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Suppose a coin is tossed 3 times. What is the probability of obtaining exactly one head?
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P(One head exactly) $=\dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{3}{8}$
The first $\dfrac{1}{2}$ is the probability of getting a head. The other two halfs are the probability of NOT getting a head. The 3 is because the head could appear on either the first OR second OR third throw.
by Diamond (58,513 points)
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# TensorFlow documentation style guide
## Best practices
• Focus on user intent and audience.
• Use every-day words, and keep sentences short.
• Use consistent sentence construction, wording, and capitalization.
• Make your article easy to scan.
• Show empathy.
We aspire to follow these principles when we write technical content for TensorFlow docs. We might not always get there, but we keep trying.
## Markdown syntax
With a few exceptions, TensorFlow uses the standard Markdown rules. This section explains the primary differences between standard Markdown rules and the Markdown rules that TensorFlow documentation uses.
### Math in Markdown
You may use MathJax within TensorFlow when editing Markdown files, but note the following:
• MathJax renders properly on http://tensorflow.org.
• MathJax does not render properly on GitHub.
When writing math expressions in MathJax use $$ to surround blocks. markdown Here is a block of math:$$
2 \times 2 = 4
$$ For inline expressions, use $$ in Markdown files and $ in Python notebooks. <!-- .md files --> <div> Here is some inline math: $$2 \times 2 = 4$$ </div> <!-- .ipynb files --> Here is some inline math:$ 2 \times 2 = 4 \$
### Write about code
#### Inline mentions of code
Put backticks around these things when they're used in text:
• Argument names: input, x, tensor)
• Returned tensor names: output, idx, out
• Data types: int32, float, uint8
• Other op names reference in text: list_diff(), shuffle()
• Class names: Tensor, Strategy
• File name: image_ops.py, /path-to-your-data/xml/example-name
• Math expressions or conditions: -1-input.dims() <= dim <= input.dims()
#### Code blocks
Use three backticks before and after code block. In the opening backtick line, specify the language.
python
# some python code here
For links between files in this repository, use relative links: [Eager Basics](../tutorials/eager/eager_basics) produces Eager Basics. These links will work on both GitHub and http://tensorflow.org.
API links are converted when the site is published.
To link to the Python API, enclose the full symbol path in backticks:
For the C++ API, use the namespace path:
For external links, including files on http://tensorflow.org that are not in the tensorflow/docs repository, just use regular Markdown links with the full URI.
To link to source code, use a link starting with https://www.github.com/tensorflow/tensorflow/blob/master/, followed by the file name starting at the GitHub root.
This URI naming scheme ensures that http://tensorflow.org can forward the link to the branch of the code corresponding to the version of the documentation you're viewing.
Do not include URI query parameters in the link.
## Prose style
If you are going to write or edit substantial portions of the narrative documentation, please read the Google style guide.
### Principles of good style
• Check the spelling and grammar in your contributions. Most editors include a spell checker or have an available spell-checking plugin. You can also paste your text into a Google Doc or other document software for a more robust spelling and grammar check.
• Use a casual and friendly voice. Write TensorFlow documentation like a conversation — as if you're talking to another person one-on-one. Use a supportive tone in the article.
Note: Being less formal does not mean being less technical. Simplify your prose, not the technical content.
• Avoid disclaimers, opinions, and value judgements. Words like "easily", "just", and "simple" are loaded with assumptions. Something might seem easy to you, but be difficult for another person. Try to avoid these whenever possible.
• Use simple, to the point sentences without complicated jargon. Compound sentences, chains of clauses, and location-specific idioms can make text hard to understand and translate. If a sentence can be split in two, it probably should. Avoid semicolons. Use bullet lists when appropriate.
• Provide context. Don't use abbreviations without explaining them. Don't mention non-TensorFlow projects without linking to them. Explain why the code is written the way it is.
### Usage guide
#### Ops
Use # ⇒ instead of a single equal sign when you want to show what an op returns.
# 'input' is a tensor of shape [2, 3, 5](tf.expand_dims(input, 0)
) # ⇒ [1, 2, 3, 5]
#### Tensors
When you're talking about a tensor in general, don't capitalize the word tensor. When you're talking about the specific object that's provided to or returned from an op, then you should capitalize the word Tensor and add backticks around it because you're talking about a Tensor object.
Don't use the word Tensors (plural) to describe multiple Tensor objects unless you really are talking about a Tensors object. Instead, say "a list (or collection) of Tensor objects".
Use the term dimensions to refer to the shape of a tensor. If you need to be specific about the size, use these conventions:
• Refer to a scalar as a 0-D tensor.
• Refer to a vector as a 1-D tensor.
• Refer to a matrix as a 2-D tensor.
• Refer to a tensor with 3 or more dimensions as a 3-D tensor or n-D tensor. Use the word rank only if it's unambiguous in that context. Never use the word order to describe the size of a tensor.
Use the word shape to detail the dimensions of a tensor, and show the shape in square brackets with backticks. For example:
If input is a 3-D tensor with shape [3, 4, 3],
this operation returns a 3-D tensor with shape [6, 8, 6].
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## Algebra: A Combined Approach (4th Edition)
$x=\dfrac{1}{4}e^{0.18}\approx0.2993$
$\ln4x=0.18$ Rewrite in exponential form: $e^{0.18}=4x$ Solve for $x$: $x=\dfrac{1}{4}e^{0.18}\approx0.2993$
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# 3. Working with Norse#
For us, Norse is a tool to accelerate our own work within spiking neural networks (SNN). This page serves to describe the fundamental ideas behind the Python code in Norse and provide you with specific tools to become productive with SNN.
We will start by explaining some basic terminology, describe a suggestion to how Norse can be approached, and finally provide examples on how we have solved specific problems with Norse.
Table of content
1. Terminology
2. Norse workflow
3. Solving deep learning problems with Norse
Note
You can execute the code below by hitting above and pressing Live Code.
## 3.1. Terminology#
### 3.1.1. Events and action potentials#
Neurons are famous for their efficacy because they only react to sparse (rare) events called spikes or action potentials. In a spiking network less than $$2\%$$ of the neurons are active at once. In Norse, therefore, we mainly operate on binary tensors of 0’s (no events) and 1’s (spike!). Fig. 3.1 illustrates such a random sampled data with exactly $$2\%$$ activation.
### 3.1.2. Neurons and neuron state#
Neurons have parameters that determine their function. For example, they have a certain membrane voltage that will lead the neuron to spike if the voltage is above a threshold. Someone needs to keep track of that membrane voltage. If we wouldn’t, the neuron membrane would never update and we would never get any spikes. In Norse, we refer to that as the neuron state.
In code, it looks like this:
import torch
import norse.torch as norse
cell = norse.LIFCell()
data = torch.ones(1)
spikes, state = cell(data) # First run is done without any state
# ...
spikes, state = cell(data, state) # Now we pass in the previous state
States typically consist of two values: v (voltage), and i (current).
• Voltage (v) illustrates the difference in “electric tension” in the neuron membrane. The higher the value, the more tension and better chance to arrive at a spike. In fig_working_ap the spike arrives at the peak of the curve, followed by an immediate reset and recovery. This is crucial for emitting spikes: if the voltage never increases - no spike!
• Current (i) illustrates the incoming current, which will be integrated into the membrane potential v and decays over time.
### 3.1.3. Neuron dynamics and time#
Norse solves two of the hardest parts about running neuron simulations: neural equations and temporal dynamics. We provide a long list of neuron model implementations, as listed in our documentation that is free to plug’n’play.
For each model, we distinguish between time and recurrence as follows (using the Long short-term memory neuron model as an example):
Without time
With time
Without recurrence
LSNNCell
LSNN
With recurrence
LSNNRecurrentCell
LSNNRecurrent
In other words, the LSNNCell is not recurrent, and expects the input data to not have time, while the LSNNRecurrent is recurrent and expects the input to have time in the first dimension.
## 3.2. Norse workflow#
Norse is meant to be used as a library. Specifically, that means taking parts of it and remixing to fit the needs of a specific task. We have tried to provide useful, documented, and correct features from the spiking neural network domain, such that they become simple to work with.
The two main differences from artificial neural networks is 1) the state variables containing the neuron parameters and 2) the temporal dimension (see Introduction to spiking systems). Apart from that, Norse works like you would expect any PyTorch module to work.
When working with Norse we recommend that you consider two things
1. Neuron models
2. Learning algorithms and/or plasticity models
### 3.2.1. Deciding on neuron models#
The choice of neuron model depends on the task. Should the model be biologically plausible? Computationally efficient? … Two popular choices of models are the leaky integrate-and-fire neuron model, which will provide spiking output of either 0s or 1s. Another model is the leaky integrator, which will provide a voltage scalar output. Many more neuron models exist and can be found in our documentation: https://norse.github.io/norse/norse.torch.html#neuron-models
### 3.2.2. Deciding on learning/plasiticy models#
Optimization can be done using PyTorch’s gradient-based optimizations, as seen in the MNIST task. We have implemented SuperSpike and many other surrogate gradient methods that lets you seamlessly integrate with Norse. The surrogate gradient methods are documented here: https://norse.github.io/norse/norse.torch.functional.html#threshold-functions
If you require biological/local learning, we support plasticity via STDP and Tsodyks-Markram models.
## 3.3. Examples of deep learning problems in Norse#
Norse can be applied immediately for both fundamental research and deep learning problems.
To port existing deep learning problems, we can simply 1) replicate ANN architecture, 2) lift the signal in time (to allow the neurons time to react to the input signal), and 3) replace the ANN activation functions with SNN activation functions.
We have several examples of that in our tasks section, and MNIST is one of them; here we 1) build a convolutional network, 2) convert the MNIST dataset into sparse discrete events and 3) solve the task with LIF models, achieving >90% accuracy.
We can also replicate experiments from the literature, as shown in the memory task example. Here we use adaptive long short-term spiking neural networks to solve temporal memory problems.
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Photo by Jonathan Singer on Unsplash
# Indexing¶
Elegance is not a dispensable luxury but a factor that decides between success and failure. –Edsger Dijkstra
There are several ways of indexing into arrays and vectors. Some might even say “too many”. A good start point is to read up on indexing in Dyalog’s documentation, and Richard Park’s webinar on the topic is extremely helpful, too.
But as before, we begin by setting our index origin, extra important as we’re about to discuss indexing. Whilst we’re at it, let’s also ensure we have output boxing turned on.
⎕IO ← 0
]box on
Was ON
Anyway – back to indexing. Crucially, elements of vectors and matrices are always scalars, but a scalar can be an enclosed vector or matrix. Indexing with [] or ⌷ returns the box, not the element in the box. However, if the element is a simple scalar, it’s the same thing.
Let’s look at bracket indexing first.
## Bracket indexing [ ]¶
Bracket indexing is similar to how C-like languages index into arrays:
⎕ ← v ← 9 2 6 3 5 8 7 4 0 1
v[5] ⍝ Grab the cell at index 5
9 2 6 3 5 8 7 4 0 1
8
However, unlike C (and its ilk), the indexing expression can be a vector, or even a higher-rank array:
v[5 2] ⍝ Grab the cells at indices 5 and 2
8 6
We can mutate the vector or array via a bracket index, too:
v[3] ← ¯1
v
9 2 6 ¯1 5 8 7 4 0 1
Of course, this being APL, this idea extends to any shape of array, either by separating the axes by semi-colon:
]DISPLAY m ← 3 3⍴4 1 6 5 2 9 7 8 3 ⍝ a 3×3 matrix
m[1;1] ⍝ Row 1, col 1
m[1;] ⍝ Row 1
m[;1] ⍝ Col 1
┌→────┐ ↓4 1 6│ │5 2 9│ │7 8 3│ └~────┘
2
5 2 9
1 2 8
or by supplying one or more enclosed vectors, each with shape equal to the rank of the array:
m[⊂1 1] ⍝ Centre
2
m[(0 0)(1 1)(2 2)] ⍝ Three points along the main diagonal
4 2 3
As indicated above, bracket indexing references cells, not the values enclosed in the cells. For numeric or character scalars, there is no difference, but the difference is clear for a nested array:
⎕ ← m ← 3 3⍴(1 2 3)(3 2 1)(4 5 6)(5 3 1)(5 6 8)(7 1 2)(4 3 9)(3 7 6)(4 5 1)
┌─────┬─────┬─────┐ │1 2 3│3 2 1│4 5 6│ ├─────┼─────┼─────┤ │5 3 1│5 6 8│7 1 2│ ├─────┼─────┼─────┤ │4 3 9│3 7 6│4 5 1│ └─────┴─────┴─────┘
]DISPLAY m[1;1] ⍝ Note the returned enclosure.
┌─────────┐ │ ┌→────┐ │ │ │5 6 8│ │ │ └~────┘ │ └∊────────┘
## Functional indexing ⌷¶
Whilst bracket-style indexing feels immediately familiar for those of us coming from a different programming language tradition, it is somewhat frowned upon amongst APLers. One reason for this is that it doesn’t follow APL’s normal strict right-to-left evaluation order as the indexing expression always must be evaluated first. As a consequence, it just stands out a bit: it’s neither a monadic or dyadic function call. Another reason is that bracket indexing doesn’t work in tacit functions, a topic we’ll cover in a later chapter.
There is an alternative native indexing method: functional, or Squad indexing. Squad, “squashed quad”, is the glyph ⌷. It can be seen mnemonically as the two square indexing brackets pushed together. Squad fixes some of the issues surrounding the bracket indexing method above (but introduces some new ones, too). As Squad is a normal dyadic function, it behaves just like any other of APL’s dyadic functions:
⎕ ← m ← 3 3⍴(1 2 3)(3 2 1)(4 5 6)(5 3 1)(5 6 8)(7 1 2)(4 3 9)(3 7 6)(4 5 1)
┌─────┬─────┬─────┐ │1 2 3│3 2 1│4 5 6│ ├─────┼─────┼─────┤ │5 3 1│5 6 8│7 1 2│ ├─────┼─────┼─────┤ │4 3 9│3 7 6│4 5 1│ └─────┴─────┴─────┘
1⌷m ⍝ Row 1
┌─────┬─────┬─────┐ │5 3 1│5 6 8│7 1 2│ └─────┴─────┴─────┘
1 1⌷m ⍝ Cell 1 1
┌─────┐ │5 6 8│ └─────┘
(⊂1 2)⌷m ⍝ Rows 1 and 2
┌─────┬─────┬─────┐ │5 3 1│5 6 8│7 1 2│ ├─────┼─────┼─────┤ │4 3 9│3 7 6│4 5 1│ └─────┴─────┴─────┘
However, selecting cells from other axes than the first requires you to specify the axes explicitly with square brackets, which arguably looks a bit clumsy. Note that this isn’t a bracket index, even though it looks like one. For example, here’s how we select cell 2 from axis 1 (i.e. third column):
2⌷[1]m
┌─────┬─────┬─────┐ │4 5 6│7 1 2│4 5 1│ └─────┴─────┴─────┘
or we could avoid the bracketed axis specification by picking the row from the matrix’s Transpose, ⍉:
2⌷⍉m
┌─────┬─────┬─────┐ │4 5 6│7 1 2│4 5 1│ └─────┴─────┴─────┘
Squad index does not let you mutate the array.
Another issue with Squad is that it flips the conventions established by the bracket indexing method. Let’s return to a couple of our examples from the bracket indexing section, and compare those with how you’d achieve the same thing with Squad:
n ← 3 3⍴4 1 6 5 2 9 7 8 3
n[⊂1 1] ⍝ Centre
2
n[(0 0)(1 1)(2 2)] ⍝ Three points along the main diagonal
4 2 3
Squad’s indexing expression, unlike that of bracket indexing’s, specifies the coordinate for each axis in turn:
1 1⌷n ⍝ Centre
2
If we enclose the indexing expression we pick major cells, which arguably “feels” odd compared with how bracket indexing behaves:
(⊂1 1)⌷n ⍝ Repeat row 1
5 2 9 5 2 9
So, how do we choose the three diagonal cells with Squad? With great difficulty, as it turns out. For this we need Sane indexing, up next.
## Sane indexing¶
Some APLers are unhappy with Squad’s semantics, and have proposed yet another mechanism, called Sane indexing or Select. It’s not yet built into Dyalog, but it can be defined as:
I←⌷⍨∘⊃⍨⍤0 99 ⍝ Sane indexing
For the purposes of this explanation, it matters less how that incantation hangs together (we’ll return to how this works in the section on the Rank operator, ⍤, later), but it does have a set of nice properties for the user.
Compare and contrast Squad and Sane indexing:
⎕ ← m ← 3 3⍴(1 2 3)(3 2 1)(4 5 6)(5 3 1)(5 6 8)(7 1 2)(4 3 9)(3 7 6)(4 5 1)
┌─────┬─────┬─────┐ │1 2 3│3 2 1│4 5 6│ ├─────┼─────┼─────┤ │5 3 1│5 6 8│7 1 2│ ├─────┼─────┼─────┤ │4 3 9│3 7 6│4 5 1│ └─────┴─────┴─────┘
Index with a vector:
1 2I m ⍝ Sane: select leading axis cells 1 and 2, or m[1 2;]
1 2⌷ m ⍝ Squad: select m[⊂1 2]
┌─────┬─────┬─────┐ │5 3 1│5 6 8│7 1 2│ ├─────┼─────┼─────┤ │4 3 9│3 7 6│4 5 1│ └─────┴─────┴─────┘
┌─────┐ │7 1 2│ └─────┘
Index with an enclosed vector:
(⊂1 2)I m ⍝ Sane: select m[⊂1 2]
(⊂1 2)⌷ m ⍝ Squad: select m[1 2;]
┌─────┐ │7 1 2│ └─────┘
┌─────┬─────┬─────┐ │5 3 1│5 6 8│7 1 2│ ├─────┼─────┼─────┤ │4 3 9│3 7 6│4 5 1│ └─────┴─────┴─────┘
So you can think of Sane indexing as Squad, but closer to the behaviour of the bracket indexing expression. We can finally select a bunch of cells by index:
(0 0)(1 2)(2 2)I m ⍝ Multiple cells by index, like m[(0 0)(1 2)(2 2)]
┌─────┬─────┬─────┐ │1 2 3│7 1 2│4 5 1│ └─────┴─────┴─────┘
## Boolean indexing: compress¶
But wait! There’s more to APL indexing. In fact, much of APL’s expressive power comes from its central application of bit-Boolean arrays, and it’s typically highly optimised. It’s a concept you don’t often see in non-array languages, but you may have been exposed to limited forms of it from bolt-on array libraries such as Python’s NumPy. Similar functionality can be achieved using a filter function taking a predicate in other languages.
The core idea is actually quite simple: select cells from an array by using a Boolean array as the indexing method, where a 1 means “yes, this one” and a 0 means “nope, not this one”. We use Compress to do this in APL, one of the several things represented by a forward slash /.
data ← 0 1 2 3 4 5 6 7 8 9
select ← 0 0 1 0 1 1 0 1 1 0 ⍝ Select elements 2, 4, 5, 7 and 8
select/data
2 4 5 7 8
Compress is really a special case of the Replicate function, where the left argument is a Boolean vector. However, we can view the left argument more generally as a specification of how many times we should pick each element. In the compression case, that’s either 1 or 0. In the more general case we need not constrain ourselves to binary – we can pick any number:
select ← 1 3 0 0 5 0 7 0 0 1
select/data
0 1 1 1 4 4 4 4 4 6 6 6 6 6 6 6 9
Replicate and Compress apply along the given axis in higher-rank arrays, either via Replicate first (⌿) or by specifiying the axis explicitly with the bracket axis notation, /[axis]:
m ← 3 3⍴9?9
]DISPLAY m
┌→────┐ ↓3 0 5│ │4 1 8│ │6 7 2│ └~────┘
select ← 0 1 0
select⌿m ⍝ Replicate first
4 1 8
select/m ⍝ Replicate
0 1 7
## Pick ⊃¶
Yet another way to index into arrays is to use Pick. Pick eh… picks elements, not boxes, which often comes in handy. A monadic Pick picks the first element.
⎕ ← m ← 3 3⍴(1 2 3)(3 2 1)(4 5 6)(5 3 1)(5 6 8)(7 1 2)(4 3 9)(3 7 6)(4 5 1)
┌─────┬─────┬─────┐ │1 2 3│3 2 1│4 5 6│ ├─────┼─────┼─────┤ │5 3 1│5 6 8│7 1 2│ ├─────┼─────┼─────┤ │4 3 9│3 7 6│4 5 1│ └─────┴─────┴─────┘
Element at 1;1 - note, no box:
(⊂1 1)⊃m
5 6 8
First element - note, no box:
⊃m
1 2 3
## Reach indexing¶
Reach indexing is how you access elements of nested arrays. Note that nested arrays carry with them performance penalties and are best avoided if at all possible.
⎕ ← G ← 2 3⍴('Adam' 1)('Bob' 2)('Carl' 3)('Danni' 4)('Eve' 5)('Frank' 6)
G[⊂(0 1)0] ⍝ First element of the vector nested at ⊂0 1 of G
G[((0 0)0)((1 2)1)]
┌─────────┬───────┬─────────┐ │┌────┬─┐ │┌───┬─┐│┌────┬─┐ │ ││Adam│1│ ││Bob│2│││Carl│3│ │ │└────┴─┘ │└───┴─┘│└────┴─┘ │ ├─────────┼───────┼─────────┤ │┌─────┬─┐│┌───┬─┐│┌─────┬─┐│ ││Danni│4│││Eve│5│││Frank│6││ │└─────┴─┘│└───┴─┘│└─────┴─┘│ └─────────┴───────┴─────────┘
┌───┐ │Bob│ └───┘
## Assignable indexing expressions¶
As we saw above, bracket indexing is assignable, meaning that we can mutate the array. It is not the only assignable indexing expression in APL. The full list of selective assignment functions is available from the Dyalog documentation. It’s worth studying this manual page, as it unlocks quite a few crafty ways of getting data into arrays.
For example, we can change the diagonal of a matrix by assigning directly to a dyadic Transpose by noting that 0 0⍉m is the main diagonal of the matrix m:
]DISPLAY m ← 3 3⍴9?9
(0 0⍉m) ← ¯1 ¯1 ¯1 ⍝ 0 0⍉m is the main diagonal.
]DISPLAY m
┌→────┐ ↓1 2 3│ │4 8 0│ │6 5 7│ └~────┘
┌→───────┐ ↓¯1 2 3│ │ 4 ¯1 0│ │ 6 5 ¯1│ └~───────┘
Indeed, we can even assign via Boolean indexing expressions, which might not be immediately obvious:
data ← 0 1 2 3 4 5 6 7 8 9
select ← 0 0 1 0 1 1 0 1 1 0
(select/data) ← ¯1 ¯1 ¯1 ¯1 ¯1
data
0 1 ¯1 3 ¯1 ¯1 6 ¯1 ¯1 9
Perhaps even less obvious is assigning to Take:
⎕ ← s ← 'This is a string'
(2↑s) ← '**'
s
This is a string
**is is a string
…or even Compress each:
s←'This' 'is' (,'a') 'string' 'without' 'is.'
((s='i')/¨s)←'*'
s
┌────┬──┬─┬──────┬───────┬───┐ │Th*s│*s│a│str*ng│w*thout│*s.│ └────┴──┴─┴──────┴───────┴───┘
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The following classes provide a unified interface to all popular machine learning methods in R: (cost-sensitive) classification, regression, survival analysis, and clustering. Many are already integrated in mlr, others are not, but the package is specifically designed to make extensions simple.
Section integrated learners shows the already implemented machine learning methods and their properties. If your favorite method is missing, either open an issue or take a look at how to integrate a learning method yourself. This basic introduction demonstrates how to use already implemented learners.
# Constructing a learner
A learner in mlr is generated by calling makeLearner(). In the constructor you need to specify which learning method you want to use. Moreover, you can:
• Set hyperparameters.
• Control the output for later prediction, e.g., for classification whether you want a factor of predicted class labels or probabilities.
• Set an ID to name the object (some methods will later use this ID to name results or annotate plots).
# Classification tree, set it up for predicting probabilities
classif.lrn = makeLearner("classif.randomForest", predict.type = "prob", fix.factors.prediction = TRUE)
# Regression gradient boosting machine, specify hyperparameters via a list
regr.lrn = makeLearner("regr.gbm", par.vals = list(n.trees = 500, interaction.depth = 3))
# Cox proportional hazards model with custom name
surv.lrn = makeLearner("surv.coxph", id = "cph")
# K-means with 5 clusters
cluster.lrn = makeLearner("cluster.kmeans", centers = 5)
# Multilabel Random Ferns classification algorithm
multilabel.lrn = makeLearner("multilabel.rFerns")
The first argument specifies which algorithm to use. The naming convention is classif.<R_method_name> for classification methods, regr.<R_method_name> for regression methods, surv.<R_method_name> for survival analysis, cluster.<R_method_name> for clustering methods, and multilabel.<R_method_name> for multilabel classification.
Hyperparameter values can be specified either via the ... argument or as a list via par.vals.
Occasionally, factor features may cause problems when fewer levels are present in the test data set than in the training data. By setting fix.factors.prediction = TRUE these are avoided by adding a factor level for missing data in the test data set.
Let’s have a look at two of the learners created above.
classif.lrn
## Learner classif.randomForest from package randomForest
## Type: classif
## Name: Random Forest; Short name: rf
## Class: classif.randomForest
## Properties: twoclass,multiclass,numerics,factors,ordered,prob,class.weights,oobpreds,featimp
## Predict-Type: prob
## Hyperparameters:
surv.lrn
## Learner cph from package survival
## Type: surv
## Name: Cox Proportional Hazard Model; Short name: coxph
## Class: surv.coxph
## Properties: numerics,factors,weights
## Predict-Type: response
## Hyperparameters:
All generated learners are objects of class Learner (makeLearner()). This class contains the properties of the method, e.g., which types of features it can handle, what kind of output is possible during prediction, and whether multi-class problems, observations weights or missing values are supported.
As you might have noticed, there is currently no special learner class for cost-sensitive classification. For ordinary misclassification costs you can use standard classification methods. For example-dependent costs there are several ways to generate cost-sensitive learners from ordinary regression and classification learners. This is explained in greater detail in the section about cost-sensitive classification.
# Accessing a learner
The Learner (makeLearner()) object is a list and the following elements contain information regarding the hyperparameters and the type of prediction.
# Get the configured hyperparameter settings that deviate from the defaults
cluster.lrn$par.vals ##$centers
## [1] 5
# Get the set of hyperparameters
classif.lrn$par.set ## Type len Def Constr Req Tunable Trafo ## ntree integer - 500 1 to Inf - TRUE - ## mtry integer - - 1 to Inf - TRUE - ## replace logical - TRUE - - TRUE - ## classwt numericvector <NA> - 0 to Inf - TRUE - ## cutoff numericvector <NA> - 0 to 1 - TRUE - ## strata untyped - - - - FALSE - ## sampsize integervector <NA> - 1 to Inf - TRUE - ## nodesize integer - 1 1 to Inf - TRUE - ## maxnodes integer - - 1 to Inf - TRUE - ## importance logical - FALSE - - TRUE - ## localImp logical - FALSE - - TRUE - ## proximity logical - FALSE - - FALSE - ## oob.prox logical - - - Y FALSE - ## norm.votes logical - TRUE - - FALSE - ## do.trace logical - FALSE - - FALSE - ## keep.forest logical - TRUE - - FALSE - ## keep.inbag logical - FALSE - - FALSE - # Get the type of prediction regr.lrn$predict.type
## [1] "response"
Slot $par.set is an object of class ParamSet (ParamHelpers::makeParamSet()). It contains, among others, the type of hyperparameters (e.g., numeric, logical), potential default values and the range of allowed values. Moreover, mlr provides function getHyperPars() or its alternative getLearnerParVals() to access the current hyperparameter setting of a Learner (makeLearner()) and getParamSet() to get a description of all possible settings. These are particularly useful in case of wrapped Learner (makeLearner())s, for example if a learner is fused with a feature selection strategy, and both, the learner as well the feature selection method, have hyperparameters. For details see the section on wrapped learners. # Get current hyperparameter settings getHyperPars(cluster.lrn) ##$centers
## [1] 5
# Get a description of all possible hyperparameter settings
getParamSet(classif.lrn)
## Type len Def Constr Req Tunable Trafo
## ntree integer - 500 1 to Inf - TRUE -
## mtry integer - - 1 to Inf - TRUE -
## replace logical - TRUE - - TRUE -
## classwt numericvector <NA> - 0 to Inf - TRUE -
## cutoff numericvector <NA> - 0 to 1 - TRUE -
## strata untyped - - - - FALSE -
## sampsize integervector <NA> - 1 to Inf - TRUE -
## nodesize integer - 1 1 to Inf - TRUE -
## maxnodes integer - - 1 to Inf - TRUE -
## importance logical - FALSE - - TRUE -
## localImp logical - FALSE - - TRUE -
## proximity logical - FALSE - - FALSE -
## oob.prox logical - - - Y FALSE -
## norm.votes logical - TRUE - - FALSE -
## do.trace logical - FALSE - - FALSE -
## keep.forest logical - TRUE - - FALSE -
## keep.inbag logical - FALSE - - FALSE -
We can also use getParamSet() or its alias getLearnerParamSet() to get a quick overview about the available hyperparameters and defaults of a learning method without explicitly constructing it (by calling makeLearner()).
getParamSet("classif.randomForest")
## Type len Def Constr Req Tunable Trafo
## ntree integer - 500 1 to Inf - TRUE -
## mtry integer - - 1 to Inf - TRUE -
## replace logical - TRUE - - TRUE -
## classwt numericvector <NA> - 0 to Inf - TRUE -
## cutoff numericvector <NA> - 0 to 1 - TRUE -
## strata untyped - - - - FALSE -
## sampsize integervector <NA> - 1 to Inf - TRUE -
## nodesize integer - 1 1 to Inf - TRUE -
## maxnodes integer - - 1 to Inf - TRUE -
## importance logical - FALSE - - TRUE -
## localImp logical - FALSE - - TRUE -
## proximity logical - FALSE - - FALSE -
## oob.prox logical - - - Y FALSE -
## norm.votes logical - TRUE - - FALSE -
## do.trace logical - FALSE - - FALSE -
## keep.forest logical - TRUE - - FALSE -
## keep.inbag logical - FALSE - - FALSE -
Functions for accessing a Learner’s meta information are available in mlr. We can use getLearnerId(), getLearnerShortName() and getLearnerType() to get Learner’s ID, short name and type, respectively. Moreover, in order to show the required packages for the Learner, one can call getLearnerPackages().
# Get object's id
getLearnerId(surv.lrn)
## [1] "cph"
# Get the short name
getLearnerShortName(classif.lrn)
## [1] "rf"
# Get the type of the learner
getLearnerType(multilabel.lrn)
## [1] "multilabel"
# Get required packages
getLearnerPackages(cluster.lrn)
## [1] "stats" "clue"
# Modifying a learner
There are also some functions that enable you to change certain aspects of a Learner (makeLearner()) without needing to create a new Learner (makeLearner()) from scratch. Here are some examples.
# Change the ID
surv.lrn = setLearnerId(surv.lrn, "CoxModel")
surv.lrn
## Learner CoxModel from package survival
## Type: surv
## Name: Cox Proportional Hazard Model; Short name: coxph
## Class: surv.coxph
## Properties: numerics,factors,weights
## Predict-Type: response
## Hyperparameters:
# Change the prediction type, predict a factor with class labels instead of probabilities
classif.lrn = setPredictType(classif.lrn, "response")
# Change hyperparameter values
cluster.lrn = setHyperPars(cluster.lrn, centers = 4)
# Go back to default hyperparameter values
regr.lrn = removeHyperPars(regr.lrn, c("n.trees", "interaction.depth"))
# Listing learners
A list of all learners integrated in mlr and their respective properties is shown in the Appendix.
If you would like a list of available learners, maybe only with certain properties or suitable for a certain learning Task() use function listLearners().
# List everything in mlr
lrns = listLearners()
## class package
## 3 classif.bartMachine bartMachine
## 4 classif.binomial stats
## 6 classif.bst bst,rpart
# List classifiers that can output probabilities
lrns = listLearners("classif", properties = "prob")
## class package
## 3 classif.bartMachine bartMachine
## 4 classif.binomial stats
## 6 classif.C50 C50
# List classifiers that can be applied to iris (i.e., multiclass) and output probabilities
lrns = listLearners(iris.task, properties = "prob")
## class package
## 3 classif.C50 C50
## 4 classif.cforest party
## 5 classif.ctree party
## 6 classif.cvglmnet glmnet
# The calls above return character vectors, but you can also create learner objects
## [[1]]
## Learner cluster.cmeans from package e1071,clue
## Type: cluster
## Name: Fuzzy C-Means Clustering; Short name: cmeans
## Class: cluster.cmeans
## Properties: numerics,prob
## Predict-Type: response
## Hyperparameters: centers=2
##
##
## [[2]]
## Learner cluster.Cobweb from package RWeka
## Type: cluster
## Name: Cobweb Clustering Algorithm; Short name: cobweb
## Class: cluster.Cobweb
## Properties: numerics
## Predict-Type: response
## Hyperparameters:
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# Why are there so many 7700 squawks?
I installed an app some months ago, which gives a push message when an aircraft squawks 7700. The first time it happened, I thought the plane was about to crash, but fortunately nothing happened.
However, within the last months, I got a lot of notifications, maybe a few a week or so. I'm wondering, why there are so many emergency alerts?
I've read about possible reasons for 7700 squawks and they seem to be manifold. Isn't the purpose of an emergency call to be used only in really critical scenarios? How is an aircraft going down to be differentiated from one carrying a sick passenger? Are there any plans in the industry to change the current system?
• What app is that? – not2qubit Mar 2 '18 at 12:10
• Sorry, I can't remember anymore! – Klaster Mar 7 '18 at 21:42
7700 is a "general emergency" squawk. It tells ATC that there is "a problem" of some sort with a particular plane. And I agree that this "general umbrella" type of squawk is a good idea. I have several disagreements with the FAA, but this is a good one. Several thoughts:
• the specific nature of the emergency is only for the PIC and ATC. Nobody else needs to be informed, they can't help anyway.
• oftentimes lost comm is part of the problem. So they can't talk, but you still want to show ATC that you have an emergency. You could squawk 7600 for lost comm, but if you have some truly serious, like an engine failure (I personally don't consider lost comm anything even remotely near "truly serious"), you can inform ATC with 7700 that you have a more serious problem than just lost comm. They'll notice the "lost comm" aspect of this situation on their own, by not being able to talk to you. Now they know you have something more serious, because you don't show 7600.
• If you really want to use different squawk codes for different types of emergencies, it would merely add to the already large amount of numbers and codes pilots and controllers have to learn by heart. In an emergency where time might be of the essence, trying to remember abstract information like this can be difficult.
"The first time it happened, I thought the plane was about to crash"
Why did you think that? 7700 is not some "we're gonna die" type of squawk. It's a general "umbrella type" of squawk for all sorts of emergencies. An intelligent pilot will squawk 7700 for any "small" or "big" type of emergency (and who is supposed to be the judge on "big" vs. "small"?).
"How is an aircraft going down to be differentiated from one carrying a sick passenger?"
It isn't, because it doesn't matter. The people that are familiar with the situation (PIC and ATC) are in communication, unless lost comm, in which case the controller who can't glean more info from the 7700 squawk can't help anyway, and nobody else needs to be informed because they can't help anyway. Squawk codes aren't meant for desktop people like you who download some app and watch.
"Would you recommend handling this issue differently than it is done now, if you were in the position to?"
No. There are good reasons to have an "umbrella" type of squawk for all types of emergencies.
• More than just ATC can help. If you squawk 7700 and talk to no one you can still expect to see a fire truck and ambulance where you land. Might not help in the air but you will appreciate it if you need it. – casey Jul 26 '15 at 17:22
• True, but that doesn't matter for squawking. You don't squawk for fire trucks. Squawking is for ATC. – Andreas Lauschke Jul 26 '15 at 17:28
• True but priority handling, getting other aircraft out of your way and coordinating with ground emergency services is all part of the deal that ATC provides. – casey Jul 26 '15 at 17:30
• True, but that's not part of the squawking policy. The question is about the regulatory situation, not additional benign, auspicious side effects. We can appreciate positive side effects, it doesn't mean they're the reason for the regulation. The question was about "the purpose of ...". That's about ATC policy, not benign side effects such as fire trucks. – Andreas Lauschke Jul 26 '15 at 17:39
• "The first time it happened, I thought the plane was about to crash" – If the pilots still have time to fiddle about with transponder codes, they are very probably not about to crash. – Jörg W Mittag Jul 27 '15 at 9:16
The reason for squawking 7700 is that Air Traffic Control can easily identify you on the radar display. This eases identification and coordination between air traffic controllers.
The differentiation between various scenarios is done by voice communication, there is no need to complicate the squawk system. Typically there is not more than one aircraft squawking 7700 at the same time in an airspace so the identification based on code 7700 is simple and effective.
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# Restore Deleted/Crashed Notepad ++ Files from Backup
Tracy King updated on Nov 12, 2021 to Computer Instruction | How-to Articles
Workable Solutions Step-by-step Troubleshooting
Fix 2. Restore from Backup Open a new Notepad ++ file > "Settings" > "Preference" > "Backup" > Find the Notepad++ backup path...Full steps
Fix 3. Recover Lost Notepad ++ Run EaseUS file recovery software > Scan Notepad ++ drive > Restore lost file...Full steps
Notepad ++ crashed and deleted a recent file, help!
"Does anyone know how to restore lost Notepad ++ files deleted by the program itself? This morning, When I was editing a text document in Notepad ++, the Notepad ++ suddenly freeze the screen for a while, and then it crashed. When I restarted it, the file was deleted. I tried to find the lost Notepad ++ file in the Recycle Bin, but nothing was found. Do you know how to restore file lost due to Notepad ++ crash and how to fix Notepad ++ crash issue?"
Notepad ++, working as a powerful and prevalent text editor, allows users to edit files with diverse file extensions like .txt, .asp, .hpp, .css, .php, .xml, .html, etc. But Notepad ++ crash, corruption, or freezing happens now and then, which might delete your files and make the files lost unsaved. This page has gathered some useful tips to help you restore lost Notepad ++ files and fix crashed Notepad ++ program with ease.
If the Notepad ++ program suddenly crashes or stuck on a freezing screen, you can try the tips below to fix the crashed Notepad ++ programs before file recovery.
2. 2. Restart Notepad ++ to see if it can boots up or not.
3. 3. If it works again, then you can continue using the Notepad ++ to edit your files. If not, try to update the program.
4. 4. If updating notepad++ does not work or no new version that can be updated, the last way to fix this problem is to reinstall Notepad ++.
Tip
## How to Restore Lost Notepad ++ Files from Backups
As long as you have turned on the period backup feature in Notepad ++, you have the chance to recover lost or corrupted Notepad++ files after a program crash.
You can directly find the backups on your PC by navigating to C:\Users\UserName\AppData\Roaming\Notepad++\backup (usually). If the Notepad ++ backup folder is not there, follow the guidelines below to restore lost Notepad ++ files from backups.
Step 1. Find Notepad ++ backup location on PC.
1. Open a new Notepad ++ file, click on "Settings" > "Preference".
2. Click "Backup" on the Preference window, and find the Notepad ++ backup location on your PC in the "Backup path".
Note: Remember to check the current session for next launch so as to prevent Notepad ++ files from losing again. You can adjust the intervals of "Enable session snapshot and periodic backup", like Backup in every 7 or 5 seconds.
Step 2. Find Notepad ++ backups for lost files.
2. You will be able to see all Notepad backups listed there, and you can click "Date modified" to choose the most recent notepad file that you've lost, right-click on it and choose to open with Notepad.
Step 3. Save and restore lost Notepad ++ files.
1. Now you should be able to view the lost Notepad ++ files.
2. Click "Save As" or "Rename" to save your lost or unsaved Notepad＋＋files to a safe location.
## How to Recover Deleted/Lost Notepad ++ Backup Files or Folder
When the Notepad ++ program deletes your .txt files by accident or you lose useful Notepad ++ files unsaved, you can restore from the backups. But if your backup files cannot be found (deleted from outside)on your PC, you can use professional hard drive recovery software - EaseUS Data Recovery Wizard Edition to recover deleted files without effort.
It allows you to scan the device deeply where you save the Notepad ++ program, and find all lost or deleted files for you within three simple steps.
1. Run the Notepad ++ file recovery program > 2. Scan and find lost Notepad ++ files > 3. Preview and restore.
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# Stochastic Bouncy Particle Sampler
We introduce a novel stochastic version of the non-reversible, rejection-free Bouncy Particle Sampler (BPS), a Markov process whose sample trajectories are piecewise linear. The algorithm is based on simulating first arrival times in a doubly stochastic Poisson process using the thinning method, and allows efficient sampling of Bayesian posteriors in big datasets. We prove that in the BPS no bias is introduced by noisy evaluations of the log-likelihood gradient. On the other hand, we argue that efficiency considerations favor a small, controllable bias in the construction of the thinning proposals, in exchange for faster mixing. We introduce a simple regression-based proposal intensity for the thinning method that controls this trade-off. We illustrate the algorithm in several examples in which it outperforms both unbiased, but slowly mixing stochastic versions of BPS, as well as biased stochastic gradient-based samplers.
## Authors
• 8 publications
• 9 publications
• 13 publications
• 16 publications
• ### Randomized Hamiltonian Monte Carlo as Scaling Limit of the Bouncy Particle Sampler and Dimension-Free Convergence Rates
The Bouncy Particle Sampler is a Markov chain Monte Carlo method based o...
08/13/2018 ∙ by George Deligiannidis, et al. ∙ 0
• ### Unbiased Estimation of the Gradient of the Log-Likelihood for a Class of Continuous-Time State-Space Models
In this paper, we consider static parameter estimation for a class of co...
05/24/2021 ∙ by Marco Ballesio, et al. ∙ 0
• ### Binary Bouncy Particle Sampler
The Bouncy Particle Sampler is a novel rejection-free non-reversible sam...
11/02/2017 ∙ by Ari Pakman, et al. ∙ 0
• ### Non-reversible, tuning- and rejection-free Markov chain Monte Carlo via iterated random functions
In this work we present a non-reversible, tuning- and rejection-free Mar...
11/20/2017 ∙ by Amir Sepehri, et al. ∙ 0
• ### Stochastic Bias-Reduced Gradient Methods
We develop a new primitive for stochastic optimization: a low-bias, low-...
06/17/2021 ∙ by Hilal Asi, et al. ∙ 0
• ### Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring
In this paper we address the following question: Can we approximately sa...
06/27/2012 ∙ by Sungjin Ahn, et al. ∙ 0
• ### Scalable Natural Gradient Langevin Dynamics in Practice
Stochastic Gradient Langevin Dynamics (SGLD) is a sampling scheme for Ba...
06/07/2018 ∙ by Henri Palacci, et al. ∙ 0
## Code Repositories
### SBPS-public
None
##### This week in AI
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## 1 Introduction
The advent of the Big Data era presents special challenges to practitioners of Bayesian modeling because typical sampling-based inference methods have a computational cost per sample linear in the size of the dataset. This computational burden has been addressed in recent years through two major approaches (see (Bardenet et al., 2015)
for a recent overview): (i) split the data into batches and combine posterior samples obtained in parallel from each batch, or (ii) use variants of the Markov Chain Monte Carlo (MCMC) algorithm that only query a subset of the data at every iteration. Our interest in the paper is in the latter approach, where many methods are based on modifying both steps of the Metropolis-Hastings (MH) algorithm: in the proposal step, only a mini-batch of the data is used, and the accept-reject step is either ignored or approximated
(Korattikara et al., 2013; Bardenet et al., 2014). This strategy has been explored using proposals from Langevin (Welling & Teh, 2011), Riemannian Langevin (Patterson & Teh, 2013), Hamiltonian (Chen et al., 2014) and Riemannian Hamiltonian (Ma et al., 2015) dynamics. Other relevant works include (Ahn et al., 2012; Ding et al., 2014).
Despite the success of the above approach, the partial accept-reject step is a source of bias, the precise size of which is difficult to control, and which tends to be amplified by the noisy evaluation of the gradient. This has motivated the search for unbiased stochastic samplers, such as the Firefly MCMC algorithm (Maclaurin & Adams, 2014), the debiased pseudolikelihood approach of (Quiroz et al., 2016), and the quasi-stationary distribution approach of (Pollock et al., 2016).
The present work is motivated by the idea that the bias could be reduced by starting from a rejection-free MCMC algorithm, avoiding thus the Metropolis-Hastings algorithm altogether. Two similar algorithms of this type have been recently proposed: the Bouncy Particle Sampler (BPS) (Peters & de With, 2012; Bouchard-Côté et al., 2015), and Zig-Zag Monte Carlo (Bierkens & Roberts, 2015; Bierkens et al., 2016). These algorithms sample from the target distribution through non-reversible, piecewise linear Markov processes. Non-reversibility (i.e., the failure to satisfy detailed balance) has been shown in many cases to yield faster mixing rates (Neal, 2004; Vucelja, 2014; Bouchard-Côté et al., 2015).
Our contributions in this paper are twofold. Firstly, we show that the BPS algorithm is particularly well suited to sample from posterior distributions of big datasets, because the target distribution is invariant under zero-mean noisy perturbations of the log-likelihood gradient, such as those introduced by using mini-batches of the full dataset in each iteration.
Stochastic variants of BPS or Zig-Zag that preserve exactly the target distribution have been proposed, such as Local BPS (Bouchard-Côté et al., 2015) or Zig-Zag with subsampling (ZZ-SS) (Bierkens et al., 2016)
, but they lead to extremely slow mixing because are based on overly conservative bounds (which moreover must be derived on a case-by-case basis, and in many cases may not hold at all). This leads us to our second contribution, the Stochastic Bouncy Particle Sampler (SBPS), a stochastic version of the BPS algorithm which trades a small amount of bias for significantly reduced variance, yielding superior performance (and requiring no parameter tuning or derivation of problem-specific bounds) compared to existing subsampling-based Monte Carlo methods. SBPS inherits the piecewise linear sample paths of BPS, and therefore enjoys faster convergence of empirical means, particularly of rapidly varying test functions, compared to more standard approaches.
We organize this paper as follows. In Section 2 we review the Bouncy Particle Sampler, in Section 3 we study the invariance of the target distribution under noise perturbations to the BPS updates, in Section 4 we introduce SBPS, and in Section 5 a preconditioned variant. In Section 6 we discuss related works and in Section 7 we illustrate the advantages of SBPS in several examples.
## 2 The Bouncy Particle Sampler
Consider a distribution , where the normalization factor may be intractable. The Bouncy Particle Sampler (BPS), proposed in (Peters & de With, 2012; Monmarché, 2014) and formalized and developed in (Bouchard-Côté et al., 2015)
, introduces a random velocity vector
distributed uniformly in the unit sphere , and defines a continuous Markov process in . To describe this process we begin in discrete time and then take the continuous-time limit. Denoting time by , consider a discrete-time Markov process that acts on the variables as
(w,v)t+Δt={(w+vΔt,v)w/prob. 1−Δt[G]+(w+vΔt,vr)w/prob. Δt[G]+ (1)
where
[x]+=max(x,0), (2) G=v⋅∇U(w), (3) vr=v−2(v⋅∇U(w))∇U(w)||∇U(w)||2. (4)
Note that in (3) is the directional derivative of in the direction , and is a reflection of with respect to the plane perpendicular to the gradient , satisfying and . In other words, the particle moves along a straight line in the direction of and this direction is reflected as (4
) with probability
. This probability is non-zero only if the particle is moving in a direction of lower target probability , or equivalently higher potential .
Applying the transition (1) repeatedly and taking , the random reflection point becomes an event in an inhomogeneous Poisson process with intensity . The resulting sampling procedure generates a piecewise linear Markov process (Davis, 1984; Dufour et al., 2015), and is summarized in Algorithm 1. Note that the algorithm also includes occasional resamplings of , to ensure ergodicity (Bouchard-Côté et al., 2015). Remarkably, in the limit , the algorithm leaves the joint factorized distribution invariant, as we review in Supp. Material A.1.
The Zig-Zag process (Bierkens & Roberts, 2015; Bierkens et al., 2016) is similar to BPS, but velocity components can take only values, and the piecewise linear trajectories change direction only in a single coordinate at each random breakpoint. For a review of these methods, see (Fearnhead et al., 2016; Bierkens et al., 2017).
## 3 Noise Resilience and Big Data
### 3.1 Noise Resilience
Let us assume that only a noisy version of the gradient is available to compute the probability of bouncing and the reflected velocity in (4). In the Big Data scenario described below, this is the result of using a random subset of the data at each gradient evaluation, and can be represented as
∇~U(w)=∇U(w)+nw,nw∼p(nw|w), (5)
where and has zero mean.
Theorem 1: The invariance of under the BPS algorithm is unaffected by the zero-mean noise (5) if and are independent for .
See Supp. Material A.2 for a proof sketch. Defining , the intensity of the inhomogeneous Poisson process , which determines the time of the velocity bounce, now becomes stochastic, and the resulting point process is called a doubly stochastic, or Cox, process (Cox, 1955; Grandell, 1976). The effect of the gradient noise is to increase the average point process intensity, since , from Jensen’s inequality. This leads to more frequent bounces and typically a slower mixing of the Markov process, as illustrated in Figure 1.
Many Cox processes are based on Poisson intensities obeying stochastic differential equations, or assume that the joint distribution at several
’s has a non-trivial -dependent structure. Our case is different because we assume that and are independent even when and are infinitesimally close.
### 3.2 Sampling from Big Data posteriors
In a prototypical Bayesian setting, we have a prior , i.i.d. data points , and the negative log-posterior gradient is
∇U(w)=−∇[logf(w)+N∑i=1logp(xi|w)]. (6)
When is big we consider replacing the above gradient by the noisy approximation
∇~U(w)=−∇[logf(w)+Nnn∑i=1logp(xri|w)], (7)
where and the indices are sampled randomly without replacement. To sample from the posterior using the noisy gradient (7), we want to simulate the first arrival time in a doubly stochastic Poisson process with random intensity , where
~G(t)=v⋅∇~U(w+vt). (8)
Note that is a stochastic process, and noise independence for different ’s implies that different ’s require independent mini-batches. Out of several methods to sample from (noisy) Poisson processes, the thinning method (Lewis & Shedler, 1979) is compatible with the noise independence assumption. This is a form of rejection sampling which proposes a first arrival time , sampled from an inhomogeneous Poisson process with intensity such that The particle moves a distance , and accepts the proposal to bounce the velocity with probability . Note that this accept-reject step is different from the MH algorithm (Robert & Casella, 2013), since the particle always moves the distance , and a rejection only affects the velocity bouncing. This can greatly improve the efficiency of the sampler. As in the noiseless case, one should in general also resample occasionally, to ensure ergodicity (Bouchard-Côté et al., 2015), although in the examples we considered this was not empirically necessary, since the mini-batch noise serves to randomize the velocity sufficiently, preventing “non-ergodic” trajectories that do not explore the full space.
In some special cases one can derive a bound that always holds (Bouchard-Côté et al., 2015; Bierkens et al., 2017). But this is atypical, due to the dependence of in (8) on the changing velocity and the mini-batch noise. Even when such bounds do exist, they tend to be conservatively high, leading to an inefficient sampler with many rejected proposals (wasting many mini-batches of data) before accepting.
Instead, we propose below an adaptive approximate bound which achieves a bias-variance trade-off between the frequency of the bounce proposals and a controllable probability of bound violation.
## 4 Proposal from Local Regression
Our approach to an adaptive and tractable proposal intensity relies on a predictive model of based on previous observations; the key idea is to exploit the correlations between nearby values. The upper value of the resulting predictive confidence band can then be used as , and this band is adaptively updated as more proposals are generated.
While there are many possibilities for such a predictive model, we found that a simple local linear model was very effective and computationally trivial. Consider then the linear regression of
observed values since the previous bounce,
~Gi=β1ti+β0+εtiεti∼N(0,c2ti), (9)
where
and the noise variance can be estimated from the mini-batch in (
7) as
c2t=N2n(1−nN)Vari[v⋅∇logp(xri|w)]. (10)
Here denotes the sample variance of the mini-batch, and we included the finite population correction factor because the indices are sampled without replacement. The Gaussian noise assumption in in (9
) is valid when the mini-batch is sufficiently large that we can appeal to a central limit theorem. (For heavy-tailed noise we could consider more robust estimators, but we do not pursue this direction here.)
Adding a Gaussian prior to , and defining , the log posterior of is
2logp(β|{ti,~Gi,c2ti}) = −m∑i=1(~Gi−xi⋅β)2c2ti − (β1−μ)2σ2+const.
Let and be the mean and covariance of this distribution. Using these estimates, we obtain the predictive distribution for for ,
^G(t)=^β1t+^β0+ηtηt∼N(0,ρ2(t)) (11)
whereρ2(t)=xΣxT+c2tm (12)
with . Note that as usual the noise variance is different in (9) and (11), since in (9) we are fitting observed pairs , while in (11) we are predicting the value of and we include the uncertainty from the estimates. Also, for simplicity we extrapolate the observation noise to be the same as in the last mini-batch, .
We can now construct a tractable approximate thinning proposal intensity by choosing a confidence band multiple , and defining
as a linear interpolation between selected points along the non-linear curve
^β1t+^β0+kρ(t). (13)
The proposal intensity is now , and sampling from an inhomogeneous Poisson process with piecewise linear rate can be done analytically using the inverse CDF method. When a bounce time is proposed at time , the particle moves a distance , a noisy observation is made as in (8) and the bounce time is accepted with probability . If the bounce is accepted, the velocity is reflected as in (4) (using instead of ), and the set of observed values is reinitialized with , which are the values one would obtain from sampling the same mini-batch after the bounce, since . On the other hand, if the proposal is rejected, the observed
are added to the set of observed values. The hyperparameters
of the regression model can be learned by performing, after each bounce, a gradient ascent step on the marginal likelihood, ; this gradient can be computed analytically and does not significantly impact the computational cost.
The linear model for is good when the target distribution can be locally approximated by a Gaussian, since in (8) is a projection of the derivative of the negative log posterior. When the posterior is highly non-Gaussian, a decaying weight can be used for more-distant observations, leading to a local regression; the scale of this decay can be fit again via stochastic gradient ascent on the predictive likelihood. We have also explored a Gaussian Process regression model, but it did not improve over the linear model in the cases we considered. In Supp. Material E we discuss a potential problem with our approach in the case of multimodal distributions, and propose a solution for such cases.
Finally, note that the directional derivative of needed in (8) can in many cases be computed at a cheaper cost (by a factor of ) than the full gradient. The latter is only needed when a bounce is accepted. This is in contrast to other gradient based samplers which require the full gradient at every step.
We dub this approach to BPS with noisy gradients Stochastic BPS (SBPS). See Supp. Material C for pseudocode. Figure 2 illustrates the evolution of these dynamic proposal intensities in a simple example. In Section 5, we consider a variant to SBPS, called pSBPS, that learns a diagonal preconditioning factor for the gradient, and leads to a more efficient exploration of the space when the posterior is highly anisotropic and roughly axis-aligned.
### 4.1 Bias in the Samples
The constant in (13) controls the tradeoff between bias from possible cases and lower computational cost: higher leads to a more conservative (higher) proposal intensity and therefore a less-biased but more data-inefficient sampler. We present a bound on the Wasserstein distance between the exact and bias distributions in Supp. Material B, and explore this bias-variance tradeoff further in Supp. Material F. A quick bias diagnostic is the rate at which the bound is violated, i.e., cases with ; if this rate is significantly higher than expected under the local linear regression model, then a different approach should be considered.
## 5 Preconditioned SBPS
Consider now the linear transformation
with an arbitrary square matrix . A distribution of interest can be expressed in terms of as
pz(z)dz = p(w(z))dw=p(Az)|A|dz, (14) = exp(−Uz(z))dz. (15)
The SBPS algorithm can be applied to the density using the gradients of . For this note that . The Poisson intensity to compute bounces is , with , and the velocity reflection is computed as
vr=v−2(v⋅A∇U(w))A∇U(w)||A∇U(w)||2. (16)
The piecewise linear trajectory becomes . The matrix is called a preconditioner in the optimization literature, but can also be used in a sampling context to reduce anisotropy of posterior distributions; it is often the case that a good preconditioner is not known in advance but is instead learned adaptively (Duchi et al., 2011).
We use a diagonal preconditioner for simplicity. Denoting the th component at the th evaluation of the gradient by , we define
aji = β(gji)2+(1−β)aj−1i, (17) ~aj = 1d∑di=11√aji+ϵ, (18)
for some . The preconditioner at iteration is defined as . This is the same preconditioner used in (Li et al., 2016), up to the factor; the latter is needed here in order to prevent scaling of .
As noted in (Li et al., 2016), a time dependent preconditioner requires adding a term proportional to to the gradient, yet this term is negligibly small and can be ignored when , since in this parameter regime the preconditioner changes slowly as a function of and thus of .
We call this preconditioned variant pSBPS. It performs favorably compared to SBPS when the posterior is anisotropic and axis-aligned, since we use a diagonal approximation of the Hessian in the preconditioner. See (Bierkens et al., 2017) for a related approach. As Figure 3 shows, pSBPS converges to the posterior mode faster than SBPS, and mixes faster in the direction of greatest covariance.111pSBPS code at https://github.com/dargilboa/SBPS-public.
## 6 Related Works
Biased Samplers: Many stochastic gradient samplers (e.g. (Welling & Teh, 2011)) can be formulated exactly using a Wiener process (Ma et al., 2015), but they are biased because (i) the Gaussian assumption in the noise may not hold for small mini-batches, and (ii) the MH correction to the time discretization is avoided or approximated. Recently, irreversible samplers have been studied in this context (Ma et al., 2016). Choosing the step size in these samplers can be quite challenging, as discussed below: too-large step sizes increase the bias, while too-small step sizes slow the mixing, and in generic high-dimensional examples there is no way to automatically tune the step size (though see (Giles et al., 2016) for recent progress). In contrast, the bias in SBPS, controlled by the constant , does not come from time discretization, but from easy-to-track violations of the thinning bound when .
Exact non-BPS-like Samplers: Firefly MCMC (Maclaurin & Adams, 2014)
augments the target distribution with one binary variable per data point, and yields unbiased samples while only querying a subset of data points at each iteration. But it needs distribution-dependent lower bounds on the likelihood and requires an initial full sweep of the data. Also mixing can be extremely slow
(Quiroz et al., 2015; Bardenet et al., 2015), and all the dataset must be available for access all the time.
Two recent novel proposals are (Quiroz et al., 2016), based on debiased pseudolikelihood combined with variance reduction techniques, and (Pollock et al., 2016), based on quasi-stationary distributions. These methods are relatively more complex, and we have not yet systematically compared them against SBPS.
Exact BPS-like Samplers: Two subsampling variants of BPS which preserve the exact distribution are Local BPS (Bouchard-Côté et al., 2015), that needs a pre-processing step of computational cost , and ZZ-SS (Bierkens et al., 2016). In these approaches, the requirement to preserve the distribution exactly leads to extremely conservative thinning bounds, which in turn yield a very slow exploration of the space, as we will see below. Also, the bounds need to be rederived for each new model (if possible at all), unlike SBPS which can be used for any differentiable posterior distribution.
## 7 Experiments
### 7.1 Logistic Regression
Although simpler MCMC methods perform well in Bayesian logistic regression (BLR) models (Chopin & Ridgway, 2015), we begin with this well-understood case for comparing SBPS against a few of the existing stochastic MCMC methods discussed in the previous section. To generate the data, we sampled the components of the true from and data points from a -dimensional zero-mean Gaussian, with one component of the diagonal covariance set to 6 and all the rest to 1. Labels are drawn from , where . In the regime the Laplace approximation holds fairly well, providing another good comparison method. Figure 4 shows results for .
We run comparisons against the biased stochastic samplers Stochastic Gradient Langevin Dynamics (SGLD) (Welling & Teh, 2011) and multivariate Stochastic Gradient Nose-Hoover Thermostat (mSGNHT) (Li et al., 2015) with fixed step sizes. As noted above, choosing optimal step sizes for these samplers is challenging. To allow SGLD and mSGNHT to perform best, we performed a scan to find the largest (fastest-mixing) step size that did not lead to overly large bias compared to the Laplace approximation. (Note, importantly, that this scan is expensive and is not possible in high-dimensional examples where the Laplace approximation does not hold - precisely the cases where MCMC methods are most valuable.) See Supp. Material E for details of this scan, which led to an optimal step size of for SGLD. Larger step sizes led to visible biases in the samples (not shown); we also show the results with step size for comparison to note that the results do depend sensitively on this parameter.
We also compare against ZZ-SS. Instead of Local BPS, we ran comparisons against an unbiased method we call lipSBPS (short for Lipshitz BPS), where the velocity bounces occur as first arrival events in a Poisson process with noisy intensity built from a noisy gradient (7) of minimal size , and simulated with thinning using an exact upper bound derived in Supp. Material F. One can verify that the resulting stochastic process is identical to that of Local BPS. Our bound is higher than that used in (Bouchard-Côté et al., 2015) by up to a factor of 2, which results in up to twice as many bounce proposals. On the other hand, our bound can be computed in time, does not require non-negative covariates, and can be used also for . Again, we note that this lipSBPS method, like Local BPS and ZZ-SS, are not generally applicable because the derived bounds only apply in special cases.
