodecEnrichedWithHListSupport in scodec.
Definition Classes
7. def :~>:[B](codecB: Codec[B])(implicit ev: =:=[Unit, B]): Codec[::[(A, B, C, D, E, F, G, H, I), HNil]]
When called on a Codec[A], returns a new codec that encodes/decodes B :: A :: HNil.
When called on a Codec[A], returns a new codec that encodes/decodes B :: A :: HNil. HList equivalent of ~>.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ValueCodecEnrichedWithHListSupport[(A, B, C, D, E, F, G, H, I)] performed by method
ValueCodecEnrichedWithHListSupport in scodec.
Definition Classes
8. final def <~[B](codecB: Codec[B])(implicit ev: =:=[Unit, B]): Codec[(A, B, C, D, E, F, G, H, I)]
Assuming B is Unit, creates a Codec[A] that: encodes the A followed by a unit; decodes an A followed by a unit and discards the decoded unit.
Assuming B is Unit, creates a Codec[A] that: encodes the A followed by a unit; decodes an A followed by a unit and discards the decoded unit.
Operator alias of dropRight.
Definition Classes
9. final def ==(arg0: Any): Boolean
Definition Classes
AnyRef → Any
10. def >>:~[L <: HList](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Codec[L]): Codec[::[(A, B, C, D, E, F, G, H, I), L]]
Creates a new codec that encodes/decodes an HList type of A :: L given a function A => Codec[L].
Creates a new codec that encodes/decodes an HList type of A :: L given a function A => Codec[L]. This allows later parts of an HList codec to be dependent on earlier values. Operator alias for
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ValueCodecEnrichedWithHListSupport[(A, B, C, D, E, F, G, H, I)] performed by method
ValueCodecEnrichedWithHListSupport in scodec.
Definition Classes
11. final def >>~[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Codec[B]): Codec[((A, B, C, D, E, F, G, H, I), B)]
Returns a new codec that encodes/decodes a value of type (A, B) where the codec of B is dependent on A.
Returns a new codec that encodes/decodes a value of type (A, B) where the codec of B is dependent on A. Operator alias for flatZip.
Definition Classes
12. val a: Tuple9Codec[A, B, C, D, E, F, G, H, I]
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ValueEnrichedWithTuplingSupport[Tuple9Codec[A, B, C, D, E, F, G, H, I]] performed by method
ValueEnrichedWithTuplingSupport in scodec.codecs.
Definition Classes
13. def as[B](implicit as: Transformer[(A, B, C, D, E, F, G, H, I), B]): Codec[B]
Transforms using implicitly available evidence that such a transformation is possible.
Transforms using implicitly available evidence that such a transformation is possible.
Typical transformations include converting:
□ an F[L] for some L <: HList to/from an F[CC] for some case class CC, where the types in the case class are aligned with the types in L
□ an F[C] for some C <: Coproduct to/from an F[SC] for some sealed class SC, where the component types in the coproduct are the leaf subtypes of the sealed class.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec
Definition Classes
14. def asDecoder: Decoder[(A, B, C, D, E, F, G, H, I)]
Gets this as a Decoder.
15. def asEncoder: Encoder[(A, B, C, D, E, F, G, H, I)]
Gets this as an Encoder.
16. final def asInstanceOf[T0]: T0
17. def clone(): AnyRef
Definition Classes
@throws( ... )
18. final def compact: Codec[(A, B, C, D, E, F, G, H, I)]
Converts this codec to a new codec that compacts the encoded bit vector before returning it.
Converts this codec to a new codec that compacts the encoded bit vector before returning it.
Definition Classes
Codec → GenCodec → Encoder
19. final def complete: Codec[(A, B, C, D, E, F, G, H, I)]
Converts this codec to a new codec that fails decoding if there are remaining bits.
Converts this codec to a new codec that fails decoding if there are remaining bits.
Definition Classes
Codec → GenCodec → Decoder
20. final def consume[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Codec[B])(g: (B) ⇒ (A, B, C, D, E, F, G, H, I)): Codec[B]
Similar to flatZip except the A type is not visible in the resulting type -- the binary effects of the Codec[A] still occur though.
Similar to flatZip except the A type is not visible in the resulting type -- the binary effects of the Codec[A] still occur though.
Example usage:
case class Flags(x: Boolean, y: Boolean, z: Boolean)
(bool :: bool :: bool :: ignore(5)).consume { flgs =>
conditional(flgs.x, uint8) :: conditional(flgs.y, uint8) :: conditional(flgs.z, uint8)
} {
case x :: y :: z :: HNil => Flags(x.isDefined, y.isDefined, z.isDefined) }
Note that when B is an HList, this method is equivalent to using flatPrepend and derive. That is, a.consume(f)(g) === a.flatPrepend(f).derive[A].from(g).
Definition Classes
21. def contramap[C](f: (C) ⇒ (A, B, C, D, E, F, G, H, I)): GenCodec[C, (A, B, C, D, E, F, G, H, I)]
Converts this GenCodec to a GenCodec[C, B] using the supplied C => A.
Converts this GenCodec to a GenCodec[C, B] using the supplied C => A.
Definition Classes
GenCodec → Encoder
Attempts to decode a value of type A from the specified bit vector.
Attempts to decode a value of type A from the specified bit vector.
bits to decode
error if value could not be decoded or the remaining bits and the decoded value
Definition Classes
Tuple9Codec → Decoder
23. def decodeOnly[AA >: (A, B, C, D, E, F, G, H, I)]: Codec[AA]
Converts this to a codec that fails encoding with an error.
Converts this to a codec that fails encoding with an error.
Definition Classes
Codec → Decoder
24. final def decodeValue(bits: BitVector): Attempt[(A, B, C, D, E, F, G, H, I)]
Attempts to decode a value of type A from the specified bit vector and discards the remaining bits.
Attempts to decode a value of type A from the specified bit vector and discards the remaining bits.
bits to decode
error if value could not be decoded or the decoded value
Definition Classes
25. final def downcast[B <: (A, B, C, D, E, F, G, H, I)](implicit tb: Typeable[B]): Codec[B]
Safely lifts this codec to a codec of a subtype.
Safely lifts this codec to a codec of a subtype.
When a supertype of B that is not a supertype of A is decoded, an decoding error is returned.
Definition Classes
26. final def dropLeft[B](codecB: Codec[B])(implicit ev: =:=[Unit, (A, B, C, D, E, F, G, H, I)]): Codec[B]
Assuming A is Unit, creates a Codec[B] that: encodes the unit followed by a B; decodes a unit followed by a B and discards the decoded unit.
Assuming A is Unit, creates a Codec[B] that: encodes the unit followed by a B; decodes a unit followed by a B and discards the decoded unit.
Definition Classes
27. final def dropRight[B](codecB: Codec[B])(implicit ev: =:=[Unit, B]): Codec[(A, B, C, D, E, F, G, H, I)]
Assuming B is Unit, creates a Codec[A] that: encodes the A followed by a unit; decodes an A followed by a unit and discards the decoded unit.
Assuming B is Unit, creates a Codec[A] that: encodes the A followed by a unit; decodes an A followed by a unit and discards the decoded unit.
Definition Classes
28. def econtramap[C](f: (C) ⇒ Attempt[(A, B, C, D, E, F, G, H, I)]): GenCodec[C, (A, B, C, D, E, F, G, H, I)]
Converts this GenCodec to a GenCodec[C, B] using the supplied C => Attempt[A].
Converts this GenCodec to a GenCodec[C, B] using the supplied C => Attempt[A].
Definition Classes
GenCodec → Encoder
29. def emap[C](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Attempt[C]): GenCodec[(A, B, C, D, E, F, G, H, I), C]
Converts this GenCodec to a GenCodec[A, C] using the supplied B => Attempt[C].
Converts this GenCodec to a GenCodec[A, C] using the supplied B => Attempt[C].
Definition Classes
GenCodec → Decoder
30. def encode(abcdefghi: (A, B, C, D, E, F, G, H, I)): Attempt[BitVector]
Attempts to encode the specified value in to a bit vector.
Attempts to encode the specified value in to a bit vector.
error or binary encoding of the value
Definition Classes
Tuple9Codec → Encoder
31. def encodeOnly: Codec[(A, B, C, D, E, F, G, H, I)]
Converts this to a codec that fails decoding with an error.
Converts this to a codec that fails decoding with an error.
Definition Classes
32. def ensuring(cond: (Tuple9Codec[A, B, C, D, E, F, G, H, I]) ⇒ Boolean, msg: ⇒ Any): Tuple9Codec[A, B, C, D, E, F, G, H, I]
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to Ensuring[Tuple9Codec[A, B, C, D, E, F, G, H, I]] performed by method Ensuring in scala.Predef.
Definition Classes
33. def ensuring(cond: (Tuple9Codec[A, B, C, D, E, F, G, H, I]) ⇒ Boolean): Tuple9Codec[A, B, C, D, E, F, G, H, I]
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to Ensuring[Tuple9Codec[A, B, C, D, E, F, G, H, I]] performed by method Ensuring in scala.Predef.
Definition Classes
34. def ensuring(cond: Boolean, msg: ⇒ Any): Tuple9Codec[A, B, C, D, E, F, G, H, I]
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to Ensuring[Tuple9Codec[A, B, C, D, E, F, G, H, I]] performed by method Ensuring in scala.Predef.
Definition Classes
35. def ensuring(cond: Boolean): Tuple9Codec[A, B, C, D, E, F, G, H, I]
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to Ensuring[Tuple9Codec[A, B, C, D, E, F, G, H, I]] performed by method Ensuring in scala.Predef.
Definition Classes
Definition Classes
AnyRef → Any
38. final def exmap[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Attempt[B], g: (B) ⇒ Attempt[(A, B, C, D, E, F, G, H, I)]): Codec[B]
Transforms using two functions, A => Attempt[B] and B => Attempt[A].
Transforms using two functions, A => Attempt[B] and B => Attempt[A].
Definition Classes
39. def exmapc[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Attempt[B])(g: (B) ⇒ Attempt[(A, B, C, D, E, F, G, H, I)]): Codec[B]
Curried version of exmap.
Curried version of exmap.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec
Definition Classes
40. def finalize(): Unit
Definition Classes
@throws( classOf[java.lang.Throwable] )
41. def flatMap[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Decoder[B]): Decoder[B]
Converts this decoder to a Decoder[B] using the supplied A => Decoder[B].
Converts this decoder to a Decoder[B] using the supplied A => Decoder[B].
Definition Classes
42. def flatPrepend[L <: HList](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Codec[L]): Codec[::[(A, B, C, D, E, F, G, H, I), L]]
Creates a new codec that encodes/decodes an HList type of A :: L given a function A => Codec[L].
Creates a new codec that encodes/decodes an HList type of A :: L given a function A => Codec[L]. This allows later parts of an HList codec to be dependent on earlier values.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ValueCodecEnrichedWithHListSupport[(A, B, C, D, E, F, G, H, I)] performed by method
ValueCodecEnrichedWithHListSupport in scodec.
Definition Classes
43. final def flatZip[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Codec[B]): Codec[((A, B, C, D, E, F, G, H, I), B)]
Returns a new codec that encodes/decodes a value of type (A, B) where the codec of B is dependent on A.
Returns a new codec that encodes/decodes a value of type (A, B) where the codec of B is dependent on A.
Definition Classes
44. def flatZipHList[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Codec[B]): Codec[::[(A, B, C, D, E, F, G, H, I), ::[B, HNil]]]
Creates a new codec that encodes/decodes an HList type of A :: B :: HNil given a function A => Codec[B].
Creates a new codec that encodes/decodes an HList type of A :: B :: HNil given a function A => Codec[B]. If B is an HList type, consider using flatPrepend instead, which avoids nested HLists.
This is the direct HList equivalent of flatZip.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ValueCodecEnrichedWithHListSupport[(A, B, C, D, E, F, G, H, I)] performed by method
ValueCodecEnrichedWithHListSupport in scodec.
Definition Classes
45. final def flattenLeftPairs(implicit f: FlattenLeftPairs[(A, B, C, D, E, F, G, H, I)]): Codec[Out]
Converts this codec to an HList based codec by flattening all left nested pairs.
Converts this codec to an HList based codec by flattening all left nested pairs. For example, flattenLeftPairs on a Codec[(((A, B), C), D)] results in a Codec[A :: B :: C :: D :: HNil]. This is
particularly useful when combined with ~, ~>, and <~.
Definition Classes
46. def formatted(fmtstr: String): String
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to StringFormat[Tuple9Codec[A, B, C, D, E, F, G, H, I]] performed by method StringFormat in
Definition Classes
47. final def fuse[AA <: (A, B, C, D, E, F, G, H, I), BB >: (A, B, C, D, E, F, G, H, I)](implicit ev: =:=[BB, AA]): Codec[BB]
Converts this generalized codec in to a non-generalized codec assuming A and B are the same type.
Converts this generalized codec in to a non-generalized codec assuming A and B are the same type.
Definition Classes
48. final def getClass(): Class[_]
Definition Classes
AnyRef → Any
49. def hashCode(): Int
Definition Classes
AnyRef → Any
50. final def hlist: Codec[::[(A, B, C, D, E, F, G, H, I), HNil]]
Lifts this codec in to a codec of a singleton hlist.
Lifts this codec in to a codec of a singleton hlist.
Definition Classes
51. final def isInstanceOf[T0]: Boolean
52. def map[C](f: ((A, B, C, D, E, F, G, H, I)) ⇒ C): GenCodec[(A, B, C, D, E, F, G, H, I), C]
Converts this GenCodec to a GenCodec[A, C] using the supplied B => C.
Converts this GenCodec to a GenCodec[A, C] using the supplied B => C.
Definition Classes
GenCodec → Decoder
53. final def narrow[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Attempt[B], g: (B) ⇒ (A, B, C, D, E, F, G, H, I)): Codec[B]
Transforms using two functions, A => Attempt[B] and B => A.
Transforms using two functions, A => Attempt[B] and B => A.
The supplied functions form an injection from B to A. Hence, this method converts from a larger to a smaller type. Hence, the name narrow.
Definition Classes
54. def narrowc[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ Attempt[B])(g: (B) ⇒ (A, B, C, D, E, F, G, H, I)): Codec[B]
Curried version of narrow.
Curried version of narrow.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec
Definition Classes
56. final def notify(): Unit
57. final def notifyAll(): Unit
58. final def pairedWith[B](codecB: Codec[B]): Codec[((A, B, C, D, E, F, G, H, I), B)]
Creates a Codec[(A, B)] that first encodes/decodes an A followed by a B.
Creates a Codec[(A, B)] that first encodes/decodes an A followed by a B.
Definition Classes
59. def pcontramap[C](f: (C) ⇒ Option[(A, B, C, D, E, F, G, H, I)]): GenCodec[C, (A, B, C, D, E, F, G, H, I)]
Converts this GenCodec to a GenCodec[C, B] using the supplied partial function from C to A.
Converts this GenCodec to a GenCodec[C, B] using the supplied partial function from C to A. The encoding will fail for any C that f maps to None.
Definition Classes
GenCodec → Encoder
60. def polyxmap[B](p: Poly, q: Poly)(implicit aToB: Aux[p.type, ::[(A, B, C, D, E, F, G, H, I), HNil], B], bToA: Aux[q.type, ::[B, HNil], (A, B, C, D, E, F, G, H, I)]): Codec[B]
Polymorphic function version of xmap.
Polymorphic function version of xmap.
When called on a Codec[A] where A is not a subytpe of HList, returns a new codec that's the result of xmapping with p and q, using p to convert from A to B and using q to convert from B to A.
polymorphic function that converts from A to B
polymorphic function that converts from B to A
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ValueCodecEnrichedWithGenericSupport[(A, B, C, D, E, F, G, H, I)] performed by method
ValueCodecEnrichedWithGenericSupport in scodec.
Definition Classes
61. def polyxmap1[B](p: Poly)(implicit aToB: Aux[p.type, ::[(A, B, C, D, E, F, G, H, I), HNil], B], bToA: Aux[p.type, ::[B, HNil], (A, B, C, D, E, F, G, H, I)]): Codec[B]
Polymorphic function version of xmap that uses a single polymorphic function in both directions.
Polymorphic function version of xmap that uses a single polymorphic function in both directions.
When called on a Codec[A] where A is not a subytpe of HList, returns a new codec that's the result of xmapping with p for both forward and reverse directions.
polymorphic function that converts from A to B and from B to A
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ValueCodecEnrichedWithGenericSupport[(A, B, C, D, E, F, G, H, I)] performed by method
ValueCodecEnrichedWithGenericSupport in scodec.
Definition Classes
62. def selectEncoder[A](implicit inj: Inject[Coproduct with (A, B, C, D, E, F, G, H, I), A]): Encoder[A]
When called on a Encoder[C] where C is a coproduct containing type A, converts to an Encoder[A].
When called on a Encoder[C] where C is a coproduct containing type A, converts to an Encoder[A].
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to EnrichedCoproductEncoder[Coproduct with (A, B, C, D, E, F, G, H, I)] performed by method
EnrichedCoproductEncoder in scodec.
Definition Classes
Provides a bound on the size of successfully encoded values.
Provides a bound on the size of successfully encoded values.
Definition Classes
Tuple9Codec → Encoder
64. final def synchronized[T0](arg0: ⇒ T0): T0
65. def toField[K]: Codec[FieldType[K, (A, B, C, D, E, F, G, H, I)]]
Lifts this codec to a codec of a shapeless field -- allowing it to be used in records and unions.
Lifts this codec to a codec of a shapeless field -- allowing it to be used in records and unions.
Definition Classes
66. def toFieldWithContext[K <: Symbol](k: K): Codec[FieldType[K, (A, B, C, D, E, F, G, H, I)]]
Lifts this codec to a codec of a shapeless field -- allowing it to be used in records and unions.
Lifts this codec to a codec of a shapeless field -- allowing it to be used in records and unions. The specified key is pushed in to the context of any errors that are returned from the resulting
Definition Classes
67. def toString(): String
68. final def unit(zero: (A, B, C, D, E, F, G, H, I)): Codec[Unit]
Converts this to a Codec[Unit] that encodes using the specified zero value and decodes a unit value when this codec decodes an A successfully.
Converts this to a Codec[Unit] that encodes using the specified zero value and decodes a unit value when this codec decodes an A successfully.
Definition Classes
69. final def upcast[B >: (A, B, C, D, E, F, G, H, I)](implicit ta: Typeable[(A, B, C, D, E, F, G, H, I)]): Codec[B]
Safely lifts this codec to a codec of a supertype.
Safely lifts this codec to a codec of a supertype.
When a subtype of B that is not a subtype of A is passed to encode, an encoding error is returned.
Definition Classes
70. final def wait(): Unit
Definition Classes
@throws( ... )
71. final def wait(arg0: Long, arg1: Int): Unit
Definition Classes
@throws( ... )
72. final def wait(arg0: Long): Unit
Definition Classes
@throws( ... )
73. final def widen[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ B, g: (B) ⇒ Attempt[(A, B, C, D, E, F, G, H, I)]): Codec[B]
Transforms using two functions, A => B and B => Attempt[A].
Transforms using two functions, A => B and B => Attempt[A].
The supplied functions form an injection from A to B. Hence, this method converts from a smaller to a larger type. Hence, the name widen.
Definition Classes
74. def widenAs[X](to: (A, B, C, D, E, F, G, H, I) ⇒ X, from: (X) ⇒ Option[(A, B, C, D, E, F, G, H, I)]): Codec[X]
75. def widenOpt[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ B, g: (B) ⇒ Option[(A, B, C, D, E, F, G, H, I)]): Codec[B]
Transforms using two functions, A => B and B => Option[A].
Transforms using two functions, A => B and B => Option[A].
Particularly useful when combined with case class apply/unapply. E.g., widenOpt(fa, Foo.apply, Foo.unapply).
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec
Definition Classes
76. def widenOptc[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ B)(g: (B) ⇒ Option[(A, B, C, D, E, F, G, H, I)]): Codec[B]
Curried version of widenOpt.
Curried version of widenOpt.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec
Definition Classes
77. def widenc[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ B)(g: (B) ⇒ Attempt[(A, B, C, D, E, F, G, H, I)]): Codec[B]
Curried version of widen.
Curried version of widen.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec
Definition Classes
78. final def withContext(context: String): Codec[(A, B, C, D, E, F, G, H, I)]
Creates a new codec that is functionally equivalent to this codec but pushes the specified context string in to any errors returned from encode or decode.
Creates a new codec that is functionally equivalent to this codec but pushes the specified context string in to any errors returned from encode or decode.
Definition Classes
79. final def withToString(str: ⇒ String): Codec[(A, B, C, D, E, F, G, H, I)]
Creates a new codec that is functionally equivalent to this codec but returns the specified string from toString.
Creates a new codec that is functionally equivalent to this codec but returns the specified string from toString.
Definition Classes
80. final def xmap[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ B, g: (B) ⇒ (A, B, C, D, E, F, G, H, I)): Codec[B]
Transforms using the isomorphism described by two functions, A => B and B => A.
Transforms using the isomorphism described by two functions, A => B and B => A.
Definition Classes
81. def xmapc[B](f: ((A, B, C, D, E, F, G, H, I)) ⇒ B)(g: (B) ⇒ (A, B, C, D, E, F, G, H, I)): Codec[B]
Curried version of xmap.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec
Definition Classes
82. final def ~[B](codecB: Codec[B]): Codec[((A, B, C, D, E, F, G, H, I), B)]
Creates a Codec[(A, B)] that first encodes/decodes an A followed by a B.
Creates a Codec[(A, B)] that first encodes/decodes an A followed by a B.
Operator alias for pairedWith.
Definition Classes
83. final def ~>[B](codecB: Codec[B])(implicit ev: =:=[Unit, (A, B, C, D, E, F, G, H, I)]): Codec[B]
Assuming A is Unit, creates a Codec[B] that: encodes the unit followed by a B; decodes a unit followed by a B and discards the decoded unit.
Assuming A is Unit, creates a Codec[B] that: encodes the unit followed by a B; decodes a unit followed by a B and discards the decoded unit.
Operator alias of dropLeft.
Definition Classes
84. def ~~[J](J: Codec[J]): Tuple10Codec[A, B, C, D, E, F, G, H, I, J]
85. def →[B](y: B): (Tuple9Codec[A, B, C, D, E, F, G, H, I], B)
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ArrowAssoc[Tuple9Codec[A, B, C, D, E, F, G, H, I]] performed by method ArrowAssoc in
Definition Classes
Transforms using two functions, A => Attempt[B] and B => Attempt[A].
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec.
This implicitly inherited member is shadowed by one or more members in this class.
To access this member you can use a type ascription:
(tuple9Codec: TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)]).exmap(f, g)
Definition Classes
Transforms using two functions, A => Attempt[B] and B => A.
The supplied functions form an injection from B to A. Hence, this method converts from a larger to a smaller type. Hence, the name narrow.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec.
This implicitly inherited member is shadowed by one or more members in this class.
To access this member you can use a type ascription:
(tuple9Codec: TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)]).narrow(f, g)
Definition Classes
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to Tuple2CodecSupport[(A, B, C, D, E, F, G, H, I)] performed by method Tuple2CodecSupport in scodec.
This implicitly inherited member is ambiguous. One or more implicitly inherited members have similar signatures, so calling this member may produce an ambiguous implicit conversion compiler
To access this member you can use a type ascription:
(tuple9Codec: Tuple2CodecSupport[(A, B, C, D, E, F, G, H, I)]).self
Definition Classes
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to EnrichedCoproductEncoder[Coproduct with (A, B, C, D, E, F, G, H, I)] performed by method
EnrichedCoproductEncoder in scodec.
This implicitly inherited member is ambiguous. One or more implicitly inherited members have similar signatures, so calling this member may produce an ambiguous implicit conversion compiler
To access this member you can use a type ascription:
(tuple9Codec: EnrichedCoproductEncoder[Coproduct with (A, B, C, D, E, F, G, H, I)]).self
Definition Classes
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ValueCodecEnrichedWithGenericSupport[(A, B, C, D, E, F, G, H, I)] performed by method
ValueCodecEnrichedWithGenericSupport in scodec.
This implicitly inherited member is ambiguous. One or more implicitly inherited members have similar signatures, so calling this member may produce an ambiguous implicit conversion compiler
To access this member you can use a type ascription:
(tuple9Codec: ValueCodecEnrichedWithGenericSupport[(A, B, C, D, E, F, G, H, I)]).self
Definition Classes
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ValueCodecEnrichedWithHListSupport[(A, B, C, D, E, F, G, H, I)] performed by method
ValueCodecEnrichedWithHListSupport in scodec.
This implicitly inherited member is ambiguous. One or more implicitly inherited members have similar signatures, so calling this member may produce an ambiguous implicit conversion compiler
To access this member you can use a type ascription:
(tuple9Codec: ValueCodecEnrichedWithHListSupport[(A, B, C, D, E, F, G, H, I)]).self
Definition Classes
Supports TransformSyntax.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec.
This implicitly inherited member is ambiguous. One or more implicitly inherited members have similar signatures, so calling this member may produce an ambiguous implicit conversion compiler
To access this member you can use a type ascription:
(tuple9Codec: TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)]).self
Definition Classes
Transforms using two functions, A => B and B => Attempt[A].
The supplied functions form an injection from A to B. Hence, this method converts from a smaller to a larger type. Hence, the name widen.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec.
This implicitly inherited member is shadowed by one or more members in this class.
To access this member you can use a type ascription:
(tuple9Codec: TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)]).widen(f, g)
Definition Classes
Transforms using the isomorphism described by two functions, A => B and B => A.
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec.
This implicitly inherited member is shadowed by one or more members in this class.
To access this member you can use a type ascription:
(tuple9Codec: TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)]).xmap(f, g)
Definition Classes
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to ValueEnrichedWithTuplingSupport[Tuple9Codec[A, B, C, D, E, F, G, H, I]] performed by method
ValueEnrichedWithTuplingSupport in scodec.codecs.
This implicitly inherited member is shadowed by one or more members in this class.
To access this member you can use a type ascription:
(tuple9Codec: ValueEnrichedWithTuplingSupport[Tuple9Codec[A, B, C, D, E, F, G, H, I]]).~(b)
Definition Classes
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to Tuple2CodecSupport[(A, B, C, D, E, F, G, H, I)] performed by method Tuple2CodecSupport in scodec.
This implicitly inherited member is shadowed by one or more members in this class.
To access this member you can use a type ascription:
(tuple9Codec: Tuple2CodecSupport[(A, B, C, D, E, F, G, H, I)]).~~(B)
Definition Classes
Transforms using two functions, A => B and B => Option[A].
Particularly useful when combined with case class apply/unapply. E.g., pxmap(fa, Foo.apply, Foo.unapply).
Implicit information
This member is added by an implicit conversion from Tuple9Codec[A, B, C, D, E, F, G, H, I] to TransformSyntax[Codec, (A, B, C, D, E, F, G, H, I)] performed by method TransformSyntax in scodec.
Definition Classes
(Since version 1.7.0) Use widenOpt instead | {"url":"http://scodec.org/api/scodec-core/1.10.0/scodec/codecs/Tuple9Codec.html","timestamp":"2024-11-09T10:53:28Z","content_type":"text/html","content_length":"331649","record_id":"<urn:uuid:bc9152bb-ce8c-4a8c-b15b-3cc5e85d173f>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00511.warc.gz"} |
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TensorFlow Transpose | How does tensorflow transpose works?
Updated March 15, 2023
Introduction to TensorFlow Transpose
Tensorflow transpose is the method or function available in the Tensorflow library or package of the Python Machine Learning domain. Whenever we pass the input to the tensorflow model, this function
helps us evaluate the transpose of the provided input. In this article, we will have a detailed look at the tensorflow transpose function, how it works, the parameters needed to pass to it, the use
of the transpose function, and also have a look at its implementation along with the help of an example. Lastly, we will conclude our statement.