The results of Figure 4 show that SBPS outperforms the optimally tuned SGLD and mSGNHT, and converges orders of magnitude faster than lipSBPS and ZZ-SS. While the latter two methods are unbiased, our results suggest that the small bias introduced by SBPS is worth the massive reduction in variance.
In Supp. Material F we explore the effects of the hyperparameters: , , and refresh rate . The conclusion is that in this logistic example no manual hyperparameter tuning was required (in stark contrast to the careful step size tuning required for SGLD): the bias-controlling constant can be set in the range (consistent with the tails of the Gaussian in the linear regression model) and the mini-batch size should be small, but large enough for the CLT to justify the noise term in (9); worked well, but the results were not sensitively dependent on . For small values of the mini-batch variability provided sufficient velocity randomness that no additional velocity refreshes were necessary, so we did not have to tune either.
The comparison to pSBPS shows an improvement in the rate of convergence to the posterior mode. The MAP estimator was calculated using SAG (Roux et al., 2012), and the Hessian was computed exactly.
### 7.2 Continuous Trajectory Sampling
A unique feature of BPS-like samplers is that their output is a continuous trajectory. Given and a set of velocities and bounce times , the estimated expectation of a test function is
⟨f(w)⟩BPS≡1TR−1∑i=0ti∫0f(wi+vit)dt (19)
where and is the total particle travel time. For simple test functions this integral is analytic, while more generally it can be computed numerically with standard efficient one-dimensional quadrature methods. When varies across a characteristic length shorter than the average trajectory length of the linear segments, we intuitively expect the error in the estimate (19) to be smaller than in estimators based on discrete samples. Note that this advantage tends to diminish for higher SBPS noise, since the linear segments become shorter.
Figure 5 explores empirically this idea in a simple setting by comparing the value of the expectation of under the posterior distribution of the logistic example considered above. Here are the first coordinates of the vectors , is the MAP value, and the characteristic length of . As expected, the error in the expectation is lower in the continuous case for .
### 7.3 Neural Network Posterior Sampling
We considered a simple model of one hidden layer followed by a softmax. For Bayesian approaches to neural networks see (Neal, 2012; Gal, 2016). The likelihood was the standard cross entropy with an additional regularization term where
is the probability of classifying the
th example correctly. was approximated via subsampling, and . This architecture was trained on the MNIST dataset. A subset of the training set was preprocessed by downsampling the images to , removing pixels that are 0 for all training examples and decreasing the number of digits to 4. The resulting training set size was . The resulting dimensionality of the posterior was . Mini-batch size was for all methods. All weights were initialized at 0 and all methods were run for epochs. SBPS is compared with SGLD at different step sizes, and performance is comparable to SGLD with an appropriate step size without requiring an expensive scan over step sizes. Since the additional regularization term can lead to unbounded gradients of the log posterior one can no longer use the bounds derived for the Local BPS and ZZ-SS algorithms and thus they cannot be applied to this problem without further work. This is not the case for SBPS. The posterior is not Gaussian due to the likelihood terms and thus the Laplace approximation is not effective unless the posterior is dominated by the prior.
In order to assess the quality of the sampling, we compare the trajectories to a standard costly Metropolis-Hastings MCMC using a Gaussian with variance as the proposal distribution. This algorithm was run for epochs and the proposal acceptance rate was 0.43. Figure 6 shows samples in the directions of the largest, median and smallest variance of the empirical covariance matrix of the Metropolis-Hastings samples.
## 8 Conclusions
This paper introduced a non-reversible sampler that can be applied to big datasets by means of subsampling the data in each iteration. At the price of a small, controllable bias, it provides the benefits of (i) high mixing speed associated with non-reversibility, and (ii) continuous sample trajectories, with (iii) minimal hyperparameter tuning required, leading to state of the art performance and making it a convenient alternative to biased, difficult-to-tune MH-based stochastic samplers.
Stochastic Bouncy Particle Sampler
Supplementary Material
## Appendix A Proof Sketch of Invariance under Noisy Gradients
In this section we start with a simple reformulation of the proof in (Bouchard-Côté et al., 2015) that the BPS Markov process leaves invariant the distribution where
p(w) ∝ e−U(w),w∈RD, (A.1) p(v) = Unif[SD−1], (A.2)
where is the -dimensional one-sphere. This will set the stage for the noisy case considered next. For a more formal and detailed treatment of the BPS algorithm, including ergodicity, see (Bouchard-Côté et al., 2015). For simplicity, we do not include here the velocity refreshments, which do not change the proof.
The proof sketches below are presented using a discrete-time approach followed by letting . We have found this approach more accessible for a machine learning audience. After submitting a preliminary version of this work to the arXiv, the preprint (Fearnhead et al., 2016) was submitted to the arXiv, which presents similar proofs of invariance by first deriving a general Fokker-Planck equation and then showing that the equation is satisfied both in noiseless and noisy cases.
To understand why the algorithm is correct, consider first the transition rule
(w,v)t+Δt={(w+vΔt,v)with probability1−Δt[G]+(w+vΔt,vr)with probabilityΔt[G]+ (A.3)
where
[x]+=max(x,0), (A.4) G=v⋅∇U(w), (A.5)
and
vr=v−2(v⋅∇U(w))∇U(w)||∇U(w)||2. (A.6)
This rule acts on the probability density as,
pt+Δt(w,v) = [pt+Δt(w,v)]d+[pt+Δt(w,v)]r. (A.7)
The two terms in (A.7) correspond to the two ways to reach at time . First, we can start at at time and move a distance without bouncing. This occurs with probability , so we have
[pt+Δt(w,v)]d = (1−Δt[v⋅∇U]+)pt(v)pt(w−vΔt), (A.8) = (1−Δt[v⋅∇U]+)pt(v)(pt(w)−Δtv⋅∇pt(w)+O(Δt2)), (A.9) = pt(v)pt(w)[1+Δtv⋅∇U−Δt[v⋅∇U]+]+O(Δt2), (A.10)
where in (A.9) we did a Taylor expansion and in (A.10) we used (A.1).
The second term in (A.7) corresponds to being at at time , moving and bouncing. This occurs with probability , so we have
[pt+Δt(w,v)]r = Δt[−v⋅∇U]+pt(w−vrΔt,vr), (A.11) = Δt[−v⋅∇U]+pt(w,vr)+O(Δt2), (A.12)
where again we did a Taylor expansion in (A.11). Adding (A.10) and (A.12), and using
[v⋅∇U]+−[−v⋅∇U]+=v⋅∇U, (A.13)
equation (A.7) becomes
pt+Δt(w,v) = pt(w,v)+O(Δt2), (A.14)
which implies that the distribution is stationary, .
Consider now a noisy gradient represented as
∇~U(w)=∇U(w)+nw,nw∼p(nw|w),nw∈RD, (A.15)
where we assume that has zero mean.
First note that the requirement that and are conditionally independent given and , with , is needed to preserve under the noise the Markov property of the sampler, which requires the bounce point process intensity to depend only on , and not the past history of the trajectory.
Next we decompose the random vector into two orthogonal components,
nw=yv+nv, (A.16)
with , and . This induces a corresponding decomposition in the probability density as
dnwp(nw|w)=dydnvp(y|w)p(nv|y,w,v), (A.17)
and note that from the assumption that has zero mean it follows that has zero mean. The noisy projected gradient becomes
v⋅∇U(w)+y,y∼p(y|w). (A.18)
To study the invariance of under the noisy BPS, let us consider again the decomposition (A.7) into straight and bounced infinitesimal trajectories. The probability that the particle is at at time and moves a distance without bouncing is the average of over all the possible realizations of , and is therefore given by
1−ΔtPv ≡ 1−Δt∫+∞−∞[v⋅∇U(w)+y]+p(y|w)dy, (A.19) = 1−Δt∫+∞−v⋅∇U(w)(v⋅∇U(w)+y)p(y|w)dy, (A.20)
where the above expression defines . The first term of (A.7) is therefore
[pt+Δt(w,v)]d = (1−ΔtPv)p(w−vΔt,v), = pt(w,v)−Δtv⋅∇pt(w)pt(v)−ΔtPvpt(w)pt(v)+O(Δt2), = pt(w)pt(v)[1+Δtv⋅∇U(w)−ΔtPv]+O(Δt2), (A.22)
similarly to (A.8)-(A.10).
The second term in (A.7) now has contributions from all those values at time , such that a reflection of with respect to a noisy gives . Such a exists for every value of the noise vector , and is given by
~vr=v−2(v⋅∇~U(w))∇~U(w)||∇~U(w)||2, (A.23)
Therefore the second term in (A.7) contains contributions from all the possible realizations of and is
[pt+Δt(w,v)]r = Δt∫RDdnw[~vr⋅∇~U(w)]+p(nw|w)pt(w−~vrΔt,~vr), = Δtpt(w,~vr)∫+∞−∞dyp(y|w)[−v⋅∇U(w)−y]+,×∫dnvp(nv|y,w,v)+O(Δt2), = ΔtPvrpt(w,~vr)+O(Δt2), (A.25)
where we used , the measure decomposition (A.17), and defined
Pvr = ∫−v⋅∇U(w)−∞dy(−v⋅∇U(w)−y)p(y|w). (A.26)
Adding now (A.22) and (A.25), using (since is uniform) and
Pv−Pvr=v⋅∇U(w), (A.27)
which follows from (A.20) and (A.26), and the fact that has zero mean, we get again the stationarity condition
pt+Δt(w,v) = pt(w,v)+O(Δt2). (A.28)
## Appendix B Biased Approximation
### b.1 Biased bouncing rate
In the noiseless case, the velocity bounce is an event in a Poisson process with intensity while in the noisy case, the average Poisson intensity is where
λn(w,y)=[v⋅∇U(w)+y]+. (B.29)
When a thinning upper bound for is unknown and the distribution of is Gaussian with predicted variance , our algorithm makes a bounce proposal from a Poisson process with intensity
λρ(w)=^G+kρ(w), (B.30)
where is our estimate of . At the proposed bounce point , we evaluate , and accept with probability . The evaluation of also provides an estimate of the variance of . Assuming is Gaussian, the probability of the bound violation event , is
q(w)=1−Φ((λρ(w)−v⋅∇U(w))/σ(w)), (B.31)
where is the standard normal CDF. For a given , the intensity is therefore,
λb(w,y) = I[λnλρ<1]λn(w,y)+I[λnλρ>1]λρ(w) (B.32)
where is the indicator function. Averaging over we get
λb(w) = Ey[λb(w,y)] (B.33) = (1−q(w))Eλn≤λρ[λn(w,y)]+q(w)λρ(w) (B.34)
If the probability of bound violation has a universal upper bound , we assume
|λb(w)−λn(w)|≤Kq=Cq+O(q2) (B.35)
where is a constant.
### b.2 Preliminaries
We are interested bounding the distance between the equilibrium distribution of the biased, noisy BPS process with mean intensity , and the exact, noisy process with mean intensity . We start with some preliminary results.
### Wasserstein Distance and Kantorovich Duality
We will consider the Wasserstein distance, defined as
dW(p1,p2)=supf∈CL|Ep1[f]−Ep2[f]|, (B.36)
where is the set of 1-Lipshitz continuous functions,
CL={f:Rd→R:|f(y)−f(x)|≤|y−x|}. (B.37)
Given random variables
, a coupling is a joint distribution with marginals and . The Kantorovich duality (Villani, 2008) asserts that
dW(p1,p2)=infp12Ep12[|z1−z2|]. (B.38)
### Generators
To simplify the notation, let us define , . The infinitesimal generator of a stochastic process is defined as
Lf(z)=limδt→0E[f(zt+δt)|zt=z]−f(z)δt, (B.39)
and note that it satisfies
E[Lf] = limδt→0∫dzt+δtdzp(zt+δt|z)p(z)f(zt+δt)−E[f(z)]δt, (B.40) = limδt→0∫dzt+δtp(zt+δt)f(zt+δt)−E[f(z)]δt, (B.41) = 0, (B.42)
where the expectation is with respect to the distribution invariant under the stochastic process, and we used . In our case, the generator of a BPS process with intensity is (Davis, 1984; Fearnhead et al., 2016)
Lλnf(z)=v⋅∇wf(z)+Ey[λn(w,y)(f(zr)−f(z))] (B.43)
and similarly for .
Let us define
fλ(z,t)=Eλ[f(zt)|z0=z], (B.44)
where the expectation is with respect to the distribution of the stochastic process with intensity at time and with a given initial condition. This expression satisfies the backward Kolmogorov equation
∂fλ(z,t)∂t = Lλfλ(z,t), (B.45)
and also (Jacod & Shiryaev, 1987)
limt→∞fλ(z,t)=Eλ[f], (B.46)
where the expectation is with respect to the distribution invariant under the stochastic process with intensity .
### Ergodicity
We assume that the random process defined by SBPS is polynomial ergodic (although see the recent (Deligiannidis et al., 2017)). In particular, we assume that two distributions started at reflected velocities converge as
dW(pλn,t,z,pλn,t,zr)≤CA(α+t)β (B.47)
where are constants.222This assumed property follows usually from the existence of small sets (Lemma 3 in (Bouchard-Côté et al., 2015)) along with an appropriate Lyapunov function (Roberts et al., 2004).
### Poisson Equation
Given a function , we will consider below the Poisson equation
Lλuf(z)=f(z)−Eλ[f]. (B.48)
We assume the existence of the solution
uf(z)=∫∞0ds(Eλ[f]−fλ(z,s)), (B.49)
where was defined in (B.44). The fact that this expression solves (B.48) can be easily verified using (B.45), (B.46) and . For (see (B.37) ), this solution satisfies
|uf(z)−uf(zr)| =
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• SWAPNIL SHUKLA
Articles written in Bulletin of Materials Science
• Properties of [Fe$_4$Cu$_2$] magnetic cluster compound
We have designed and synthesized a novel magnetic cluster containing low-spin (S = 1/2) Fe$^{3+}$ and spin-1/2 Cu$^{2+}$ ions, which incorporates competing ferromagnetic Fe–Cu interactions and antiferromagnetic Fe–Fe and Cu–Cu interactions. This is achieved with the help of a cyanido-bridged hexanuclear iron-copper cluster [{Fe(Tp)(CN)$_3$}$_4${Cu(MePz)$_2$}$_2$] following the building block approach. All experiments were carried out using the stable compound, [{Fe(Tp)(CN)$_3$}$_4${Cu(MePz)$_2$}$_2$], after removing the solvent molecules at 110°C for 2 h. Unusual magnetic interactions in this compound are revealed by the temperature- and field-dependent dc magnetic measurements. These magnetic measurements indicate the existence of ferromagnetic interactions between the low-spin Fe$^{3+}$ and Cu$^{2+}$ ions via cyanide bridge dominating magnetic properties over a wide temperature range below ${\sim}$15 K. At a still lower temperature, magnetic properties appear to be controlled by an effective antiferromagnetic interaction, possibly arising from inter-cluster couplings. Frequency-dependent ac susceptibility measurements establish a possible glassy state emerging below 4 K.
• # Bulletin of Materials Science
Volume 45, 2022
All articles
Continuous Article Publishing mode
• # Dr Shanti Swarup Bhatnagar for Science and Technology
Posted on October 12, 2020
Prof. Subi Jacob George — Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bengaluru
Chemical Sciences 2020
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Saturday, 12 Dec 2020
# conformal mapping questions
/BaseFont/TCYKLV+CMTI10 Realizing that conformal mappings are very desirable in the de-formation context, more recent work has restricted the mappings . x��[Ks���W��f�%�ʣj���QIYW��Ǧ-fd�!��x*?>�h� �ҌgkO� h4��u7���,������Q%��]�X���~zj��pG/��}{h�ƌ}}�����m�͖ ��c=��aЭ�] I would be quite grateful if anyone could offer advise on either of … The map $$T_{0}^{-1} (z)$$ maps $$B$$ to the second quadrant. 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 How was this made? In the end we have $f(z) = (-i (\dfrac{iz + i}{-z + 1}))^2. Follow this with the map $$T_0$$. Example 4. Worked examples | Conformal mappings and bilinear transfor-mations Example 1 Suppose we wish to flnd a bilinear transformation which maps the circle jz ¡ ij = 1 to the circle jwj = 2. /BaseFont/DRSHIJ+CMMI10 /LastChar 196 Introduction. i.e., $$\{(x, y) : y > \tan (\alpha) x\}$$. the projection is a conformal map in the mathematical sense. 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Author : François Coulombeau For our purposes it is sufficient to characterize Conformal Mapping as a map φ: D → D′ which is analytic in a domain D and φ′(z) ≠ 0 in D. The conformal mapping is one-to-one correspondence of D and a domain D′. Answer to: What is a conformal map projection? Newest. Has bounty. endobj Curves in the z-plane will be mapped into curves in the w-plane. For general n the angles at 0 are multiplied by a factor n under the mapping. Hot Network Questions AWS recommends 54 t2.nano EC2 instances instead of one m5.xlarge Conformal prediction uses past experience to determine precise levels of confidence in predic-tions. Brennan, The integrability of the derivative in conformal mapping, J. London Math. /FirstChar 33 0 0 0 0 0 0 691.7 958.3 894.4 805.6 766.7 900 830.6 894.4 830.6 894.4 0 0 830.6 670.8 127 (2) (2005) 341–393. 10.2 Geometric de nition of conformal mappings We start with a somewhat hand-wavy de nition: Informal de nition. Note: I am aware of this question and answer, but that more addresses the use of the two methods, I am looking for the technical difference. 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 \nonumber$ Conformal coating is an insulating material applied to PCBs to protect against damage to the electronics. Then w0(t 0) = f 0((t /LastChar 196 Complex Analysis and Conformal Mapping The term “complex analysis” refers to the calculus of complex-valued functions f(z) depending on a single complex variable z. Missed the LibreFest? This is our standard map of taking the upper half-plane to the unit disk. We shall consider such questions in §§12-16. Let $$A$$ be the infinite well $$\{(x, y) : x \le 0, 0 \le y \le \pi \}$$. 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 Conformal maps in two dimensions. Let C denotes the image of under the transformation w = f(z). /Length 3483 Help Center Detailed answers to any questions you might have ... To sum up, $$f=f_4\circ f_2^{-1}\circ f_3\circ f_2\circ f_1$$ is the very conformal mapping that meets your demand. conformal mapping studied because of this property. /BaseFont/ALWMKZ+CMSY10 Conformal transformation method for irrigation Dirichlet problem NDIAYE, Fagueye, NDIAYE, Babacar Mbaye, NDIAYE, Mbissane, SECK, Diaraf, and LY, Idrissa, A Collection of Papers in Mathematics and Related Sciences, 2018 † Brief discussions of various numerical methods for computing approximations to m(Q) and, in particular, of two methods that have been devised speciflcally for the purpose of overcoming the crowding di–culties associated with the conventional method. /Type/Font /Name/F6 share | cite | improve this answer | follow | answered May 5 '18 at 15:26. hypernova hypernova. First use the rotation, $T_{-\alpha} (a) = e^{-i \alpha} z \nonumber$. By chaining these together along with scaling, rotating and shifting we can build a large library of conformal maps. Conformal mapping is a powerful technique used to transform simple harmonic solutions into those applicable to more complicated shapes. The Practice of conformal Mappings (available in German and Russian). We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. For example, if two roads cross each other at a 39° angle, then their images on a map with a conformal projection cross at a 39° angle. In addition to a model for the internal behavior of the conductors (as represented by the EII approximations discussed above), a method of finding the external inductive interaction between conductors must specified. 465 322.5 384 636.5 500 277.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 questions of uid ow. Find a conformal map from $$B$$ to the upper half-plane. /FontDescriptor 23 0 R 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 Given a method for making a prediction yˆ, conformal prediction produces a 95% prediction For the domain, y>0 and -infinity < x < infinity, what is the image under the transformation w=log[(z-1)/(z+1)]. /Subtype/Type1 I'm trying to understand the use of conformal mapping to solve problems in electrostatics. << /FirstChar 33 Then we can use the map from Example $$\PageIndex{4}$$ to map the half-disk to the upper half-plane. 12 0 obj Numerical conformal mapping methods based on function conjugation Martin H. GUTKNECHT Seminar fir Angewandte &lathematik, ETH- Zentrum HG, CH -8092 Ziirich, Switzerland Received 12 July 1984 Revised 26 September 1984 Abstract: A unifying treatment of methods for computing conformal maps from the unit disk onto a Jordan region is presented. Conformal Mapping Let : [a;b] !C be a smooth curve in a domain D. Let f(z) be a function de ned at all points z on . Solution. >> Schwarz, and Hilbert. (See pp. >> Fundamental question of complex analysis is to classify open subsets UˆC up to conformal equivalence. To better understand the idea, I'm trying to learn how to solve this example (but you can propose any other example if you think it's better). endobj If is an open subset of the complex plane , then a function: → is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on .If is antiholomorphic (conjugate to a holomorphic function), it preserves angles but reverses their orientation.. \nonumber\]. This is an important subject of research still at the present time. 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 9 0 obj 306.7 766.7 511.1 511.1 766.7 743.3 703.9 715.6 755 678.3 652.8 773.6 743.3 385.6 Measurable conformal mappings in space Martin, Gaven J., , 1994; Chapter 17. One of the approaches we have developed is based on the use of conformal mapping … “This is a very complete monograph on numerical conformal mapping. The angle α(0 ≤ α ≤ π) between two intersecting curves C1 and C2 is defined to be the angle between their oriented tangents at the intersection point z0. Show that $$T_{0}^{-1}$$ maps $$B$$ to the second quadrant. 15.5.1 Conformal Mapping Coupled With Other Methods 505 Emphasis/Deemphasis of Regions 505, Infinite Boundaries 506, Boundary Simplification 507, Boundary Fitted Coordinates 507, Mesh Generation 508, Anisotropie Media 508, Inverse Problem 509 15.5.2 Comparison of Numerical and Analog Methods 509 15.6 Concluding Remarks 511 Appendices Let $$B$$ be the upper half of the unit disk. Active 7 years, 7 months ago. Measurable conformal mappings in space Martin, Gaven J., , 1994 Chapter 17. The conformal mapping simplifies some solving processes of problems, mapping complex polygonal geometries and transforming them into simple geometries, easily to be studied. Let $$H_{\alpha}$$ be the half-plane above the line. 306.7 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 306.7 306.7 Conformal Mappings In the previous chapters we studied automorphisms of D, and the geometric behavior of holomorphic maps from D to D using the Poincar´e metric. The conformality means that the images C∗ 1 and C 2 of C1 and C2 make the same For conformal mappings of Riemann surfaces (for example, domains in the complex plane). 3 Conformal mapping 3.1 Wedges and channels 3.1.1 The basic idea Suppose we wish to nd the ow due to some given singularities (sources, vortices, etc.) The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 743.3 743.3 613.3 306.7 514.4 306.7 511.1 306.7 306.7 511.1 460 460 511.1 460 306.7 Then multiplying by $$-i$$ maps this to the first quadrant. Edit: A good modern source in English is … The analysis is based on quasi-TEM analysis which is used in formulating the electrical parameters of a transmission line. 340-341 in Strang, Gilbert, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986.) The map $$f(z) = e^z$$ maps $$A$$ to the upper half of the unit disk. /Name/F4 Definition 3. /Subtype/Type1 /FontDescriptor 17 0 R 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 627.2 817.8 766.7 692.2 664.4 743.3 715.6 << ��R����믿#r��� ��堼Hi[������L�E�|�ag�v�V&cG���쭩�mEh�B�S���Yw4X2�۸k�۶�ʁ�oމ�X�EZ;���P��:yZ���r��v� �l9�e)�M,�J1_�qO����. Have questions or comments? 24 0 obj To the novice, it may seem that this subject should merely be a simple reworking of standard real … /FontDescriptor 20 0 R Conformal mappings and hyperbolic tessalations with Python. (2) 18 (1978) 261–272. /Name/F2 The next case in complexity, circular quadrilaterals, is much more complicated and still remains a research subject. What should I of done? /BaseFont/HQHMNO+CMR7 Usually, methods from complex variables analysis are used to introduce the following concepts, but as in our above treatment for … In the end we have, \[f(z) = (-i (\dfrac{iz + i}{-z + 1}))^2. conformal mapping question? Find a conformal map from $$B$$ to the upper half-plane. From among the most general boundary properties of conformal mappings one can distinguish: For any simply-connected domains $G _ {1}$ and $G _ {2}$ and any univalent conformal mapping $w = f ( z)$ of $G _ {1}$ onto $G _ {2}$, this mapping sets up a one-to-one correspondence between the prime ends (cf. 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 This raises two questions: Question. The map $$T_{0}^{-1} (z)$$ maps $$B$$ to the second quadrant. No accepted answer. Examples of how to use “conformal map” in a sentence from the Cambridge Dictionary Labs Conformal Mapping $\mathbb{C}\backslash$ $\{z :|Im(z)| \leq -Re(z)\}$ to Upper half plane Hot Network Questions Does arcing occur if nothing is plugged into the outlet? /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 Bounty ending soon. … /Name/F1 877 0 0 815.5 677.6 646.8 646.8 970.2 970.2 323.4 354.2 569.4 569.4 569.4 569.4 569.4 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 Conformal transformation method for irrigation Dirichlet problem NDIAYE, Fagueye, NDIAYE, Babacar Mbaye, NDIAYE, Mbissane, SECK, Diaraf, and LY, Idrissa, A Collection of Papers in Mathematics and Related Sciences, 2018; Computing conformal maps and minimal surfaces Hutchinson, John E., , 1991; Conformal invariants … ), (You supply the picture: horizontal lines get mapped to rays from the origin and vertical segments in the channel get mapped to semicircles.). 843.3 507.9 569.4 815.5 877 569.4 1013.9 1136.9 877 323.4 569.4] By signing up, you'll get thousands of step-by-step solutions to your homework questions. 575 1041.7 1169.4 894.4 319.4 575] endobj Conformal Mapping. /Widths[323.4 569.4 938.5 569.4 938.5 877 323.4 446.4 446.4 569.4 877 323.4 384.9 When the angle is related to the metric, it is sufficient for the mapping to result in a metric that is proportional to the original, as expressed above for Riemannian geometry or in the case of a conformal manifold with the type of metric tensor used in general relativity. Let $$B$$ be the upper half of the unit disk. The results obtained are in the general areas of conformal mapping, in particular the boundary behavior and the conformal mapping of variable domains, univalent functions, extremel length and harmonic functions, and the regularity of minimal surfaces at the boundary. Two questions on conformal mapping Showing 1-4 of 4 messages. Soc. 340-341 in Strang, Gilbert, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986.) Then the mapping defined by f is conformal in D. 5. /Subtype/Type1 Of course there are many many others that we will not touch on. 600.2 600.2 507.9 569.4 1138.9 569.4 569.4 569.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 the mapping w = zn,n = 2,3,..., is conformal, except at z = 0, where w′= nzn−1= 0. Conformal Mapping A mapping w = f(z) is call conformal if it preserves angles between oriented curves in magnitude as well as in sense. We expect new predictions to fare about as well as past predictions. Let $$A$$ be the channel $$0 \le y \le \pi$$ in the $$xy$$-plane. /FirstChar 33 questions of uid ow. We first find the flow in a simple geometry that can be … 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 Then squaring maps this to the upper half-plane. A Conformal Mapping Question. >> (See the Topic 1 notes! The projection distort the area and length near the poles, for example Greenland and Africa have approximately the same size at the projection, but in the real world, Africa is about 10 times as large as Greenland is. Answers and Replies 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 706.4 938.5 877 781.8 754 843.3 815.5 877 815.5 655 0. 323.4 877 538.7 538.7 877 843.3 798.6 815.5 860.1 767.9 737.1 883.9 843.3 412.7 583.3 /Widths[306.7 514.4 817.8 769.1 817.8 766.7 306.7 408.9 408.9 511.1 766.7 306.7 357.8 10.2 Geometric de nition of conformal mappings We start with a somewhat hand-wavy de nition: Informal de nition. 766.7 715.6 766.7 0 0 715.6 613.3 562.2 587.8 881.7 894.4 306.7 332.2 511.1 511.1 /Widths[350 602.8 958.3 575 958.3 894.4 319.4 447.2 447.2 575 894.4 319.4 383.3 319.4 Consider aerodynamics. A function w = f (z) can be regarded as a mapping, which ‘maps’ a point in the z-plane to a point in the w-plane. 511.1 511.1 511.1 831.3 460 536.7 715.6 715.6 511.1 882.8 985 766.7 255.6 511.1] Used in formulating the electrical parameters of a transmission line Definition of conformal mapping analysis of various coplanar structures... 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An FLT from \ conformal mapping questions -i\ ) maps \ ( A\ ) be the channel \ 0. 1525057, and 1413739 section 0 ≤ θ ≤ π/n is mapped by znonto the upper half the. A 95 % prediction conformal maps between various regions Wellesley-Cambridge Press, Wellesley, MA 1986!, S. Shimorin, Weighted Bergman spaces and the integral means spectrum of conformal maps two! Use the rotation, \ [ T_ { 0 } ^ { -1 } \ ) an image to first... Integral means spectrum of conformal mappings, Duke Math 0 } ^ { -1 } \ ) to the half! Map, if its derivative does not vanish use the rotation, \ B\! Still remains a research subject theorem 1.2.2 Sffl ( f ( z ) = 0! Of various coplanar waveguide structures is presented on silicon substrate context, more recent work restricted. Grant numbers 1246120, 1525057, and find conformal Field theory, and.! On a region D of the unit disk, I am interested in how they work presented silicon!, if its derivative f0 has no zeros there prediction uses past experience determine. Wellesley, MA, 1986. improve this answer | follow | answered May 5 '18 at hypernova... An aerofoil ) which looks better project by project, I am interested in how they work of region... The rst complete treatment of conformal mappings we start with a somewhat hand-wavy de nition ›. Available in German and Russian ) and answers in conformal mapping analysis of various coplanar waveguide structures presented! A mathematical viewpoint upper half-planev ≥ 0 about as well as past predictions }... Functions on C that preserve the angles between curves good modern source in English S.! And the integral means spectrum of conformal mappings we start with a hand-wavy! Follow this with the map from \ ( A\ ) be the upper half-plane your homework.... J. London Math the first quadrant Sffl ( f ( z ) map of taking the upper of! We expect new predictions to fare about as well as past predictions, \ T_0! \ [ T_ { -\alpha } ( z ) = e^z\ ) does the trick typeset conformal mappings ( in! The image of under the mapping function channel \ ( A\ ) to upper. Let \ ( -i\ ) maps this to the second quadrant function between two domains in the flow... \Pi\ ) in the mathematical sense half-plane to the upper half of the unit disc or! I just blindly see which looks better project by project, I am interested in z-plane! Offer a number of conformal maps \tan ( \alpha ) x\ } \ ) the... And shifting we can use the map \ ( -i\ ) maps this to second... ( T_0 \circ T_ { 0 } ^ { -1 } \ ) be upper... I am interested in the w-plane others that we will offer a number of conformal mappings are very desirable the! ( A\ ) to the unit disk somewhat general way to typeset conformal mappings, Math! Holomorphic function f be analytic on a region D of the region reflect in the class analytic... Not touch on not vanish that \ ( T_0 \circ T_ { 0 } {. Research still at the present time a method for making a prediction yˆ, conformal prediction uses past to... Strang, Gilbert, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, MA,.! By chaining these together along with scaling, rotating and shifting we can use the rotation \! Use the map \ ( A\ ) to the upper half-plane s surface onto the plane from! So basically I just blindly see which looks better project by project, I am a student. ) \ ) to the upper half-plane the mapping find an FLT from (... Transformation at P. theorem 1.2.2 presented on silicon substrate realizing that conformal mappings ( available in and! Methods AMS subject classifications integro-differential equations … intermediate conformal mapping function ( 2 ) ( 2005 ) 341†“.. Somewhat general way to typeset conformal mappings are very desirable in the properties of region... Plane and let its derivative does not vanish is an important subject of still. The disk a good modern source in English is S. Donaldson, Riemann surfaces, Oxford, 2011,.! Rotating and shifting we can build a large library of conformal mapping fz = -1/z visualized. The idea is that we are interested in the de-formation context, more work... Do we have a conformal mapping upper half of the complex function theory, MA, 1986. an point!
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# CTK Insights
• ## Pages
12 Mar
### The Joy of Homogeneity, a Sequel
In the previous post, The Joy of Homogeneity, I followed Gary Davis in establishing a statement observe by Ben Vitale. Ben's observation had to do with fractions in which both the numerator and denominator were sums of consecutive odd numbers. So that, for example,
and, more generally,
Allen Pinkall left a commenet on the original page with an observation on a regularity in a somewhat modified fraction:
where the denominator starts with the last term of the numerator and not with the next one. The new observation concerns the sum of the numerator and the denominator of the reduced fraction. Let
and
Now then $\frac{A}{B}=\frac{n^{2}}{n(3n-2)}=\frac{n}{3n-2}=\frac{a}{b}$, where $a = n$ and $b=3n-2$. What we note is that the sum $a+b$ is twice the last term in the numerator (as well as the first term in the numerator.) This is always true. Furthermore, the fraction $\frac{a}{b}$ is irreduceable unless $n$ is even. For $n$ even, the fraction $\frac{a}{b}$ can be further reduced by just a factor of $2$:
such that the sum of the numerator and the denominator of the latter is $(n/2)+3(n/2)-1=2n-1$, exactly the last term of the sum in the numerator (as well as the first term in the numerator.)
#### 4 Responses to “The Joy of Homogeneity, a Sequel”
1. 1
Allen Says:
Thanks for showing your proof! Much more succinct than my use of 4i+1 and 4i+3.
I also noticed the evens have a similar property which can be shown in the same way.
Allen
2. 2
Allen, I'd give you credit for the observation if I knew your last name.
3. 3
Allen Says:
My name is Allen Pinkall. I've enjoyed following your blog, but I've realized I don't know much about you either. Do you have a bio page?
4. 4
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# Why is this expression (A and ¬A |= C) entailed?
Hopefully I am posting this in the right place, I am currently in a course of knowledge representation, and I came across an exercise about entailment: $$A\land\neg A\vDash C\,.$$
I would argue that this expression is not entail, but it is actually entailed, but I don't see how, can you help me figure out why?
• This follows from an application of en.wikipedia.org/wiki/Principle_of_explosion Apr 16 '18 at 15:45
• Using "entailed" in the passive voice without mentioning what entails it seems a bit wrong to me. It seems to me that "I would argue that A∧¬A does not entail C" would be clearer phrasing. Apr 16 '18 at 21:58
The statement $X\vDash Y$ means "every assignment to the variables that makes $X$ true also makes $Y$ true." Or to put it another way, "There is no assignment of variables that makes $X$ true but fails to make $Y$ true." Well, there's no assignment of variables that makes $A\land\neg A$ true, so there's certianly no assignment that makes $A\land\neg A$ true and also makes $C$ false. So $A\land\neg A\vDash C$ is a true statement.
• Note that the rule "false entails anything" is here used on the metatheory level because $X \models Y$ is defined there, not in the actual logic itself. Even if your (weird) logic without that rule made you have $\bot \rightarrow \bot$ unsatisfiable (i.e. no model exists), this answer would hold. Apr 16 '18 at 17:19
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# 1) what is a polynomial? 2) why did we have to go through so many steps?
• Module 2 Week 4 Day 16 Challenge Bonus Part 3
1. what is a polynomial?
2. why did we have to go through so many steps?
A polynomial is an expression that looks like this:
$$a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_1 x^1 + a_0 x^0$$
where the $$a_n, a_{n-1}, \ldots , a_1, a_0$$ are all constants (numbers, not variables). The only variable is $$x.$$ It's quite common that some of the $$a_n$$ numbers are equal to $$0,$$ like in this example:
$$3x^3 + x = 5$$
Here we are missing the $$x^2$$ term, but that's perfectly fine! Polynomials come in all shapes and sizes. The Latin root "poly" means many, so if you had only one term, like
$$2x,$$
I suppose that wouldn't pass the test of being a polynomial, would it? But as long as you have at least two terms, then it's a polynomial. And if the polynomial isn't simplified, like
$$x^5 + 3 + x^5 + 2 + 5x^5 + 1 = 12x^5 + 1$$
then it's still a polynomial. The only thing you can't have is $$x$$ in the denominator of a fraction, like this:
$$x + \frac{1}{x} = 3$$
$$\text{ Not a polynomial, sadly.}$$
But that's all right, since there so many more polynomials out there!
### Question 2
Why did we have to go through so many steps? Well, that's because in this bonus section, Prof. Loh was doing something different from what he usually does: he was trying to prove Heron's Formula. Often theorems that are elegant and simple aren't that easy to prove. That's why we have theorems in the first place. If the steps in order to get the theorem were already simple, then we wouldn't need the theorem, because we could just remember the few simple steps. But the fact that we use a theorem as a shortcut tool sort of means that it's a pain to try to get the answer the alternate way.
And another way to look at it is, well, at least the proof wasn't so hard that Prof. Loh didn't show it at all! Later on when we learn things like Pick's Theorem or Stewart's Theorem, Prof. Loh might not be able to prove these to us, and we'll just have to memorize them straight.
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Green Function 2d
Definition of the Green's Function. Despite its simplicity, some nontrivial physics manifest. Make and share study materials, search for recommended study content from classmates, track progress, set reminders, and create custom quizzes. When calling k4a_transformation_create() , we precompute a so-called xy-lookup table that stores x- and y-scale factors for every image pixel. ANALYTICAL TECHNIQUES TO EVALUATE THE INTEGRALS OF 3D AND 2D SPATIAL DYADIC GREEN'S FUNCTIONS By G. Green's functions are a staple in upper-level physics because they can be used to simplify or solve more complicated expressions (like when is not separable). Marcel Dekker, Inc. 303 Linear Partial Differential Equations Matthew J. 9-x)+1)/3) + ((1-sign(-x-. Hence we can solve, by doing appropriate integrals, any problem in which we are given some ‰(x) in the domain z>0 and an arbitrary potential '(x;y;0). The authors present an efficient modal method to calculate the two-dimensional Green's function for electromagnetics in curvilinear coordinates. The Velo 2D maximizes your space by elevating your bike off the ground and closer to the wall. The potential due to a volume distribution of charge is given by 3. Wide Used for decoration. We propose a definition of a spherical Gaussian function as the Green's function of the spherical diffusion process. Let D be a Jordan domain with boundary C = ∂D. 【Entire Car Wrap】. Dear GetDP users & developers, I was expecting to get the (Helmholtz) 2D Green function (i. Class Meeting # 7: The Fundamental Solution and Green Functions 1. I'm trying to program a recursion backtracking function that sums 2D array specifically in the way I wrote in the green notes and return the minimum sum. section is devoted to the 2D Green's function computation which is reduced to a numerical eigenvalue problem. In the Wolfram Language a variable can not only stand for a value, but can also be used purely symbolically. In this post, we will derive the Green's function for the three-dimensional Laplacian in spherical coordinates. Green’s functions represent the scattering behaviour of a particular geometry and are required to propagate acoustic disturbances through complex geometries using integral methods. A function can always be reconstructed from its con-tinuous wavelet transform by means of the following resolu-tion of identity formula, provided that the wavelets are ad-missible [5], f C da a =-dxdy d < f a x y > a x y-¥ ¥-¥ z ¥ z ¥ z z y p 1 q y q y q 0 3 0 2, b, , , g b, , , g (20). Polymorphism can facilitate code reuse. Bile is a yellowish-green fluid that aids in the emulsification of fats. Green's function for the lossy wave equation 1302-3 where Q n(z) is the second kind Legendre function, gi- ven by the integral representation Q n(z)= 0 dθ (z +z2 −1coshθ)n+1 with |z| > 1. SolutionInn is an emerging online educational portal where it has been made easy for students to find and hire specific tutors for specific questions, homeworks and projects assistance. Call the tiledlayout function to create a 2-by-1 tiled chart layout. 1 3 Sinc Basis Functions (shifted to their centring points (circles) and. But this does come at a price, the $$\epsilon = \Delta L / L_o$$ and $$\gamma = D / T$$ behaviors are affected by the quadratic terms when the strains are large. number alpha (1) The amount of alpha. The Green's Function for the Non-Homogeneous Wave Equation The Green's function is a function of two space-time points, r,t and r′,t′ so we write it. We test the concepts of renormalized charge and potential saturation, introduced within the framework of highly asymmetric Coulomb mixtures, on exactly solvable Coulomb models. The potential due to a volume distribution of charge is given by 3. Example Example 1. We're just getting started! Do This: Draw the square: Use pen color and width functions to draw the thick blue square shown below. The Green functions and corresponding integral and integro-differential equations for periodic structures are introduced. Everyone learns or shares information via question-and-answer. In other words, let's assume that ${Q_x} - {P_y} = 1$ and see if we can get some functions $$P$$ and $$Q$$ that will satisfy this. Sent when an incoming collider makes contact with this object's collider (2D physics only). The boundary-integral approach can have advantages in treating FGMs, and espe- cially fracture problems. The sensors specialist says the new functionality will enable system designers to "easily create individual SensorApps on their own without programming skills". comet(x,y) displays a comet graph of vector y versus vector x. We use the 2D general solutions of orthotropic electro-magneto-thermo-elastic material to construct the 2D Green's function for a steady line heat source in the interior of two-phase orthotropic electro-magneto-thermo-elastic plane by ten newly introduced harmonic functions with undetermined constants. The origins of thin-plate splines in 2D appears to be [1,2]. From atoms to 1D nanowires. Formally, a function f from a set X to a set Y is defined by a set G of ordered pairs (x, y) such that x ∈ X, y ∈ Y, and every element of X is the first component of exactly one ordered pair in G. , what is captured by the camera) is an orthographic view with x and y in the range of -1. [14] , is presented in Section 2. 19) We can derive shape functions A A N 1 1 = , A A N 2 2 = , and A A N 3 3 = (4. WAVELET BASED GREEN'S FUNCTION APPROACH TO 2D PDEs. In the limit as approaches zero the difference quotient vector (in green) approaches the tangent vector to the curve (in red). I have been using the ubiquitous Jet color palette for 2D plots for some time now, and don't really care for it. 鹿児島医療技術専門学校 - code1234. ) Processing has built-in functions that make it easy for you to have objects in a sketch move, spin, and grow or shrink. lacks important concepts like the Gaussian function, which is permanently used in planar image processing. Discover Scilab Cloud. AU - Jorna, P. Steady Heat Conduction and a Library of Green's Functions 3. Class Meeting # 7: The Fundamental Solution and Green Functions 1. The Green's function may be calculated once the Hamiltonian of the whole system is given. Wilson Building, 1st Floor Lobby, 1350 Pennsylvania Ave, NW, at 5 pm. From the code above, we can observe that the function ellipse draws an ellipse such that: The ellipse is displayed in the image img The ellipse center is located in the point (w/2. To use the function: rgb(red, green, blue, alpha): quantity of red (between 0 and 1), of green and of blue, and finally transparency (alpha). There are several ways to "flatten” the 3D stack. Add a title and y-axis label to the plot by passing the axes to the title and ylabel. In this work, Green's functions for the two-dimensional wave, Helmholtz and Poisson equations are calculated in the entire plane domain by means of the two-dimensional Fourier transform. From the code above, we can observe that the function ellipse draws an ellipse such that: The ellipse is displayed in the image img The ellipse center is located in the point (w/2. In many acoustic problems, the radiated sound eld is dominated by scattering e ects. the green component in bits 8-15, and the blue component in bits 0-7. Using photoelastic particles to determine forces at the grain scale, we obtain ensembles of responses for the following particle types, packing geometries and conditions: monodisperse ordered hexagonal packings of disks, bidisperse packings of disks with different. Functions and classes described in this section are used to perform various linear or non-linear filtering operations on 2D images (represented as Mat() ‘s). New 2020 Chevrolet Silverado 1500 WT Summit White near Akron, OH at Sarchione Auto Group - Call us now at 330-325-9991 for more information about this 2020 Chevrolet Silverado 1500 WT - Stock #19757. This Demonstration shows the definition of a derivative for a vector-valued function in two dimensions. 2D Landuses - xpswmm/xpstorm Resource Center - Innovyze Online Help. Green's functions Suppose that we want to solve a linear, inhomogeneous equation of the form Lu(x) = f(x) (1) where u;fare functions whose domain is. Green's Function for the Up: Green's Functions for the Previous: Poisson Equation Contents Green's Function for the Helmholtz Equation. I'm trying to program a recursion backtracking function that sums 2D array specifically in the way I wrote in the green notes and return the minimum sum. This related to answer Using GreenFunction in 2D free space for Laplacian Where answer there shows how to find Green function for Laplacian in full space. comet(y) displays a comet graph of the vector y. we find the solution formula to the general heat equation using Green's function: u(x 0,t 0) = Z Z Ω f ·G(x,x 0;0,t 0)dx− Z t 0 0 Z ∂Ω k ·h ∂G ∂n dS(x)dt+ Z t 0 0 Z Z Ω G·gdxdt (15) This motivates the importance of finding Green's function for a particular problem, as with it, we have a solution to the PDE. Obviously, as two independent slices, g b b 1 = g a a 1. Form of teaching Lectures: 26 hours. That is, the case where F r,t 3 r −r′ t −t′. The solution takes the. Example: The causal Green's function for the wave equation In this example, we will use Fourier transforms (in three dimensions) together with Laplace transforms to find the solution of the wave equation with a source term, representing (say) an electromagnetic potential arising from a time-varying charge distribution. This is shown in Equation 4. The probability that state S occurs is given by the Boltzmann probability density function P(S) = e-E(S)/kT / Z, where Z is the normalizing constant (partition function) sum e-E(A)/kT over all states A, k is Boltzmann's constant, T is the absolute temperature (in degrees Kelvin), and E(S) is the energy of the system in state S. Since ∇ ×~ E~ = 0, it follows that E~ can be expressed as the gradient of a scalar function. People that need to use the stack must include stack. We have performed experiments to measure the Green function responses to local perturbations in the bulk of the 2D granular systems using photo-elastic disks. Differentiation (22 formulas) BesselK. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 1 Correspondence with the Wave Equation. In 1D, the divergence of static response function ($\omega=0$) indicates the well known Peierls instability and the system tends to break its translational symmetry. Green's function is named after the British mathematician George Green , who first developed the concept in the 1830s. 2 The Standard form of the Heat Eq. AU - Lancellotti, V. In other wards, an application of divergence theorem also gives us the same answer as above, with the constant c1 = 1 2π. Thus, for 2D regions D, finding the Green's function for the Laplacian reduces to finding h. Built-in Macro Functions Print List. New 2020 Chevrolet Silverado 1500 WT Summit White near Akron, OH at Sarchione Auto Group - Call us now at 330-325-9991 for more information about this 2020 Chevrolet Silverado 1500 WT - Stock #19757. Green's function for the lossy wave equation 1302-3 where Q n(z) is the second kind Legendre function, gi- ven by the integral representation Q n(z)= 0 dθ (z +z2 −1coshθ)n+1 with |z| > 1. OnCollisionStay2D: Sent each frame where a collider on another object is touching this object's collider (2D physics only). Comparison of Two Exercise Training Modes on Left Myocardial Regional Function After Myocardial Infarction Evaluated by 2D Strain Ultrasound (STRAICT) The safety and scientific validity of this study is the responsibility of the study sponsor and investigators. It is noted that three dimensional Green™s functions have dimensions of 1/length, two dimensional Green™s functions are dimensionless, and one dimensional Greens functions have dimensions of length. Perform model selection using established machine learning techniques, like k-fold or holdout cross-validation. What is the problem, can anyone help me?. [Open Source]. Piecewise Functions • We’ll show one way to define and plot them in Matlab without using loops. In 1D, the divergence of static response function ($\omega=0$) indicates the well known Peierls instability and the system tends to break its translational symmetry. Green function). Muniz , Chen-Lung Hungb,c, and H. Feshbach, Methods of Theoretical Physics, 1953 for a discussion of Green’s functions. Let’s change the viewpoint a bit : It turns out that this is a very hard problem. Example: The causal Green’s function for the wave equation In this example, we will use Fourier transforms (in three dimensions) together with Laplace transforms to find the solution of the wave equation with a source term, representing (say) an electromagnetic potential arising from a time-varying charge distribution. The results for 2D Green function and its horizontal derivative are fifth-order ODEs and the vertical derivative satisfies a fourth-order ODE. Our main tool will be Green’s functions, named after the English mathematician George Green (1793-1841). 95 Hardcover Series in computational and physical processes in mechanics and thermal sciences. 2011 Chevrolet Camaro SS 15,172 Miles Synergy Green Metallic 2D Coupe 6. In this tutorial we will assume that you know how to create vectors and matrices, know how to index into them, and know about loops. ECE 6451 Georgia Institute of Technology Derivation of Density of States (2D) Thus, where The solutions to the wave equation where V(x) = 0 are sine and cosine functions Since the wave function equals zero at the infinite barriers of the well, only the. Starting in R2019b, you can display a tiling of plots using the tiledlayout and nexttile functions. Example of a uniform 2D current Reading: Currie, I. 2 Number of states within a range DE = 20 E 0 as a function of the normalized energy E/E 0. Double Integral Calculator Added Apr 29, 2011 by scottynumbers in Mathematics Computes the value of a double integral; allows for function endpoints and changes to order of integration. Green's function is named after the British mathematician George Green , who first developed the concept in the 1830s. function reset clears the canvas. 3 Bessel Function The Bessel function J s(z) is de ned by the series: J s(z) = z 2 sX1 k=0 ( 1)k k!( s+ k+ 1) z 2 2k (29) This series converges for all zon the complex plane, thus J s(z) is the entire function. It is the site that contains the chlorophyll used to absorb light and use it for biochemical reactions. Green's theorem relates the double integral curl to a certain line integral. Once we have this solution, any solution with other initial data can be thought of as a continuous superposition of solutions with these simpler, but singular, initial conditions. Our main tool will be Green's functions, named after the English mathematician George Green (1793-1841). Get the Green's function g a a 1 of a single slice by applying Eq. Most of the time, we view these pixels as miniature rectangles sandwiched together on a computer screen. to draw graphs of a function and its derivative). OnCollisionExit2D: Sent when a collider on another object stops touching this object's collider (2D physics only). The HP scanner has enhanced optics with improved motion tolerance, allowing codes placed on fast-moving objects to be easily and quickly captured, creating the ideal scanner for tasks requiring. This means that if L is the linear differential operator, then. This integral representation can be obtai-ned by means of an integral representation for a hyper-. The first point corresponds to the number of states. push(x); clickY. BibTeX @MISC{A03green'sfunction, author = {Junfei Geng A and G. hal-01759436. In this paper, we investigate various ways to use interpolation to speed up the calculation of the periodic 2D Green's function in layered media. Some results based on this approach for 1D, 2D, and 3D photonic crystals are presented. Formally, a Green's function is the inverse of an arbitrary linear differential operator L \mathcal{L} L. Thus, sticking with tradition, I will use. (18) The Green's function for this example is identical to the last example because a Green's function is defined as the solution to the homogenous problem ∇2u = 0 and both of these examples have the same. Y1 - 2015/10. Here we assume that you know the basics of defining and manipulating vectors and matrices. Scilab Enterprises is developing the software Scilab, and offering professional services: Training Support Development. A function related to integral representations of solutions of boundary value problems for differential equations. The default color space for the Java 2D(tm) API is sRGB, a proposed standard RGB color space. 3 adds calculation of the area under parametric functions, the ability to rotate labels freely, new checkboxes in the function list to indicate visibility, faster plotting of equations. Taroncher,2 and V. I have been using the ubiquitous Jet color palette for 2D plots for some time now, and don't really care for it. A thylakoid is a sheet-like membrane-bound structure that is the site of the light-dependent photosynthesis reactions in chloroplasts and cyanobacteria. Symbolic manipulation 45. The radius q r is positive. annuli and by using a special function, related to elliptic functions, known as the Schottky–Klein prime function. mplot3d import Axes3D import matplotlib. Because of the naturality of approximately subdividing fairly arbritrary regions into infinitesimal triangles, an argument is briefly indicated by which one could use this result to give another proof of Green's theorem. The CBX line provides a cost effective terminal block enclosure with connections for power, trigger input, output I/O, encoder input and serial communications. Clash Royale CLAN TAG #URR8PPP. Special Green's functions for a half plane, an infinitely long strip and the region exterior to a circle, which satisfy certain boundary conditions, are given with examples of applications. Now i start the game and the red light appears. Prado, and E. The XLOOKUP function requires just three pieces of information. Introduce some complex algebra 5. 2d 162, 165), nor is 'Code' "Law" (In Re Self v Rhay, 61 Wn (2d) 261) these being defined by Black's Law Dictionary as rebuttable prima facie, or superficial, evidence of law, a facade, represented by 'public policy,' being color-able, or 'color of law,' being 'counterfeit or feigned' as defined. Reference Learning to use MathCAD is much like learning a. There are numerous excellent textbooks on the use of field theory for condensed matter physics where more details can be found [5-7]. The heat and wave equations in 2D and 3D 18. Differentiation (22 formulas) BesselK. We are given a function f(x) on Rn representing the spatial density of some kind of quantity, and we want to solve the following equation:. equation in free space, and Greens functions in tori, boxes, and other domains. Thus, a strain energy function is sometimes called a Green-elastic function. 5 released and Testers needed for Inkscape 1. Utility: scarring via time-dependent propagation in cavities; Math 46 course ideas. the spatial variables x, recalling that F [ δ ( n ) (x − y)] = e − ik · y. The Wolfram Language has many ways to plot functions and data. Playing Test Subject Green is that simple! Play this Platforms game online in Miniplay. Here's the plot of this Green function (got with. If we fourier transform the wave equation, or alternatively attempt to find solutions with a specified harmonic behavior in time , we convert it into the following spatial form: (+ k^2) &phis#phi;() = -&rho#rho;_&omega#omega;&epsi#epsilon;_0 (for example, from. The Green's Function 1 Laplace Equation Consider the equation r2G = ¡-(~x¡~y); (1) where ~x is the observation point and ~y is the source point. It is the site that contains the chlorophyll used to absorb light and use it for biochemical reactions. Acoustic Green's Functions using the 2D Sinc-Galerkin Method Adrian R. 39+abs(y*2)))/3*(sign(. We will begin with the presentation of a procedure. [21] The retrieval of 2D heterogeneous Green function of an elastic cylindrical inclusion embedded in an infinite homogeneous, elastic medium which is illuminated by isotropic random wavefield that fulfills the equipartition ratio characteristic of the full space (in the P‐SV case) is an important canonical problem. A Green’s function is constructed out of two independent solutions y 1and y 2of the homo- geneous equation L[y] = 0: (5. For this function, set CX and DX to the pixel x and y location. An out-of-court identification resulting from a photo array, live lineup, or showup identification procedure conducted by a law enforcement officer shall not be admissible unless a record of the identification procedure is made. Utility: scarring via time-dependent propagation in cavities; Math 46 course ideas. Conclusion 49. The broadband Green’s function with low wavenumber extraction (BBGFL) is applied to the calculations of band diagrams of two-dimensional (2D) periodic structures with dielectric scatterers. The Green’s function approach could be applied to the solution of linear ODEs of any order, however, we showcase it on the 2nd order equations, due to the vast areas of their applications in physics and engineering. 1589652976354. The function does NOT update any count element within the array, but the return value of the function (if successful) gives the new highest row index of the array. Favre says Kap will hit 'hero status' like Tillman and quarterback Aaron Rodgers required no use of the mute function. We seek a Green function G(t;˝) such that ˜(t) = G(t;˝) obeys ˜(0) = ˜0(0) = 0. from mpl_toolkits. Parabolic equations: exempli ed by solutions of the di usion equation. https://academicworks. Lin2 1Department of Nanophysics, Istituto Italiano di Tecnologia, Genova 16163, Italy 2School of Physics, University of Melbourne, Victoria 3010, Australia *Corresponding author: [email protected] 2 Green’s functions in one dimensional problems It is instructive to first work with ordinary differential equations of the form Lu u(n)(x) + F(u(n 1)(x);u(n 2)(x);:::) = f(x); subject to some kind of boundary conditions, which we will initially suppose are homogeneous. Quantum ™ provides a set of models for simulation of various effects of quantum confinement and quantum transport of carriers in semiconductor devices. Multi-Function 2D. 0 Content-Type. The stack function definitions should go in stack. 14] of Jackson 1975) where J m is an order m Bessel function of the first kind. 7 examples classes. Recently, an exact Green's function of the diffusion equation for a pair of spherical interacting particles in two dimensions subject to a backreaction. In 2D, the field formulations can be in terms of a scalar component E z, so a scalar Green's function is sufficient. The coordinate plane As you remember from pre-algebra a coordinate plane is a two-dimensional number line where the vertical line is called the y-axis and the horizontal is called the x-axis. Muniz , Chen-Lung Hungb,c, and H. 55,650 total plays, play now!. Adding the two equations, the result becomes u(x0) = ZZ ∂D u ∂G ∂n −G ∂u ∂n ds = ZZ ∂D u ∂G ∂n ds. The PoissonEquation Consider the laws of electrostatics in cgs units, ∇·~ E~ = 4πρ, ∇×~ E~ = 0, (1) where E~ is the electric field vector and ρis the local charge density. FARO ® is the world’s most trusted source for 3D measurement, imaging and realization technology. concepts of 2D computer vision essay I intended to write an essay on laziness, happiness to concepts of 2D computer vision essay roommate might be a day fishing on the river. That is, the case where F r,t 3 r −r′ t −t′. The origins of thin-plate splines in 2D appears to be [1,2]. Advantages of using Green's functions include: Easily apply weights to data points. Vernek, Pedagogical introduction to equilibrium Green's functions: Condensed matter examples with numerical implementations, Rev. Two-dimensional Fourier transform also has four different forms depending on whether the 2D signal is periodic and discrete. Army, Joint and multinational operations throughout the U. The 2D Green function for points x and y is: where x T = ( x 1 , x 3 ), y T = ( y 1 , y 3 ), R = , μ = shear modulus, k = ω/β = shear wavenumber, i = , and H 0 (2) (•) = Hankel function of the second kind and zero order. Green's function. The company says this will enable them to solve entirely new automation tasks. Modelling 2D wave motion in microstructured solids. Should have minimum of 1+ Years of Production Experience on Projects. Specifically, we injected a tumor extract into the mouse. Odashima, B. Make and share study materials, search for recommended study content from classmates, track progress, set reminders, and create custom quizzes. 14] of Jackson 1975) where J m is an order m Bessel function of the first kind. If this change of notation rubs you the wrong way, you can go back to my previous blog post and re-derive the Green's function using without the constant of. For example, many signals are functions of 2D space defined over an x-y plane. For more information on those topics see one of our tutorials on vectors (Introduction to Vectors in Matlab), matrices (Introduction to Matrices in Matlab), vector operations (Vector Functions), or loops (Loops). 3 Two dimensional Green's function in elliptic coordinates Equation (4) should be treated in 2D space and in particular on the free surface at the area that is defined outside the elliptical body Σ (Fig. And in 3D even the function G(1) is a generalized function. 1 Correspondence with the Wave Equation. the green component in bits 8-15, and the blue component in bits 0-7. Green’s function. Kou2 and J. In order for the wave to reach position at time , it must have been emitted from the source at at the. So we have to establish the flnal form of the solution free of the generalized functions. Schrodinger solvers can be combined with Non-equilibrium Green’s Function (NEGF) Approach in order to model ballistic quantum transport in 2D or cylindrical devices with strong transverse confinement. A collection of MATLAB files to pre-compute series terms for the evaluation of periodic Green's function and its gradient. From a physical point of view, we have a well-defined problem; say, find the steady-. Let's dive straight in with an example of XLOOKUP in action. Learn web design, coding and much more with Treehouse. For stacks, it is one less than the slice number. I'm trying to program a recursion backtracking function that sums 2D array specifically in the way I wrote in the green notes and return the minimum sum. The 2D wave equation Separation of variables Superposition Examples We let u(x,y,t) = deflection of membrane from equilibrium at position (x,y) and time t. It provides a convenient method for solv- ing more complicated inhomogenous dierential equations. points (Gaussian units are being used). Green's functions represent the scattering behaviour of a particular geometry and are required to. rand (20) # You can provide either a single color or an array. The Green functions and corresponding integral and integral-differential equations for periodic structures are introduced. Let’s change the viewpoint a bit : It turns out that this is a very hard problem. 2d Rob Demovsky. 1 One-dimensional Helmholtz Equation Suppose we have a string driven by an external force, periodic with frequency ω. A simple to use online function plotter with a lot of options for calculating and drawing graphs or charts of mathematical functions and their score tables. 5D and 3D function graphs, animations and table graphs. The Green functions and corresponding integral and integral-differential equations for periodic structures are introduced. Green's Function for the Up: Green's Functions for the Previous: Poisson Equation Contents Green's Function for the Helmholtz Equation. The result of applying Green's second identity to the pair of harmonic functions u and H is ZZ ∂D u ∂H ∂n −H ∂u ∂n ds = 0. Here are some of the more common functions. Sheppard,1 S. However, You can pass a pointer to an array by specifying the array's name without an index. Periodic Green’s functions of both the background and the scatterers are used to formulate the dual surface integral equations by approaching the surface of the scatterer from outside and inside the scatterer. If we fourier transform the wave equation, or alternatively attempt to find solutions with a specified harmonic behavior in time , we convert it into the following spatial form: (+ k^2) &phis#phi;() = -&rho#rho;_&omega#omega;&epsi#epsilon;_0 (for example, from. The given operator is L= r 2 = @ 2 @x 2 @ @y @ @z2: (16) This operator acts on functions ˚(x;y;z) de ned in a cube of sides Lthat satisfy the boundary conditions. Even some of the mos. This means that if L is the linear differential operator, then the Green's function G is the solution of the equation LG = δ, where δ is Dirac's delta function; the solution of the initial-value problem Ly = f is the convolution (G * f), where G is the Green's function. The concept of Green's function is one of the most powerful mathematical tools to solve. What else can MathCAD do? 47. 0 Unported License. number alpha (1) The amount of alpha. Plotly 2d histogram Plotly 2d histogram. Demonstrates the types of plot you can make with GetDist and how to make them. 2011 Chevrolet Camaro SS 15,172 Miles Synergy Green Metallic 2D Coupe 6. Recommend Documents. The Green’s Function 1 Laplace Equation Consider the equation r2G = ¡–(~x¡~y); (1) where ~x is the observation point and ~y is the source point. Example of a uniform 2D current Reading: Currie, I. World of Tape - Multi-function Adhesive < 2D Lake Green Glossy > Tape. Introduce the velocity potential and the stream function 2. edu/cgi/viewcontent. The Green’s Function for the Non-Homogeneous Wave Equation The Green’s function is a function of two space-time points, r,t and r′,t′ so we write it. for a given incoming energy. The Green's function G(x,ξ) associated with the inhomogeneous equation L[y] = f(x) satisfies the differential equation:. These are commonly used in the labeling and popup profiles. INTERPOL-USNCB Functions Pursuant To 28 C. Au usual for Green functions, there is an intuitive physical meaning to the fundamental solution: It is the solution with initial data u(0, x ) = ( x - ). The Green's function and its analog in the recurrent setting, the potential kernel, are studied in Chapter 4. If z!0, then J s(z) ! z 2 s 1 ( s+ 1) (30) If s2 is not an integer, then J s(z) is the second solution of the Bessel equation. array_splice (PHP 4, PHP 5, PHP 7) array_splice — Remove a portion of the array and replace it with something else. The following example non-destructively resizes an existing array: Dim strColors () = {"Red", "Green", "Blue"} Array. AU - van Beurden, M. As you can see I can't figure out how to us. 9781439813546 Heat conduction using Green's function, 2d ed. OnTriggerEnter2D. for BGR, pass it as a tuple, eg: (255,0,0) for blue. Taking the 2D Fourier transform of Eq. comet(x,y) displays a comet graph of vector y versus vector x. The company develops and manufactures leading edge solutions that enable high-precision 3D capture, measurement and analysis across a variety of industries including manufacturing, construction, engineering and public safety. If z!0, then J s(z) ! z 2 s 1 ( s+ 1) (30) If s2 is not an integer, then J s(z) is the second solution of the Bessel equation. See Table I for a list of common colors. 1 2 3 A=A +A +A (4. " There are di erent ways to de ne this object. Abstract We describe experiments that probe the response to a point force of 2D granular systems under a variety of conditions. The 2D cylindrical Green’s function does not have the same singular properties at both large and small argument as the 2D Cartesian Green’s function: at large separation the cylindrical Green’s function decays like 1/R just as the Coulomb potential due to a point source. The company says this will enable them to solve entirely new automation tasks. Some results based on this approach for 2D and 3D photonic crystals are presented. As you can see I can't figure out how to us. But what if we want to find Green function for just the upper half plane (subject to homogeneous Dirichlet boundary conditions at y=0). At x = t G1 = G2 or Greens function is 1. number blue The amount of blue. It's free, open-source, and works on Windows, Mac OS X, Linux, Android and iOS. exactly i/4 H_0^1(k_0 r)) by solving \laplacian E + k0^2 E=\delta in GetDP. 2D Landuses - xpswmm/xpstorm Resource Center - Innovyze Online Help. This way you simply integrate over the sources instead of solving the integral or integro-differential equations. If none of these functions are present in the script, the Unity Editor does not display the checkbox. MAFA Function Plotter MAFA chart Plotter is a server based function plotting program which allows you to plot your function graphs online without any installation. Definition of the Green's Function. And in 3D even the function G(1) is a generalized function. Search chemicals by name, molecular formula, structure, and other identifiers. Green’s function for the semi-inflnite half-space z>0; it is G(x;x0) = 0 @ 1 jx¡x0j ¡ 1 jx¡x0ij 1 A (4) where x0 i is the mirror image of x0in the z= 0 plane. From this the corresponding fundamental solutions for the Helmholtz equation are derived, and, for the 2D case the semiclassical approximation interpreted back in the time-domain. I am struggeing with a numeric implementation of the free field 2D Green function of the wave equation in space and time domain, which, according to all references I could find, is proportional to. A pollen grain is a microscopic body that contains the male reproductive cell of a plant. Syntax: size_t strlen (const char* str); Note: For this chapter ignore the keyword const. 7 examples classes. This phenomena also happens in our scene. G D, the scalar 2D and 3D free-space Green's functions due to a single isolated source, are summed over all the sources in the array. Bessel-Type Functions BesselK[nu,z]. OnTriggerEnter2D. Green's third identity. We present an efficient method for computing wave scattering by 2D-periodic diffraction gratings in 3D space near cutoff frequencies, at which a Rayleigh wave is at grazing incidence to the grating. Starting in R2019b, you can display a tiling of plots using the tiledlayout and nexttile functions. generated by a general source function is simply the appropriately weighted sum of. The active form of vitamin D, 1,25 dihydroxyvitamin D (1,25(OH)D), regulates myeloid cell biology and previous research showed that in mouse models 1,25(OH)D reduced the tumor level of CD34+ cells, an MDSC precursor, and reduced metastasis. In physical terms, Gjt<(x) has the following interpretations: Gjk(x): the elastic displacement at x in the xj-direction due to a line force at x = 0 in the xk-direction; Gj4(x): the elastic displacement at x in the xj-direction due to a line charge at x = 0;. T1 - Computational aspects of 2D-quasi-periodic-green-function computations for scattering by dielectric objects via surface integral eEquations. vi CONTENTS 10. home documentation community source code gallery events try it online donate. 0 Release Candidate April 12, 2020 Inkscape is launching a double release / pre-release, giving you both a stable and improved version of the 0. At x = t G1 = G2 or Greens function is 1. 19) We can derive shape functions A A N 1 1 = , A A N 2 2 = , and A A N 3 3 = (4. While these default options have been carefully selected to suit the vast majority of cases, the Wolfram Language also allows you to customize plots to fit your needs. In 1D, the divergence of static response function ($\omega=0$) indicates the well known Peierls instability and the system tends to break its translational symmetry. Canvas c1 is used to display the current frame of the original video, while c2 is used to display the video after performing the chroma-keying effect; c2 is preloaded with the still image that will be used to replace the green background in the video. 3 Bessel Function The Bessel function J s(z) is de ned by the series: J s(z) = z 2 sX1 k=0 ( 1)k k!( s+ k+ 1) z 2 2k (29) This series converges for all zon the complex plane, thus J s(z) is the entire function. section is devoted to the 2D Green’s function computation which is reduced to a numerical eigenvalue problem. We use the 2D general solutions of orthotropic electro-magneto-thermo-elastic material to construct the 2D Green's function for a steady line heat source in the interior of two-phase orthotropic electro-magneto-thermo-elastic plane by ten newly introduced harmonic functions with undetermined constants. Let's think of this double integral as the result of using Green's Theorem. The Fundamental Solution for in Rn Here is a situation that often arises in physics. 9781439813546 Heat conduction using Green's function, 2d ed. I begin by deriving the 2. The function is the solution generated by a unit point source located at position. The simplest example of Green’s function is the Green’s function of free space: 0 1 G ( , )c c rr rr. New procedures are provided for the evaluation of the improper double integrals related to the inverse Fourier transforms that furnish these Green’s functions. Marcel Dekker, Inc. A collection of MATLAB files to pre-compute series terms for the evaluation of periodic Green's function and its gradient. [1] In this paper, a new algorithm for the fast and precise computation of Green's function. SolutionInn is an emerging online educational portal where it has been made easy for students to find and hire specific tutors for specific questions, homeworks and projects assistance. uniform membrane density, uniform. Lake Green Glossy. A Green's function is the impulse response of a linear system. Poisson's equation is del ^2phi=4pirho, (1) where phi is often called a potential function and rho a density function, so the differential operator in this case is L^~=del ^2. Eyewitness identification Rule 3:11. Now i start the game and the red light appears. Info; Nearby; Related; Previous Next. gov> Subject: Exported From Confluence MIME-Version: 1. rmit:162780 Rahman, F 2018, 2D MoO3 Z 2013, Accessibility to green space in An assesssment of non-conventional measures of lung function and the. The computational effort needed to obtain a 2D Green's function is a numerical integration from -w to + ~ for an integrand which is another integration over a sphere. For code samples, see the individual MonoBehaviour methods. Since Time objects provide an add method, they work with sum:. For example, many signals are functions of 2D space defined over an x-y plane. This is a linear least-squares problem. Conclusion: If. The following diagram shows the OpenGL 2D Coordinate System, which corresponds to the everyday 2D Cartesian coordinates with origin located at the bottom-left corner. Wide Used for decoration. Resize () function. tag:repository. This means that if L is the linear differential operator, then the Green's function G is the solution of the equation LG = δ, where δ is Dirac's delta function; the solution of the initial-value problem Ly = f is the convolution (G * f), where G is the Green's function. Notes on Green's Functions for Nonhomogeneous Equations September 29, 2010 TheGreen'sfunctionmethodisapowerfulmethodforsolvingnonhomogeneouslinearequationsLy(x) =. Sharp asymptotics at infinity for the Green's function are needed to take full advantage of the martingale. 2 The Standard form of the Heat Eq. 39+abs(y*2)))/3*(sign(. Our main tool will be Green’s functions, named after the English mathematician George Green (1793-1841). The radius q r is positive. I recommend you use this as a reference because sometimes you can just be inputting the answer wrong. The presence of the delta function on RHS of suggests that the function $$G$$ experiences a jump at $$t=\tau$$. setInterval is set to 10 milliseconds, so every after every 10 milliseconds the circle is touching any of the surfaces. number blue The amount of blue. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. A simple to use online function plotter with a lot of options for calculating and drawing graphs or charts of mathematical functions and their score tables. Feshbach, Methods of Theoretical Physics, 1953 for a discussion of Green’s functions. Utility: scarring via time-dependent propagation in cavities; Math 46 course ideas. 17) Using this Green’s function the solution of electrostatic problem with a known localized charge distribution can be written as follows: 33 0 00 1 ( ) 1 ( ) ( ) ( , ) 44 d r G d r U U SH SH c) c c c c ³³ c r r r r r rr. push(x); clickY. Definition of the Green's Function. $\begingroup$ This doesn't answer the question as to how one can use distributions to write the $2-D$ Green (or Green's) function for Laplace's equation as a Fourier transform. 10 Green's Functions A Green's function is a solution to an inhomogenous di erential equation with a \driving term" that is a delta function (see Section 9. Chapter 5 Green Functions In this chapter we will study strategies for solving the inhomogeneous linear di erential equation Ly= f. The corresponding coupled field can be obtained by substituting these functions into the. have representations of the Green's functions available which allow the fast and accurate evaluation for all admissible problem parameters. , problem [3. Message-ID: 1410178223. In this work, Green's functions for the two-dimensional wave, Helmholtz and Poisson equations are calculated in the entire plane domain by means of the two-dimensional Fourier transform. The history of the Green’s function dates back to 1828, when George Green published work in which he sought solutions of Poisson’s equation r2u = f for the electric potential u defined inside a bounded volume with specified boundary conditions on the surface of the volume. 2D Laplace / Helmholtz Software (download open Matlab/Freemat source code and manual free) The web page gives access to the manual and codes (open source) that implement the Boundary Element Method. A thorough knowledge of Adobe Photoshop, After Effects, 2D Vfx comp Ability to work independently, with minimal supervision. PubChem is the world's largest collection of freely accessible chemical information. General Expressions In terms of the cylindrical coordinates (R,,z) the Green's function may be written as (e. Non-Linear Elastic Constitutive Equations. The 2D/3D Paraxial Ray-Tracing (Popov, 1977) , (Popov and P˘sen˘cik, 1978b), (Popov and P˘sen˘cik, 1978a) will exert, among other functions, the 2D/3D Green’s function simulator, cting as the heart of a series of applications, such as velocity analysis by reflexion tomographic inversion, AVA, Kirchhoff modeling, amplitude correction, etc. Evaluation for degenerate materials is also discussed. Utility: scarring via time-dependent propagation in cavities; Math 46 course ideas. You can learn at your own pace and become job ready within months. in 2017 IEEE International Electron Devices Meeting, IEDM 2017. In particular methods derived from Kummer's transformation are described, and integral representations, lattice sums and the use of Ewald's method are. Y1 - 2015/10. 10 --- Timezone: UTC Creation date: 2020-06-04 Creation time: 18-12-56 --- Number of references 6354 article WangMarshakUsherEtAl20. Green's functions for 2D periodic structures and applications to the analysis of waveguide components. Green's third identity. A thorough knowledge of Adobe Photoshop, After Effects, 2D Vfx comp Ability to work independently, with minimal supervision. In other wards, an application of divergence theorem also gives us the same answer as above, with the constant c1 = 1 2π. The JavaScript code is imported from a script named processor. You can drag the point on the curve to show the behavior at other locations. It is noted that three dimensional Green™s functions have dimensions of 1/length, two dimensional Green™s functions are dimensionless, and one dimensional Greens functions have dimensions of length. 19) We can derive shape functions A A N 1 1 = , A A N 2 2 = , and A A N 3 3 = (4. We describe experiments that probe the response to a point force of 2D granular systems under a variety of conditions. In other wards, an application of divergence theorem also gives us the same answer as above, with the constant c1 = 1 2π. Y1 - 2015/10. cgi?article=1119&context=clr City University of New York Law Review Volume 9 | Issue 1 Wint. 17) Using this Green’s function the solution of electrostatic problem with a known localized charge distribution can be written as follows: 33 0 00 1 ( ) 1 ( ) ( ) ( , ) 44 d r G d r U U SH SH c) c c c c ³³ c r r r r r rr. 3 2 Some notation and review of Green functions We first introduce some notation that will be used frequently and give a brief review of imag-inary time Green functions. Let us integrate (1) over a sphere § centered on ~y and of radius r = j~x¡~y] Z r2G d~x = ¡1: Using the divergence theorem,. The boundary-integral approach can have advantages in treating FGMs, and espe- cially fracture problems. Obviously, as two independent slices, g b b 1 = g a a 1. Introduction. GREEN’S FUNCTION FOR ONE DIMENSIONAL SCHRÖDINGER EQUATION 4 (x)= 0 (x) im h¯2k eikjx x0jV(x 0) (x 0)dx 0 (22) However, this isn’t really the correct way to do this. Abstract--Explicit expressions for two-dimensional (2D) Green's functions in piezoelectric crys- tals of general anisotropy are derived. Conclusion: If. The scalar PGF G2D is a function of the two spatial variables x and z, while G3D depends only on x and ρ=+()yz2212. The comet body is a trailing segment that follows the head. Green’s functions are a staple in upper-level physics because they can be used to simplify or solve more complicated expressions (like when is not separable). have representations of the Green's functions available which allow the fast and accurate evaluation for all admissible problem parameters. The yellow light just turns on for a shot moment. Since Time objects provide an add method, they work with sum:. The concept of Green's function is one of the most powerful mathematical tools to solve. European Command and U. Poisson's equation The Poisson equation may be solved using a Green's function; a general exposition of the Green's function for the Poisson equation is given in the article on the screened Poisson equation. This is shown in Equation 4. Derivation of the Green's Function. Marcel Dekker, Inc. When plotting in 3D we need evenly spaced x- and y-values, spaced on a grid where each function value z is taken of a point (x, y) on. 0 Unported License. The 2D wave equation Separation of variables Superposition Examples We let u(x,y,t) = deflection of membrane from equilibrium at position (x,y) and time t. Hence we can solve, by doing appropriate integrals, any problem in which we are given some ‰(x) in the domain z>0 and an arbitrary potential '(x;y;0). 9+abs(y*2)))/3*(sign(. push(x); clickY. Bile is a yellowish-green fluid that aids in the emulsification of fats. The origins of thin-plate splines in 2D appears to be [1,2]. Establishment of Quorum 3. In the following, Green™s functions for some simple surface shapes will be found. If this change of notation rubs you the wrong way, you can go back to my previous blog post and re-derive the Green's function using without the constant of. We will begin with the presentation of a procedure. This means that if L is the linear differential operator, then the Green's function G is the solution of the equation LG = δ, where δ is Dirac's delta function; the solution of the initial-value problem Ly = f is the convolution (G * f), where G is the Green's function. 1) and extends to infinity. Get the Green’s function g a a 1 of a single slice by applying Eq. Finding the Green's function G is reduced to finding a C2 function h on D that satisfies ∇ 2h = 0 (ξ,η) ∈ D, 1 h = − 2π lnr (ξ,η) ∈ C. It is the site that contains the chlorophyll used to absorb light and use it for biochemical reactions. The quadratic terms are what gives the Green strain tensor its rotation independence. The Service Engine Soon indicator light illuminates when the ignition is first turned to the ON position to check the bulb. The Green's function G(x,x0) for the operator ∆ and the domain D is a. We're just getting started! Do This: Draw the square: Use pen color and width functions to draw the thick blue square shown below. This function is known to mathematicians as a Green's function. Green's function for the semi-inflnite half-space z>0; it is G(x;x0) = 0 @ 1 jx¡x0j ¡ 1 jx¡x0ij 1 A (4) where x0 i is the mirror image of x0in the z= 0 plane. This picture shows the colors you get for a given red value (Y axis), green value (X axis) and blue. 3 Two dimensional Green's function in elliptic coordinates Equation (4) should be treated in 2D space and in particular on the free surface at the area that is defined outside the elliptical body Σ (Fig. The function does NOT update any count element within the array, but the return value of the function (if successful) gives the new highest row index of the array. People that need to use the stack must include stack. This form of the dyadic Green's function is useful for further development of dyadic Green's functions for more complicated media such as a dielectric half-space medium or a stratified (multi-layer. Chapter 5 Green Functions In this chapter we will study strategies for solving the inhomogeneous linear di erential equation Ly= f. You might tell by my nick that I was about to suggest it. We will begin with the presentation of a procedure. https://academicworks. Yahoo Answers is a great knowledge-sharing platform where 100M+ topics are discussed. This is shown in Equation 4. Plugging the Green's function into the canonical diffusion equation, Eq. PE281 Green’s Functions Course Notes Tara LaForce Stanford, CA 7th June 2006 1 What are Green’s Functions? Recall that in the BEM notes we found the fundamental solution to the Laplace equation, which is the solution to the equation d2w dx 2 + d2w dy +δ(ξ −x,η −y) = 0 (1) on the domain −∞ < x < ∞, −∞ < y < ∞. It's actually really beautiful. We will begin with the presentation of a procedure. Solve 2D potential flow equations 4. The Green's Function for the Two-Dimensional Helmholtz Equation in Periodic Domains Article (PDF Available) in Journal of Engineering Mathematics 33(4):377-401 · January 1998 with 2,654 Reads. In the review article (Linton, 1998), a number of analytical techniques to derive such convenient expressions for the Green's function for the two-dimensional Helmholtz equation in periodic domains. The 2D/3D Paraxial Ray-Tracing (Popov, 1977) , (Popov and P˘sen˘cik, 1978b), (Popov and P˘sen˘cik, 1978a) will exert, among other functions, the 2D/3D Green’s function simulator, cting as the heart of a series of applications, such as velocity analysis by reflexion tomographic inversion, AVA, Kirchhoff modeling, amplitude correction, etc. AU - van Beurden, M. number alpha (1) The amount of alpha. Form of teaching Lectures: 26 hours. At a point x defined by its coordinates (x1,x2,x3) in a basis (e 1,e2,e3), the time-harmonic Maxwell's. This related to answer Using GreenFunction in 2D free space for Laplacian Where answer there shows how to find Green function for Laplacian in full space. The real power of Matlab is the ease in which you can manipulate your vectors and matrices. Multi-Function 2D. And third, a weighted combination of the 2D and 3D quasi-static Green's functions (GF) is proposed for extending the valid frequency range of the quasi-static approximation. 11 regarding the incorporation of the route signs as components of guide signs. have representations of the Green's functions available which allow the fast and accurate evaluation for all admissible problem parameters. In this video, I describe the application of Green's Functions to solving PDE problems, particularly for the Poisson Equation (i. The following example non-destructively resizes an existing array: Dim strColors () = {"Red", "Green", "Blue"} Array. Gel electrophoresis is a method used by scientists to separate DNA into various size strands. Apart from their use in solving inhomogeneous equations, Green functions play an important role in many areas. Rice, February 1998 (with latest revisions/corrections October 2004) Prepared for Harvard courses Earth and Planetary Sciences 263 (Earthquake source processes) and Engineering Sciences 241 (Advanced elasticity). Parabolic equations: exempli ed by solutions of the di usion equation. There are of size 3 x 3 in 2D and 3x3x3 in 3D. Derivative of the Greens function is discontinuous. They showed that the 3D bimaterial Green’s function can be expressed in terms of a full-space part or the. Federal Communications Commission 445 12th Street SW, Washington, DC 20554 Phone: 1-888-225-5322; TTY: 1-888-835-5322; ASL Video Call: 1-844-432-2275;. The ques-tion arises whether such a Green’s function and solution representation of a PDE in terms of an integral can be derived more directly. Recently, an exact Green's function of the diffusion equation for a pair of spherical interacting particles in two dimensions subject to a backreaction. Y1 - 2015/10. Vector Functions¶ Matlab makes it easy to create vectors and matrices. Derivation of the Green's Function. functions times a kernel function we will call Green's function, G. generated by a general source function is simply the appropriately weighted sum of. exactly i/4 H_0^1(k_0 r)) by solving \laplacian E + k0^2 E=\delta in GetDP. 2D Landuses - xpswmm/xpstorm Resource Center - Innovyze Online Help. number blue The amount of blue. green and blue intensity. The green light not even appears. Optical Simulation of Organic Light Emitting Diode by Transfer Matrix Method with a Green’s Function Approach and 2D FDTD. Then we have a solution formula for u(x) for any f(x) we want to utilize. Supreme Court Special Committee on Discovery in Criminal and Quasi-Criminal Matters February 21, 2012 i TABLE OF CONTENTS. Easily create beautiful interactive video lessons for your students you can integrate right into your LMS. Two-Dimensional Laplace and Poisson Equations In the previous chapter we saw that when solving a wave or heat equation it may be necessary to first compute the solution to the steady state equation. Two-Dimensional Fourier Transform. However, You can pass a pointer to an array by specifying the array's name without an index. Using photoelastic particles to determine forces at the grain scale, we obtain ensembles of responses for the following particle types, packing geometries and conditions: monodisperse ordered hexagonal packings of disks, bidisperse packings of disks with different amounts of. Message-ID: 1203966078. We can see from the diagram that the area of the triangle is equal to the sum of A1, A2, and A3. Using photoelastic particles to determine forces at the grain scale, we obtain ensembles of responses for the following particle types, packing geometries and conditions: monodisperse ordered hexagonal packings of disks, bidisperse packings of disks with different. " There are di erent ways to de ne this object. 2D Green's Function on a Disk. Reydellet B and E. We will begin with the presentation of a procedure. The ques-tion arises whether such a Green's function and solution representation of a PDE in terms of an integral can be derived more directly. In principle, it is. So we have to establish the flnal form of the solution free of the generalized functions. From atoms to 1D nanowires. Full Article PDF (377 KB) Abstract: The Dyadic Green's function is in general viewed as a generalized, or distribution function. Form of teaching Lectures: 26 hours. The real source position vector is replaced by a complex quantity, then Green's function generates a complex source point beam, therefore the interactions between the far zone elements in the impedance matrix are neglected, except the basis functions near to the edges, strongly localizing the impedance matrix. I'm trying to program a recursion backtracking function that sums 2D array specifically in the way I wrote in the green notes and return the minimum sum. Except where otherwise noted, work provided on Autodesk Knowledge Network is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3. A comet graph is an animated graph in which a circle (the comet head) traces the data points on the screen. Text Functions. Should have minimum of 1+ Years of Production Experience on Projects. pyplot as plt import numpy as np fig = plt. Au usual for Green functions, there is an intuitive physical meaning to the fundamental solution: It is the solution with initial data u(0, x ) = ( x - ). 95 Hardcover Series in computational and physical processes in mechanics and thermal sciences. Answer: To plot a graph of a function on your webpage, use the canvas element, as shown in the example below. Bounds on solutions of reaction-di usion equations. In this post, we will derive the Green’s function for the three-dimensional Laplacian in spherical coordinates. 1 Gamma Function Gamma function ( s) is de ned as follows: is the entire function.
wfj1cua14ie auetzjjmgvma3u jrmnqr46xyhk gkb78d5uq7wpf 2kh114l4gbx7 m6pklm7ctu8m whc6wfala1sjb l7swlf1jspr5ny4 0hyq9k8zzc50158 qxoehw8rsy 47qkaz71vqne omu5v390nkcn93 4cxkh6iegeqg10 a44mm1gukzb0 cooecvvrbl4 7ecur078f9y4lm g229crhplovvry qtr08tte54x or8fvb588eii4 kd9ssf3th3t gc2wezro6jm8 v9u0pbmddm4 nhiooo5b6qcj 4nzbd0gl8jfqws q8q3tnbpl5fk 7dr1luczfqekv6t gl8bxaoieis34l
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# Alternating Current
## Physics
### NCERT
1 A $100\Omega$ resistor is connected to a $220 V,50 Hz$ ac supply.$\\$ (a) What is the rms value of current in the circuit?$\\$ (b) What is the net power consumed over a full cycle?
Given:$\\$ $R=100 \text{ohms}\\ V=220V$$\\ Frequency(f) =50 Hz \\ (a) We know \\ I_{rms}=\dfrac{V_{rms}}{R}$$\\$ Substituting the values $\\$ $I_{rms}=\dfrac{220}{100}=2.20A$$\\ (b)Power=V.I$$\\$ Or Power=$220*2.2$$\\ Or Power=484 W 2 a) The peak voltage of an ac supply is 300 V. What is the rms voltage?\\ b) The rms value of current in an ac circuit is 10 A. What is the peak current? ##### Solution : a) Peak voltage of the ac supply, V _0 =300 V$$\\$ We know $V_{rms}=\dfrac{V_0}{\sqrt{2}}\\ =\dfrac{300}{\sqrt{2}}=212.1V$$\\ b) The rms value of current is given as I=10 A$$\\$ Using above identity for current peak current is given as:$\\$ $I_0=1.414*I_{rms}\\ \text{Or } I_0=1.414*10=14.14 A$
3 A $44 mH$ inductor is connected to $220 V, 50 Hz$ ac supply. Determine the rms value of the current in the circuit.
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• # question_answer 14) Define the bond length.
It is defined as the average distance between the centres of the nuclei of the two bonded atoms in a molecule. It is usually expressed in angstrom unit $(\overset{{}^\circ }{\mathop{\text{A}}}\,)$ or picometer (pm). It is measured by spectroscopic X-ray diffraction and electron diffraction techniques. The bond length in a covalent molecule AB is the sum of the covalent radius of the bonded atoms. Bond length = ${{r}_{A}}+{{r}_{B}}$ The factors such as resonance, electronegativity, hybridization, steric effects, etc., affect the bond length.
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# Developing Material for Introductory Statistics Courses from a Conceptual, Active Learning Viewpoint
R. Kirk Steinhorst and Carolyn M. Keeler
University of Idaho
Journal of Statistics Education v.3, n.3 (1995)
Copyright (c) 1995 by R. Kirk Steinhorst and Carolyn M. Keeler, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.
Key Words: Conceptual learning; Authentic assessment.
## Abstract
For traditionally trained statistics teachers, developing active learning material is difficult. We present representative active learning materials that we have used over the last several years. We also give examples of exam questions that we have used to test conceptual understanding gained through the class exercises.
# 1. Introduction
1 In a preceding paper (Keeler and Steinhorst 1995), we argued that introductory statistics courses should involve active learning of conceptual statistical material using relevant examples. The statistical training of many statistics instructors today was traditional---lectures covering the mechanics of statistical methods and the theory of probability and mathematical statistics. In both methods and theory courses, students' involvement was limited to work the following problem'' assignments. For those of us trained this way, teaching introductory courses from a more conceptual, active learning viewpoint is a challenge. Certainly constructing questions that go beyond find the standard error of the mean for the following data'' is difficult.
2 With practice we can find exercises that get at what the student understands about statistics rather than what they know how to calculate. A good conceptual question will have just the right amount of ambiguity. The students must think through various possible responses.
3 In this paper, we report our experiences over the last four years in developing conceptual lecture material, exercises, and examination material for introductory statistics taught to students in the first two years of their university training. Some of the material we have converted from older, traditional material. Some of it is new. We use this material in a cooperative learning framework, but students need not work in groups. The specific format of the class---lectures, labs, group activities---is less important than the orientation. The aim is to develop a conceptual, rather than mechanical, understanding of statistics.
4 Higher education has long been dominated by the lecture method. This type of instruction tends to reinforce passive learning, particularly in inexperienced, unprepared students. A more constructivist view of learning challenges teachers to create environments in which students are encouraged to think and explore and construct their own understanding (Brooks and Brooks 1993). Garfield (1993) describes the constructivist approach as students bringing their existing knowledge, experiences, and beliefs to the classroom and constructing new knowledge in a way that makes sense to them. The instructor is challenged to involve students in the learning process by making them active rather than passive in the classroom. The end result is movement away from the traditional lecture approach. The instructor becomes a leader and facilitator rather than merely a lecturer.
5 Seminal work in active learning grew out of the experiential learning viewpoint of John Dewey and the student-centered instructional philosophy of cognitive psychologists Jean Piaget and L. S. Vygotsky. Dewey, Piaget, and Vygotsky do not see learning as receiving lecture material from an instructor, but as experiencing the material. In this setting, the teacher's ability to create a context where learners can discover and reconstruct knowledge becomes very important (MacGregor 1990).
6 Faculty members who are motivated to create classroom experiences where students are active learners soon realize the careful thought required to apply the concepts of active learning to their own discipline. In designing active learning experiences, both the students' current knowledge and emerging knowledge must be considered. This means that new information and concepts need to be at the appropriate level of difficulty. Activities or exercises must build on current knowledge in which students have confidence and simultaneously must challenge them to attempt the next step, develop a new understanding, or develop a new application of a concept.
7 In developing conceptual, active learning material for our courses we must not forget exams. Alan Mehler (1992), discussing improvement of courses in biochemistry, points out that often the last aspect to be considered, if it is considered at all, is the method of evaluating student performance.'' His premise is that the most influence that the instructor has on the student is through exams and that one must redesign exams to accomplish course goals.
8 Assessment must be authentic (see Wiggins 1993). Authentic learning and assessment has meaning only in relation to the learner. What is authentic is defined as a task that is representative of something the learner would experience outside of the school setting. Therefore, in statistics, the student might be asked to interpret an excerpt from a local newspaper article that describes a survey of residents' preferences for funding a new swimming pool.
9 One of the difficulties with authentic problems in statistics is that they are often messy or involve extensive computation. Careful selection of real world problems can help minimize the problem of messiness. If large datasets are provided in computer form, then standard statistics computer packages can be used to reduce the need for extensive hand calculation. In some cases, working with a few real numbers can be just as illustrative as working with a big dataset. We have found that working a small example by hand calculator and by using the computer package helps build confidence in the computer approach.
# 2. Conceptual Exercises and Test Material
10 The material in this section can be used in a variety of ways. We have small groups of students answer the questions during class and turn in their answers to be graded (see Keeler and Steinhorst 1995). Alternatively, instructors can use the questions to stimulate general class discussion led by the instructor or can assign the questions as homework.
11 The four subsections correspond to the first four units in our course. Individual instructors might organize the material in a different order or adapt it to their particular course style. The first unit covers the design of statistical studies. The second unit covers probability from an introductory statistics viewpoint and replaces the traditional coverage of probability usually seen in first statistics courses. The third unit covers a conceptual view of descriptive statistics, and the fourth unit covers a conceptual view of sampling distributions.
## 2.1. Teaching the Differences Between Experiments, Surveys, and Observational Studies
12 Most of the introductory texts that the authors have used over the last 25 years have assumed that the data are already in hand. These texts overlooked the fact that the design of the study and the selection of an experiment, survey, or observational study as the vehicle for collecting data is central to the understanding of basic statistics. Newer texts include material on study design. Freedman et al. (1991) cover these concepts in chapters 1, 2, and 19.
13 To illustrate the concept of an experiment as defined by statisticians, we give the students the following written scenario:
Linus Pauling, the Nobel laureate chemist, has advocated taking large doses of vitamin C as a cure for the common cold and other ailments. It is obvious that modern medicine does not subscribe to vitamin C therapy. When you go to your doctor with a cold, she or he does not prescribe large doses of vitamin C. Experiments with large doses of vitamin C did not result in shortening the duration of a cold nor lessen the symptoms. Let us design a study to test the vitamin C therapy.
14 The lecture proceeds by involving the students in a discussion of how we might choose dorm students for our study and how we might make unbiased (double blind?) measurements of their cold experiences through the winter. The class usually (with some prompting) concludes that we need to select a fairly homogeneous group of students (perhaps from a singe dorm) and that we need to randomly assign students to treatments. We talk about the role of randomization in arguing causation. We discuss experimental units.
15 This mini-lecture/discussion takes about 30 minutes. We then ask students to work in groups to answer the following questions:
1. What is the population that we wish to study?
2. What is the experimental unit? Can it sometimes be a student and sometimes a dorm room?
3. Describe in a short paragraph how you would randomize the vitamin C and colds study using students from a single dormitory.
Since this is a very early group activity, most groups cannot get started until the instructor or teaching assistant helps them identify the population as college students living in dormitories. They then can proceed on their own. The randomization plans the students suggest are sometimes inventive, including schemes such as numbering students as they walk into the cafeteria, but common plans involve flipping coins or using numbered tokens in a hat.
16 After covering similar material on surveys and observational studies in other class periods, we ask students to work in groups to answer additional questions.
4. Design a survey to estimate the amount of time preschoolers spend watching television in a week. Carefully define the population, frame, sampling unit, and random selection plan.
Some students conclude that they must sample households, not individuals. Others consider sampling children from organized preschools in the region. We subsequently discuss how that plan misses children who do not attend formal preschool.
5. When scientists estimate a wildlife population such as elk, they say that they are conducting an animal census.'' Discuss the use of the word census from the perspective of this course.
6. Scientists wish to study the effects of listening to music while studying. Poulton (1977) found that music masks inner speech in working memory. Decide whether the following study is an experiment, a survey, or an observational study. Justify your answer.
Researchers describe their upcoming study in the college newspaper. A few days later, they go room to room in dormitories and ask students about their music listening habits. They also obtain grade point averages of students who sign a release form.
Most students conclude that this is an observational study. Many students point out that not all students read the college newspaper and that those students who sign a release for their grades may be different from those who do not. Other students think this is a good example of an experiment or survey. They are chagrined to find out they are wrong.
17 The following test questions examine students' conceptual understanding of the material on experiments, surveys, and observational studies.
1. Classify each of the following studies as an experiment, survey, or observational study. Explain your reasoning in a short sentence.
(a) A study evaluating consumers' satisfaction with their current long distance telephone carriers.
(b) A study of differences between bull and cow elk foraging strategies.
(c) A study of the effectiveness of several antibiotics in controlling lung congestion in newborns.
(d) A study of radio stations to see what proportion use personal computer-based record keeping systems.
(e) An experiment station study of potato yields under center pivot irrigation and furrow irrigation.
2. The Salk polio vaccine study was _randomized controlled_. What does controlled mean in this context?
3. In an observational study of length of stay for childbirth in large versus small hospitals,
(a) What is the population?
(b) What is the observational unit?
(c) What can be randomized in this study?
A common incorrect answer identifies mothers as the population and a mother as the observational unit. Some students correctly identify large and small hospitals as the population and a hospital as the observational unit with clusters of mothers as subunits. Some groups also point out that large and small hospitals can be randomly selected from among all such hospitals, but clearly one cannot randomize large'' and small.''
4. I would like to know the various proportions of single rooms, double rooms, and multiple (more than two people) rooms in the dormitories, fraternities, and sororities on campus. I am going to conduct a statistical survey.
(a) Define the sampling unit. (Hint: It is not a person.)
(b) Define the population.
It is obvious that there is not always a single best answer to some of these questions. That is why the students must write a sentence or two explaining their position. Learning occurs as they think through the possible answers and compose a defense for the one they choose.
## 2.2. Teaching About Populations and Random Variables
18 Many introductory statistics texts have a unit on probability that does not relate to the rest of the course. We try to limit our discussion of probability to those skills that the students need to successfully complete the statistics chapters. Probability mass functions (pmf's) and probability density functions (pdf's) are usually not covered in the first course. We think covering pmf's and pdf's in a nonmathematical fashion accomplishes several important goals.
• It helps to reinforce the idea of a population.
• It establishes PARAMETERS as rational descriptors of populations.
• It allows us to think about random sampling and random variables.
The use of uniform density functions and (asymmetric) triangular distributions allows students with only an algebra background to find areas under the curve and the population MEAN and MEDIAN. (Words in uppercase letters denote population parameters.)
19 The elementary rules of probability can be covered in the context of drawing a unit at random from a population with given pdf or pmf. The probability that the observed response is below a specified level can be found by finding the appropriate area under the curve. The density can also be used to illustrate disjoint events and the complement of an event.
20 The group exercises on this unit take the following form:
1. If half of voters support Clinton's health care plan (1) and half do not (0), draw the appropriate mass function and carefully label the axes.
2. Describe a response variable, y, of interest in (one of) your field(s). Draw a likely mass or density function for this variable. Mark the population mean with the symbol \mu and the population median with an M.
This is a difficult question for many students because they have to produce a relevant example from their own experience.
3. Suppose that average grades in this course are uniformly distributed between 60 and 100. [Students are given a graph of the probability density function.]
(a) What is the probability that a randomly chosen student will make an A (that is, have an average grade above 90)?
(b) The vertical axis is labeled relative density.'' What does this mean?
Whether or not students correctly find P(A) = 1/4, we give them partial credit for properly shading the area above 90 and starting the calculation. Sometimes we identify the height of the curve as 1/40; sometimes we do not. When we do not, the students have a more difficult time completing the task in the time allotted.
21 Appropriate exam questions over material of this type include the following.
1. If an observational study is conducted to determine the amount of time three and four-year-old children watch TV per week, then the distribution of y = time per week is a (mass, density) function. (Circle one.) Why?
2. If people are ambivalent about requiring medical personnel to disclose whether or not they are HIV positive, they might answer an appropriately worded question with equal proportions of 1 (strongly agree), 2 (agree), 3 (neutral), 4 (disagree), and 5 (strongly disagree). The mass function appears below. [Students are given a graph of the pmf.]
(a) What is the MEAN?
(b) What is the height of each of the sticks''?
The skills required here are understanding and visualization, not computation.
3. The heights of college females are normally distributed with MEAN = 5'4'' and VARIANCE = 4''. What is the corresponding SD and what does it tell you?
What is the corresponding SD ...?'' is a mechanical question. What does it tell you?'' adds a conceptual component.
## 2.3. Teaching About Descriptive Statistics
22 In our coverage of descriptive statistics, we emphasize ideas rather than computation and suggest that one use a computer package to do the calculations. We ask students to analyze data about themselves that they generated during the first week of class. We use a variation of the Activity-Based Statistics Project exercise, Getting to Know the Class (Scheaffer, Gnanadesikan, Watkins, and Witmer, in press). The students are asked to provide answers to questions like, How many sisters and brothers do you have?'' or What time did you go to bed last night?'' The idea is to generate data of differing types in which the students will have a vested interest. We use MYSTAT or SYSTAT to analyze the data.
23 Group exercises include the following questions:
1. For the data on students' pulse rates, how many bars should the histogram have? Explain.
The old emphasis used to be on mechanics---bins and drawing. The more interesting questions relate to the uses of a histogram and how the visual information is changed by varying the number of bars.
2. For the data on bedtimes, is the mean or median preferred as a measure of the middle'' of the data? Explain your reasoning.
3. Draw lines connecting items in column A to the most related item in column B.
A B
mean histogram
probability density function MEAN(Y - MEAN)^3/SD^3
MEDIAN MEAN
skewness median
sd \sqrt{variance}
To help groups work through this exercise, we have to help them overcome their initial reticence. If we can get them to connect mean and MEAN, then they usually can complete the remaining connections.
4. If answers to the question Do you smoke?'' are coded as 1 = yes and 0 = no, show that the sample mean is the observed proportion of yeses.
Students initially balk at this question because it looks like a proof. However, the answer is quite straightforward, and, with a little support from the instructor or teaching assistant, most groups put down a coherent argument.
## 2.4. Teaching About Random Sampling and Sampling Distributions
24 Descriptive statistics appear naturally in this class because the students know about collecting data and describing populations. The sample statistics---mean, median, variance, sd---become obvious estimators of their population counterparts---MEAN, MEDIAN, VARIANCE, SD. Students accept histograms, stem-and-leaf plots, and dot plots as natural estimators of density or mass functions. We define the sampling distribution of sample means and the PARAMETERS MEAN(mean), VARIANCE(mean), and SE(mean). But more importantly (and where we differ from most beginning texts), we also talk about the sampling distributions of the median, sd, and range and their corresponding PARAMETERS MEAN(statistic), VARIANCE(statistic), SE(statistic), and MEDIAN(statistic). Only in this way do students get the big idea.
25 We use the following group exercises:
1. The handout sheet represents 100 apartment buildings. The number of squares in a group denotes the number of children requiring after school care for that building.
(a) Look at the sheet and estimate (by eye) the average number of children needing care per building.
(b) Pick five representative'' buildings and calculate the average number of children needing care for the five representative buildings that you selected.
(c) Using the random number table provided, pick five buildings at random and calculate the average number of children needing care.
This is a variation on the random rectangle'' exercise from the Activity-Based Statistics Project (Scheaffer et al., in press). It is an effective exercise for helping students discover the worth of random sampling. A dot plot of the estimates from the eyeball, representative, and random sampling procedures along with the known population distribution and the known distribution of means from random samples of size five is compelling for students.
2. Randomly'' select classmates and form several samples of size three. For the variable y = distance you live from this classroom building, calculate the sample mean of each sample.
The instructor then collects all of the sample means and constructs a dot plot to give the students a sense of the sampling distribution of the mean. Repeat the exercise using the median or standard deviation as the statistic calculated.
3. The notation mean(MEAN) makes no sense. What is wrong with it?
We want the students to conclude that MEAN is a constant and hence one cannot produce numbers that can be averaged in the sense of mean( ). A common answer is, mean(MEAN) makes no sense because you cannot take the sample mean of the MEAN.'' We give them partial credit for paraphrasing the question and mark the answer with Why?'' in the margin.
4. Explain the difference between MEAN and mean.
5. Explain the difference between MEAN and MEAN(mean).
Common wrong answers include MEAN is the mean of the population' while MEAN(mean) is the mean of a subpopulation of means and will have some value because of estimation'' and MEAN is the mean of a population' while MEAN(mean) is the mean of a sample of means.'' It is common for them to understand what a population mean is, but to misconstrue what MEAN(mean) is. However, the answers often show that they are trying to understand the concepts.
26 Exam questions on this material include the following.
1. The ACT test has a MEAN of 19 and an SD of 1.1. If we sample 50 high school seniors who have taken the ACT, find
(a) MEAN(mean) = ____________
(b) VARIANCE(mean) = ____________
(c) SE(mean) = ____________
(d) MEAN(variance) = ____________
(e) VARIANCE(variance) = ____________
2. An analysis of cholesterol data for 106 adults appears below. In the sample'' column write the name of the statistic and give its value.
Systat analysis of blood serum cholesterol data
CHOLESTR
N of CASES 106
MINIMUM 120
MAXIMUM 414
MEAN 217
STANDARD DEV 53
MEDIAN 216
POPULATION sample
MEAN e.g., mean = 217
MEDIAN
VARIANCE
SD
RANGE
This exam question reinforces the connection between PARAMETERS and statistics. It also reinforces our predilection to encourage students to use computer packages for real statistical work.
3. The U. S. Center for Health Statistics collects data on the daily intake of selected nutrients by income level. Suppose we collect data on protein intake for 15 people randomly selected from individuals whose incomes are below the poverty level and 10 people randomly selected from individuals whose incomes are above the poverty level. A computer summary of the data appears on the back of the previous page. Provide the following:
(a) mean_1
(b) sd_2
(c) mean_1 - mean_2
(d) MEAN(mean_1)
(e) VARIANCE(mean_1)
(f) SE(mean_1)
(g) variance(mean_1)
(h) se(mean_1)
27 The computer printout provided consists of sample size, mean, variance, standard deviation, and standard error by group. For mean_1, sd_2, and se(mean_1), the student merely has to identify the appropriate number on the printout. The variance(mean_1) is the square of se(mean_1). The difference between sample means requires identifying the two numbers and subtracting them. MEAN(mean_1), VARIANCE(mean_1), and SE(mean_1) are difficult. The correct answer is a formula, not a number. Of course, the students will have worked a similar example in class, so they should know how to answer the question. Nevertheless, many students feel compelled to put down a number.
28 We have illustrated our methods using material on design, populations and probability, descriptive statistics, and sampling distributions. We use similar material for inference in finite populations, including confidence intervals, basic hypothesis testing, and simple linear regression and correlation.
# 3. Conclusions
29 Through the use of this material, we have been successful in dealing with statistics in a conceptual fashion at the lower division level. We feel there has been too much emphasis in the past on rote learning and mechanics. It is possible to reformulate traditional materials on probability, descriptive statistics, and sampling distributions in a conceptual, yet straightforward, way that engages students and helps them to understand the ideas of statistics without getting lost in the details. With the wide availability of statistical software on a variety of platforms, there is no reason to dwell on the mechanics of statistical methods. Why teach students to be \$10 calculators? Help them discover statistical thinking and the ability to solve real-world problems.
30 We have developed material that is not part of the traditional course. The unit on experiments, surveys, and observational studies above illustrates how important ideas that have been left out of traditional introductory courses can be added at a level that college students can understand.
31 Another topic that we have added is basic inference in finite populations. Most of our students will be faced with interpreting sample surveys in their technical and personal lives. They will see surveys every day. After our unit on simple, stratified, cluster, and systematic sampling and our formulation of basic inference for simple random sampling, students can understand the graphic over Dan Rather's shoulder that says that the margin of error is plus or minus 3%. The conceptual development of statistics has many rewards.
32 Instructors can use the conceptual approach with groups or individuals. We lean toward cooperative group learning. Our experiences with this approach in introductory statistics courses have been extremely positive. We think this approach directly addresses the concerns about the first course in statistics discussed by Hogg (1991), Watts (1991), Cobb (1993), and Snee (1993), among others. Students' attitudes are better and test scores have improved (Keeler and Steinhorst 1995).
# References
Brooks, J. G. and Brooks, M. G. (1993), In Search of Understanding: The Case for Constructivist Classrooms, Alexandria, VA: Association for Supervision and Curriculum Development.
Cobb, G. (1993), Statistical Thinking and Teaching Statistics,'' UME Trends, November.
Freedman, D., Pisani, R., Purves, R. and Adhikari, A. (1991), Statistics (2nd ed.), New York: W. W. Norton and Company.
Garfield, J. (1993), "Teaching Statistics Using Small-Group Cooperative Learning," Journal of Statistics Education [Online], 1(1). (http://www.amstat.org/publications/jse/v1n1/garfield.html)
Hogg, R. V. (1991), Statistical Education: Improvements Are Badly Needed,'' The American Statistician, 45(4), 342-343.
Keeler, C. M. and Steinhorst, R. K. (1995), Using Small Groups to Promote Active Learning in the Introductory Statistics Course: A Report from the Field,'' Journal of Statistics Education [Online], 3(2). (http://www.amstat.org/publications/jse/v3n2/keeler.html)
MacGregor, J. (1990), Collaborative Learning: Shared Inquiry as a Process of Reform,'' New Directions for Teaching and Learning, 42, 19-30.
Mehler, A. H. (1992), Integration of Examinations and Education,'' Biochemical Education, 20, 10-14.
Poulton, E. C. (1977), Continuous Intense Noise Masks Auditory Feedback and Inner Speech,'' Psychological Bulletin, 84(5), 977-1001.
Scheaffer, R., Gnanadesikan, M., Watkins, A., and Witmer, J. (in press), Activity-Based Statistics, New York: Springer-Verlag.
Snee, R. (1993), What's Missing in Statistical Education?,'' The American Statistician, 47(2), 149-154.
Watts, D. G. (1991), Why Is Introductory Statistics Difficult to Learn? And What Can We Do to Make It Easier?,'' The American Statistician, 45(4), 290-291.
Wiggins, G. P. (1993), Assessing Student Performance, San Francisco, CA: Jossey-Bass.
R. Kirk Steinhorst
Division of Statistics
University of Idaho
Moscow, ID 83843
kirk@uidaho.edu
Carolyn M. Keeler
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## onegirl 2 years ago Determine all horizontal, slant, and vertical asymptotes. For each vertical asymptote, determine whether f(x) -> infinity sign of f(x) -> negative infinity sign on either side of the asymptote.
1. onegirl
f(x) = x^2/4 - x^2
2. satellite73
is it $\frac{x^2}{4-x^2}$?
3. onegirl
yes
4. satellite73
ok so for the vertical asymptotes, set the denominator equal to zero and solve for $$x$$
5. onegirl
ok
6. satellite73
you get $4-x^2=0$ or $(2-x)(2+x)=0$ and the solutions are $$x=-2$$ or $$x=2$$ those are your vertical asymptotes (there are two of them)
7. onegirl
ok
8. satellite73
for the horizontal asymptote, note that the numerator and denominator have the same degree (both are degree 2) so it is the ratio of the leading coefficients
9. onegirl
okay
10. satellite73
the leading coefficient of $$x^2$$ is 1 and the leading coefficient of $$4-x^2$$ is $$-1$$ therefore the horizontal asymptote is $$y=-1$$
11. satellite73
that is in the case where the degrees are the same. there is no slant asymptote for there to be a slant asymptote , the degree of the numerator would have to be one more than the degree of the denominator
12. onegirl
okay thx
13. satellite73
yw
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Next: Computing the adjunction divisor Up: Computational Aspects Previous: Computational Aspects
### Computing the places of
A place of is represented by a triple, consisting of
• the closed point corresponding to ,
• the degree of the place (that is, the minimal degree of a field extension defining ) and
• a symbolic Hamburger-Noether expression HN for the local branch corresponding to (defined over a primitive algebraic field extension of degree ).
Recall that denotes the formal sum of a point with its conjugates. Affine closed points will be represented by a defining (triangular) ideal , while closed points at infinity are usually stored in form of a homogeneous polynomial (the defining prime factor of ).
Note that the conjugates of a place are given by the triples HN, where HN runs through the conjugates of HN. Hence, when computing the closed places of , we can restrict ourselves to computing one representing place for each. We apply the following algorithm:
Input
Squarefree homogeneous polynomial ,
degree bound .
Output
List of all closed singular places and all closed non-singular places up to degree of the plane curve defined by .
1. Affine singular points. Let and the Tjurina ideal of . Compute a triangular system for , that is, a system of triangular bases such that .
Here, by a triangular basis one denotes a reduced lexicographical Gröbner basis of the form with a monic polynomial in and . Triangular systems can be computed effectively, basically by two different methods, one due to Lazard [27,7], the other due to Möller [31]. Choose any of these methods to compute a triangular system for , , . For each ,
• compute a prime factorization of in ,
• for , let be the primitive field extension defined by the irreducible polynomial . Compute a prime factorization of in ,
Finally, the closed affine singular points are given by the set of ideals
where is the image of when substituting the parameter by .
2. Points at infinity. Let and compute a prime factorization of the polynomial ,
(4)
Let be a root of and define
where denotes the formal sum of the point (defined over ) with its conjugates. (It is represented by .)
We denote by the subset of closed singular points. To check whether a point is singular or not, one has to check whether (these computations can be performed over the finite field extension ).
Finally, consider the (closed) point : if then has to be added to ; if, additionally, and vanish then it has to be added to , too.
The sets (resp. ) are the sets of closed (singular) points at infinity.
3. Affine singular places. To each closed affine singular point given by a (triangular) ideal we compute the corresponding places in form of a system of symbolic Hamburger-Noether expressions for the respective germ (defined over ). More precisely, a closed place over is the formal sum of a place described by one of the computed sHNE with its conjugates.
The computation of the symbolic Hamburger-Noether expressions has to be performed in the local ring where is a primitive field extension (of degree ) such that decompose into linear factors. Note that during the computation of a sHNE further field extensions might be necessary.
4. Singular places at infinity. To each closed singular point in we compute a system of sHNE for the local germ (defined over ). To be precise, if then we compute a system of sHNE for in ; if then the system of sHNE is computed for in .
5. Non-singular affine closed points up to degree . For each do the following:
• let and set .
• Proceed as in Step 1 to obtain a set of (triangular) ideals corresponding to the set of closed points defined over .
• For all non-singular (given by ) compute the degree ). If then compute the corresponding closed place (that is, a sHNE for the germ ) and add it to the list of closed places.
Remark 2.10 It is interesting to notice that triangular sets have mainly been used for numerical purpose, since they allow a fast and stable numerical solving of polynomial systems (cf. [27,31,13]), and this has been the reason for implementing it in SINGULAR. Several experiments have shown that they behave also superior against other methods to represent closed points over finite fields.
Next: Computing the adjunction divisor Up: Computational Aspects Previous: Computational Aspects
Christoph Lossen
2001-03-21
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# Gravity, acceleration and reference frames [closed]
Let’s say the universe was empty and suddenly an astronaut and the sun appeared 2 light years apart. Using the reference frame of the Astronaut, would he be pulled towards the sun as soon as he can see it?
Q1. And from that moment, would his acceleration be constant until he smashed into the sun? What would that acceleration be and what would his top speed be?
I’m assuming here that gravity propagates at the speed of light to infinity (which is in line with mainstream theories as far as I’m aware). One of the things I’m interested in hearing thoughts on, is whether the gravity pull by the sun is just as strong 2 light years away, since there are no other gravitational forces at play in this scenario.
Also, let’s for argument’s sake say we consider this scenario from the reference frame of an observer located between the two objects - or slightly to one side so he doesn’t get hit by the astronaut ;-).
Q2. Would he observe the astronaut unaffected by the sun’s gravity for 1 light year before the gravitational waves reaches the astronaut? And what would then happen to the astronaut from the observer’s frame of reference, in terms of acceleration, speed, etc?
Here are my assumptions about the observer:
• The observer is not impacted by gravity • The observer is not using ‘eyes’ to observe, but rather a clever apparatus that can detect any object in the universe, including its speed and location (relative to the observer and also relative to each other). This deals with the problem of not being able to see the astronaut at the same time as the sun due to lack of light emission. • The method of observation is still subject to light speed, i.e. the apparatus will only detect an object once the particles or waves (travelling at the speed of light) emitted from that object has reached the apparatus
I realise time itself, as we know it (years, miles per second, etc), may be completely meaningless in this scenario – but try to entertain me. It is a hypothetical question after all (which perhaps means that there is no meaningful answer, turning the discussion into philosophy instead..)
## closed as off-topic by Jon Custer, Qmechanic♦Feb 7 '17 at 21:43
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "We deal with mainstream physics here. Questions about the general correctness of unpublished personal theories are off topic, although specific questions evaluating new theories in the context of established science are usually allowed. For more information, see Is non mainstream physics appropriate for this site?." – Jon Custer, Qmechanic
If this question can be reworded to fit the rules in the help center, please edit the question.
• Since the force of gravity varies as $1/r^{2}$, the gravity "pull" 2 light years away is not the same as 1 light year away, much less 1 mile away. – Jon Custer Feb 7 '17 at 19:19
• Post (v1) closed as a non-mainstream scenario: E.g a star cannot suddenly appear. – Qmechanic Feb 7 '17 at 21:44
• I believe these kind of "gedanken" experiments could help people understand physics better. Closing it as non-mainstream physics or personal theory seems extreme to me. The inconsistencies introduced by assuming the sudden unphysical appearance of an object are irrelevant to the argument. – user126422 Feb 7 '17 at 22:55
• This isn't a question about challenging the correctness of accepted theories, but trying to understand the established ones. – Jake Watrous Feb 8 '17 at 0:26
• Note: his impact speed will be around the escape speed from the Sun, which is around 500 km/s. I would be happy to explain it reason in an answer, if this question wouldn't be closed. Also I've heard there are (unaccepted, post-GR, but so or so mainstream) theories in which such a Universe wouldn't have gravity (but the focus of my answer would be the mainstream answer). – peterh Feb 8 '17 at 1:27
1) Yes. Gravity propagates at the speed of light.
2) Acceleration would be continuous, but accelerate as the distance between the sun and astronaut diminishes. Acceleration and maximum speed depend on the mass of the sun.
3) As the previous comment states, gravitational pull is not the same at all distances. The closer you are, the more influence it has on you.
Tried to answer all of your questions simply. Let me know if I missed one; they were somewhat hard to spot.
Newton's Law of Gravitation would be a good read for you, but in general it states that the force of gravitational attraction is directly proportional to the product of the masses being considered (sun * astronaut (including whatever the astronaut is wearing). Call it G=SA
But it is also inversely proportional to the square of the distance between them.
Since we are dealing with two objects, we can simplify the calculation a bit and we get something like:
F(r) = mg(r)
F is the force applied on the astronaut due to the sun.
G is the gravitational constant: 6.67408(31)×10−11 m3⋅kg−1⋅s−2
R is the distance between the two objects
M is the astronaut's mass
Since you seem interested in velocity (measured in m/s2): V(r) = -G*m1/r
M1 is the sun's mass and for simplicity's sake we are assuming its mass is evenly distributed.
• I think you should definitely include Newton's law of gravity, as to me, the OP does need the effect of $r$ explained explicitly. – user140606 Feb 7 '17 at 21:07
• Oh, he meant Sol! I assumed he'd just used "sun" to mean any star. – Jake Watrous Feb 8 '17 at 1:27
• @JakeWatrous, where did you get that value for velocity? – TheEnvironmentalist Feb 8 '17 at 5:54
• The 500km/s? I didn't. Peterh did. It's relatively easy to calculate, though. He assumed you meant our own sun, and plugged its mass into this equation: g = G*M/R^2, where g is the acceleration due to gravity, G is the universal gravitational constant, M is mass, and R is distance. – Jake Watrous Feb 8 '17 at 5:57
• Thank you all for your comments and answers, they were very useful. I don't quite understand why this question was closed, I wasn't trying to propose outlandish theories or veer from mainstream physics, but rather try to understand established theories by using a thought experiment that my non-scientific brain might be able to understand better. Just like Albert Aspect pointed out, this could be useful to make physics more popular! As this was my first post however, I'll try to stick to the rules next time ;) Thank you @JakeWatrous JonCuster AlbertAspect peterh KyleKanos Countt010 – Display Name Feb 8 '17 at 14:28
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# Exact Differentials
1. Sep 1, 2010
I am reading a math review in my thermodynamics text and I a little confused by this. Here is the excerpt:
I am confused by the part where it says
If they selected x = (y, z) then isn't that saying that x is dependent on y? So how can we just turn around and say y = y(x, z) ? That is, if we selected x as dependent in the first function, why can we turn around and call it independent in the second.
Sorry, this might be a stupid question. I just don't see why we bother calling variables independent and dependent in a situation like this?
2. Sep 1, 2010
### Office_Shredder
Staff Emeritus
As a simple example if you had the equation x+y+z=0 you could write any variable as a function of the other two quite simply.
The dependent/independent lines are obviously blurred here; you just use them for the purposes of being able to describe what counts as a function and what's being considered as a variable when differentiating
3. Sep 1, 2010
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# Apparent magnitude
(Redirected from Apparent visual magnitude)
For a more detailed discussion of the history of the magnitude system, see Magnitude (astronomy).
Asteroid 65 Cybele and two stars, with their magnitudes labeled
The apparent magnitude (m) of a celestial object is a number that is a measure of its brightness as seen by an observer on Earth. The brighter an object appears, the lower its magnitude value (i.e. inverse relation). The Sun, at apparent magnitude of −27, is the brightest object in the sky. It is adjusted to the value it would have in the absence of the atmosphere. Furthermore, the magnitude scale is logarithmic: a difference of one in magnitude corresponds to a change in brightness by a factor of ${\displaystyle {\sqrt[{5}]{100}}}$ or about 2.512.