What is TensorFlow transpose?
Tensorflow transpose is the function provided in the tensorflow, which helps to find out the transpose of the input. Transpose means to flip or reverse the value. For example, IF the matrix is the
input, then the transpose of the matrix input will provide the flipping of the values along the diagonal. This causes the switch in the columns and rows of the matrix.
The location of the tensorflow transpose function is found in tensorflow/python/ops/array_ops.py. In tensorflow, the transpose function can be used by using the below-mentioned syntax –
sampleTF.transpose ( sampleValue, perm = None , name = ‘transpose’ , conjugate = False)
In the above syntax, the terminologies involved will be described in the below list –
• sampleValue – It is the input value that is to be transposed. In case of the value of the conjugate is set to true, then the sampleValue dtype can have the value of either complex128 or
complex64, and the sampleValue is the transposed and conjugated value.
• Perm – It is the parameter that is responsible for allowing the dimensions of the input.
• Returned Output – The output of the transpose function will be the matrix as per the dimensions permitted by perm[n] and corresponding to the dimensions of the input matrix. By default, the perm
value is considered to be (m-1….0) when not specified. Here the value of the m is nothing but the rank of the matrix of tensorflow provided in the input. The default transpose operations are
carried out on a 2-dimensional input tensor provided.
How does tensorflow transpose works?
The working of transpose is similar to the flipping of the row and column values in a diagonal manner. Let us consider one sample input matrix –
[ 21, 22, 23],
[ 24, 25, 26],
[ 27, 28, 29],
[30, 31, 32]
Will be transposed to –
[ 21, 24, 27, 30],
[ 22, 25, 28, 31],
[ 23, 26, 29, 32]
We can observe that the rows and columns are interchanged.
Tensorflow Transpose Function
Tensorflow transpose function allows you to flip the tensor matrix passed as the input. This function is defined inside the file whose directory is inside the path tensorflow/python/ops/array_ops.py.
If you pass a matrix that contains the dimension [m, n] where m and are the number of rows and columns, respectively. Then the transpose function will flip the tensor’s input matrix, leading to the
interchange of rows and columns by flipping them through a diagonal. The output matrix will be diagonal [n, m].
The transpose function can be called by using the syntax –
sampleTF.transpose(sampleValue, perm = None, name = ‘transpose’, conjugate = False)
whose parameters and terminologies are already described previously.
TensorFlow Transpose Examples
Let us consider one example of a python program where we will be implementing and using the transpose function –
Let us understand how the transpose of the input tensor works considering one sample matrix of tensorflow for input. Suppose that we call the transpose function of tensorflow by using the below
statement –
sample = sampleTFObject. Constant ( [[11,12,13], [14,15,16]])
sampleTFObject. Transpose (sample)
sampleTFObject. Transpose (sample, perm = [1,0])
The output of either of the above two statements will lead to the conversion of the sample matrix to the following where the rows and columns are interchanged –
[[11,14], [12, 15], [13, 16]]
Let us consider one example where our input matrix will be a complex matrix involving imaginary numbers. If we set the property of conjugate to true, then it will provide us the transpose of the
input matrix –
Sample = sampleTFObject. Constant ([[11 + 11j, 12 + 12j, 13 + 13j],
[14 + 14j, 15 + 15j, 16 + 16j]])
sampleTFObject. Transpose (sample, conjugate = True)
The output of the above command will lead to the following transpose –
# [[11 - 11j, 14 - 14j],
# [12 - 12j, 15 - 15j],
# [13 - 13j, 16 - 16j]]
Let us consider one more example where we will use perm, which helps specify the permutation of the dimensions of the tensor matrix.
sample = sampleTFObject .constant([[[ 21, 22, 23],
[ 24, 25, 26]],
[[ 27, 28, 29],
[30, 31, 32]]])
We have taken a matrix of 0 dimensions which is the shorthand of ‘linalg.transpose’.
Perm value will be [0, 2, 1]
sampleTFObject. transpose (sample, perm=[0, 2, 1])
[[[21, 24],
[22, 25],
[23, 26]],
[[27, 30],
[28, 31],
[29, 32]]]
In the above example that we considered, the sampleTFObject is the tensor object, and a perm is the permutation of the required and specified dimensions of the sample. A name can be any name of the
object we want to perform. The conjugate parameter will be an optional Boolean value. If we set the value of this parameter to true, then the equivalent mathematical will be to the sampleTFObject.
conj (sampleTFObject. transpose (input)). The returned value will be a tensor that is completely transposed and flipped.
The compatibility of NumPy transpose with TensorFlow transpose is not supporting the stride functionality. The numpy transpose is the efficient operation for memory and has constant time as it gives
the same output with the new view of the passed data. It just adjusts the strides in the output view.
The transpose function of tensorflow helps flip the input tensor, leading to the matrix’s interchange of rows and columns.
Recommended Articles
This is a guide to TensorFlow Transpose. Here we discuss the Introduction, What and How does tensorflow transpose works? Examples with code implementation. You may also have a look at the following
articles to learn more – | {"url":"https://www.educba.com/tensorflow-transpose/","timestamp":"2024-11-13T09:41:33Z","content_type":"text/html","content_length":"312511","record_id":"<urn:uuid:e211b45b-685a-4437-a592-4cf3651357c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00255.warc.gz"} |
An Elementary Humanomics Approach to Boundedly Rational Quadratic Models
Publication Date
We take a refreshing new look at boundedly rational quadratic models in economics using some elementary modeling of the principles put forward in the book Humanomics by Vernon L. Smith and Bart J.
Wilson. A simple model is introduced built on the fundamental Humanomics principles of gratitude/resentment felt and the corresponding action responses of reward /punishment in the form of higher/
lower payoff transfers. There are two timescales: one for strictly self-interested action, as in economic equilibrium, and another governed by feelings of gratitude/resentment. One of three timescale
scenarios is investigated: one where gratitude /resentment changes much more slowly than economic equilibrium (“quenched model”). Another model, in which economic equilibrium occurs over a much
slower time than gratitude /resentment evolution (“annealed” model) is set up, but not investigated. The quenched model with homogeneous interactions turns out to be a non-frustrated spin-glass
model. A two-agent quenched model with heterogeneous aligning (ferromagnetic) interactions is analyzed and yields new insights into the critical quenched probability p (1 - p) that represents the
frequency of opportunity for agent i to take action for the benefit (hurt) of other that invokes mutual gratitude (resentment). A critical quenched probability p[i]*, i = 1,2, exists for each agent.
When p < p[i]*, agent i will choose action in their self-interest. When p > p[i]*, agent i will take action sensitive to their interpersonal feelings of gratitude/resentment and thus reward/punish
the initiating benefit/hurt. We find that the p[i]* are greater than one-half, which implies agents are averse to resentful behavior and punishment. This was not built into the model, but is a result
of its properties, and consistent with Axiom 4 in Humanomics about the asymmetry of gratitude and resentment. Furthermore, the agent who receives less payoff is more averse to resentful behavior;
i.e., has a higher critical quenched probability. For this particular model, the Nash equilibrium has no predictive power of Humanomics properties since the rewards are the same for self-interested
behavior, resentful behavior, and gratitude behavior. Accordingly, we see that the boundedly rational Gibbs equilibrium does indeed lead to richer properties.
ESI Working Paper 20-35
This paper later underwent peer review and was published as:
Campbell, M. J., & Smith, V. L. (2020). An elementary humanomics approach to boundedly rational quadratic models. Physica A, 562, 125309. https://doi.org/10.1016/j.physa.2020.125309 | {"url":"https://digitalcommons.chapman.edu/esi_working_papers/330/","timestamp":"2024-11-13T15:03:54Z","content_type":"text/html","content_length":"43517","record_id":"<urn:uuid:4d53497c-d046-46b2-87b9-408ff28e9289>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00182.warc.gz"} |
Modular multiplicative inverse calculator
This calculator calculates the modular multiplicative inverse of a given integer a under modulo m:
\[x\equiv a^{-1} \pmod{m}\]
Modular multiplicative Inverse
The inverse of an element \(x\) is another element \(y\) such that \(x\circ y = e\), where \(e\) is the neutral element. For example:
\[ \begin{array}{rl} x + y &= 0\Rightarrow y = -x\\ x \cdot y &= 1\Rightarrow y = x^{-1}\\ \end{array} \]
In number theory and encryption often the inverse is needed under a modular ring. Less formal spoken, how can one divide a number under a modular relation? Here the multiplicative inverse comes in.
The multiplicative inverse of an integer \(a\) modulo \(m\) exists if and only if \(a\) and \(m\) are coprime (i.e., if \(\gcd(a, m) = 1\)) and is an integer \(x\) such that
\[a x\equiv 1 \pmod{m}\]
Dividing both sides by \(a\) gives
\[x\equiv a^{-1} \pmod{m}\]
The solution can be found with the euclidean algorithm as follows. Lets take
This is a linear diophantine equation with two unknowns, which solution should be a multiple of \(\gcd(a,b)\)
To calculate the modular inverse, the calculator uses this idea to find solutions to the Bezout identity using the EGCD: | {"url":"https://raw.org/tool/modular-inverse-calculator/","timestamp":"2024-11-02T11:37:49Z","content_type":"text/html","content_length":"29194","record_id":"<urn:uuid:20fa91b0-b5e6-4a93-836a-b5c44f0bc045>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00771.warc.gz"} |
Fluctuations of linear statistics of fr
SciPost Submission Page
Fluctuations of linear statistics of free fermions in a harmonic trap at finite temperature
by Aurélien Grabsch, Satya N. Majumdar, Grégory Schehr, Christophe Texier
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Aurélien Grabsch · Gregory Schehr
Submission information
Preprint Link: http://arxiv.org/abs/1711.07770v1 (pdf)
Date submitted: 2017-11-22 01:00
Submitted by: Grabsch, Aurélien
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
• Atomic, Molecular and Optical Physics - Theory
Specialties: • Quantum Physics
• Statistical and Soft Matter Physics
Approach: Theoretical
We study a system of 1D non interacting spinless fermions in a confining trap at finite temperature. We first derive a useful and general relation for the fluctuations of the occupation numbers valid
for arbitrary confining trap, as well as for both canonical and grand canonical ensembles. Using this relation, we obtain compact expressions, in the case of the harmonic trap, for the variance of
linear statistics $\mathcal{L}=\sum_n h(x_n)$, where $h$ is an arbitrary function of the fermion coordinates $\{ x_n \}$. As anticipated, we demonstrate explicitly that these fluctuations do depend
on the ensemble in the thermodynamic limit, as opposed to averaged quantities, which are ensemble independent. We have applied our general formalism to compute the fluctuations of the number of
fermions $\mathcal{N}_+$ on the positive axis at finite temperature. Our analytical results are compared to numerical simulations.
Current status:
Has been resubmitted
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2018-1-9 (Invited Report)
• Cite as: Anonymous, Report on arXiv:1711.07770v1, delivered 2018-01-09, doi: 10.21468/SciPost.Report.320
1) Very neat presentation of original results
2) Calculations and logic easy to follow
1) Focus only on the harmonic potential
The authors develop a general method to calculate the variance of linear statistics in a free one-dimensional Fermi gas, with a confining potential (Eq.(38) in particular). They apply the formalism
to determine the variance of the number of fermions in the positive semi-line (index) both in the canonical and grand canonical ensembles. The analysis is limited to the harmonic potential.
The authors also recover the variance of the potential energy previously derived in Ref. [27]. They conclude with a numerical simulation of the determinantal process associated to the fermion
positions in an harmonic potential that further confirms their findings in the grand canonical ensemble.
I found the paper pedagogically written and interesting. Since moreover to my best knowledge the results presented in Sec.2.2, 3.2.1 and 4-5 are original and correct, I recommend the paper to be
published after the authors will address the minor points below.
Requested changes
1)The formalism allows calculating in particular the variance of N_+ in the Ground State for an arbitrary potential. Does this show a Log(N) term as in Eq.(5)? If yes, is this term universal (i.e.
potential independent)?
2)At pag. 13: There is a typo in the penultimate sentence of sec. 3.1
3)At pag. 22: kinetic->potential, (although they are equivalent seems more consistent with the text to continue discussing potential energy)
4)At pag. 22: formula (113) an "f_y" is missing after the second sum
5)At pag. 23: Formulas (119)-(120). I think "c" has to be replaced with "g", saying perhaps that what is calculated in (120) is the variance in the canonical ensemble with N replaced by \bar{N}_g.
6)It seems that the Bose factor that appears in Eq.(110) is independent from the choice of the potential. Perhaps the authors can stress this fact and its implications for the asymptotics (111) in
the quantum regime. Are then these asymptotics expected to be potential independent?
answer to question
We are grateful to the referee for his/her very careful reading of the manuscript and his/her positive report.
1) and 6) The referee has raised the same interesting points as Referee 1. Here is our answer: I) Concerning the zero temperature result: although the referee is right about the universality of the
dominant logarithmic term of the variance, we stress that the subleading constant is potential dependent. For a hard box confinement, the zero temperature result for $\mathcal{N}_+$ can be found for
example in Eq. (40) of EPL 98, 20003 (2012) and reads
$$ \mathrm{Var}(\mathcal{N}+) |^\text{Box} = \frac{1}{2\pi^2} \ln N + \frac{1+\gamma+2\ln 2}{2\pi^2} \:, $$
which differs from Eq. (5) by a constant term
$\ln 2/(2\pi^2)$
II) Nevertheless, at finite temperature, stimulated by the question of the referee we have investigated further the universality of our formulae. We have now added a new section (5.5) and extended
the discussion of Appendix A. The main outcome is: - In the quantum regime the thermal fluctuations involve excitations around the Fermi level. For a one dimensional smooth confining potential, the
spectrum is regular and can be linearlised near the Fermi level. The fluctuations are thus described by the same function $F_Q$ as for the harmonic oscillator. - In the thermal regime all the
spectrum contributes and therefore the results are not universal.
2-5) We thank the referee for pointing out these typos.
Report #1 by Anonymous (Referee 1) on 2018-1-8 (Invited Report)
• Cite as: Anonymous, Report on arXiv:1711.07770v1, delivered 2018-01-08, doi: 10.21468/SciPost.Report.319
1. simple problem with physically interesting question
2. clear presentation
1. results could have been further discussed
The authors study fluctuations properties of a free Fermi gas
trapped in a harmonic potential at finite temperature.
More precisely, they consider the so-called linear statistics,
i.e. the sum of an arbitrary function of the fermion coordinates.
The main result of the manuscript is an explicit formula for
the variance of the linear statistics, Eq. (38), which is valid
for both canonical and grand canonical ensembles.
This result is then applied to calculate the fluctuations
of the particle number in the right half of the trap.
The scaling functions in the two different, quantum and thermal
regimes are explicitly calculated. In the quantum regime the
analytical result is compared against numerical simulations
with a good agreement.
I believe that, despite the simple, textbook-like calculations,
the manuscript deals with a physically interesting problem.
In particular, the finite temperature results on the particle
fluctuation are new, and the general result on linear statistics
possibly has a broader range of applicability.
Therefore I recommend the publication of the manuscript after the
authors have considered the issue raised below.
Requested changes
What I was missing a bit is a discussion about the role of the
potential. In fact, one could have equally well asked about the
fluctuations in one half of a finite box with no external potential.
At zero temperature this question has already been studied, see
EPL 98, 20003 (2012) and Phys. Rev. B 82, 012405 (2010).
Remarkably, it turns out that the box-result at $T=0$ is
identical to Eq. (5) of the manuscript, i.e. the role of the
potential is irrelevant. This naturally raises the question,
what is the situation at finite temperature? Since the precise
form of the eigenfunctions enters only through integrals in
Eqs. (90-91), one could expect some universality also at finite T.
It would be nice, if the authors could comment about this.
answer to question
We are thankful to the referee for his/her careful reading of the manuscript and his/her positive comments.
The referee has raised an interesting question about the universality of our results, which were derived for a harmonic confinement.
I) Concerning the zero temperature result: although the referee is right about the universality of the dominant logarithmic term of the variance, we stress that the subleading constant is potential
dependent. For a hard box confinement, the zero temperature result for $\mathcal{N}_+$ can be found for example in Eq. (40) of EPL 98, 20003 (2012) and reads
$$ \mathrm{Var}(\mathcal{N}+) |^\text{Box} = \frac{1}{2\pi^2} \ln N + \frac{1+\gamma+2\ln 2}{2\pi^2} \:, $$
which differs from Eq. (5) by a constant term
$\ln 2/(2\pi^2)$
II) Nevertheless, at finite temperature, stimulated by the question of the referee we have investigated further the universality of our formulae. We have now added a new section (5.5) and extended
the discussion of Appendix A. The main outcome is: - In the quantum regime the thermal fluctuations involve excitations around the Fermi level. For a one dimensional smooth confining potential, the
spectrum is regular and can be linearlised near the Fermi level. The fluctuations are thus described by the same function $F_Q$ as for the harmonic oscillator. - In the thermal regime all the
spectrum contributes and therefore the results are not universal. | {"url":"https://scipost.org/submissions/1711.07770v1/","timestamp":"2024-11-10T02:49:10Z","content_type":"text/html","content_length":"45138","record_id":"<urn:uuid:d5093e0e-a300-48cd-9152-34235ab943af>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00363.warc.gz"} |
Lipschitz constant log{<i>n</i>} almost surely suffices for mapping <i>n</i> grid points onto a cube
Kaluža, Kopecká and the author have shown that the best Lipschitz constant for mappings taking a given n^d-element set in the integer lattice ℤ^d, with n∈ℕ, surjectively to the regular n times n grid
{1,…,n}^d may be arbitrarily large. However, there remain no known, non-trivial asymptotic bounds, either from above or below, on how this best Lipschitz constant grows with n. We approach this
problem from a probabilistic point of view. More precisely, we consider the random configuration of n^d points inside a given finite lattice and establish almost sure, asymptotic upper bounds of
order log n on the best Lipschitz constant of mappings taking this set surjectively to the regular n times n grid {1,…,n}^d.
Bibliographical note
18 pages
• math.FA
• 51F99, 51M05, 52C99, 26B10
Dive into the research topics of 'Lipschitz constant log{n} almost surely suffices for mapping n grid points onto a cube'. Together they form a unique fingerprint. | {"url":"https://research.birmingham.ac.uk/en/publications/lipschitz-constant-logini-almost-surely-suffices-for-mapping-ini-","timestamp":"2024-11-05T22:00:20Z","content_type":"text/html","content_length":"54650","record_id":"<urn:uuid:3e6a2513-6e72-48ab-8232-586578047b47>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00286.warc.gz"} |
Nominal-accidental chain
Jump to navigation Jump to search
"Sharp" and "flat" redirect here. For the temperaments, see Sharp (temperament) and Flat (temperament).
This is a neologism for the common pattern in notating microtonal pitch systems. These are analogous extensions of basic Western musical notation.
Nominals are pitch elements that have specific names. In Western musical notation, these names are the seven letters A, B, C, D, E, F, and G (historically, H has also been used). In a pentatonic
notation, there would be only five names.
Accidentals are additional pitches that arise as modifications of the nominals. Unmodified pitches are natural notes. In diatonic circle-of-fifths notation, the additional pitches are denoted by
adding sharps or flats to the natural notes. The sharp accidental denotes a pitch raise by a chromatic semitone, equivalent to a raise by 7 fifths minus 4 octaves. Conversely, the flat accidental
denotes a pitch drop by the same amount. In equal temperaments, the number of steps this interval is mapped to is called the sharpness.
These pitches form a chain, with each one separated from the next by a specific interval. This interval can be said to generate the notation, or the notation can be said to be based on this interval.
In diatonic circle-of-fifths notation, this interval has been a just or near-just 3/2. Other intervals are possible, and even desirable for certain edos like 13, 18 and 23.
Equivalence may arise from this approach. This is when you have multiple names for the same pitch. The equivalence that occurs in 12edo is enharmonic equivalence. C-sharp is enharmonically equivalent
to D-flat, but only in 12edo, 24edo, 36edo, etc. The same term is sometimes used to refer to equivalence in general, but each edo technically has its own equivalence. 7edo has the type of equivalence
that could be called chromatic equivalence, for example.
Specific notation schemes
Related topics
See also | {"url":"https://en.xen.wiki/w/Nominal-Accidental_Chains","timestamp":"2024-11-13T07:51:00Z","content_type":"text/html","content_length":"28473","record_id":"<urn:uuid:0b42cb0b-7a05-408c-a72c-e2c66bee6ada>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00429.warc.gz"} |
Modification of the motion characteristics of a one-mass linear vibratory conveyor
Linear vibratory conveyors are common equipment to convey goods. They are used in different industries such as for example the building, mining and food industry. The systems are often used for the
supply of bulk goods into further processing operations. The goods to be conveyed, typically requesting a heavy-duty design of the conveyor and eccentric excitation drives with relatively high
torque. Often the motion of these conveyors is not ideal with reference to the conveying process of the bulk goods. Motion speed and amplitude of the goods are not constant and blocking effects as
well as separation effects can occur. The target of this article is to determine a dynamic model for the targeted displacement of the centre of elasticity of a one-mass conveyor, enabling an optimal
motion of the conveyor by an optimized set of springs, connecting the conveying element to the frame.
1. Introduction
Linear vibratory conveyors are mainly built to forward bulk material for the mining, building and food industry, such as rock, grit and flour. The assembly of these vibratory conveyors consists of
two principal parts, the conveying element build as a conveying trough and a supporting element, build of steel elements and fixed to the ground by heavy-duty screws [1, 2]. Often, the conveying
element is linked to the frame just be a set of springs. Such a system represents an efficient method for the generation of a micro-throw motion to the goods to be conveyed. Often, conveyors of this
type are not intended to generate a constant motion regarding conveying speed and amplitude over the full length of the conveying element. This is mainly caused by the situation of the mass centre
point and the centre of elasticity of such a conveyor. To eliminate this disadvantage, some linear vibratory conveyors are additionally equipped with a set of guiding levers between conveying element
and Frame. Fig. 1 shows a typical one-mass vibratory conveyor without any guiding mechanism.
Fig. 1One-mass vibratory conveyor without guiding levers [own source]
2. Applied methods
The theoretical solution is done in two steps. At first, a dynamic model is designed and then the dynamic parameters are determined.
2.1. Determination of the conveyors dynamic model
One-mass linear vibratory conveyors corresponding to the shown variant consisting of a conveying element which is connected to the base frame by four helical springs. Two eccentric drives are
connected directly to the conveying element under a certain angle to the horizontal line. The eccentric dives turning clockwise (left drive in direction of material flow) and anti-clockwise (right
drive in direction of material flow). These drives are self-centring and so a motion across the main forwarding direction is eliminated [3, 4].
Corresponding to the conveyor as shown in Fig. 1, the following general mechanical model can be of this conveyor type can be generated [5]:
Fig. 2Mechanical model of the vibratory conveyor [own source]
2.2. Calculation of dynamic parameters
Assuming ${\xi }_{A}={\xi }_{B}$, Eq. (1) can be converted to:
${J}_{z}=m\bullet {\xi }_{A}\bullet {\xi }_{B},$
${J}_{z}=m\bullet {\xi }^{2}.$
Based on this assumption, the following equations can be set up:
$m\stackrel{¨}{x}+\left({k}_{Ax}+{k}_{Bx}\right)\bullet z+\left[\left({k}_{Ax}-{k}_{Bx}\right)\frac{\xi }{2}\right]\bullet \phi ={F}_{0}\bullet \mathrm{cos}\left(\omega Ft\right),$
$m\stackrel{¨}{z}+\left({k}_{Az}+{k}_{Bz}\right)\bullet z+\left[\left({k}_{Az}-{k}_{Bz}\right)\frac{\xi }{2}\right]\bullet \phi ={F}_{0}\bullet \mathrm{sin}\left(\omega Ft\right),$
${J}_{z}\stackrel{¨}{\phi }+\left[\left({k}_{Az}-{k}_{Bz}\right)\frac{\xi }{2}\right]\bullet z+\left[\left({k}_{Az}-{k}_{Bz}\right)\frac{{\xi }^{2}}{4}\right]\bullet \phi$$=-{F}_{0}\mathrm{c}\mathrm
{o}\mathrm{s}\beta F\bullet \rho PF\mathrm{s}\mathrm{i}\mathrm{n}\beta PF+{F}_{0}\mathrm{s}\mathrm{i}\mathrm{n}\beta F\bullet \rho PF\mathrm{c}\mathrm{o}\mathrm{s}\beta PF.$
Converted into the matrix form, the following matrices can be built.
For the mass:
$M=\left[\begin{array}{ccc}m& 0& 0\\ 0& m& 0\\ 0& 0& {J}_{z}\end{array}\right].$
For the stiffness of the springs:
$K=\left[\begin{array}{ccc}{k}_{Ax}+{k}_{Bx}& 0& {-{\eta }_{A}k}_{Ax}-{\eta }_{B}{k}_{Bx}\\ 0& {k}_{Az}+{k}_{Bz}& {{\xi }_{A}k}_{Az}+{\xi }_{B}{k}_{Bz}\\ {-{\eta }_{A}k}_{Ax}-{\eta }_{B}{k}_{Bx}& {{\
xi }_{A}k}_{Az}+{\xi }_{B}{k}_{Bz}& {{\eta }_{A}^{2}k}_{Ax}+{\eta }_{B}^{2}{k}_{Bx}+{{\xi }_{A}^{2}k}_{Az}+{\xi }_{B}^{2}{k}_{Bz}\end{array}\right].$
The mechanical model described in Fig. 2, enables three degrees of freedom, i.e. a longitudinal motion in $x$-direction, a second longitudinal motion in $z$-direction and a rotation around the mass
centre point, depending on the situation and effective direction of the excitation force and the centre of elasticity of the system. To ensure a pure linear motion under the intended excitation
angle, it is necessary to avoid any kind of rotation around the mass centre point on this type of linear vibratory conveyor. As soon as a rotation takes place, the motion of the conveyed goods will
be influenced in an unintended way, causing a suboptimal motion of the goods and a potential malfunction of the conveyor [6].
A mayor influence in this model comes from the damping conditions. Neglecting the horizontal components of the general model, the following equations can be formed for the values in $z$-direction.
For the displacement:
$z={z}_{0}\bullet \mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right).$
For the velocity:
$\stackrel{˙}{z}={z}_{0}\omega \bullet \mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right),$
and for the acceleration:
$\stackrel{¨}{z}=-{z}_{0}{\omega }^{2}\bullet \mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right).$
The velocity which is linked to the damping in the dynamic calculation of the system, represents the first derivation of the displacement and is the only component including a cosine function. For
systems with very low damping, as given on a linear vibratory conveyor of the variant $A$, the following equations then can be formed:
${k}_{Az}{z}_{0}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)+{k}_{Bz}{z}_{0}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)-m{x}_{0}{\omega }^{2}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega
t\right)={F}_{0}\bullet \mathrm{s}\mathrm{i}\mathrm{n}\left(\omega Ft\right),$
${k}_{Az}{z}_{0}+{k}_{Bz}{z}_{0}-m{z}_{0}{\omega }^{2}={F}_{0},$
${k}_{Az}{\xi }_{A}+{k}_{Bz}{\xi }_{B}-{F}_{0}\rho =0.$
By the use of a numerical model, based on the Eqs. (11) to (13), a graph can be designed, showing the area of no-rotation. A valley between the stiffness ${k}_{AZ}$ and ${k}_{BZ}$ occurs and the
intended amplitudes of the system can be selected by following the particular lines through the valley.
The following graph Fig. 4 for amplitude over stiffness in $z$-direction will be received.
Transmitting the requested design conditions for pure linear motion into the general model shown in Fig. 2, a conveyor with pure linear motion and no rotation around its mass centre point can be
3. Results and discussions
To verify and validate the possibilities of modification on linear vibratory conveyors build on the mechanical model shown in Fig. 2, an existing conveyor is analysed and modified by changing the
springs of the system. The stiffness of the springs is calculated based on the general Eqs. (3)-(5).
Fig. 3Amplitude over stiffness 3-D graph of a linear vibratory conveyor
Fig. 4Rotation component of the amplitude in point A of the conveyor
In addition, the damping of the system has to be considered on the real conveyor. This is done by extending the equations by the terms corresponding to the damping of the system as follows:
$m\stackrel{¨}{x}+\left({k}_{Ax}+{k}_{Bx}\right)\bullet x+\left[\left({k}_{Ax}-{k}_{Bx}\right){\xi }_{A}+\left({k}_{Ax}-{k}_{Bx}\right){\xi }_{B}\right]\bullet \phi +\left({b}_{Ax}+{b}_{Bx}\right)\
bullet \stackrel{˙}{x}$$+\left[\left({b}_{Ax}-{b}_{Bx}\right){\xi }_{A}+\left({b}_{Ax}-{b}_{Bx}\right){\xi }_{B}\right]\bullet \stackrel{˙}{\phi }={F}_{0}\bullet \mathrm{cos}\left(\omega Ft\right),$
$m\stackrel{¨}{z}+\left({k}_{Az}+{k}_{Bz}\right)\bullet z+\left[\left({k}_{Az}-{k}_{Bz}\right){\xi }_{A}+\left({k}_{Az}-{k}_{Bz}\right){\xi }_{B}\right]\bullet \stackrel{˙}{\phi }+\left({b}_{Az}+{b}_
{Bz}\right)\bullet \stackrel{˙}{z}$$+\left[\left({b}_{Az}-{b}_{Bz}\right){\xi }_{A}+\left({b}_{Az}-{b}_{Bz}\right){\xi }_{B}\right]\bullet \stackrel{˙}{\phi }={F}_{0}\bullet \mathrm{sin}\left(\omega
${J}_{z}\stackrel{¨}{\phi }+\left[\left({k}_{Az}-{k}_{Bz}\right){\xi }_{A}+\left({k}_{Az}+{k}_{Bz}\right){\xi }_{B}\right]\bullet z+\left[\left({b}_{Az}+{b}_{Bz}\right){\xi }_{A}+\left({b}_{Az}+{b}_
{Bz}\right){\xi }_{B}\right]\bullet \stackrel{˙}{z}$$+\left[\left({k}_{Az}-{k}_{Bz}\right){\xi }_{A}^{2}+\left({k}_{Az}-{k}_{Bz}\right){\xi }_{B}^{2}\right]\bullet \phi +\left[\left({b}_{Az}+{b}_{Bz}
\right){\xi }_{A}^{2}+\left({b}_{Az}+{b}_{Bz}\right){\xi }_{B}^{2}\right]$$=-{F}_{0}\mathrm{c}\mathrm{o}\mathrm{s}\beta F\bullet \rho PF\mathrm{s}\mathrm{i}\mathrm{n}\beta PF++{F}_{0}\mathrm{s}\
mathrm{i}\mathrm{n}\beta F\bullet \rho PF\mathrm{c}\mathrm{o}\mathrm{s}\beta PF.$
Finally, all equations are converted into matrix form. For the mass matrix and the damping matrix, the equations will stay identical to Eqs. (6) and (7). For the damping, the following matrix has to
be created:
$B=\left[\begin{array}{ccc}{b}_{Ax}+{b}_{Bx}& 0& {-{\eta }_{A}b}_{Ax}-{\eta }_{B}{b}_{Bx}\\ 0& {b}_{Az}+{b}_{Bz}& {{\xi }_{A}b}_{Az}+{\xi }_{B}{b}_{Bz}\\ {-{\eta }_{A}b}_{Ax}-{\eta }_{B}{b}_{Bx}& {{\
xi }_{A}b}_{Az}+{\xi }_{B}{b}_{Bz}& {{\eta }_{A}^{2}b}_{Ax}+{\eta }_{B}^{2}{b}_{Bx}+{{\xi }_{A}^{2}b}_{Az}+{\xi }_{B}^{2}{b}_{Bx}\end{array}\right].$
3.1. Result of the analysis of the existing system
Following the graph in Fig. 4, ${k}_{Bx}$ is situated in the area of no rotation while the amplitude for ${k}_{Ax}$ corresponds to an amplitude of approximately 1.00 mm. That means, the rotating
component of the motion in point A has an additional effect of 0.90 mm compared to point B.
Calculation and graph confirming the experience made during testing the system. So, the equations are verified as useful tool for the general analysis of linear vibratory conveyors of the analysed
design. In a further step, the calculation model will be used for the determination of spring pairs, elimination the rotation component of the conveyor.
3.2. Calculation of the optimized conditions
To keep the compact and rigid design of the linear vibratory conveyor, the best possible solution to modify the conveyors motion would be a re-location of the centre of elasticity.
Keeping the mass conditions of the conveying element untouched and ensuring the damping conditions staying competitive to the presently existing once, a solution will be found by changing the
stiffness conditions of the springs in the points A and B of the conveyor.
Focused on this modification possibility, three options of influencing the situation of the mass centre point with a simultaneous elimination of the rotation part of the motion are generally
• Changing the stiffness ${k}_{Az}$ by replacing the springs in point A,
• Changing the stiffness ${k}_{Bz}$ by replacing the springs in point B,
• Changing the stiffness ${k}_{Az}$ and the stiffness ${k}_{Bz}$ by replacing all springs in points A and B.
Based on this numerical calculation, a test series for the determination of the new situation of the centre of elasticity and the operation conditions is realised and the described modification
possibilities are verified.
Fig. 5Corresponding stiffness kAz and kBz for the selection of test springs-different possibilities
4. Conclusions
In this paper, a dynamic model of a one-mass linear vibratory conveyor, conducted with the calculation of kinematic parameters for the model is determined. The result of the elaboration is a
mechanical model, which can be used as base for the modification of the conveyors motion by situating the centre of elasticity optimal.
General equations for the model are developed and the final numerical system to calculate specific stiffness of springs for the targeted situation of the centre of elasticity of linear vibratory
conveyors is introduced.
Based on the results of the paper, optimized parameters for the design of one-mass linear vibratory conveyors can be defined and simulated.
• Harris C. M. Shock and Vibration Handbook. 5th edition, McGraw-Hill, New York, 2005.
• Buja H. O. Praxishandbuch Ramm- und Vibrationstechnik. Bauwerk Verlag Berlin, 2007.
• Dresig H., Holzweißig F. Maschinendynamik 8. Auflage, Springer Verlag, Berlin, 2008.
• Knaebel M., Jäger H., Roland Mastel Technische Schwingungslehre. 7. Auflage, Vieweg+Teubner Verlag, Wiesbaden, 2009.
• Pešík M. Function and Performance Optimization of Vibration Conveyors. Ph.D. Thesis, Liberec, 2013.
• Griemert R., Römisch P. Fördertechnik. Springer Fachmedien Wiesbaden, 2015.
About this article
Mechanical vibrations and applications
vibratory conveyor
spring stiffness
resonance frequencies
dynamic model
centre of elasticity
Copyright © 2018 Martin Sturm.
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Linear response time-dependent density-functional theory (LR-TDDFT)
Linear response time-dependent density-functional theory (LR-TDDFT)
Tim Zuehlsdorff, Imperial College London
Linear Response TDDFT
The linear response TDDFT (LR-TDDFT) functionality in ONETEP allows the calculation of the low energy excited states of a system in linear scaling effort. In contrast to time-evolution TDDFT, where
the density matrix of the system is propagated explicitly in time, LR-TDDFT recasts the problem of finding TDDFT excitation energies into an effective non-hermitian eigenvalue equation of the form:
\[\begin{split}\begin{pmatrix} \textbf{A} & \textbf{B} \\ \textbf{B} & \textbf{A}\end{pmatrix}\begin{pmatrix} \vec{\textbf{X}} \\ \vec{\textbf{Y}} \end{pmatrix} = \omega \begin{pmatrix} 1 & 0 \\ 0 &
-1 \end{pmatrix}\begin{pmatrix} \vec{\textbf{X}} \\ \vec{\textbf{Y}} \end{pmatrix}\end{split}\]
where the elements of the block matrices \(\textbf{A}\) and \(\textbf{B}\) can be expressed in canonical Kohn-Sham representation as
\[\begin{split}\begin{aligned} A_{cv,c'v'}&=&\delta_{c,c'}\delta_{v,v'}(\epsilon^{\textrm{\scriptsize{KS}}}_{c}-\epsilon^{\textrm{\scriptsize{KS}}}_{v})+K_{cv,c'v'} \\ B_{cv,c'v'}&=&K_{cv, v'c'}\end
Here, \(c\) and \(v\) denote Kohn-Sham conduction and valence states and K is the coupling matrix with elements given by
\[\begin{split}\begin{aligned} \nonumber K_{cv,c'v'}=2\int \mathrm{d} ^3 r \mathrm{d} ^3 r'\left[\frac{1}{|\textbf{r}-\textbf{r}'|}+\left. \frac{\delta^2 E_{\textrm{\scriptsize{xc}}}}{\delta\rho(\
textbf{r})\delta\rho(\textbf{r}')}\right|_{\rho^{\{0\}}}\right] \\ \times \psi^{\textrm{\scriptsize{KS}}*}_{c}(\textbf{r})\psi^{\textrm{\scriptsize{KS}}}_{v}(\textbf{r})\psi^{\textrm{\scriptsize{KS}}
with \(E_{\textrm{\scriptsize{xc}}}\) being the exchange-correlation energy. Its second derivative, evaluated at the ground-state density \(\rho^{\{0\}}\) of the system, is normally referred to as
the exchange-correlation kernel.
The above equation can be understood as an effective 2-particle Hamiltonian consisting of a diagonal part of conduction-valence eigenvalue differences and a coupling term \(K_{cv,c'v'}\) connecting
individual Kohn-Sham excitations.
In ONETEP, LR-TDDFT is implemented both in terms of the full TDDFT eigenvalue equation (Eqn. [full_tddft]) and in the Tamm-Dancoff approximation, a commonly used simplification to the full
non-hermitian eigenvalue equation, where the off diagonal elements \(\textbf{B}\) are set to zero. The problem of calculating the TDDFT excitation energies thus becomes equivalent to solving the
hermitian eigenvalue equation
\[\textbf{A}\vec{\textbf{X}}=\omega \vec{\textbf{X}}\]
The Tamm-Dancoff approximation violates time-reversal symmetry and oscillator strength sum rules and can blue-shift strong peaks in the spectrum by up to 0.3 eV, however, dark states are typically
left almost unaltered from their corresponding states in the Tamm-Dancoff approximation.
In the ONETEP code, the Tamm-Dancoff eigenvalue equation is re-expressed in terms of two sets of NGWFs, one optimised for the valence space (denoted as \(\{ \phi_\alpha\}\)) and one optimised for a
low energy subspace of the conduction manifold (denoted as \(\{\chi_\beta \}\), see the documentation of the conduction NGWF optimiation functionality). Furthermore, the eigenvalue equation is solved
iteratively for the lowest few eigenvalues using a conjugate gradient method. In order to do so we define the action \(\textbf{q}\) of operator \(\textbf{A}\) acting \(\vec{\textbf{X}}\) in
conduction-valence NGWF space as
\[(q^{\chi\phi})^{\alpha\beta}=(P^{\{\mathrm{c}\}} H^{\chi}P^{\{1\}}-P^{\{1\}} H^{\phi}P^{\{\mathrm{v}\}})^{\alpha\beta} +(P^{\{\mathrm{c}\}} V^{\{1\}\chi\phi}_{\textrm{\scriptsize{SCF}}}P^{\{\mathrm
where \(\textbf{H}^{\chi}\) and \(\textbf{H}^\phi\) are the Hamiltonians in conduction and valence NGWF representation respectively, \(\textbf{P}^{\{c\}}\) and \(\textbf{P}^{\{v\}}\) denote the
conduction and valence density matrices and \(\textbf{P}^{\{1\}}\) is the response density matrix, a representation of the trial vector \(\vec{\textbf{X}}\) in conduction-valence NGWF space. \(V^{\{1
\}}_{\textrm{\scriptsize{SCF}}}\) is the first order response of the system due to the density \(\rho^{\{1\}}(\textbf{r})\) associated with \(\textbf{P}^{\{1\}}\). Under this redefinition of the
action \(\textbf{A}\) in conduction-valence NGWF space, finding the lowest \(N_\omega\) excitation energies is equivalent to minimising
\[\Omega=\sum_i^{N_\omega}\omega_i=\sum_i^{N_{\omega}}\left[ \frac{\textrm{Tr}\left[\textbf{P}^{\{1\}\dagger}_i\textbf{S}^{\chi}\textbf{q}^{\chi\phi}_i\textbf{S}^\phi\right]}{\textrm{Tr}\left[\textbf
with respect to \(\left\{ \textbf{P}^{\{1\}}_i\right\}\) under the constraint
\[ \textrm{Tr}\left[\textbf{P}^{\{1\}\dagger}_i\textbf{S}^{\chi}\textbf{P}^{\{1\}}_j\textbf{S}^\phi\right]=\delta_{ij}.\]
If all density matrices involved in the above expressions, ie. \(\textbf{P}^{\{1\}}\), \(\textbf{P}^{\{c\}}\) and \(\textbf{P}^{\{v\}}\) are truncated and thus become sparse, the algorithm scales as
\(O(N)\) with system size for a fixed number of excitation energies \(N_\omega\) and as \(O(N_\omega^2)\) with the number of excitation energies required.
A similar algorithm can be derived for the full TDDFT eigenvalue equation, where we make use of the change of variables \(\textbf{p}=\vec{\textbf{X}}+\vec{\textbf{Y}}\) and \(\textbf{q}=\vec{\textbf
{X}}-\vec{\textbf{Y}}\). Each TDDFT excitation then has two effective density matrices, \(\textbf{P}^{\{p\}}\) and \(\textbf{P}^{\{q\}}\), associated with it that have the same structure as \(\textbf
{P}^{\{1\}}\) in the Tamm-Dancoff approximation. The density matrices do obey an updated orthonormality constraint of the form
\[ \frac{1}{2}\left(\textrm{Tr}\left[\textbf{P}^{\{p\}\dagger}_i\textbf{S}^{\chi}\textbf{P}^{\{q\}}_j\textbf{S}^\phi\right]+ \textrm{Tr}\left[\textbf{P}^{\{q\}\dagger}_i\textbf{S}^{\chi}\textbf{P}^{\
and an analogous expression for the total energy \(\Omega\) in full TDDFT can be derived.
Performing a LR-TDDFT calculation
The LR-TDDFT calculation in ONETEP is enabled by setting the task flag to TASK=LR_TDDFT. The LR-TDDFT calculation mode reads in the density kernels and NGWFs of a converged ground state and
conduction state calculation, so the .dkn, .dkn_cond, .tightbox_ngwfs and .tightbox_ngwfs_cond files all need to be present. The most important keywords in a TDDFT calculation are:
• \(\tt{lr\_tddft\_RPA}\): T/F.
Boolean, default \(\tt{lr\_tddft\_RPA}\)=F. If set to T, the code performs a full TDDFT calculation without relying on the simplified Tamm-Dancoff approximation.
• \(\tt{lr\_tddft\_num\_states}\): n
Integer, default \(\tt{lr\_tddft\_num\_states}=1\).
The keyword specifies how many excitations we want to converge. If set to a positive integer n, the TDDFT algorithm will converge the n lowest excitations of the system.
• \(\tt{lr\_tddft\_cg\_threshold}\): x
Real, default \(\tt{lr\_tddft\_cg\_threshold}=10^{-6}\).
The keyword specifies the convergence tolerance on the sum of the n TDDFT excitation energies. If the sum of excitation energies changes by less than x in two consecutive iterations, the
calculation is taken to be converged.
• \(\tt{lr\_tddft\_maxit\_cg}\): n
Integer, default \(\tt{lr\_tddft\_maxit\_cg}=60\).
The maximum number of conjugate gradient iterations the algorithm will perform.
• \(\tt{lr\_tddft\_triplet}\): T/F
Boolean, default \(\tt{lr\_tddft\_triplet}=F\).
Flag that decides whether the \(\tt{lr\_tddft\_num\_states}=n\) states to be converged are singlet or triplet states.
• \(\tt{lr\_tddft\_write\_kernels}\): T/F
Boolean, default \(\tt{lr\_tddft\_write\_kernels}=T\).
If the flag is set to T, the TDDFT response density kernels are printed out at every conjugate gradient iteration. These files are necessary to restart a LR_TDDFT calculation.
• \(\tt{lr\_tddft\_restart}\): T/F
Boolean, default \(\tt{lr\_tddft\_trestart}=F\).
If the flag is set to T, the algorithm reads in \(\tt{lr\_tddft\_num\_states}=n\) response density kernels in .dkn format and uses them as initial trial vectors for a restarted LR_TDDFT
• \(\tt{lr\_tddft\_restart\_from\_TDA}\): T/F
Boolean, default \(\tt{lr\_tddft\_trestart\_from\_TDA}=F\).
If the flag is set to T and \(\tt{lr\_tddft\_RPA}\): T, the code will read in already converged density kernels \(\left\{\textbf{P}^{\{1\}}_i\right\}\) and use them as a starting guess for a full
TDDFT calculation such that \(\textbf{P}^{\{p\}}_i=\textbf{P}^{\{q\}}_i=\textbf{P}^{\{1\}}\). In many cases, the full TDDFT results are similar to the Tamm-Dancoff results and this strategy of
starting the full TDDFT calculation leads to a rapid convergence.
• \(\tt{lr\_tddft\_init\_random}\) T/F
Boolean, default \(\tt{lr\_tddft\_init\_random}\) T.
By default, the initial TDDFT eigenvector guesses are initialised to random matrices. This yields an unbiased convergence of the TDDFT algorithm but can mean that one starts the optimisation
relatively far away from the minimum. If \(\tt{lr\_tddft\_init\_random}\)=F, the code instead computes the \(n\) minimum energy pure Kohn-Sham transitions in linear-scaling effort and initialises
the \(n\) TDDFT response density matrices to the pure Kohn-Sham transition density matrices. In many small to medium sized systems this leads to initial states much closer to the TDDFT minimum
and rapid convergence. In large extended systems this can yield states that are spurious charge transfer states that are not ideal, especially if a more advanced density matrix truncation scheme
is used. In this case it is possible to set the keyword \(\tt{lr\_tddft\_init\_max\_overlap}\) T, in which, rather than choosing the lowest Kohn-Sham transitions, the code picks the lowest few
transitions that also have a maximum overlap of electron and hole densities.
• \(\tt{lr\_tddft\_kernel\_cutoff}\): x
Real, default \(\tt{lr\_tddft\_kernel\_cutoff}=1000 a_0\).
Keyword sets a truncation radius on all response density kernels in order to achieve linear scaling computational effort with system size.
While the LR_TDDFT calculation can be made to scale linearly for a fixed number of excitations converged, it should be kept in mind that the algorithm needs to perform orthonormalisation procedures
and thus scales as \(O(N^2)\) with \(\tt{lr\_tddft\_num\_states}\).
Truncation of the Response density matrix
To run a fully linear scaling TDDFT calculation the response density matrix has to be truncated by setting \(\tt{lr\_tddft\_kernel\_cutoff}\). This truncation introduces numerical errors into the
calculation, which mainly manifest themselves in the form that the response density matrices do no longer exactly obey a first order idempotency constraint that is placed on them. The idempotency
constraint can be written in form of an invariance equation:
To measure the degree to which the invariance relation is violated we make use of a penalty functional \(Q\left[ \textbf{P}^{\{1\}}\right]\) given by:
\[Q\left[\textbf{P}^{\{1\}}\right]=\textrm{Tr}\left[\left(\textbf{P}^{\{1\}\dagger}\textbf{S}^{\chi}\textbf{P}^{\{1\}}\textbf{S}^{\phi}-\textbf{P}^{\{1\}' \dagger}\textbf{S}^{\chi}\textbf{P}^{\{1\}'}
\textbf{S}^{\phi} \right)^2\right].\]
For truncated \(\textbf{P}^{\{1\}}\), \(Q\left[\textbf{P}^{\{1\}}\right]\neq 0\) which can lead to problems in the convergence of the conjugate gradient algorithm. In order to avoid these issues, the
TDDFT routines perform the minimisation of the energy in an analogous form to the LNV method in ground-state calculations: The auxiliary density kernel \(\textbf{P}^{\{1\}'}\) is used instead of \(\
textbf{P}^{\{1\}}\) for the minimisation of \(\Omega\). While \(\textbf{P}^{\{1\}'}\) is much less sparse than \(\textbf{P}^{\{1\}}\) it preserves idempotency to the same degree as the conduction and
valence density kernel, yielding a stabilised convergence.
However, should \(Q\left[\textbf{P}^{\{1\}}\right]\) diverge significantly from 0 during the calculation, there are routines in place similar to the kernel purification schemes in ground state DFT
that force the kernel towards obeying its idempotency constraint. The keyword controlling these routines are given below:
• \(\tt{lr\_tddft\_penalty\_tol}\): x
Real, default \(\tt{lr\_tddft\_penalty\_tol}=10^{-8}\).
Keyword sets a tolerance for the penalty functional. If \(Q\left[\textbf{P}^{\{1\}}\right]\) is larger than \(\tt{lr\_tddft\_penalty\_tol}\) the algorithm will perform purification iterations in
order to decrease the penalty value and force \(\textbf{P}^{\{1\}}\) towards the correct idempotency behaviour.
• \(\tt{lr\_tddft\_maxit\_pen}\): n
Integer, default \(\tt{lr\_tddft\_maxit\_pen}=20\).
The maximum number purification iterations performed per conjugate gradient step.
More advanced TDDFT kernel truncation schemes
There are many situations where physical intuition allows one to specify a more sophisticated sparsity pattern than a uniform spherical kernel cutoff on \(\textbf{P}^{\{1\}}\) (or \(\textbf{P}^{\{p
\}}\) and \(\textbf{P}^{\{q\}}\) for full TDDFT). For example, in pigment-protein complexes the excitations of interest retain a relative localisation on the pigment and one would ideally converge
these states directly, without obtaining any spurious charge transfer states from the pigment to far away regions of the protein, that can arise due to failures in semi-local exchange correlation
functionals. This can be achieved by introducing a new block into the input file of the form
%block species_tddft_kernel
label1 label 2 label3 ...
label5 ...
%endblock species_tddft_kernel
where the labels refer to atom labels. As an example, consider a pigment protein complex, where the pigment atoms are labelled H1, C1 etc. while the protein atoms are labelled H, C, etc. Then we can
force the excitations of the system to be fully localised on the pigment by including
%block species_tddft_kernel
C1 H1 ...
%endblock species_tddft_kernel
This has the effect of setting all elements of \(\textbf{P}^{\{1\}}\) to zero that correspond to conduction or valence NGWFs centered on atoms of the environment. In this way the electrostatic
effects of the environment are treated fully quantum mechanically, while no delocalisation into the protein is allowed. If one would like to introduce a coupling to the environment but wants to
suppress any charge transfer coupling between the pigment and its environment, it is possible to specify
%block species_tddft_kernel
C1 H1 ...
C H ...
%endblock species_tddft_kernel
It is possible to specify an arbitrary number of subregions in the system in this way. It is also possible to list the same species in different lines, allowing for charge transfer interactions
between some atom types of two regions but not others.
Rather than having the off-diagonal charge-transfer blocks defined in %block species_tddft_kernel set exactly to zero, it is also possible to give these blocks a more realistic sparsity pattern, for
example that of the overlap matrix. While this process still suppresses any significant amount of charge transfer between TDDFT regions, it can be used to allow overlapping NGWFs from different TDDFT
regions to contribute to the TDDFT transition density. In order to do so, set the block
%block species_tddft_ct
C1 H1 ...
C2 H2 ...
%endblock species_tddft_ct
and set lr_tddft_ct_length to a chosen cutoff length for the charge-transfer interaction between the specified blocks. For example, if the off-diagonal blocks of the response density matrix
(corresponding to charge-transfer excitations between TDDFT regions) should have the same sparsity pattern as the overlap matrix, set lr_tddft_ct_length to twice the NGWF localisation radius.
The TDDFT eigenvalue problem is generally ill-conditioned, which can lead to a relatively slow convergence. For this reason, it is possible to precondition the eigenvalue problem, which is achieved
by solving a linear system iteratively to a certain tolerance at each conjugate gradient step. Solving the linear system only requires matrix-matrix multiplications and is very cheap for small and
medium sized systems, however, it can get more costly for very large systems, especially when no kernel truncation is used. In these cases, it can be necessary to reduce the number of default
iterations of the preconditioner. The main keywords controlling the preconditioner are
• \(\tt{lr\_tddft\_precond}\): T/F
Boolean, default \(\tt{lr\_tddft\_precond}=T\).
Flag that decides whether the preconditioner is switched on or off.
• \(\tt{lr\_tddft\_precond\_iter}\): n
Integer, default \(\tt{lr\_tddft\_precond\_iter}=20\).
Maximum number of iterations in the linear system solver applying the preconditioner.
• \(\tt{lr\_tddft\_precond\_tol}\): x
Real, default \(\tt{lr\_tddft\_precond\_tol}=10^{-8}\).
The tolerance to which the linear system is solved in the preconditioner. Choosing a large tolerance means that the preconditioner is only applied approximately during each iteration.
Representation of the unoccupied subspace
In the LR_TDDFT method as implemented in ONETEP, the user has two options regarding the representation of the unoccupied subspace. The first option is to define the active unoccupied subspace of the
calculation to only contain the Kohn-Sham states that were explicitly optimised in the COND calculation. The other is to make use of a projector onto the entire unoccupied subspace, where we redefine
the conduction density matrix as:
\[\textbf{P}^{\{\textrm{c}\}}=\left(\left(\textbf{S}^{\chi}\right)^{-1} -\left(\textbf{S}^{\chi}\right)^{-1}\textbf{S}^{\chi\phi}\textbf{P}^{\{\textrm{v}\}}\left(\textbf{S}^{\chi\phi}\right)^\dagger
\left(\textbf{S}^{\chi}\right)^{-1}\right) .\]
The first option has the advantage that we only include states for which the NGWFs are well optimised, but has the drawback that some excitations converge very slowly with the size of the unoccupied
subspace and thus a good convergence with the number of conduction states optimised is hard to reach. The second method implicitly includes the entire unoccupied subspace (to the extent that it is
representable by a small, localised NGWF representation), but has the disadvantage that now states are included in the calculation for which the NGWFs are not optimised. Furthermore, the density
matrix defined above is no longer strictly idempotent, leading to violations of the idempotency condition and thus a non-vanishing penalty functional \(Q\left[\textbf{P}^{\{1\}}\right]\), requiring
kernel purification iterations as described in the previous section.
The problem of loss of idempotency can be avoided by using the joint NGWF set to represent the conduction space when using the projector. While this increases the computational cost of the LR_TDDFT
calculation by a factor of 2, it preserves the idempotency of \(\textbf{P}^{\{\textrm{c}\}}\) and is the recommended option when using the projector onto the unoccupied subspace.
The keywords controlling the use of the projector are
• \(\tt{lr\_tddft\_projector}\): T/F
Boolean, default \(\tt{lr\_tddft\_projector}=T\).
If the flag is set to T, the conduction density matrix \(\textbf{P}^{\{\textrm{c}\}}\) is redefined to be a projector onto the entire unoccupied subspace.
• \(\tt{lr\_tddft\_joint\_set}\): T/F
Boolean, default \(\tt{lr\_tddft\_joint\_set}=T\).
If the flag is set to T, the joint NGWF set is used to represent the conduction space in the LR_TDDFT calculation.