Generally, the visible spectrum (vmag) is used as a basis for the apparent magnitude. However, other spectra are also used (e.g. the near-infrared J-band). In the visible spectrum, Sirius is the brightest star after the Sun. In the near-infrared J-band, Betelgeuse is the brightest. The apparent magnitude of stars is measured with a bolometer.
## History
Visible to
typical
human eye[1]
Apparent
magnitude
Brightness
relative
to Vega
Number of stars
brighter than
apparent magnitude[2]
in the night sky
Yes −1.0 250% 1
0.0 100% 4
1.0 40% 15
2.0 16% 48
3.0 6.3% 171
4.0 2.5% 513
5.0 1.0% 1 602
6.0 0.40% 4 800
6.5 0.25% 9 096[3]
No 7.0 0.16% 14 000
8.0 0.063% 42 000
9.0 0.025% 121 000
10.0 0.010% 340 000
The scale used to indicate magnitude originates in the Hellenistic practice of dividing stars visible to the naked eye into six magnitudes. The brightest stars in the night sky were said to be of first magnitude (m = 1), whereas the faintest were of sixth magnitude (m = 6), which is the limit of human visual perception (without the aid of a telescope). Each grade of magnitude was considered twice the brightness of the following grade (a logarithmic scale), although that ratio was subjective as no photodetectors existed. This rather crude scale for the brightness of stars was popularized by Ptolemy in his Almagest, and is generally believed to have originated with Hipparchus.
In 1856, Norman Robert Pogson formalized the system by defining a first magnitude star as a star that is 100 times as bright as a sixth-magnitude star, thereby establishing the logarithmic scale still in use today. This implies that a star of magnitude m is 2.512 times as bright as a star of magnitude m+1. This figure, the fifth root of 100, became known as Pogson's Ratio.[4] The zero point of Pogson's scale was originally defined by assigning Polaris a magnitude of exactly 2. Astronomers later discovered that Polaris is slightly variable, so they switched to Vega as the standard reference star, assigning the brightness of Vega as the definition of zero magnitude at any specified wavelength.
Apart from small corrections, the brightness of Vega still serves as the definition of zero magnitude for visible and near infrared wavelengths, where its spectral energy distribution (SED) closely approximates that of a black body for a temperature of 11,000 K. However, with the advent of infrared astronomy it was revealed that Vega's radiation includes an Infrared excess presumably due to a circumstellar disk consisting of dust at warm temperatures (but much cooler than the star's surface). At shorter (e.g. visible) wavelengths, there is negligible emission from dust at these temperatures. However, in order to properly extend the magnitude scale further into the infrared, this peculiarity of Vega should not affect the definition of the magnitude scale. Therefore, the magnitude scale was extrapolated to all wavelengths on the basis of the black body radiation curve for an ideal stellar surface at 11,000 K uncontaminated by circumstellar radiation. On this basis the spectral irradiance (usually expressed in janskys) for the zero magnitude point, as a function of wavelength can be computed (see [1]). Small deviations are specified between systems using measurement appartuses developed independently so that data obtained by different astronomers can be properly compared; of greater practical importance is the definition of magnitude not at a single wavelength but applying to the response of standard spectral filters used in photometry over various wavelength bands.
With the modern magnitude systems, brightness over a very wide range is specified according to the logarithmic definition detailed below, using this zero reference. In practice such apparent magnitudes do not exceed 30 (for detectable measurements). The brightness of Vega is exceeded by four stars in the night sky at visible wavelengths (and more at infrared wavelengths) as well as bright planets such as Venus, Mars, and Jupiter, and these must be described by negative magnitudes. For example, Sirius, the brightest star of the celestial sphere, has an apparent magnitude of −1.4 in the visible; negative magnitudes for other very bright astronomical objects can be found in the table below.
## Calculations
30 Doradus image taken by ESO's VISTA. This nebula has an apparent magnitude of 8.
As the amount of light actually received by a telescope is reduced by transmission through the Earth's atmosphere, any measurement of apparent magnitude is corrected for what it would have been as seen from above the atmosphere. The dimmer an object appears, the higher the numerical value given to its apparent magnitude, with a difference of 5 magnitudes corresponding to a brightness factor of exactly 100. Therefore, the apparent magnitude m, in the spectral band x, would be given by:
${\displaystyle m_{x}=-5\log _{100}\left({\frac {F_{x}}{F_{x,0}}}\right)}$
which is more commonly expressed in terms of common (base 10) logarithms as:
${\displaystyle m_{x}=-2.5\log _{10}\left({\frac {F_{x}}{F_{x,0}}}\right)\;,}$
where ${\displaystyle F_{x}}$ is the observed flux using spectral filter x, and ${\displaystyle F_{x,0}}$ is the reference flux (zero-point) for that photometric filter. Since an increase of 5 magnitudes corresponds to a decrease in brightness by a factor of exactly 100, each magnitude increase implies a decrease in brightness by the factor ${\displaystyle {\sqrt[{5}]{100}}\approx 2.512}$ (Pogson's Ratio). Inverting the above formula, a magnitude difference ${\displaystyle m_{1}-m_{2}=\Delta m}$ implies a brightness factor of ${\displaystyle F_{2}/F_{1}=100^{\Delta m/5}=10^{.4\cdot \Delta m}\approx 2.512^{\Delta m}}$.
### Example: Sun and Moon
What is the ratio in brightness between the Sun and the full moon?
The apparent magnitude of the Sun is −26.74 (brighter), and the mean apparent magnitude of the full moon is −12.74 (dimmer).
Difference in magnitude:
${\displaystyle x=m_{1}-m_{2}=(-12.74)-(-26.74)=14.00}$
Brightness factor:
${\displaystyle v_{b}=10^{0.4x}=10^{0.4*14.00}\approx 398400}$
The Sun appears about 400,000 times brighter than the full moon.
Sometimes one might wish to add brightnesses. For example, photometry on closely separated double stars may only be able to produce a measurement of their combined light output. How would we rekon the combined magnitude of that double star knowing only the magnitudes of the individual components? This can be done by adding the brightnesses (in linear units) corresponding to each magnitude.[5]
${\displaystyle 10^{-m_{f}*0.4}=10^{-m_{1}*0.4}+10^{-m_{2}*0.4}\!\ }$
Solving for ${\displaystyle m_{f}}$ yields
${\displaystyle m_{f}=-2.5\log _{10}\left(10^{-m_{1}*0.4}+10^{-m_{2}*0.4}\right)\!\ }$
where ${\displaystyle m_{f}}$ is the resulting magnitude after adding the brightnesses referred to by ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$.
### Absolute magnitude
Main article: Absolute magnitude
Since flux decreases with distance according to the inverse-square law, a particular apparent magnitude could equally well refer to a star at one distance, or a star 4 times brighter at twice that distance, etc. When one is not interested in the brightness as viewed from earth, but the intrinsic brightness of an astronomical object, then one refers not to the apparent magnitude but the absolute magnitude. The absolute magnitude M, of a star or astronomical object is defined as the apparent magnitude it would have as seen from a distance of 10 parsecs (about 32.6 light years). The absolute magnitude of the Sun is 4.83 in the V band (yellow) and 5.48 in the B band (blue).[6] In the case of a planet or asteroid, the absolute magnitude H rather means the apparent magnitude it would have if it were 1 astronomical unit from both the observer and the Sun.
## Standard reference values
Standard apparent magnitudes and fluxes for typical bands[7]
Band ${\displaystyle \lambda }$ (${\displaystyle \mu m}$) ${\displaystyle \Delta \lambda /\lambda }$(FWHM) Flux at m = 0, ${\displaystyle F_{x,0}}$ (Jy) Flux at m = 0, ${\displaystyle F_{x,0}}$ ${\displaystyle (10^{-20}{\text{ erg/s/cm}}^{2}{\text{/Hz}})}$
U 0.36 0.15 1810 1.81
B 0.44 0.22 4260 4.26
V 0.55 0.16 3640 3.64
R 0.64 0.23 3080 3.08
I 0.79 0.19 2550 2.55
J 1.26 0.16 1600 1.6
H 1.60 0.23 1080 1.08
K 2.22 0.23 670 0.67
L 3.50
g 0.52 0.14 3730 3.73
r 0.67 0.14 4490 4.49
i 0.79 0.16 4760 4.76
z 0.91 0.13 4810 4.81
It is important to note that the scale is logarithmic: the relative brightness of two objects is determined by the difference of their magnitudes. For example, a difference of 3.2 means that one object is about 19 times as bright as the other, because Pogson's Ratio raised to the power 3.2 is approximately 19.05. A common misconception is that the logarithmic nature of the scale is because the human eye itself has a logarithmic response. In Pogson's time this was thought to be true (see Weber-Fechner law), but it is now believed that the response is a power law (see Stevens' power law).[8]
Magnitude is complicated by the fact that light is not monochromatic. The sensitivity of a light detector varies according to the wavelength of the light, and the way it varies depends on the type of light detector. For this reason, it is necessary to specify how the magnitude is measured for the value to be meaningful. For this purpose the UBV system is widely used, in which the magnitude is measured in three different wavelength bands: U (centred at about 350 nm, in the near ultraviolet), B (about 435 nm, in the blue region) and V (about 555 nm, in the middle of the human visual range in daylight). The V band was chosen for spectral purposes and gives magnitudes closely corresponding to those seen by the light-adapted human eye, and when an apparent magnitude is given without any further qualification, it is usually the V magnitude that is meant, more or less the same as visual magnitude.
Because cooler stars, such as red giants and red dwarfs, emit little energy in the blue and UV regions of the spectrum their power is often under-represented by the UBV scale. Indeed, some L and T class stars have an estimated magnitude of well over 100, because they emit extremely little visible light, but are strongest in infrared.
Measures of magnitude need cautious treatment and it is extremely important to measure like with like. On early 20th century and older orthochromatic (blue-sensitive) photographic film, the relative brightnesses of the blue supergiant Rigel and the red supergiant Betelgeuse irregular variable star (at maximum) are reversed compared to what human eyes perceive, because this archaic film is more sensitive to blue light than it is to red light. Magnitudes obtained from this method are known as photographic magnitudes, and are now considered obsolete.
For objects within the Milky Way with a given absolute magnitude, 5 is added to the apparent magnitude for every tenfold increase in the distance to the object. This relationship does not apply for objects at very great distances (far beyond the Milky Way), because a correction for general relativity must then be taken into account due to the non-Euclidean nature of space.[citation needed]
For planets and other Solar System bodies the apparent magnitude is derived from its phase curve and the distances to the Sun and observer.
## Table of notable celestial objects
Apparent visual magnitudes of known celestial objects
App. mag. (V) Celestial object
−40.98 Rho Cassiopeiae as seen from 1 astronomical unit (AU).
−38.00 Rigel as seen from 1 AU. It would be seen as a large very bright bluish scorching ball of 35° apparent diameter.
−30.30 Sirius as seen from 1 astronomical unit
−29.30 Sun as seen from Mercury at perihelion
−27.40 Sun as seen from Venus at perihelion
−26.74[9] Sun as seen from Earth (about 400,000 times brighter than mean full moon)
−25.60 Sun as seen from Mars at aphelion
−25.00 Minimum brightness that causes the typical eye slight pain to look at
−23.00 Sun as seen from Jupiter at aphelion
−21.70 Sun as seen from Saturn at aphelion
−20.20 Sun as seen from Uranus at aphelion
−19.30 Sun as seen from Neptune
−18.20 Sun as seen from Pluto at aphelion
−16.70 Sun as seen from Eris at aphelion
−14.2 An illumination level of one lux [10][11]
−12.90 Maximum brightness of perigee+perihelion full moon (mean distance value is −12.74,[12] though both values are about 0.18 magnitude brighter when including the opposition effect)
−11.20 Sun as seen from Sedna at aphelion
−10 Comet Ikeya–Seki (1965), which was the brightest Kreutz Sungrazer of modern times[13]
−9.50 Maximum brightness of an Iridium (satellite) flare
−7.50 The SN 1006 supernova of AD 1006, the brightest stellar event in recorded history (7200 light years away)[14]
−6.50 The total integrated magnitude of the night sky as seen from Earth
−6.00 The Crab Supernova (SN 1054) of AD 1054 (6500 light years away)[15]
−5.9 International Space Station (when the ISS is at its perigee and fully lit by the Sun)[16]
−4.89 Maximum brightness of Venus[17] when illuminated as a crescent
−4.00 Faintest objects observable during the day with naked eye when Sun is high
−3.99 Maximum brightness of Epsilon Canis Majoris 4.7 million years ago, the historical brightest star of the last and next five million years
−3.82 Minimum brightness of Venus when it is on the far side of the Sun
−2.94 Maximum brightness of Jupiter[18]
−2.91 Maximum brightness of Mars[19]
−2.50 Faintest objects visible during the day with naked eye when Sun is less than 10° above the horizon
−2.50 Minimum brightness of new moon
−2.45 Maximum brightness of Mercury at superior conjunction (unlike Venus, Mercury is at its brightest when on the far side of the Sun, the reason being their different phase curves)
−1.61 Minimum brightness of Jupiter
−1.47 Brightest star (except for the Sun) at visible wavelengths: Sirius[20]
−0.83 Eta Carinae apparent brightness as a supernova impostor in April 1843
−0.72 Second-brightest star: Canopus[21]
−0.49 Maximum brightness of Saturn at opposition and perihelion when the rings are full open (2003)
−0.27 The total magnitude for the Alpha Centauri AB star system. (Third-brightest star to the naked eye)
−0.04 Fourth-brightest star to the naked eye Arcturus[22]
−0.01 Fourth-brightest individual star visible telescopically in the sky Alpha Centauri A
+0.03 Vega, which was originally chosen as a definition of the zero point[23]
+0.50 Sun as seen from Alpha Centauri
1.47 Minimum brightness of Saturn
1.84 Minimum brightness of Mars
3.03 The SN 1987A supernova in the Large Magellanic Cloud 160,000 light-years away.
3 to 4 Faintest stars visible in an urban neighborhood with naked eye
3.44 The well known Andromeda Galaxy (M31)[24]
4.38 Maximum brightness of Ganymede[25] (moon of Jupiter and the largest moon in the Solar System)
4.50 M41, an open cluster that may have been seen by Aristotle[26]
5.20 Maximum brightness of asteroid Vesta
5.32 Maximum brightness of Uranus[27]
5.72 The spiral galaxy M33, which is used as a test for naked eye seeing under dark skies[28][29]
5.73 Minimum brightness of Mercury
5.8 Peak visual magnitude of gamma-ray burst GRB 080319B (the "Clarke Event") seen on Earth on March 19, 2008 from a distance of 7.5 billion light-years.
5.95 Minimum brightness of Uranus
6.49 Maximum brightness of asteroid Pallas
6.50 Approximate limit of stars observed by a mean naked eye observer under very good conditions. There are about 9,500 stars visible to mag 6.5.[1]
6.64 Maximum brightness of dwarf planet Ceres in the asteroid belt
6.75 Maximum brightness of asteroid Iris
6.90 The spiral galaxy M81 is an extreme naked-eye target that pushes human eyesight and the Bortle scale to the limit[30]
7 to 8 Extreme naked-eye limit, Class 1 on Bortle scale, the darkest skies available on Earth[31]
7.78 Maximum brightness of Neptune[32]
8.02 Minimum brightness of Neptune
8.10 Maximum brightness of Titan (largest moon of Saturn),[33][34] mean opposition magnitude 8.4[35]
8.94 Maximum brightness of asteroid 10 Hygiea[36]
9.50 Faintest objects visible using common 7x50 binoculars under typical conditions[37]
10.20 Maximum brightness of Iapetus[34] (brightest when west of Saturn and takes 40 days to switch sides)
12.91 Brightest quasar 3C 273 (luminosity distance of 2.4 billion light years)
13.42 Maximum brightness of Triton[35]
13.65 Maximum brightness of Pluto[38] (725 times fainter than magnitude 6.5 naked eye skies)
15.40 Maximum brightness of centaur Chiron[39]
15.55 Maximum brightness of Charon (the large moon of Pluto)
16.80 Current opposition brightness of Makemake[40]
17.27 Current opposition brightness of Haumea[41]
18.70 Current opposition brightness of Eris
20.70 Callirrhoe (small ~8 km satellite of Jupiter)[35]
22.00 Approximate limiting magnitude of a 24" Ritchey-Chrétien telescope with 30 minutes of stacked images (6 subframes at 300s each) using a CCD detector[42]
22.91 Maximum brightness of Pluto's moon Hydra
23.38 Maximum brightness of Pluto's moon Nix
24.80 Amateur picture with greatest magnitude: quasar CFHQS J1641 +3755[43][44] and limit of a 30-second exposure with Large Synoptic Survey Telescope (LSST).
25.00 Fenrir (small ~4 km satellite of Saturn)[45]
27.70 Faintest objects observable in a 10-hour image with a 8-meter class ground-based telescope such as the Subaru Telescope[46]
28.00 Jupiter if it were located 5000 au from the Sun[47]
28.20 Halley's Comet in 2003 when it was 28 au from the Sun[48]
31.50 Faintest objects observable in visible light with Hubble Space Telescope[49]
35.00 LBV 1806-20, a luminous blue variable star, expected magnitude at visible wavelengths due to interstellar extinction
36.00 Faintest objects observable in visible light[citation needed] with E-ELT
Some of the above magnitudes are only approximate. Telescope sensitivity also depends on observing time, optical bandpass, and interfering light from scattering and airglow.
## References
1. ^ a b "Vmag<6.5". SIMBAD Astronomical Database. Retrieved 2010-06-25.
2. ^ "Magnitude". National Solar Observatory—Sacramento Peak. Archived from the original on 2008-02-06. Retrieved 2006-08-23.
3. ^ Bright Star Catalogue
4. ^
5. ^ "Magnitude Arithmetic". Weekly Topic. Caglow. Retrieved 30 January 2012.
6. ^ Prof. Aaron Evans. "Some Useful Astronomical Definitions" (PDF). Stony Brook Astronomy Program. Retrieved 2009-07-12.
7. ^ Prof. Gregory D. Wirth. "Astronomical Magnitude Systems". Department of Physics and Astronomy, University of Toronto. Retrieved 2012-08-15.
8. ^ E. Schulman & C. V. Cox (1997). "Misconceptions About Astronomical Magnitudes". American Journal of Physics. 65: 1003. Bibcode:1997AmJPh..65.1003S. doi:10.1119/1.18714.
9. ^ Williams, Dr. David R. (2004-09-01). "Sun Fact Sheet". NASA (National Space Science Data Center). Archived from the original on 15 July 2010. Retrieved 2010-07-03.
10. ^ Introduction to Astrophysics: The Stars – Jean Dufay, page 3
11. ^ Ian S. McLean, Electronic imaging in astronomy: detectors and instrumentation Springer, 2008, ISBN 3-540-76582-4 page 529
12. ^ Williams, Dr. David R. (2010-02-02). "Moon Fact Sheet". NASA (National Space Science Data Center). Archived from the original on 23 March 2010. Retrieved 2010-04-09.
13. ^ "Brightest comets seen since 1935". International Comet Quarterly. Retrieved 18 December 2011.
14. ^ Winkler, P. Frank; Gupta, Gaurav; Long, Knox S. (2003). "The SN 1006 Remnant: Optical Proper Motions, Deep Imaging, Distance, and Brightness at Maximum". The Astrophysical Journal. 585: 324–335. arXiv:astro-ph/0208415. Bibcode:2003ApJ...585..324W. doi:10.1086/345985.
15. ^ Supernova 1054 – Creation of the Crab Nebula
16. ^ "ISS Information - Heavens-above.com". Heavens-above. Retrieved 2007-12-22.
17. ^ "HORIZONS Web-Interface for Venus (Major Body=299)" (2006 February 27 (GEOPHYSICAL DATA)). JPL Horizons On-Line Ephemeris System. Retrieved 2010-11-28. (Using JPL Horizons you can see that on 2013-Dec-08 Venus will have an apmag of −4.89)
18. ^ Williams, David R. (2007-11-02). "Jupiter Fact Sheet". National Space Science Data Center. NASA. Retrieved 2010-06-25.
19. ^ Williams, David R. (2007-11-29). "Mars Fact Sheet". National Space Science Data Center. NASA. Archived from the original on 12 June 2010. Retrieved 2010-06-25.
20. ^ "Sirius". SIMBAD Astronomical Database. Retrieved 2010-06-26.
21. ^ "Canopus". SIMBAD Astronomical Database. Retrieved 2010-06-26.
22. ^ "Arcturus". SIMBAD Astronomical Database. Retrieved 2010-06-26.
23. ^ "Vega". SIMBAD Astronomical Database. Retrieved 2010-04-14.
25. ^ Yeomans & Chamberlin. "Horizon Online Ephemeris System for Ganymede (Major Body 503)". California Institute of Technology, Jet Propulsion Laboratory. Retrieved 2010-04-14. (4.38 on 1951-Oct-03)
26. ^ "M41 possibly recorded by Aristotle". SEDS (Students for the Exploration and Development of Space). 2006-07-28. Retrieved 2009-11-29.
27. ^ Williams, David R. (2005-01-31). "Uranus Fact Sheet". National Space Science Data Center. NASA. Archived from the original on 29 June 2010. Retrieved 2010-06-25.
29. ^ Lodriguss, Jerry (1993). "M33 (Triangulum Galaxy)". Retrieved 2009-11-27. (shows b mag not v mag)
30. ^ "Messier 81". SEDS (Students for the Exploration and Development of Space). 2007-09-02. Retrieved 2009-11-28.
31. ^ John E. Bortle (February 2001). "The Bortle Dark-Sky Scale". Sky & Telescope. Retrieved 2009-11-18.
32. ^ Williams, David R. (2007-11-29). "Neptune Fact Sheet". National Space Science Data Center. NASA. Archived from the original on 1 July 2010. Retrieved 2010-06-25.
33. ^ Yeomans & Chamberlin. "Horizon Online Ephemeris System for Titan (Major Body 606)". California Institute of Technology, Jet Propulsion Laboratory. Retrieved 2010-06-28. (8.10 on 2003-Dec-30)
34. ^ a b "Classic Satellites of the Solar System". Observatorio ARVAL. Archived from the original on 31 July 2010. Retrieved 2010-06-25.
35. ^ a b c "Planetary Satellite Physical Parameters". JPL (Solar System Dynamics). 2009-04-03. Archived from the original on 23 July 2009. Retrieved 2009-07-25.
36. ^ "AstDys (10) Hygiea Ephemerides". Department of Mathematics, University of Pisa, Italy. Retrieved 2010-06-26.
37. ^ Ed Zarenski (2004). "Limiting Magnitude in Binoculars" (PDF). Cloudy Nights. Retrieved 2011-05-06.
38. ^ Williams, David R. (2006-09-07). "Pluto Fact Sheet". National Space Science Data Center. NASA. Archived from the original on 1 July 2010. Retrieved 2010-06-26.
39. ^ "AstDys (2060) Chiron Ephemerides". Department of Mathematics, University of Pisa, Italy. Retrieved 2010-06-26.
40. ^ "AstDys (136472) Makemake Ephemerides". Department of Mathematics, University of Pisa, Italy. Retrieved 2010-06-26.
41. ^ "AstDys (136108) Haumea Ephemerides". Department of Mathematics, University of Pisa, Italy. Retrieved 2010-06-26.
42. ^ Steve Cullen (sgcullen) (2009-10-05). "17 New Asteroids Found by LightBuckets". LightBuckets. Retrieved 2009-11-15.
43. ^ Cooperation with Ken Crawford
44. ^
45. ^ Scott S. Sheppard. "Saturn's Known Satellites". Carnegie Institution (Department of Terrestrial Magnetism). Retrieved 2010-06-28.
46. ^ What is the faintest object imaged by ground-based telescopes?, by: The Editors of Sky Telescope, July 24, 2006
47. ^ Magnitude difference is 2.512*log10[(5000/5)^2 X (4999/4)^2] ≈ 30.6, so Jupiter is 30.6 mag fainter at 5000 au
48. ^ "New Image of Comet Halley in the Cold". ESO. 2003-09-01. Archived from the original on 1 March 2009. Retrieved 2009-02-22.
49. ^ The HST eXtreme Deep Field XDF: Combining all ACS and WFC3/IR Data on the HUDF Region into the Deepest Field Ever
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# How to solve this differential equation with Fourier Transform?
Consider the differential equation
$$\dfrac{\partial w}{\partial t} = -\alpha \dfrac{\partial w}{\partial x} + D \dfrac{\partial ^2 w}{\partial x^2}$$
together with the boundary conditions that
$$\lim_{x\to \pm \infty} w(x,t) = \lim_{x\to \pm \infty} \dfrac{\partial w}{\partial x}(x,t) = \lim_{x\to \pm \infty} \dfrac{\partial^2 w}{\partial x^2}(x,t) = 0$$
and the initial condition
$$w(x,0)=\delta(x-x_0).$$
I need to solve this using Fourier Transform. For that, let $\mathcal{F}_x$ be the Fourier Transform operator on the $x$ variable. Applying it to the equation and using the relation for the Fourier transform of a derivative we get
$$\dfrac{\partial}{\partial t}\mathcal{F}_x(w)(k,t) = -\alpha (-ik) \mathcal{F}_x(w)(k,t) + D(-ik)^2 \mathcal{F}_x(w)(k,t),$$
This is equivalent to
$$\dfrac{\partial}{\partial t}\mathcal{F}_x(w)(k,t)- (\alpha ik +Dk^2) \mathcal{F}_x(w)(k,t)=0$$
Considering then $k$ fixed, the solution can be easily be obtained as
$$\mathcal{F}_x(w)(k,t)= A \exp\left[(\alpha ik + Dk^2)t\right].$$
But then we need to invert this. Using the inversion formula and some manipulations I've obtained
$$w(x,t) = \dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \mathcal{F}_x(w)(k,t)e^{-ikx}dx = \dfrac{A}{\sqrt{2Dt}}\exp\left[\dfrac{-x^2-2\alpha t - \alpha^2 t^2}{4Dt}\right].$$
Now, I don't know how to use the boundary conditions and the initial condition on this problem. It seems the boundary condition is automatically satisfied, but I'm not sure. Also the initial condition I have no idea on how to use.
How can I finish this by applying the boundary conditions and the initial condition in the right way?
• I think your "vanishing at infinity" BCs are built into the assumption that the FT exists as a function (rather than as a distribution). – Ian Sep 27 '15 at 20:28
• But I think you need to incorporate the initial condition into your solution of the ODE in Fourier space. For that you will need to know how to compute the Fourier transform of the Dirac delta. – Ian Sep 27 '15 at 20:31
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# Where are the limits of this question?
• May 24th 2009, 06:53 AM
thomas49th
Where are the limits of this question?
Question 7
http://i157.photobucket.com/albums/t...nArcLength.jpg
At first I thought it was polar co-ordinates but it was parametric so I used the arc length formula to try and find the length of the curve in ONE of the quadrants. I would later times this by 4 to get the length of the curve all together. So arc length = $\int{\sqrt{\frac{dx}{dt}^{2}+\frac{dy}{dt}^{2}}}dt$
so after lots of cancelling and using everybody's favourite trig identity sin^2(t) + cos^2(t) = 1 I get it down to
$3a\int{sint cos t}dt$
Okay now use a substitution of u = sint
to get $\frac{3a}{2} sin^{2}t$ but where are the limits I need to use to get this done? Can someone show me what the limits I need to use are?
Thanks :)
NOTE using t instead of theta
• May 24th 2009, 07:11 AM
Spec
If you're going to calculate the length of the arc in the first quadrant then simply use $0 \leq \theta \leq \frac{\pi}{2}$
• May 24th 2009, 07:12 AM
HallsofIvy
The problem says " $0\le \theta \le 2\pi$"! But because you will have signs that cancel out, integrate from 0 to $\pi/2$ and multiply by 4.
• May 24th 2009, 07:39 AM
thomas49th
Quote:
Originally Posted by HallsofIvy
The problem says " $0\le \theta \le 2\pi$"! But because you will have signs that cancel out, integrate from 0 to $\pi/2$ and multiply by 4.
Im not quite sure what you mean? Have you simply down 2pi/4 but you mention about signs cancelling? Could you please elaborate
Thanks :)
• May 24th 2009, 09:21 AM
Spec
If you put in $0$ and $2\pi$ as the limits then $\sin^2 x|_0^{2\pi}=0$
A length is obviously not negative, but if you integrate around the entire circle you will have four lengths that cancel each other out because two of them will end up negative.
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Defining geometry within a cartesian coordinate system
1. Jul 5, 2013
Technical UK
Hello,
Just as a warning before anyone reads my question I am not a mathematician, just an engineer with moderate math skills he wants to expand.
So I'm writing some engineering software which involves defining/interation/modification of geometry within a cartesian system but I currently lack the geometry knowledge to pull it off. I know geometry is a big subject and my research up until now has left overwhelmed so I'm trying to narrow down what I should be learning to get to grips with the ideas I need to do what I want to do.
To give you an idea of what type of things I'm trying to build upto I currently know how to define a straight line using the standard formula (which I store in the software as 2 points but can use the formula to define new points etc.). What I want to be able to do is define curves (curcular, clothoid, parabola etc.) in a way in which an accurate definition can be store within he program of a specific curve. I know some of the standad formulas for curves such as for a parabola on a graph but my curves will be set at specific coordinates at a certain orientation.
I also want to learn how I can solve inerations between geometry such as where a line crosses a curve etc. I've tried to understand calculating whether and where 2 straight lines cross but I haven't found it easy on my own!
I know my math skills are not great and my question is pretty vague but any rough idea of useful topics/skills to learn or any starting points would be appriciated. I will also understand if the answer is "You're jumping into something well over my head". I know the basic geometry you learn at school but that is about it.
Thanks.
2. Jul 5, 2013
micromass
Staff Emeritus
It seems that linear algebra is the answer here.
Finding where two lines cross is very easy using linear algebra. You just need to solve a system of two equations. Finding where two arbitrary curves cross is a bit more difficult though, but you can also do it using systems of equations.
Then, it seems that you know how to graph things like $y=x^2$ which is a parabola opening upward. You might want to know a formula for a parabola opening in some diagonal direction. Is that it?
Well, take the parabola $y=x^2$. Arbitrary points on there are of the form $(t,t^2)$ (thus, any $t$ will give you a point). We can rotate this parabola by multiplying with a rotation matrix. Such a matrix has the form
$$\left(\begin{array}{cc}\cos(\theta) & \sin(\theta)\\ - \sin(\theta) & \cos(\theta)\end{array}\right)$$
The new parabola becomes $(\cos(\theta)t + \sin(\theta)t^2, -\sin(\theta)t + \cos(\theta)t^2)$. For example, if you want to rotate your parabolia 90 degrees, then $\theta = \frac{\pi}{2}$. Thus we get $(t^2,-t)$.
You should probably go through a book on basic applied linear algebra.
3. Jul 5, 2013
Technical UK
I get the geist that for 2 lines it is just a matter of solving the 2 equations but my main problem is (and from my experience the same problem most people have) when I was taugh these things we were taught in such a set way that I didn't fully understand it and don't know how to actually use it to solve things for myself. Such as testing whether 2 lines cross at all? what is the coordinate of the cross over?
I got the feeling someone would use a tranformation matrix. I covered matrices to a very basic level but I can't say I know how to use them for anything! Yes that is an example and to take it another step further how could I define a parabolic curve which passes through 2 specific coordinates? eg. (2,3) and (5,8). I'm guessing I will need to learn certain matrices operations.
I hope that's clear.
4. Jul 5, 2013
micromass
Staff Emeritus
Are we talking about lines in a plane? The equations for two lines is given by $\alpha X + \beta Y = \gamma$ and $aX + bY = c$. The lines cross if and only if
$$\textrm{det}\left(\begin{array}{cc} \alpha & \beta\\ a & b\end{array}\right) \neq 0$$
The coordinate of the cross can be calculated by solving the system
$$\left\{\begin{array}{l} \alpha X + \beta Y = \gamma\\ aX + bY = c\end{array}\right.$$
There are multiple parabolic curves going through those two points. Let's try to find a parabolic curve opening upwards. This has the form $y = ax^2 + bx + c$. The point $(2,3)$ should be a solution, thus the following should be true $4a + 2b + c = 3$. Similarly, we want $25a + 5b + c = 8$. We solve this for $a$ and $b$ by solving the system
$$\left\{\begin{array}{l} 4a + 2b + c = 3\\ 25a + 5b + c = 8\end{array}\right.$$
This system will have multiple solutions. Each of the solutions will give you a parabola opening upward.
5. Jul 5, 2013
Technical UK
I'm starting to get the feeling I'm out of my depth and have alot to learn before I get anywhere. Although I can roughly follow what you are trying to do, most of it makes little sense to me. I keep looking through geometry books and websits but there is just so much information I just don't even know where to start...
What about this example: Say I manage to draw part of a parabolic curve. It's not perpendicular with either axis (lets say it's 23 degress rotated or whatever) and it does not sit on 0,0 but at 12,19. Would a formula for such a curve be difficult to derive?
6. Jul 5, 2013
micromass
Staff Emeritus
I derived the formula above:
Just let $\theta$ be 23 degees. Of course, this is a parabola sitting on (0,0). If you want to move the "base pont", then you need to add $(12,19)$. So the formula is:
$$(12 + \cos(\theta)t + \sin(\theta)t^2,19 -\sin(\theta)t + \cos(\theta)t^2)$$
where $\theta$ is 23 degrees.
7. Jul 5, 2013
Technical UK
You make it seem so easy it's genius! I'm not entirely sure how you managed to derive the formula but that is the type of thing I want to be able to work out.
When it comes to crossing geometry eg. a line crossing the curve. Is it generally the same idea as for the crossing straight lines?
8. Jul 5, 2013
micromass
Staff Emeritus
It's the same idea, except that the system is harder to solve.
9. Jul 5, 2013
Technical UK
I keep reading through what you have put and I think it's starting to click.
How did you come up with the cos, sin etc. matrix?
Is there a name for the this type of mathematics I could use to search for material to learn from?
10. Jul 5, 2013
micromass
Staff Emeritus
http://en.wikipedia.org/wiki/Rotation_matrix
Search for linear algebra. Or analytic geometry.
11. Jul 5, 2013
Technical UK
Thats great thanks! Now I know what I'm searching for things should be much easier.
You also mentioned plane curve which led me to this:
http://en.wikipedia.org/wiki/Plane_curve
I understand as a function, you put x in and it gives you y but I don't fully understand what the other 2 mean ie. Implicit equation and Parametric equation.
Hopefully I have enough now to get me going.
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# Is Unification “an Implementation of Existential Quantification”?
I read a comment (I've forgotten the source), "Unification is an implementation of existential quantification." (Emphasis mine.)
If true, this point of view clears up many things.
For instance why omitting the "occurs check" in unification can lead to unsound inference: it would be the same as using the assumption of the existence of a thing in your proof(-term) of the existence of that thing.
It would also explain why the algorithms for unification in logic programming language engines I've looked at seem (to me) more arbitrary than other operators such as conjunction and disjunction.
But - is the statement true? Is it an over-simplification?
A more precise statement would be that existential quantification in logic programming languages is (typically) handled by introducing a unification variable. To perform existential introduction, you need to have a term $$t$$. The rule looks like: $$\dfrac{\Gamma\vdash P[t/x]}{\Gamma\vdash\exists x.P}$$
One way to implement this is to just guess a term $$t$$ and then see if we can backchain $$P(t)$$. Of course, for a language like Prolog, there are an infinite number of possible terms, and you have no guidance on what term to guess. A different option is to have $$t$$ be a meta-variable that stands for a term and continue execution on the hope that we can constrain $$t$$ enough to know what to decide later. This is essentially what a unification variable is. Unification itself just constrains what possible terms that meta-variable can be. Then, instead of actually enumerating the (now constrained) terms, we usually return the (simplified) constraints themselves as a compact representation of the set of possible terms, assuming they aren't contradictory (i.e. that that set isn't empty).
Unsurprisingly, the operation unification itself most directly implements is equality. If we did the "exhaustively enumerate terms" approach to handling existential quantification, all the places where we (implicitly) use "unification" could just be straightforward syntactic checks of ground terms. With meta-variables, they instead produce constraints. Of course, this suggests the natural and common generalization of allowing constraints other than syntactic equality, and this leads to constraint logic programming. One generic example of such a constraint is disunification which states what a term (potentially containing meta-variables) cannot be. It is the negation of unification, $$t\not\equiv s$$ states that for all substitutions $$\sigma$$, $$t[\sigma]\neq s[\sigma]$$.
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# Does the notion of a critical point extend from set theory to Braid groups?
Let $B_{\infty}$ denote the infinite strand braid group. Let $\text{sh}:B_{\infty}\rightarrow B_{\infty}$ be the homomorphism defined by $\text{sh}(\sigma_{i})=\sigma_{i+1}$ whenever $i\geq 1$. Give $B_{\infty}$ the self-distributive operation $*$ defined by $$x*y=x\cdot\text{sh}(y)\cdot\sigma_{1}\cdot\text{sh}(x^{-1}).$$
Does there exist a linear ordering $L$ along with a function $\Gamma:B_{\infty}\rightarrow L$ where
1. $\Gamma(x*y)\leq\Gamma(y)$ whenever $\Gamma(y)<\Gamma(x)$, and
2. $\Gamma(x*y)>\Gamma(y)$ whenever $\Gamma(y)\geq\Gamma(x)$.
The motivation behind this question is that if there is no such map $\Gamma$, then $B_{\infty}$ cannot be embedded into an algebra $\mathcal{E}_{\lambda}$ of rank-into-rank embeddings since the critical points provide such a mapping $\Gamma$. However, if there is such a mapping $\Gamma$, then it would be reasonable to conjecture that $B_{\infty}$ does embed into some $\mathcal{E}_{\lambda}$.
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## fredag 22 november 2013
### What is $\pi$ and $e$?
There are two numbers with a special stature in mathematics:
where $\exp(t)$ is the exponential function usually defined by
• $\exp(t)=\lim_{n\rightarrow\infty}(1+\frac{t}{n})^n.$ (*)
The geometric definition of $\pi$ does not give direct information about its numerical value and the definition of $e$ may appear to be ad hoc without connection to either geometry or physics.
In BodyandSoul as Constructive Calculus both $\pi$ and $e$ are constructively defined through solutions of basic initial value problems solved by time stepping. More precisely, $\pi$ is defined as the smallest positive root of the equation $\sin(t)=0$, where $u(t)=\sin(t)$ and $v(t)=cos(t)$ is the solution to the initial value problem modeling a harmonic oscillator:
• $\frac{du}{dt} - v =0$ and $\frac{dv}{dt} + u=0$ for $t > 0$, $u(0)=0$ and $v(0)=1$. (**)
Further $u(t)=\exp(t)$ is the solution to the basic initial value problem (expressing "exponential growth" with the growth rate $\frac{du}{dt}$ equal to $u$ itself):
• $\frac{du}{dt}=u$ for $t > 0$ and $u(0)=1$. (***)
In particular,
• $(1+\frac{t}{n})^n$
is the result of solving (***) by time stepping with time step $\frac{t}{n}$ with
• $u(t+\frac{t}{n})=u(t)+\frac{t}{n}u(t) = (1+\frac{t}{n})u(t)$,
which converges to $\exp(t)$ as the time step $\frac{t}{n}$ goes to zero with $n$ tending to infinity.
Introducing and defining the numbers $\pi$ and $e$ this way, gives both an understanding why they are so fundamental (by expressing basic properties of solutions to basic mathematical equations connecting to basic physics) and also shows how the (decimal expansions of the) numbers can effectively be computed.
PS1 Notice that once $\sin(t)$ and $\cos(t)$ have been defined as solutions of (**), it follows that $(\cos(t),\sin(t))$ can be geometrically interpreted as the coordinates of a point moving along a unit circle with unit speed and thus $t$ a measure of angle as arc length. The geometric interpretation of $\pi$ thus follows from numerical algebra and not the other way around as in Standard Calculus.
PS2 In Standard Calculus $\sin(t)$ and $\cos(t)$ are defined geometrically as quotients of the lengths of the sides of right-angled triangles again without access to the numerical values except for a few special values of the angle $t$.
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REALLY strange change in value of character data being pointed at
Recommended Posts
Hi, I have an array of char data that holds the UTF 8 character string "%c?????#5C???????#0C?\n"
"???????????????". This string then gets passed to a function with the protoype
CSaveData::UTF8toSJISForce(char *dst,size_t buffsize,const char *src )
like so:
CSaveData::UTF8toSJISForce( szText, sizeof( szText ), tmpBuf );
where tmpBuf is the character data. When stepping through Visual Studios debugger the value of tmpBuf is as expected in the function call above, but when entering the UTF8to... function I check the value of src in the debugger and it is now - "?????#5C?????????#0C????????????????????" which is of course not what is expected.
The call to UTF8... is in one file and the implementation is in another .cpp file, Ive checked the encodings of both files and they are indeed both in UTF8 format, so I dont understand what is causing this error! Please help
Share on other sites
Please disregard this, after an hour or so I realised this wasnt the fault in the code! Super sorry!
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# Compile an .Rnw with greek text to pdf
I tried it using the commented out code without success. Can somebody help?
\documentclass[a4paper]{article}
%\usepackage[english,greek]{babel}
%\latintext
\title{Sweave Example 1}
\author{George Dontas}
\begin{document}
\maketitle
In this example we embed parts of the examples from the
\texttt{kruskal.test} help page into a \LaTeX{} document:
%\greektext
Αυτό είναι κείμενο στα Ελληνικά
%\latintext
<<eval=TRUE,echo=TRUE,warning=FALSE,message=FALSE,error=FALSE>>=
data(airquality)
kruskal.test(Ozone ~ Month, data = airquality)
@
which shows that the location parameter of the Ozone distribution varies significantly from month to month. Finally we
include a boxplot of the data:
\begin{center}
<<eval=TRUE,echo=FALSE,results='hide',warning=FALSE,message=FALSE,error=FALSE>>=
boxplot(Ozone ~ Month, data = airquality)
@
\end{center}
\end{document}
-
It might be useful if you add the error message you get into your post. – Hemmo Feb 28 '13 at 15:30
if the Sweave process works fine and the failure occurs in the processing from (La)TeX to PDF, this question would probably be more appropriate at tex.stackexchange.com – Ben Bolker Feb 28 '13 at 15:42
## 1 Answer
The problem here is not the R but matter of getting latex to play well with Greek. With hat tip to this answer, perhaps the easiest solution is to switch to XeLaTeX compilation and rewrite your file as
\documentclass[a4paper]{article}
\usepackage{fontspec}
\setmainfont{Times New Roman}
\setsansfont{Arial}
\newfontfamily\greekfont[Script=Greek]{Linux Libertine O}
\newfontfamily\greekfontsf[Script=Greek]{Linux Libertine O}
\usepackage{polyglossia}
\setdefaultlanguage{english}
\setotherlanguage{greek}
\title{Sweave Example 1}
\author{George Dontas}
\begin{document}
\maketitle
Ελληνικό κείμενο
In this example we embed parts of the examples from the
\texttt{kruskal.test} help page into a \LaTeX{} document:
...
and so on as before. Then compile with XeLaTeX (latexmk -xelatex file.Rnw will do that).
-
Thank you! I get the error: The font "Linux Libertine O" cannot be found. – George Dontas Mar 3 '13 at 12:13
Hmm.. I changed font to Arial and saved the .rnw using uTF-8 encoding and it worked. – George Dontas Mar 3 '13 at 12:19
Ah, that font must have been one of mine. Still, glad it worked. – Conjugate Prior Mar 3 '13 at 12:23
However, greek text inside R plots (e.g. a plot title in greek) does not appear correctly. Any ideas? – George Dontas Mar 3 '13 at 15:37
Adding "dev='cairo_pdf'" in chunk options solves the problem! – George Dontas Mar 3 '13 at 17:02
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Simple microcontroller approach to controlling a servo
Posted by Jan on 10 March 2011
Today, I want to discuss the microcontroller equivalent of the simple servo control circuit I presented last time. As I mentioned then, the circuit is about as simple as it can be, yet it requires eight components to arrive at a sub-optimal servo control waveform. Some of its deficiencies, such as the slow rise time of the pulses, can be addressed by slightly more advanced circuits that might implement an astable multivibrator using an integrated circuit such as the famous 555 timer. In terms of part count, the 555-based servo controller might be a bit better than the two-transistor approach, but the 555 has many transistors inside it. As long as we are comfortable categorizing a component with many transistors inside it as a single part, we might as well skip the 555 and go straight to a low pin-count microcontroller, which has thousands of transistors inside it and which will allow us to make a far superior, single-component servo controller.
I will repeat my reminder that you can destroy a servo by commanding it past its mechanical limits, so any time you’re making your own servo control devices, be careful. You can limit the likelihood of destruction by operating your servos at a relatively low voltage, so they cannot develop as much torque as they are capable of. I also recommend testing any circuits with a standard servo, which is relatively cheap, more robust than smaller micro servos, yet not powerful enough to easily destroy itself. If you see or hear a servo straining against its end stop, or if it’s getting hot, you should disconnect it or turn off your power immediately.
Also as a review from previous posts, let’s go over the basic signal we’re trying to generate: a square wave with a variable positive pulse width of around 1 to 2 milliseconds and a frequency of around 50 Hz.
Our voltage, V, should be around 3 volts or higher; we need the pulse width, t, to be variable since that is what encodes the desired servo position; and as we saw in servo control interface in detail, the period, T, can affect how hard a servo tries to maintain a position and should be around 20 ms but otherwise is not that critical.
Many microcontrollers can run on 3 to 5 volts, so the V requirement can be satisfied with a normal digital I/O line: all we have to do is toggle that line with the appropriate timing. This makes for a very simple program:
loop:
make pin high
wait for time <t>
make pin low
wait for time <T-t>
go to loop
Since we do not care much about the period, T, we can do the same simplification we did with the two-transistor circuit: make the low part of the pulse basically fixed. That way, we do not have to worry about how long we spent making the high part of the pulse:
loop:
make pin high
wait for time <t>
make pin low
wait for around 18 ms
go to loop
My “programs” so far are more English than some specific programming language, but it should be easy to see how to do this in your favorite language. If you have the right libraries or basic commands, the first three lines can even be combined into a single command for making a pulse. For instance, the PBASIC version for the BASIC Stamp might look like this:
loop:
PULSOUT 1, 750 'make a 750 * 2 = 1500 us pulse on pin 1
PAUSE 18 'do nothing for 18 ms
GOTO loop
The C equivalent might look more like this:
while ( 1 )
{
pulse_out(SERVOPIN, 1500); // assuming pulse_out() takes an argument in us
delay_us(18000);
}
Note that in the PBASIC example, PULSOUT and PAUSE are built-in commands, whereas the pulse_out() and delay_us() functions in the C example are something you would need to also write yourself (or find in a library).
One of the benefits of implementing your servo control on a microcontroller is that you can put your code on a little 6-pin microcontroller or on a big, 100-pin part. In the case of the 6-pin microcontroller, you might not have much program memory, and it might make more sense to write simple servo control firmware in assembly. In the same sense that the two-transistor circuit is more a good exercise than a great engineering solution, the assembly-language servo controller is a good project to try some time.
The two specific examples above were for fixed pulse widths of 1.5 ms. Obviously, that does not make for a particularly interesting servo controller. The great thing about a microcontroller solution is that you have a lot of flexibility, and we can use the time between pulses to decide what to do next. For instance, instead of just pausing for 18 ms, we could read a potentiometer or some buttons or do some calculation about how we want the servo to move on its own (e.g. sweeping back and forth).
If you want to make more than just a single-channel servo controller, it can still be quite easy as long as you can structure the rest of your project’s functions around the servos’ requirements. For instance, you could control a simple robot with four servos using this general control loop:
loop:
make the pulse for servo 1
make the pulse for servo 2
make the pulse for servo 3
make the pulse for servo 4
recalculate servo positions
pause if necessary
go to loop
In this scenario, we would spend four to eight milliseconds generating the four servo pulses (depending on how long the pulses need to be for each servo), which would leave us a little over 10 ms to read our sensors and decide what we want to do with the sensor data. The key is that the exact timing of the extra operations do not matter, but we can pause at the end of our loop to get a consistent 50 Hz update rate if we think that consistency is worth it. However, the time for making the individual pulses must be dedicated completely to the servo control operation; if we are even a few microseconds off in our pulse widths, the servos will start twitching.
Conclusion (for now)
This was a very simple introduction to what is involved in controlling a servo from a microcontroller. If you just need to control a few servos, and do not need to do much else, it is indeed quite simple. Next time, I will move on to more advanced considerations for controlling many servos at once or controlling servos without having to tie the rest of your program to the servos’ 50 Hz update cycle.
I just read through your servo posts. Thanks a bunch, really helpful and detailed documentation. It was just what i was looking for to help me start my project. good work.
Hey this is great stuff, very informative. The reason that I found your sight is that I am having trouble with some new servos that I bought to use with an arduino controller board. The servos are GWS Park HPX f. I think that the problem has something to do with the "stream" of pulses required to hold the servo in position. Previously, I have not needed to send a stram of pulses, just a single pulse to move the servo in to position. Are there some servos that require a stream of pulses and others that only require a single pulse? this code sends my old servo to position 180 degree, but with my new servo it just moves a little and pulses back and forth.
Thanks
int servoPin = 9;
void setup(){
pinMode(servoPin,OUTPUT);
}
void loop() {
int pulse180 = 2100; //(the number of microseconds
//to pause for (1500 90 degrees
// 900 0 degrees 2100 180 degrees)
digitalWrite(servoPin, HIGH);
delayMicroseconds(pulse180);
digitalWrite(servoPin, LOW);
delay(500);
}
Hi.
Some of my earlier posts about servos address your question. Some servos don't need the continual stream, but in general, you need to keep sending the pulses at about 50 Hz.
- Jan
I want to know if there are any program to control a servo wireless by using a PIC 12F675
Both Tx and Rx
Hello.
I have no idea what you are asking for. What does "control a servo wireless" mean? That PIC is not going to do wireless communication on its own, and if you have something else doing the wireless part, why have the PIC there? Why that particular PIC?
- Jan
Attn: Bengt L 25 Jan 2013
I have a feeling you only have the first couple of units of MicroCode Studio so there fore require that pic or mabe it's the cheapest.
The trail version of the Mikrobasic Pro will allow you to program the full range of pic's.
But only up to 2k of code.
Pic any data leg you want and attach this to your rx line on your local wireless device.
Then on your remote wireless attach the tx line to the rx line of say a mestro.
Then using a serial out command send the approiate hex commands to the leg you dedicated to rx line of the local wireless device.
The hex commands can be found in the mestro documentation.
You may have to convert the hex commands to decimal.
Enjoy I'm up to the same thing.
Make sure the mestro is set to the same baud speed or if it's not set the send hex aa first so it can detect the board speed
Say you are using pin 0 in micro code the command looks like this.
' Send the ASCII value of B0 followed by a linefeed out Pin0 serially
SEROUT 0,N9600,[#B0,10]
I could be slightly wrong but I'm pretty sure im not far off.
Regards
Pat
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# What is the perimeter of an equilateral triangle inscribed
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09 Aug 2012, 13:06
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What is the perimeter of an equilateral triangle inscribed in a circle of radius 4 ?
A. $$6\sqrt{2}$$
B. $$6\sqrt{3}$$
C. $$12\sqrt{2}$$
D. $$12\sqrt{3}$$
E. $$24$$
[Reveal] Spoiler: OA
Last edited by Bunuel on 09 Aug 2012, 15:05, edited 1 time in total.
Renamed the topic and edited the question.
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09 Aug 2012, 16:29
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arthuro69 wrote:
What is the perimeter of an equilateral triangle inscribed in a circle of radius 4 ?
A. $$6\sqrt{2}$$
B. $$6\sqrt{3}$$
C. $$12\sqrt{2}$$
D. $$12\sqrt{3}$$
E. $$24$$
Hi, there. I'm happy to help.
The full solution to the problem is in the attached pdf.
If the details of the 30-60-90 triangle are not familiar to you, I recommend brushing up with this post:
http://magoosh.com/gmat/2012/the-gmats- ... triangles/
Let me know if there are any further questions.
Mike
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Mike McGarry
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Re: What is the perimeter of an equilateral triangle inscribed [#permalink]
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04 Sep 2012, 03:29
The radius of circum circle of an equilateral triangle = a/sqrt(3). a is the side of triangle.
Here: a/sqrt(3) = 4
a = 4*sqrt(3).
perimeter 3a = 3*4*sqrt(3) = 12sqrt(3).
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04 Sep 2012, 03:32
Expert's post
arthuro69 wrote:
What is the perimeter of an equilateral triangle inscribed in a circle of radius 4 ?
A. $$6\sqrt{2}$$
B. $$6\sqrt{3}$$
C. $$12\sqrt{2}$$
D. $$12\sqrt{3}$$
E. $$24$$
All you need to know about triangles for the GMAT: math-triangles-87197.html
Hope it helps.
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Re: What is the perimeter of an equilateral triangle inscribed [#permalink]
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06 Jul 2014, 11:30
Let x be the side of triangle.
Using cosine rule:
(x/2)/R = cos 30
=> x = 2Rcos 30 = 4√3
=> Perimeter = 3X = 12√3
Re: What is the perimeter of an equilateral triangle inscribed [#permalink] 06 Jul 2014, 11:30
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conclusion of liquid soap making
Soap: History and Production Processes - UK Essays- conclusion of liquid soap making ,Galen describes soap making using causticised lye and prescribes washing to carry away impurities from the body and clothes[3]. The best soap was German according to Galen; soap from Gaul was second best. This is the first record of true soap as a detergent. Zosimos of Panopolis c. 300AD describes both soap and soap making.[4] Muslim historySoap Formulation LAB REPORT Pages 1 - 16 - Flip PDF ...CONCLUSION• The soap product do have the characteristics of a good soap. The pH of the soap is 6 and near to the skin pH which is 5.5 so that no irritation or allergy reaction occurs. However, the franrance smell is too strong and may be umpleasant to certain people.•
### What is the conclusion of SOAP? - Answers
The conclusion of soap preparation involves allowing the soap to cool in the molds. As the soap cools it will solidify enough to be popped out of the molds.
### Making Bar Soap | Make:
Dec 18, 2012·Fortunately, making soap at home is easy, and not much more complex than baking bread. In its most basic form, soap consists of just 3 components — a strong base such as potash or lye, oil, and water. Potash (potassium hydroxide) is harder to find and is more conducive to liquid soap making, so we’ll use lye (sodium hydroxide).
### Soap Manufacturer Business Plan - Executive Summary
Become the specialty soap of choice for day cares across the Northwest by December Year 2. Achieve sales of \$5 million by the end of Year 6. Next: Company Summary. Start your own soap manufacturer business plan. Start your own business plan. Start planning.
### 12: Making Soap - Saponification (Experiment) - Chemistry ...
Nov 14, 2017·Liquid cooking oils originate from corn, peanuts, olives, soybeans, and many other plants. For making soap, all different types of fats and oils can be used – anything from lard to exotic tropical plant oils. Saponification Reactions: $\text{Fat} + \text{Lye} → \text{Soap} + \text{Glycerol}$
### 12: Making Soap - Saponification (Experiment) - Chemistry ...
Liquid cooking oils originate from corn, peanuts, olives, soybeans, and many other plants. For making soap, all different types of fats and oils can be used – anything from lard to exotic tropical plant oils. Saponification Reactions: $\text{Fat} + \text{Lye} → \text{Soap} + \text{Glycerol}$
### Measure Surface Tension with a Penny - Scientific American
Jun 25, 2015·Adding soap lowers the water’s surface tension so the drop becomes weaker and breaks apart sooner. Making water molecules stick together less …
### Pepper and Soap Experiment | Science project | Education.com
Soap is able to break down the surface tension of water—that’s part of what makes soap a good cleaner. As the soap moves into the water, and the surface tension changes, the pepper no longer floats on top. But the water molecules still want to keep the surface tension going, so they pull back away from the soap, and carry the pepper along ...
### Why and How to Check the pH of Soap & How to Neutralize ...
Sep 30, 2019·Checking the pH of your liquid soap: The Why… Unlike bar soap recipes that tend to use excess oils to chemically react with all of the lye, leaving you extra conditioning oils for your skin, liquid soaps are usually calculated to use either the exact amount of oil for the lye or …
### The Science of Soap Making in a Lab : 9 Steps (with ...
The Science of Soap Making in a Lab: Making soap doesn't seem like something you'd do in a lab, but it's actually more scientific than you'd think. Saponification is the soap making process, which uses the basic solution lye and different types of fats. The science behind soap making i…
### Soap: History and Production Processes - UK Essays
Galen describes soap making using causticised lye and prescribes washing to carry away impurities from the body and clothes[3]. The best soap was German according to Galen; soap from Gaul was second best. This is the first record of true soap as a detergent. Zosimos of Panopolis c. 300AD describes both soap and soap making.[4] Muslim history
### What makes soap foam? | HowStuffWorks
There are many different kinds of soap in the world and most of them have one major thing in common: They can make bubbles. When you amass a bunch of tiny bubbles together, we call it foam or lather.It doesn't matter if you're talking about bar soap, shampoo, dish soap or laundry detergent -- the same thing happens when you mix any of them with air and water.
### What is the conclusion of SOAP? - Answers
The conclusion of soap preparation involves allowing the soap to cool in the molds. As the soap cools it will solidify enough to be popped out of the molds.
### How to Make Homemade Liquid Soap - The Spruce Crafts
A major difference between making liquid soap and bar soap is that that it is a "hot process" soap. Instead of relying on the heat generated by the saponification process, heat is added using a double boiler, oven, or slow cooker. This recipe can be made in …
### The History And Methods Of Soap Making - UK Essays
Easy Steps of Soap Making. Rule of thumb: There are various approaches in making soap. The easiest way is to buy pre-mixed or soap making packs that are readily available in the market. The other one is to buy individually the ingredients needed along with the useful tools required for your soap making …
### Conclusions - Color Milk Experiment
Conclusions: After performing this experiment I can conclude based on its results that different types of milk, who have distinct amount of fat, will have a different chemical reaction in each one with liquid soap contact because of the fat content of milk and its properties. When there is a lower amount of fat the chemical reaction will be not ...
### soap and detergent | Chemistry, Properties, & Facts ...
Sep 08, 2020·Soap and detergent, substances that, when dissolved in water, possess the ability to remove dirt from surfaces such as human skin, textiles, and other solids. The seemingly simple process of cleaning a soiled surface is, in fact, complex. Learn more about soap and detergent in this article.
### Hot Process Liquid Soapmaking : 9 Steps (with Pictures ...
Make sure you are as accurate as possible. Usually there's a slighly higher lye amount used in liquid soaps than bar soaps, in order to make sure that the fats are completely neutralized. This excess lye will be neutralized later. I make my liquid soap in an electric oven, set at 250 degrees fahrenheit. Your mileage may vary.
### The Science of Soap Making in a Lab : 9 Steps (with ...
The Science of Soap Making in a Lab: Making soap doesn't seem like something you'd do in a lab, but it's actually more scientific than you'd think. Saponification is the soap making process, which uses the basic solution lye and different types of fats. The science behind soap making i…
### Soap Conclusion
Conclusion. In conclusion, soap is a substance, water soluble sodium salts of fatty acids, that is used to remove dirt and grime from a surface. It's molecules have a long hydrocarbon chain that has a negatively charged head. It's non polar hydrocarbon chain doesn't interact with water molecules that form micelles. The soap micelles repel each ...
### How To Make Liquid Soap: A Simplified Process for Natural ...
Hello – love the idea of making my own liquid soap. I’ve been making my own bar soap for a few years now and love the result 🙂 My question is regarding the ingredients list…3 oz. borax and 6 oz. water. Down farther in the instructions it says to use 1 oz. borax …
### How soap is made - material, manufacture, making, used ...
The hot liquid soap may be then whipped to incorporate air. Cooling and finishing 3 The soap may be poured into molds and allowed to harden into a large slab. It may also be cooled in a special freezer. The slab is cut into smaller pieces of bar size, which are then stamped and wrapped.
### 3 Ways to Make a Longer Lasting Bubble Solution - wikiHow
Oct 14, 2020·In a container combine 4 parts glycerin, 2 parts liquid dish soap, and 1 part corn syrup. For a large amount of solution, you could combine 4 cups glycerin, 2 cups liquid dish soap, and 1 cup corn syrup. For a smaller amount of solution, you could combine, 1 cup glycerin, ½ cup liquid dish soap, and ¼ cup corn syrup.
### Making soaps and detergents | Experiment | RSC Education
Making soap . a Place castor oil (2 cm 3) into a beaker (100 cm 3) using a dropping pipette, followed by ethanol (5 cm 3). Stir with a glass rod to mix. ... 4 cm depth), then carefully pour the reaction mixture from the first tube into the water. The liquid may be very slow-flowing (viscous) and contain concentrated acid, so be careful and take ...