Calculations in implicit solvent
A TDDFT calculation in implicit solvent is performed in an analogous way to the implicit solvent calculation in combination with a conduction optimisation (see the documentation of the conduction
optimisation for further details). By default, the implicit solvent only acts on the ground state of the system and thus influences the conduction and valence Kohn-Sham states mixed into the TDDFT
calculation. However, a screening of the response density due to a dynamic dielectric constant is not included in the calculation. In order to activate dynamic screening effects in TDDFT, the user
can set the keyword \(\tt{lr\_optical\_permittivity}\) to the effective dynamic dielectric constant \(\epsilon_\infty\) of the system in question.
The LR_TDDFT calculation will produce a number of outputs. At the end of the calculation, the individual excitation energies and oscillator strengths will be computed and printed in the main ONETEP
output file. Furthermore, the energies and oscillator strengths are used to generate a excitation spectrum written to the textfile rootname.tddft_spectrum. The peaks in the spectrum are Gaussians of
a width controlled by \(\tt{lr\_tddft\_spectrum\_smear}\). Furthermore, by default, density cube files of the response density, the electron and the hole density for each excitation are printed out.
The LR_TDDFT code can also perform an analysis of individual excitations, where the response density matrix is decomposed into dominant Kohn-Sham transitions. Since this analysis requires the
Kohn-Sham eigenstates and thus a diagonalisation of the Hamiltonian, it scales as \(O(N^3)\) and should not be performed for very large system sizes.
The keywords controlling these outputs are:
• \(\tt{lr\_tddft\_write\_densities}\): T/F
Boolean, default \(\tt{lr\_tddft\_write\_densities}=T\).
If the flag is set to T, the response density, electron density and hole density for each excitation is computed and written into a .cube file.
• \(\tt{lr\_tddft\_analysis}\): T/F
Boolean, default \(\tt{lr\_tddft\_analysis}=F\).
If the flag is set to T, a full \(O(N^3)\) analysis of each TDDFT excitation is performed in which the response density is decomposed into dominant Kohn-Sham transitions.
Good practices and common problems
• The quality of the TDDFT excitation energies critically depends on the representation of the conduction space manifold. Any excitation that has a large contribution from an unoccupied state that
is not explicitly optimised in the COND calculation is not expected to be represented correctly in the LR_TDDFT calculation. In general it is advisable to optimise as many conduction states as
possible. However, high energy conduction states are often very delocalised and only representable if the conduction NGWF radius is increased significantly, thus leading to poor computational
efficiency. In practice, there is a tradeoff between computational efficiency and the representation of the conduction state manifold (see also the documentation on conduction state optimisation
on this issue). Generally, TDDFT excitations should be converged with respect to both the conduction NGWF radius and the number of conduction states explicitly optimised.
• Since the ground state and conduction density kernels are used as projectors onto the occupied and unoccupied subspace in LR_TDDFT, one often finds that the inner loop of the SINGLEPOINT and COND
optimisation has to be converged to a higher degree of accuracy to achieve well behaved TDDFT results. It is therefore recommended to increase MAXIT_LNV and MINIT_LNV from their default value in
the SINGLEPOINT and COND calculation. If no density kernel cutoff is used, the penalty functional value in the LR_TDDFT calculation should be vanishingly small. If the number increases
significantly during a calculation or if the code begins to perform penalty optimisation steps, that is a clear sign that the initial conduction and valence density kernels are not converged well
• In order to perform a LR_TDDFT calculation that scales fully linearly with system size, all density matrices involved have to be sparse and thus a KERNEL_CUTOFF has to be set for both the
SINGLEPOINT and COND calculation. Using a density matrix truncation on the conduction states can sometimes be difficult depending on how the subspace of optimised conduction states is chosen and
care has to be taken to prevent unphysical results.
• When running calculations in full linear scaling mode, the ground state and conduction density kernels are no longer strictly idempotent, which means that the penalty functional in LR_TDDFT will
no longer be strictly zero. The code might perform penalty functional optimisation steps to keep the idempotency error small. However, these idempotency corrections can cause the conjugate
gradient algorithm to stagnate and can even cause the energy to increase. If this happens, it is an indication that the minimum energy and maximum level of convergence for this truncation of the
density kernel has been reached.
• When placing a truncation onto the the response density kernels it should be kept in mind that this may cause the optimisation to miss certain low energy excitations completely. Very long range
charge-transfer type excitations cannot be represented by a truncated response density kernel and will thus be missing from the spectrum of excitations converged. However, well localised
excitations should be unaffected. In a similar way, if the TDDFT kernel is limited to a certain region, it should be checked whether increasing the region leads to a smooth convergence of the
energy of the localised state within the region.
For further background regarding the theory behind the LR_TDDFT method in ONETEP, as well as a number of benchmark tests, see
• Linear-scaling time-dependent density-functional theory in the linear response formalism, T. J. Zuehlsdorff, N. D. M. Hine, J. S. Spencer, N. M. Harrison, D. J. Riley, and P. D. Haynes, J. Chem.
Phys. 139, 064104 (2013) | {"url":"https://docs.onetep.org/lr_tddft.html","timestamp":"2024-11-14T21:51:36Z","content_type":"text/html","content_length":"48852","record_id":"<urn:uuid:d2c26a2c-9230-4480-bd3b-355d8583b4b5>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00299.warc.gz"} |
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Steam Pipes - Online Pressure drop Calculator
Sponsored Links
The pressure drop in a saturated steam distribution pipe line can be calculated in metric units as
dp = 0.6753 10^6 q^2 l (1 + 91.4/d) / ρ d^5 (1)
dp = pressure drop (Pa)
q = steam flow rate (kg/h)
l = length of pipe (m)
d = pipe inside diameter (mm)
ρ = steam density (kg/m^3)
The Unwin formula can also be expressed in imperial units as
dp = 0.0001306 q^2 l (1 + 3.6/d) / (3600 ρ d^5)^ (2)
dp = pressure drop (psi)
q = steam flow rate (lb/hr)
l = length of pipe (ft)
d = pipe inside diameter (inches)
ρ = steam density (lb/ft^3)
Use values for the specific volume corresponding to the average pressure if the pressure drop exceeds 10 - 15 % of the initial absolute pressure.
Note that for elevated velocities the Unwin's formula is known to give pressure drops higher than the actual.
Example - Pressure Drop in a Steam Pipe
A steam flow 4000 kg/h with pressure 10 bar (1000 kPa) and density 5.15 kg/m^3 flows through a schedule 40 ND 100 steel pipe with inside diameter 102 mm and length 100 m. The pressure drop can be
calculated as
dp = 0.6753 10^6 (4000 kg/h)^2 (100 m) (1 + 91.4/(102 mm)) / ((5.15 kg/m^3) (102 mm)^5 )
= 36030 Pa
= 36 kPa
Steam Pressure Drop Calculator - Metric Units
Steam Pressure Drop Calculator - Imperial Units
Pressure Drop Diagrams - metric units
The diagrams below can be used to indicate pressure drop in steam pipes at different pressures. The calculations are made for schedule 40 pipe.
• 1 bar = 100 kPa = 10,197 kp/m^2 = 10.20 m H[2]O = 0.9869 atm = 14.50 psi (lb[f]/in^2)
• 1 kg/h = 2.778x10^-4 kg/s = 3.67x10^-2 lb/min
Steam - pressure 1 bar gauge
Steam - pressure 3 bar gauge
Steam - pressure 7 bar gauge
Steam - pressure 10 bar gauge
Sponsored Links
Related Topics
Dimensions of steam and condensate pipe lines. Calculate pressure losses, recommended velocities, capacities and more.
Design of steam & condensate systems with properties, capacities, sizing of pipe lines, system configuration and more.
Related Documents
Pipe sizes, inside and outside diameters, wall thickness, schedules, weight and weight of pipe filled with water - Metric Units.
Introduction to pressure - online pressure units converter.
Saturated Steam Table with steam properties as specific volume, density, specific enthalpy and specific entropy.
The steam velocity in a steam distribution system should be within certain limits to avoid excessive wear and tear
Steam table with sensible, latent and total heat, and specific volume at different gauge pressures and temperatures.
Saturated Steam Table with properties like boiling point, specific volume, density, specific enthalpy, specific heat and latent heat of vaporization.
Steam is a compressible gas where pipe line mass flow capacity depends on steam pressure.
Steam is a compressible gas where the capacity of a pipe line depends on the size of the pipe and the steam pressure.
Steam consumption and condensate generation when heating liquid or gas flows
Steam flow through orifices - for steam pressures ranging 2 - 300 psi
Flow rate (lb/h) and pressure drop per 100 feet of pipe.
An introduction to the basic design of steam heating systems.
Pressure drop and maximum allowable flow in steam pipes.
Steam pipes and pressure drop diagrams - imperial and metric units.
Sizing of steam pipe lines - major and minor loss in steam distribution systems.
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Python Numerical Simulation Projects
Python Numerical Simulation assistance are provided by us drop a message to matlabprojects.org we will help you with results. Explore the utilization of Python for numerical analysis we guide you
with our comprehensive tools and libraries. Reach out to us for professional support and to achieve the best outcomes. Numerical simulation is examined as both an intricate and significant process
that must be carried out by following specific guidelines. To conduct this process efficiently, we suggest a few prevalent algorithms and parameters, along with brief outlines:
1. Finite Difference Method (FDM)
• Algorithm: Through estimating derivatives with finite differences, the differential equations can be resolved by this method.
• Relevant Parameters:
• Grid spacing (Δx, Δt)
• Initial conditions
• Boundary conditions
• Time step size
2. Finite Element Method (FEM)
• Algorithm: For complicated boundary and initial value issues, this algorithm is widely employed. It assists to segment the field into various components.
• Relevant Parameters:
• Element shape
• Mesh size
• Initial conditions
• Boundary conditions
• Material properties
3. Monte Carlo Simulation
• Algorithm: This method is mostly utilized for probabilistic and statistical issues. In order to acquire numerical outcomes, it employs random sampling.
• Relevant Parameters:
• Number of simulations
• Probability distributions
• Random seed
• Sample size
4. Runge-Kutta Methods
• Algorithm: To resolve ODEs (ordinary differential equations) with more preciseness, the Runge-Kutta methods are highly suitable.
• Relevant Parameters:
• Initial conditions
• Order of the method (for instance: 4th order Runge-Kutta)
• Time step size
5. Particle Swarm Optimization (PSO)
• Algorithm: PSO is referred to as an optimization approach, which is specifically derived from the fish or birds’ social activity.
• Relevant Parameters:
• Number of particles
• Maximum velocity
• Cognitive and social coefficients
• Inertia weight
6. Genetic Algorithms
• Algorithm: It is an optimization method, which is facilitated by natural selection.
• Relevant Parameters:
• Number of generations
• Population size
• Mutation rate
• Crossover rate
7. Gradient Descent
• Algorithm: For identifying the least of a function, this optimization approach is appropriate.
• Relevant Parameters:
• Convergence criteria
• Batch size (for stochastic gradient descent)
• Learning rate
• Number of iterations
Python Libraries and Tools
• NumPy: It is highly ideal for array management and numerical operations.
• SciPy: Scientific calculation and incorporation can be facilitated by SciPy.
• Matplotlib: This tool is more suitable for visualization and plotting.
• SymPy: Make use of SymPy for symbolic mathematics.
• Pyomo: It is generally utilized for optimization issues.
• SimPy: For discrete-event simulation, it is more useful.
Python numerical simulation projects
The process of resolving mathematical models of physical events with computational approaches is included in the numerical simulation. For numerical simulations, Python is an appropriate and robust
tool because of having a wide range of libraries like Matplotlib, SciPy, and NumPy. By encompassing various topics, we list out numerous Python project plans related to numerical simulation:
1. Heat Equation Simulation
• Explanation: To analyze heat diffusion periodically, the 1D or 2D heat equation has to be resolved with finite difference methods.
• Major Characteristics: Stability exploration, explicit and implicit techniques, and visualization of temperature diffusion.
2. Projectile Motion Simulation
• Explanation: In the impact of air resistance and gravity, the movement of a projectile must be simulated.
• Major Characteristics: Impact of air resistance, equations of motion, and trajectory plotting.
3. Fluid Flow in a Pipe
• Explanation: Across a pipe, the laminar and turbulent fluid flow should be simulated. For that, we utilize the Navier-Stokes equations.
• Major Characteristics: Pressure loss analysis, velocity profiles, and Reynolds number estimation.
4. Pendulum Dynamics
• Explanation: By means of ordinary differential equations (ODEs), the movement of a double and simple pendulum has to be simulated.
• Major Characteristics: Chaotic activity in double pendulums, phase space plots, and nonlinear dynamics.
5. Spring-Mass-Damper System
• Explanation: The motions of a spring-mass-damper framework must be designed and simulated.
• Major Characteristics: Resonance exploration, forced vibrations, and damped and undamped oscillations.
6. Population Growth Models
• Explanation: Various population growth models have to be applied and simulated. It could include logistic growth, exponential growth, and predator-prey models.
• Major Characteristics: Long-term activity, stability exploration, and differential equations.
7. Traffic Flow Simulation
• Explanation: Make use of the cell transmission model or other major traffic flow models to simulate the flow of traffic on a highway.
• Major Characteristics: Congestion exploration, flow and speed connections, and traffic load.
8. Wave Equation Simulation
• Explanation: To simulate wave distribution, we plan to resolve the 1D or 2D wave equation.
• Major Characteristics: Visualization of wave movement, finite difference methods, and reflection and distribution at boundaries.
9. Epidemic Spread Modeling
• Explanation: By utilizing compartmental models such as SEIR or SIR, the distribution of an infectious disease should be simulated.
• Major Characteristics: Differential equations, intervention policies, recovery rates, and contact rates.
10. Orbital Mechanics Simulation
• Explanation: Through Newton’s law of gravitation, the paths of satellites or planets have to be simulated.
• Major Characteristics: Trajectory plotting, Kepler’s laws, n-body problem, and two-body problem.
Instance: Heat Equation Simulation
For resolving the 1D heat equation, we offer an instance, which employs the explicit finite difference method.
Step 1: Import Required Libraries
import numpy as np
import matplotlib.pyplot as plt
Step 2: Specify the Parameters and Initial Conditions
# Parameters
L = 1.0 # Length of the rod
T = 0.5 # Total time
nx = 50 # Number of spatial points
nt = 1000 # Number of time steps
alpha = 0.01 # Thermal diffusivity
# Discretization
dx = L / (nx – 1)
dt = T / nt
# Stability condition
if dt > dx**2 / (2 * alpha):
raise ValueError(“Stability condition violated: dt must be less than dx^2 / (2 * alpha)”)
# Initial condition: u(x,0) = sin(pi * x)
x = np.linspace(0, L, nx)
u = np.sin(np.pi * x)
# Boundary conditions: u(0,t) = u(L,t) = 0
u[0] = u[-1] = 0
Step 3: Time-Stepping Loop
# Time-stepping loop
u_new = np.zeros(nx)
for n in range(nt):
for i in range(1, nx-1):
u_new[i] = u[i] + alpha * dt / dx**2 * (u[i+1] – 2*u[i] + u[i-1])
u = u_new.copy()
Step 4: Plot the Outcomes
plt.plot(x, u, label=’t = 0.5′)
plt.title(‘1D Heat Equation’)
A simple simulation of the 1D heat equation is depicted in the above instance. It indicates the preliminary temperature distribution as u(x,0)=sin(πx)u(x,0) = \sin(\pi x)u(x,0)=sin(πx). On the basis
of the heat equation, the temperature emerges periodically.
Instance: Projectile Motion Simulation
In order to simulate projectile motion using air resistance, an explicit instance is provided by us.
Step 1: Import Essential Libraries
import numpy as np
import matplotlib.pyplot as plt
Step 2: Specify the Parameters and Initial Conditions
# Parameters
g = 9.81 # Acceleration due to gravity (m/s^2)
m = 0.145 # Mass of the projectile (kg)
C_d = 0.47 # Drag coefficient
rho = 1.225 # Air density (kg/m^3)
A = 0.0042 # Cross-sectional area of the projectile (m^2)
v0 = 30 # Initial velocity (m/s)
theta = 45 # Launch angle (degrees)
# Initial conditions
v0_x = v0 * np.cos(np.radians(theta))
v0_y = v0 * np.sin(np.radians(theta))
# Time settings
dt = 0.01 # Time step (s)
t_max = 5 # Maximum simulation time (s)
Step 3: Specify the Equations of Motion
# Equations of motion with air resistance
def equations_of_motion(v, t):
v_x, v_y = v
speed = np.sqrt(v_x**2 + v_y**2)
F_d = 0.5 * C_d * rho * A * speed**2
F_d_x = F_d * (v_x / speed)
F_d_y = F_d * (v_y / speed)
dv_xdt = – (F_d_x / m)
dv_ydt = – g – (F_d_y / m)
return np.array([dv_xdt, dv_ydt])
Step 4: Time-Stepping Loop
# Time-stepping loop
t = np.arange(0, t_max, dt)
v = np.array([v0_x, v0_y])
positions = []
for ti in t:
positions.append(v * ti)
v = v + equations_of_motion(v, ti) * dt
positions = np.array(positions)
x = positions[:, 0]
y = positions[:, 1]
Step 5: Plot the Outcomes
plt.plot(x, y, label=’Projectile Path’)
plt.xlabel(‘x (m)’)
plt.ylabel(‘y (m)’)
plt.title(‘Projectile Motion with Air Resistance’)
Across the impact of air resistance and gravity, we simulate the movement of a projectile in this instance. The drag force has to be explained in the equations of motion. It is important to plot the
path of the projectile.
Expanding the Simulation
Focus on the below specified improvements to expand these instances:
1. Higher Dimensions: The simulations have to be expanded to 3D.
2. Variable Properties: It is approachable to consider variable air density, thermal diffusivity, and others.
3. Advanced Solvers: For ODEs, innovative numerical techniques such as Runge-Kutta must be utilized.
4. Interactive Visualization: By employing libraries such as Dash or Plotly, we have to develop communicative visualizations.
5. Optimization: In order to acquire anticipated results, identify the optimal parameters by applying optimization methods.
For performing numerical simulations in an efficient manner, various significant algorithms and parameters are recommended by us. Relevant to numerical simulations, we proposed several Python project
plans, along with concise explanations, major characteristics, and explicit instances.
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This post will discuss area and how it applies to concreting. It will also explain how to calculate the area of different shapes which can be applied practically when working.
Area and Concreting
Almost everything in concreting is to do with area. We construct, or lay, “areas of concrete”. A concrete slab is an area where concrete is laid.
Many of the materials we use to construct concrete areas will be calculated or thought about in terms of area.
Understanding how to find the area of different shapes will make it faster, easier and more accurate to price, start and finish a concreting job.
When we work out the area of a shape, we usually will talk about the area size in square metres.
square metres is written as: m2, m^2 or sqm
A true square’s area can be found by raising the value of square’s side Length by the power of 2.
A rectangle’s area can be found by multiplying the rectangle’s Length by its Width.
Length * Width
A triangle’s area can be found by multiplying the Base by the Perpendicular Height and then multiplying by 0.5
Math Tip
Multiplying by 0.5 halves a value.
Base * P. Height * 0.5
The radius of a circle is the distance from the center to the edge of the circle. It is half the value of the circle’s diameter.
A circle’s area can be found by multiplying Pi (3.14) by the circle’s Radius raised by the power of 2.
3.14 * Radius^2
3.14 * Radius * Radius
A trapezoid is a square or a rectangle with 1 or 2 angled sides.
By understanding and using the trapezoid’s area formula it saves you calculating the triangles separately.
(Length + Width) * P. Height * 0.5
Waffle Pod – Square
A waffle pod is 110cm on both its length and width. The area that a waffle pod takes up can be found using the Area of a Square Formula.
If we square a number, it means we multiply it by itself.
This is also called raising by the power of 2 or to the power of 2.
When we write this operation, it looks like: ^2
We can raise to the power of different numbers as well. So to the power of is indicated by the ^ character.
On a calculator you will see a button that says x^y.
This button is the to the power of button.
The “to the power of” button on Windows Calculator.
Length = 110cm
= 12100cm2
Now change the Length into metres instead of centimetres...
Length = 1.1m
= 1.1^2
= 1.21m2
The area of a 110cm waffle pod in square metres is 1.21m2
Viscrine (Black Plastic) – Rectangle
A roll of viscrine is 50m long. It folds out to be 4m wide. Let’s use the Area of a Rectangle Formula to find how much area a whole roll can cover:
Length * Width
Length = 50m, Width = 4m
Length * Width
= 50 * 4
= 200m2
The total area a full roll of viscrine will cover is 200m2
Mesh – Triangle
We are making someone’s driveway wider.
The shape will be a triangle and the mesh needs to be cut to suit the shape. The mesh size needs to be a triangle with one side 2.2m and the other side is 4.8m. Work out the area that the mesh piece
will be:
Base * P. Height * 0.5
The Base will be the Length. The Perpendicular Height (P. Height) will be the Width. Therefore:
Base = Length, P. Height = Width
Length = 4.8m, Width = 2.2m
Base * P. Height * 0.5
= 4.8 * 2.2 * 0.5
= 5.28m2
The triangular piece of mesh will have an area of 5.28m2.
Your uncle wants to have a patio laid at his house. The area he wants to make the patio is a square shape with sides that are 6m.
What is the area of the proposed patio?
Find the area of the shape shown above. Hint: you can use rectangle & triangle formulas, or use the trapezoid formula.
The plan above shows a round-about. The inner circle has a 5m radius and the outer circle has a 15m radius. If we are going to concrete the roundabout (like a donut!) How many square metres will it
The 5m radius circle will not be concrete. | {"url":"http://www.beardeddonkey.com/2022/08/03/area-formulas/","timestamp":"2024-11-12T13:52:05Z","content_type":"text/html","content_length":"74222","record_id":"<urn:uuid:e906d5bf-2406-41fd-8f03-5508d78f28ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00363.warc.gz"} |
State Space Models
Next | Prev | Up | Top | Index | JOS Index | JOS Pubs | JOS Home | Search
As introduced in Book II [452, Appendix G], in the linear, time-invariant case, a discrete-time state-space model looks like a vector first-order finite-difference model:
where state vector at discrete time state transition matrix, and it determines the dynamics of the system (its poles, or modal resonant frequencies and damping).
The state-space representation is especially powerful for multi-input, multi-output (MIMO) linear systems, and also for time-varying linear systems (in which case any or all of the matrices in Eq.(
1.8) may have time subscripts 221].
To cast the previous force-driven mass example in state-space form, we may first observe that the state of the mass is specified by its velocity ^2.9Thus, to Eq.(1.5) we may add the explicit
difference equation
which, in canonical state-space form, becomes (letting
General features of this example are that the entire physical state of the system is collected together into a single vector, and the elements of the sampling interval, in the discrete-time case).
The parameters may also vary with time (time-varying systems), or be functions of the state (nonlinear systems).
The general procedure for building a state-space model is to label all the state variables and collect them into a vector output gains spring, capacitor, inductor), and one for each sample of delay
in sampled distributed systems. After that, various equivalent (but numerically preferable) forms can be generated by means of similarity transformations [452, pp. 360-374]. We will make sparing use
of state-space models in this book, because they can be linear-algebra intensive, and therefore rarely used in practical real-time signal processing systems for music and audio effects. However, the
state-space framework is an important general-purpose tool that should be kept in mind [221], and there is extensive support for state-space models in the matlab (``matrix laboratory'') language and
its libraries. We will use it mainly as an analytical tool from time to time.
As noted earlier, a point mass only requires a first-order model:
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How to Calculate a Spearman Correlation in Excel - 3 Methods - ExcelDemy
What Is the Spearman Correlation?
The Spearman Correlation is a derivative of the Pearson Correlation Coefficient in nonparametric form.
This value determines the linear correlation between two sets of data, often denoted by r[s] or ⲣ.
The Pearson Product Moment Correlation determines the linear relationship between continuous variables. The general expression of Pearson Correlation is:
R[X] and R[Y] are the values that are ranked and the standard deviation of the datasets.
The Spearman Correlation evaluates the monotonic relationship between the values.
The complete form of the Spearman Coefficient is
This version is a slightly modified version of Pearson’s equation. Here,
• R[x] and R[y] denote the rank of the x and y variables.
• R̅(x) and R̅(Y) are the mean ranks.
Use the Spearman correlation:
• If your data has outliers that can influence the result.
• If the data is in a non-linear relationship or not fully distributed.
• If one of the variables is ordinal.
The range of the Spearman Correlation coefficient value ranges from +1 to 1.
• 1 indicates a perfect correlation. Both datasets are matched.
• -1 indicates negative correlated data.
• 0 shows no correlation between data.
The sample dataset showcases two arrays.
Method 1 – Using an Excel Formula to Calculate the Spearman Correlation
A simple approximation to the Spearman Correlation is the following:
d[i] is the difference between a pair of ranks
And n is the number of observations.
This formulation won’t work if there is tied value in ranking.
• To rank the value of the columns Math and Economics, enter the following formula in E5 and press enter:
• Drag down the Fill Handle to see the result in the rest of the cells.
The values in E5:E14 are ranked.
• To rank D5:D14, enter the following formula in F5 and press enter:
• Drag down the Fill Handle to see the result in the rest of the cells.
The values in F5:F14 are ranked.
The ranks of both Math and Economics column values does not contain any tied value, there is no value with the same rank.
To calculate the Spearman correlation in the Excel worksheet:
• Find the difference between the ranked value in each row.
• Enter the following formula in G5 and press enter:
• Drag down the Fill Handle to see the result in the rest of the cells.
The difference between the ranked values in each row is displayed in G5:G14.
• To find the square of the difference between the ranked value in each row, calculated in D5:D14, enter the following formula in H5 and press enter:
• Drag down the Fill Handle to see the result in the rest of the cells.
The square of the difference between all ranked values in each row is displayed in H5:H14.
• To get the sum of H5:H15, enter the following formula in H15:
• Enter the number of entries in E16, here, 10.
• Enter the following formula in E17:
You will get the Spearman Correlation.
The output is a negative value, which indicates a negative correlation between the two ranked data columns.
If the value of one column increases, the value in the other column will not increase, and vice versa.
Read More: How to Find Spearman Rank Correlation Coefficient in Excel
Method 2 – Inserting the CORREL Function to Compute the Spearman Correlation
• To rank the value of the columns Math and Economics, enter the following formula in E5 and press enter:
• Drag down the Fill Handle to see the result in the rest of the cells.
The values in E5:E14 are ranked.
• To rank the range F5:F14, enter the following formula in F5 and press enter:
• Drag down the Fill Handle to see the result in the rest of the cells.
The values in F5:F14 are ranked.
• Select D17 and enter the following formula:
The Spearman correlation is displayed in D17.
Read More: How to Calculate P Value for Spearman Correlation in Excel
Method 3 – Calculating the Spearman Correlation Using a Graph in Excel
• To rank the value of the columns Math and Economics, enter the following formula in E5 and press enter:
• Drag down the Fill Handle to see the result in the rest of the cells.
The values in E5:E14 are ranked.
• To rank F5:F14, enter the following formula in F5 and press enter:
• Drag down the Fill Handle to see the result in the rest of the cells.
The values in F5:F14 are ranked.
• Create a scatter plot with the ranked columns: select both R[Math] and R[Economics].
• In the Insert tab, go to Scatter in Charts and click Scatter plot.
• The chart will display the R[Math ]column values in the X-axis and the R[Economics] values in the Y-axis.
• Click Chart Elements.
• Check Trendline.
• A downward-facing Trendline will be displayed in the chart.
• Double-click the chart.
• In the new window, click Trendline options.
• Select the Trendline you created.
• Click the histogram-shaped icon.
• Check Display R squared value on chart.
• The R-value is displayed.
• Note it down.
• Select E16 and enter the value of R^2.