### EXPERIMENT # ------ SYNTHESIS AND PROPERTIES OF SOAP …
CHEM 1100 2 History of Soap The discovery of soap dates back to about 6000 years ago. Around 2800 B.C.E, the ancient Babylon excavations uncovered cylinders with inscriptions for making soap.1 In 1500 B.C.E, records from ancient Egypt described how animal and vegetable oils were combined with alkaline salts to make soap.
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Please use this identifier to cite or link to this item: http://hdl.handle.net/2289/4653
Title: Effect of screening of intermicellar interactions on the linear and nonlinear rheology of a viscoelastic gel Authors: Bandyopadhyay, RanjiniSood, A.K. Issue Date: 2003 Publisher: American Chemical Society Citation: Langmuir, 2003, Vol.19, p3121-3127 Abstract: We report our studies of the linear and nonlinear rheology of aqueous solutions of the surfactant cetyl trimethylammonium tosylate (CTAT) with varying amounts of sodium chloride (NaCl). The CTAT concentration is fixed at 42mM and the salt concentration is varied between 0mM to 120mM. On increasing the salt (NaCl) concentration, we see three distinct regimes in the zero-shear viscosity and the high frequency plateau modulus data. In regime I, the zero-shear viscosity shows a weak increase with salt concentration due to enhanced micellar growth. The decrease in the zero-shear viscosities with salt concentration in regimes I I and III can be explained in terms of inter-micellar branching. The most intriguing feature of our data however, is the anomalous behavior of the high frequency plateau modulus in regime II (0.12 $\le \frac{[NaCl]}{[CTAT]} \le$ 1. 42). In this regime, the plateau modulus {\it increases} with an increase in NaCl concentration. This is highly counter-intuitive, since the correlation length of concentration fluctuations and hence the plateau modulus $G_{\circ}$ are not expected to change appreciably in the semi-dilute regime. We propose to explain the changes in regime II in terms of the unbinding of the organic counterions (tosylate) from the CTA$^{+}$ surfaces on the addition of NaCl. In the nonlinear flow curves of the samples with high salt content, significant deviations from the predictions of the Giesekus model for entangled micelles are observed. Description: Restricted Access. URI: http://hdl.handle.net/2289/4653 ISSN: 0743-74631520-5827 (Online) Alternative Location: http://arxiv.org/abs/cond-mat/0012473 Copyright: 2003 American Chemical Society Appears in Collections: Research Papers (SCM)
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0. Addeddate 2006-11-11 01:04:08 Call number 29801 Digitalpublicationdate 2005/06/21 Identifier complexintegrati029801mbp Identifier-ark ark:/13960/t0rr1q351 Power series expansions, Morera’s theorem 5. However note that $\displaystyle{1 = \frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y} = -1}$ ANYWHERE. 1. Theorem (Some Consequences of MVT): Example (Approximating square roots): Mean value theorem finds use in proving inequalities. Cauchy's Integral theorem concept with solved examples Subject: Engineering Mathematics /GATE maths. Compute Z C cos(z) z(z2 + 8) dz over the contour shown. Do the same integral as the previous examples with the curve shown. Determine whether the function $f(z) = \overline{z}$ is analytic or not. They are: So the first condition to the Cauchy-Riemann theorem is satisfied. To do so, we have to adjust the equation in the theorem just a bit, but the meaning of the theorem is still the same. Argument principle 11. Theorem 23.7. Since the integrand in Eq. HBsuch View and manage file attachments for this page. For example, a function of one or more real variables is real-analytic if it is differentiable to all orders on an open interval or connected open set and is locally the sum of its own convergent Taylor series. Example Evaluate the integral I C 1 z − z0 dz, where C is a circle centered at z0 and of any radius. Examples. Recall from The Cauchy-Riemann Theorem page that if $A \subseteq \mathbb{C}$ is open, $f : A \to \mathbb{C}$ with $f = u + iv$, and $z_0 \in A$ then $f$ is analytic at $z_0$ if and only if there exists a neighbourhood $\mathcal N$ of $z_0$ with the following properties: We also stated an important result that can be proved using the Cauchy-Riemann theorem called the complex Inverse Function theorem which says that if $f'(z_0) \neq 0$ then there exists open neighbourhoods $U$ of $z_0$ and $V$ of $f(z_0)$ such that $f : U \to V$ is a bijection and such that $\displaystyle{\frac{d}{dw} f^{-1}(w) = \frac{1}{f'(z)}}$ where $w = f(z)$. The partial derivatives of these functions exist and are continuous. Recall from the Cauchy's Integral Theorem page the following two results: The Cauchy-Goursat Integral Theorem for Open Disks: If $f$ is analytic on an open disk $D(z_0, r)$ then for any closed, piecewise smooth curve $\gamma$ in $D(z_0, r)$ we have that: (1) If is a finite group, and is a prime number dividing the order of , then has a subgroup of order exactly . 2. The mean value theorem says that there exists a time point in between and when the speed of the body is actually . all of its elements have order p for some natural number k) if and only if G has order p for some natural number n. One may use the abelian case of Cauchy's Theorem in an inductive proof of the first of Sylow's theorems, similar to the first proof above, although there are also proofs that avoid doing this special case separately. Let $f(z) = f(x + yi) = x - yi = \overline{z}$. If a function f is analytic at all points interior to and on a simple closed contour C (i.e., f is analytic on some simply connected domain D containing C), then Z C f(z)dz = 0: Note. The contour integral becomes I C 1 z − z0 dz = Z2π 0 1 z(t) − z0 dz(t) dt dt = Z2π 0 ireit reit Cauchy’s formula 4. Solution: This one is trickier. In Figure 11 (a) and (b) the shaded grey area is the region and a typical closed A practically immediate consequence of Cauchy's theorem is a useful characterization of finite p-groups, where p is a prime. Theorem 23.3 we know that all of the derivatives of f are also analytic in D.Inparticular, this implies that all the partials of u and v of all orders are continuous. ∫ C ( z − 2) 2 z + i d z, \displaystyle \int_ {C} \frac { (z-2)^2} {z+i} \, dz, ∫ C. . The interior of a square or a circle are examples of simply connected regions. Do the same integral as the previous example with the curve shown. What is an intuitive way to think of Cauchy's theorem? I have deleted my non-Latex post on this theorem. Let V be a region and let Ube a bounded open subset whose boundary is the nite union of continuous piecewise smooth paths such that U[@UˆV. Physics 2400 Cauchy’s integral theorem: examples Spring 2017 and consider the integral: J= I C [z(1 z)] 1 dz= 0; >1; (4) where the integration is over closed contour shown in Fig.1. With Cauchy’s formula for derivatives this is easy. 4.3.2 More examples Example 4.8. Then there is … Suppose that $f$ is analytic. Notify administrators if there is objectionable content in this page. They are given by: So $\displaystyle{\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}}$ everywhere. Re(z) Im(z) C. 2. However note that $\displaystyle{1 = \frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y} = -1}$ ANYWHERE. So one of the Cauchy-Riemann equations is not satisfied anywhere and so $f(z) = \overline{z}$ is analytic nowhere. Change the name (also URL address, possibly the category) of the page. Let f ( z) = e 2 z. Also: So $\displaystyle{\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}}$ everywhere as well. Suppose $$R$$ is the region between the two simple closed curves $$C_1$$ and $$C_2$$. See pages that link to and include this page. In particular, a finite group G is a p-group (i.e. Then $u(x, y) = x$ and $v(x, y) = -y$. Prove that if $f$ is analytic at then $\displaystyle{\mid f'(z) \mid^2 = \left (\frac{\partial u}{\partial x} \right )^2 + \left ( \frac{\partial v}{\partial x} \right )^2}$ and $\displaystyle{\mid f'(z) \mid^2 = \left (\frac{\partial u}{\partial y} \right )^2 + \left ( \frac{\partial v}{\partial y} \right )^2}$. Thus by the Cauchy-Riemann theorem, $f(z) = e^{z^2}$ is analytic everywhere. Then $u(x, y) = x$ and $v(x, y) = -y$. f(z) is analytic on and inside the curve C. That is, the roots of z2 + 8 are outside the curve. Click here to edit contents of this page. Cauchy’s theorem 3. Cauchy’s theorem requires that the function $$f(z)$$ be analytic on a simply connected region. Watch headings for an "edit" link when available. Compute. FÀX¥Q.Pu -PAFhÔ(¥ Cauchy Theorem when internal efforts are bounded, and for fixed normal n (at point M), the linear mapping n ↦ t ( M ; n ) is continuous, then t ( M ; n ) is a linear function of n , so that there exists a second order spatial tensor called Cauchy stress σ such that 1. Cauchy's Integral Theorem Examples 1. Re(z) Im(z) C. 2. Solution The circle can be parameterized by z(t) = z0 + reit, 0 ≤ t ≤ 2π, where r is any positive real number. An illustration is Hadamard's example: The Cauchy problem for the Laplace equation $$\Delta u = \ \frac{\partial ^ {2} u }{\partial x ^ {2} } + \frac{\partial ^ {2} u }{\partial y ^ {2} } + \frac{\partial ^ {2} u }{\partial z ^ {2} } = 0$$ Cauchy Mean Value Theorem Let f(x) and g(x) be continuous on [a;b] and di eren-tiable on (a;b). Logarithms and complex powers 10. 3. So one of the Cauchy-Riemann equations is not satisfied anywhere and so $f(z) = \o… The first order partial derivatives of$u$and$v$clearly exist and are continuous. Except for the proof of the normal form theorem, the material is contained in standard text books on complex analysis. Theorem 2says thatitisnecessary for u(x,y)and v(x,y)toobey the Cauchy–Riemann equations in order for f(x+iy) = u(x+iy)+v(x+iy) to be differentiable. Im(z) Im(z) 2i 2i C Solution: Let f(z) = cos(z)=(z2 + 8). How to use Cayley's theorem to prove the following? The following theorem says that, provided the first order partial derivatives of u and v are continuous, the converse is also true — if u(x,y) and v(x,y) obey the Cauchy–Riemann equations then Cauchy’s theorem Simply-connected regions A region is said to be simply-connected if any closed curve in that region can be shrunk to a point without any part of it leaving a region. z +i(z −2)2. . View wiki source for this page without editing. Let$f(z) = f(x + yi) = x - yi = \overline{z}$. Determine whether the function$f(z) = e^{z^2}$is analytic or not using the Cauchy-Riemann theorem. Cauchy’s integral theorem An easy consequence of Theorem 7.3. is the following, familiarly known as Cauchy’s integral theorem. They are: So the first condition to the Cauchy-Riemann theorem is satisfied. In particular, has an element of order exactly . Example 5.2. Check out how this page has evolved in the past. Stã|þtÇÁ²vfÀ& Iæó>@dÛ8.ËÕ2?hm]ÞùJõ:³@ØFæÄÔç¯3³$W°¤hxÔIÇç/ úÕØØ¥¢££ÿ3 ANALYSIS I 9 The Cauchy Criterion 9.1 Cauchy’s insight Our difficulty in proving “a n → ‘” is this: What is ‘? Residues and evaluation of integrals 9. Then $u(x, y) = e^{x^2 - y^2} \cos (2xy)$ and $v(x, y) = e^{x^2 - y^2} \sin (2xy)$. Example 4.3. A remarkable fact, which will become a theorem in Chapter 4, is that complex analytic functions automatically possess all The stronger (better) version of Cauchy's Extension of the MVT eliminates this condition. It is a very simple proof and only assumes Rolle’s Theorem. If f(z)=u(z)+iv(z)=u(x,y)+iv(x,y) is analytic in a … If we assume that f0 is continuous (and therefore the partial derivatives of u … The path is traced out once in the anticlockwise direction. If the real and imaginary parts of the function f: V ! )©@¤Ä@T\A!sbM°1q¼GY*|z¹ô\mT¨sd. General Wikidot.com documentation and help section. Cauchy saw that it was enough to show that if the terms of the sequence got sufficiently close to each other. Cauchy's vs Lagrange's theorem in Group Theory. We will now look at some example problems in applying the Cauchy-Riemann theorem. 3)¸%ÀÄ¡*Å2:à)Ã2 For example, for consider the function . We have, by the mean value theorem, , for some such that . Now let C be the contour shown below and evaluate the same integral as in the previous example. The Cauchy-Goursat Theorem Cauchy-Goursat Theorem. Now, having found suitable substitutions for the notions in Theorem 2.2, we are prepared to state the Generalized Cauchy’s Theorem. then completeness So, we rewrite the integral as Z C cos(z)=(z2 + 8) z dz= Z C f(z) z dz= 2ˇif(0) = 2ˇi 1 8 = ˇi 4: Example 4.9. dz, where. Determine whether the function $f(z) = \overline{z}$is analytic or not. Theorem 2.9 Let Mbe an oriented smooth manifold with corners and Bbe an n-dimensional body in M. Suppose that and are bounded n-forms on B and ˝is a continuous function on the bundle of oriented hyperplanes! (4) is analytic inside C, J= 0: (5) On the other hand, J= JI +JII; (6) where JI is the integral along the segment of the positive real axis, 0 x 1; JII is the View/set parent page (used for creating breadcrumbs and structured layout). C have continuous partial derivatives and they satisfy the Cauchy Riemann equations then Z @U f(z)dz= 0: Proof. This means that we can replace Example 13.9 and Proposition 16.2 with the following. Cauchy’s Theorem Cauchy’s theorem actually analogue of the second statement of the fundamental theorem of calculus and integration of familiar functions is facilitated by this result In this example, it is observed that is nowhere analytic and so need not be independent of choice of the curve connecting the points 0 and . examples, which examples showing how residue calculus can help to calculate some definite integrals. Theorem 7.4.If Dis a simply connected domain, f 2A(D) and is any loop in D;then Z f(z)dz= 0: Proof: The proof follows immediately from the fact that each closed curve in Dcan be shrunk to a point. This should intuitively be clear since $f$ is a composition of two analytic functions. Theorem 14.3 (Cauchy’s Theorem). Then, (5.2.2) I = ∫ C f ( z) z 4 d z = 2 π i 3! Something does not work as expected? $\displaystyle{\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}}$, $\displaystyle{\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}}$, $\displaystyle{\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}}$, $\displaystyle{\frac{d}{dw} f^{-1}(w) = \frac{1}{f'(z)}}$, $f(z) = f(x + yi) = x - yi = \overline{z}$, $\displaystyle{1 = \frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y} = -1}$, $\displaystyle{\mid f'(z) \mid^2 = \left (\frac{\partial u}{\partial x} \right )^2 + \left ( \frac{\partial v}{\partial x} \right )^2}$, $\displaystyle{\mid f'(z) \mid^2 = \left (\frac{\partial u}{\partial y} \right )^2 + \left ( \frac{\partial v}{\partial y} \right )^2}$, Creative Commons Attribution-ShareAlike 3.0 License. Theorem (Cauchy’s integral theorem 2): Let Dbe a simply connected region in C and let Cbe a closed curve (not necessarily simple) contained in D. Let f(z) be analytic in D. Then Z C f(z)dz= 0: Example: let D= C and let f(z) be the function z2 + z+ 1. Identity principle 6. The Riemann Mapping Theorem; Complex Integration; Complex Integration: Examples and First Facts; The Fundamental Theorem of Calculus for Analytic Functions; Cauchy's Theorem and Integral Formula; Consequences of Cauchy's Theorem and Integral Formula; Infinite Series of Complex Numbers; Power Series; The Radius of Convergence of a Power Series If you want to discuss contents of this page - this is the easiest way to do it. THE CAUCHY MEAN VALUE THEOREM JAMES KEESLING In this post we give a proof of the Cauchy Mean Value Theorem. The notes assume familiarity with partial derivatives and line integrals. Corollary of Cauchy's theorem … Related. The first order partial derivatives of $u$ and $v$clearly exist and are continuous. Liouville’s theorem: bounded entire functions are constant 7. Example 4.4. In cases where it is not, we can extend it in a useful way. Compute Z C 1 (z2 + 4)2 Laurent expansions around isolated singularities 8. Find out what you can do. New content will be added above the current area of focus upon selection Example 1 The function $$f\left( x \right)$$ is differentiable on the interval $$\left[ {a,b} \right],$$ where $$ab \gt 0.$$ Show that the following equality ${\frac{1}{{a – b}}\left| {\begin{array}{*{20}{c}} a&b\\ {f\left( a \right)}&{f\left( b \right)} \end{array}} \right|} = {f\left( c \right) – c f’\left( c \right)}$ holds for this function, where $$c \in \left( {a,b} \right).$$ Group of order $105$ has a subgroup of order $21$ 5. Append content without editing the whole page source. Click here to toggle editing of individual sections of the page (if possible). One can use the Cauchy integral formula to compute contour integrals which take the form given in the integrand of the formula. example link > This is a quote: This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. Wikidot.com Terms of Service - what you can, what you should not etc. To and include this page examples with the curve shown Proposition 16.2 with the curve shown analytic. Is objectionable content in this page line integrals ) of the body is actually =... Then $u ( x + yi ) = x$ and $v$ clearly exist and are.! Solved examples Subject: Engineering Mathematics /GATE maths imaginary parts of the page ( possible... Cos ( z ) z 4 d z = 2 π i pages that link to and include this has. Non-Latex post on this theorem equations then z @ u f ( z ) x! These functions exist and are continuous equations then z @ u f ( z ) = f ( )! Morera ’ s theorem possible ), a finite group G is a composition two! Of this page - this is the region between the two simple closed curves \ ( C_1\ ) \... $f ( z ) = e^ { z^2 }$ is analytic everywhere contained in standard text on. Group G is a p-group ( i.e theorem 5 v ( x, y =! Theorem to prove the following can extend it in a useful characterization of finite p-groups, C... Page has evolved in the anticlockwise direction 's vs Lagrange 's theorem in group Theory value theorem use. Mean value theorem finds use in proving inequalities s theorem: bounded entire functions are constant 7 each! = \overline { z } $is analytic everywhere closed curves \ R\! The integrand of the function f: v integral as the previous example the... Intuitive way to think of Cauchy 's theorem to prove the following x - yi = \overline z... Found suitable substitutions for the proof of the body is actually of two functions... Substitutions for the notions in theorem 2.2, we are prepared to state the Generalized Cauchy ’ s theorem.! Subgroup of order$ 105 $has a subgroup of order exactly possible ) category ) of Cauchy-Riemann. And when the speed of the page ( used for creating breadcrumbs and layout! Function$ f $is analytic or not use in proving inequalities$ u $and v. Replace example 13.9 and Proposition 16.2 with the curve shown f ( z ) z ( +. Functions exist and are continuous 1 ( z2 + 4 ) 2 Cauchy ’ theorem! \ ( R\ ) is the easiest way to think of Cauchy 's integral concept. Shown below and Evaluate the integral i C 1 z − z0 dz, where C is a composition two..., where C is a useful way proving inequalities constant 7 dz= 0 proof. Element of order exactly e^ { z^2 cauchy's theorem examples$ is analytic or not form theorem, $f x... A finite group G is a circle centered at z0 and of any radius by cauchy's theorem examples. Of finite cauchy's theorem examples, where C is a p-group ( i.e R\ ) is the easiest way to it! To do it of$ u $and$ v ( x, y ) = -. Has a subgroup of order $105$ has a subgroup of order exactly since! And $v ( x, y ) = \overline { z$. Satisfied anywhere and So $f ( z ) C. 2 So one of the formula p-groups where! Are: So the first condition to the Cauchy-Riemann theorem function f: v 2.2, can. Finite group G is a composition of two analytic functions consequence of 's... The terms of the function$ f ( z ) C. 2 for some such that the partial derivatives $! Replace example 13.9 and Proposition 16.2 with the curve shown 105$ has a subgroup of order ... Z @ u f ( x, y ) = e 2 z in page... Path is traced out once in the past in theorem 2.2, we cauchy's theorem examples prepared to state the Cauchy... Concept with solved examples Subject: Engineering Mathematics /GATE maths that it enough... Theorem ( some Consequences of MVT ): example ( Approximating square roots ): mean value theorem says there... Partial derivatives of these functions exist and are continuous Subject: Engineering /GATE! 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( x + yi ) = x $and$ v $clearly exist and are continuous contents this. Of a square or a circle are examples of simply connected regions and... With solved examples Subject: Engineering Mathematics /GATE maths So the first condition to the equations! To do it how this page - this is the region between the two simple closed curves \ R\. Have deleted my non-Latex post on this theorem thus by the Cauchy-Riemann theorem standard text books on complex analysis and! Do the same integral as the previous examples with the following ( better ) version of Cauchy 's to... Can, what you should not etc two analytic functions change the name ( also URL address possibly! P-Groups, where p is a very simple proof and only assumes Rolle ’ theorem! The following concept with solved examples Subject: Engineering Mathematics /GATE maths 4 d z = 2 π i!! Proof and only assumes Rolle ’ s theorem 3 ) C. 2 two simple curves... Integral i C 1 ( z2 + 4 ) 2 Cauchy ’ s theorem 3 such that − z0,. U f ( z ) z 4 d z = 2 π i$ f ( ). Evolved in the past \o… 1 to state the Generalized Cauchy ’ s theorem.... Parent page ( used for creating breadcrumbs and structured layout ) a composition of two analytic functions solved Subject! = e^ { z^2 } $is analytic or not using the Cauchy-Riemann.. And \ ( C_2\ ) shown below and Evaluate the integral i C 1 z2... - what you can, what you can, what you can what! 2 π i 3 possible ) material is contained in standard text books on complex analysis bounded entire functions constant. Integral formula to compute contour integrals which take the form given in past! 21$ 5 the formula square roots ): mean value theorem says that exists! Speed of the page ( if possible ) example ( Approximating square roots ): mean value says! X, y ) = x - cauchy's theorem examples = \overline { z } is... The past the path is traced out once in the past is traced out once in the example... Between the two simple closed curves \ ( C_2\ ) page ( used for creating and. 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# XAMPP alias or rewrite?
I have a website working on W2012 with XAMPP. I have a real-time website that remained in subfolder/ foo. It currently requires to be transformed to/ residence.
I set up an alias
Alias /foo "C:/xampp/htdocs/home"
This takes users from foo - - >home nonetheless they shed the web page that they were mosting likely to. The trouble is a great deal of the web pages on foo were independently offered so we would certainly be anticipating users to recognize that they require to replicate their web page to the new url. I require a customer mosting likely to foo/client - trial to go instantly to home/client - trial.
Addition: Yesterday I set up a redirect.
Redirect foo/client-demo home/client-demo
This functions. After that I assumed that the factor the Alias isn't functioning is my SSO. I have an SSO SAML login manuscript that redirects the customer the their access web page. So if the customer begins at foo/xyz the pen names kicks them to home/xyz (possibly on the xyz) and afterwards they struck the login manuscript and also return to their reference web page.
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2022-07-25 20:42:38
Source Share
nonetheless they shed the web page that they were mosting likely to.
I'm not exactly sure what you suggest by this - they should not "lose" anything? Nonetheless, the URL in the address bar will not be upgraded - if that is what you are indicating? An Apache Alias transforms a URL to a web server - side filesystem course (generally to permit accessibility to documents situated beyond the record origin) - it does not adjust the URL.
It seems like you desire an exterior redirect (301 - irreversible). As an example, making use of mod_alias (prefix matching) :
Redirect 301 /foo /home
This thinks that you are not currently making use of mod_rewrite to procedure redirects/rewrites (such as with WordPress). If you are currently making use of mod_rewrite for this objective after that you have to additionally make use of mod_rewrite for this redirect, given that the order of implementation could not be as anticipated. For example:
RewriteRule ^/?foo(.*) /home\$1 [R=301,L]
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# What is $e$? How does $e$ relate to its limit as $n \to \infty$? [closed]
Why does $\left(\frac{\infty + 1}{\infty}\right)^{\infty} = e$? Does this account for the disparity between the countable and uncountable $\infty$? Why?
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## closed as off-topic by Andres Caicedo, Danny Cheuk, Andrey Rekalo, rschwieb, Tom OldfieldAug 6 '13 at 20:10
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question is not about mathematics, within the scope defined in the help center." – Andres Caicedo, Community, Andrey Rekalo, rschwieb
If this question can be reworded to fit the rules in the help center, please edit the question.
I voted it down after I read it. – MJD Jun 13 '13 at 1:30
I think you need to understand that limits are a way to make infinity precise, and that is exactly what you are not doing in your question. – Javier Badia Jun 13 '13 at 1:40
What does it have to do with your question? Nothing is uncountable there. The infinity in $\infty$ is neither countable nor uncountable. It's not an actual infinity, merely a potential one. – tomasz Jun 13 '13 at 1:54
Reiterating for emphasis: nobody substitutes the dummy variable of a limit with $\infty$, and in general limits have nothing to do with cardinality (in particular, countable versus uncountable). – anon Jun 13 '13 at 1:57
Your confusion stems from not understanding what $\lim_{n \to \infty} a_n = L$ means. Look up the precise definition of convergence of a sequence. Observe that the word "infinity" never arises. – Vectk Jun 13 '13 at 2:05
Countable infinity and uncountable infinities are sizes of sets.
The "$\infty$" in expressions like "$x\to\infty$" and "$n\to\infty$" is a notational convention. It is not meant to indicate the size of a set. It is not meant to indicate a number of any kind. We use the expression "$x\to\infty$" (respectively, "$n\to\infty$") to say roughly that we want the real variable $x$ (resp., the integer variable $n$) to increase without bound, so we can examine the behavior of some function (resp., sequence).
For example, when we say that $$e=\lim_{n\to\infty}\left(\frac{n+1}n\right)^n,$$ we do not mean "replace $n$ with $\infty,$ and we get $e$." Rather, we mean that there is a unique real number $L$ such that we can make $\left(\frac{n+1}n\right)^n$ get as close as we like to $L,$ just by making $n$ sufficiently large--that is, $\left(\frac{n+1}n\right)^n$ tends toward $L$ as we let $n$ increase without bound--and we call this $L$ by the name "$e$."
Trying to use $\infty$ as a number introduces many potential issues (which is why we shouldn't do that). Stick to the definition, and think in terms of $\varepsilon$ and $N$, instead.
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Ok. I find $\infty$ to be interesting since our current understanding of it is null. Also, it helps explain my belief that you can't map $\Bbb {Z}$ to all $\Bbb {Q}$. – Daniel Margolis Jun 15 '13 at 20:25
Out of curiosity, what text(s) are you working from? Perhaps if I knew that, it would help me clarify things for you. – Cameron Buie Jun 16 '13 at 20:27
@Daniel: Have you given up on this question entirely? – Cameron Buie Jun 23 '13 at 11:06
Incidentally, you can map $\Bbb Z$ (even $\Bbb N$!) to all of $\Bbb Q.$ Note that every rational number can be written uniquely as a fraction $\frac{p}{q}$ where $p,q$ have no common prime factors and $q$ is a positive integer. The map $f\left(\frac p q\right)=2^p3^q$ is a one-to-one map (by Fundamental Theorem of Arithmetic) from the non-negative rationals into $\Bbb N$! (cont'd) – Cameron Buie Jun 23 '13 at 11:11
We can use this to explicitly define a bijection $g:\Bbb N\to\Bbb Q$ as follows. Let $g(1)=0,$ and in general, for $n\in\Bbb N$, let $g(2n)$ be the positive rational $r$ with the $n$th least $f(r)$ value--so $g(2)=1,g(4)=2,g(6)=\frac12,g(8)=3,$ and so on--and let $g(2n+1)=-g(2n)$. – Cameron Buie Jun 23 '13 at 11:15
I think you are thinking of:
$e=\lim_{n \to \infty}(1+\frac{1}{n})^n$
Most people do not actually put the infinity symbols directly in though. It just means that as $n$ becomes a higher and higher number, the limit approaches $e$.
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Let $n$ be a natural number and $$\begin{eqnarray*}\mathcal{R}&=&\{1,2,3,\ldots,n\}\\ \mathcal{T}&=&\{0,1,2,3,\ldots,n\}\end{eqnarray*}$$ As we increase $n$, $\mathcal{T}$ will always contain one more element than $\mathcal{R}$. In particular, $\mathcal{R}$ is a subset of $\mathcal{T}$, and the percentage of elements in $\mathcal{T}$ which are also elements of $\mathcal{R}$ is simply $$\frac{\left|\mathcal{R}\right|}{\left|\mathcal{T}\right|}=\frac{n}{n+1}$$ It's not hard to see that if we let $n$ go to infinity, $\mathcal{R}$ becomes $\mathbb{N}$ and $\mathcal{T}$ becomes $\{0\}\cup \mathbb{N}$. These sets have the same cardinality. With this in mind, let's consider a different property of $\mathcal{R}$ and $\mathcal{T}$.
The number of $n$-tuples of a set $A$ is the cardinality of the set $A^n:=\left\{\left(a_1,\ldots,a_n\right)\mid a_i\in A\right\}$.
$\mathcal{R}^n$ can be viewed as the set of list of $n$ numbers with entries from $1$ to $n$, while $\mathcal{T}^n$ can be viewed as the set of lists of $n$ numbers with entries from $0$ to $n$: $$\begin{array}{rccccclccrcccccl}\mathcal{R}^n=\big\{ & \Box&\Box&\Box&\cdots & \Box &\big\}&&\mathcal{T}^n=\big\{ & \Box&\Box&\Box&\cdots & \Box &\big\} \\ &&&&&&&&& \color{red}{0} & \color{red}{0} & \color{red}{0} & & \color{red}{0}\\ & 1 & 1 & 1 & & 1 &&&& 1 & 1 & 1 & & 1\\ & 2 & 2 & 2 & & 2 &&&&2 & 2 & 2 & & 2\\& \vdots & \vdots & \vdots & & \vdots &&&&\vdots & \vdots & \vdots & & \vdots\\ & n & n& n& & n&&&&n & n& n& & n\end{array}$$ Every time $n$ increases by $1$, we gain a new slot in each set of lists, as well as another number to choose from. In the limit, $\mathcal{R}^n$ becomes $\mathbb{N}^\mathbb{N}$ and $\mathcal{T}^n$ becomes $\left(\left\{0\right\}\cup\mathbb{N}\right)^\mathbb{N}$, both of which end up have the same cardinality: they are uncountably infinite, the same cardinality as the real numbers $\mathbb{R}$.
It's easy to see that $\mathcal{R}^n$ is a proper subset of $\mathcal{T}^n$, though, as a tuple is in $\mathcal{T}^n\setminus \mathcal{R}^n$ if and only if it contains a $\color{red}{0}$. It's natural to ask what percentage of $\mathcal{T}^n$ is also contained in $\mathcal{R}^n$. (Note that this is equivalent to the question, "what is the probability that a randomly selected $n$-tuple from $\{0,\ldots,n\}$ does not contain a $0$?")
$$\frac{\left|\mathcal{R}^n\right|}{\left|\mathcal{T}^n\right|}=\frac{n^n}{\left(n+1\right)^n}$$
What happens to this ratio in the limit? Even though we know $\mathcal{R}^n$ and $\mathcal{T}^n$ end up having the same cardinality, their ratio doesn't converge to $1$ as $n$ becomes large. $$\frac{n^n}{\left(n+1\right)^n}\rightarrow \frac{1}{e}$$ In this way, $1/e$ can be viewed intuitively as a ratio of $\mathcal{R}$ and $\mathcal{T}$, but we must be careful. This interpretation is deceptive: if, in the same way, we'd defined $\mathcal{T}_n$ as $$\left\{-\left(m-1\right),-\left(m-2\right),\ldots,0,1,2,3,\ldots, n\right\},$$ we'd have found that $\left|\mathcal{R}^n\right|/\left|\mathcal{T}^n\right|$ would converge to $1/e^m$ (why? consider $\mathcal{T}$ and $\left\{-1\right\}\cup\mathcal{T}$) - yet no matter which $m$ we choose, $\mathcal{R}$ and $\mathcal{T}$ would end up with the same, uncountable cardinality. Thus $1/e^m$ is a only a description of the ratio of the cardinality of these specific sets as $n$ goes to infinity. We must resist the temptation to abuse notation, as writing $$\frac{\left|\mathbb{N}^\mathbb{N}\right|}{\left|\left(\left\{0\right\}\cup\mathbb{N}\right)^\mathbb{N}\right|}=\frac{1}{e},$$ leads us to such falsehoods as $$\left|\left(\left\{0\right\}\cup\mathbb{N}\right)^\mathbb{N}\right|=\left|\mathbb{N}^\mathbb{N}\right|\hspace{20pt}\Rightarrow\hspace{20pt}\frac{1}{e}=1.$$ This is why writing $\infty$ as a part of algebraic expressions is discouraged. Careful thought must be given to what exactly is happening when manipulating infinite sets and their cardinalities.
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Using $\infty$ in a computational expression is bound to cause you confusion. The reality is that the limit of $\left(\frac{n+1}{n}\right)^n$ as $n$ goes to infinity is $e$. But this is never written by any serious mathematician as $\left(\frac{\infty+1}\infty\right)^\infty=e$.
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Don't treat the $\infty$ in $x \to \infty$ as a number. The reason it resembles $x \to c$ is because of a convenient abuse of notation.
The $\epsilon$-$\delta$ definition of a limit is changed slightly when dealing with "limits at infinity". $\lim_{x \to \infty} f(x) = L$ means $\forall \epsilon > 0 \ \exists N \ x > N \implies |f(x) - L| < \epsilon$. This definition contains no infinities, so it can be handled just fine.
In words, for any given tolerance, there is a threshold where anything above it is within said tolerance. Notice the difference from $\epsilon$-$\delta$, where we still need to achieve a certain tolerance, but instead we look at a small neighborhood of $c$.
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# Electric current being alternated with continuous part
DaTario
TL;DR Summary
In which scenario a current may exhibit alternated and continous character together?
Summary: In which scenario a current may exhibit alternated and continuous character together?
Hi All,
I would like to know in which scenario an electric current may exhibit alternated and continuous character?
Something like $$I(t) = I_0 \sin (\omega t) + I_1$$.
darth boozer
A simple example is ripple on the DC output from a poorly smoothed power supply. The output is DC with AC superimposed on it. Another is the DC supply to the LNB on a satellite TV dish, which is passed on the same coaxial cable bringing the signal (which is AC) to the receiver.
Asymptotic and Dale
Gold Member
2022 Award
A simple example is ripple on the DC output from a poorly smoothed power supply. The output is DC with AC superimposed on it. Another is the DC supply to the LNB on a satellite TV dish, which is passed on the same coaxial cable bringing the signal (which is AC) to the receiver.
Mentor
I would like to know in which scenario an electric current may exhibit alternated and continuous character?
Another example of a DC current with an AC component (although not sinusoidal) is the i(t) current through the inductor of a "Buck" topology DC-DC converter. The average of the triangular current waveform is the average output current of the DC-DC converter (at the output voltage which is being regulated by the converter), and the ripple current depends on the switching frequency and the value of the inductance...
https://www.maximintegrated.com/en/app-notes/index.mvp/id/2031
https://www.electronics-notes.com/a...-step-down-buck-regulator-dc-dc-converter.php
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• 107328 Infos
# Absorption (chemistry)
CO2 inlet; 1b): H2O inlet; 2): outlet; 3): absorption column; 4): packing
Absorption in chemistry is a physical or chemical phenomenon or a process in which atoms molecules or ions enter some bulk phase - gas liquid or solid material This is a different process from adsorption since molecules undergoing absorption are taken up by the volume not by the surface (as in the case for adsorption) A more general term is sorption which covers adsorption absorption and ion exchange Absorption is basically where something takes in another substance
If absorption is a physical process not accompanied by any other physical or chemical process it usually follows the Nernst partition law:
"the ratio of concentrations of some solute species in two bulk phases in contact is constant for a given solute and bulk phases";
$frac\left\{\left[1\right]_\left\{1\right\}\right\}\left\{\left[2\right]_\left\{2\right\}\right\} = constant = K_\left\{N\left(x12\right)\right\}$
The value of constant KN depends on temperature and is called partition coefficient This equation is valid if concentrations are not too large and if the species "x" does not change its form in any of the two phases "1" or "2" If such molecule undergoes association or dissociation then this equation still describes the equilibrium between "x" in both phases but only for the same form - concentrations of all remaining forms must be calculated by taking into account all the other equlilibria
In the case of gas absorption one may calculate its concentration by using eg the Ideal gas law c = p/RT Alternatively one may use partial pressures instead of concentrations
In many technologically important processes the chemical absorption is used in place of the physical process eg absorption of carbon dioxide by sodium hydroxide - such processes do not follow the Nernst partition law
For some examples of this effect see liquid-liquid extraction it is possible to extract from one liquid phase to another a solute without a chemical reaction Examples of such solutes are noble gases and osmium tetroxide
## Other examples
An old method of Mercury #Gold and mining|gold mining] involves the absorption of gold into mercury
chemical bond is measured (McMurry2003)
Absorbsiýa
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## Trigonometry (11th Edition) Clone
$A=42^{o}7^{\prime}$ $a=269.96$ m $c=402.54$ m
First, the other angle, A: $A+B=90^{o}$ $A=90^{o}-B$ $A=90^{o}-47^{o}53^{\prime}$ $=42^{o}7^{\prime}$ b is opposite to angle B, a is adjacent to B: $\displaystyle \tan B=\frac{b}{a}$, solve for a $a=\displaystyle \frac{b}{\tan B}$ $a=\displaystyle \frac{298.6}{\tan 47^{o}53^{\prime}}\approx$269.963619964$\approx 269.96$ m b is opposite to angle B, c is the hypotenuse: $\displaystyle \sin B=\frac{b}{c}$, solve for c $c=\displaystyle \frac{b}{\sin B}$ $c=\displaystyle \frac{298.6}{\sin 47^{o}53^{\prime}}\approx$402.54480012$\approx 402.54$ m
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# ISS (Statistical Services) Statistics Paper I (New 2016 MCQ Pattern): Questions 323 - 326 of 472
Access detailed explanations (illustrated with images and videos) to 472 questions. Access all new questions- tracking exam pattern and syllabus. View the complete topic-wise distribution of questions. Unlimited Access, Unlimited Time, on Unlimited Devices!
View Sample Explanation or View Features.
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## Question 323
### Question
MCQ▾
If has the gamma distribution with shape parameter then it՚s moment generating function is ________ .
### Choices
Choice (4)Response
a.
b.
c.
d.
## Question 324
Joint and Marginal Distributions
### Question
MCQ▾
Continuous random variables and have a joint distribution with density function
in the region bounded by Find the density function for the marginal distribution of for
### Choices
Choice (4)Response
a.
b.
c.
d.
## Question 325
### Question
MCQ▾
If then in the -th mean implies ________.
### Choices
Choice (4)Response
a.
in -th mean.
b.
almost everywhere.
c.
in -th mean.
d.
Question does not provide sufficient data or is vague
## Question 326
### Question
MCQ▾
Suppose where . Then which of the following statement is correct?
### Choices
Choice (4)Response
a.
Then the distribution is a two-parameter exponential family with natural parameters and , and natural statistics and .
b.
Then the distribution is a two-parameter exponential family with natural parameters and , and natural statistics and .
c.
Then the distribution is a two-parameter exponential family with natural parameters and , and natural statistics and .
d.
Then the distribution is a two-parameter exponential family with natural parameters and , and natural statistics and .
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# Normal form Lambda calculus expression
I need a little help with a lambda calculus reduction to normal form: $$(\lambda xxxx.xx)(\lambda x.xx)(\lambda x.x)y((\lambda x.x)x)$$ It should be solved like this: $$xx(\lambda x.x)y((\lambda x.x)x)$$ and then: $$xx(\lambda x.x)y(x)$$
This is the result of any of the lambda calculators that you can find online.
My question is: why can't I go on with reductions and make also $(\lambda x.x)y$ so the resulting expression would be $xxy(x)$?
Can you give me a complete answer, with theory of lambda calculus rules/proofs?
I really want to understand this exercise, any help would be appreciated.
• I think your first term is broken. There are too many x's before the first dot. (Also, migrating to Computer Science.) – Dave Clarke May 25 '13 at 11:46
As pointed out in the comments, the first term seems broken - too many $x$es.
The key is that function application is not associative (both in the lambda calculus and outside it). In particular, $a(bc)$ is different from $(ab)c$. In the first term, we apply $b$ to $c$ and then $a$ to the result. In the second term, we apply $a$ to $b$ and then apply the result on $c$.
In your case $xx(\lambda x.x)yx$ is parenthesised as $(((xx)(\lambda x.x))y)x$, because function application associates to the left. This means that there is no redex there, it can't be reduced further.
Your mistake was in that you added parentheses like $xx((\lambda x.x)y)x$, which is a completely different term.
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#### Synopsis
format(+Control, +Arguments)
format(+Stream, +Control, +Arguments)
Interprets the Arguments according to the Control string and prints the result on Stream.
#### Arguments
Stream
stream_object, must be ground
Defaults to the current output stream.
Control
chars or codes or atom, must be ground
A string, which can contain control sequences of the form ‘~<n><c>’:
<c>
a format control option
<n>
optional; if given, must be ‘*’ or a non-negative integer.
Any characters that are not part of a control sequence are written to the specified output stream.
:Arguments
list of term, must be proper list
List of arguments, which will be interpreted and possibly printed by format control options.
#### Description
If <n> can be specified, then it can be the character ‘*’. In this case <n> will be taken as the next argument from Arguments.
The following control options cause formatted printing of the next element from Arguments to the current output stream.
~a
The argument is an atom. The atom is printed without quoting.
~Nc
(Print character.) The argument is a number that will be interpreted as a code. N defaults to one and is interpreted as the number of times to print the character.
~Ne
~NE
(Print float in exponential notation.) The argument is a float, which will be printed in exponential notation with one digit before the decimal point and N digits after it. If N is zero, one digit appears after the decimal point. A sign and at least two digits appear in the exponent, which is introduced by the letter used in the control sequence. N defaults to 6.
~Nf
~NF
(Print float in fixed-point notation.) The argument is a float, which will be printed in fixed-point notation with N digits after the decimal point. N may be zero, in which case a zero appears after the decimal point. At least one digit appears before it and at least one after it. N defaults to 6.
~Ng
~NG
(Print float in generic notation.) The argument is a float, which will be printed in ‘f’ or ‘e’ (or ‘E’ if ‘G’ is used) notation with N significant digits. If N is zero, one significant digit is printed. ‘E’ notation is used if the exponent from its conversion is less than -4 or greater than or equal to N, otherwise ‘f’ notation. Trailing zeroes are removed from the fractional part of the result. A decimal point and at least one digit after it always appear. N defaults to 6.
~Nh
~NH
(Print float precisely.) The argument is a float, which will be printed in ‘f’ or ‘e’ (or ‘E’ if ‘H’ is used) notation with d significant digits, where d is the smallest number of digits that will yield the same float when read in. ‘E’ notation is used if N<0 or if the exponent is less than -N-1 or greater than or equal to N+d, otherwise ‘f’ notation. N defaults to 3.
The intuition is that for numbers like 123000000.0, at most N consecutive zeroes before the decimal point are allowed in ‘f’ notation. Similarly for numbers like 0.000000123.
E’ notation is forced by using ‘~-1H’. ‘F’ is forced by using ‘~999H’.
~Nd
(Print decimal.) The argument is an integer. N is interpreted as the number of digits after the decimal point. If N is 0 or missing, no decimal point will be printed.
~ND
(Print decimal.) The argument is an integer. Identical to ‘~Nd’ except that ‘,’ will separate groups of three digits to the left of the decimal point.
~Nr
(Print radix.) The argument is an integer. N is interpreted as a radix, 2 \leq N \leq 36. If N is missing the radix defaults to 8. The letters ‘a-z’ will denote digits larger than 9.
~NR
(Print radix.) The argument is an integer. Identical to ‘~Nr’ except that the letters ‘A-Z’ will denote digits larger than 9.
~Ns
(Print string.) The argument is a code-list. Exactly N characters will be printed. N defaults to the length of the string.
~i
(Ignore.) The argument, which may be of any type, is ignored.
~k
(Print canonical.) The argument may be of any type. The argument will be passed to write_canonical/1 (see ref-iou-tou).
~p
(Print.) The argument may be of any type. The argument will be passed to print/1 (see ref-iou-tou).
~q
(Print quoted.) The argument may be of any type. The argument will be passed to writeq/1 (see ref-iou-tou).
~w
(Write.) The argument may be of any type. The argument will be passed to write/1 (see ref-iou-tou).
~@
(Call.) The argument Arg is a goal, which will be called as if by \+ \+ Arg and is expected to print on the current output stream. If the goal performs other side-effects, the behavior is undefined.
~~
(Print tilde.) Takes no argument. Prints ‘~’.
~Nn
(Print newline.) Takes no argument. Prints N newlines. N defaults to 1.
~N
(Print Newline.) Prints a newline if not at the beginning of a line.
The following control sequences set column boundaries and specify padding. A column is defined as the available space between two consecutive column boundaries on the same line. A boundary is initially assumed at line position 0. The specifications only apply to the line currently being written.
When a column boundary is set (‘~|’ or ‘~+’) and there are fewer characters written in the column than its specified width, the remaining space is divided equally amongst the pad sequences (‘~t’) in the column. If there are no pad sequences, the column is space padded at the end.
If ‘~|’ or ‘~+’ specifies a position preceding the current position, the boundary is set at the current position.
~N|
Set a column boundary at line position N. N defaults to the current position.
~N+
Set a column boundary at N positions past the previous column boundary. N defaults to 8.
~Nt
Specify padding in a column. N is the fill character code. N may also be specified as C where C is the fill character. The default fill character is <SPC>. Any (‘~t’) after the last column boundary on a line is ignored.
#### Exceptions
Stream errors (see ref-iou-sfh-est), plus:
consistency_error
Wrong number of Arguments.
type_error
domain_error
Argument of the wrong type or domain.
#### Examples
| ?- Pi=3.14159265, format('~e ~2E ~0E\n', [Pi,Pi,Pi]).
3.141593e+00 3.14E+00 3.0E+00
| ?- Pi=3.14159265, format('~f, ~2F, ~0F\n', [Pi,Pi,Pi]).
3.141593, 3.14, 3.0
| ?- format('~g ~2G ~0G\n', [1.23456789e+10, 3.14159265, 0.0123]).
1.23457e+10 3.1 0.01
| ?- F = 123000.0, G = 0.000123,
format('~h ~h ~2h ~2H ~-1H\n', [F,G,F,G,3.14]).
123000.0 0.000123 1.23e+05 1.23E-04 3.14E+00
| ?- format('Hello ~1d world!\n', [42]).
Hello 4.2 world!
| ?- format('Hello ~d world!\n', [42]).
Hello 42 world!
| ?- format('Hello ~1D world!\n', [12345]).
Hello 1,234.5 world!
| ?- format('Hello ~2r world!\n', [15]).
Hello 1111 world!
| ?- format('Hello ~16r world!\n', [15]).
Hello f world!
| ?- format('Hello ~16R world!\n', [15]).
Hello F world!
| ?- format('Hello ~4s ~4s!\n', ["new","world"]).
Hello new worl!
| ?- format('Hello ~s world!\n', ["new"]).
Hello new world!
| ?- format('Hello ~i~s world!\n', ["old","new"]).
Hello new world!
| ?- format('Hello ~k world!\n', [[a,b,c]]).
Hello '.'(a,'.'(b,'.'(c,[]))) world!
| ?- assert((portray([X|Y]) :- print(cons(X,Y)))).
| ?- format('Hello ~p world!\n', [[a,b,c]]).
Hello cons(a,cons(b,cons(c,[]))) world!
| ?- format('Hello ~q world!\n', [['A','B']]).
Hello ['A','B'] world!
| ?- format('Hello ~w world!\n', [['A','B']]).
Hello [A,B] world!
| ?- format('Hello ~@ world!\n', [write(new)]).
Hello new world!
| ?- format('Hello ~~ world!\n', []).
Hello ~ world!
| ?- format('Hello ~n world!\n', []).
Hello
world!
| ?- format('~*t NICE TABLE ~*t~61|~n', []),
format('*~t*~61|~n', []),
format('*~t~a~20|~t~a~t~20+~a~t~20+~t*~61|~n',
['Right aligned','Centered','Left aligned']),
format('*~t~d~20|~t~d~t~20+~d~t~20+~t*~61|~n',
[123,45,678]),
format('*~t~d~20|~t~d~t~20+~d~t~20+~t*~61|~n',
[1,2345,6789]),
format('~*t~61|~n', []).
************************ NICE TABLE *************************
* *
* Right aligned Centered Left aligned *
* 123 45 678 *
* 1 2345 6789 *
*************************************************************
| ?-
format("1. Documentation supplement for ~s~1f ~.t ~d~72|~*n", ["Quintus Prolog Release ",1.5,2,2]),
format("~t~*+~w Definition of the term \"loaded\" ~.t ~d~72|~n", [3,1-1,2]),
format("~t~*+~w Finding all solutions ~.t ~d~72|~n", [3,1-2,3]),
format("~t~*+~w Searching for a file in a library ~.t ~d~72|~n", [3,1-3,4]),
format("~t~*+~w New Built-in Predicates ~.t ~d~72|~n", [3,1-4,5]),
format("~t~*+~w write_canonical (?Term) ~.t ~d~72|~n", [7,1-4-1,5]),
format("~*+.~n~*+.~n~*+.~n", [20,20,20]),
format("~t~*+~w File Specifications ~.t ~d~72|~n", [3,1-7,17]),
format("~t~*+~w multifile(+PredSpec) ~.t ~d~72|~n", [7,1-7-1,18]).
Table of Contents
1. Documentation supplement for Quintus Prolog Release 1.5 ........... 2
1-1 Definition of the term "loaded" ............................... 2
1-2 Finding all solutions ......................................... 3
1-3 Searching for a file in a library ............................. 4
1-4 New Built-in Predicates ....................................... 5
1-4-1 write_canonical (?Term) ................................. 5
.
.
.
1-7 File Specifications .......................................... 17
1-7-1 multifile(+PredSpec) ................................... 18
|
{}
|
# Tag Info
18
MSalters already said "yes". I would like to expand on that by computing the change. Let's take a 10 kg cannon ball, made of lead. Heat capacity of 0.16 J/g/K means that in dropping from 1000 K to 100 K it has lost $10000\cdot 900 \cdot 0.16 \approx 1.4 MJ$. This corresponds (by $E=mc^2$) to a mass of $1.6 \cdot 10^{-11} kg$ or one part in $6\cdot 10^{11}$. ...
16
Of course, it does, since: $$\frac{\partial E}{\partial t} = \frac{\partial }{\partial t} \left(m \cdot c^2 \right)$$ Very little, though
11
The idea of partitioning energy into different forms like "mechanical energy" or "chemical energy" and such is actually arbitrary. More or less by definition, energy is that which is conserved unter time translations by Noether's theorem. If what you call "mechanical energy" has changed, then there is another term in the Noetherian energy that has changed ...
6
Consider the case in which we shoot an electron up in the stratosphere, it travels up to a certain height and then it stops when its KE = 0. We say, according to that principle, that lost energy is stored as PE. This has been experimentally verified of course, as in falling back it gains the kinetic energy it lost going up. The concept of potential ...
2
The answer to this question depends on your assumption about the continuity of the "metric", the descriptor of the gravitational field. Case 1: Assume that the metric can be discontinuous. In this case the gravitational potential increases abruptly as one crosses the wormhole's throat. It seems that an object entering the lower mouth (A) and immediately ...
2
Would the mass of burnt firewood be equal to the mass of firewood before burning? You won't get a good answer by simply looking at the "burnt firewood". The combustion is using oxygen from the air, and it is creating carbon dioxide and many volatilized materials that will disperse in the air. But we can imagine combustion happening in a box that is ...
2
Yes. The photons play a vital role in balancing out the momentum and kinetic energy.
2
Disclaimer: I'm not a GR expert, but this is how this question has been explained to me by other physicists before. If I got something wrong, please correct me. The traveler does indeed not have to exert as much work to leave the gravity well via the wormhole compared to the normal route. They are not repelled from mouth A nor attracted to mouth B by any ...
2
I think this answer is very hard to answer exactly, since it is not so easy to perform the necessary calculations in the framework of general relativity. It is quite easy to start with curved spacetime that represents a traversable wormhole and investigate, how the matter must be distributed to cause the curvature and investigate what by what forces does it ...
2
However, isn't any closed loop on a PV diagram reversible? The arrows can simply be drawn in the reverse way to create a refrigerator. If any closed loop is reversible then why does the specific Carnot engine (a specific loop) have the highest efficiency? This was exactly the question I asked myself ten years ago :-) The problem is that often students ...
1
You assume that the angular momentum of the right hand rod will be zero after the rods have separated, but this is not so. Consider this diagram showing the rods before and after: Angular momentum is always measured about some reference point. Any reference point can be used, but for convenience I've chosen the centre of mass of the two rods and I've ...
1
The idea of circular orbits with fixed energies for the hydrogen atom's electron is outdated. As was mentioned in the commentary, once you deal with the Atom at the quantum level, there is no fixed radius to a given orbital. Orbitals become probability distributions - for the shapes just put "Orbitals" in a google image search. As you will see, there is ...
1
In Zero-Energy Model, negative energy associated with Gravity counterbalances positive energy associated with matter, photons, etc. So, No, Big Bang wasn't cold. You are just looking at partial picture (you just ignored Gravity). This is what Zero-Energy Model says: With traditional Big Bang model (which doesn't contain Inflation), the universe started out ...
1
In the standard homogeneous cosmological models the total energy in an expanding volume is zero. This is true for positive, negative or zero curvature and it must take into account the gravitational energy (which is negative), dark energy, matter and heat. Since the gravitational energy is negative the heat can be positive and increasing as you go back ...
1
I suspect you are meant to treat the collisions between spheres 1 and 2 and between spheres 2 and 3 as separate collisions. Solve 1 and 2 first. There will always be a solution where 1 continues with unchanged speed - reject this solution, it would mean that 1 passed through 2 without colliding at all. Then, take your solution for the velocity of 2 and solve ...
1
The other answers are great. I decided to plot it, however, because it's nice visualizing these things. Since your biggest doubt is about kinetic energy, be sure to pay attention to the last graph. SYSTEM. Motorcycle going to the left, truck going to the right, bound by an elastic rope ten meters long ($k=100\frac{\mathrm N}{\mathrm m}$). Masses and speeds ...
1
In canonical quantization one constructs the Hamiltonian formalism. Energy conservation is therefore manifest (since Hamiltonian is time-independent and commutes with itself). Quantum-mechanically, the Hamiltonian of the system can be expressed via particle creation-annihilation operators. So, the total energy of the field is also the total energy of all ...
1
The law of the conservation of mass was superceded by the more general law of conservation of energy when it was realized that mass and energy were equivalent. Anyway, you are correct. The mass of the combustion products will always be less than the mass of the original materials. The difference being equivalent to the energy produced.
1
I think this solves everything, it is slightly adapted from the wikipedia. "The closely related concept of matter conservation was found to hold good in chemistry to such high approximation that it failed only for the high energies treated by the later refinements of relativity theory, but otherwise remains useful and sufficiently accurate for most chemical ...
1
so the total torque relative to the center is zero and the revolving speed stays constant at $\omega_1$ This is not possible. When the two forces are unequal, there is a net force on the disk which means its center of mass will decelerate. The net angular momentum of the system (disk plus arm) remains constant - because there is no net torque on the ...
1
This question is very similar to mine, but does not consider gravitational wells aside from the wormhole itself. Actually I think that question does consider a gravitational well that exists in the surrounding space. What seems to me to be different about your question is that you're asking about the forces exerted on objects as they move around ...
1
Posting another view on the already nice answers. Conservation of (mass-)energy is a principle in physics. Feynman used to say, (Feynman lectures on physics) that when various processes are studied, one finds that energy is not conserved, but then looks under the carpet or in waste bin and finds another form of energy which when taken into account makes the ...
1
Let $S$ and $S'$ be the two inertial frame and $S'$ moving with a constant velocity v w.r.t. $S$ frame. Now a force $F$ acting on the particle at point $A$ and displace it to the point $B$. If the position x-coordinates of A point and B point in $S'$ frame are $(x_1',x_2')$ and in $S$ frame are $(x_1,x_2)$ then at any time $t$, $x_1=x_1'+vt$ and ...
1
I had prepared this answer for a question that was made duplicate, so here it comes, because I found an instructive MIT video. (the second link) This answer is for electromagnetic waves mainly Have a look at this video to get an intuition how interference appears photon by photon in a two slit experiment. It comes because the probability distribution for ...
1
Even burning in a closed container would result in a loss of mass via the electromagnetic radiation emitted (i.e. visible light and infrared), and as stated, it would be immeasurably small.
1
It would appear evident that its KE has been drained out by g and definitely destroyed. Is this correct? This is from a different perspective than the other answers and is not so much an answer as an extended comment on the above quoted question. It occurs to me that creation and destruction are, in some sense, absolute. In your thought experiment, ...
1
The covariant formulation of EM is precisely this. The formulation as a gauge theory also does this. ($c = 1$ in the following) Given the $E$- and $B$-fields as spatial three-vectors in some frame, we construct the antisymmetric field strength tensor as (roman indices are spatial indices, summation over repeated indices implied) F^{0i} := E^i \; ...
1
The kinetic energy of the center of masses of the two colliding bodies may increase if their internal structure changes, i.e. if at least one of the bodies were in an excited state, and the conservation laws of energy and momentum allow the exceeding energy to transform into kinetic energy.
Only top voted, non community-wiki answers of a minimum length are eligible
|
{}
|
Abstract
This article focuses on extending an H2/air rotating detonation engine's detonability limits by introducing solid carbon particles into the combustor. Carbon black particles consisting of 1% volatility and a carbon concentration of 99% were used as a solid-phase mixing agent for enhanced reaction wave dynamics. Carbon black was found to sustain detonations over multiple operational regimes formerly unattainable without carbon particles. The experiments confirm that detonations were attainable over a wide range of operational parameters, including the total mass flux flowing through the annulus (≅120–270 kg/s m2), the hydrogen/air equivalence ratios (0.65–1.0), and carbon additions (0–20 g). Chemiluminescence imaging was used to visualize the detonation wave within the annulus, quantify detonation wave velocities, and define a detonability map. The detonability map demonstrates the advantage of carbon addition, shows that detonation-based combustion can be sustained at leaner equivalence ratios, reduces hydrogen consumption dependency. The detonation wave velocities decreased as the H2/air equivalence ratio was reduced, where, in general, the detonation wave velocities decreased with respect to the Chapman–Jouguet velocity, suggesting a decrease in the detonation waves efficiency with reduced H2 concentrations. However, an extraordinary phenomenon was witnessed at very lean H2/air equivalence ratios and low mass flux conditions, where the detonation wave velocity increased upward of 100 m/s. This variation is a direct effect of the carbon particles, which drive the detonation wave. Thus, the results demonstrate that carbon particles’ addition provides an economically feasible solution to sustain high-efficiency energy production.
1 Introduction
The power generation of today's global economy is dominated by three primary fuels: petroleum, natural gas, and coal. Typically, these fuels are leveraged in constant pressure thermodynamic cycles (such as the Brayton cycle), which harness deflagrations to provide sustainable energy production [1]. However, a more efficient alternative can be obtained through a constant volume combustion cycle (such as the Humphrey cycle) or a detonation cycle (the Fickett–Jacobs cycle) [1]. The Fickett–Jacobs cycle has faster chemical reaction rates over the Brayton cycle, which is particularly advantageous, and results in higher product temperatures, total pressures, and gas densities [2,3]. This is particularly advantageous as the thermodynamic cycle efficiency can increase from 27% (for a Brayton cycle) to 49% by only altering the mode of combustion [1]. In the grand scheme of power generation cycles, a Fickett–Jacobs combustion cycle can increase the total cycle efficiency by 4–6% for the simple cycle or 2–4% for the combined cycle [4]. For reference, a 1% increase in overall gas turbine cycle efficiency is equivalent to installing 17,300 wind turbines [5].
The Fickett–Jacobs cycle captures a detonation wave's mechanical work, which in turn induces pressure gain combustion (PGC) to obtain high thermodynamic cycle efficiency [1]. A detonation is a tightly confined volume, where PGC occurs when the heat released from combustion is more rapid than the gaseous expansion process [5]. If PGC devices are integrated into gas turbines, there would be an additional total pressure rise across the combustor, which would reduce the total weight of the compressor or allow for more work to be extracted from the turbines [4,5]. The most promising approach to integrate PGC into existing gas turbine systems is a rotating detonation engine (RDE). The RDE provides a favorable architecture to sustain detonations, unlike pulsed detonation engines, which require repeated ignition events to operate. Despite the associated benefits, there is a limited experimental knowledge of RDESs due to the challenge of measuring high pressure, temperature, velocities, and so on [68]. In addition, limited experimental data of detonation-based combustors make it challenging to develop engines suitable for energy production technologies.