• To square root the R^2 value and get the value of Spearman Correlation, enter the following formula in E18:
• Mind the slope of the trendline: if it is downward, alter the sign in E18. If the slope is upward, no change is needed. Here, the trendline is downward. The sign in E18 is changed from 0.41821 to
This is the final value of the Spearman Correlation.
The negative value shows a negative correlation between the data columns.
Read More: How to Find Correlation Coefficient in Excel Scatter Plot
Download Practice Workbook
Download the practice workbook.
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2 Comments
1. Thanks for the tutorial. my question is how i can find or calculate the critical values to assess its significance?
2. Thanks for your question. Actually, you can approach this problem in two separate ways. One is directly calculating the significant value or using a chart. For the chart, you need to calculate
the T value first, and then you will calculate the p-value. Using them, you can calculate the critical value from a chart available online.
1. The formula for calculating the T value is, t=r_s×√((n-2)/(1-r_s^2 ))
Where r_s is the Spearman correlation value.
n is the no of entry
The Excel formula would be in our case =E17*SQRT((E16-2)/(1-E17^2))
2. The formula for significant value, p =T.DIST.2T(ABS(calculated t value),n-2)
3. To calculate the critical value, you need to have a critical value chart. Using the p-value and the n (Number of entries), from the chart, you need to get the critical value.
4. You may be needed to interpolate the critical values as you may not have the exact p or n values. If your correlation value>critical value, then there is a significant correlation between the
values. In other words, the correlation result is significant.
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What does it mean to say that the gravity of the Earth is 9.8 m/s2? | Socratic
What does it mean to say that the gravity of the Earth is 9.8 m/s2?
2 Answers
The acceleration of gravity (also referred to as the gravitational field strength) at the surface of the earth has an average of $9.807 \frac{m}{s} ^ 2$, which means that an object dropped near
earth's surface will accelerate downward at that rate.
Gravity is a force, and according to Newton's Second Law, a force acting on an object will cause it to accelerate:
$F = m a$
Acceleration is a rate of change of speed (or velocity, if working with vectors). Speed is measured in $\frac{m}{s}$, so a rate of change of speed is measured in $\frac{\frac{m}{s}}{s}$ or $\frac{m}
{s} ^ 2$.
An object dropped near Earth's surface will accelerate downwards at about $9.8 \frac{m}{s} ^ 2$ due to the force of gravity, regardless of size, if air resistance is minimal.
Since a large object will feel a large force of gravity and a small object will feel a small force of gravity, we can't really talk about the "force of gravity" being a constant. We can talk about
the "gravitational field strength" in terms of the amount of gravitational force per kg of mass $\left(9.8 \frac{N}{k g}\right)$, but it turns out that the Newton (N) is a derived unit such that $1 N
= 1 k g \cdot \frac{m}{s} ^ 2$, so $\frac{N}{k g}$ is really the same thing as $\frac{m}{s} ^ 2$ anyway.
It should be noted that the strength of gravity is not a constant - as you get farther from the centre of the Earth, gravity gets weaker. It is not even a constant at the surface, as it varies from
~9.83 at the poles to ~9.78 at the equator. This is why we use the average value of 9.8, or sometimes 9.81.
It means that any object is attracted by the earth towards its center with a Force $F = m \times g$, where $m$ is the mass of the body and $g$ acceleration due to gravity, stated in the question.
As per Law of Universal Gravitation the force of attraction between two bodies is directly proportional to the product of masses of the two bodies. it is also inversely proportional to the square of
the distance between the two. That is the force of gravity follows inverse square law.
${F}_{G} \propto {M}_{1.} {M}_{2}$
Also ${F}_{G} \propto \frac{1}{r} ^ 2$
Combining the two we obtain the proportionality expression
${F}_{G} \propto \frac{{M}_{1.} {M}_{2}}{r} ^ 2$
Follows that
${F}_{G} = G \frac{{M}_{1.} {M}_{2}}{r} ^ 2$
Where $G$ is the proportionality constant.
It has the value $6.67408 \times {10}^{-} 11 {m}^{3} k {g}^{-} 1 {s}^{-} 2$
$r$ is the mean radius of earth and taken as $6.371 \times {10}^{6} m$
Mass of earth is $5.972 \times {10}^{24} k g$
If one of the body is earth the equation becomes
${F}_{G} = \left(G \frac{{M}_{e}}{r} ^ 2\right) . m$
See this has reduced to $F = m g$
Were $g = G \frac{{M}_{e}}{r} ^ 2$
Inserting the values
$g = 6.67408 \times {10}^{-} 11 \frac{5.972 \times {10}^{24}}{6.371 \times {10}^{6}} ^ 2$
Simplifying we obtain
$g \approx 9.8 m / {s}^{2}$
In other words if an object is dropped from a height $h$ above the earth's surface, the object will fall towards earth with constant acceleration of $g = 9.8 m / {s}^{2}$
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Finding the Order of A Number Modulo 17
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Finding the Order of A Number Modulo 17
I'm trying to write a code to find a number in the field of integers Mod 17 which has order 16. This is my first time using Mathematica and I feel really lost in the syntax (my file is attached).
My approach is to start with the number 2 and just see if 2^2 (mod 17) is equivalent to 1. If not, I increase the power by one and check it again. I keep doing this until 2^p (mod 17) is equivalent
to 1, and then I check to see if p == 16. If not, I just start the whole process over again with 3, and I do this until I find a number which is equivalent to 1 for the first time when taken to the
power 16. Where am I going wrong with my code?
Edit: I've uploaded a more current file but I'm still not sure what is wrong.
3 Replies
Yes, k needs to be incremented whether we print something or not, also p needs to go back to 2 for each new k.
k = 2;
While[k < 16, p = 2; While[Mod[k^p, 17] =!= 1, p++];
If[p == 16, Print[k]]; k++]
A simpler way might be
In[2]:= Select[Range[16], FreeQ[Mod[#^Range[15], 17], 1] &]
Out[2]= {3, 5, 6, 7, 10, 11, 12, 14}
or just use the built-in function
In[3]:= PrimitiveRootList[17]
Out[3]= {3, 5, 6, 7, 10, 11, 12, 14}
Where am I going wrong with my code?
You've got lots of basic syntax error. May be you should start with small example first in order to learn the basic syntax., You need ";" to separate statements in M. You can't write
k = 2;
p = 2
While[k < 17,
p = 1
While[GF[17][{k}]^p =!= 1,
p ++
If[p == 16,
Print [k]
notice there is no ";" after the p=1 and no ";" after the ] there., You also need ";" after the Print.
It is important to know the difference between "," and ";" in Mathematica. I have not tried to run your code, just an observation. Also the GF[17][{k}]^p looks strange but I never used this function.
Have you tried to test this on its own first to make sure you get the syntax correct?
My new file has fixes to a lot of these syntax errors. The new code I have looks like:
k = 2;
p = 2;
While[k < 17,
While[(GF[17][{k}])^p =!= GF[17][{1}], p ++];
If[p == 16, Print [k], k++];
This seems to be stuck in an infinite loop because it started printing out a bunch of 3s (and I do know that 3 is the first number with order 16 mod 17). I figure this is because the (k++) is in the
"else" part of the If statement, but I'm not sure where to put this (k++) in the code so that it will end appropriately.
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Find the rate of increase of the surface area (S = 4\pi r^2) with respect to the radius r when r is (a) 1 ft, (b) 2 ft, … Ask your homework questions to teachers and professors, meet other students,
and be entered to win $600 or an Xbox Series X 🎉 Join our Discord! 1 dag sedan · A Spherical Balloon Is Being Inflated By A Mechanical Pump. Let R(t) Denote The Radius Of The Balloon, In Centimeters,
T Seconds After Starting Inflation. The Balloon Starts Un-inflated, So R(0) = 0. We Give A Table Of Values For R'(t). Suppose That R’(t) Is Positive And Decreasing. | {"url":"https://forsaljningavaktierwlhd.web.app/13655/63092.html","timestamp":"2024-11-06T06:20:33Z","content_type":"text/html","content_length":"9881","record_id":"<urn:uuid:1204aeb2-8129-4000-8e25-0f1242f5dab8>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00187.warc.gz"} |
Partial Derivative Calculator
Partial derivative calculator with steps
Partial derivative calculator is used to calculate the derivatives of the functions w.r.t several variables with steps. This partial derivative solver differentiates the given constant, linear, or
polynomial functions multiple times. It is a subtype of the derivative calculator.
What is a partial derivative?
“In differential calculus, the differentiation of a function of multivariable w.r.t change in just one of its variables is known as a partial derivative.”
The equations of partial derivatives can be written as:
• \(\frac{\partial }{\partial x}\left(f\left(x,y,z\right)\right)\)
The above equation is used for a multivariable function to calculate the partial differential w.r.t “x”.
• \(\frac{\partial }{\partial y}\left(f\left(x,y,z\right)\right)\)
This equation used, if a function has to be solved w.r.t “y”.
• \(\frac{\partial }{\partial z}\left(f\left(x,y,z\right)\right)\)
This equation is used when the partial derivative of a function has to be solved w.r.t “z”
Use our partial derivative calculator xyz for getting the results of multivariable problems.
How to use this first partial derivative calculator?
To use this first partial differentiation calculator, follow the steps below.
• Input the multivariable function like f(x, y, z).
• Choose one variable from x, y, and z while other variables remain constant.
• Use the keypad icon
• Hit the calculate button to get the result of the given input function.
• Press the show more button to get the step-by-step calculations.
• If you want to calculate another problem hit the reset button next to the calculate button.
How to evaluate the problems of partial derivatives?
Following are two examples of the 3 variables function evaluated by our multivariable derivative calculator.
Example 1
Find the partial derivative of 3xyz w.r.t “x”.
Step 1: Write the function with the partial differentiation notation.
\(\frac{\partial }{\partial x}\left(3xyz\right)\)
Step 2: Now calculate the partial derivative of 3xyz w.r.t “x” while y & z remains constant.
\(\frac{\partial }{\partial x}\left(3xyz\right)=3yz\frac{\partial }{\partial x}\left(x\right)\)
\(\frac{\partial }{\partial x}\left(3xyz\right)=3yz\left(1\right)\)
\(\frac{\partial }{\partial x}\left(3xyz\right)=3yz\)
Step 3: Similarly, the partial derivative of 3xyz w.r.t y & z are:
\(\frac{\partial }{\partial y}\left(3xyz\right)=3xz\)
\(\frac{\partial }{\partial z}\left(3xyz\right)=3xy\)
Example 2
Find the partial derivative of \(3x^2y+4xyz-9xy\) w.r.t “y”.
Step 1: Write the given function along with the partial derivative notation.
\(\frac{\partial }{\partial y}\left(3x^2y+4xyz-9xy\right)\)
Step 2: Now apply the notation separately.
\(\frac{\partial }{\partial y}\left(3x^2y+4xyz-9xy\right)=\frac{\partial }{\partial y}\left(3x^2y\right)+\frac{\partial }{\partial y}\left(4xyz\right)-\frac{\partial }{\partial y}\left(9xy\right)\)
Step 3: Now calculate the partial derivative of \(3x^2y+4xyz-9xy\) w.r.t “y” while x & z remains constant.
\( \frac{\partial }{\partial y}\left(3x^2y+4xyz-9xy\right)=3x^2\frac{\partial }{\partial y}\left(y\right)+4xz\frac{\partial }{\partial y}\left(y\right)-9x\frac{\partial }{\partial y}\left(y\right)\)
\( \frac{\partial }{\partial y}\left(3x^2y+4xyz-9xy\right)=3x^2\left(1\right)+4xz\left(1\right)-9x\left(1\right)\)
\( \frac{\partial }{\partial y}\left(3x^2y+4xyz-9xy\right)=3x^2+4xz-9x\)
Step 4: Similarly, the partial derivative of given function w.r.t x & z are:
\( \frac{\partial }{\partial x}\left(3x^2y+4xyz-9xy\right)=6xy+4yz-9y\)
\( \frac{\partial }{\partial z}\left(3x^2y+4xyz-9xy\right)=4xy\)
Table of partial derivatives of functions
Following are questions and answers of partial derivatives solved by this partial derivatives calculator.
Questions Answers
Partial derivative of xy w.r.t x y
Partial derivative of xy w.r.t y x
Partial derivative of xy w.r.t z 0
Partial derivative of e^xy w.r.t x \(ye^{xy}\)
Partial derivative of e^xyz w.r.t x \(yze^{xyz}\)
Partial derivative of sqrt(xy) w.r.t x \(\frac{y}{2\sqrt{xy}\:}\) | {"url":"https://www.limitcalculator.online/partial-derivative-calculator","timestamp":"2024-11-14T18:47:03Z","content_type":"text/html","content_length":"68990","record_id":"<urn:uuid:75cbc31a-9f75-402f-8c55-d05b179342d5>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00368.warc.gz"} |
include/media/v4l2-vp9.h - third_party/kernel - Git at Google
/* SPDX-License-Identifier: GPL-2.0-or-later */
* Helper functions for vp9 codecs.
* Copyright (c) 2021 Collabora, Ltd.
* Author: Andrzej Pietrasiewicz <andrzej.p@collabora.com>
#ifndef _MEDIA_V4L2_VP9_H
#define _MEDIA_V4L2_VP9_H
#include <media/v4l2-ctrls.h>
* struct v4l2_vp9_frame_mv_context - motion vector-related probabilities
* @joint: motion vector joint probabilities.
* @sign: motion vector sign probabilities.
* @classes: motion vector class probabilities.
* @class0_bit: motion vector class0 bit probabilities.
* @bits: motion vector bits probabilities.
* @class0_fr: motion vector class0 fractional bit probabilities.
* @fr: motion vector fractional bit probabilities.
* @class0_hp: motion vector class0 high precision fractional bit probabilities.
* @hp: motion vector high precision fractional bit probabilities.
* A member of v4l2_vp9_frame_context.
struct v4l2_vp9_frame_mv_context {
u8 joint[3];
u8 sign[2];
u8 classes[2][10];
u8 class0_bit[2];
u8 bits[2][10];
u8 class0_fr[2][2][3];
u8 fr[2][3];
u8 class0_hp[2];
u8 hp[2];
* struct v4l2_vp9_frame_context - frame probabilities, including motion-vector related
* @tx8: TX 8x8 probabilities.
* @tx16: TX 16x16 probabilities.
* @tx32: TX 32x32 probabilities.
* @coef: coefficient probabilities.
* @skip: skip probabilities.
* @inter_mode: inter mode probabilities.
* @interp_filter: interpolation filter probabilities.
* @is_inter: is inter-block probabilities.
* @comp_mode: compound prediction mode probabilities.
* @single_ref: single ref probabilities.
* @comp_ref: compound ref probabilities.
* @y_mode: Y prediction mode probabilities.
* @uv_mode: UV prediction mode probabilities.
* @partition: partition probabilities.
* @mv: motion vector probabilities.
* Drivers which need to keep track of frame context(s) can use this struct.
* The members correspond to probability tables, which are specified only implicitly in the
* vp9 spec. Section 10.5 "Default probability tables" contains all the types of involved
* tables, i.e. the actual tables are of the same kind, and when they are reset (which is
* mandated by the spec sometimes) they are overwritten with values from the default tables.
struct v4l2_vp9_frame_context {
u8 tx8[2][1];
u8 tx16[2][2];
u8 tx32[2][3];
u8 coef[4][2][2][6][6][3];
u8 skip[3];
u8 inter_mode[7][3];
u8 interp_filter[4][2];
u8 is_inter[4];
u8 comp_mode[5];
u8 single_ref[5][2];
u8 comp_ref[5];
u8 y_mode[4][9];
u8 uv_mode[10][9];
u8 partition[16][3];
struct v4l2_vp9_frame_mv_context mv;
* struct v4l2_vp9_frame_symbol_counts - pointers to arrays of symbol counts
* @partition: partition counts.
* @skip: skip counts.
* @intra_inter: is inter-block counts.
* @tx32p: TX32 counts.
* @tx16p: TX16 counts.
* @tx8p: TX8 counts.
* @y_mode: Y prediction mode counts.
* @uv_mode: UV prediction mode counts.
* @comp: compound prediction mode counts.
* @comp_ref: compound ref counts.
* @single_ref: single ref counts.
* @mv_mode: inter mode counts.
* @filter: interpolation filter counts.
* @mv_joint: motion vector joint counts.
* @sign: motion vector sign counts.
* @classes: motion vector class counts.
* @class0: motion vector class0 bit counts.
* @bits: motion vector bits counts.
* @class0_fp: motion vector class0 fractional bit counts.
* @fp: motion vector fractional bit counts.
* @class0_hp: motion vector class0 high precision fractional bit counts.
* @hp: motion vector high precision fractional bit counts.
* @coeff: coefficient counts.
* @eob: eob counts
* The fields correspond to what is specified in section 8.3 "Clear counts process" of the spec.
* Different pieces of hardware can report the counts in different order, so we cannot rely on
* simply overlaying a struct on a relevant block of memory. Instead we provide pointers to
* arrays or array of pointers to arrays in case of coeff, or array of pointers for eob.
struct v4l2_vp9_frame_symbol_counts {
u32 (*partition)[16][4];
u32 (*skip)[3][2];
u32 (*intra_inter)[4][2];
u32 (*tx32p)[2][4];
u32 (*tx16p)[2][4];
u32 (*tx8p)[2][2];
u32 (*y_mode)[4][10];
u32 (*uv_mode)[10][10];
u32 (*comp)[5][2];
u32 (*comp_ref)[5][2];
u32 (*single_ref)[5][2][2];
u32 (*mv_mode)[7][4];
u32 (*filter)[4][3];
u32 (*mv_joint)[4];
u32 (*sign)[2][2];
u32 (*classes)[2][11];
u32 (*class0)[2][2];
u32 (*bits)[2][10][2];
u32 (*class0_fp)[2][2][4];
u32 (*fp)[2][4];
u32 (*class0_hp)[2][2];
u32 (*hp)[2][2];
u32 (*coeff[4][2][2][6][6])[3];
u32 *eob[4][2][2][6][6][2];
extern const u8 v4l2_vp9_kf_y_mode_prob[10][10][9]; /* Section 10.4 of the spec */
extern const u8 v4l2_vp9_kf_partition_probs[16][3]; /* Section 10.4 of the spec */
extern const u8 v4l2_vp9_kf_uv_mode_prob[10][9]; /* Section 10.4 of the spec */
extern const struct v4l2_vp9_frame_context v4l2_vp9_default_probs; /* Section 10.5 of the spec */
* v4l2_vp9_fw_update_probs() - Perform forward update of vp9 probabilities
* @probs: current probabilities values
* @deltas: delta values from compressed header
* @dec_params: vp9 frame decoding parameters
* This function performs forward updates of probabilities for the vp9 boolean decoder.
* The frame header can contain a directive to update the probabilities (deltas), if so, then
* the deltas are provided in the header, too. The userspace parses those and passes the said
* deltas struct to the kernel.
void v4l2_vp9_fw_update_probs(struct v4l2_vp9_frame_context *probs,
const struct v4l2_ctrl_vp9_compressed_hdr *deltas,
const struct v4l2_ctrl_vp9_frame *dec_params);
* v4l2_vp9_reset_frame_ctx() - Reset appropriate frame context
* @dec_params: vp9 frame decoding parameters
* @frame_context: array of the 4 frame contexts
* This function resets appropriate frame contexts, based on what's in dec_params.
* Returns the frame context index after the update, which might be reset to zero if
* mandated by the spec.
u8 v4l2_vp9_reset_frame_ctx(const struct v4l2_ctrl_vp9_frame *dec_params,
struct v4l2_vp9_frame_context *frame_context);
* v4l2_vp9_adapt_coef_probs() - Perform backward update of vp9 coefficients probabilities
* @probs: current probabilities values
* @counts: values of symbol counts after the current frame has been decoded
* @use_128: flag to request that 128 is used as update factor if true, otherwise 112 is used
* @frame_is_intra: flag indicating that FrameIsIntra is true
* This function performs backward updates of coefficients probabilities for the vp9 boolean
* decoder. After a frame has been decoded the counts of how many times a given symbol has
* occurred are known and are used to update the probability of each symbol.
void v4l2_vp9_adapt_coef_probs(struct v4l2_vp9_frame_context *probs,
struct v4l2_vp9_frame_symbol_counts *counts,
bool use_128,
bool frame_is_intra);
* v4l2_vp9_adapt_noncoef_probs() - Perform backward update of vp9 non-coefficients probabilities
* @probs: current probabilities values
* @counts: values of symbol counts after the current frame has been decoded
* @reference_mode: specifies the type of inter prediction to be used. See
* &v4l2_vp9_reference_mode for more details
* @interpolation_filter: specifies the filter selection used for performing inter prediction.
* See &v4l2_vp9_interpolation_filter for more details
* @tx_mode: specifies the TX mode. See &v4l2_vp9_tx_mode for more details
* @flags: combination of V4L2_VP9_FRAME_FLAG_* flags
* This function performs backward updates of non-coefficients probabilities for the vp9 boolean
* decoder. After a frame has been decoded the counts of how many times a given symbol has
* occurred are known and are used to update the probability of each symbol.
void v4l2_vp9_adapt_noncoef_probs(struct v4l2_vp9_frame_context *probs,
struct v4l2_vp9_frame_symbol_counts *counts,
u8 reference_mode, u8 interpolation_filter, u8 tx_mode,
u32 flags);
* v4l2_vp9_seg_feat_enabled() - Check if a segmentation feature is enabled
* @feature_enabled: array of 8-bit flags (for all segments)
* @feature: id of the feature to check
* @segid: id of the segment to look up
* This function returns true if a given feature is active in a given segment.
v4l2_vp9_seg_feat_enabled(const u8 *feature_enabled,
unsigned int feature,
unsigned int segid);
#endif /* _MEDIA_V4L2_VP9_H */ | {"url":"https://cos.googlesource.com/third_party/kernel/+/5384c5015506de785504eb25ef668d31431786e5/include/media/v4l2-vp9.h","timestamp":"2024-11-08T12:38:01Z","content_type":"text/html","content_length":"71782","record_id":"<urn:uuid:ddfa8cc1-4069-40b5-a56b-a0aa1873208d>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00054.warc.gz"} |
Artis REIT
Sound bite for Twitter and StockTwits is: Dividend Growth REIT. The stock price is probably reasonable. The dividends are being raised again after being cut, so not a good dividend growth REIT. I
worry about the Liquidity Ratio because it is so low. See my spreadsheet on
Artis REIT
Is it a good company at a reasonable price? The stock price is probably reasonable. It is not much of a dividend growth stock. Dividends have been flat and decreased as well as being increased. So,
there is no real consistency. You will probably not lose money on this stock. See below what has happened over the past 10 years.
I do not own this stock of Artis REIT (TSX-AX.UN, OTC-ARESF). Early in 2013, this company was mentioned as a good REIT to own. A number of people I correspond with mentioned this REIT. However, my
first view of it is not positive. It is also not a dividend growth stock.
When I was updating my spreadsheet, I noticed Revenue expected was $461M and it came in at $419.5M. The Adjusted Funds from Operations (AFFO) expected was $1.02, and it came in at $0.96. The Funds
from Operations (FFO) $1.38 and it came in at $1.34. Basically, why EPS went up was because of “Fair value gain (loss) on investment properties”. For REITs the better measurement than EPS is the AFFO
and FFO values.
If you had invested in this company in December 2011, $1007.28 you would have bought 72 shares at $13.99 per share. In December 2021, after 10 years you would have received $661.25 in dividends. The
stock would be worth $859.68. Your total return would have been $1,520.93.
Cost Tot. Cost Shares Years Dividends Stock Val Tot Ret
$13.99 $1,007.28 72 10 $661.25 $859.68 $1,520.93
The dividend yields are good with dividend growth restarting. The current dividend yield is good (5% and 6% ranges) at 5.30%. The 5 year median dividend yield is also good at 5.62%. The 10 year and
historical median dividend yields are high (7% and above) at 7.14% and 7.15%. The dividends were flat between 2009 and 2017. The dividends were decreased by 50% in 2018. They were raised in 2021 by
10%. There has been no raise so far in 2022, but analysts do expect an increase in 2022 or 2023.
The Dividend Payout Ratios (DPR) are fine (with DPR from AFFO and FFO the best measure). The DPR for EPS for 2021 is 21% with 5 year coverage at 64%. The DPR for Adjusted Funds from Operations (AFFO)
for 2021 is 61% with 5 year coverage at 59%. The DPR for Funds from Operations (FFO) for 2021 is 44% with 5 year coverage at 43%. The DPR for Cash Flow per Share (CFPS) for 2021 is 46% with 5 year
coverage at 54%. The DPR for Free Cash Flow (FCF) for 2021 is 62% with 5 year coverage at 61%.
Most Debt Ratios are fine, but I worry about the Liquidity Ratio. The Long Term Debt/Market Cap Ratio for 2021 is 0.70 and is fine. The Debt Ratio is good at 2.16. The Leverage and Debt/Equity Ratios
are good at 1.86 and 0.86. The Liquidity Ratio at 0.36 is awful. If you add in Cash Flow after dividends, you are only up to 0.46. If this is below 1.00, it means that current assets cannot cover
current liabilities. Only if you add back in the debt payable and credit facilities do you get a good number (2.85). However, if debt and credit facilities cannot be rolled over, then the company
could be in trouble. This can sometime happen in recessions.
The Total Return per year is shown below for years of 5 to 17 to the end of 2021. Under the Capital Gain column is the portion of the Total Return attributable to capital gains. Under the Dividend
column is the portion of the Total Return attributable to dividends. See chart below.
From Years Div. Gth Tot Ret Cap Gain Div.
2016 5 -11.42% 4.99% -1.23% 2.88%
2011 10 -5.88% 5.64% -1.57% 2.15%
2006 15 -3.78% 4.83% -2.07% 2.53%
2004 17 -1.47% 17.33% 3.98% 2.21%
The 5 year low, median, and high median Price/Earnings per Share Ratios are 10.06, 13.19 and 16.33. The corresponding 10 year ratios are 10.23, 12.27 and 13.24. The corresponding historical ratios
are 4.07, 4.46 and 4.85. The current P/E Ratio is 2.66 based on a stock price of $11.32 and EPS for the last 12 months of $4.26. This ratio is below the 10 year median ratio. This stock price testing
suggests that the stock price is relatively cheap.
I also have Adjusted Funds from Operations (AFFO) data. The 5 year low, median, and high median P/AFFO are 9.62,11.67 and 13.40. The corresponding 10 year ratios are 10.15, 11.85 and 13.34. The
current P/AFFO Ratio is 11.10 based on AFFO estimate for 2022 of $1.02 and a stock price of $11.32. The current ratio is between the low and median of the 10 year median ratios. This stock price
testing suggests that the stock price is relatively reasonable and below the median.
I also have Funds from Operations (FFO) data. The 5 year low, median, and high median P/FFO are 7.08,836 and 9.70. The corresponding 10 year ratios are 7.82, 9.23 and 10.25. The current P/FFO Ratio
is 7.92 based on FFO estimate for 2022 of $1.43 and a stock price of $11.32. The current ratio is between the low and median of the 10 year median ratios. This stock price testing suggests that the
stock price is relatively reasonable and below the median.
I get a Graham Price of $42.21. The 10 year low, median, and high median Price/Graham Price Ratios are 0.55, 0.65 and 0.74. The current P/GP Ratio is 0.27 based on a stock price of $11.32. The
current ratio is below the low ratio of the 10 year median ratios. This stock price testing suggests that the stock price is relatively cheap.
I also do the Graham Price calculation using AFFO in the formula instead of EPS. Here I get a Graham Price of $20.65. The 10 year low, median, and high median Price/Graham Price Ratios are 054, 0.64
and 0.74. The current P/GP Ratio is 0.55 based on a stock price of $11.32. The current ratio is between the low and median ratios of the 10 year median ratios. This stock price testing suggests that
the stock price is relatively reasonable and below the median.
I get a 10 year median Price/Book Value per Share Ratio of 0.83. The current P/B Ratio is 0.61 based on a Book Value of $2,296M, Book Value per Share of $18.59 and a stock price of $11.32. The
current ratio is 27% below the 10 year median ratio. This stock price testing suggests that the stock price is relatively cheap.
I get a 10 year median Price/Cash Flow per Share Ratio of 8.43. The current P/CF Ratio is 8.53 based on last 12 month Cash Flow of $164M, Cash Flow per Share of $1.33 and a stock price of $11.32. The
current ratio is 1% above the 10 year median ratio. This stock price testing suggests that the stock price is relatively reasonable but above the median.
I get an historical median dividend yield of 7.15%. The current dividend yield is 5.30% based on dividends of $0.60 and a stock price of $11.32. The current dividend yield is 26% below the historical
dividend yield. This stock price testing suggests that the stock price is relatively expensive.
I get a 10 median dividend yield of 7.14%. The current dividend yield is 5.30% based on dividends of $0.60 and a stock price of $11.32. The current dividend yield is 26% below the 10 year dividend
yield. This stock price testing suggests that the stock price is relatively expensive.
The 10 year median Price/Sales (Revenue) Ratio is 3.49. The current P/S Ratio is 3.48 based on a stock price of $11.32, Revenue estimate for 2022 of $402 and Revenue per Share of $3.25. The current
ratio is 0.4% below the 10 year median ratio. This stock price testing suggests that the stock price is relatively reasonable and below the median.
Results of stock price testing is that the stock price is probably reasonable based on the P/S Ratio test. The problem with the dividend yield test is the declining dividend. (However, declining
dividends are never a good sign.) The P/AFFO test and P/FFO test are better tests than the P/E Test (because this is a REIT). These tests also show a reasonable stock price. Other tests are showing
stock from cheap to reasonable.
When I look at analysts’ recommendations, I find Buy (1) and Hold (7). The consensus would be a Hold. The 12 month stock price consensus is $13.38. This implies a total return of 23.50% with 18.20%
from capital gains and 5.30% from dividends.
Some, but not all analysts on
Stock Chase
like this stock. It is not on the Money Sense list. Stock chase gives this company 4 stars out of 5. Adam Othman on
Motley Fool
says it is an undervalued dividend stock. Adam Othman on
Motley Fool
says this stock is a bargain. He has been recommending this stock since at least July 2021. The company released their fourth quarterly 2021 results on
. The company released on
their first quarter of 2022 results.
Simply Wall Street on
Yahoo Finance
reviews this stock. They like the recent insider buying. Simply Wall Street has two risk warnings of debt is not well covered by operating cash flow and large one-off items impacting financial
Artis Real Estate Investment Trust is an unincorporated closed-end REIT based in Canada. Artis REIT's portfolio comprises properties located in Central and Western Canada and select markets
throughout the United States, including regions such as Alberta, British Columbia, Manitoba, Ontario, Saskatchewan, Arizona, Minnesota, Colorado, New York, and Wisconsin. The properties are divided
into three categories: office, retail, and industrial. Its web site is here
Artis REIT
The last stock I wrote about was about was Obsidian Energy Ltd (TSX-OBE, OTC-OBELF) ...
learn more
. The next stock I will write about will be Dorel Industries Inc (TSX-DII.B, OTC-DIIBF) ...
learn more
on Wednesday, July 20, 2022 around 5 pm. Tomorrow on my other blog I will write about Life Insurance....
learn more
on Tuesday, July 19, 2022 around 5 pm.
This blog is meant for educational purposes only and is not to provide investment advice. Before making any investment decision, you should always do your own research or consult an investment
professional. I do research for my own edification and I am willing to share. I write what I think and I may or may not be correct.
See my website for
stocks followed
investment notes
. I have three blogs. The first talks only about specific stocks and is called
Investment Talk
. The second one contains information on mostly investing and is called
Investing Economics Mostly
. My last blog is for my book reviews and it is called
Non-Fiction Mostly
. Follow me on
. I am on
. Or you can just Google #walktoronto spbrunner8166 to see my pictures. | {"url":"https://spbrunner.blogspot.com/2022/07/artis-reit.html","timestamp":"2024-11-06T06:10:00Z","content_type":"text/html","content_length":"116911","record_id":"<urn:uuid:5c4e2332-2333-4888-aaff-a15a03821124>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00729.warc.gz"} |
Fraction Symbols ¼ | Copy & Paste Calculation Icons
Discover and use a variety of unique fraction symbols with ease. Perfect for social media, design projects, and more. No hassle, just copy and paste!
Fraction symbols are useful for expressing numerical values that aren't whole numbers. Use these fraction symbols in your documents, designs, or math projects by simply copying and pasting them.
Symbol Description Unicode
¼ One Quarter U+00BC
½ One Half U+00BD
¾ Three Quarters U+00BE
⁄ Fraction Slash U+2044
⅞ Seven Eighths U+215E
⅟ One Part Per U+215F
↉ Zero Thirds (Non-existent) U+2189
⅒ One Tenth U+2152
⅓ One Third U+2153
⅝ Five Eighths U+215D
⅖ Two Fifths U+2156
⅗ Three Fifths U+2157
⅘ Four Fifths U+2158
⅙ One Sixth U+2159
⅔ Two Thirds U+2154
⅕ One Fifth U+2155
⅐ One Seventh U+2150
⅑ One Ninth U+2151
⅚ Five Sixths U+215A
⅛ One Eighth U+215B
⅜ Three Eighths U+215C
1. Browse: Look through the table to find the symbol you need.
2. Click to Copy: Click on the desired symbol to copy it to your clipboard.
3. Paste: Insert the symbol wherever you need it in your document or design. | {"url":"https://thecoolsymbols.com/fraction.html","timestamp":"2024-11-10T07:45:24Z","content_type":"text/html","content_length":"29995","record_id":"<urn:uuid:93302977-8112-4fab-a9f7-27648035b087>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00844.warc.gz"} |
15. Artificial Intelligence
• What is Intelligence?: Nobody knows, but there are lots of ideas.
• So if we don’t know what intelligence is, how can we write programs that are intelligent ?: AI researchers guess at how parts of the brain work, then try to reproduce the effect in a program.
• A Myth is that artificial programs use intelligence to guide themselves, but they are not as stupid as traditional programs.
• Most AI methods take ONE aspect of human thinking, and then model it, such as,
• There are many older, and newer AI topics of interest,
• Each of the AI techniques is distinct in which applications they are useful for, and how they are applied.
• In manufacturing there are many problems which are difficult to solve using computers, such as trying to get a computer to ‘listen for a particular PING when doing inspection’. But, AI can help
solve some of these problems.
• AI is not magic, you must still understand the problem before an AI system will help you solve it.
• The List below says a few words about some popular AI topics, and how they relate to applications,
Expert Systems: These systems use exact rules with true or false conditions, and results. “If the engine stops, the car is out of gas”
Fuzzy Logic: This method still uses rules which do not need true or false conditions. “When I want the engine to go slower, I give it less gas” where slower, and less are somewhat arbitrary values.
Neural Networks: This methods is equivalent to learning by example. “I don’t know why, I just know to go faster, I push the accelerator that much”.
• For the three methods above, how well the problem can be described determines which is too be used. In effect, if the problem is very logical, use expert systems, if you can still make rules, but
nothing is true or false use fuzzy logic. Finally, if you can’t define rules for solving a problem, but can do it by intuition, use neural networks.
15.1 Expert Systems
• To implement an expert system, a knowledge engineer will talk to ‘experts’ about how they solve a problem. The Knowledge engineer will then try to develop a set of rules for solving the problem.
After the rules are done, they are entered into Expert System Software ($0: $20,000). Expert system software will then ask questions (or check sensors, or look at data files) to compare rules to
conditions, and see the results.
• The rules are in the form shown on the other page
• There are two ways to search rules,
Forward Chaining: Consider what you know now, then check rules to see if all conditions are satisfied. The Results of the rule give you a new conclusion. The rules are all checked again using the
Results from the previous rule. This is often used for choosing an action.
Backward Chaining: This method will backtrack from a set of consequents to find which conditions caused them. This method is often used for determining how something was done.
15.2 Fuzzy Logic
• Rules are created which make sense to humans, for example when driving a car some acceleration rules may be,
if LOUD_NOISE and FAST_SPEED then SLOW_SPEED
if QUIET_NOISE and FAST_SPEED then SAME_SPEED
• Each or the rule conditions, and results, can be represented with a 1 dimension matrix,
LOUD_NOISE = { 0.0 0.1 0.5 1.0 }
QUIET_NOISE = {1.0 0.5 0.1 0.0 }
FAST_SPEED = { 0.0 0.2 0.7 1.0 }
SLOW_SPEED = { 1.0 0.5 0.2 0.0 }
SAME_SPEED = { 1.0 1.0 1.0 1.0 }
The sets are considered normalized from minimum to maximum. For example is the NOISE sets were from 20dB to 100dB, then a noise level of 80dB would result in a value of 0.5, or a 50% membership. This
way you can say if the noise is ‘absolutely loud’, giving a value of 1.0, or ‘a bit loud’ giving a value of 0.5.
• The matrices are combined using the rules to get a result matrix
• Because the conditions (like LOUD_NOISE) are defined with a sort of weight, or memebership rules are easier to make up.
• Fuzzy logic controllers have been very successful at solving control problems.
15.2.1 NEURAL NETWORKS
15.2.1.1 - Neural Network Calculation of Inverse Kinematics
• Objectives: To give insight into the neural network solution of the inverse kinematics problem.
15.2.1.2 - Inverse Kinematics
• Forward kinematics for a 3-link manipulator
• Inverse Kinematics for a three link manipulator
• Inverse Kinematics techniques
explicit, with exact solution (as for the 3 link manipulator)
iterative, for use when an infinite number of solutions exist.
• Problems that occur when doing inverse kinematics with these methods are,
both methods require a computer capable of mathematical calculations
the methods do not adapt to compensate for damage, calibration errors, wear, etc.
solutions may be slow, especially for iterative solutions
solutions are valid only for a specific robot
• advantages of using these methods are,
both solutions will yield exact answers
properties of both of these methods are well known
15.2.2 Feed Forward Neural Networks
• A feed forward neural network was used with a sigmoidal activation function
• the back propagation learning technique was used.
• disadvantage of these network is
unpredictable errors occur in the solution
discontinuous problem spaces cause problems for the networks
these networks are not well understood
neuro-computers are not commonly available
the architecture is fault tolerant
can be adjusted for changes in the robot configuration
the controllers are not specific to a single robot
15.2.3 The Neural Network Setup
• The figure below shows how the neural network was configured to solve the problem
• The first neural network estimates the proper inverse kinematics. This will contain a small error, therefore a second net is used to estimate the errors. Additional correction networks can be added
after this.
• The networks were generally connected with
10, 20 or 40 neurons in the hidden layer
a bias input was connected to each neuron
the layers were all fully connected
there were runs with one, and two hidden layers
15.2.4 The Training Set
• The problem is reduced using either left or right arm configurations, the solution is also constrained to elbow up or elbow down.
• Discontinuities were avoided by not training the neural network in the region above the origin. The elbow straight configuration is also a minor singularity problem.
• Training points were evenly distributed throughout the robot workspace
• Only a quarter of the robot workspace was used because of the robot symmetry.
• The general protocol for training was,
apply the desired position to the input, and train for the desired joint angles.
When accuracy was high enough, the first correction net was trained by comparing the actual errors, and the desired values. Additional correction networks were also trained in some cases.
the error was measured by using an RMS measure of the differences
• A list of results are provided below,
• The results in the table were obtained for a variety of network configurations
• A visual picture of the network configurations is shown below, and on subsequent pages. These are based on a set of test points that lie in a plane of the workspace.
********** Add figure of network point locations, and test conditions
• As seen in the experimental results, there are distortions that occur near the origin, and the edges of the workspace, as would be expected with the singularities found there.
• The errors also increased near the training boundaries
********* Add in more of the results figures
15.2.5 Results
• The mathematical singularities caused by Cartesian coordinates, and the +/- 180 degrees singularity could be eliminated by selecting another set of coordinates for space and the arm.
• The best results were about 1 degree RMS. | {"url":"https://engineeronadisk.com/V3/engineeronadisk-208.html","timestamp":"2024-11-13T18:40:55Z","content_type":"text/html","content_length":"16121","record_id":"<urn:uuid:b5d6a328-c080-4083-9e36-bc6d01aa856a>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00074.warc.gz"} |
Homework 6 - Academia Essay Writers
0 comments
Complete all parts and questions below. Some starter code is supplied, along with hints throughout the instructions. If you cannot complete part 12, the remainder of the assignment can be completed
without including the “LASSO model” for partial credit.
Part Zero
• Download hw6_starter.Rmd and hw6_spotify.RData and save them in the same folder.
• Knit hw6_starter.Rmd — note, it will take a few minutes to knit this document the first time (but the results are cached)
• Edit hw6_starter.Rmd to answer the following questions…
Part 1 – One-Factor Inference (8pts)
1. Build side-by-side Boxplots with overlayed means comparing the tempo of songs to the recorded key signature (key_mode in the data). Make sure the plot is properly labeled and titled. Does the
distribution of tempo appear different for these key signatures? (3pts)
2. Perform a One-Way ANOVA test to compare the mean tempo as a function of key_mode. Make sure to properly state the outcome of the results – No need to check assumption or perform follow-up multiple
comparisons. (2pts)
3. Does the ANOVA result agree with your analysis of the Boxplots? Discuss. (1pt)
4. Read sections 1 and 2 of the journal article “The p-value you can’t buy” (pdf). Based on that article and the analysis in #2, #3 and #4 above, describe/discuss the limitations of using an ANOVA in
this setting. (2pts)
Part 2 – EDA of Popularity (8pts)
5. Construct and describe a histogram (with binwidth=1) of the popularity scores for all observations in music_for_training. (3pts)
6. Calculate the mean, standard deviation and 5-number summary for the popularity scores in music_for_training and display in a well-constructed table. (3pts)
7. Describe the overall shape and behavior of the popularity scores and what implications this may have on linear regression modeling. (2pt)
Part 3 – Model Fitting and Assessment (14pts)
8. The provided Markdown file includes a model pre-built using the music_for_training data (full_model), do not edit it. Look at the residual plots provided for the full_model, describe any concerns
you may have and discuss possible transformations that may be helpful (note, you do NOT need to build a Box-Cox plot). (2pts)
9. Fit a modified full_model where the response variable has been transformed by a cube-root plus 1 transformation, that is, (popularity+1)^(1/3). No need for residual analysis. (2pt)
10. Look at the summary output from this model, what do you notice about the overall F-test and marginal t-tests. Do you think these results are particularly meaningful given the large sample size,
reference Part 1 of this assignment in your discussion. (2pts)
11. Perform stepwise backward variable selection on the model from #9 (the model with a cubed root response). What modifications does it suggest? Is this surprising given the summary output and
discussion in #10? (2pts)
12. Your instructors ran a LASSO regression (see Section 9.3 of the textbook for an introduction, if interested) and it suggests we do the following:
• to remove the instrumentalness variable
• Combine the key_mode “G major” and “A major” levels, call the new level “AG major” (hint: use case_when() )
• Create a new binary variable indicating if timing_signature is 3 is used or not (hint: use ifelse() )
Build a new model based on these changes and provide the summary() output, what do you notice compared to your other models. (4pts)
13. Build a table that reports the adjusted R-squared, AIC and BIC for the full_model, the transformed response model in #9 and the LASSO-variable selection model build in #12. (1pt)
14. You should note that the AIC and BIC values for the transformed models are substantially smaller than the full_model, discuss why it is unfair to compare the AIC and BIC of the models with a
transformed response (cube root of response) to the others (not transformed)? Hint: it has to do with the RSS; see Module 09 and Section 6.4.2 of the text. (2pts)
15. Based on the two models with a transformed response, which model appears to be the best fit? Justify with a brief discussion. (2pts)
Part 4 – Model Validation and out-of-sample Prediction (10pts)
16. Use the 3 models from Part 3 (the full_model, the transformed response model and the LASSO-based model you built in #12) to predict the popularity scores for the 10,000 songs in the
music_for_testing data. Note: you will need to perform the same mutations to music_for_testing as you did for music_for_training in part 12. (3pts)
17. Calculate the root mean squared error (RMSE) for each of the three models (make sure to “un”-transform the response), which model appears best at predicting popularity scores? (2pts)
18. Compare/contrast the RMSE values in #17 to the standard deviation calculated in #6 and the residual standard error of the full_model fit in #9. What does this imply about these models to predict
popularity scores? (1pt)
19. Use the best model of the five to predict the popularity scores for the 5,000 songs in the music_to_predict data. Determine which 10 tracks have the highest predicted popularity scores. (2pts)
20. Using the best model from #17, explore the distribution of your predicted popularity scores. Discuss why the predicted popularity scores behave as they do. Do these predictions appear surprising
given the distribution of the popularity scores? (2pts)
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Functional stochastic differential equations – The Dan MacKinlay stable of variably-well-consider’d enterprises
Functional stochastic differential equations
SDEs taking values in some function space
July 10, 2024 — July 10, 2024
Banach space
dynamical systems
Hilbert space
Lévy processes
signal processing
stochastic processes
time series
Placeholder, for the infinite-dimensional version of SDEs. | {"url":"https://danmackinlay.name/notebook/stochastic_differential_equations_functional.html","timestamp":"2024-11-08T21:28:15Z","content_type":"application/xhtml+xml","content_length":"32585","record_id":"<urn:uuid:432a4869-2bc4-4650-b923-9a78da6b410b>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00580.warc.gz"} |
Quantum Physics with Tensors
by M. Collura (4 credits, type C)
(image credits: Wikipedia)
The course aims to give a general understanding of different methods based on Tensor Network representation of the many-body wave function. I’m assuming an introduction to DMRG has been already given
in the course “Computer Simulation of Condensed Matter”. Here we focus on the special network structure of such methods in order to exploit all theoretical/numerical advantages. Special attention
will be paid on practical aspects of the algorithm implementation (mainly in C++).
PREPARATORY (from the “Computer Simulation of Condensed Matter” course)
1. Basic Knowledge on Linear algebra (SVD, etc.)
2. 2º quantization formalism: if necessary, we will remind it.
3. Basic introduction of DMRG.
MAIN TOPICS (may vary)
1. Diagrammatic representation for Tensor Networks (TNs):
□ From Tensor to Networks
□ Connection to Quantum Computing Notation
□ Physical State admitting exact TN representation
2. Tensor Network in one dimension: Matrix Product States (MPS)
□ Theoretical grounding
□ Canonical Representations
□ Vidal Representation
3. Matrix Product Operator
4. Working with MPO and MPS
□ Entanglement, local observable, correlation functions, Loschmit echo, etc.
5. What about 2D? Basics on PEPS and TTN.
6. Variational algorithms: from ground-state to excited states
7. Real and Imaginary time evolution algorithms
8. Open Systems with Tensor Networks
□ Non-Unitary dynamics
□ Projective Measurements
9. Symmetries in TNs:
□ Translational Invariance: iMPS and iPEPS
□ iDMRG and iTEBD algorithms
□ Abelian global symmetries
10. Long-range interactions: MPO vs TDVP
PRACTICAL INFO: As main reference, I have in mind lattice spin models. However, I will discuss about bosonic and fermionic systems; the latter being tricky due to the “counting” problem. In general,
I will try to give tips&tricks to implement the algorithms in C++. Many functions I will suggest to use are those of lapack. I will encourage Objected-Oriented programming.
FINAL EXAM: Implementation of an algorithm to solve an interesting many-body problem. Discussion about the implementation and results. | {"url":"https://www.statphys.sissa.it/wordpress/?page_id=7995","timestamp":"2024-11-09T07:03:51Z","content_type":"text/html","content_length":"17603","record_id":"<urn:uuid:fc6a4709-0831-46c7-9ea5-ed8b38bdc0c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00293.warc.gz"} |
Number Starters:
A Very Strange Game: Four different actions depending on the number which appears.
Abundant Buses: A game based around the concept of factors and abundant numbers.
Add 'em: Add up a sequence of consecutive numbers. Can you find a quick way to do it?
All The Nines: Add up all the multiples of nine in an elegant way.
Ancient Mysteries: This activity requires students to memorise fifteen numbers in a three by five grid.
Aunt Sophie's Post Office: Work out the number of stamps needed to post a parcel.
Birthday Clues: Work out the date Will was born by answering some number questions.
Can You Decide?: Recognise odd, even, square, prime and triangular numbers.
Christmas Bells: If all the bells ring together at noon, at what time will they next all ring together? This problem requires the use of LCM.
Christmas Eve: Is there a pattern in the number of palindromic numbers to be found less than powers of 10?
ClockEquate: Can you use the digits on the left of this clock along with any mathematical operations to equal the digits on the right?
Coins in Envelopes: Fifteen pennies are placed in four envelopes and the envelopes are sealed. It is possible to pay someone any amount from 1p to 15p by giving them one or more envelopes.
Consecutive Squares: What do you notice about the difference between the squares of consecutive numbers?
Dancing: Work out how many people were at the dance from the clues given.
Dice Nets: Determine whether the given nets would fold to produce a dice.
Dimidiate: Arrange the digits from 1 to 9 in alphabetical order. How many times can this number be halved?
Divided Age: How old is a person if when her age is divided by certain numbers, the calculator display ending are as shown.
Double Trouble: Begin with one, double it, double it again and so on. How many numbers in this sequence can you write down before the register has been called?
Factuples: Spot the factors and the multiples amongst the numbers in the grid.
Flabbergasted: If each number in a sequence must be a factor or multiple of the previous number what is the longest sequence that can be made from the given numbers?
Flowchart: Use the flowchart to generate a sequence of numbers. Which number will reach 1 the fastest?
Four Factors: Find four single digit numbers that multiply together to give 120. How many different ways are there of answering this question?
Four Problems: For mathematical questions to get everyone thinking at the beginning of the lesson.
Halve it: Start with 512. Halve it to get 256. Halve it to get 128. Continue as far as possible.
Handshakes: If all the students in this room shook hands with each other, how many handshakes would there be altogether?
Hot Numbers: Move the numbered cards to form five 2 digit numbers which are all multiples of three.
Hotel Digital: A puzzle about the lifts in a hotel which serve floors based on the day of the week.
House Numbers: The numbers on five houses next to each other add up to 70. What are those five numbers?
Inbetween Table: Write down as many multiples of 3.5 as possible in 3.5 minutes.
Last Day: The 31st of December is the last day of the year. What mathematical lasts do you know?
Leap Year: A question about the birthdays of a child born on the 29th February.
Letters in a Number: Questions about the number of letters in numbers.
Meta Products: Which numbers when multiplied by the number of letters in the word(s) of the number give square numbers?
Missing Terms: Find the missing terms from these linear sequences.
Name Again: Work out what the nth letter will be in a recurring pattern of letters in a person's name
Negative Numbers: Perform calculations involving negative numbers
No Partner: Find which numbers in a given list do not combine with other numbers on the list to make a given sum.
Number Riddles: Can you work out the numbers from the given clues.
Numbers in words: Write out in words some numbers writen as digits (optional pirate theme)
Odd One Out: From the numbers given, find the one that is the odd one out.
Only One Number: Find other numbers that can be changed to 1 on a calculator using only the 4 key and any operation.
Pears Make Squares: Find three numbers such that each pair of numbers adds up to a square number.
Perfect Numbers: Six is a perfect number as it is the sum of its factors. Can you find any other perfect numbers?
Plane Numbers: Arrange numbers on the plane shaped grid to produce the given totals
Pyramid Puzzle: Arrange numbers at the bottom of the pyramid which will give the largest total at the top.
Register: When the register is called answer with a multiple of 7.
Ropey Snowballs: Arrange the numbers on the snowballs so that no two consecutive numbers are directly connected by rope.
Satisfaction: Rearrange the numbers, row and column headings so that this table is mathematically correct.
Scaramouche: Can you work out from the five clues given what the mystery number is?
Seeing Squares: How many square numbers can be found in the grid of digits.
Sign Sequences: Continue the sequences if you can work out the rule.
Simple Nim: The classic game of Nim played with a group of pens and pencils. The game can be extended to the multi-pile version.
Small Satisfaction: Arrange the digits one to nine in the grid so that they obey the row and column headings.
Square and Even: Arrange the numbers on the cards so that each of the three digit numbers formed horizontally are square numbers and each of the three digit numbers formed vertically are even.
Square Angles: Find a trapezium, a triangle and a quadrilateral where all of the angles are square numbers.
Square Christmas Tree: Draw a picture of a Christmas tree using only square numbers.
Square Pairs: Arrange the numbered trees so that adjacent sums are square numbers.
Square Sequence: Write out as many square numbers as possible in 4 minutes.
Squigits: A challenge to find numbers which have each of their digits as square numbers.
The Power of Christmas: Find a power of 2 and a power of 3 that are consecutive numbers.
The story of ...: Be creative and come up with as many facts about a number as you can think of.
Tindice: How can you put the dice into the tins so that there is an odd number of dice in each tin?
Twelve Days: A Maths puzzle based on the 12 Days of Christmas song.
Two Numbers: Find the two numbers whose sum and product are given.
Upside Number: Work out the phone number from the clues given.
Venn Diagram: Arrange numbers on the Venn Diagram according to their properties.
What are they?: A starter about sums, products, differences, ratios, square and prime numbers.
Small images of these Starters :: Index of Starters
Number Advanced Starters:
Back To The Factory: Find all the numbers below 1000 which have exactly 20 factors
Barmy BIDMAS: A misleading way of stating the answer to a simple calculation.
Calendar Riddle: Work out the date of my birthday from the clues in rhyme.
Car Inequalities: Solve three simultaneous inequalities to find how many cars I own.
Cube Ages: Calculate the mean age of the two fathers and two sons with the given clues.
Difference Cipher: Find the mathematical word from the cipher
Divisible by 11: Can you prove that a three digit number whose first and third digits add up to the value of the second digit must be divisible by eleven?
HCF and LCM given: If given the HCF, LCM can you find the numbers?
Is it a Number?: A mathematical object is a number if it ...
Key Eleven: Prove that a four digit number constructed in a certain way will be a multiple of eleven.
Nine Digit Numbers: How many different nine digit numbers are their that contain each of the digits from one to nine?
Penny Bags: Can you place 63 pennies in bags in such a way that you can give away any amount of money (from 1p to 63p) by giving a selection of these prepacked bags?
Unlucky Seven Eleven: Follow the instructions to multiply a chosen number then explain the result you get.
Zero Even: Prove that zero is an even number.
Index of Advanced Starters
Counting Quest
Try this exercise to master counting up to 100 items with precision and confidence.
The short web address is:
Curriculum for Number:
Year 5
Pupils should be taught to read, write, order and compare numbers to at least 1 000 000 and determine the value of each digit more...
Pupils should be taught to know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers more...
Pupils should be taught to establish whether a number up to 100 is prime and recall prime numbers up to 19 more...
Pupils should be taught to read Roman numerals to 1000 (M) and recognise years written in Roman numerals more...
Year 6
Pupils should be taught to read, write, order and compare numbers up to 10 000 000 and determine the value of each digit more...
Pupils should be taught to solve number and practical problems that involve other recently learnt mathematical skills more...
Pupils should be taught to identify common factors, common multiples and prime numbers more...
Years 7 to 9
Pupils should be taught to use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime
factorisation, including using product notation and the unique factorisation property more...
Pupils should be taught to appreciate the infinite nature of the sets of integers, real and rational numbers. more...
International Baccalaureate
See the Number and Algebra sub-topics, syllabus statements, exam-style questions and learning resources for the IB AA course here.
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"An absolutely brilliant resource. Only recently been discovered but is used daily with all my classes. It is particularly useful when things can be saved for further use. Thank you!"