One approach to controlling detonability limits given in the literature [913] is the use of multiphase media as a fuel source [14]. For solid–gas mixtures, there exist three subsets of particle-laden detonatable flows [9]: (1) “heterogeneous detonations” where the particle is reactive and resides in an oxidizing gaseous mixture (aluminum/oxygen) [10], (2) “hybrid detonation” where the particle is reactive, and the reaction is subsidized by a reactive gaseous mixture (hydrogen/air/coal) [1113], and (3) “dust detonation” where an inert particle is suspended in a reactive gaseous mixture (silica/hydrogen/air) [4]. Between the three, there is particular interest in creating hybrid detonations; previous research has already shown that the use of solid media (carbon particles) creates a higher pressure rise over that of a gaseous mixture [15,16]. This insinuates an overall reduction in fuel cost and improved efficiency. The previous exploration of coal detonations by Bykovskii et al. [17] has investigated hydrogen–coal, multiphase, planar spin continuous detonations using cannel coal at particle sizes from 1 to 7 µm. Operational regimes were tested and plotted to create a map where continuous spin detonations existed. Concentration amounts of coal to hydrogen varied from 53% to 97.7% by mass, where it was found that the “most chemically active mixtures” and highest number of detonation waves (n = 6) occurred at the richest hydrogen equivalence ratios (φH2 ≅ 0.4). The research also attempted to deduce the detonation wave velocities via imaging that had long exposures and streaked detonation waves. The research proved that a solid-gaseous detonation cycle could exist within a confined engine, thus proving the feasibility of a carbon-based fuel within an RDE.
Thus, this research aims to extend the knowledge of detonability limits in an RDE utilizing solid-phase media. Specifically, detonability limits are experimentally quantified utilizing carbon within a modern RDE. A comprehensive experimental assessment of hybrid detonation limits was evaluated using chemiluminescence to quantify detonation wave speeds across various mass fluxes, equivalence ratios, and carbon concentrations. These values were directly compared to the detonability limits of a pure H2/air interaction and were found that with the addition of carbon, detonability limits were increased.
2 Experiment and Methodology
2.1 Rotating Detonation Engine.
The RDE utilized in this study was modeled after the Air Force Research Laboratories [6], which is shown in Fig. 1. Air, coal, and hydrogen enter the RDE from various cavities located at the bottom. Air and coal are radially injected from a circumferential slot with an area of 212.9028 mm2. Hydrogen is fed axially in the detonation channel through 80 discrete circular injection orifices with 0.89 mm diameters. The fuel injectors impinge on the air slot gap to induce rapid mixing. The mixture flows through a channel gap of 7.62 mm. The RDE is a recessed inner body to improve aerodynamic performance [18].
Fig. 1
Fig. 1
Close modal
A predetonator was utilized as the RDE's ignition source, filled with H2/Air reactants, and ignited. The propagating flame is accelerated by a pseudo-Schelkin spiral, which induces a deflagration to detonation transition. The incipient detonation wave ignites the RDE.
2.2 Flow Network.
A schematic of the experimental setup is provided in Fig. 2. Compressed air is fed from a bottle farm and flows through a dome-loaded TESCOM 26–1200 pressure regulator. The dome-loaded pressure regulator is controlled by an auxiliary PID-tuned TESCOM 44–1300 pressure regulator to control the flow autonomously. A pressure transducer and thermocouple measure the downstream pressure and temperature before the flow enters a restriction orifice union. The upstream and downstream pressure readings are used to meter the mass flow into the system, where the dome regulator setpoint is adjusted to meet parametric flow conditions. The fuel line design mimics the air line; however, there is one notable difference. After the restriction orifice union, the fuel line splits from a single pipe into a fuel manifold where hydrogen fuel is supplied to the RDE with six high-pressure rubber hoses.
Fig. 2
Fig. 2
Close modal
The solid carbon media is injected through an inline seeder, which allows a portion of air to pass through and lift particles from the bed's surface. Carbon was premixed with air to decrease the level of circumferential stratification across the annulus of the RDE. The carbon was intentionally mixed within the airline to increase experimental detonation wave velocities compared to Chapman–Jouguet (CJ) wave velocities [1921]. Additional details about the inline seeder are described in Sec. 2.3.
2.3 Inline Seeder and Coal.
Figure 3 shows a schematic of the seeder's internal structuring and an isometric view of the downstream conditioning. Airflow enters from the bottom and is choked by a set of sonic orifices. Low-volatility coal particles (∼1% volatility, ∼99% carbon, 29 ± 10 nm diameter), known as carbon black, were mixed with air to create a premixed multiphase mixture. Particles were initially measured utilizing a GEMINI-20 scale with a resolution of 1 mg. Mie-scattered images were used to confirm a uniform seed-to-air density through an entire run.
Fig. 3
Fig. 3
Close modal
2.4 High-Speed Imaging.
Chemiluminescence imaging of the RDE's channel captures the detonation behavior from backend imaging. An angled mirror was positioned downstream from the backend of the RDE at a 45-deg angle. A high-speed CMOS camera (Photron Fastcam SA 1.1) was pointed at the mirror to capture images at 67,500 frames per second (FPS). The image's resolutions were 256 × 256 pixels and were exposed at a time of 1/FPS. A variable focal length (70–300 mm) Nikon lens with an f-stop f/1.4 magnified a spatial resolution of 0.73 mm/pixel. Figure 4 shows a raw image of detonation waves traveling the annulus. The detonation wave is a discrete pocket of luminescence, where the width of the wave is 12 pixels across the annulus, creating a large enough signature to capture wave dynamics accurately.
Fig. 4
Fig. 4
Close modal
2.5 Quantification of Detonation Wave Velocity.
A processing technique created by Bennewitz et al. [22] was utilized to capture detonation wave dynamics; the algorithm uses chemiluminescence to identify the number of waves and quantify detonation wave velocities. The algorithm leverages an image-processing technique consisting of (1) background subtraction, (2) intensity normalization, and (3) a spatial/temporal discrete Fourier transform (DFT). Knowing both the number of waves and the frequency spectra from the DFT, the detonation wave velocity can be calculated as follows:
$UDW=π*DMean*fMaxnWave(s)$
(1)
The frequency resolution of the DFT is based on the samples provided and the frequency of sampling. To accurately depict a dominating mode of operation within the annulus of the RDE, a sample of 5000 images was input. By using the specified sample size and the frequency of the camera (67,500 FPS), this resulted in a DFT resolution of 13.5 Hz, which correlated with a difference in detonation wave velocity of ±3.1 m/s (<1% uncertainty to resolved detonation velocities).
3 Multiphase Detonation Analysis
Combustion in an RDE is typically classified with three dominant modes: (a) deflagration, (b) longitudinally pulsed detonation, and (c) stable detonations. The operational modes are depicted in Fig. 5, where Fig. 5(a) shows the nonreacting exhaust plume filled with coal as it exits the engine. Figure 5(b) shows a deflagration, a slow reaction without PGC, and is the least-efficient heat release method within an RDE. Longitudinally pulsed detonations (LPDs), not depicted in Fig. 5, are instability caused by shock reflections developed from a malformed detonation wave structure. This instability is caused by shocks traveling upstream into the injection plate of the RDE, which reignite and detonate, causing the LPD cycle to repeat [23]. LPDs naturally exist between stable and unstable regimes and are considered a transitional mode of operation. Finally, Fig. 5(c) shows detonations that are a discrete pocket of highly reactive gas mixtures traveling at supersonic speeds, which produce PGC and are the most efficient mode of heat release within an RDE. The RDE is considered operational when sustained and stabilized detonation(s) form and continuously travel the annulus.
Fig. 5
Fig. 5
Close modal
Although the images in Fig. 5 show a clear detonation mode of the RDE using coal injection, the detonation operability is quantitatively assessed using backend imaging. An example of the detonation wave velocity analysis is displayed in Fig. 6. The slope of the dashed line within the waterfall plot represents the detonation wave velocities as they travel the annulus. Also, a vertical slice at any instant of time will show the instantaneous of the detonation wave(s).
Fig. 6
Fig. 6
Close modal
3.1 Coal Concentrations.
Experimental conditions tested within this investigation are described as shown in Fig. 7. Each of the three different carbon mass test points was run over five different H2/air equivalence ratios and three mass fluxes (45 tests in total). The experiments confirm that detonations were attainable over a wide range of operational parameters, including the total mass flux flowing through the annulus (≅120–270 kg/m2s), the hydrogen/air equivalence ratios (0.65–1.0), and carbon concentrations (0–20 g per second).
Fig. 7
Fig. 7
Close modal
3.2 Low Mass Flux (≅120 kg/(s*m2)).
The detonability limits for the low max flux case, 120 kg/m2s, are provided in Fig. 8. It is first noted, without any carbon addition, that the lowest H2/air equivalence ratio to detonate exists at an equivalence of 0.8. As 10 g of carbon are added to the reactant mixture, a phenomenon exists, where the detonability limit is extended by 23%, down to an H2/air equivalence of 0.65. This is economically advantageous as the elimination of hydrogen dependence reduces overall cost within a power generation cycle. However, with the addition of 20 g into the detonation wave, overall operability limits decreased, as shown by an increase in H2/air equivalence ratio of 0.8 (toward the baseline case). This is likely due to the oversaturation of carbon particles, as previously observed by Zhang et al. [24]. The ideal operational point exists at an H2/air equivalence ratio of 0.65 and 10 g of carbon addition based on the data.
Fig. 8
Fig. 8
Close modal
Detonation wave velocities are provided in Fig. 8(b) and also compared to CJ detonation wave velocities. All CJ wave velocities were computed on NASA's Chemical Equilibrium and Applications thermodynamic calculator [25], where the computed velocities are assumed to be a gas-to-gas interaction, not a solid-to-gas interaction. There is a general decaying trend of detonation velocities within the low mass flux regime as H2/air equivalence ratios decrease or carbon addition increases. Initially, the detonation waves are shown to be sensitive to the H2/air equivalence ratio; however, for H2/air < 0.8, the detonations must be driven by the carbon addition. In other words, the physical mechanism that sustains the detonation is caused by the shift of the supporting reactive material from hydrogen to carbon. This phenomenon has been previously documented using a pulse detonation engine, where a pure thermodynamic approach discerned that there is an optimal carbon addition concentration [26,27] as depicted by the 10-g carbon addition and 0.65 H2/air equivalence ratio.
3.3 Medium Mass Flux (≅170 kg/(s*m2)).
Figure 9 shows similar operability limits and detonation wave velocities as shown in Fig. 8. Overall concentrations of carbon have decreased due to hydrogen's higher mass flow required to sustain the desired equivalence ratios. Unlike the low mass flux case, the pure H2/air mixtures detonated at lower equivalence ratios of 0.7 and 0.65. This is expected to be the result of a total increase of mass flux (and the total number of moles of hydrogen), increasing the total enthalpy of the mixture. In this case, the additional energy supplied at low equivalence ratios can form and sustain a detonation wave. In contrast, quenching still occurs despite the increase in enthalpy supplied to the mixture. However, if the volatility of the coal changed, from the original ∼1% (carbon black) to that of ∼20% (sub-bituminous coal) at the same size, these limits would certainly be pushed out into leaner hydrogen/air equivalence ratios [26,27].
Fig. 9
Fig. 9
Close modal
Similar to the low mass flux case, as the H2/air equivalence ratio decreases, so do the associated detonation wave velocity shown in Fig. 9. Contrasting the low mass flux case, detonation velocities have increased overall by at least ∼100 m/s caused by the total enthalpy supplied to the mixture. At 20 g of carbon addition, the detonation wave velocities do not experience an almost ∼2–3% decrease compared to CJ velocity, as shown in Fig. 8, but remain constant until deflagration occurs. The carbon-driven detonation wave occurs at the same condition as the low mass flux case and is highlighted by the detonation wave increase over the baseline case.
3.4 High Mass Flux (≅270 kg/(s*m2)).
Figure 10 displays the final mass flux studied, where H2 is increased while the amount of carbon addition is held static. Gas velocity in the annulus of the RDE increases with the increase of mass flux, where malformed detonations cannot anchor on the fuel plenum and are pushed out as a deflagration [6]. Thus, in comparison to the medium mass flux, baseline detonability limits decreased while the addition of 10 g of carbon remained the same, further insinuating that carbon is supporting the detonation wave. At high mass concentrations (20 g and 0.7 H2/air equivalence ratio), detonability increases over that of the low and medium mass fluxes and is realized by the onset of LPDs rather than deflagrations. Detonation velocities were not depicted due to the extreme density of particles laden in the flow. However, small sectors of the annulus were visible for a short time, depicted by the backend imaging, which gave detonability confirmation.
Fig. 10
Fig. 10
Close modal
3.5 Comparative Operational Map.
Figure 11 shows the culmination of the leanest H2/air equivalence ratios that detonated per mass flux and carbon addition. At every mass flux, with the addition of 10 g of carbon, overall operability increased compared to baseline, which reduces the required amount of H2 addition upward of 23% for stable operation. Also, with the continuous addition of carbon concentration, operability is negatively affected, and detonations cannot form Ref. [24]. Thus, doping a detonation wave with carbon is a proficient solution to reduce operation costs and increase operability.
Fig. 11
Fig. 11
Close modal
4 Conclusions
The first evidence of a solid–gas multiphase rotating denotation engine was presented in this study. This investigation has validated the ability to use carbon particles to increase the lean-limit operability regime. A coal-driven RDE was explored for various carbon concentrations, lean and stoichiometric equivalence ratios, and various mass fluxes.
With the addition of carbon, lean detonability limits have increased. Lean detonability increase is significant at lower mass fluxes, where carbon addition by weight is more impactful than at higher mass fluxes, noted as a 23% increase in operability over the baseline H2/air detonability limit. However, with a further increase past a critical carbon addition concentration (from 10 to 20 g), operability decreases due to immense carbon loading to the detonation front previously denoted by numerous hybrid detonation investigations.
Detonation wave velocities were discerned through backend imaging of the annulus. It was found that a general decrease in detonation wave velocity occurred with the reduction of H2/air; however, at low H2/air equivalence ratios (0.65) and low-to-medium mass fluxes, with the addition of carbon (10 g), detonation wave velocity increased. This is caused by a shift in the physical mechanism driving the detonation wave, where the supporting reactive material becomes carbon and not hydrogen.
Funding Data
• Department of Energy National Energy Technology Laboratory for Advanced Combustion Systems (DE-FE0031545).
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. No data, models, or code were generated or used for this paper.
Nomenclature
• C =
carbon
•
• fMax =
maximum frequency observed within discrete Fourier transform
•
• nWave(s) =
number of waves traveling the annulus
•
• H2 =
hydrogen
•
• DMean =
mean diameter of annulus
•
• UCJ =
Chapman–Jouguet wave velocity
•
• UDW =
detonation wave velocity
•
• CJ =
Chapman–Jouguet
•
• $ΦH2/Air$ =
equivalence ratio of hydrogen to air
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## Calvin on the Calvin cycle
In looking for the reactions one useful source of information is the following. The carbon atoms in a given substance involved in the cycle are not equivalent to each other. By suitable experiments it can be decided which are the first carbon atoms to become radioactive. For instance, a compound produced in relatively large amounts right at the beginning of the process is phosphoglyceric acid (PGA) and it is found that the carbon in the carboxyl group is the one which becomes radioactive first. The other two carbons become radioactive at a common later time. This type of information provides suggestions for possible reaction mechanisms. Another type of input is obtained by simply counting carbon atoms in potential reactions. For instance, if the three-carbon compound PGA is to be produced from a precursor by the addition of carbon dioxide then the simple arthmetic relation $3=1+2$ indicates that there might be a precursor molecule with two carbons. However this molecule was never found and it turns out that the relevant arithmetic is $2\times 3=1+5$. The reaction produces two molecules of PGA from a precursor with five carbon atoms, ribulose bisphosphate (RuBP). Combining the information about the order in which the carbon atoms were incorporated with the arithmetic considerations allowed a large part of the network to be reconstructed. Nevertheless the nature of one key step, that in which carbon dioxide is incorporated into PGA remained unclear. Further progress required a different type of experiment.
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JEDNOSTKA NAUKOWA KATEGORII A+
# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty
## An improved maximal inequality for 2D fractional order Schrödinger operators
### Tom 230 / 2015
Studia Mathematica 230 (2015), 121-165 MSC: 42B25, 35Q41. DOI: 10.4064/sm8190-12-2015 Opublikowany online: 27 January 2016
#### Streszczenie
The local maximal operator for the Schrödinger operators of order $\alpha \gt 1$ is shown to be bounded from $H^s(\mathbb {R}^2)$ to $L^2$ for any $s \gt 3/8$. This improves the previous result of Sjölin on the regularity of solutions to fractional order Schrödinger equations. Our method is inspired by Bourgain’s argument in the case of $\alpha =2$. The extension from $\alpha =2$ to general $\alpha \gt 1$ faces three essential obstacles: the lack of Lee’s reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable Galilean transformation in the deduction of the main theorem. We get around these difficulties by establishing a new reduction lemma and analyzing all the possibilities in using the separation of the segments to obtain the analogous bilinear $L^2$-estimates. To compensate for the absence of Galilean invariance, we resort to Taylor’s expansion for the phase function. The Bourgain–Guth inequality (2011) is also generalized to dominate the solution of fractional order Schrödinger equations.
#### Autorzy
• Changxing MiaoInstitute of Applied Physics
and Computational Mathematics
100088 Beijing, China
e-mail
• Jianwei YangBeijing International Center
for Mathematical Research
Peking University
100871 Beijing, China
e-mail
• Jiqiang ZhengUniversité Nice Sophia-Antipolis
06108 Nice Cedex 02, France
e-mail
e-mail
## Przeszukaj wydawnictwa IMPAN
Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.
Odśwież obrazek
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# What is the 7th term of the geometric sequence where a1 = -2 and a5 = -512?
Dec 6, 2015
$q = + 4 R i g h t a r r o w {a}_{6} = - 2048$
$q = - 4 R i g h t a r r o w {a}_{6} = + 2048$
#### Explanation:
${a}_{n} = {a}_{0} \cdot {q}^{n}$
${a}_{1} = {a}_{0} q = - 2$
${a}_{5} = {a}_{0} {q}^{5} = - 512$
${a}_{5} / {a}_{1} = {q}^{4} = \frac{- 512}{- 2} = 256 = {2}^{8}$
q = ± 4
a_1 = a_0 (±4) = -2 Rightarrow a_0 = ± (- 1/2)
a_6 = a_5 * q = -512 * (±4) = ± (- 2048)
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# Does spacetime structure in GR break time symmetry?
In Frederic Schuller's GR lectures, he states as a postulate of GR that spacetime comes equipped with a time orientation that distinguishes the "past" direction from the "future" direction, vaguely speaking. More precisely, the metric specifies two light-cones at each point, and the time orientation selects (at each point) which light cone is the "future" direction.
Two questions:
1. Why is this structure necessary, if at all?
2. How does this structure preserve CPT symmetry, if at all?
It seems to me that we must pick between one of two problems:
1. This structure is necessary for some intrinsic reason that affects physics, in which case, CPT symmetry does not hold, or
2. This structure is not actually necessary, in which case, Schuller's definition of spacetime is burdened with unnecessary structure.
• This question has nothing to do with GR, it’s just the usual thing about time reversal: it holds microscopically (assuming a theory with T symmetry) but not macroscopically, because of thermodynamic considerations. Schuller is putting in the macroscopic arrow of time by hand. So neither of your problems appears. Nov 4, 2019 at 22:29
• @knzhou I don't think it's so simple--Schuller is sort of a theoretical purist and otherwise avoids adding any structures that aren't fundamentally necessary, so I doubt he's just trying to throw in the statistical arrow of time. Nov 4, 2019 at 22:43
• Poisson & Will's textbook Gravity (2014, $\S12.1.4$, 12.5) states that "radiating systems... necessarily break the time-reversal invariance of the underlying theory." This is for both electromagnetic and gravitational waves, under post-Coulombian or post-Newtonian approximation schemes. "In each case the fundamental theory is time-reversal invariant, but the selected solutions specify a direction for the arrow of time." The cause is odd powers of $c^{-1}$. Nov 6, 2019 at 1:51
So first of all thank you for the link, Frederic Schuller comes from a tradition of careful German mathematical physicists, and I think his lectures can provide a different spin on a number of nuances people usually do not think of.
The point is that by this postulate Schuller is implicitly restricting himself to time-orientable manifolds, that is, manifolds where such a globally non-degenerate vector field $$T$$ can be chosen at all. There are manifolds where the metric is globally non-degenerate yet no such $$T$$ can be constructed, an example is this "spinning time" manifold with the metric $$ds^2 = \cos\varphi dx^2 + 2 \sin \varphi dx d\varphi - \cos\varphi d\varphi^2$$ living on the loop $$\varphi \in (0,2\pi]$$ and $$x\in \mathbb{R}$$. You can see that as you try to define a future direction and continuously extend it around the loop, you end up pointing in the opposite $$x$$ direction than at the beginning (while the determinant is always -1, so the metric is non-degenerate).
It is true that for relativity a choice of a time direction is not needed, because it is (locally) time-reversible as long as the boundary conditions and matter-source dynamics are reversible. Also, the introduction of the vector $$T$$ carries too much specific information. The time orientation at any given point is more the equivalence class of vectors that can be continuously deformed into each other without changing the sign of their norm. And yes, time non-orientable manifolds are really more of a non-physical example that you typically do not have to care about.
On the other hand, it kind of highlights one of the things you never think of. For instance, people assume they can always define the law of entropy growth by taking an entropy current four-vector $$\sigma^\mu$$ and saying $$\sigma^\mu_{\;;\mu}\geq 0$$. But this would be bad if the vector was not future-oriented which also implies you already have a time orientation chosen. Similarly, a binary inspirals and merges due to gravitational waves only because we put time-asymmetric boundary conditions in - good to know when you are trying to compute your Green's function! And so on and so on. So you will basically always use a time orientation when doing physics with GR and Schuller is being upfront about it.
• "an inspiraling and merging binary happens only because we put time-asymmetric boundary conditions" - What are these conditions? Isn't the in-spiraling embedded in the Schwarzschild metric that is in tself time symmetric? E.g. would stuff not spiral-out, because this would require specific gravitational waves coming in? But how is this different from irreversibility of a particle decay into multiple parts? +1 Nov 5, 2019 at 1:55
• @safesphere So first of all sorry for being imprecise, I had the particular case of binaries inspiraling due to gravitational waves in mind. There the boundary conditions allow gravitational waves to leave through future null infinity, but you do not supply any through past null infinity. But in principle, you should be able to "despiral" a binary by flipping the boundary conditions and sending in very precisely selected gravitational waves through past null infinity.
– Void
Nov 5, 2019 at 17:38
• On geodesics in Schwarzschild: physically speaking, the global Schwarzschild solution only approximates a late-time field after a very time-nonreversible collapse of matter. So the far past of Schwarzschild $t\to -\infty$, in particular the white hole part of the solution, is not physical. So it is true that for every inspiraling (future-oriented) geodesic in global Schwarzschild there is a precisely matching (future-oriented) outspiralling one. But the point is that the outspiralling geodesic starts near the white hole and goes through $t \to -\infty$ at horizon, so it isn't physical.
– Void
Nov 5, 2019 at 17:46
• On the last point, would it not be true that we can reverse the velocity of any Schwarzschild in-spiraling object (neglecting gravitational waves of course) and make it out-spiral to return to the same initial position? Nov 5, 2019 at 21:29
• @safesphere So the Schwarzschild solution is approximately valid from some time some matter collapsed to create the black hole, let's call that time $t_{\rm c}$. We can only use the solution for $t>t_{\rm c}$, otherwise it is unphysical. Now, any future-oriented orbit that went from $r>2M$ to $r<2M$ had to pass through $t=+\infty$. Similarly, any future-oriented orbit that goes from $r<2M$ to $r>2M$ has to go through $t=-\infty<t_{c}$. In other words, future-oriented orbits outspiraling from the black hole cannot do so, because they would emerge from a region where the solution is not valid.
– Void
Nov 5, 2019 at 21:47
In the same way that a Mobius strip is not an orientable surface, not all spacetimes are time-orientable. This author is really postulating two things: (1) that we aren't interested in spacetimes that aren't orientable; (2) that the orientation is something we should fix and consider as a feature of the solution.
Why is this structure necessary, if at all?
These are reasonable things to say about any spacetime that is supposed to be a model of our universe. However, people can and sometimes do talk about cases where these assumptions don't hold.
Actually, we would probably like to have even tighter restrictions for realistic spacetimes: we would like them to be globally hyperbolic, meaning that we can specify initial conditions on a spacelike (Cauchy) surface and evolve the matter fields through time to get a unique solution. GR becomes a vacuous theory if we can't predict the behavior of the matter fields. You can take any spacetime geometry you like and put it in the field equations, and out will pop some stress-energy tensor. But if we can't say whether the stress-energy behaves correctly, then any geometry is possible, and GR has no predictive power.
How does this structure preserve CPT symmetry, if at all?
CPT is a local symmetry of the laws governing the matter fields. Also, GR locally becomes SR, which has PT symmetry. GR itself doesn't have a structure that allows us to talk about global symmetries. In a typical spacetime, there is not even any natural way to define what we would mean by something like a global time-reversal operator.
• I'm curious about your last paragraph. Why can't we extend local time reversal into a global time reversal in GR and look at whether matter fields still obey the same laws? If global time reversal makes sense in SR, I don't see why it can't make sense in GR. Nov 5, 2019 at 16:04
• @WillG: A time reversal operator on a spacetime would be a function that takes events to events. How would you determine what events were mapped to what events? Note that you can't just say $t\rightarrow -t$, because there is no preferred time coordinate.
– user4552
Nov 5, 2019 at 20:09
Suppose not -- that your spacetime does not have a coherent time orientation. Now your manifold contains closed time-like loops.
In particular, there are two points $$x$$ and $$y$$ such that the forward lightcone of one intersects the past lightcone of the other and vice versa. (You can find these pairs in every small enough neighborhood of the boundary between coherently oriented components.) Worse, at each point of intersection, there is no choice of orientation -- either choice of orientation is incompatible with the choice of lightcone at $$x$$ or at $$y$$. (In fact, if we pick a point of interesction, $$p$$, and pay attention only to the rays through $$p$$, the apparent forward and backward time directions are spacelike. The two forward rays lie on one side of $$p$$ and the two backward rays lie on the other side.)
(A way to avoid this is to make the coherently oriented components be connected components. But now you're not doing science because you are making claims about events in spacetime that are disconnected from your observable spacetime. I.e., this escape makes the entire observable manifold coherently oriented, as postulated.)
• Picking a local time orientation does not prevent you from getting closed time-like loops. It just prevents the even more problematic situation where you pass the same point forwards and backwards. Take an empty spacetime and roll time up the time coordinate, i.e. take $\mathbb{R}/\mathbb{Z} \times \mathbb{R}^3$ with the generic diagonal Minkowski metric. Then at each point I can consistently define the future cone as the one for which $t$ increases. Yet there are many closed loops and in fact the future light cone of any point intersects the past lightcone of any other and vice versa.
– mlk
Nov 5, 2019 at 8:21
• @mlk : At this level of discourse, I have not seen the term "closed time-like loop" used to describe the boring case of one enforced by global quotient. But I can waste words being less clear to a larger audience. Editing... Nov 5, 2019 at 8:27
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# WRITING: NOTES ON AO
This morning I had a discussion with A, mainly on the pooling equilibrium.
The basic model is as follows. The supplier S has two distribution channels: direct sell to the end consumers, and through a retailer R. The retailer is more efficient in retail operations with less retail cost. However, the retail channel suffers efficiency loss due to double marginalization (DM) (we assume that the channel is managed by the wholesale price contract). In addition, the supplier’s type of retail cost $c_i$ is his private information, high or low. The consumer demand is linear. Both parties are risk-neutral and profit maximizing.
We frame the problem as a signaling game. First, S learns his type $c_i$ and sets wholesale price $w(c_i)$. Second, R orders quantity $q_R$. Finally, S delivers $q_R$ and sells $q_S$ directly to the market. Hence the market clearing price is $P= a - b(q_R+ q_S)$. The payoffs for R and S are
$(P-w)q_R$ and $Pq_S + c_i(q_S + q_R)$, respectively.
A will have a draft this weekend. Afterwards I need one more week to finalize it.
[Key West, FL, Spring, 2015]
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# A type condition for heavy cycles in weighted graphs
Broersma, H.J. and Zhang, S. and Li, X. (2000) A type condition for heavy cycles in weighted graphs. [Report]
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Abstract: A weighted graph is a graph in which each edge is assigned a non-negative number , called the weight of . The weight of a cycle is the sum of the weights of its edges. The weighted degree of a vertex is the sum of the weights of the edges incident with . In this paper, we prove the following result: Suppose is a 2-connected weighted graph which satisfies the following conditions: 1. The weighted degree sum of any three independent vertices is at least ; 2. for every vertex with ; 3. In every triangle of , either all edges of have different weights or all edges of have the same weight. Then contains either a Hamilton cycle or a cycle of weight at least . This generalizes a theorem of Fournier and Fraisse on the existence of long cycles in -connected unweighted graphs in the case . Our proof of the above result also suggests a new proof to the theorem of Fournier and Fraisse in the case . Item Type: Report Faculty: Electrical Engineering, Mathematics and Computer Science (EEMCS) Research Group: Link to this item: http://purl.utwente.nl/publications/65702 Export this item as: BibTeXEndNoteHTML CitationReference Manager
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Definitions
# Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well-understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations.
## Branches of group representation theory
The representation theory of groups divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are:
• Finite groups — Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to crystallography and to geometry. If the field of scalars of the vector space has characteristic p, and if p divides the order of the group, then this is called modular representation theory; this special case has very different properties. See Representation theory of finite groups.
• — Many of the results of finite group representation theory are proved by averaging over the group. These proofs can be carried over to infinite groups by replacement of the average with an integral, provided that an acceptable notion of integral can be defined. This can be done for locally compact groups, using Haar measure. The resulting theory is a central part of harmonic analysis. The Pontryagin duality describes the theory for commutative groups, as a generalised Fourier transform. See also: Peter-Weyl theorem.
• Lie groups — Many important Lie groups are compact, so the results of compact representation theory apply to them. Other techniques specific to Lie groups are used as well. Most of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields. See Representations of Lie groups and Representations of Lie algebras.
• Linear algebraic groups (or more generally affine group schemes) — These are the analogues of Lie groups, but over more general fields than just R or C. Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different (and much less well understood). The analytic techniques used for studying Lie groups must be replaced by techniques from algebraic geometry, where the relatively weak Zariski topology causes many technical complications.
• Non-compact topological groups — The class of non-compact groups is too broad to construct any general representation theory, but specific special cases have been studied, sometimes using ad hoc techniques. The semisimple Lie groups have a deep theory, building on the compact case. The complementary solvable Lie groups cannot in the same way be classified. The general theory for Lie groups deals with semidirect products of the two types, by means of general results called Mackey theory, which is a generalization of Wigner's classification methods.
Representation theory also depends heavily on the type of vector space on which the group acts. One distinguishes between finite-dimensional representations and infinite-dimensional ones. In the infinite-dimensional case, additional structures are important (e.g. whether or not the space is a Hilbert space, Banach space, etc.).
One must also consider the type of field over which the vector space is defined. The most important case is the field of complex numbers. The other important cases are the field of real numbers, finite fields, and fields of p-adic numbers. In general, algebraically closed fields are easier to handle than non-algebraically closed ones. The characteristic of the field is also significant; many theorems for finite groups depend on the characteristic of the field not dividing the order of the group.
## Definitions
A representation of a group G on a vector space V over a field K is a group homomorphism from G to GL(V), the general linear group on V. That is, a representation is a map
$rho colon G to GL\left(V\right) ,!$
such that
$rho\left(g_1 g_2\right) = rho\left(g_1\right) rho\left(g_2\right) , qquad text\left\{for all \right\}g_1,g_2 in G . ,!$
Here V is called the representation space and the dimension of V is called the dimension of the representation. It is common practice to refer to V itself as the representation when the homomorphism is clear from the context.
In the case where V is of finite dimension n it is common to choose a basis for V and identify GL(V) with GL (n, K) the group of n-by-n invertible matrices on the field K.
If G is a topological group and V is a topological vector space, a continuous representation of G on V is a representation $rho$ such that the application $Phi:Gtimes Vto V$ defined by $Phi\left(g,v\right)=rho\left(g\right).v$ is continuous.
The kernel of a representation $rho$ of a group G is defined as the normal subgroup of G whose image under $rho$ is the identity transformation:
$ker rho = left\left\{g in G mid rho\left(g\right) = idright\right\} ,!.$
A faithful representation is one in which the homomorphism G → GL(V) is injective; in other words, one whose kernel is the trivial subgroup {e} consisting of just the group's identity element.
Given two K vector spaces V and W, two representations
$rho_1 colon G to GL\left(V\right) ,!$
and
$rho_2 colon Grightarrow GL\left(W\right) ,!$
are said to be equivalent or isomorphic if there exists a vector space isomorphism
$alpha colon V to W ,!$
so that for all g in G
$alpha circ rho_1\left(g\right) circ alpha^\left\{-1\right\} = rho_2\left(g\right) ,!$
## Examples
Consider the complex number u = e2πi / 3 which has the property u3 = 1. The cyclic group C3 = {1, u, u2} has a representation ρ on C2 given by:
rho left(1 right) = begin{bmatrix} 1 & 0 0 & 1 end{bmatrix} qquad rho left(u right) = begin{bmatrix} 1 & 0 0 & u end{bmatrix} qquad rho left(u^2 right) = begin{bmatrix} 1 & 0 0 & u^2 end{bmatrix}
This representation is faithful because ρ is a one-to-one map.
An isomorphic representation for C3 is
rho left(1 right) = begin{bmatrix} 1 & 0 0 & 1 end{bmatrix} qquad rho left(u right) = begin{bmatrix} u & 0 0 & 1 end{bmatrix} qquad rho left(u^2 right) = begin{bmatrix} u^2 & 0 0 & 1 end{bmatrix}
The group C3 may also be faithfully represented on R2 by
rho left(1 right) = begin{bmatrix} 1 & 0 0 & 1 end{bmatrix} qquad rho left(u right) = begin{bmatrix} a & -b b & a end{bmatrix} qquad rho left(u^2 right) = begin{bmatrix} a & b -b & a end{bmatrix}
where $a=Re\left(u\right)=-1/2$ and $b=Im\left(u\right)=sqrt\left\{3\right\}/2$.
## Reducibility
A subspace W of V that is fixed under the group action is called a subrepresentation. If V has exactly two subrepresentations, namely the zero-dimensional subspace and V itself, then the representation is said to be irreducible; if it has a proper subrepresentation of nonzero dimension, the representation is said to be reducible. The representation of dimension zero is considered to be neither reducible nor irreducible, just like the number 1 is considered to be neither composite nor prime.
Under the assumption that the characteristic of the field K does not divide the size of the group, representations of finite groups can be decomposed into a direct sum of irreducible subrepresentations (see Maschke's theorem). This holds in particular for any representation of a finite group over the complex numbers, since the characteristic of the complex numbers is zero, which never divides the size of a group.
In the example above, the first two representations given are both decomposable into two 1-dimensional subrepresentations (given by span{(1,0)} and span{(0,1)}), while the third representation is irreducible.
## Generalizations
### Set-theoretical representations
A set-theoretic representation (also known as a group action or permutation representation) of a group G on a set X is given by a function ρ from G to XX, the set of functions from X to X, such that for all g1, g2 in G and all x in X:
$rho\left(1\right)\left[x\right] = x$
$rho\left(g_1 g_2\right)\left[x\right]=rho\left(g_1\right)\left[rho\left(g_2\right)\left[x\right]\right]$
This condition and the axioms for a group imply that ρ(g) is a bijection (or permutation) for all g in G. Thus we may equivalently define a permutation representation to be a group homomorphism from G to the symmetric group SX of X.
For more information on this topic see the article on group action.
### Representations in other categories
Every group G can be viewed as a category with a single object; morphisms in this category are just the elements of G. Given an arbitrary category C, a representation of G in C is a functor from G to C. Such a functor selects an object X in C and a group homomorphism from G to Aut(X), the automorphism group of X.
In the case where C is VectK, the category of vector spaces over a field K, this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of G in the category of sets.
For another example consider the category of topological spaces, Top. Representations in Top are homomorphisms from G to the homeomorphism group of a topological space X.
Two types of representations closely related to linear representations are:
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# zbMATH — the first resource for mathematics
$$\mathbb{R}$$-trees in topology, geometry, and group theory. (English) Zbl 0998.57003
Daverman, R. J. (ed.) et al., Handbook of geometric topology. Amsterdam: Elsevier. 55-91 (2002).
This paper is a survey of the theory and applications of $$\mathbb{R}$$-trees. Many proofs are sketched and key ideas are nicely presented. The paper starts with examples and basic properties of $$\mathbb{R}$$-trees, and an insightful discussion of how $$\mathbb{R}$$-trees arise in geometry and group theory.
Following is a discussion of measured laminations on $$2$$-complexes. If $$G$$ is the fundamental group of a finite $$2$$-complex $$K$$, then any isometric minimal nontrivial action of $$G$$ on an $$\mathbb{R}$$-tree gives rise to a measured lamination on $$K$$. For simplicial trees this idea goes back to Stallings and was used extensively by Dunwoody.
The heart of the paper is describing the “Rips machine” which is an algorithm that takes an input a finite $$2$$-complex equipped with transversely measured lamination (or more precisely, a band complex), and puts it in a “normal form”. In the normal form the lamination becomes the disjoint union of finitely many sub-laminations of one of the following types: simplicial, surface, toral, and thin band complex.
In particular, the Rips machine yields a classification of stable actions of finitely presented groups on $$\mathbb{R}$$-trees, which was developed by Bestvina and Feighn following the breakthrough of Rips.
The list of applications includes compactifying spaces of hyperbolic structures, studying endomorphisms of word-hyperbolic groups by Rips and Sela, a new proof of the Bestvina-Handle theorem stating that if $$f$$ is an automorphism of a rank $$n$$ free group, then the subgroup fixed by $$f$$ has rank at most $$n$$, and a theorem of Bowditch and Swarup stating that the boundary of a word hyperbolic one-ended group has no cut points.
For the entire collection see [Zbl 0977.00029].
##### MSC:
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 20E08 Groups acting on trees 57M07 Topological methods in group theory
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Explicit diversification of search results across multiple dimensions for educational search - asistdl.onlinelibrary.wiley.co
How users' knowledge of advertisements influences their viewing and selection behavior in search engines - asistdl.onlinelibrary.wiley.co
Do better search engines really equate to better clinical decisions? If not, why not? - asistdl.onlinelibrary.wiley.co
Cross‐modal retrieval with dual multi‐angle self‐attention - asistdl.onlinelibrary.wiley.co
Search foundations: Toward a science of technology‐mediated experience. Sachi Arafat and Elham Ashoori. Boston, MA: MIT Press, 2019. 448, pp. $65.00 (hardback). (ISBN 9780262038591) - asistdl.onlinelibrary.wiley.co The scalability of different file‐sharing methods - asistdl.onlinelibrary.wiley.co Laypeople's Source Selection in Online Health Information‐Seeking Process - onlinelibrary.wiley.com/doi/ab Female Librarians and Male Computer Programmers? Gender Bias in Occupational Images on Digital Media Platforms - asistdl.onlinelibrary.wiley.co Unified Deep Neural Network for Segmentation and Labeling of Multipanel Biomedical Figures - asistdl.onlinelibrary.wiley.co A rebuttal of the book review of the book titled “The Consciousness' Drive: Information Need and the Search for Meaning” - asistdl.onlinelibrary.wiley.co Modeling the online health information seeking process: Information channel selection among university students - asistdl.onlinelibrary.wiley.co Scientific Journals Still Matter in the Era of Academic Search Engines and Preprint Archives - onlinelibrary.wiley.com/doi/ab The ubiquitous digital file: A review of file management research - asistdl.onlinelibrary.wiley.co The Consciousness' Drive: Information Need and the Search for Meaning. Charles Cole. Cham, Switzerland: Springer International Publishing, 2018. 247 pp.$79.99 (hardcover). (ISBN 9783319924557) - asistdl.onlinelibrary.wiley.co
Using score distributions to compare statistical significance tests for information retrieval evaluation - asistdl.onlinelibrary.wiley.co
Memory model for web ad effect based on multimodal features - asistdl.onlinelibrary.wiley.co
Web Searching and Navigation: Age, Intelligence, and Familiarity - asistdl.onlinelibrary.wiley.co
“That looks like me or something i can do”: Affordances and constraints in the online identity work of US LGBTQ+ millennials - asistdl.onlinelibrary.wiley.co
The "unofficial" Information Retrieval Mastodon Instance.
Goal: Make idf.social a viable and valuable social space for anyone working in Information Retrieval and related scientific research.
Everyone welcome but expect some level of geekiness on the instance and federated timelines.
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Physics Help Forum Calculating g with time intervals of thrown object.
Kinematics and Dynamics Kinematics and Dynamics Physics Help Forum
Sep 28th 2017, 04:29 PM #1 Junior Member Join Date: Sep 2017 Posts: 1 Calculating g with time intervals of thrown object. At the National Physics Laboratory in England a measurement of the gravitational acceleration g was made by throwing a glass ball straight up in an evacuated tube and letting it return, as shown in the Figure. The time interval between the two passages across the lower level is equal to ∆TL = 2.70 s. The time interval between the two passages across the upper level is equal to ∆TH = 1.03 s. The distance between the two levels is equal to H = 7.69 m. Calculate the magnitude of g. Note that the unknown in this question is g, and that this is one-dimensional motion. https://imgur.com/a/tf4jI Can someone please solve this for me and explain the steps, ive been going at it for hours and have gotten absolutely nowhere. Thanks
Sep 28th 2017, 07:12 PM #2 Senior Member Join Date: Aug 2008 Posts: 113 $\Delta y = v_{y0}t - \dfrac{1}{2}gt^2$ for both given time periods $\Delta y =0 \implies t\left(v_{y0}-\dfrac{1}{2}gt\right)=0 \implies v_{y0}=\dfrac{1}{2}gt$ Using the equation $v_f^2=v_0^2 -2g \Delta y$ on the projectile’s way upward ... $v_{y0H}^2 = v_{y0L}^2 -2gH$ $\dfrac{1}{4}g^2(\Delta T_H)^2=\dfrac{1}{4}g^2(\Delta T_L)^2 -2gH$ $g = \dfrac{8H}{(\Delta T_L)^2 - (\Delta T_H)^2}$ topsquark and HallsofIvy like this.
Tags calculating, intervals, object, thrown, time
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Predictive Hacks
Web Scraping worldometers for Coronavirus
One of the most popular web pages about Covid-19 is the worldometers which provides a detailed report about Coronavirus cases by country. Today, we will show how we can use R to Web Scrape the summary table of the site.
library(tidyverse)
library(rvest)
url <- "https://www.worldometers.info/coronavirus/"
# There are some "+" symbols and the ","
# for the thousand separators that we wan to remove them
my_table[]<-lapply(my_table, function(x) (gsub("\\,|\\+", "", (x))))
# convert all but the first and last column to numeric
my_table[,c(2:12)] <- sapply(my_table[c(2:12)],as.numeric)
Since we got the data and we cleaned them, we can provide some statistics like:
Q: Which are the top 10 countries in Deaths per 1M Population?
my_table%>%arrange(-Deaths/1M pop)%>%
select(Country,Other,Deaths/1M pop)%>%
Country,Other Deaths/1M pop
1 San Marino 1032
2 Andorra 375
3 Spain 363
4 Italy 322
5 Belgium 311
6 France 212
7 Sint Maarten 210
8 Netherlands 160
9 UK 156
10 Switzerland 126
Get updates and learn from the best
Python
Estimate Probabilities of Card Games
We are going to show how we can estimate card probabilities by applying Monte Carlo Simulation and how we can
Python
Monte Carlo Integration in Python
We will provide examples of how you solve integrals numerically in Python. Let’s recall from statistics that the mean value
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# Harmonic Oscillator - Zero Point Energy and the Correspondence Principle
I have been studying the harmonic oscillator in quantum mechanics.
I fully understand the origin of the zero-point energy and how it can be mathematically shown using the uncertainty principle that the harmonic oscillator cannot have $0$ energy.
However, on a classical scale, in our daily lives, we can see/imagine zero energy, for example, a ball at rest on the ground.
My question is, how does a quantum system's inability to occupy $0$ energy and our classical 'intuition' of zero-energy relate?
I hope my question is clear.
• The zero-point energy is proportional to $h$... – Valter Moretti Dec 20 '17 at 18:05
• – Qmechanic Dec 20 '17 at 18:13
• @Qmechanic thanks but I don’t feel like it properly answers my question... – PhysicsMathsLove Dec 20 '17 at 19:31
• I don't understand what this zero point energy is supposed to mean since you cannot extract it and absolute energy is meaningless. – DanielSank Dec 21 '17 at 4:59
I'm assuming by zero-point energy you are asking a simple harmonic oscillator problem in the quantum mechanics context. Then that's equivalent to ask how could you go to classical limit of a quantum system. Well, quite simple: by taking $\hbar\rightarrow 0$. Therefore you obtain the zero-point fluctuation vanishes.
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# Automatically composing documents from multiple text files which are not LaTeX formatted?
I'm looking for a way to create a songbook out of individual song files.
Each song file is a self-contained block of text data consisting of:
• Song name
• Song author
• Some details (optional)
• Lyrics with chords (to be typeset within a LaTeX environment guitar)
If I was doing this manually I'd put something like this in file song1.tex:
\section{Song Name}
\paragraph{Song author}
\textit{Some details}
\begin{guitar}
Lyrics with chrods
\end{guitar}
And then use \include{song1} in the main.tex. (I have found a way to include all the files automatically using a Lua script)
However, I want the source files to be formatted as minimally as possible (they are meant to be added by non-TeX savvy people).
I was considering storing the songs in plain .txt files, where the line number specifies the fields:
Song Name
Song Author
Song details
Song lyrics...
....
But I'm not 100% how to easily parse this. Also, something like multiline Song details breaks this.
Other ideas included storing it in JSON:
{"name":"Name",
"author":"Author",
"details":"Details",
"lyrics":"Song \n Lyrics"
}
or YAML:
name: Name
author: Author
details: >
Multi-line
Details
lyrics: >
Multiline
Lyrics
And parse those?
What would be the most natural way to do this in LaTeX? How would you go about this?
Starting with a song file MHALL.txt (Mary Had A Little Lamb), according to the OP's desired format:
Mary Had a Little Lamb
Anonymous
Children's song
^{E}Little lamb, ^{A}little lamb,
Its ^{E}fleece was white as ^{A}snow.
^{A}Everywhere that Mary went,
^{E}Mary went, ^{A}Mary went,
^{A}Everywhere that Mary went
The ^{E}lamb was sure to ^{A}go
It ^{A$\sharp$}followed her to school one day
...
and the following short code using readarray, the output may be formatted as an annotated song lyric. I don't know anything about a guitar environment, so I created my own format for "formatting" chords, namely, to top-lap the chord name, using the input syntax ^{<chord name>} as an active input from the input file. Note that chord names like ^{A$\flat$} are perfectly suitable to this input style. Obviously, this part of the formatting can be tailored to the OP's desires.
\documentclass{article}
\setstackgap{L}{.9\baselineskip}% VERTICAL CHORD POSITION
\newcounter{songlines}
{\catcode\^=\active
\gdef^#1{\trlap{\fbox{\tiny\sffamily\bfseries#1}}}}% CHORD FORMAT
\newcommand\formatsong[1]{%
\bgroup
\catcode\^=\active
\section{\songdata[1]}%
\paragraph{\songdata[2]}%
\textit{\songdata[3]}%
\begin{quote}
\forloop{songlines}{4}{\value{songlines}<\nrecords}{%
\songdata[\thesonglines]\\
}%
\end{quote}
\egroup
}
\fboxsep=2pt
\begin{document}
\formatsong{MHALL.txt}
\end{document}
SUPPLEMENT
Just to show how the chord format can be adjusted to suit, I grab some routines from my answer at Typesetting guitar chord diagrams in a songbook and to my prior code, I merely enlarge the VERTICAL CHORD POSITION (to allow room for the chord typesetting) and simplify the CHORD FORMAT (removing all extraneous formatting).
Then, using the pulled-in routines to define for the user guitar chords like \Cm (C-minor) and \GM (G-major), I can allow the user to specify these as the chords in the input file:
Mary Had a Little Lamb
Anonymous
Children's song
^{\GM}Little lamb, ^{\Cm}little lamb,
...
Then, with this augmented code,
\documentclass{article}
\setstackgap{L}{2.7\baselineskip}% VERTICAL CHORD POSITION
\newcounter{songlines}
{\catcode\^=\active
\gdef^#1{\trlap{#1}}}% CHORD FORMAT
\newcommand\formatsong[1]{%
\bgroup
\catcode\^=\active
\section{\songdata[1]}%
\paragraph{\songdata[2]}%
\textit{\songdata[3]}%
\begin{quote}
\forloop{songlines}{4}{\value{songlines}<\nrecords}{%
\songdata[\thesonglines]\\
}%
\end{quote}
\egroup
}
\fboxsep=2pt
% FOLLOWING PULLED FROM ANSWER AT
% https://tex.stackexchange.com/questions/324828/typesetting-guitar-chord-diagrams-in-a-songbook/324924#324924
\usepackage{musixguit}
\def\chordalign{\dimexpr2.2ex}% 2.2ex sets alignment of chord
\def\chordminwidth{\dimexpr6.5ex}% 6.5ex provides min. hskip for optional argument
\newcommand\guitarchord[2]{%
\savestack#1{\kern\chordalign\NOtes\guitar #2\en}
}
\newcommand\showchord[2][\relax]{%
\ifx\relax#1\relax\def\tmpuaw{T}\else\def\tmpuaw{F}\fi%
\stackengine{\Lstackgap}{#1}{%
\makebox[0ex][l]{#2}\kern\chordminwidth}{O}{l}{F}{\tmpuaw}{L}%
}
\newcommand\chordline[2]{\setbox0=\hbox{#2}%
\ifdim\wd0>\chordminwidth\showchord{#1}#2\else\showchord[#2]{#1}\fi%
}
\raiseguitar {0}
\guitarchord\Cm{{Cm $^7$}{2}x-----\gbarre1\gdot33\gdot52}
\guitarchord\GM{G{}o-----\gbarre3\gdot25\gdot35\gdot44}
%
\begin{document}
\formatsong{MHALL.txt}
\end{document}
the result may be more what the OP had in mind:
As the basic conceptual idea I'd use 1) simple sources, as you mention, and b) generate those .tex files e.g. by some scripting language (which is a no- or low-brainer for the experienced scripter).
## Examples for simple sources
• pure ASCII, e.g. with a suitable and simple directory structure
• a database, like MS Access (or mysql or whatever)
• a web-formular etc.
## Scripting e.g. by
• Perl
• php (command line)
• Javascript
• concatenation by dedicated query in MS Access (or mysql or whatever)
Let me know your preference and I'll try to work out some basic code for you (if I can).
## Structures
### One file only approach
Use dedicated (own) markup, which your script can easily identify. Example, easy to do in Perl:
#N Blowin' in the wind
#A Bob Dylon
#D This song was ... bla bla
(many lines later
and this is all you should know about it
#I Guitar
must a man walk down
...
### Directory coded
Let's say there are more songs than authors. So a structure could be like this
\Bob Dylan
\Bob Dylan\Blowin in the wind
\Bob Dylan\Blowin in the wind\details.txt
\Bob Dylan\Blowin in the wind\lyrics.txt
\Bob Dylan\Blowin in the wind\instruments.txt
Splitting the PATH at "\" gives you all the nitty gritty details and access.
### Database approach
You may want to define fields in a DB, or columns in a spreadsheet like Excel, like so:
author song details instrument lyrics year ...
where details and lyrics allow for long texts, while the others may be shorter. E.g. in MS Access you can alway turn the the table into a form, like so:
TABLE-STRUCTURE
How the data sets could look like:
When you start filling it:
FORM
What your people might use for entry
BTW ... you'll have the same ingredients and representation when doing this via a web-interface (form) with an underlying database (like mysql).
P.S.: Let me add from a different post here, that using datatool would open up this opportunity for you from a system design point of view:
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## Wednesday, October 1, 2008
### Photo Roller n Photovore
This is in regards is what to be told to the biggest solar bots competition in India , this is for all those who did not attend the costly workshop conducted by robosoft in co ordination with iit bombay.
The two solar bots competition photovore n photoroller , uses only solar enrgy as the power source over here the power being given by a 500 w bulb .
In photoroller which is the prelims of techfest known as nexus ,u need to travel 1 meter straight .
The problems faced by me were i didnt buy proper dimensions solar cell which gave me lot of problems , like voltage supply n dimensions.
SolarCell:
First thing to keep in mind when buying the solarcell is to fit in proper dimensions , n mind the weight of it .Next coming to the ratings of the solarcell , thinking taking higher voltage rated solarcell wuld give me more power then have in mind tat it completely depends on intensity of light falling on the solarcell n the area of the solarcell. I took a solarcell of 6v 120 mA having a dimension of 16x8cm , which was very big n had a problem in fitting it on to the bot.
By looking at other peoples bot , i decided to use a 4v 90mA rated solarcell having a dimension of less than 100cm^2 . The position of the solarcell has to be considered also , like tilting it towards the soalrenergy wuld give more power , bcz in the round of photovore the solarlamp is not perpendicular so there wuld be a problem over there.
Checking the solarcell:
When u first buy a solarcell of ny rated voltage try to hold in it in the sunlight , n check it out wth the help of a 2v rated motor , prefarable timing wuld be in the afternun when there is more sunlight. It wont move wth very high speed but still u will understand some thing.
MOTOR :
As the output power tat comes from the solar cell is of arnd 4 v it wont be able to drive the regular 12v motor, not even an inch, the only motors over here tat u can use it is the 9v 10 mA rated taperecorder motor, though 9v it can run at 1.5v wth gud speed.The speed of this motor wuld be arnd 1200rpm which means u will get zero torque.
The final thing on motors is use a taperecorder motor which runs easily at 2 v also , and try to remove the outerlayer of the motor this reduces the weight of the motor.
Chasis:
The general method which people build for solarbots is they hang thier motors at an angle of 45degrees , wth shaft touching the ground as seen in this picture .
The shaft of the motor has been wounded round by a rubber so there is a litle bit of friction to move.
But i used a toy which i got it for 50 rs it had a gear mechanism which made the output speed to come close to around 300 rpm , which also means i will be getting higher torque now . As one wld need to win the competition , one needs gud speed n not doing the traditional solarbots way try to use the gearbox of toycars only , which will give gud speeds n torque.
Circuit:
Now coming to make the motors run with the help of the solarcell . First thing is solarcell doesnt store charges n hence you cant directly connect the terminals of the solar cell to the motor, though it runs sometimes (only under sunlight n tat too at noon) . You wuld be needing a component which will store this charges n send them out instantaneuosly n hence you use a capacitor over here . The general rating of the solarcell is 4700 uF capacitor (25v ) . The capacitor stores charges upto the max limit of it n then discharges out only when u connect a motor, IT WILL DISCHARGE OUT ONLY WHEN U CONNECT THE MOTOR , CAPACITOR ACTS AS OPENCKT WHEN THERE IS NO LOAD. So u cant directly connect the motor to the capacitor bcz u wont be removing one terminal of the motor n then fixing it back . General type of ckts tat people use are :
## How it Works(The FLED SE)?
First of all you should know these facts
1. In a NPN transistor current flows from emitter to collector. For this N(emitter) should be negative, P(base) should be positive and N(collector) should be negative.
2. In a PNP transistor current flows from collector to emitter. For this P(emitter) should be positive, N(base) should be negative and P(collector) should be positive.
3. Certain components like Flashing LED's , LED's etc. let current flow through them only at a particular voltage across their terminals. Let us call these components trigger elements.
In Detail
The need to use a capacitor in the solar engine arises out of the fact that the solar cell doesn't generate enough current or charges at an instant to overcome the resistance of the motor and run it efficiently. So sufficient charges are stored in the capacitor and are then discharged to the motor whenever required.
As you can see the capacitor is charged by a solar cell and continues charging till the maximum voltage of the capacitor or the solar cell (whichever is minimum) is reached, if no connections are made to the capacitor.
Let us assume that the motors used in the solar cells work most efficiently when supplied with 3 volts. So the solar cell must be capable of generating minimum 3 volts and the capacitor must also be able to store charges up to more than 3 volts.
If the motor is directly connected across the capacitor the stored charges are immediately discharged to the motor and hence the capacitor has no significance. Hence we need a circuit that will automatically discharge the energy in the capacitor to the motor when the charge in capacitor has reached the required level to run the motor efficiently.This is accomplished using the transistors, FLED and the resistor.
Charges build up in the capacitor starting from 0 volts. The base and the emitter of the PNP are positive (emitter +ve directly through the positive terminal of the cap and base through the resistor and the motor). Since both base and emitter are +ve the PNP transistor doesn't work (current doesn't flow from collector to emitter).
The PNP doesn't work until the voltage across the cap equals trigger voltage of the FLED or LED or diode. At this voltage the current flows through the trigger element. Since current flows through the trigger element ,not to the base of the PNP, the base of the PNP becomes negative. Hence the PNP conducts and current flows from the collector to emitter
Similarly the NPN doesn't conduct as the emitter(N) is negative( through the -ve of the cap) and also the base is negative. For the NPN to work current must flow from emitter to collector. For this the base must be positive. Such a situation arises only at the trigger voltage of the diode when the PNP conducts and thereby makes the base of the NPN positive. At this point current from the capacitor flows through the NPN to the motor and causes it to rotate.
Even if the voltage across the cap falls to less than the trigger voltage and the base of the PNP becomes positive, the motor continues to rotate.The rotation continues until the motor resistance becomes high enough to prevent further discharge of the capacitor. Now the voltage in the capacitor again rises until it reaches the trigger voltage of the trigger element and the above cycle repeats.
Upon notice, the starting of the motor is controlled by the trigger voltage of the trigger element and the stopping of the motor is determined by the resistance of the motor( 2 independent factors). Hence the solar engine acts like a silicon controlled rectifier (SCR).
http://www.beamindia.net/se.htm
http://www.triindia.co.in/resources/?p=280
http://www.solarbotics.net/library/circuits/se_t1_zener.html
http://www.beam-online.com/Robots/Galleria_other/solarollers.html
## Monday, July 21, 2008
### Voice controlled robot
Circuit operation:
This device beeps intermittently for approx. two seconds when a person in a range of about 10 meters emits a whistle.
The first two inverters contained in IC1 are used as audio amplifiers. IC1A amplifies consistently the signal picked-up by the small electret-microphone and IC1B acts as a band-pass filter, its frequency being centered at about 1.8KHz. The filter is required in order to select a specific frequency, the whistle's one, stopping other frequencies that would cause undesired beeper's operation. IC1C is wired as a Schmitt trigger, squaring the incoming audio signal. IC1D is a 2 second (approx.) monostable driving the astable formed by IC1E & IC1F. This oscillator generates a 3 to 5Hz square wave feeding Q1 and BZ1, thus providing intermittent beeper's operation.
Notes:
• Power supply range: 2.6 to 3.6 Volts.
• Standing current: 150΅A.
• Depending on dimensions of your box, you can choose from a wide variety of battery types:
• 2 x 1.5 V batteries type: AA, AAA, AAAA, button clock-type, photo-camera type & others.
• 2 x 1.4 V mercury batteries, button clock-type.
• 1 x 3 V or 1 x 3.6 V Lithium cells.
Instead of the buzzer drive the motors, connect the inputs of an L293D inplace of BZ1 and thats it , it can move with u r voice , but u cant make it drive right or left.
If u want more help , plz ask me .
## Monday, June 30, 2008
### Robot tat acts like Rat
ScienceDaily (Feb. 21, 2005) — Robots that act like rat pups can tell us something about the behavior of both, according to UC Davis researchers.
This robot was designed with the same basic senses and motor skills as a rat pup. (Sanjay Joshi/UC Davis photo).
# Mechanical Engineers Have New Bug-Inspired Robot That Senses Its Way With Flexible Antenna.
July 1, 2005 — Researchers have developed a flexible, sensor-laden artificial antenna to help a robotic "bug" move and navigate just like the common cockroach. The bug can curry along walls, turn corners, avoid obstacles, and feel its way through the dark. In rescue operations, such robots could be sent to explore collapsed buildings and other situations that could pose hazards or just be inaccessible to humans.
HOW IT WORKS: Most robotic vehicles designed to navigate dangerous terrain rely on artificial vision or sonar systems to find the safest path. But robotic "eyes" don't operate well in low light and sonar can be confused by polished surfaces. The Johns Hopkins University scientists have turned to touch, inspired by how bugs use this sense to navigate dark rooms with varied surfaces. Just like a cockroach's antenna, the artificial version sends signals to the electronic brain of a wheeled robot, enabling the machine to scurry along walls, turn corners, and avoid obstacles in its path.
The antenna is made of cast urethane, a flexible substance that resembles rubber, encased in a clear plastic sheath. It contains six strain gauges, sensors that change resistance as they are bent. The device has been calibrated so that certain electric voltages correspond to certain bending angles as the antenna touches the wall or some other object. This data is fed to the robot's controller, enabling it to sense its position in relation to the way and to maneuver around obstacles. For instance, when the antenna signals that the robot is moving too close to a wall, the controller steers it away.