Comment recorded on the 19 October 'Starter of the Day' page by E Pollard, Huddersfield:
"I used this with my bottom set in year 9. To engage them I used their name and favorite football team (or pop group) instead of the school name. For homework, I asked each student to find a
definition for the key words they had been given (once they had fun trying to guess the answer) and they presented their findings to the rest of the class the following day. They felt really special
because the key words came from their own personal information."
Comment recorded on the 11 January 'Starter of the Day' page by S Johnson, The King John School:
"We recently had an afternoon on accelerated learning.This linked really well and prompted a discussion about learning styles and short term memory."
Comment recorded on the 9 April 'Starter of the Day' page by Jan, South Canterbury:
"Thank you for sharing such a great resource. I was about to try and get together a bank of starters but time is always required elsewhere, so thank you."
Comment recorded on the 5 April 'Starter of the Day' page by Mr Stoner, St George's College of Technology:
"This resource has made a great deal of difference to the standard of starters for all of our lessons. Thank you for being so creative and imaginative."
Comment recorded on the 3 October 'Starter of the Day' page by S Mirza, Park High School, Colne:
"Very good starters, help pupils settle very well in maths classroom."
Comment recorded on the 3 October 'Starter of the Day' page by Mrs Johnstone, 7Je:
"I think this is a brilliant website as all the students enjoy doing the puzzles and it is a brilliant way to start a lesson."
Comment recorded on the i asp?ID_Top 'Starter of the Day' page by Ros, Belize:
"A really awesome website! Teachers and students are learning in such a fun way! Keep it up..."
Comment recorded on the 24 May 'Starter of the Day' page by Ruth Seward, Hagley Park Sports College:
"Find the starters wonderful; students enjoy them and often want to use the idea generated by the starter in other parts of the lesson. Keep up the good work"
Comment recorded on the 7 December 'Starter of the Day' page by Cathryn Aldridge, Pells Primary:
"I use Starter of the Day as a registration and warm-up activity for my Year 6 class. The range of questioning provided is excellent as are some of the images.
I rate this site as a 5!"
Comment recorded on the 6 May 'Starter of the Day' page by Natalie, London:
"I am thankful for providing such wonderful starters. They are of immence help and the students enjoy them very much. These starters have saved my time and have made my lessons enjoyable."
Comment recorded on the 23 September 'Starter of the Day' page by Judy, Chatsmore CHS:
"This triangle starter is excellent. I have used it with all of my ks3 and ks4 classes and they are all totally focused when counting the triangles."
Comment recorded on the 14 September 'Starter of the Day' page by Trish Bailey, Kingstone School:
"This is a great memory aid which could be used for formulae or key facts etc - in any subject area. The PICTURE is such an aid to remembering where each number or group of numbers is - my pupils
love it!
Comment recorded on the 19 November 'Starter of the Day' page by Lesley Sewell, Ysgol Aberconwy, Wales:
"A Maths colleague introduced me to your web site and I love to use it. The questions are so varied I can use them with all of my classes, I even let year 13 have a go at some of them. I like being
able to access Starters for the whole month so I can use favourites with classes I see at different times of the week. Thanks."
Mount School York, Twitter
Wednesday, October 18, 2017
Mr Gray, Twitter
Wednesday, November 1, 2017 | {"url":"https://transum.org/Software/sw/Starter_of_the_day/Similar.asp?ID_Topic=27","timestamp":"2024-11-07T14:15:50Z","content_type":"text/html","content_length":"83690","record_id":"<urn:uuid:ae817882-5556-492e-b628-b88a52700c6c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00222.warc.gz"} |
Feature Column from the AMS
Colorful Mathematics: Part IV
3. Resolving conflicts (II)
We have seen how, using a graph-coloring model, we can assist in the scheduling of committees to avoid scheduling committee members in two places at once in the minimum number of time slots. Just as
in "theoretical" mathematics, applied mathematicians seek ways to generalize or specialize results that they have proven, building on success in solving a problem to try to solve similar problems in
related situations.
We seek situations where we can use a graph model to represent objects and join two of these objects when we want to "avoid a conflict." For example, the scheduling of examinations at high schools
and colleges. Our goal is to schedule exams so that no student has two exams at the same time. There are some practical considerations, such as sections of a course having a common final exam.
Graph-coloring models have proved to be valuable in practice here. Typically one might set in advance the number of time slots one wants to achieve and see if one can find a coloring of the conflict
graph with this number of colors. If no solution exists, one might try to find a coloring that minimizes the number of conflicts, in some precisely definable sense.
Other examples of the way that coloring the vertices of a graph can be used to schedule or resolve conflict are:
a. Scheduling the use of tracks by railroads
b. Assigning radio frequencies
When radio stations are geographically too close to each other and broadcast on the same or nearby frequencies they can interfere with each other. Draw a graph whose vertices are the stations and
join them with an edge if they are are within a certain distance. Coloring the vertices of the graph where the colors correspond to frequencies gives an assignment where two stations with the same
frequency will not interfere with each other.
c. Allocating resources
Suppose that there is a collection of basic tasks to perform which can be accomplished by allocating resources. The time it takes to perform a task is a fixed constant amount. Furthermore, we have
the list for each task T[i] of resources R[i] that need to be assigned to get the task done. We construct a graph as follows. For each task we create a vertex T[i] and we join two tasks T[i] and T[j]
by an edge whenever the lists R[i] and R[j] have at least one element in common. The reason for doing this is that we want to "tag" the fact that we can not do T[i] and T[j] at the same time because
they need overlapping resource allocations. A coloring of the graph assures that tasks which get the same color can be performed simultaneously because the resources for them are simultaneously
available. The shortest (total) time to get the tasks done will be accomplished when we have colored the vertices of the graph with the chromatic number of colors for the graph.
d. Organizing animals in zoos and pet stores
A pet store (zoo) wants to assign fish to aquaria (enclosures) so that fish (animals) that are not compatible (e.g. eat or attack each other) go into separate tanks (enclosures). For the pet store
setting, if we create a vertex of a graph for each species of fish and join two vertices if the species they represent are compatible, then finding the chromatic number of the resulting graph will
give the most "efficient" way to store the fish. | {"url":"https://www.ams.org/publicoutreach/feature-column/fcarc-colorapp3","timestamp":"2024-11-11T00:36:14Z","content_type":"text/html","content_length":"48894","record_id":"<urn:uuid:08f0bd54-146b-4454-99e5-3a468308b7ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00587.warc.gz"} |
Neural Machine Translation by Jointly Learning to Align and Translate (Sep 2014)
These are my notes from the paper Neural Machine Translation by Jointly Learning to Align and Translate (2014) by Dzmitry Bahdanau, Kyunghyun Cho, Yoshua Bengio.
This paper proposed an improvement to the RNN Encoder-Decoder ^1 network architecture, introducing an "attention mechanism" to the decoder, which significantly improved performance over longer
sentences. The concept of attention went on to become extremely influential in Machine Learning.
At the time, neural networks had emerged as a promising approach to machine translation, where researchers were aiming for an end-to-end translation model, in contrast to the state-of-the-art
statistical phrase-based translation methods, which involved many individually trained components. The RNN Encoder-Decoder approach would encode an input sentence into a fixed-length context vector;
a decoder would then output a translation using the context vector. The encoder and decoder are jointly trained on a dataset of text pairs, where the goal is to maximise the probability of the target
given the input.
However, this approach struggles with longer sentences, as the encoder has to drop information to compress it into a fixed-length context vector.
The authors proposed modifying the encoder to output a sequence with one hidden representation per input word, then adding a search mechanism to the decoder, allowing it to find the most relevant
information in the input sequence to predict each word in the output sequence.
They likened the modification to the human notion of "attention", calling it an Attention Mechanism. Though not the first Machine Learning paper to propose applying human-like attention to model
architectures ^2, this approach was very influential in NLP, leading to a lot of research eventually converging on an entirely attention-based architecture called the Transformer.
Architecture Details
The authors propose an RNNSearch model: an Encoder / Decoder model with an attention mechanism. For comparison, they train RNNencdec, which follows the standard RNN Encoder / Decoder architecture ^2
with the encoder returning a fixed-length context vector.
To demonstrate the ability to handle longer sequences, they train each model twice:
• First, with sentences of length up to 30 words: RNNencdec-30, RNNsearch-30
• Next, with sentences of size up to 50 words: RNNencdec-50, RNNsearch-50
For the RNN, they use a Bidirectional RNN: a Gated Recurrent Unit (GRU).
Each input token is fed into an embedding layer, $x_i$, and then a GRU encodes into a forward and backward "annotation" per token, concatenated to make a single representation, $h_i$.
The idea is to allow each annotation to summarise the preceding and the following words, providing the most possible representation for the attention mechanism.
Figure 1: The graphical illustration of the proposed model trying to generate the t-th target word $y_t$ given a source sentence ( $x_1, x_2, \ldots, x_T$ )
For the decoder, they use a uni-directional Gated Recurrent Unit.
The initial hidden state $s_0$ is computed as an initialisation layer, which comprises a linear layer followed by a tanh activation function.
$s_0 = \tanh \left( W_s \overleftarrow{h}_1 \right)$ where $W_s \in \mathbb{R}^{n \times n}$.
For each prediction step, they calculate the word probability as:
$p(y_i|y_1, \ldots, y_{i−1}, x) = g(y_{i−1}, s_i, c_i)$
• $y_{i-1}$ is the embedding of the token from the previous step.
• $s_i$ is the hidden state output from the previous layer.
• $c_i$ is the context vector.
The context vector, $c_i$ is calculated at each step as follows:
$c_i = \sum\limits_{j=1}^{T_x}\alpha_{ij}h_j$
The weights, $\alpha_{ij}$, are calculated by the alignment (Attention) model.
Alignment Model (Attention)
The alignment scores are calculated by combining a projection of the decoder's previous state and a projection of the encoder output, then applying tanh activation followed by a linear combination
with another weight vector.
$e_{ij} = v_a^{T} \tanh(W_as_{i-1} + U_{a}h_{j})$
This function gives us an output score for each token in the input sequence. Finally, we can perform a Softmax calculation to convert the weights to probability distribution:
$\sigma_{ij} = \frac{\exp(e_{ij})}{\sum_{k=1}^{T_x}\exp(e_{ij})}$
The final layer, which returns the probabilities for each word, uses a Maxout layer to generate the final probabilities. A Maxout layer acts as a form of regularisation. It projects the input vector
onto multiple "buckets" and selects the maximum value from each bucket. This process introduces non-linearity and helps prevent overfitting, akin to dropout.
Training Params
• Algorithm: Stochasic Gradient Descent (SGD)
• Optimiser: Adadelta (Zeiler, 2012)
• Batch Size: 80 sentences
• Training Duration: Approximately 5 days per model
• Inference method: Beam search
• Task: English-to-French translation
• Dataset: bilingual, parallel corpora provided by ACL WMT 14.
□ Word count: 850 (reduced to 348M)
□ Components:
☆ Europarl (61M words)
☆ News Commentary (5.5M)
☆ UN (421M)
☆ two crawled corpora of 90M and 272.5M words, respectively
• Metric: BLEU Score.
• Tokeniser: from open-source machine translation package Moses. They shortlist the most frequent 30k words and map everything else to [UNK].
• Comparisons: They compare RNNsearch with a standard RNN Encoder-Decoder, RNNenc and Moses, the state-of-the-art translation package.
• Test Set: For the test set, they evaluate news-test-2014 from WMT'14, which contains 3003 sentences not in training
• Valid Set: They concat news-test-2012 and news-test-2013.
• Initialisation: Orthogonal for recurrent weights, Gaussian ($0, 0.01^2$) for feed-forward weights, zeros for biases.
They record results on all test data with only examples that don't contain unknown tokens.
The RNNsearch-50 model achieved a BLEU score of 34.16 on sentences with unknown tokens excluded, significantly outperforming the RNNencdec-50 model, which scored 26.71 and training RNNsearch-50 to
convergence beat the state-of-the-art Moses. However, when unknown tokens are included, the model performs considerably worse.
RNNsearch was much better at longer sentences than RNNenc.
Figure 2: The BLEU scores of the generated translations on the test set with respect to the lengths of the sentences.
Model All No UNK
RNNencdec-30 13.93 24.19
RNNsearch-30 21.50 31.44
RNNencdec-50 17.82 26.71
RNNsearch-50 26.75 34.16
RNNsearch-50* 28.45 36.15
Moses 33.30 35.63
*Note: RNNsearch-50* was trained much longer until the performance on the development set stopped improving.
Interpreting Attention
One benefit of calculating attention weights for each output word is that they are interpretable, allowing us to visualise word alignments.
Figure 3. Four sample alignments that were found by RNNsearch-50.
As we can see, typically, words are aligned to similarly positioned words in a sentence, but not always.
1. Cho, K., van Merrienboer, B., Gulcehre, C., Bahdanau, D., Bougares, F., Schwenk, H., & Bengio, Y. (2014). Learning phrase representations using RNN encoder-decoder for statistical machine
translation. arXiv. https://arxiv.org/abs/1406.1078 ↩
2. Brauwers, G., & Frasincar, F. (2023). A general survey on attention mechanisms in deep learning. IEEE Transactions on Knowledge and Data Engineering, 35(4), 3279–3298. https://doi.org/10.1109/
tkde.2021.3126456 ↩↩ | {"url":"https://notesbylex.com/neural-machine-translation-by-jointly-learning-to-align-and-translate-sep-2014","timestamp":"2024-11-07T12:27:30Z","content_type":"text/html","content_length":"55155","record_id":"<urn:uuid:b13025e7-bdc4-4f30-981d-d572f8eb3b52>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00377.warc.gz"} |
The Egoroff Property and Related Properties in the Theory of Riesz Spaces
Holbrook, John Arthur Rankin (1965) The Egoroff Property and Related Properties in the Theory of Riesz Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BN4B-9460. https:/
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. A Riesz space L is said to be Egoroff, if, whenever [...] and [...], there is a
sequence [...] in L such that [...] and, for each n,m, there exists an index k(n,m) such that [...]. This notion was introduced, in rather a different form, by Nakano. Banach function spaces are
Egoroff, and Lorentz showed that, for any function seminorm [...], the maximal seminorm [...] among those which are dominated by [...] and which are [...] (a monotone seminorm [...] is [...] if
[...]) is precisely the "Lorentz seminorm" [...], where [...]. In this thesis the extent to which [...] holds in general Riesz spaces is determined. In fact, [...] for every monotone seminorm [...]
on a Riesz space L if, and only if, L is "almost-Egoroff". The almost-Egoroff property is closely related to the Egoroff property and, indeed, coincides with it in the case of Archimedean spaces.
Analogous theorems for Boolean algebras are discussed. The almost-Egoroff property is shown to yield a number of results which ensure that, under certain conditions, a monotone seminorm is [...] when
restricted to an appropriate super order dense ideal. Riesz spaces L possessing an integral, Riesz norm [...](i.e., a Riesz norm such that [...] are considered also, since in many cases these are
known to be Egoroff. In particular if [...] is normal on L (i.e., [...] a directed system, [...] ), then L is Egoroff. In this connection, a pathological space, possessing an integral Riesz norm
which is nowhere normal, is constructed.
Item Type: Thesis (Dissertation (Ph.D.))
Subject Keywords: (Mathematics)
Degree Grantor: California Institute of Technology
Division: Physics, Mathematics and Astronomy
Major Option: Mathematics
Thesis Availability: Public (worldwide access)
Research Advisor(s): • Luxemburg, W. A. J.
Thesis Committee: • Unknown, Unknown
Defense Date: 5 April 1965
Record Number: CaltechETD:etd-01142003-101748
Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-01142003-101748
DOI: 10.7907/BN4B-9460
Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code: 157
Collection: CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On: 15 Jan 2003
Last Modified: 08 Feb 2024 22:06
Thesis Files
PDF (Holbrook_jar_1965.pdf) - Final Version
See Usage Policy.
Repository Staff Only: item control page | {"url":"https://thesis.library.caltech.edu/157/","timestamp":"2024-11-01T23:27:23Z","content_type":"application/xhtml+xml","content_length":"26202","record_id":"<urn:uuid:96e72374-c4b7-4b5b-80b9-3e5e1cf64bfb>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00320.warc.gz"} |
Lifting the lid on quantum computing - Openforum
Lifting the lid on quantum computing
Andrew Trounson
| November 24, 2018
For many of us, the fast-evolving pace of technology means we are increasingly surrounded by one black box after another. We may have a rudimentary idea of how things work, but not enough to do
anything more than understand the instructions.
To really understand, you have to be able to open up the black box, actually see how it works and then put it back together. The latter point is one reason I leave my car engine well alone. The same
goes for my laptop – I’m happy to leave that to the techies and the coders.
But even if I was computer savvy, how am I supposed to get my head around the quantum computer revolution heading our way, when it’s impossible to look inside while it’s running?
One of the many quirks of quantum computing is that it relies on the strange interaction of atoms and subatomic particles, and then there’s the small matter that the whole fragile quantum state
collapses once you try and look at what’s actually going on.
Seeing quantum computing in action
“A quantum computer is the ultimate black box,” smiles quantum physicist Professor Lloyd Hollenberg who heads the University of Melbourne’s first ever course on quantum computer programming.
He’s smiling because even after just 15 minutes, he is pleased a dullard like me is starting to show some rudimentary understanding of how quantum computing works. And it’s all thanks to a quantum
computer simulator he and his colleagues have developed that basically lets you operate a quantum computer… with the lid off.
“To see how a quantum computer works you want to crack it open, but in the process, you collapse the quantum state, so what do you do? Our simulator was designed to solve that problem, and in terms
of its ease of use and what it tells you, it’s unique,” says Professor Hollenberg.
There are already opportunities for people to access online a few of the small-scale quantum computers that have so far been developed, but generally programmers will only get back the final output
from their coding – they won’t be able to ‘see’ how it works.
It’s this ability to see inside that Professor Hollenberg says is crucial to help students learn by actually doing, and for professionals to debug their quantum code.
Qubits and pieces
The simulator – the Quantum User Interface or QUI – is software that lets you click and drag logic instructions that operate on quantum bits (known as qubits) in order to write a program.
A remote cluster of computers at the University runs the program on a simulated quantum computer and sends back the results in real time so the user can inspect and visualise all aspects of the
quantum computer’s state at every stage in the program.
A qubit is simply the quantum version of a classical computer ‘bit’ – the basic unit of computer data that exists in one of two states, which in programming we know as 0 or 1. This is the basis of
all computer coding, and in a classical computer the 0s or 1s are usually represented by the different voltages that run through its transistor.
But in a quantum computer the bits, or qubits, are quantum objects, like an electron in an atom, which can be in one of two states that we can likewise label for convenience as 0 or 1.
What these quantum objects actually are varies across different quantum computer systems, but for the computer programmer that isn’t so important. What is important is the 0 and 1 generated by the
quantum objects enables us to use the objects for coding.
Getting past the weird physics
What’s different about qubits is that because of the weird physics that exists at the atomic scale, each qubit can be in an unresolved ‘quantum superposition’ state of 1 and 0. When observed (which,
remember, collapses the quantum state) each will have some probability of being 0 or 1, depending on how the quantum superposition was formed.
Qubits can also be made to influence each other through a property called entanglement so that if a qubit is resolved as 0, another might automatically be 1.
It is these peculiarities of quantum physics that promise to make quantum computers much more powerful than classical computers and, therefore, able to address difficult problems – like optimising
complex routes or systems from weather forecasting to finance, designing new materials, or aiding machine learning.
Unlike a classical computer that laboriously computes all possibilities before finding the right answer, a quantum computer uses the wave-like properties of data structures and numbers to narrow down
the probability of the correct answer for problems that, theoretically, our current computers have no hope of matching.
For the students accustomed to classic computer programming – it’s like learning from scratch.
New rules
“You have to think really differently with quantum programming,” says electrical engineering student Fenella McAndrew, who is doing the course.
“We’re basically going back to the start of the programming process, like working at the level of one circuit in conventional programming.”
And the rules of how numbers are processed in a quantum computer is, for the uninitiated, mind boggling.
“Teaching the material is a challenge, especially without a quantum mechanics background,” says Professor Hollenberg. “But there is a clear demand from students and professionals to learn more about
the technology and get themselves ‘quantum ready’.”
It was this need to make quantum computing more accessible to people with no physics background that was the genesis of developing QUI. The system allows programmers to see each phase of the
operation and exactly what the quantum computer is doing – in particular how the quantum data is being manipulated to produce the output of the program.
This is critical information for a would-be quantum programmer to understand.
For students grappling with quantum theory that even experts struggle to explain, it’s reassuring to get started using the QUI and actually see quantum computing at work.
“Quantum software design is such a new concept, and it feels more abstract than conventional computer coding, so being able to see what is happening when we design a function really helps,” says
Daniel Johnston, a maths student taking the course.
Solving problems instantly
Professor Hollenberg’s co-teacher, physicist Dr Charles Hill, says QUI means students are actually doing quantum computing themselves from the outset of the course.
“People learning quantum computing need to understand how bits of information are manipulated with the unique rules that are different from classic computing, and then write their programs in a way
that solves the problem they are looking at.
“We’ve found that the QUI is easy and intuitive to use. It takes the beginner no more than five minutes to get started with the system and writing programs,” Dr Hill says.
According to Professor Hollenberg, further versions of QUI are now in the works.
“We are working on improvements and add-ons which will not only enhance the user experience, but also the uptake of the system more broadly in teaching and research,” he says.
Budding quantum software programmers, or indeed anyone interested in quantum computing, can view the introductory video and try the system at QUIspace.org.
In addition to Professor Hollenberg and Dr Hill, the QUI development team comprised Aidan Dang, Alex Zable and Matt Davis; IT experts Dr Melissa Makin and Dr Uli Felzmann; and research students Sam
Tonetto, Gary Mooney and Greg White. This article was published by Pursuit.
Andrew Trounson is a senior journalist for the University of Melbourne’s Pursuit magazine. He was previously an education correspondent for The Australian. | {"url":"https://www.openforum.com.au/lifting-the-lid-on-quantum-computing/","timestamp":"2024-11-11T14:06:05Z","content_type":"application/xhtml+xml","content_length":"176981","record_id":"<urn:uuid:56c81296-9f82-4da7-ab3a-a4c44c81df1c>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00643.warc.gz"} |
What exactly is the kind "*" in Haskell?
In Haskell, the kind "*" represents the type of all ordinary values or data types. It is often referred to as the "type of types." When we write a type signature, like "Int" or "[Bool]", we are
referring to values of type Int or [Bool], which have the kind *.
The kind * is the base kind, or the kind of "base types." It is associated with concrete types that can have values. For example, the type Int has the kind * because it represents concrete values
like 1, 2, 3, etc.
Haskell also supports higher-kinded types, which have kinds other than . For example, the kind " -> " represents type constructors that take one type argument, like Maybe or [] (list). The kind " ->
* -> *" represents type constructors that take two type arguments, like Either or (,).
Overall, the kind "*" represents the type of all values in Haskell, and it is often used to refer to concrete types. | {"url":"https://devhubby.com/thread/what-exactly-is-the-kind-in-haskell","timestamp":"2024-11-07T16:19:01Z","content_type":"text/html","content_length":"108920","record_id":"<urn:uuid:3ae779a9-f2f1-4744-87b0-31e7e1610fec>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00120.warc.gz"} |
Metric Space
A metric space is a set with a global distance function that for every two of the set's points gives the distance between them as a nonnegative real number.
Metric space is a college-level concept that would be first encountered in a topology course.
Metric: A metric is a nonnegative function describing the distance between neighboring points for a given set.
Vector Space: A vector space is a set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space.
Classroom Articles on Topology (Up to College Level)
Closed Set Projective Plane
Dimension Projective Space
Homeomorphism Subspace
Möbius Strip Topological Space
Neighborhood Topology
Open Set Torus
Point-Set Topology | {"url":"https://mathworld.wolfram.com/classroom/MetricSpace.html","timestamp":"2024-11-12T18:50:15Z","content_type":"text/html","content_length":"47625","record_id":"<urn:uuid:f1f374ab-2c05-4ca8-bd91-b5c093eb4a6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00767.warc.gz"} |
timers – Chicken Bit
Timers are a usual feature/requirement in several applications of digital design. Fundamentally, timers are implemented as counters that increment/decrement at the rate of provided clock. It’s very
common to find a large number of different timers distributed throughout design, each ticking with base clock.
There’s a nice opportunity for area saving in such cases. The idea is very simple, let me illustrate with example:
Imagine, your clock’s base frequency is 4MHz (a period of 0.25us). In the design, three timers are required for 1us, 2us and 16us. If you were to implement these timers by clocking them at 4MHz, that
would need 14 (3+4+7) bits.
Here’s the idea, you divide the base 4 MHz clock down to 1 MHz, and then use divided clock for implementing timers. Now, you’ll need 8 (1+2+5) bits. Now of course, we would also need some additional
bits for clock-division (in this example 2 more)
The area savings can be significant if a large number of timers can be brought into this divided-clock domain, and especially if the base frequency is high. Also, consider the power savings from
reduced switching activity…So whenever possible, it can be a good practice to try and identify as many timers as possible that have the potential to be moved under a shared divided clock domain. | {"url":"https://chickenbit.net/tag/timers/","timestamp":"2024-11-06T23:29:26Z","content_type":"text/html","content_length":"16485","record_id":"<urn:uuid:e16b9edf-79f9-41eb-903f-a84984891966>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00144.warc.gz"} |
American Mathematical Society
Dimension and Trace of the Kauffman Bracket Skein Algebra
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by Charles Frohman, Joanna Kania-Bartoszynska and Thang Lê;
Trans. Amer. Math. Soc. Ser. B 8 (2021), 510-547
DOI: https://doi.org/10.1090/btran/69
Published electronically: July 7, 2021
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Let $F$ be a finite type surface and $\zeta$ a complex root of unity. The Kauffman bracket skein algebra $K_\zeta (F)$ is an important object in both classical and quantum topology as it has
relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of $K_\zeta (F)$
over its center, and we extend a theorem of the first and second authors in [Math. Z. 289 (2018), pp. 889–920] which says the skein algebra has a splitting coming from two pants decompositions of
$F$. Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society, Series B with MSC (2020): 57K31
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Bibliographic Information
• Charles Frohman
• Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa
• MR Author ID: 234056
• ORCID: 0000-0003-0202-5351
• Email: charles-frohman@uiowa.edu
• Joanna Kania-Bartoszynska
• Affiliation: Division of Mathematical Sciences, The National Science Foundation, Alexandria, Virginia
• MR Author ID: 239347
• Email: jkaniaba@nsf.gov
• Thang Lê
• Affiliation: Department of Mathematics, Georgia Institute of Technology, Atlanta, Georgia
• ORCID: 0000-0003-4551-0285
• Email: letu@math.gatech.edu
• Received by editor(s): February 5, 2019
• Received by editor(s) in revised form: January 15, 2021
• Published electronically: July 7, 2021
• Additional Notes: This material is based upon work supported by and while serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
• © Copyright 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
• Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 510-547
• MSC (2020): Primary 57K31
• DOI: https://doi.org/10.1090/btran/69
• MathSciNet review: 4282692 | {"url":"https://www.ams.org/journals/btran/2021-08-18/S2330-0000-2021-00069-2/?active=current","timestamp":"2024-11-09T23:16:33Z","content_type":"text/html","content_length":"84866","record_id":"<urn:uuid:4a9edf65-7785-48a8-9d00-d3e1a84aeb2f>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00816.warc.gz"} |
The Taub Faculty of Computer Science
The Taub Faculty of Computer Science Events and Talks
Merav Parter (Weizmann Institute of Science)
Wednesday, 18.03.2015, 12:30
We study the topological properties of wireless communication maps and their usability in algorithmic design. We consider the SINR model, which compares the received power of a signal at a receiver
against the sum of strengths of other interfering signals plus background noise. To model the reception regions, we use the convenient representation of an \emph{SINR diagram}, introduced in \cite
{Avin10journal}, which partitions the plane into $n$ reception zones, one per station, and a complementary region where no station can be heard. The topology and geometry of SINR diagrams was studied
in Avin et al. in the relatively simple setting of {\em uniform power}, where all stations transmit with the same power level. It was shown therein that uniform SINR diagrams assume a particularly
convenient form: the reception zone of each station is convex, fat and strictly contained inside the corresponding Voronoi cell.