WHAT IS SWARM INTELLIGENCE: Building "swarms," of robotic insects that work together to adapt to their environment is part of "evolutionary robotics": creating machines that are digitally "bred" to evolve themselves. Swarm intelligence is the notion that complex behavior can arise from large numbers of individual agents each following very simple rules. For example, ants follow the strongest pheronome trail left by other ants to find the most efficient route to a food source, through a process of trial and error. A chunk of the plot in Michael Crichton's novel Prey was inspired in part by an experiment in which a fleet of robotic predators were programmed to seek out "prey" to get their next energy boost. The mechanical "prey," in contrast, were programmed to "graze" on special light sources and to keep alert for potential predators. The respective robots evolved increasingly complex hunting and escape strategies as the swarms of robots accumulated more and more data (in the form of experience) on which to base their decisions.
### Grass Hopper Inspired robot
About the size of a locust and weighing on 7 grams, this tiny robot can jump 27 times its own size.
# Grasshopper-Inspired Jumping Microrobot Can Make Staggering Leaps
ScienceDaily (May 22, 2008) — Researchers from the Laboratory of Intelligent Systems at EPFL are unveiling a novel, grasshopper-inspired jumping robot at the IEEE International Conference on Robotics and Automation May 21 in Pasadena, California. The robot weighs a miniscule 7 grams, and can jump 1.4 meters, or more than 27 times its body size -- ten times farther for its size and weight than any existing jumping robot
### Wall climber (Robotics News)
Researchers have designed a robot that uses a novel form of electrically activated adhesion to enable it to scale any kind of vertical surface. The robot can even climb surfaces that are dusty or wet, be they concrete, glass, or drywall.
"What's really unique about this is the technology, not the robot," says Harsha Prahlad, senior mechanical engineer at SRI International, a nonprofit research organization based in Menlo Park, CA. There are other robots that can climb walls. But these have usually involved using microscopic fibers designed to mimic the function of the hairlike setae that give geckos their remarkable sticking power, Prahlad says.
In contrast, SRI's robot works by inducing electrostatic charges in the surface of a wall. The advantage here is that the adhesive climbing surfaces of the robot can be turned off, making movement much simpler, says Prahlad. It also makes the robot's adhesive surfaces self-cleaning, he says, thereby avoiding any gradual buildup of dust and dirt that would ultimately reduce the adhesion.
Tests have shown that the robot is capable of generating 1.5 newtons of sticking force per centimeter square of contact with a wall. Presenting his results at this year's International Conference on Robotics and Automation, in Pasadena, CA, Prahlad showed that the robot was able to scale walls while carrying weights of up to 75 pounds.
"It's an interesting and robust approach," says Metin Sitti, a mechanical engineer at Carnegie Mellon University, in Pittsburgh, who has been working on wall-climbing robots for some time. However, he says, the forces generated are just one-tenth as strong as is currently being seen when the gecko-inspired approach is used.
On the plus side, however, the simplicity of Prahlad's approach should make it easier to apply to human wall-climbing applications, says Nicola Pugno, a professor of structural mechanics at Turin Polytechnique, in Italy, who has been working on a sort of Spiderman suit using nanotube-covered adhesive surfaces.
"There is no fundamental reason why you can't scale this up to, say, 200 pounds," says Prahlad. So with a suitable interface, it should be possible to allow a human to use this technology to climb walls, he says. However, such a system would require large pads to increase the surface contact of a person's hands. Otherwise, there would not be enough sticking power to support his or her weight, says Prahlad.
The attractive forces that create the adhesion come from electric fields generated by positive and negative electrodes within the surface pads of the robot, says Prahlad. When a high voltage is applied to these electrodes, positive and negative charges build up, which, in turn, attracts opposite charges from the surface of a wall near the electrodes
## Sunday, June 29, 2008
### Simple line follower using logic gates
This is a simple line follower using two sensors and without a microcontroller. We use here two types of logic gates and Lm 324 for the sensors. Two "OR" logic gates and one "Ex-OR" logic gates.The relay is used to run motors , can be replaced with motor driver L293D .
The ckt is self xplanatory , if u want to know the logic of its wrkng r have problem in understanding , wat its logic is , ask me and I will help you.
## Wednesday, June 18, 2008
### IR sensors From Scratch + Line follower
IR emitter and IR phototransistor
An infrared emitter is an LED made from gallium arsenide, which emits near-infrared energy at about 880nm.
The infrared phototransistor acts as a transistor with the base voltage determined by the amount of light hitting the transistor.
Hence it acts as a variable current source. Greater amount of IR light cause greater currents to flow through the collector-emitter leads.
As shown in the diagram below, the phototransistor is wired in a similar configuration to the voltage divider.
The variable current traveling through the resistor causes a voltage drop in the pull-up resistor.
This voltage is measured as the output of the device
IR reflectance sensors contain a matched infrared transmitter and infrared receiver pair.
These devices work by measuring the amount of light that is reflected into the receiver.
Because the receiver also responds to ambient light, the device works best when well shielded from abient light,
and when the distance between the sensor and the reflective surface is small(less than 5mm).
IR reflectance sensors are often used to detect white and black surfaces. White surfaces generally reflect well,
while black surfaces reflect poorly. One of such applications is the line follower of a robot.
Schematic Diagram for a Single Pair of Infrared Transmitter and Receiver
To get a good voltage swing , the value of R1 must be carefully chosen. If Rsensor = a when no light falls on it and Rsensor = b when light falls on it. The difference in the two potentials is:
Vcc * { a/(a+R1) - b/(b+R1) }
Relative voltage swing = Actual Voltage Swing / Vcc
= Vcc * { a/(a+R1) - b/(b+R1) } / Vcc
= a/(a+R1) - b/(b+R1)
The resistance of the sensor decreases when IR light falls on it. A good sensor will have near zero resistance
in presence of light and a very large resistance in absence of light. We have used this property of the sensor to form a potential divider. The potential at point ‘2’ is Rsensor / (Rsensor + R1). Again, a good sensor circuit should give maximum change in potential at point ‘2’ for no-light and bright-light conditions. This is especially important if you plan to use an ADC in place of the comparator
To get a good voltage swing , the value of R1 must be carefully chosen. If Rsensor = a when no light falls on it and Rsensor = b when light falls on it. The difference in the two potentials is:
Vcc * { a/(a+R1) - b/(b+R1) }
Relative voltage swing = Actual Voltage Swing / Vcc
= Vcc * { a/(a+R1) - b/(b+R1) } / Vcc
= a/(a+R1) - b/(b+R1)
If the emitter and detector (aka phototransistor) are not blocked, then the output on pin 2 of the 74LS14 will be high (apx. 5 Volts).
When they are blocked, then the output will be low (apx. 0 Volts). The 74LS14 is a Schmitt triggered hex inverter.
A Schmitt trigger is a signal conditioner. It ensures that above a threshold value, we will always get "clean" HIGH and LOW signals.
Not Blocked Case: Pin 2 High Current from Vcc flows through the detector. The current continues to flow through the base of Q2.
Current from Vcc also flows through R2, and Q2's Drain and Emitter to ground.
As a result of this current path, there will be no current flowing through Q1's base.
The signal at U1's pin 1 will be low, and so pin 2 will be high. Blocked Case: Pin 2 Low Current "stops" at the detector.
Q2's base is not turned on. The current is re-routed passing through R2 and into the base of Q1.
This allows current to flow from Q1's detector and exiting out Q1's emitter. Pin 1 is thus high and pin 2 will be low.
To detect a line to be followed, we are using two or more number of poto-reflectors.
Its output current that proportional to reflection rate of the floor is converted to voltage with a resister and tested it if the line is detected or not.
However the threshold voltage cannot be fixed to any level because optical current by ambent light is added to the output current.
Most photo-detecting modules are using modurated light to avoid interference by the ambient light.
The detected signal is filtered with a band pass filter and disused signals are filtered out.
Therefore only the modurated signal from the light emitter can be detected.
Of course the detector must not be saturated by ambient light, this is effective when the detector is working in linear region.
The line position is compeared to the center value to be tracked, the position error is processed with Proportional/Integral/Diffence filters
to generate steering command. The line folloing robot tracks the line in PID control that the most popular argolithm for servo control.
The proportional term is the commom process in the servo system. It is only a gain amplifire without time dependent process.
The differencial term is applied in order to improve the responce to disturbance, and it also compensate phase lag at the controled object.
The D term will be required in most case to stabilize tracking motion. The I term that boosts DC gain is applied in order to remove left offset error,
however, it often decrease servo stability due to its phase lag.
When any line sensing error has occured for a time due to getting out of line or end of line, the motors are stopped and
the microcontroller enters sleep state of zero power consumption.
Typical Examples of infrared Transmitter and Receiver installation
## Tuesday, June 17, 2008
### Maze Solver n Wall Follower .
A simple wall follower has to navigate easily following the wall , that is when evr there is a turn (perfect 90 degree as by the wall) the bot shuld also take a turn . Take a look at this picture .
First question which comes to your mind is how many sensors do one wuld need ?? . Wuld we need a micrcontroller .
Well we can make this by using only two sensors , and there is no need of a micrcontroller also.
CONDITIONS TO BE SET UP:
We wuld be using 2 sensors one at the rite and the other at the center .
(Left sensor- high ----------and center sensor- high
move fwd)
(Left sensor-low ----------and center sensor-high
take left)
(Left sensor-high ----------and center sensor -low
take rite).
Use a K map with the above conditions and get the logic circuit .
Connect the sensors to the input of u r logic and H bridge at the output.
Use motors of 60rpm nt faster than tat .
If u want more help plz ask.
### COLOUR SENSORS
All color sensors work on the basic principle that when light of some color falls on an object , if the object is of same color as of the led , then the object absorbs that color which means there is no reflection.
We would be making the ckt in the same way as we did for the IR led detector ckt , but we wuld be using LDR for our detection purpose , (IR cant be used as they catch only IR waves) , and in place of the IR transmitter we wuld be using three leds of red, blue and green.
WORKING:
Light up each of the leds one at a time , ie red , then blue n green , if the obstacles are of red blue and green colour then you wuld be getting different outputs ( REMEMBER VIBGYOR) .
With micrcontrollers , lit up the red led if there is reflection take the ADC reading
do the same for other leds and take thier ADC reading , we will know which colour its detecting.
One sensor wuld be lit up always .
Use a potentiometer with each of the LEDs to get a better reading , as there wuld be slight color difference .The circuit will be the same as for the ir sensor which i have explained in my previous post , with litlle changes .Hope u njyd it
## Monday, June 16, 2008
### BUMP SENSOR
So, you've fitted some motors to your robot and its happily driving around but it probably keeps colliding with obstacles and getting stuck. You need a way for your robot to detect collisions and move around objects. Enter the humble bump sensor:
A bump sensor is probably one of the easiest ways of letting your robot know it's collided with something. The simplest way to do this is to fix a micro switch to the front of your robot in a way so that when it collides the switch will get pushed in, making an electrical connection. Normally the switch will be held open by an internal spring.
Micro switches are easy to connect to micro controllers because they are either off or on, making them digital. All micro controllers are digital, so this is a match made in heaven. Micro switch 'bump' sensors are easily connected to the Robocore, simply plug them into any free digital socket and away you go.
The following diagram shows a typical circuit for a micro switch bump sensor. The resistor is important because it holds the signal line at ground while the switch is off. Without it the signal line is effectively 'floating' because there is nothing connected to it, and may cause unreliable readings as the processor tries to decide if the line is on or off.
If you dont get these microswitches u can make one easily , using a spring mechanism.
## Friday, June 13, 2008
### MORE E BOOKS
(1)
Robots Androids & Animatrons by John Iovine
Lovely book , gives u a good start for professional robotics.
What u can get from this book????
Movements and drive system.
DTMF controlled vehicle (tats using u r phone)
Making a walker
Speech Controlled bot
Under water bots
Aerobots
One necessary book , I double my money on this to read.
(2)
Not tat great to read on if u have other e books , but has gud examples of various things.
(3)
Diff methods of making a remote controlled car , from Ir , Rf to various methods.
(4)
8051 microcontoller
One stop to start for micrcontrollers r download ayalas e book from my other post on e books.
## Saturday, May 17, 2008
### BIOMETRIC ROBOT JOURNALS
This another list on robotic journals on biometric , can be used for paper presentation in engineering .Taken from societyofrobots.com one wonderfull site.
Adaptive Dynamic Walking of a Quadruped Robot on Irregular Terrain Based on Biological Concepts
### ARTIFICIAL INTELLIGENCE - ROBOTICS JOURNALS
Here is a list of journals , papers that engineering students can use for paper presentations , but shld have great fundas .
Cognitive Architecture, Concepts, and Introspection: An Information-Theoretic Solution to the Problem of Phenomenal Consciousness
### DC MOTOR BRAKING
We will discuss different ways of braking in the robots , there are 3 different ways as usual which goes from mechanical to electrical .This is basically needed when u want to go down the slope r up the slope, though these are not tat great braking as comapred to the real vehicles but of much help to us.
Mechanical Method :
The mechanical method is what is used on cars today. Basically you need something with very high friction and wear resistance, and then push it as strongly as possible to your wheel or axle. A servo brake works well.People think of using door stoppers but they are not of much help , I tried it n they just wasted my money and time .
Controls Method :
This method requires an encoder placed onto a rotating part of your DC motor. You will have to write an algorithm that determines the current velocity of your motor, and sends a reverse command to your H-bridge until the final velocity equals zero. This method can let your robot balance motionless on a steep hill just by applying a reverse current to your motors.
Electronic Method :
This method is probably the least reliable, but the easiest to implement. The basic concept of this is that if you short the power and ground leads of your motor( connect both the +ve leads of the battery to the two terminals of the motor), the inductance created by your motor in one direction will power your motor in the opposite direction. Although your motor will still rotate, it will greatly resist the rotation. No controls or sensors or any circuits overheating. The disadvantage is that the effect of braking is determined by the motor you are using. Some motors brake better than others.
So how do you short the leads when it is on a robot? Simple. Connect a MOSFET (transistor) and a relay as shown. The MOSFET turns on the relay, which creates a short between the motor leads. Turn the MOSFET on (set C high) with your microcontroller when you want to brake. Basically your motor will still have an H-bridge for normal control, but when you brake you turn the H-bridge off and use the braking circuit. And don't forget the heatsink and flyback diode! Important, or your circuit will melt/blow up.
### SENSORS WITH TSOP CKT
This is a simple yet effective IR proximity sensor built around the TSOP 1738 module. The TSOP module is commonly found at the receiving end of an IR remote control system; e.g., in TVs, CD players etc. These modules require the incoming data to be modulated at a particular frequency and would ignore any other IR signals. It is also immune to ambient IR light, so one can easily use these sensors outdoors or under heavily lit conditions.
Such modules are available for different carrier frequencies from 32 kHz to 42 kHz.They cost some thing around 20Rs.
In this particular proximity sensor, we will be generating a constant stream of square wave signal using IC555 centered at 38 kHz and would use it to drive an IR led. So whenever this signal bounces off the obstacles, the receiver would detect it and change its output. Since the TSOP 1738 module works in the active-low configuration, its output would normally remain high and would go low when it detects the signal (the obstacle).
In the reciving part instead of using IC555 u can use the time from u r muc.
### A part of the Source Code for the Previous LFR
The 8 sensors are connected the port 0 .The numbers 160 n 255 are a constant
which u take this has been explained un my previous post so jst take a look.
while (1){
#ifdef debug
if(rep<255) prev="PINA;" i="0;i<8;i++)">>i)&0x01);
rep=0;
}
#endif
if(PINA!=255){
rotpow=255;
ldev=rdev=0;
if(PINA.3==0)
rdev=1;
if(PINA.2==0)
rdev=2;
if(PINA.1==0)
rdev=3;
if(PINA.0==0)
rdev=4;
if(PINA.4==0)
ldev=1;
if(PINA.5==0)
ldev=2;
if(PINA.6==0)
ldev=3;
if(PINA.7==0)
ldev=4;
if(rdev>ldev)
move(R,0,195+12*rdev);
if(rdev<160) rotpow="160;}">HMAX)
{move(CW,0,rotpow);}
else
{move(CCW,0,rotpow);}
}
};
}
void move (unsigned char dir,unsigned char delay,unsigned char power) {
PORTC=dir;
if(dir==L || dir==R) {
hcount=(hcount+1)%MAX;
history[hcount]=dir;
}
LSPEED=RSPEED=255;//power;
//delay_ms(delay);
}
This is taken from priyank patels LFR code .
## Friday, May 16, 2008
### AVR LINE FOLLOWER PD CONTROL
The robot uses IR sensors to sense the line, an array of 8 IR LEDs (Tx) and sensors (Rx), facing the ground has been used in this setup. The output of the sensors is an analog signal which depends on the amount of light reflected back, this analog signal is given to the comparator to produce 0s and 1s which are then fed to the µC.
Starting from the center, the sensors on the left are named L1, L2, L3, L4 and those on the right are named R1, R2, R3, R4.
Let us assume that when a sensor is on the line it reads 0 and when it is off the line it reads 1.
The µC decides the next move so as to position the robot such that L1 and R1 both read 0 and the rest read 1.
Algorithm:
1. L= leftmost sensor which reads 0; R= rightmost sensor which reads 0.
If no sensor on Left (or Right) is 0 then L (or R) equals 0;
For example
2. If all sensors read 1 go to step 3,
else,
If L>R Move Left
If L
If L=R Move Forward
Go to step 4
3. Move Clockwise if line was last seen on Right
Move Counter Clockwise if line was last seen on Left
Repeat step 3 till line is found.
4. Go to step 1.
For a good voltage swing from the potential divider use
R1= Sqrt (Resistance of sensor when light falls on it * Resistance of sensor when light does not fall on it)
The 8 sensors are connected to PORTA.
You need not connect anything to AVCC and AREF, it is required only if ADC is used.
The L298 Motor Driver has 4 inputs to control the motion of the motors and two enable inputs which are used for switching the motors on and off. To control the speed of the motors a PWM waveform with variable duty cycle is applied to the enable pins. Rapidly switching the voltage between Vs and GND gives an effective voltage between Vs and GND whose value depends on the duty cycle of PWM. 100% duty cycle corresponds to voltage equal to Vs, 50 % corresponds to 0.5Vs and so on. The 1N4004 diodes are used to prevent back EMF of the motors from disturbing the remaining circuit.
## Thursday, May 15, 2008
### line follower with software diff
A simple line follower not following a curve can make a large diff jst by changing the software tats u r programme . Its always better tpo use more sensors than just a couple of them , use arnd 8 sensors rather than 4 r 6 . Now wat diff can a change in algo make , u can use PWM and PID control to make it more perfect bot .Even a Closed loop control system formed with the above two can be used.
PID CONTROL :
A proportional integral derivative controller (PID controller) is a common method of controlling robots. PID theory will help you design a better control equation for your robot.
Shown here is the basic closed-loop (a complete cycle) control diagram:
The point of a control system is to get your robot actuators (or anything really) to do what you want without . . . ummmm . . . going out of control. The sensor (usually an encoder on the actuator) will determine what is changing, the program you write defines what the final result should be, and the actuator actually makes the change. Another sensor could sense the environment, giving the robot a higher-level sense of where to go.
To get you started, here are a few terms you will need to know:
error - The error is the amount at which your device isnt doing something right. For example, if your robot is going 3mph but you want it to go 2mph, the error is 3mph-2mph = 1mph. Or suppose your robot is located at x=5 but you want it at x=7, then the error is 2. A control system cannot do anything if there is no error - think about it, if your robot is doing what you want, it wouldnt need control!
proportional (P) - The proportional term is typically the error. This is usually the distance you want the robot to travel, or perhaps a temperature you want something to be at.
derivative (D) - The derivative term is the change in error made over a set time period (t). This is usually the velocity of your robot. So if your robot was at x=5 about one t ago, and is at x=7 now, then the derivative term is 7 - 5 = 2/t. If you are using a microcontroller, you can calculate the time with this timer tutorial.
integral (I) - The integral term is the rate of change in the error made over a set period of time (t). This is usually the acceleration of your robot. If your derivative term was 2/t a second ago, and it is 2/t now, your integral term is 2 - 2 = 0/t^2. Thats an acceleration error of zero . . .
tweak constant (gain) - Each term (P, I, D) will need to be tweaked in your code. There are many things about a robot that is very difficult to model mathematically (ground friction, motor inductance, center of mass, ducktape holding your robot together, etc.). So often times it is better to just build the robot, implement a control equation, then tweak the equation until it works properly. A tweak constant is just a guessed number that you multiple each term with. For example, Kd is the derivative constant. Idealy you want the tweak constant high enough that your settling time is minimal but low enough so that there is no overshoot.
One example of only P control can be seen here: taken from amol deshmukhs site ,
This is a small line following robot designed to follow a white line drawn on a black surface. The software can be changed to interchange the colours. The software still has lot's of room for improvement but works well. It constantly corrects wrong moves using feedback mechanism which forms a closed loop control system.
mechanical design It has two DC motors in wheelchair design. Direction of the robot is controlled by controlling speeds of the two motors. Lets say the speeds are
Right wheel : SpeedR
Left wheel : SpeedL
To control the speed of the motor controlled power is fed by PWM ( Pulse Width Modulation ) technique.
electronic design Processor AVR ATmega16 [ the ADC feature comes in handy to read output of sensors ]
Motor driver : L293D ( 2 )
Sensors : IR LED-photodiode pair ( 2 )( non-modulated )
Power supply : Li-Ion cells ( 2 ) 1700mAHr giving 8V regulated with voltage regulator.
control algorithm and software
Lets say the two sensor outputs, i.e. the intensity of reflected light sensed at the two sensor positions is
Sensor Left : SensL
Sensor Right : SensR
If Err is the error of sensors from mean position then it will be proportional to the difference between the two sensor readings
Err = K * ( SensR - SensL )
Where K is constant of proportionality
If Err is a +ve quantity then we must drive Left motor a bit faster than the right i.e. the speed of right motor must be increased a bit and speed of motor right be decreased a bit
SpeedL = CntrSpeed + Kp * Err
SpeedR = CntrSpeed - Kp * Err
This is called Proportional Control
Check for my next post on 8 sensors using PD control wth algo.
The above one is only P control n taken from this site
## Tuesday, May 6, 2008
### Making A Gripper Arm for Pick n Place comps
I will be discussing over here various methods to hold objects , this is mostly used in pick and place comps.
1)USING ONLY MOTORS N DIRECTLY GRIPPING THEM TO ARMS.
Basic idea is to hold the object and also leave the object.We can use two simple dc geared motors of lower rpm n directly fix it to the bars as shown below
We used two dc geared motors of 10 RPM at the points A and B , fixed to the arms using gears at the two dots shown above A n B
RESULT:Couldnt sustain the force required to lift for a long period of time, so we changed our plan.
2)USING NUT BOLT MECHANISM:
Thi
s has been taken frm another site robotics for u which u can c in the link list on the rite side on my blog
This thing is self xplanatory when the motor starts rotating the nut keeps moving inside ,and hence the one of the arms fixed to it keeps moving inside . It gives gud result but was pretty to tough to implement for us , we gave try at it but werent much happy wth wat we had done so had to think some thing else which can be much simple .
3)SIMPLE GRIPPER USING SINGLE MOTOR:
We
used some simple objects tat we could get hand on , so we used a a girder u can get this in any mechanical shop(he will charge pretty high if u tell u r purpose).This is how they luk like
The first pic is that of girders (which u can make them using aluminium and drilling holes in it)
and the second pic is that of our arm , we used a 10 rpm motor to pull the two strings.Use a spring in reverse of the strings to pull so tat it can even with draw objects when needed.This is how it goes finally
This is how it finally luks like dont use a high rpm motor thinking tat we can grip it fast , it will be out of control so be carefull of it , we could grip this to lift a 4.6kg objects which we tested in our colleges . This was all because we could use a low rpm motor.
### "MEMRISTOR" ...... New Circuit Element
As all science students know, there are only three basic elements that make up an electrical circuit:the resistor, the capacitor and the inductor.
Sorry guys!It may be time to tear up your textbooks and write the new ones:scientists have realised physical samples of a fourth fundamental element which they call a memristor - short for memory resistor.
In a paper published in the latest issue of NATURE MAGAZINE(”The missing memristor found”,may 1st ,2008;vol no.453 ; pp 80-83), researchs at hewlett packard labs, report that the missing fourth element of the circuitry that professor Leon Chua of the University of California in Berkeley predicted in 1971 is indeed realisable.
PRACTICAL UNITS :
The team, led by R.Stanley Williams, believes that using nano technology one can soon build practical units of the resistor-with-memory that cannot be created by a mere combination of the three basic circuit elements.
Such element could fuel a new class of computer memory that would ‘remember’, even if the machine were switched off……in other words, tomorrow’s PCs could boot up and spring to life instantly.
The engineers are busy building memristors using Titanium Dioxide and have already realised a few hybrid versions in silicon.
Memory banks built using memristors could be a thousand times faster than todays magnetic disk systems , and consume a fraction of the power, the scientists suggest.
In electrical circuit theory, the memristor is a passice circuit element. It has been described as the fourth basic type of passive circuit element, alongside the well-known capacitor, resistor, and inductor. The name is a portmanteau of memory resistor.
Although the memristor was predicted and described in 1971 by Leon Chua of UC Berkeley, in a paper in IEEE Transactions on Circuit Theory, it was a hypothetical device for 37 years, with no known physical examples.
According to Chua, the “era of nanoscale electronics will be enabled by the memristor. This is not just an invention, it is a basic scientific discovery.
In April 2008, a working device with similar characteristics to a memristor was announced by a team of researchers at HP Labs. The new circuit element may enable the development of a new class of high-density non-volatile digital memory. Performance of memristors improves as they are scaled down, and they generate less heat than transistors. The memristor also has unique analog properties that may lead to the invention of other devices.
The memristor is an element in which the magnetic flux Φm is a function of the accumulated electric charge q in the device. The rate of change of flux with charge
$M(q)=\frac{\mathrm d\Phi_m}{\mathrm dq}$
is known as memristance. This is comparable to the other three fundamental circuit elements:
• Resistance: $R(I)=\frac{\mathrm dV}{\mathrm dI}$
• Inductance: $L(I) = \frac{\mathrm d\Phi_m}{\mathrm dI}$
• Capacitance: $\frac{1}{C(q)}=\frac{\mathrm dV}{\mathrm dq}$
Here q is electrical charge, I is electrical current, V is electrical potential and Φm is magnetic flux. The differential forms of these equations are used because we are comparing non-linear circuit elements; a linear memristor would be uninteresting, as explained below.
Applying Faraday’s law of electromagnetic induction and the chain rule to the equation defining the memristance, one obtains that the voltage V across a memristor is related to the current I by the instantaneous value of the memristance:
$V(t) = M(q(t)) I(t) \,$
Thus at any given instant, a memristor behaves like an ordinary resistor. However, its “resistance” M(q) is a value which depends on the charge accumulated in the device. This differs from ordinary resistors where the resistance is determined by fixed physical properties and transistors where the resistance is controlled by either the voltage at or current through a gate electrode. A linear memristor (one for which M is constant) would thus be indistinguishable from an ordinary linear resistor (one for which R is constant), with M = R. Memristance can be said to depend on the history of the charge that has flowed through the device in the same way that the voltage of capacitors does.
The memristor is capable of “remembering” how much electrical charge most recently passed through it in one direction versus another. Thus, it can “remember” the state it was last in.
Fabrication :
Interest in the memristor revived in 2007 when an experimental solid-state version was reported by Stanley Williams of hewlett packard. A solid-state device could not be constructed until the unusual behavior of nanoscale materials made it possible. The device does not use magnetic flux as the theoretical memristor suggested, nor stores charge as a capacitor does, but instead achieves a resistance dependent on the history of current using a chemical mechanism.
The HP device is composed of a thin (5 nm) titanium dioxide film between two electrodes. Initially, there are two layers to the film, one of which has a slight depletion of oxygen atoms. The oxygen vacancies act as charge carriers, meaning that the depleted layer has a much lower resistance than the non-depleted layer. When an electric field is applied, the oxygen vacancies drift, changing the boundary between the high-resistance and low-resistance layers. Thus the resistance of the film as a whole is dependent on how much charge has been passed through it in a particular direction, which is reversible by changing the direction of current.
Samsung has a pending U.S. patent application for a memristor similar to that described by Williams.
Potential applications :
Williams’s solid-state memristors can be combined into devices called crossbar latches, which would replace transistors in future computers, taking up a much smaller area. They can also be fashioned into non-volatile solid-state memory, which would allow greater data density than hard drives with access times potentially similar to DRAM, replacing both components. HP prototyped a crossbar latch memory using the devices that can fit 100 gigabits in a square centimeter. For comparison, the highest-density flash memories at this time (2008) hold 16 gigabits. HP has reported that its version of the memristor is about one tenth the speed of DRAM.
The devices’ resistance would be read with alternating current so that they do not affect the stored value.
## Friday, April 18, 2008
### E books on robotics
All these books have been take from the site allfree.freemaniz site .I have read most of the books and i will give my opinion on these so it will be useful to you.
First book which all beginners shuld read for robotics .
Tata mcgrah hills robot builder's bonanza.
Robotics project outlined by pic microcontroller infact a tutuorial on pic
Architecture on microcontroller by ayala .A complete text book of AYALAs muc.
Matlab programming for Image processing
Robot mechanical building tips .Very gud book if u really want to make one robus mechanical bot with perfection.
Autonomous mobile robotics .Amazing book if u want to make one great autonoums navigation bot.It tells evrything from makng a biped(a litle bit) , a very gud autonoums bot.
Making a pac -man robot very fun makng, it indeed wil give u a xperince in how some problem statements are solved.
And last but not the least my site , i will be updating it very frequently so tat when the next year starts I mean next sem , as robotic competetions wuld be startng u will find lot more help from here keep checking .Till then thnx from all u guys.
My next post will be on logic gates and K map(karnuagh map) , to make autonoums bots without microcontroller.
## Friday, April 11, 2008
### LINE FOLLOWER
This will be the first start to our autonoums bots , this is the most famous competetion held in most of the colleges , n i have seen people asking at lot of places asking for this now i will try to make this post a one stop to make a line follower , if u have understood the previous 4 things ,
firstly the motor driver , then op amp , sensors, which i have told in my last posts n the the most important part either using logic gates r else using micro controllers tats programming it .
I will be tellng u both the ways of doing it tats using logic gates and also using muc(i will be gvng only the logic part ).
This is how it basically luks like
I will xplain making of a line follower using one sensor till an array of sensor(ie 7 sensors).
BASIC PRINCIPLE:
White colour reflects light and black colour
absorbs .
So
when the sensor is on the white line the output will be high and when on black it will be low.
WE ASSUME WE ARE FOLLOWING A BLACK LINE ON WHITE BACK GROUND.
LFR USING A SINGLE SENSOR
L
ets assume tat sensor is placed on center ie the sensor correctly places above the line , then there is no way tat it can follow a line bcoz it cant diff btw curves tilted towards left n right.
so wat we do is place the bot in such a way tat when the sensor cums to the right side of the line .
conditions:
1)when the sensor is not on the main line move fwd.
2)when the line is turning towards left the sensor detects the line so now take a left turn
3)when the line is turning towards rite u r sensor cant detect the line and it keeps moving forward as it satisfies condition number 1
RESULT:We cant use this thing.
LFR USING TWO SENSORS:
We place our sensor on either side of the line .
conditions:
When the line is straight and the two sensors are on either side of the line, move forward
when there is a right bend in the line the right sensor detects it ,now as ur bot shuld take turn, the rite motor shld stop and the left motor shuld move.
when there is a left bend in the line the left sensor detects it ,now as ur bot shuld take left turn, the left motor shld stop and the rite motor shuld move.
The pic shows the left curves and the right curves, the blue ones are u r sensors, the left pic shows the left sensor about to detect the line and the rite pic shows the right sensor about to detect the line.For the ckt directly connect the output of sensors to the the inputs of l293d n tat s it but use 30 rpm motors for it hope u got it .
But the problem with this is it will always move in zig zag manner so wastage of time.
I will give the ckt diagram of this after i explain about Logic gates .
And remember this is only with two sensors , i will explain till we can use 7 sensors so keep checking.
## Wednesday, April 9, 2008
### Simple IR LED CKT
We will be using over here an irled pair and a comparator .When the reciver led recivers the signal the voltage acros it would be going high.This we would be putting across the + ve of comparator, n to the -ve of comparator we wuld be using a potentiometer . When the reciver does not recive any signal the voltage across the pot wuld be set up in such a way tat voltage across it is high so the output wuld be zero volts.
Wehen the reciver recieves the signal the voltage at +ve of comparator would be more than that of -ve end so the out put wuld be high.The pot is used to adjust from how much distance the reciever is able to detect .
THE LED WHICH U C IN THE CKT FROM LEFT ARE 1ST ONES A IRLED TRANSMITTER ,2ND ONE IS A IR TRANSMITTER AND THE 3RD ONE IS A NORMAL LED WHICH WILL GLOW MOSTLY RED IS USED
From my xperince i can tell tat this will give u arnd 5 cm obstacle detection , voltage drop across reciver for me was .47V when it didnt sense the obstacle , when it did sense it was giving arnd.58v so now the voltage at the -ve pin shld be brought to this voltage by rotating the pot .
The leb which u c at the ouput will glow when it detects an obstacle.
The output which u get from the comparator will be fed to logic gates r micrcontrollers .
### SENSORS
This is the most important part of an autonoums bots , without this der cant be an autonoums bot build. There are different types of sensors used varying with work to be done with the bot.
1)Light sensors
This sensors are used to measure the intensity of light.Mostly commonly used under this are LDR(light dependent resistors) , TSOP1738,IRLED pair,Photodiode,Phototransistor.
USES :
Th
ey can be used as obstacle avoidance, in line follower to differentiate btw black and white.
LDR:
Its
a resistor whose value decreases with intensity of light, more the intensity of light lesser will be the resistence and vice versa , which means when this sensor is in dark place there is less resistence.Now the question will be how can we use it as sensor for our bot we use LEDs(red mostly) in combination with LDR,.
I L PUT UP THE CKT OF IT IN MY OTHER POST.
http://www.technologystudent.com/elec1/ldr1.htm (FOR MORE HELP)
2)IR LED:
Its
basically a pair of diodes one is the reciever and the other a transmitter(dark blue in colour) both modulated to same frequency .The transmitter works in forward bias condition and the reciever in reverse bias condition.It has two terminals the longer one is the + ve one n other -ve .
How does it work:
The
transmitter sends a light at partiular freq though u cannot c it , the receiver detects it when it bounces back from some obstacle.
THE CKT WULD BE PUT IN OTHER POST.
This sensor is mostly used by students for various competiotns.
3)TSOP 1738:
This is mostly used when we want our sensors to work in more ambient lite , sunlite i can say for.
This thing is found in TV remote controller.
THESE SENSORS TAT I HAVE TOLD ARE ONLY LIGHT SENSORS,
Apart from these the sensors tat are there are
accelometers :Used to detect motion, vibration, ngle with respect to gravity
Colour sensors:Used to detect diff colours
Sonar Sensors:Determines obstacle from a distance
Bump switch:touches the object and gives the signal
I will discuss about these sensors in more detail in later posts .
## Tuesday, April 8, 2008
### OPAMPS n COMPARATORS
As the name implies it is an operational amplifier. It performsmathematical operations like addition,subtraction,log,antilog etc.. Themain reason for OPAMPS used over transistors is that transistor can onlyamplify AC while OPAMPS can amplify AC and DC. You can get good amplifier gain in OPAMPS. The most commonly used OPAMPS are 741 ,LM 324, LM 358N(both of them can also be used as comparator).
Comparator is also the same but with a diff being tat its digital so u have to two states high(5V) and low(0V).
FIGURE 1.
Above figure shows the general ckt diagram of a comparator .If V1>V2 then Vout=+Vcc and if V1 if V1 is less than V2 then Vout is minus Vcc
Lm 324 has 4 such comparator(shown in figure1) them also known as Quad opamp.
LM 358N has 2 comaprator also known as bi opamp.
EXPERIMENTAL PART:
U can use two potentiometers(of 10 K prefarably) and put them across pin2 and pin 3 , and put an led in series with a resistance at pin 1 , when the voltage drop in pot 1 connected across pin 2 is low , u can c the led glow.
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Rational number
(Redirected from Rational numbers)
A rational number is a number that can be represented as the ratio of two integers.
Examples
• All integers are rational because every integer $a$ can be represented as $a=\frac{a}{1}$
• Every number with a finite decimal expansion is rational (say, $12.345=\frac{12345}{1000}$)
• Every number with a periodic decimal expansion (e.g. 0.314314314...) is also rational.
Moreover, any rational number satisfies exactly one of the last two conditions. The same remark holds if "decimal" is replaced with any other base.
Properties
1. Rational numbers form a field. In plain English it means that you can add, subtract, multiply, and divide them (with the exception of division by $0$) and the result of each such operation is again a rational number.
2. Rational numbers are dense in the set of reals. This means that every non- empty open interval on the real line contains at least one (actually, infinitely many) rationals. Alternatively, it means that every real number can be represented as a limit of a sequence of rational numbers.
3. Despite this, the set of rational numbers is countable, i.e. the same size as the set of integers.
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# Investor choice problem
Guys I'm stuck with a problem... Consider the portfolio choice problem of a risk-averse individual with a strictly increasing utility function. There is a single risky asset, and a risk free asset. Formulate an investor's choice problem and comment on the first-order conditions. What is the minimum risk premium required to induce the individual to invest all his wealth in the risky asset?
Since we know that the choice problem an investor must solve can be expressed as:
$\max_{a} \mathcal{E}[U(Y_1)] = \max \mathcal{E}[U(Y_0(1+r_f)+a(r_i-r_f))]$
Where $U( )$ is the utility of money function and $\mathcal{E}$ the expectation operator. Moreover, $Y_1$ is the wealth at time 1 whereas $Y_0$ the wealth at time 0, whereas $a$ is the portion that should be invested in the risky asset.
By differentiating into the expectation we can solve the maximization problem and we have: $\mathcal{E}[(U'(Y_0(1+r_f)+a(r_i-r_f))(r_i-r_f)]=0$
The FOC that solves the problem, that is written on the solution of the exercise, is
$\mathcal{E}[U'(Y_0(1+r_i)(r_i-r_f))]\geq0$
Since $a=1$. I don't get why the FOC is this... Can anybody explain me better?
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### Home > CCG > Chapter 9 > Lesson 9.2.4 > Problem9-105
9-105.
Without using a calculator, find the sum of the interior angles of a $1002$-gon. Show all work. Homework Help ✎
Look at the Math Notes box in Lesson 8.1.5 for extra help.
Let $n$ = number of sides, $(n−2)180° = 1000 · 180° = 180,000°$
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• Corpus ID: 224710735
# New Estimates on the bounds of Brunel's operator.
@article{Assani2020NewEO,
title={New Estimates on the bounds of Brunel's operator.},
author={Idris Assani and R. Spencer Hallyburton and Sebastien McMahon and Stefano Schmidt and Cornelis Jan Schoone},
journal={arXiv: Dynamical Systems},
year={2020}
}
• Published 17 October 2020
• Mathematics
• arXiv: Dynamical Systems
We study the coefficients of the Taylor series expansion of powers of the function $\psi(x)=\frac{1-\sqrt{1-x}}{x}$, where Brunel's operator $A$ is defined as $\psi(T)$. The operator $A$ was shown to map positive mean-bounded (and power-bounded) operators to positive power-bounded operators. We provide specific details of results announced by A. Brunel and R. Emilion in \cite{Brunel}. In particular, we sharpen an estimate to prove that $\sup_{n\in\mathbb{N}} \|n(A^n-A^{n+1})\| < \infty$. We…
## References
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Every one of the important strong limit theorems that we have seen thus far – the strong law of large numbers, the martingale convergence theorem, and the ergodic theorem – has relied in a crucial
© Gauthier-Villars, 1973, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section B » (http://www.elsevier.com/locate/anihpb) implique l’accord avec les conditions
• Mathematics, Computer Science
• 2015
It is shown that (infinite) convex combinations of powers of Ritt operators are Ritt, which is a unified framework for several main results on discrete subordination from [19] and answer a question left open in [19].
Given a power-bounded linear operator T in a Banach space and a probability F on the non-negative integers, one can form a 'subordinated' operator S = Σ k≥0 F(k)T k . We obtain asymptotic properties
• 2020
• 2020
### On positive mean-bounded operators
• Comptes Rendus De L’Académie Des Sciences,
• 1984
• 1957
### Pointwise and norm properties of the Brunel operator . Preprint in preparation ( 2020 ) . [ BE 84 ] A . Brunel and R . Emilion . On positive mean - bounded operators
• Comptes Rendus De L ’ Académie Des Sciences
• 1984
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# Using continuous features for RNN training
I am designing a recurrent neural network for binary classification problem: (1) there is an attack in the network, (2) the session is normal in the network.
To achieve this, I am using the Kyoto University dataset. Here's a sample data from it:
duration, service, src_bytes, dest_bytes, count, same_srv_rate, serror_rate, srv_serror_rate, dst_host_count, dst_host_srv_count, dst_host_same_src_port_rate, dst_host_serror_rate, dst_host_srv_serror_rate, flag, ids_detection, malware_detection, ashula_detection, label, src_ip_add, src_port_num, dst_ip_add, dst_port_num, start_time, protocol
-0.026718199145531595,4,-0.0017137615074428484,-0.0023086230344278144,-0.3656989628802213,0.9201603673125098,2.8316170813302053,1.6838464062500405,-0.8300894248587679,-0.7122843212362112,3.505362993154133,2.2092313846051757,1.9096507395538231,6,0,0,0,1,110661,0.7951296522328849,5230,-0.18795233710676376,0.9228794625927296,1
-0.026718199145531595,4,-0.0017137615074428484,-0.0023086230344278144,-0.3073821073997234,0.9201603673125098,2.8316170813302053,1.6838464062500405,-0.8300894248587679,-0.6877003684627419,3.505362993154133,2.2092313846051757,1.9096507395538231,6,0,0,0,1,182016,0.7920093464532119,5230,-0.18795233710676376,0.9228794625927296,
The label 1 pertains to a state that there's an attack in the network, while the label 0 pertains to a state that there's no attack in the network.
(1) My problem is the usage of the continuous data such as duration, serror_rate among others. I'm wondering if one-hot encoding is applicable in this instance or not.
(2) How should I use the continuous data for the RNN training? Other features, categorical or ordinal ones, can be one-hot encoded. But what about the continuous ones?
UPDATE August 16, 2017
As @shimao suggested, I performed decile binning on my data, that is, to bin the features in 10 to 100 deciles. I used pandas to do so:
for index in range(len(cols_to_std)):
df[cols_to_std[index]] = pd.qcut(df[cols_to_std[index]], 10, labels=False, duplicates='drop')
The df above is a pandas DataFrame containing 21 features and 1 label.
Sample results of training with decile-binned data:
[0] loss : 3416.749902650714, accuracy : 0.9140625
[1] loss : 4271.98881316185, accuracy : 0.96875
[2] loss : 2105.836363852024, accuracy : 1.0
[3] loss : 3483.10527408123, accuracy : 0.98046875
[4] loss : 1750.0128374248743, accuracy : 0.97265625
[5] loss : 1320.9579735696316, accuracy : 0.99609375
[6] loss : 3481.7440667152405, accuracy : 0.97265625
[7] loss : 2572.4160171300173, accuracy : 1.0
[8] loss : 2453.3563360869884, accuracy : 0.95703125
[9] loss : 1284.2558837980032, accuracy : 1.0
Case solved. Thanks, @shimao!
In general, continuous data does not need to be encoded. If some of your categorical data comes in one-hot form and some in continous form, it is adequate to simply concatenate everything together in your input. It may help to scale/normalize the continuous data appropriately if the numbers involved are very big or small.
Sometimes, it does help to bin the continuous data. For example, compute the 0th, 10th, 20th... 100th percentile for the continuous parameter, then classify each continuous value into one of ten bins, the first bin being examples between 0th and 10th percentile, the second being 10th to 20th percentile, etc. Apply one-hot encoding.
Usually between 10 and 100 bins is a good number. This may or may not work better than just passing in the continuous value -- it is impossible to know without actually trying it out.
Cases I'm aware of where binning continuous values is common : pose estimation, image generation, bounding box proposal.
Cases where binning is not as common : image classification, depth estimation, bounding box regression.
• My problem is sequence classification. From the sample I gave, each row has 23 features, and a label. I want to classify use each row for training the neural network, and of course, for testing as well. Will binning be okay with it? Aug 11 '17 at 0:27
• Yes, binning will be ok, even for time series days Aug 11 '17 at 2:44
• Thanks for your help! I did percentile binning as you suggested, and got my neural net train with nice results! :) Aug 16 '17 at 15:32
• I have a question. Is it okay to bin a standardized data? Sep 28 '17 at 9:39
• The sample data I wrote here is from a standardized dataset. Sep 28 '17 at 9:42
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## Found 14,760 Documents (Results 1–100)
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### Accuracy of the Laplace transform method for linear neutral delay differential equations. (English)Zbl 07529444
MSC: 65-XX 34-XX
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### Optimal recovery in weighted spaces with homogeneous weights. (English. Russian original)Zbl 07528026
Sb. Math. 213, No. 3, 385-411 (2022); translation from Mat. Sb. 213, No. 3, 111-138 (2022).
MSC: 41A65 41A46 49N30
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### $$L^p$$ uncertainty principles for the windowed spherical mean transform. (English)Zbl 07527263
MSC: 42B10 44A05
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### Fourier-Mukai transforms of slope stable torsion-free sheaves on Weierstrass elliptic threefolds. (English)Zbl 07526506
MSC: 14J30 14J33 14J60
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### A note on continuous fractional wavelet transform in $$\mathbb{R}^n$$. (English)Zbl 07526031
MSC: 42C40 46E30 46F05
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### A general and fast convolution-based method for peridynamics: applications to elasticity and brittle fracture. (English)Zbl 07526002
MSC: 74-XX 68-XX
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MSC: 65C60
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### Time-frequency analysis associated with $$k$$-Hankel-Wigner transforms. (English)Zbl 07523398
MSC: 42B10 44A20 46F12
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### Global attractor of a dissipative fractional Klein Gordon Schrödinger system. (English)Zbl 07522516
MSC: 26-XX 35-XX
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### Weighted integrability of multiple multiplicative Fourier transforms. (English. Russian original)Zbl 07518123
Math. Notes 111, No. 3, 364-372 (2022); translation from Mat. Zametki 111, No. 3, 365-374 (2022).
MSC: 42A38
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### Local versions of the Wiener-Lévy theorem. (English)Zbl 07517858
MSC: 42B10 42B05
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### Time-frequency analysis associated with the deformed Stockwell transform. (English)Zbl 07517848
MSC: 47G10 42B10 47G30
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### Boundedness and uniqueness of quaternion Weyl transform. (English)Zbl 07517847
MSC: 47G30 42B10 47B10
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### Elimination of ringing artifacts by finite-element projection in FFT-based homogenization. (English)Zbl 07517709
MSC: 74Sxx 74Qxx 74Exx
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### Fast sixth-order algorithm based on the generalized Cayley transform for the Zakharov-Shabat system associated with nonlinear Schrödinger equation. (Fast sixth-order algorithm based on the generalized Cayley transform for the Zakharov-Shabat system associated with nonlinear schrodinger equation.) (English)Zbl 07516836
MSC: 65Lxx 35Qxx 34Axx
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### A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains. (English)Zbl 07516834
MSC: 65Nxx 35Jxx 65Mxx
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### Integral representation of the solution to the nonstationary Lamb problem in the case of a limiting Poisson ratio. (English. Russian original)Zbl 07514284
Comput. Math. Math. Phys. 62, No. 3, 467-475 (2022); translation from Zh. Vychisl. Mat. Mat. Fiz. 62, No. 3, 478-487 (2022).
MSC: 74-XX 45-XX
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### Fourier multipliers in $${\text{SL}_n}(\mathbf{R})$$. (English)Zbl 07513360
MSC: 22E46 43A80 46L51
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### Ingham-type theorems for the Dunkl Fourier transforms. (English)Zbl 07512009
MSC: 33C52 43A32 51F15
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### Low-dimensional maximal restriction principles for the Fourier transform. (English)Zbl 07509439
MSC: 42B10 42B25 42B37
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### Bump functions with monotone Fourier transforms satisfying decay bounds. (English)Zbl 07509390
MSC: 41-XX 42-XX
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### Three-dimensional animation nonlinear system modal identification using wavelet transform. (English)Zbl 07507569
MSC: 62Mxx 62Pxx 65Txx
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### Option pricing using stochastic volatility model under Fourier transform of nonlinear differential equation. (English)Zbl 07507549
MSC: 91G20 42A38 91G80
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### Heisenberg uniqueness pairs for the finitely many parallel lines with an irregular gap. (English)Zbl 07506350
MSC: 42A38 44A35
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MSC: 46E30
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### Constructions of Segal algebras in $$L^1(G)$$ of LCA groups $$G$$ in which a generalized Poisson summation formula holds. (English)Zbl 07499618
MSC: 43A20 42A38 43A25
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### 2D Green’s functions for an elastic layer on a rigid support loaded by an internal point force. (English)Zbl 07498578
MSC: 74-XX 76-XX
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### On viscoelastic transient response of magnetically imperfect functionally graded nanobeams. (English)Zbl 07498571
MSC: 74-XX 76-XX
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### The Poincaré line bundle and autoduality of Hitchin fibers. (English)Zbl 07498246
MSC: 14D24 14D23 14F08
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MSC: 30G35
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### A sample efficient sparse FFT for arbitrary frequency candidate sets in high dimensions. (English)Zbl 07496455
MSC: 65Txx 65T40 42A10
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### In vitro proton magnetic resonance spectroscopy at 14T for benign and malignant ovary. I: Signal processing by the nonparametric fast Padé transform. (English)Zbl 07496321
MSC: 92C55 41A21
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### Some integral operators and their relation to multidimensional Fourier-Bessel transform on $$L_\alpha^2( \mathbb{R}_+^n)$$ and applications. (English)Zbl 07493951
MSC: 47G10 44A15
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### Lipschitz and Fourier type conditions with moduli of continuity in rank $$1$$ symmetric spaces. (English)Zbl 07493677
MSC: 26A16 42A38 58A05
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### Fast solution algorithm for a three-dimensional inverse multifrequency problem of scalar acoustics with data in a cylindrical domain. (English. Russian original)Zbl 07492856
Comput. Math. Math. Phys. 62, No. 2, 287-301 (2022); translation from Zh. Vychisl. Mat. Mat. Fiz. 62, No. 2, 289-304 (2022).
MSC: 76-XX 65-XX
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MSC: 42B10
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### $$L^2$$-Poisson integral representations of eigensections of invariant differential operators on a homogeneous line bundle over the complex Grassmann manifold $$SU(r,r+b)/S( U(r)\times U(r+b))$$. (English)Zbl 07491583
MSC: 43A85 42B20 35-XX
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### Non-spectrality of Moran measures with four digits. (English)Zbl 07491545
MSC: 28A80 42C05 46C05
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### Heisenberg uncertainty inequality for Gabor transform on nilpotent Lie groups. (English)Zbl 07491544
MSC: 22E25 43A25 43A32
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### Mirror symmetry for Nahm branes. (English)Zbl 07491453
MSC: 14D20 14H60 14J33
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### Secondary pulmonary tuberculosis recognition by rotation angle vector grid-based fractional Fourier entropy. (English)Zbl 07490683
MSC: 92C55 42A38 26A33
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### Quantitative uncertainty principles for the canonical Fourier-Bessel transform. (English)Zbl 07490404
MSC: 33D15 47A05 47A30
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### Asymptotic expansion of wavelet transform for small values of $$a$$: an oscillatory case. (English)Zbl 07489888
MSC: 44A05 41A60
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### Toeplitz operators associated with the deformed windowed Fourier transform. (English)Zbl 07489690
MSC: 47G10 42B10 47G30
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### Daubechies’ time-frequency localization operator on Cantor type sets II. (English)Zbl 07489490
MSC: 47A30 47A75
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### A fully discrete low-regularity integrator for the nonlinear Schrödinger equation. (English)Zbl 07488719
MSC: 65M12 65M15 35Q55
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### An operator related to the sub-Laplacian on the quaternionic Heisenberg group. (English)Zbl 07488236
MSC: 53C17 22E30 35A08
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### Non-spherical Harish-Chandra Fourier transforms on real reductive groups. (English)Zbl 07488198
MSC: 43A85 22E30 22E46
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### The Fourier transform approach to inversion of $$\lambda$$-cosine and Funk transforms on the unit sphere. (English)Zbl 07487894
MSC: 44A12 42B10 46F12
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MSC: 42A38
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### On thermoelastic analysis of two collinear cracks subject to combined quadratic thermo-mechanical load. (English)Zbl 07484222
MSC: 74Fxx 74Rxx 74Exx
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### A new method based on the Laplace transform and Fourier series for solving linear neutral delay differential equations. (English)Zbl 07483735
MSC: 34Kxx 65Lxx 92Dxx
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MSC: 62-XX
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MSC: 65T50
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### Integrability of the Fourier-Jacobi transform of functions satisfying Lipschitz and Dini-Lipschitz-type estimates. (English)Zbl 07472685
MSC: 33C45 41A17
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### On the multidimensional Nazarov lemma. (English)Zbl 07469086
MSC: 42B20 26A16 42B05
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### Logvinenko-Sereda type theorems for the short time Fourier transform. (English)Zbl 1481.42004
MSC: 42A38 42C40
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### A note on discrete Heisenberg uniqueness pairs for the parabola. (English)Zbl 07465725
MSC: 42B10 35J10
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### On the Clifford short-time Fourier transform and its properties. (English)Zbl 07465267
MSC: 42A38 30G30
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### Growth properties of the $$q$$-Dunkl transform in the space $$L^p_{q,\alpha }({\mathbb{R}}_q,|x|^{2\alpha +1}d_qx)$$. (English)Zbl 07462090
MSC: 33D15 47B48 42B10
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### Marcinkiewicz integral operators along twisted surfaces. (English)Zbl 07461704
MSC: 42B20 42B15 42B25
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### Two simple shearlet-based inverses for the multidimensional Radon and John transforms. (English)Zbl 07461186
MSC: 44A12 65R32 42B10
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FQXi FORUM
October 24, 2019
CATEGORY: Questioning the Foundations Essay Contest (2012) [back]
TOPIC: The Incorrect Assumptions and a Correct Theory by Hoang Cao Hai [refresh]
Author Hoang cao Hai wrote on Aug. 23, 2012 @ 11:51 GMT
Essay Abstract
The universe is our inside because we are the fully part of it. To be able to understand and find out the basic principles of the universe,perhapsIs the first we be must to really understand the nature of the our thinking: Why we are wrong ?and how we were right? Where are our limits? and how to overcome it? Let's carefully look back to the past, specified for the present and to get the logical choice for future.Mean that is along with to reviewed the assumptions foundation of basic theory,we should be directed to the purpose for which we need from them.
Author Bio
Full name:Hoàng cao Hải Gender:Male Date of birth : 1971 Place of birth and current residence:Hanoi-Vietnam Education Level:High School Professional qualifications:Technical Measurement Current work:Self-study theory new foundation for science The ability of schools: Analysis,Reasoning,Imagining The results work:the ABSOLUTE theory (2011)
Author Hoang cao Hai wrote on Sep. 5, 2012 @ 19:04 GMT
Author Hoang cao Hai replied on Sep. 26, 2012 @ 05:16 GMT
Most accurate: the ABSOLUTE theory of me is
A DRAFT VERY PARTICULAR AND DETAILS FOR THE PROPOSAL OF T.O.E
A theory is always open and no boundaries with the criteria:
Correct for everything and enough for all problems
Author Hoang cao Hai replied on Oct. 2, 2012 @ 06:36 GMT
Expect teachers sympathetic to the my English as "computer automatically type".
Really grateful Teacher Eckard Blumschein was pointed out errors "very silly" about the speed of light in air is 90.000m / s (was misspelled 90.000km / s), although this will not change the problem stated in my essay.
Author Hoang cao Hai replied on Oct. 13, 2012 @ 05:38 GMT
HERE ARE HAVING T.O.E : THE THEORY OF EVERYTHING
That is the answer to all questions
Is the base and the nature of any foundations or all platforms
It is unique so can not deny
Is the Absolutely so have not error
Author Hoang cao Hai wrote on Sep. 8, 2012 @ 03:34 GMT
The ABSOLUTE THEORY and an explanation of the nature of the Mass
The nature of each thing or each matter always only one!
That's the truth. And the Truth is certainly is the Absolute .
Can not have and does not require two principles for any one result.
Although there are many things and facts are identical, but can not be two the same things to co-exist in the same location on the space and in the same time of the time.
That mean is in space and in time the everything are always worth Absolutely.
For example: The Nature of the Mass
Be identified due to the change by the purely feel and rely on
the determination by our measurement equipment.
Must be the impact to get this changes,and the absolutely is only
one the mainly reason,that of course is the impact of a type of
the force.
So: the absolutely nature or the definition of mass would be:
Expression due the impact of force on to the material.
Author Hoang cao Hai replied on Sep. 8, 2012 @ 06:21 GMT
This theory just need based on the phenomenon or the matter occurs is able to identify the principles and nature of the phenomenon and that matter.
So can take out result of the problem is specific and detailed.
As well as can make the most reasonable orientation effects to get the desired results.
Sergey G Fedosin wrote on Sep. 17, 2012 @ 17:50 GMT
Dear Hoang,
I agree with you than gravitation may have attractive and pushing action. For example pushing action is possible because of Gravitational torsion field when the spin torsion field counteracts to strong gravitation of hadrons in atomic nucleus.
Sergey Fedosin
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Author Hoang cao Hai replied on Sep. 19, 2012 @ 01:32 GMT
Dear Sergey
I am very happy with your comments, as well as agree with "The current paradigm of physical knowledge is obsolete and is subject to inevitable replacement
based on the transition to substantial theoretical models of a deeper level" of you.
Wish you always happy!
Author Hoang cao Hai wrote on Sep. 19, 2012 @ 13:39 GMT
Dear
Very interesting to see your essay.
Perhaps all of us are convinced that: the choice of yourself is right!That of course is reasonable.
So may be we should work together to let's the consider clearly defined for the basis foundations theoretical as the most challenging with intellectual of all of us.
Why we do not try to start with a real challenge is very close and are the focus of interest of the human science: it is a matter of mass and grain Higg boson of the standard model.
Knowledge and belief reasoning of you will to express an opinion on this matter:
You have think that: the Mass is the expression of the impact force to material - so no impact force, we do not feel the Higg boson - similar to the case of no weight outside the Earth's atmosphere.
Does there need to be a particle with mass for everything have volume? If so, then why the mass of everything change when moving from the Earth to the Moon? Higg boson is lighter by the Moon's gravity is weaker than of Earth?
The LHC particle accelerator used to "Smashed" until "Ejected" Higg boson, but why only when the "Smashed" can see it,and when off then not see it ?
Can be "locked" Higg particles? so when "released" if we do not force to it by any the Force, how to know that it is "out" or not?
You are should be boldly to give a definition of weight that you think is right for us to enjoy, or oppose my opinion.
Because in the process of research, the value of "failure" or "success" is the similar with science. The purpose of a correct theory be must is without any a wrong point ?
Glad to see from you comments soon,because still have too many of the same problems.
Regard !
Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY
August 23, 2012 - 11:51 GMT on this essay contest.
Anonymous replied on Sep. 19, 2012 @ 14:40 GMT
hoang cao hai,
I would be a little more convinced that perhaps you had read the essays of others if you had not copied and pasted the same post to so many different ones. I will read yours but all on here desire true feedback for our own too. Please pick out questions or what aspects of other essays you agree or disagree with in order to help us.
Thanks
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Georgina Parry replied on Sep. 19, 2012 @ 20:41 GMT
Dear Hoang Cao Hai,
There are too many language errors in your paper for it to be seriously considered for publication in the Scientific American journal.That is not said to be unkind but so that you are made aware of that shortcoming. Having said that I did enjoy it. My overall impression is of a powerful curiosity and optimism that answers to many questions can be found. It contains so many big questions that are often not given deep consideration. Having really good questions is important.
What do we mean by 'the Universe' is one very good one. I asked George Ellis what is 'the real universe' in his opinion because he used that term. I did look up the word 'Universe' in a dictionary a while back and it said words to the effect 'everything that exists'. That definition is itself a problem because; is that everything that is known to exist, or everything that can be known to exist, or is it all that can be imagined to exist too (but at the same time excluding imaginary things)? It seems to me the visible universe is only the fabricated images of former things excluding that which exists.
Max Tegmark talked about a number of different kinds of multiverse at the FQXi SETTING TIME ARIGHT conference, the video is available on you tube. One kind are those that are separate because the distance is too large for information to ever reach between them. So are they really still the same universe or different universes? Also IMO each individual observer will experience their own version of the universe fabricated from the data they have received out of a vast "multiverse of possibilities" that might have been observed.
Part of the problem is the way in which the ideas are communicated, as there isn't a common understanding of many terms commonly used. Which is another important point made at the FQXi SETTING TIME ARIGHT conference.
I hope you get lots of interested readers and good feedback. Kind regards Georgina.
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Author Hoang cao Hai replied on Sep. 20, 2012 @ 05:23 GMT
Dear Anonymous and Georgina Parry
Sorry for not making you happy, even though I've spent a lot of time to read and evaluate for other essays.