Here we consider the more general (and common) case where transmission energies are arbitrary (or non-uniform). Under that setting, the reception zones are not necessarily convex or even connected.
This poses the algorithmic challenge of designing efficient algorithmic techniques for the non-uniform setting, as well as the theoretical challenge of understanding the geometry of SINR diagrams
(e.g., the maximal number of connected components they might have). We achieve several results in both directions. One of our key results concerns the behavior of a $(d+1)$-dimensional map, i.e., a
map in one dimension higher than the dimension in which stations are embedded.
Specifically, although the $d$-dimensional map might be highly fractured, drawing the map in one dimension higher ``heals'' the zones, which become connected (in fact hyperbolically connected). In
addition, we establish the {\em minimum principle} for the SINR function, and utilize it as a discretization technique for solving two-dimensional problems in the SINR model. This approach is shown
to be useful for handling optimization problems over two dimensions (e.g., power control, energy minimization); in providing tight bounds on the number of null-cells in the reception map; and in
approximating geometrical and topological properties of the wireless reception map (e.g., maximum inscribed sphere). Essentially, the minimum principle allows us to reduce the dimension of the
optimization domain \emph{without} losing anything in the accuracy or quality of the solution.
Joint work with Erez Kantor, Zvi Lotker and David Peleg. | {"url":"https://cs.technion.ac.il/events/view-event.php?evid=2229","timestamp":"2024-11-07T06:52:51Z","content_type":"text/html","content_length":"16478","record_id":"<urn:uuid:921c13c5-b37f-41b6-bc43-bd3043de1b74>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00427.warc.gz"} |
IntroductionRecent approaches for optimizing the parameters of antennasThe Proposed MethodologyData PreprocessingDescription of the metamaterial dataset featuresDistribution of bandwidth feature in the metamaterial datasetCorrelation matric of features in the metamaterial antenna datasetLSTM neural network architectureAdaptive Dynamic TechniqueResultsPerformance evaluation metricsThe values of the configuration parameters of the adaptive PSO-guided WOA algorithmThe updating of the groups of exploitation and exploration using AD-PSO-Guided WOA algorithm [22]Green color refers to actual values, and red color refers to predicted values using the proposed modelHeat map and QQ plot of the prediction results using the proposed modelResidual and homoscedasticity plots of the prediction results using the proposed modelROC and RMSE for the prediction results of the proposed modelHistogram of the RMSE values measured from the bandwidth predictionsComparison of the proposed model and the other traditional modelsResults of the ANOVA test for predicting the antenna bandwidth using the proposed modelThe descriptive statistics of the proposed model compared with the other modelsWilcoxon signed-rank test of bandwidth prediction results using the proposed modelConclusionReferences
CMC CMC CMC Computers, Materials & Continua 1546-2226 1546-2218 Tech Science Press USA 28550 10.32604/cmc.2022.028550 Article Improved Prediction of Metamaterial Antenna Bandwidth Using Adaptive
Optimization of LSTM Improved Prediction of Metamaterial Antenna Bandwidth Using Adaptive Optimization of LSTM Improved Prediction of Metamaterial Antenna Bandwidth Using Adaptive Optimization of
LSTM KhafagaDoaa Sami1 AlhussanAmel Ali1AAAlHussan@pnu.edu.sa El-kenawyEl-Sayed M.2 3 IbrahimAbdelhameed4 Abd ElkhalikSaid H.3 El-MashadShady Y.5 AbdelhamidAbdelaziz A.6 7 Department of Computer
Sciences, College of Computer and Information Sciences, Princess Nourah Bint Abdulrahman University, Riyadh, 11671, Saudi Arabia Department of Communications and Electronics, Delta Higher Institute
of Engineering and Technology, Mansoura, 35111, Egypt Faculty of Artificial Intelligence, Delta University for Science and Technology, Mansoura, 35712, Egypt Computer Engineering and Control Systems
Department, Faculty of Engineering, Mansoura University, Mansoura, 35516, Egypt Department of Computer Systems Engineering, Faculty of Engineering at Shoubra, Benha University, Egypt Department of
Computer Science, Faculty of Computer and Information Sciences, Ain Shams University, Cairo,11566, Egypt Department of Computer Science, College of Computing and Information Technology, Shaqra
University, 11961,Saudi Arabia Corresponding Author: Amel Ali Alhussan. Email: AAAlHussan@pnu.edu.sa 16 05 2022 73 1 865 881 1222022 1432022 © 2022 Khafaga et al. 2022 Khafaga et al. This work is
licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The design of an antenna requires a careful selection of its parameters to retain the desired performance. However, this task is time-consuming when the traditional approaches are employed, which
represents a significant challenge. On the other hand, machine learning presents an effective solution to this challenge through a set of regression models that can robustly assist antenna designers
to find out the best set of design parameters to achieve the intended performance. In this paper, we propose a novel approach for accurately predicting the bandwidth of metamaterial antenna. The
proposed approach is based on employing the recently emerged guided whale optimization algorithm using adaptive particle swarm optimization to optimize the parameters of the long-short-term memory
(LSTM) deep network. This optimized network is used to retrieve the metamaterial bandwidth given a set of features. In addition, the superiority of the proposed approach is examined in terms of a
comparison with the traditional multilayer perceptron (ML), K-nearest neighbors (K-NN), and the basic LSTM in terms of several evaluation criteria such as root mean square error (RMSE), mean absolute
error (MAE), and mean bias error (MBE). Experimental results show that the proposed approach could achieve RMSE of (0.003018), MAE of (0.001871), and MBE of (0.000205). These values are better than
those of the other competing models.
Metamaterial antenna long short term memory (LSTM) guided whale optimization algorithm (Guided WOA) adaptive dynamic particle swarm algorithm (AD-PSO)
Machine learning is an active research field that is widely used in data analysis for numerous domains. On the other hand, the design of antennas becomes complex every day; which encourages antenna
designers to benefit from the powerful capabilities of machine learning to build reliable models to be used for intelligent and fast optimization of antenna designs. Optimization techniques based on
machine learning algorithms; such as particle swarm optimization and genetic algorithms are used recently to optimize the structure of antennas while retaining their significant target performance.
These optimization techniques are essential especially when the structure of the antenna is complex. In this case, the machine learning algorithms are effective in discovering the best antenna design
parameters. In addition, the tools used in the simulation and analysis of electromagnetic (EM), which can be used to measure the fitness of antennas, require a lot of effort and time by antenna
designers to get the proper results from the optimization process performed by these tools. The solution to this bottleneck is offered by machine learning that can be used to overcome the limitations
of these simulation tools. The advantage of machine learning is embedded in its ability to provide approximate and proper antenna structures without the need to use EM simulations [1,2].
Machine learning techniques; such as K-nearest neighbors (KNN), artificial neural network (ANN), and least absolute shrinkage and selection operator (LASSO) were used recently in integrated and
unified frameworks to optimize the parameters of antennas and to discover their optimal design parameters [3]. One of the key benefits of machine learning is the reduction of the computation time
observed in EM approaches. This benefit is obvious when optimizing several parameters or when it is needed to build a large model structure. The geometries of microstrip antennas, particularly
complicated geometrical antennas, or the new model structures are still challenging to be handled efficiently using the established theories of antennas. This can be shown when given some of these
geometries have low accuracy. In real-time, machine learning may be used to model and forecast scattering problems as well as evaluate and improve antennas performance. In the literature, many
research efforts employed machine learning models for optimizing the parameters of antennas. The most commonly used model to perform this optimization is the neural network. However, the antenna
structure determines the number of geometrical parameters needed in the optimization process, and as this number increases, it becomes hard for the model to derive a better relationship among these
parameters [4].
Many antennas are designed with variable performance and usage. The rectangular antenna is considered the simplest design that can be simply designed based on machine learning. The relevant
parameters are the length and width of the rectangular patch, the dielectric, the height of the substrate, and the resonant frequency. A circular patch antenna is another essential type of antenna.
The parameters of this design are similar to the rectangular patch antenna but with a radius of the patch antenna instead of the length and width of the rectangular type. An elliptical patch antenna
is a special design of a circular antenna, where the shape is an ellipse. Similar parameters to the circular antenna are specified to be optimized using machine learning. In addition, in a fractal
antenna, the perimeter (outer structure or inner sections of antenna) can be increased, and the effective length can be maximized based on the self-similar design structures/fractals of the material
that can transmit or receive electromagnetic radiation within a given volume or a total surface area. This type of antenna consists of several parameters to be optimized, requiring a suitable dataset
with enough samples to predict some of these parameters efficiently. Another type of microstrip antenna is the monopole antenna. The class of monopole antennas consists of a conductor in the form of
a straight rod shape. In addition, there is a type of conductive surface, namely, a ground plane used to hold the conductor perpendicularly. When this antenna works as a receiver, the output signal
is taken between the ground plane and the lower end of the monopole. Moreover, to the lower end of the monopole, one side of a feedline is attached to the ground plane, and the other side is attached
to the antenna. There are other types of microstrip antennas that are designed with particular configurations. These types include substrate integrated waveguide, planar inverted-F antenna,
reflect-array, and other special patch designs. Each type of these design has a set of configuration parameters that can be optimized and adjusted using machine learning [5].
One of the main outcomes of the field of computational electromagnetics is known as metamaterial antenna. This type of antenna has a significant advantage which is the possibility of designing this
type of antenna using computations and optimization methods without relying on the modeling of its broader type [6]. The field of study of metamaterial antenna is based on customizing the light
molecule interactions using complex geometric shapes. However, the other types of conventional antennas use Maxwell’s equations in their inherent design. Therefore, the metamaterial antenna is the
best antenna that can benefit from the current advances in machine learning. The material of metamaterial antenna has special properties; which can not be regenerated using the natural material. This
type of material is used in many other fields; such as microwave components, revolutionary electronics, and compact antennas. The design of antennas is considered the most important application of
this material [7].
Recently, many optimization approaches have emerged in the literature with promising performance [8–11]. These methods are necessary for the tasks where the optimization process grows with time and
is thus complex. When the conventional techniques cannot solve complex problems, the metaheuristic approaches are the best choice to handle these cases. The metaheuristic approaches are based on the
idea of starting with a random population then selecting the best candidate solution to be passed to the next generation. This dynamic nature of metaheuristic algorithms makes it a proper choice for
optimizing the parameters of metamaterial antenna.
Tab. 1 presents some of the key achievements in the literature in which machine learning approaches are employed for optimizing the parameters of antennas. As shown in the table, it can be noted that
the most common method used for parameter optimization is neural networks in the form of MLP or SVR that can be harnessed with other approaches in ensembles. In addition, most of the parameters
targeted by the parameter optimization are the resonance frequency given the geometrical information of the antenna as an input. Moreover, the table shows the achieved results of each approach in
optimizing the target parameters.
Paper Approach Input Output Results
[12] RBF + MLP, resilient Length and width, and substrate’s dielectric constant r and Resonant frequency Error=3.5×10^–14
backpropagation thickness
[13] SVR + kernel configurations Height and width Resonant frequency Error=3dB
[14] Gaussian kernel + SVR Length and width Voltage standing wave ratio (VSWR), gain, and resonant NA
[15] SVR Height and length Input impedance Rn, bandwidth (BW), and resonant Error are 1.21% for RF, 2.15% for BW, and
frequency RF. 0.2% for Rn
[16] RBF Permittivity, height, and resonance frequency Width and length of the patch Error=0.91%
[17] MLP Permittivity, height and resonance frequency Width and length of the patch Error=3.47%
Metaheuristic algorithms are capable of handling problems with unexpected behavior. These algorithms are distinguished by their intelligence which is based on the random search methodology. In
addition, these algorithms can efficiently avoid local perfections as usually referred to as simple and flexible approaches. This type of algorithm has two main processes namely exploitation and
exploration that are running while searching the population for the optimal solution. In the exploration process, the search space is examined thoroughly to find the best solution. In recent decades,
many global optimization approaches were developed based on observing natural objects in the real life. One of these global optimization approaches is known as population-based metaheuristic
algorithms that can be used in a variety of situations. Two types of metaheuristic approaches are widely spread in the field namely metaphor and non-metaphor base approaches. The metaphor approach is
based on observing human behavior or a natural phenomenon in its inherent details [18].
In this paper, we propose a novel approach for predicting the bandwidth of the metamaterial antenna based on optimized long short-term memory (LSTM). The optimization of LSTM is performed in terms of
the recently emerged algorithm referred to as adaptive dynamic particle swarm algorithm (AD-PSO) with a guided whale optimization algorithm (Guided WOA) [19,20]. The resulting optimized LSTM with
optimum hyperparameters are used to robustly predict the bandwidth of the metamaterial antenna.
The rest of this paper is organized as follows. The description of the dataset along with a detailed discussion of the proposed methodology are explained in Section 2. The achieved results are
comparisons are presented and investigated in Section 3. Then, the conclusions are examined in Section 4.
The proposed methodology used to improve the prediction of the metamaterial antenna bandwidth is discussed in this section. The section starts with presenting the dataset employed in this research,
followed by discussing the main components of the proposed approach.
The metamaterial dataset used in this work contains eleven features, which are described in Tab. 2. Using Kaggle [21], we were able to access the dataset. There are 572 records in this collection,
which is rather large. Each record in the dataset represents the optimum design of a metamaterial antenna. The columns of this dataset represent the features that describe the configurations of the
antenna. The bandwidth of the antenna is adopted as the target feature that we need to robustly predict given the other features. A short description of these features is presented in Tab. 2. To get
a better understanding of these features, Fig. 1 shows the distribution of the bandwidth from the dataset records. As shown in the figure, the majority of samples in the dataset have a bandwidth in
the range from 0.9 to 1.0. This means that the changes in the bandwidth among the samples are very small; which requires a robust predictor to predict the bandwidth value of a given set of parameters
No. Feature Description
1 S11 Return loss
2 Bandwidth The bandwidth of the antenna
3 VSWR Standing wave ration voltage of the antenna
4 Gain Antenna Gain
5 Ya Split ring resonator cell distance
6 Xa Array-patch antenna distance
7 SRR_num Number of cells in the split ring resonator
8 Tm Rings’ width
9 Dm Rings’ distance
10 W0m Rings’ gap
11 Wm Dimensions of the split ring resonator
As the dataset is collected from electromagnetic simulation tools, it has been noted that some samples were recorded in the dataset with missing values. To deal with the missing values (denoted by
null in the dataset) in a certain feature column, we replaced these values with the average of the surrounding not-null values in the same column. In addition, it is noted that the values of the
features in the dataset do not lay in the same range; which might affect the performance of the predictor. Therefore, we applied the min-max normalization. After applying this step, each feature in
the dataset has the lowest value of zero and the greatest value of one. Fig. 2 depicts the correlation matrix of the features of the dataset after dealing with the null values and normalizing the
feature. As shown in the figure, the values of Tm and Wm are highly correlated with the metamaterial antenna bandwidth.
As explained in [22], the LSTM model is an improvement on the ANN model and may be used to solve a variety of issues. Long-term memory retention is one of LSTM’s key advantages. Fig. 3 depicts the
LSTM design. In the LSTM model, the initial step is to determine which data from the cell state should be disregarded.
After the optimization process has been initialized, and for each solution in the population, a fitness value is calculated. To get the highest possible fitness value, the optimization algorithm
selects the optimum agent for the situation (solution) [23]. To initiate the adaptive dynamic process, the optimization algorithm divides the population of agents into two groups, which are referred
to as the exploitation group and the exploration group. Folks in the exploitation group are mostly concerned with moving toward the ideal or best answer, while individuals in the exploration group
are primarily concerned with searching the region surrounding the leaders. The exchange of information (update) between the agents of the various population groupings occurs dynamically. The
optimization method is started with a (50/50) population to establish a balance between the exploitation group and the exploration group. The process is then iterated over and over again until it
finds the best or optimum solution [24].
The Guided WOA refers to a modified version of the original WOA algorithm [25], which is used in this research. By changing the search strategy via a single agent, the guideline-based WOA overcomes
one of the disadvantages of the original WOA. Using a modified algorithm, the agents are directed toward prey or the optimal solution depending on the input from more than one agent. In the original
WOA algorithm, agents are forced to roam randomly around each other to get the results of the global search. Agents are forced to follow three random agents instead of just one when they are using
the Guided WOA algorithm, which results in a more efficient exploration process. More exploration may be obtained by requiring agents not to be impacted by a single leader position. The updating
positions mechanism of the algorithm three random solutions of Xo1,Xo2 and Xo3. These solutions are updated every iteration to enhance the algorithm performance and get the optimal solution. X(t+1)=
A variety of swarming patterns in nature, such as those shown by birds, are represented by the PSO algorithm, which replicates their social behavior. By shifting their places, the agents in the PSO
algorithm look for the optimal solution or meal based on the most recent velocity. The following algorithm shows the detailed steps of the AD-PSO-Guided WOA algorithm employed in this research.
A set of experiments were conducted to evaluate the proposed model and compare it with the other competing models in the literature. These models are the Multilayer Perceptron (MLP), K-Nearest
Neighbors (KNN), LSTM, in addition to the proposed AD-PSO-Guided WOA for the LSTM model which is used to select the optimal hyperparameters value of the LSTM network for predicting the bandwidth of
metamaterial antenna. The evaluation of these models is based on the criteria listed in Tab. 3. These metrics are the mean absolute percentage error (MAPE), the root mean square error (RMSE), the
relative root mean square error (RRMSE), and Pearson’s correlation coefficient (R). In addition, the modified agreement index (d) was employed to determine agreement (WI), where M is the number of
observations in the subset; Ym^ and Ym are the mth estimated and observed PV power values, and Ym^¯ and Ym¯ are the arithmetic means of the estimated and observed values.
Metric Value
MAPE 1M∑m=1M|Ym^−YmYm|×100
RMSE 1M∑m=1M[Ym^−Ym]2
RRMSE RMSEYm¯×100
R ∑m=1M(Ym^−Ym^¯)(Ym−Ym¯)[∑m=1M(Ym^−Ym^¯)2][∑m=1M(Ym−Ym¯)2]
WI 1−∑m=1M|Ym^−Ym|∑m=1M|Ym−Ym¯|+|Ym^−Ym^¯|,0<d≤1
It was necessary to utilize around 80 percent of the dataset obtained at each site for training, with the remaining 20 percent being used to test for the dependability of the models that had been
constructed. A uniform distribution was used to provide random sampling and assignment of data to the two subgroups, rather than using a chronological partitioning method. This is done to lessen the
reliance of the produced models on the particular data that was used in the fitting process and to guarantee that the models behave in the same manner when dealinsg with other datasets. In addition,
the calculation of validation metrics is performed to present non-dimensional error estimates.
After preparing the training and testing sets, a set of values were selected for the configuration parameters of the adaptive PSO-guided WOA algorithm to train the LSTM model. The settings of these
values are presented in Tab. 4. As shown in the table, the initial parameters are chosen as follows. The maximum number of iterations is set to 20 and the number of runs is chosen as 20 for the
training set. The initial size of the population is selected as 20. In addition, the values of Wmin and Wmax that represent the main parameters of the PSO algorithm are set to 0.6 and 0.9. Moreover,
the values of α and β as assigned to 0.99 and 0.1, respectively.
Parameter Value
Number of iterations 50
Number of runs 20
Number of whales 20
Dimension Number of features
Inertia Wmin, Wmax [0.6, 0.9]
Acceleration constants C1, C2 [2, 2]
α for Fn 0.99
β for Fn 0.01
The parameters listed in Tab. 4 are used in the running of the optimization process of the LSTM parameters. The progress of the training process follows the behavior depicted in Fig. 4. In this
figure, the model optimization reaches its best values near the 20^th iteration. During the optimization process, the exploration and exploitation operations vary depending on the recorded optimized
values until reaching the saturation level, then the recorded values of the optimized LSTM parameters are saved to be used in predicting the metamaterial antenna bandwidth.
The prediction results of the metamaterial antenna bandwidth using the proposed optimized model are shown in Fig. 5. In this figure, the actual values of the bandwidth are shown in green color and
the predicted values are presented in red color. As shown in the figure, the predicted values are too close to the actual values, which indicates the effectiveness of the proposed model in accurately
predicting the bandwidth of metamaterial antenna.
Fig. 6 shows the heat map and quantile-quantile (QQ) plot of the prediction results. As shown in the figure, the points of the QQ plot are distributed in a way that fits the diagonal line between the
predicted and the actual residuals. This fitting confirms the performance of the optimized LSTM model in predicting the bandwidth of the metamaterial antenna.
The plots depicted in Fig. 7 show the residuals and the homoscedasticity of the prediction results using the proposed optimized LSTM model. As shown in the figure, the values of the residual error
are close to zero which confirms the effectiveness of the proposed approach. In addition, a homoscedasticity plot is useful to show if the independent variables have the same error term, which is
confirmed from the plot values.
Moreover, the plots of Fig. 8 show the receiver operating characteristics (ROC) of the proposed optimized LSTM. The figure shows also the RMSE values of the proposed LSTM and the other competing
models. As shown in the figure, the area under the curve is high for the proposed model with the minimum value of RMSE. These values emphasize the superiority of the proposed approach.
Fig. 9 depicts the histogram of the RMSE values resulting from the prediction of the bandwidth using the proposed model and the other three competing approaches. In this figure, the RMSE values
approximately follow a straight line with a few peaks, which reflects the stability of the proposed approach in predicting the bandwidth of the metamaterial antenna.
The evaluation criteria listed in Tab. 2 are used to measure the performance of the proposed approach along with the other competing approaches. Tab. 5 presents the evaluations of these criteria
based on the metamaterial antenna test set. As shown in the table, the results achieved by the proposed model are RMSE of (0.003108), MAE of (0.001871), MBE of 0.000205), and R of (0.999888). These
values outperform those of the other competing approaches (MLP, KNN, and LSTM). More evaluation criteria are presented in the table with their measured values which are also much better than the
values achieved by the other models.
MLP KNN LSTM Proposed model
RMSE 0.051860371 0.027929868 0.020169502 0.003108
MAE 0.045904674 0.023111993 0.015496072 0.001871
MBE −0.042390451 0.000227761 0.004493794 0.000205
R 0.991067787 0.992886219 0.995676454 0.999888
R^2 0.982215359 0.985823043 0.991371601 0.999776
RRMSE 8.723646595 4.698198039 11.57581043 0.522864
NSE 0.936988554 0.981723798 0.990468998 0.999774
WI 0.873977428 0.93655041 0.957458475 0.994862
The ANOVA test, using the prediction results achieved by the proposed model, is presented in Tab. 6. The values presented in these tables indicate the stability of the proposed approach. In addition,
Tab. 7 shows the descriptive statistics of the proposed approach over 20 runs. The calculated values of these statistics using the proposed approach is promising when compared with the corresponding
values calculated from the results achieved by the other approaches, as shown in the table. These values confirm the efficiency and superiority of the proposed model in predicting the bandwidth of
metamaterial antenna.
ANOVA table SS DF MS F (DFn, DFd) P value
Treatment (between columns) 0.0177 3 0.005901 F (3, 52) = 203.5 P<0.0001
Residual (within columns) 0.001508 52 0.000029
Total 0.01921 55
Descriptive statistics MLP KNN LSTM Proposed model
Number of values 14 14 14 14
Minimum 0.03186 0.02093 0.01817 0.003071
25% Percentile 0.05186 0.02793 0.02017 0.003108
Median 0.05186 0.02793 0.02017 0.003108
75% Percentile 0.05186 0.02793 0.02017 0.003108
Maximum 0.08186 0.03793 0.02817 0.003138
Range 0.05 0.017 0.01 6.72E−05
10% Percentile 0.04186 0.02443 0.01917 0.003081
90% Percentile 0.06686 0.03361 0.02417 0.003128
95% CI of median
Actual confidence level 98.71% 98.71% 98.71% 98.71%
Lower confidence limit 0.05186 0.02793 0.02017 0.003108
Upper confidence limit 0.05186 0.02793 0.02017 0.003108
Mean 0.05257 0.02824 0.02060 0.003107
Std. Deviation 0.00997 0.00339 0.00224 1.44E−05
Std. Error of Mean 0.002665 0.000906 0.0006 3.84E−06
Lower 95% CI of mean 0.04682 0.02628 0.0193 0.003099
Upper 95% CI of mean 0.05833 0.0302 0.02189 0.003116
Coefficient of variation 18.97% 12.01% 10.89% 0.4619%
Geometric mean 0.05175 0.02806 0.0205 0.003107
Geometric SD factor 1.203 1.124 1.1 1.005
Lower 95% CI of geo. mean 0.0465 0.02622 0.0194 0.003099
Upper 95% CI of geo. mean 0.05758 0.03003 0.02167 0.003116
Harmonic mean 0.05091 0.02788 0.02042 0.003107
Lower 95% CI of harm. mean 0.04571 0.0261 0.01947 0.003099
Upper 95% CI of harm. mean 0.05744 0.02993 0.02147 0.003116
Quadratic mean 0.05345 0.02843 0.02071 0.003107
Lower 95% CI of quad. mean 0.04656 0.02627 0.01916 0.003099
Upper 95% CI of quad. mean 0.05954 0.03044 0.02215 0.003116
Skewness 1.468 1.195 3.329 −0.6237
Kurtosis 7.539 6.955 12.21 4.044
Sum 0.736 0.3954 0.2884 0.0435
The Wilcoxon signed-rank test is performed on the metamaterial test set and the results are recorded in Tab. 8. As shown in the table, the proposed approach achieves the lowest discrepancy values
over the test set when compared with the other methods. This proves the stability of the proposed approach in predicting the bandwidth of the metamaterial antenna.
MLP KNN LSTM Proposed LSTM
Theoretical mean 0 0 0 0
Actual mean 0.05257 0.02824 0.0206 0.003107
Number of values 14 14 14 14
One sample t-test
t, df t=19.73, df=13 t=31.16, df=13 t=34.35, df=13 t=810.1, df=13
P-value (two tailed) <0.0001 <0.0001 <0.0001 <0.0001
P-value summary **** **** **** ****
Significant (alpha=0.05)? Yes Yes Yes Yes
How big is the discrepancy?
Discrepancy 0.05257 0.02824 0.0206 0.003107
SD of discrepancy 0.009972 0.003391 0.002243 0.00001435
SEM of discrepancy 0.002665 0.000906 0.0005996 0.000003836
95% confidence interval 0.0468:0.0583 0.0263:0.0302 0.0193:0.0219 0.00310:0.00312
R squared (partial eta squared) 0.9677 0.9868 0.9891 1
Machine learning is considered the backbone of the ongoing research efforts which contributes significantly to many fields of today’s technology. The choice of a machine learning model usually
affects the accuracy of the predictions of the task results. In this paper, we adopted the application of the recently emerged optimization algorithm, which is referred to as Adaptive PSO-Guided WOA,
to search for the best parameters of the LSTM deep network. This network is then used to predict the bandwidth of the metamaterial antenna. The choice of metamaterial antenna in this work was due to
its capability to overcome the bandwidth constraints imposed by tiny sizes of antennas. On the other hand, the interesting features and capabilities of deep learning allow it to be widely used in
almost all fields of science. LSTM is one of the most significant types of deep networks which is optimized using the adaptive guided whale algorithm to fit the task of predicting the bandwidth. To
emphasize the superiority of the proposed approach, other competing models were incorporated in the conducted experiments. The findings of this work indicate that the prediction accuracy using the
proposed approach outperforms the standard LSTM, MLP, and KNN models.
Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R308), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Funding Statement: Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R308), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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