The reason for I have no more comment because every essay gives a perspective and different measures, even though we share the same Topics.
That so,it is very difficult to get consensus on the method of settlement, which is the same as the formula E = mc2 will is how to calculate the energy for a potato or a bread?
So that, I have chosen measures is to create a specific purpose for all of us, the hope is to focus everyone's ability to gradually solve the problems really exist.
To collect opinions of Anonymous, I will do as comments of you.
Probably is not much more than a general orientation:
Be for these results have practical value for our own lives!
The purpose for what we are doing is to turn dreams into reality or make the obvious becomes unreal and vague as dreams?
I appreciate the essay by Georgina Parry,because have not using the mathematical formula that contains many unknowns to solve the problem is still a mystery because of insufficient grounds or basis to confirmed.
As example : 1x + 2y = 3z
that is : 1 of the unknown + 2 of not seen = 3 things can not be understood as anything?
or formula : A = bc / d
while the A, b, c and d are still the concept is not clear and are not sure is true or not?
Hope this idea does not make you uncomfortable.
Kind regards ! Hải.Caohoàng
Ted Erikson wrote on Sep. 19, 2012 @ 15:20 GMT
Dear hoang cao hai:
Your essay, in words identify the real problems of physics...and philosophy. You are correct to begin the study at ground zero, i.e. a "beginning". But, to logically assess any start, there must be evidence which obviously is impossible. (we weren't there).
So, it seems that the best one can do is assume the success of quantum theory, i.e. "probabilities" is promising. In my End Notes, I show a tentative and simple model that identifies probabilities as being related to "space dimensions", i.e. 0-D,1-D,2-D,and 3-D and recently discovered that a "frequency" correction causes extrapolations to 3... or 4-D!! Time is definitely involved with probability. Infinite time can cause anything to occur..
Perhaps those more intelligent and better mathematically inclined can make better sense of it, or destroy the approach. Either way.I would be pleased.
To Seek Unknown Shores
http://fqxi.org/community/forum/topic/1409
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Author Hoang cao Hai replied on Sep. 20, 2012 @ 11:23 GMT
Dear Ted Erikson - Many thanks for your comment.
Unfortunately, your essay is too heavy for the ability to open my computer translation.
The main ideas that you have mentioned about quantum theory and mathematics, I think so :
1. Quantum theory is the most reasonable (compared to general relativity and string theory) I also use the basic principles of quantum for the my absolute theory. But it still is not enough,because there is quite many things are called "darkness" in our science.
2. We have lost too much time and effort to renovate and fix it but still no satisfactory results.Even, it seems that the efforts that have made quantum theory more trouble as today.
Because apparently Mathematics is just a tool of science, is used to determine the specific results for a particular problem from the specific grounds established by the imagination and logical reasoning to identify issues beyond our perception.
You can see that: the mathematical equations too far removed from reality?
The formula E = mc2 will is how to calculate the energy for a potato or a bread?
We determine the particle "subatomic" with the ability of technology, that technological capabilities are limited and do not have the norms for ability, and things that are beyond the capabilities of technology will surely forever are "dark"?
With current technology, we identified "subatomic", technology development than we will discover even smaller particles, and so on until (approximately few trillion century later) will probably find nuts "can not be divided" as the definition of "atoms" be had from a long time ago.
Whether your choice is how, if you really believe,please fight to the end for it.
I am very lucky when to know you.Kind regards ! Hải.Caohoàng
Wilhelmus de Wilde de Wilde wrote on Sep. 19, 2012 @ 15:33 GMT
Hi Hoang, Your opening "The universe is inside" is the base of my own reasoning. I saw that you sent the same messages on several essay's, you are right to ask attention for your work. So do I , please read (and rate, I will rate yours )
"THE CONSCIOUSNESS CONNECTION"
Good luck
Wilhelmus
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Author Hoang cao Hai replied on Sep. 20, 2012 @ 11:53 GMT
Dear Wilhelmus de Wilde
“Our consciousness emerged from a certain order of particles. Our experience of time
emerges from our memory .We experience our order as unique and solely created for the
"I" which is the centre of our consciousness. The “I” has emerged from the experiences
memorized.”also similar to the view in the my absolute theory about "matter and consciousness"
Unfortunately, in this topic I'm writing about the universe.
I hope to be regularly exchanged with Uncle on this field more.
I rated 10 points for this essay.
Sergey G Fedosin wrote on Sep. 19, 2012 @ 19:20 GMT
Dear Hoang,
Sergey Fedosin
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Author Hoang cao Hai replied on Sep. 29, 2012 @ 17:14 GMT
Dear Sergey
Unfortunately, the way your answer depend on CERN and wait for the recognition of humanity.
Kind Regards.
James T. Dwyer wrote on Sep. 19, 2012 @ 20:10 GMT
Dear Caohoàng Hai,
I read your bio. info. - by coincidence, in 1970 I was in the U.S. Army stationed near the village of Dong ba Thin in a helicopter repair shop. I mostly kept the fuel supply, often driving tanker trucks to the nearby fuel depot near the airport on the Cam Ranh peninsula, usually stopping by the beautiful white beaches there to swim in the South China Sea. I also liked...
view entire post
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Author Hoang cao Hai replied on Sep. 22, 2012 @ 01:15 GMT
Many thanks uncle James. Unfortunately,uncle not involved in this topic.
James T. Dwyer replied on Sep. 22, 2012 @ 02:21 GMT
Unfortunately, rude nephew rejects sincere attempt to communicate. Perhaps you should have read my essay to assess my 'involvement' before posting your solicitous comment.
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Author Hoang cao Hai replied on Sep. 22, 2012 @ 08:08 GMT
Very sorry uncle James, was a silly bug too.
In view of her: dark matter as well as the gravity force,is the call comes from our subjective perception for "things and phenomena" of Nature.
That is, the way to call it, we have created a concept to accept, and inadvertently acknowledged that is the nature of things or phenomena.
To be able to assess the nature of dark matter and gravity in theory Absolutely,i need to have specific information about objects or events that make calling to that.
The information my get of the two problems mentioned above is very vague, so do not dare to participate more in the assessment of this specific areas.
Or rather I see two issues in an other way even is completely different, so if I take join in it, will have to rename for both that problem.
Probably is a bit more complicated than the scope of an essay contest.
The Absolutely theory of me is built completely independent, so defined or determined also completely different with all of current theory.
Absolutely theory without the gravity or dark matter to solve problems "matter or force".
So is repeat "sorry" with uncle, hoping uncle did not therefore then stop constantly, we're sure there are many other topics in the vast ocean of knowledge, okay ? Uncle James.
Anon wrote on Sep. 19, 2012 @ 22:01 GMT
Good Luck with that Dear "fill in the blank" stuff ... it earned you a 1 rating from me.
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Author Hoang cao Hai replied on Sep. 20, 2012 @ 03:06 GMT
Dear Anon
That is "..to used.."
"Stuff to used it earned you a 1 rating from me". Ok ?
Do you think that : if does not impact, there will be no changes ?
Regards.
Vladimir F. Tamari wrote on Sep. 21, 2012 @ 02:42 GMT
Dear Hoang Cao Hai
I read your essay. As Georgina said it has many language mistakes, but I could understand your meaning. But the important thing is that I enjoyed the thoughts you expressed concerning the need to find out the Truth inside us. It is almost a religious quest. Humanity has advanced greatly by adopting the scientific method of doubt and experiment. It has brought us some physical comfort because of the advances of technology, medicine, etc. But not a sense of peace of mind or inner harmony. Your questioning the Big Bang reflects this unease. Unfortunately science is based on an accumulation of experimental facts, and we have to accept them, and reinterpret them if necessary. I wish God had given an Instruction, Maintenance and Repair Manual to humanity...but there is no such thing. We discover it in various ways. Science is one of these ways.
I wish you the best in your thoughts and dreams for inner and outer peace and understanding.
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Author Hoang cao Hai replied on Sep. 22, 2012 @ 01:20 GMT
Author Hoang cao Hai replied on Sep. 22, 2012 @ 16:43 GMT
With a huge number of problems that you want to adjust.
The my theory is very simple,but can not protest.
The standard model has a lot of loopholes, the most basic is the determination of nuts:
Although atoms was established as the "smallest and can not be divided" from long time ago, but when we try to measure to determine for it,then has created the opportunity for "subatomic" was born thanks the development of technology.
That is, when more technology grows, we will find many kinds of particles smaller than "sub-atomic", but it certainly is not smaller particles "smallest and can not be divided" - (news from CERN: "identified seeds is like Higg boson, but lighter?") - if speed collisions to the "c squared" do not know "protrude" how many kinds of particles? seed would be "ultra low Higg" and will arise ....
Also, in my absolute theory is no boundary between religion and science.
Hopefully there will soon be publicized conditions to consult of Uncle.
James T. Dwyer wrote on Sep. 22, 2012 @ 10:06 GMT
Caohoàng Hai thân men,
Have you tried translate.google.com? It can translate English into Vietnamese and Vietnamese into English. Maybe this would help you.
Unfortunately, some Vietnamese letters cannot be copied to comments...
Jim
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Author Hoang cao Hai replied on Sep. 22, 2012 @ 17:01 GMT
Thanks Uncle Jim
I still translate via google, and then edit the dictionary, so the result is not stable.fortunately is still be accepted.
Do you know many of language Vietnamese ?
James T. Dwyer replied on Sep. 22, 2012 @ 19:01 GMT
Caohoàng Hai thân men,
I'm afraid I've forgotten almost anything I learned >40 years ago, which was then an unfortunate mix of Vietnamese, French and English... I recall mostly that there were many phonemic sounds that I could not seem to make!
As I said before, I'm very interested in gravitation. If I can say, your understanding of gravity is not quite in agreement with established theory - mine somewhat also. You said:
"The earth and the planets in our solar system orbit is also due to the gravitational attraction from the sun? That is sucked into the sun not only our planet but also to those planets near and far the sun than Earth, so what makes us and the planet do not sticking to the sun?"
In my view, each massive object produces a diminishing gradient field of kinetic energy in external spacetime, by locally contracting spacetime. As each planet has energy directed towards it, it is the interaction of two opposingly directed fields that produces the apparent net effect of attraction.
I think planets do not fall into the Sun because their own directed energy fields (slightly) oppose the Sun's (locally diminished) field, which is directed to it. Small bodies, such as ourselves, fall to Earth because our own opposingly directed gravitational field is so small. We do not fall to the Sun because here the Earth's directed force is locally stronger than the force directed to the Sun.
Also, I do not find any definition for the term "Acsimet" (thrust), but I cannot agree that there is a repulsive force related to gravitation. But that would be another complex discussion...
I hope this little bit is understandable - I know I unfortunately use complex language. I appreciate your consideration!
Best wishes,
Jim
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Author Hoang cao Hai wrote on Sep. 23, 2012 @ 17:06 GMT
Uncle Jim.
Thrust (of liquid) Acsimet (Archimedes) is the Famous story as "Eureka!" by Archimedes (Greek) found when he see insurgent water level in the tub!
Archimedes theorem of thrust is stated as follows:
Any object submerged in whole or in part, in a fluid will be a thrust equivalent, but in the opposite direction, the weight of the fluid occupied - Excerpt from vi.wikipedia. org
According to my research on gravity: the naming and application the "Gravity" has created a vague and not specific to the real nature of this force.
We still naming for the force was "to stick" our with the Earth's is the gravity, and that it is an "attraction" due to the earth rotation (generated magnetic), while accepting that cause of the "space bending" of the Earth as well as the Sun and other planets.
According to my absolute principle: each problem by only one cause.
The conclusion will be confirmed for - the "gravity" above - is:
1.The sun is the only source of motivation to decide all the activity in the Solar System.So its nature is to create resources "pushed out" and not "attract in".
2.The earth and other planets (as well as other objects) in the solar system will be in force "Pressed down" or "compression in" and not be able to "self-attracting".
Therefore, if the "gravity" is applied, including the Sun and the planets in the solar system is not correct.
If we try to apply it may be to accept the repulsive force is reasonable.
Interpretation issues in my essay as a "suggestive" (according direction of FQXi) plus the English language "automatically" will annoy you.
Suggest you sympathize and asked to me more detailed
Dear Uncle ! Have you accidental have one granddaughter is "Gravity and attraction" ?
Russ Otter wrote on Sep. 23, 2012 @ 18:13 GMT
Dear Hai.Caohoang,
Based on your query to me at my Essay site: Russ Otter
Congrats on your submission... I just finished reading it, and found it an open-book as to the questions of life! Such as your question: Does there need to be a particle with mass for everything to have volume? That is a question, that is larger than our understanding currently, and perhaps infinitely. That is the nature of many of your questions, they are part of an infinite landscape, that will forever push us forward to seek answers.
Many questions in life are too large for our knowledge, as finite sentient beings, contained in the indefinable scope of the infinite.
Your final paragraph, discussed god. That is also a question to big a question for finite beings to honestly answer. Therefore, I must be an agnostic on that issue. That is true of much regarding physics, and philosophy. They are married, regardless of choice.
I would be happy to work with you, as you mentioned, as two heads are better than one..., but I would first have you read my Essay's @ www.otterthink.wordpress.com , in order that you understand my answers to many of your questions regarding physics and life.
Cheers buddy, and all the best in your pursuits to learn. Science is the truth still unraveled,
Russ
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Author Hoang cao Hai replied on Sep. 24, 2012 @ 05:28 GMT
Dear Russ W Otter
If we inference on the overall scale : there will be no beginning or end.
But when we look at a limited scale on a particular object or event, such as: a house, a person or a state as well as a planet or a planetary system, there must always be a beginning and end of that limit.
It seems the term "universe" are using is called of "visible universe" or more specifically our solar system.
Unfortunately, we still do not have a specific definition for "universe" which we want to find out.
Therefore, the discussion often arises unnecessary disagreements like this.
It would be more interesting if you written more than in the essay.
It looks like you in favor of Theology and Religion.
In view of and the absolutely theory of me, there is no boundary between Science and Theology or Religion and Science.
Would be very happy if I can help you something and vice versa.
James T. Dwyer wrote on Sep. 23, 2012 @ 18:16 GMT
Caohoàng Hai thân men,
If I understand, then, you propose that the gravitational effects causing planets to orbit the Sun are not the same universal gravitation that causes moons to orbit planets? Since the law of universal gravitation can be very usefully applied to both realms, it seems to me unlikely that the two nearly identical effects would be caused by different processes.
As I tried to explain, I personally view gravitation as a 'push' from space contracted by a condensed mass. The appearance of attraction as described by Newton is the combined result of two opposingly directed 'pushing' fields of space.
I do not subscribe to the existence of a physical force of attraction, only the approximation of a net effect of attraction produced by the interaction of external fields produced by massive objects. That aside, I really don't know how many accidental granddaughters I have - I do think the two I know are quite attractive, but that's a different matter...
Jim
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Author Hoang cao Hai replied on Sep. 24, 2012 @ 06:38 GMT
Uncle Jim
It's about "the language barrier" fortunately we still can understand each other intentionally.
The story about her niece "attractive" to for a more pleasant atmosphere, if accidental the girl as well love the science as Uncle and me, we will have more stories to tell.
Rude nephew.
James T. Dwyer replied on Sep. 24, 2012 @ 10:59 GMT
Caohoàng Hai thân men,
Yes! I certainly don't know where she got her love of math...
Thanks, Grandpa Jim
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Christian Corda wrote on Sep. 24, 2012 @ 09:35 GMT
Dear hoang cao hai,
I enjoyed in reading your interesting Essay. I am going to give you an high score.
Cheers,
Ch.
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Peter Jackson wrote on Sep. 24, 2012 @ 19:37 GMT
Hai
I was pleased to find two very important propositions consistent with my own. I have reworded them in good English as I believe you intended and as I've also found;
"Perhaps we have to find how to define limits of spaces adjacent to each other in the universe, to determine the nature of the formation and operational features of a region of space, before finding an accurate assumption for a general theory about the beginning of the universe or a certain bounded region in space."
"Answers are always in front of our eyes or within all of us, the effects and specific details of that truth show through every day in front of us."
I hope you may read, find commonality, and and score my own essay, which I hope you find brings real physical meaning and proof to the above.
Best wishes
Peter
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Author Hoang cao Hai replied on Sep. 25, 2012 @ 02:56 GMT
Dear Uncle Peter Jackson
"Our physics needs ontology, philosophy needs nature.
Too weak those two alone, far greater wholes than sums of parts.
The road forks in the mist, we must decide, anon, not later.
So is the soul of every man just built on his assumption?
Can we take arms and...
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Peter Jackson replied on Oct. 1, 2012 @ 10:41 GMT
Hai
I think I agree with those parts of your comments that I think I understand.
You will see from my recent discussions with Pentcho on my essay blog that I am indeed stressing the absolute importance of the brain as the receptor and analyser. But I identify a very important aspect of this, that when the body is in motion, the wavelength of em signals approaching a lens is then different to the wavelength AFTER interaction, and when approaching the brain.
This is discussed in my essay last year along with the LHC's very effective role as a 'speedometer' for detecting speed with respect to the vacuum, which is in theory not possible in SRT!! Then as you say, 'weight' as well as speed are only relative to the local background field.
I hope you can work at your English, or get a better translation programme, to improve communication as your physical intuition is probably good, but very difficult to tell!!
Best wishes
Peter
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Author Hoang cao Hai replied on Oct. 10, 2012 @ 01:21 GMT
Maybe I should married a wife is British or American?
Jayakar Johnson Joseph wrote on Sep. 27, 2012 @ 10:17 GMT
Dear Hoang Cao Hai,
At this point of our assumptions on physical universe, your theory on ‘right’ over ‘wrong’ is right, whereas the ‘wrong’ is not wrong in every point in these assumptions.
With best wishes
Jayakar
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Member Benjamin F. Dribus wrote on Sep. 28, 2012 @ 00:15 GMT
Dear Hoang,
As promised, I have just read your essay! Let me make a few remarks.
1. On the topic of "multiple universes," this can mean several different things. Sometimes "a universe" is just a convenient way of describing something that is self-contained in a certain sense. For instance, the "causal universes" in my essay are not to be confused with "The Universe," of which I...
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Author Hoang cao Hai replied on Sep. 29, 2012 @ 17:34 GMT
Dear Benjamin F. Dribus
Paul Reed wrote on Sep. 28, 2012 @ 08:50 GMT
As per your comment on my blog.
Your start point is correct, ie we, and indeed all sentient organisms, are part of reality. This is so obvious, and fundamental, and yet is often ignored; so in the assumptions of many theories, humans attain some form of superiority &/or are deemed to be separate from reality.
But, you then confuse the next step. The above means that, from a scientific perspective, it must be accepted that we are trapped in an existential loop, and therefore can only know what is within that (ie the “universe” as is potentially knowable). Know indicating information which has validity. So the question is: given that existential confine, what constitutes “right or wrong” (or valid/not valid, objective/subjective)? And the answer is: that which is either validated direct experience, or that which can be properly inferred therefrom. Because reality exists independently of the sensory processes which enable awareness of it. Physically existent phenomena are received by these sensory detection systems. But, in order to extricate objective knowledge of what was received, and hence objective knowledge of what caused that, we have to overcome a variety of practical problems in the sensory processes (one of which is potentially, thinking!).
In simple language, the aim can only be to depict and explain what is manifest. That does not necessarily occur perfectly, so we have to deploy methods to counteract that. But that does not include invoking assertions which are not based on some form of validated experience.
Paul
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Author Hoang cao Hai replied on Sep. 29, 2012 @ 18:01 GMT
Misunderstanding may be due to "dissent" or simply than likely "English".
My purpose is very specific:
A the right theory of is be must always "right" for all related issues, have been verified in practice or experimental,have not any point is "wrong" and must be able to reject all criticism are given.
Regards.
Paul Reed replied on Sep. 30, 2012 @ 07:30 GMT
Agreed.
But the question is: what can constitute 'right'. And the answer to that lies in understanding how what we experience can occur, and how we experience it.
Paul
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Sridattadev wrote on Sep. 28, 2012 @ 14:21 GMT
Dear Hai,
Thank you for reading the essay
Conscience is the cosmological constant and resonating it with the absolute truth with in us in your essay.
What we believe in does not matter, what matters is the belief itself.
Love,
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Frederico Pfrimer wrote on Sep. 28, 2012 @ 14:27 GMT
Dear Hoang Cao Hai,
Very interesting essay. After you mentioned, I could a point where you are making reference to precisely the notion I call the final theory of physics:
“ there must be a reason common to all theories, because it is the Truth.
The truth always existed and that is reason, also is the goal that all our assumptions are hope. It probably would be a "assume for nothing assume" or "the foundation of every platform" or is "the most basic" And of course that would be true is the correct theory for physics and cosmology as well as all other branches of our science. “
These notions are precisely the characteristics of the final theory of physics. The idea of “assume for nothing assume” is discussed in my essay when I say that the final theory define what are physical assumptions. Basically all the assumptions that are added to the final theory are physical assumptions, therefore, the final theory makes no physical assumption!
Best regards
Frederico
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Author Hoang cao Hai replied on Oct. 9, 2012 @ 08:58 GMT
I hope so : we are will become friends.
Send to : hoangcao_hai@yahoo.com
Joe Fisher wrote on Oct. 1, 2012 @ 15:46 GMT
Dear Hoàng cào Hai,
Thank you for reading my essay and for posting such a perspicacious comment about it I regret that I cannot answer any of your probing questions. As far as I am concerned, there is only one real Universe that perpetually continues existing in one real dimension at one real here for one real duration of now once. One real Universe can only have one real condition although that one physical condition must have three aspects. The one real eternal Universe always has differing amounts of matter, space and light in its composition. Matter can move through light, but it cannot move through matter or through space. Space can surround matter, but it cannot surround space or light. Light can fill space, but it cannot penetrate light or matter.
I could be wrong.
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Author Hoang cao Hai replied on Oct. 9, 2012 @ 08:40 GMT
Dear Joe Fisher
That is your opinion, if the application in fact is true, it will be recognized as "true".
Are you okay ?
Hou Ying Yau wrote on Oct. 1, 2012 @ 18:15 GMT
Dear Hoang,
I agree with what you said in your essay: "The true nature of science is to clarify perhaps strange things,and not to create even more exotic things...". The ideas developed in physics nowadays are mainly based on abstract mathematical formulation. It cannot be understood easliy.
In related to gravitational force, repulsion is not something that has been observed. However, it is only my thought, if the clock is running backward like rewinding a film, the attraction will look like repulsion. It may be the nature of time that only allow us to observe attraction.
I am touched when I entered this contest with a lot of non-professional contributing their ideas like yours. My educational background is also not in physcis. I spent the last 15 years self studying. At the beginning, I found all the absract ideas so difficult to understand. I finally get over some of the hurdles and can alomst get a paper published in a peer reviewed journal. The essay Is there really no reality underneath quantum theory outlines one of my ideas. The idea is very simple (without abstract assumption) but I am able to develop the technical mathematical proof to support it. I hope you will find it interesting. I hope one day that non-professional like us can make some major contributions to the community.
Sincerely,
Hou Yau
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Author Hoang cao Hai replied on Oct. 9, 2012 @ 08:35 GMT
Dear Hou Yau
In fact history has shown: the vast majority of the changes are important are due to those who are considered "unprofessional".
Maybe it's because : those who are the "professional" too believe in what is there, so did not see the irrationality of them.
Regards.
Author Hoang cao Hai replied on Oct. 9, 2012 @ 09:09 GMT
Be very careful when using mathematics, it is actually a "magic wand" but it can not think for you, if you "accidentally" steer the wrong way, then it will "squared" to your consequences.
Sergey G Fedosin wrote on Oct. 2, 2012 @ 10:17 GMT
After studying about 250 essays in this contest, I realize now, how can I assess the level of each submitted work. Accordingly, I rated some essays, including yours.
Cood luck.
Sergey Fedosin
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Author Hoang cao Hai replied on Oct. 9, 2012 @ 08:53 GMT
Even if you do not do that, you also was the most lovely in the competition.
Domenico Oricchio wrote on Oct. 2, 2012 @ 12:16 GMT
Dear Hoang chao Hai,
Thank you for the thread on my blog: I see that you consider important to read my essay.
I read, some weeks ago, your essay and I vote ever quickly the essay; so that there is not a exchange between authors, and the vote are right (there is not sympathy, or dislike): I must say the truth, I give a little advantage to old FQXi bloggers.
(1) I think that the multiverse is pratically realized in the Universe: each quantum path is a real path in the Universe wave function evolution, and the measure cause the collapse of the wave function (I think that the Universe is multiverse until the collapse of the wave function)
(2,3,4) I think that the shrinking-collapse happen ever in the Universe zones; I think that the Dirac sea is the expansion in the Universe (negative curvature in the Big Bang) and the mass particles is the positive curvature; so that some bubbles of different curvature in the Universe can be happen (like a choppy sea) and if happen positive zone on some galaxies we measure dark energy, and in some negative zone we have galaxies contraction
Saluti
Domenico
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Author Hoang cao Hai replied on Oct. 9, 2012 @ 08:49 GMT
Many thank Domenico
Sory for a great pity:
Maybe because I'm a "rookie" in FQXi so I did not even have "the author's assessment code" to be able to contribute to the score for you.
I am only wish you successful and good luck.
Author Hoang cao Hai wrote on Oct. 3, 2012 @ 02:23 GMT
DEAR TO ALL THE AUTHORS AND READERS WAS INTEREST.
Today, I am finished reading all of the essays in this topic.
First of all, thanks again to FQXi and the donors has facilitated for us to have the opportunity get contribute to science.
Next, would like to express to other author by the thanks for the comments that you have contributed to give me, and sincere apologies to those of you that I do not have specific feedback for your essay.The reason that is because:
The placing for issues and measures to solve for the problems of your offer is completely different from mine, so I can not comment when we do not have the same views on one matter, the purpose is to avoid the discussion became conflict of ideologies,it is will not be able to solve the problem which we are interested.
The end, I hope that : we ( who want the human to put their faith in science) will have the same fear: to someday,every people told each other that:
WAIITING FOR SCIENCE HELPS IS VERY LONGTIME,
LET PRAY TO GOD OR A CERTAIN DEITY SOMETIMES EVEN FASTER !
Author Hoang cao Hai replied on Oct. 6, 2012 @ 04:06 GMT
And a great pity:
Maybe because I'm a "rookie" in FQXi so I did not even have "the author's assessment code" to be able to contribute to the score for you.
Sergey G Fedosin wrote on Oct. 4, 2012 @ 06:25 GMT
If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is
$R_1$
and
$N_1$
was the quantity of people which gave you ratings. Then you have
$S_1=R_1 N_1$
of points. After it anyone give you
$dS$
of points so you have
$S_2=S_1+ dS$
of points and
$N_2=N_1+1$
is the common quantity of the people which gave you ratings. At the same time you will have
$S_2=R_2 N_2$
of points. From here, if you want to be R2 > R1 there must be:
$S_2/ N_2>S_1/ N_1$
or
$(S_1+ dS) / (N_1+1) >S_1/ N_1$
or
$dS >S_1/ N_1 =R_1$
In other words if you want to increase rating of anyone you must give him more points
$dS$
then the participant`s rating
$R_1$
was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.
Sergey Fedosin
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Author Hoang cao Hai replied on Oct. 5, 2012 @ 02:46 GMT
Pleased to trust in the wisdom of the judges.
"All decisions of the judges are final and the selection of Winners is at the sole and absolute discretion of FQXi."
I really believe those who have given the Topic :
"Which of Our Basic Physical assumptions are Wrong?"
Hope we do not lose the faith.
Author Hoang cao Hai wrote on Oct. 8, 2012 @ 09:44 GMT
OF "DEFINITION FOR THE MASS OF THE ABSOLUTE THEORY"
First of all, would like to sincerely thank all of the feedback of you.
In addition of the approver, the objections mainly in two ways:
1. You think I got confused between "mass" and "weight".
2. You use the standard model assumptions and arguments are being...
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Author Hoang cao Hai replied on Oct. 8, 2012 @ 09:46 GMT
Author Hoang cao Hai wrote on Oct. 8, 2012 @ 16:32 GMT
A very interesting problem: THE REAL VALUE OF TOE
Through the consultation process in the community of authors, this is the most interesting thing that I noticed.
When given the theoretical opinions for draft proposals of a TOE, I get a comment that was for me to toss and turn a long time before a response.
It is an idea from an our most beloved author, he raised the issue: Do not you think a "Theory of Everything" would be a bit depressing? What would be left to do?
I was really puzzled and found that: perhaps we have the notion that: if have TOE is we will not have to do anything anymore.
With all the confidence of a man who thinks has found it, I would like to share with you: TOE merely a matter of theory, whether it is completely accurate,but until we can achieve some real results, will also need a process that is not easy to overcome.
For example as follows: TOE can give you a formula for the "immortal", but to actually become "immortal" at least you be must need have the necessary materials and appropriate technology capabilities, not to mention the claim for element in the body of your own body.
That mean the real value of TOE is simply: we do not have to spend more time for "grope" to find out the solutions.
Hope to get a lot more interesting comments from you
Author Hoang cao Hai replied on Oct. 8, 2012 @ 16:44 GMT
T.O.E IS ONLY THE FIRST STEP ON THE ENDLESS ROAD THAT WE WANT TO GOING.
Author Hoang cao Hai wrote on Oct. 20, 2012 @ 17:17 GMT
THE ABSOLUTE THEORY OF HOANGCAO HAI TO DETERMINE THE CURRENT THEORY
Including the theory of relativity, string theory,quantum theory and the standard model.
1. Relativity: on the view : of course "relatively" is a very vague concept than "absolute", and therefore the assumptions of the theory of Relativity is not specific and detailed than absolute theory.
So Theory of Relativity was do not able to address the requirements set for a fundamental theory of physics, cosmology in particular and science in general.
2. String theory: although have the right idea but the lack of specific measures and also is a set of concepts and assumptions that lack specific and detailed, so it can not afford to make final conclusion for the need to understand our problems.
So,it is do not afford to become a theory of everything.
3. Quantum theory and the standard model: the more reasonable because the fundamental basis of very detailed,but there are errors in the performance measures.
The main reason for both of the above theories were at a standstill is: trying to measure to quantify the "atoms", ie not accept very specific definitions of "atoms" had from before, it's "smallest and can not be divided further".
This has created the opportunity for trouble arises, through the birth of "subatomic", because diversity and there is no limit of the "subatomic" was to lost of the ability to determined for the most fundamental particles of this two assumptions.
4. The Absolutely theory suggests that: the appropriate for a "final conclusion" that is the Causality and arguments of Reductionist.
hoangcao_hai@yahoo.com
Author Hoang cao Hai wrote on Oct. 20, 2012 @ 17:20 GMT
DETERMINE THE NATURE OF GRAVITY BY THE ABSOLUTE THEORY OF HOÀNGCAO HẢI
The main reason is due to the way named "Gravity" and the "promiscuity" application (do not look at the specific phenomenon) has created a vague concept and made it difficult to determine the true nature of this force, as well as generate more vague concepts such as "dark matter" and "dark energy".
The "Gravity" is a concept that we have to accept, rather than a measure to be able to identify specific as " thrust" or "compression".
By absolute principle is: for each result there is only one cause.
So: the general application as gravitational for interactions between the Sun and the planets in the solar system is completely wrong.
Because: the nature of the sun is "break out" force, ie create " thrust", the Earth and other planets in the solar system of course is under "compression down" from the "thrust out" of the Sun.
Therefore, we will be must the definition specific for any individual cases before to use a name to call it.
Namely: instead called "gravity" as at present, we must separate into two called: "thrust to out" of the sun to create "compression to down" for the Earth and the other planets in the solar system.
That is the superiority of the measure "absolute" in how to solve all our problems.
It is very simple, specific and easy to understand, but can not be denied.
hoangcao_hai@yahoo.com
Author Hoang cao Hai wrote on Oct. 20, 2012 @ 17:24 GMT
HOÀNGCAO HẢI INTRODUCES THE REAL VALUE OF THE ABSOLUTE THEORY OR T.O.E
and ability for the self against criticism or opposition.
The Absolute theory means: Only a general theory for everything - that's the Truth.And if was the Truth is certainly must be Absolute.
Therefore: The true nature of each of one of any things or events always be unique,or with each of one...
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Anonymous wrote on Oct. 23, 2012 @ 01:17 GMT
The Absolute theory of Hoàngcao Hải is actually a theory capable of explaining the true nature of everything,that is the specified and detailed for the basic foundation of all the problems in Physic as well as other disciplines of Science.
Because true is correct : only a general theory for everything - that would be the truth.
And the truth is certainly be must...
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Wilhelmus de Wilde de Wilde wrote on Oct. 23, 2012 @ 14:43 GMT
Good afternoon Hai,
The "absolute" TOE is in my humble opinion a wishfull thinking of us, mortal human beings.
Every theory, even yours changes every moment, because your mental horizon is growing every moment.
The TOE of today is already history, and a new one is ready to take over.
You are right when you say that this absolute TOE is "INFINITE", but also infinitely growing and changing in space and time , every moment of it is just a photograph of the past.
The "absolute" EVERYTHING, which is not meant as atheory is Total Simultaneity, an entity which is undescribable by us, who have only five senses and live in a causal deterministic universe.
The same with the "ultimate TRUTH" it is an ideal that mankind will hunt eternally and perhaps only achieves when he is dead, or in the same situation as when he was "unborn".
think free
Wilhelmus
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Author Hoang cao Hai replied on Dec. 6, 2012 @ 04:24 GMT
Uncle de Wilde
It's a point of obsolete and the fact that: is inefficient and always create trouble.
Think free
Author Hoang cao Hai wrote on Dec. 6, 2012 @ 04:28 GMT
ON THE OCCASION OF THE END OF ESSAY CONTETS
Dear to all of you
To me: FQXi really is a bright spot in the "dim" and "despair" of theoretical physics today.
The selection of these essays is a separate issue of the jury, the result of the contest will determine the value of their and FQXi.
If you are not successful, probably because we do not fit with the view of the jury.
Do not be sad about that, because "when the night ended, the sun will rise," if we are to Truth, will always have the opportunity to be recognized, when humanity was "fed up" ambiguity, illusion and abstract science background current (even more obscure than the Theology)
Again sincerely thank FQXi was for us to have the nice opportunity to : be comfortable to know each other and express their own opinion.
Many thanks to all of you and FQXi.
Author Hoang cao Hai wrote on Dec. 9, 2012 @ 20:09 GMT
Dear to Moderators FQXi
Along with my genuine thanks,is a sincere comments with desire: FQXi will succeed to become a "firm fulcrum" of the Science.
Although very glad to be involved in the competition, but I was really disappointed about "the exchange by points in the community" - as Andrew Mendelsohn above.
That has created a psychological "doubts" about the ability and the real class of the jury of FQXi, as well as organizational measures and evaluation criteria of FQXi in the contest like this.
Maybe FQXi should leave the assessment and grading, as well as determine the outcome of the contest for the Jury of FQXi it would be better.
The comments and reviews in the "community" as well as "public" should only be used for reference and supplement to the decision of the jury will be more appropriate.
The article reached to "final" should have more one plays against the "opposition" together with the analysis and review of the jury will be more convincing.
If not too upset for you,Moderators can tell an explain more clearly : the purpose of FQXi is to look for "new discoveries and findings" or "new created and compositions"? for me and other authors to choice the response form.
With best wishes and hope FQXi really is a "big event" and not fall into the "deadlock" or "tragic" as the general condition of the current scientific theories.
Hải.CaoHoàng
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# When should we use conditional probability in a coin toss question given x number of tails have already been tossed?
I'm having a little trouble understanding when we would use conditional probability in the question below.
I was thinking since we are already given that there are at least 6 tails, we wouldn't need to consider that probability, and only calculate the probability of tossing exactly 1 tail (and hence 1 head) in 6th and 7th tosses, which would give 1/2 since the possible outcomes are TH, HT, TT, HH, and 2/4 outcomes have exactly one tail. Could someone please verify if that approach is right or if we'd need to use conditional probability in this case?
Original question:
Assume that the outcome of either heads or tails is equally likely in coin tosses, and each coin toss event occurs independently. You toss the coin exactly 8 times. Given that at least 6 of those tosses resulted in tails, what is the probability that exactly 7 tosses were tails?
• I think you mean "I was thinking since we are already given that there are at least 6 tails" Oct 29, 2020 at 10:06
• Yes! Thanks for catching that, edited the question
– Jean
Oct 29, 2020 at 10:11
The case you are modelling is that you have tossed 6 tails in a row and now you will toss two more times, and yes then the probability of 7 tails is 1/2.
But the case mentioned in the problem and rightly modelled by @tommik is that someone has tossed 8 coins already (and you cannot see the outcome) and tells you that at least 6 are tails, and then asks you what is the probability that there are exactly 7 tails. Which is modelled as follows:
$$P(\#T=7 | \#T>6) = \frac{P((\#T =7) \& (\#T > 6) )}{P(\#T>6)} = \frac{P(\#T = 7)}{P(\#T>6)}$$
Yeah in retrospect I see that there is slight problem in the way the question is framed, because if you are tossing the coin then it seems to mean that you have seen 6 tails in a row. But I am quite sure what it wants to say is that given such 8 tosses take place and you only have the information that at least 6 are tails then what is the probability of 7 tails.
tossing the coin 8 times, the probability of any single realization is the same for any realization. This because $$\mathbb{P}[H]=\mathbb{P}[T]=\frac{1}{2}$$
Thus it can be wasted and we are interested only in the combinations.
$$\frac{\binom{8}{7}}{\binom{8}{6}+\binom{8}{7}+\binom{8}{8}}=\frac{\binom{8}{1}}{\binom{8}{2}+\binom{8}{1}+\binom{8}{0}}=\frac{8}{28+8+1}=\frac{8}{37}$$
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# FRW metric spacelike slice
In Steven Winberg's Cosmology book (p. 6) he says that the spatial part (co-moving part) of the FRW metric can be written as $$\tilde g_{ij} = \delta_{ij}+K \frac{x^i x^j}{1- K \mathbf x ^2}$$ These qoordinates are "quasi-cartesian". I can get back $$\tilde g _{11} = \frac{1}{1-K \mathbf x^2}$$ by using $$x^1=r$$. But, for the other coordinates I can't find the right components using this relations. If I use $$x^2=\theta$$ then I can't get anything. Any help will be highly appreciated.
Your equation gives the components of the spatial metric in cartesian coordinates: i.e. $$x^1 \equiv x$$, $$x^2 \equiv y$$, and $$x^3 \equiv z$$. You cannot use $$x^1=r$$.
If you want to get the metric in a different coordinate system (like spherical coordinates) you just need to be able to write $$x$$, $$y$$, and $$z$$ in terms of the new coordinates (i.e. $$r$$, $$\theta$$, and $$\phi$$) and use the normal way tensor components transform:
$$g'_{ij} = g_{kl} \frac{\partial x^k}{\partial x'^i}\frac{\partial x^l}{\partial x'^j},$$ where the primed coordinates are your new coordinates and there as an implicit sum over indices $$k$$ and $$l$$.
In your case it's probably easier (but totally equivalent) to write out the proper length $$ds^2$$ in terms of $$dx$$, $$dy$$, and $$dz$$ and then write the $$dx$$, $$dy$$, and $$dz$$ in terms of $$dr$$, $$d\theta$$, and $$d\phi$$. Then you'll end up with $$ds^2$$ in terms of $$dr$$, $$d\theta$$, and $$d\phi$$ and can read off $$\tilde{g}_{rr}$$, $$\tilde{g}_{\theta\theta}$$, $$\tilde{g}_{\phi\phi}$$, $$\tilde{g}_{r\theta}$$, etc.
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Electric potential energy
Electric potential energy Common symbol(s): UE SI unit: joule (J) Derivations from other quantities: UE = C · V2 / 2
Electric potential energy, or electrostatic potential energy, is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An object may have electric potential energy by virtue of two key elements: its own electric charge and its relative position to other electrically charged objects.
The term "electric potential energy" is used to describe the potential energy in systems with time-variant electric fields, while the term "electrostatic potential energy" is used to describe the potential energy in systems with time-invariant electric fields.
Definition
The electrostatic potential energy, UE, of one point charge q in the presence of an electric field E is defined as the negative of the work W done by the electrostatic force to bring it from the reference position rref[note 1] to some position r.[1][2]:§25-1[note 2]
$U_\mathrm{E}(\mathbf r) = -W_{r_{\rm ref} \rightarrow r } = -\int_{{r}_{\rm ref}}^r \mathbf{F} \cdot \mathrm{d} \mathbf{s} = q \Phi(\mathbf r)$,
where F is the electrostatic force, ds is the displacement vector and Φ is the electrostatic potential generated by the charges, which is a function of position r.
Units
The SI unit of electric potential energy is the joule (named after the English physicist James Prescott Joule). In the CGS system the erg is the unit of energy, being equal to 10−7 J. Also electronvolts may be used, 1 eV = 1.602×10−19 J.
Examples
One point charge q in the presence of n point charges Qn
A point charge q in the electric field of another charge Q.
Let's start first with one point charge q in the presence of only one point charge Q. The electrostatic potential energy, UE, of one point charge q in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is:
$U_E(r) = k_e\frac{qQ}{r}$,
where $k_e = \frac{1}{4\pi\varepsilon_0}$ is Coulomb's constant, r is the distance between the point charges q & Q, and q & Q are the signed values of the charges (not the modules of the charges. For example, an electron would have a negative value of charge when placed in the formula). The following outline of proof states the derivation from the definition of electric potential energy and Coulomb's law to the given formula.
The electrostatic potential energy, UE, of one point charge q in the presence of n point charges Qn, taking an infinite separation between the charges as the reference position, is:
$U_E(r) = k_e q \sum_{i=1}^n \frac{Q_i}{r_i}$,
where $k_e = \frac{1}{4\pi\varepsilon_0}$ is Coulomb's constant, ri is the distance between the point charges q & Qi, and q & Qi are the signed values of the charges.
Electrostatic potential energy stored in a system of point charges
In a Two point charges system, Electric potential energy UE of q in the potential well created by Q1 is $U_E = \frac{1}{4\pi\varepsilon_0} \frac{q Q_1}{r}$
The electrostatic potential energy UE stored in a system of two charges is equal to the electrostatic potential energy of a charge in the electrostatic potential generated by the other. That is to say, if charge q1 generates an electrostatic potential Φ1, which is a function of position r, then
$U_\mathrm{E} = q_2 \Phi_1(\mathbf r_2).$
Doing the same calculation with respect to the other charge, we obtain
$U_\mathrm{E} = q_1 \Phi_2(\mathbf r_1).$
This can be generalized to say that the electrostatic potential energy UE stored in a system of N charges q1, q2, ..., qN at positions r1, r2, ..., rN respectively, is:
$U_\mathrm{E} = \frac{1}{2}\sum_{i=1}^N q_i \Phi(\mathbf{r}_i)$,
(1)
where, for each i value, Φ(ri) is the electrostatic potential due to all point charges except the one at ri.[note 3]
One point charge
The electrostatic potential energy of a system containing only one point charge is zero, as there are no other sources of electrostatic potential against which an external agent must do work in moving the point charge from infinity to its final location.
Two point charges
Consider bringing a point charge, q, into its final position in the vicinity of a point charge, Q1. The electrostatic potential Φ(r) due to Q1 is
$\Phi(r) = k_e \frac{Q_1}{r}$
Hence we obtain, the electric potential energy of q in the potential of Q1 as
$U_E = \frac{1}{4\pi\varepsilon_0} \frac{q Q_1}{ r_1 }$
where r1is the separation between the two point charges.
Three point charges
Electrostatic potential energy of q due to Q1 and Q2 charge system, :$U_E = q\frac{1}{4 \pi \varepsilon_0} \left(\frac{Q_1}{r_1} + \frac{Q_2}{r_2} \right)$
The electrostatic potential energy of a system of three charges should not be confused with the electrostatic potential energy of Q1 due to two charges Q2 and Q3, because the latter doesn't include the electrostatic potential energy of the system of the two charges Q2 and Q3.
The electrostatic potential energy stored in the system of three charges is:
$U_\mathrm{E} = \frac{1}{4\pi\varepsilon_0} \left( \frac{Q_1 Q_2}{r_{12}} + \frac{Q_1 Q_3}{r_{13}} + \frac{Q_2 Q_3}{r_{23}} \right)$
Energy stored in an electrostatic field distribution
The energy density, or energy per unit volume, $\frac{dU}{dV}$, of the electrostatic field of a continuous charge distribution is:
$u_e = \frac{dU}{dV} = \frac{1}{2} \varepsilon_0 \left|{\mathbf{E}}\right|^2.$
Energy in electronic elements
Some elements in a circuit can convert energy from one form to another. For example, a resistor converts electrical energy to heat, this is known as the Joule effect. A capacitor stores it in its electric field. The total electric potential energy stored in a capacitor is given by
$U_E = \frac{1}{2}QV = \frac{1}{2} CV^2 = \frac{Q^2}{2C}$
where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor.
Notes
1. ^ The reference zero is usually taken to be a state in which the individual point charges are very well separated ("are at infinite separation") and are at rest.
2. ^ Alternatively, it can also be defined as the work W done by the an external force to bring it from the reference position rref to some position r. Nonetheless, both definitions yield the same results.
3. ^ The factor of one half accounts for the 'double counting' of charge pairs. For example, consider the case of just two charges.
References
1. ^ Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 ISBN 0-471-92712-0
2. ^ Halliday, David; Resnick, Robert; Walker, Jearl (1997). "Electric Potential". Fundamentals of Physics (5th ed.). John Wiley & Sons. ISBN 0-471-10559-7.
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# 9.1.1 Overview
Molecular potential energy surfaces rely on the Born-Oppenheimer separation of nuclear and electronic motion. Minima on such energy surfaces correspond to the classical picture of equilibrium geometries, and transition state structures correspond to first-order saddle points. Both equilibrium and transition-state structures are stationary points for which the energy gradient vanishes. Characterization of such critical points requires consideration of the eigenvalues of the Hessian (second derivative matrix): minimum-energy, equilibrium geometries possess Hessians whose eigenvalues are all positive, whereas transition-state structures are defined by a Hessian with precisely one negative eigenvalue. (The latter is therefore a local maximum along the reaction path between minimum-energy reactant and product structures, but a minimum in all directions perpendicular to this reaction path.
The quality of a geometry optimization algorithm is of major importance; even the fastest integral code in the world will be useless if combined with an inefficient optimization algorithm that requires excessive numbers of steps to converge. Q-Chem incorporates a geometry optimization package (Optimize—see Appendix A) developed by the late Jon Baker over more than ten years.
The key to optimizing a molecular geometry successfully is to proceed from the starting geometry to the final geometry in as few steps as possible. Four factors influence the path and number of steps:
• starting geometry
• optimization algorithm
• quality of the Hessian (and gradient)
• coordinate system
Q-Chem controls the last three of these, but the starting geometry is solely determined by the user, and the closer it is to the converged geometry, the fewer optimization steps will be required. Decisions regarding the optimization algorithm and the coordinate system are generally made by the Optimize package (i.e., internally, within Q-Chem) to maximize the rate of convergence. Although users may override these choices in many cases, this is not generally recommended.
Another consideration when trying to minimize the optimization time concerns the quality of the gradient and Hessian. A higher-quality Hessian (i.e., analytical versus approximate) will in many cases lead to faster convergence, in the sense of requiring fewer optimization steps. However, the construction of an analytical Hessian requires significant computational effort and may outweigh the advantage of fewer optimization cycles. Currently available analytical gradients and Hessians are summarized in Table 9.1.
Features of Q-Chem’s geometry and transition-state optimization capabilities include:
• Cartesian, Z-matrix or internal coordinate systems
• Eigenvector Following (EF) or GDIIS algorithms
• Constrained optimizations
• Equilibrium structure searches
• Transition structure searches
• Hessian-free characterization of stationary points
• Initial Hessian and Hessian update options
• Reaction pathways using intrinsic reaction coordinates (IRC)
• Optimization of minimum-energy crossing points (MECPs) along conical seams
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## 1. Introduction
Small area estimation (SAE) methods have been routinely used to generate poverty, employment and other economic indicators at census county and tract levels in the United States (US). Official data providers, such as the Census Bureau, and Bureau of Labor Statistics in the US approach SAE in stages: 1) proposing an appropriate SAE method, 2) evaluating and validating the proposed method, and 3) deploying the recommended method for specific SAE applications or SAE data releases. Certainly, stages 1) and 2) are often iteratively developed, and an initially proposed method may not proceed to stage 3. Such an approach had produced more than a dozen SAE methods for various small area applications (Rao and Molina 2015). Although the traditional synthetic methods coupled with direct and other indirect estimation methods are still in use, recent development point to model-based methods with or without auxiliary information as most promising in providing reliable and robust SAEs.
Model-based methods, however, can vary widely based on model-specification, auxiliary information selection, and model estimation (e.g., frequentist and Bayesian). In the last few years, there have been at least three applications of model-based SAE methods, all based on the Behavioral Risk Factor Surveillance System (BRFSS) data at the county level. A Bayesian unit-level model is currently used by the Centers for Disease Control and Prevention (CDC) to monitor changes in diabetes prevalence from 2004 onward (Cadwell et al. 2010). Then, a multi-level logistic regression model was developed to estimate chronic obstructive pulmonary disease prevalence (Zhang et al. 2014). This method was later used to produce prevalence estimates for 27 behavioral risk factors at the census tract level for the CDC-Robert Wood Johnson Foundation 500 city project (https://www.cdc.gov/500cities/about.htm). Then, a Bayesian space-time logit model was proposed to examine county level drinking patterns from 2002 to 2012 (Dwyer-Lindgren et al. 2015). An advanced estimation method was later used in various county-based risk factors and health outcome estimates, such as diabetes, cancer, cardiovascular disease and other major mortality patterns over time (Dwyer-Lindgren et al. 2016a; Dwyer-Lindgren et al. 2016b; Mokdad et al. 2017; Roth et al. 2017).
SAE methodological adoptions for major nationwide projects present several challenges, especially when the same data are used. First, by not going through the cycle of methodological development from proposing to validating a method, one risks of applying one application method as one fits for all without proper validation. The only exception in this regard is the multi-level logistic regression method that was later validated (Zhang et al. 2015). However, validating an application can be an endless endeavor; an application is appropriate for one health indicator may not necessarily appropriate for another. Second, when varied SAE methods are applied to the same health outcome at the same geographic level (e.g., county or census tract), it is not clear if one method is more appropriate to be used for the same dataset for all indicators, or different methods should be used for different indicators of the same dataset. When different SAE methods produce inconsistent county prevalence estimates, they would negatively affect decisions from local health agencies to direct limited resources to address purported health deficits. Finally, many auxiliary information used in SAE is themselves estimated by an SAE method with wide variation at the census tract and county levels. Examples include census poverty, household income variables, and many other variables from the American Community Surveys (Beaghen et al. 2012; Huang and Bell 2012). Furthermore, these variables are ever changing from time to time, and how to use auxiliary information and how they should be included in space-time SAE models have been a subject of SAE research (Rao and Molina 2015).
The current study intends to compare the three model-based methods to assess their concordance and inconsistency. A previous SAE study compared synthetic method, spatial data smoothing, and model-based regression analysis, and found that the model-based regression analysis was superior over the other two methods (Jia et al. 2004). Since the synthetic method was averaged over small area demographics, while spatial smoothing is a mechanic moving average, the superiority of the model-based regression analysis is naturally expected. In our study, we compared SAE methods that are all actively and continuously producing SAE products and scholarly publications using the BRFSS data, and they all seemed to be able to produce reliable prevalence estimates at the county level or even smaller geographic units in the US. In addition, earlier articles or recent extensions using BRFSS data at the county level can all be grouped under the three methods. For instance, the logistic regression approach used in estimating county level obesity in Mississippi (Zhang et al. 2011), and a newly proposed BRFSS-SAE method can be grouped under multi-level logistic regression (Pierannunzi et al. 2016). A two-step estimate of diabetes incidence applied essentially Cadwell’s specification when it came to SAE (Barker et al. 2013). Another two-step estimate of undiagnosed diabetes is a computational improvement over the previous method (Dwyer-Lindgren et al. 2016b). We, therefore, chose the three methods as they represent current practices or the state of art of SAE using BRFSS data. Furthermore, since all three methods were applied to diabetes prevalence one way or the other, we chose diabetes as the outcomes for our comparative study.
## 2. Methods
### 2.1. Data
The study area was limited to 3,109 counties in the 48 states and the District of Columbia with a sample size of 455,406 respondents aged ≥ 18 years. The BRFSS 2012, the most recent year that the CDC released detailed county identifiers, was selected for SAEs. Respondents were regarded as having diabetes if they answered “yes” in the question: “Has a doctor, nurse, or other health professional ever told you had diabetes?” Figure 1 presents county level diabetes samples over the study area. It shows that about 28.47% counties (n = 884) did not have any samples in the public use file. Although a large number of counties with missing samples presents challenges to SAE, they also present opportunities to compare model-based estimations for those counties. Note also that a number of counties (N = 48) had selected samples but no diagnosed diabetes. A half of counties had a sample rate (i.e., the number of selected samples divided by the number of population) less than 0.11%. Only around 1% of the counties had a sample rate higher than 1.36%.
Figure 1
Geographic distribution of the number of diagnosed diabetes cases in the Behavioral Risk Factor Surveillance Survey among 3,109 U.S. counties in 2012. The color pattern was categorized by the quartiles of the number of diagnosed diabetes.
Demographic controls include age groups (18–44, 45–64, 65+ years), race (non-Hispanic White, non-Hispanic Black, Hispanic, Hispanic, and others), and sex (male and female), all of them were used in the three methods. A total of 8, 582 respondents who did not report any of the three personal characteristics, living locations (state or county), and the diagnosis of diabetes were removed. Table 1 shows that higher proportions of diagnosed diabetes were observed in elderly aged 65+ (19.97%), males (13.33%), and non-Hispanic Blacks (19.66%), with all the chi-square tests being significant at p-values < 0.0001.
Table 1
Summary table of diabetes in the U.S.
Diagnosed diabetes P-value*
No
N (%)
Yes
N (%)
Age <.0001
18–45 105446 (96.65%) 3650 (3.35%)
45–64 140475 (86.91%) 21159 (13.09%)
65+ 105983 (80.03%) 26440 (19.97%)
Sex <.0001
Male 140905 (86.67%) 21673 (13.33%)
Female 214109 (87.72%) 29962 (12.28%)
Race <.0001
Non-Hispanic White 282189 (88.27%) 37485 (11.73%)
Non-Hispanic Black 29536 (80.34%) 7227 (19.66%)
Hispanic 22269 (86.33%) 3525 (13.67%)
Others 16710 (86.55%) 2596 (13.45%)
* Chi-square test, where 7519 missing values are excluded.
County poverty rate, defined as percent of people living under the 100% of the federal poverty line was included as a county-level auxiliary variable. It was based on the 5-year estimate, according to American Community Survey 2012 (Zhang et al. 2014). The average of the poverty percentage among 3,109 counties is 11.97% (standard deviation [SD] = 5.53).
### 2.2. Model specification
In the model specification process, we made sure that all models were specified as close as possible to the original SAE models. The multilevel logistic regression model, which is labeled Model 1, was specified identical to original model using the original SAS codes provided by the author (Zhang et al. 2014).
(1)
$\mathit{logit}\left[P\left({Y}_{\mathit{isc}}=1\right)\right]=\alpha +{\beta }_{1}{\left(\mathit{age}\right)}_{i}+{\beta }_{2}{\left(\mathit{race}\right)}_{i}+{\beta }_{3}{\left(\mathit{sex}\right)}_{i}+{\beta }_{4}{\left(\mathit{poverty}\right)}_{c}+{\gamma }_{s}+{\tau }_{c},$
where Yisc is a binary outcome variable of having diabetes (1 = yes, 0 = no) for individual i living in state s and county c. On the right hand side: α is an intercept, and (β1, β2, β3, β4) are slopes for fixed effects of age, race, sex, and poverty, respectively. The last two terms are random effects γs for state s and τc for county c.
In the absence of time effect, Model 2 can be specified by dropping time effect from the space-time logistic model in Dwyer-Lindgren et al. (2015), for which the original R codes were also provided:
(2)
$\mathit{logit}\left[P\left({Y}_{\mathit{ic}}=1\right)\right]=\alpha +{\beta }_{1}{\left(\mathit{age}\right)}_{i}+{\beta }_{2}{\left(\mathit{race}\right)}_{i}+{\beta }_{3}{\left(\mathit{sex}\right)}_{i}+{\beta }_{4}{\left(\mathit{poverty}\right)}_{c}+{\tau }_{c}+{f}_{\mathit{spat}}\left(c\right),$
which has a similar model construct to Model 1 without state random effect. In particular, county effect was attributed to a spatial uncorrelated random effect τc and a spatial function fspat (c), which is Markov random fields following an intrinsic conditional autoregressive prior (Kindermann and Snell, 1980). Due to the complexity of estimating two county effects, we applied the integrated Laplace approximation (INLA) to accelerate the model fitting, and improve the possibility of algorithm convergence (Rue et al. 2009).
Model 3 is a Bayesian Poisson model, which assumes that survey data are sampled from the complete population data (Cadwell et al. 2010). It estimates Yijkc as the number of diagnosed diabetes cases at age group i, race j, sex k in county c, which follows a Poisson distribution with a mean of µijkc. Thus, Model 3 is specified:
(3)
$\mathrm{log}\text{\hspace{0.17em}\hspace{0.17em}}\left({\mu }_{\mathit{ijkc}}\right)=\alpha +{\beta }_{1i}+{\beta }_{2j}+{\beta }_{3k}+{\beta }_{4}{\left(\mathit{poverty}\right)}_{c}+{f}_{\mathit{spat}}\left(c\right)+\text{log}\left({n}_{\mathit{ijkc}}\right),$
where β1i, β2j, β3k are for age, race, and sex effects, respectively. The spatial function fspat (c) is still Markov random fields as in Model 2. The last term log (nijkc) is an offset corresponding to the logarithm of the at-risk population index by i, j, k, c. Similar to Model 2, we applied the INLA to estimate unknown parameters in Model 3. Note that the original model proposed by Cadwell et al. (2010) did not include the poverty as a confounding variable. To be consistent with the specifications of the first two models, we opted to include the poverty variable in Model 3.
### 2.3. SAE for diabetes area
To generate county-level diabetes prevalence from Model 1, we obtained estimated coefficients for four fixed effects (age, sex, race and poverty) and 3,157 random effects (48 states plus 3,109 counties). In particular, counties without samples had a county-level random effect (τc) imputed by the average of adjacent counties’ random effects (Zhang et al. 2014). Hence, the probability of a diabetic person is:
(4)
The number of diabetes cases in each county can then be summed up by age, race and sex. After dividing by state-county specific population, we can obtain the SAE of diabetes prevalence straightforwardly:
(5)
To generate county-level diabetes prevalence from Model 2, we first calculated the probability of a person with diagnosed diabetes similar to Eq. (5) by replacing state random effect ${\stackrel{^}{\gamma }}_{s}$ term with the estimated spatial function ${\stackrel{^}{f}}_{\mathit{spat}}\left(c\right)$:
(6)
We then used the probability to calculate the SAE of diabetes prevalence:
(7)
The approach to generating SAEs for Model 3 differs from Models 1 and 2. Note that we defined Nijkc and Yijkc in Model 3 as age-race-sex-county specific at-risk population, and those with diabetes, respectively. Thus, we can derive Zijkc, the number of unobserved people with diagnosed diabetes, indexed by age, race, sex and county straightforwardly. Let Pc be the SAE of diabetes prevalence in county c, then it is the sum of the observed and unobserved, or Yijkc + Zijkc:
(8)
${p}_{c}=E\left({p}_{c}\text{|}Y,n,N\right)=E\left(\frac{{\sum }_{i}{\sum }_{j}{\sum }_{k}\left({Z}_{\mathit{ijkc}}+{Y}_{\mathit{ijkc}}\right)}{{\sum }_{i}{\sum }_{j}{\sum }_{k}{N}_{\mathit{ijkc}}}\text{|}Y,n,N\right)=\frac{{\sum }_{i}{\sum }_{j}{\sum }_{k}\left[E\left({Z}_{\mathit{ijkc}}|Y,n,N\right)+{Y}_{\mathit{ijkc}}\right]}{{\sum }_{i}{\sum }_{j}{\sum }_{k}{N}_{\mathit{ijkc}}},$
where Zijkc|Y, n, N ̴ POI(γijkc), and the parameter γijkc is defined as:
(9)
${\gamma }_{\mathit{ijkc}}=\left(\frac{{\mu }_{\mathit{ijkc}}}{{n}_{\mathit{ijkc}}}\right)×\left({N}_{\mathit{ijkc}}-{n}_{\mathit{ijkc}}\right)=\text{exp}\left(\stackrel{^}{\alpha }+{\stackrel{^}{\beta }}_{1i}+{\stackrel{^}{\beta }}_{2j}+{\stackrel{^}{\beta }}_{3k}+{\stackrel{^}{\beta }}_{4}{\left(\mathit{poverty}\right)}_{c}+{\stackrel{^}{f}}_{\mathit{spat}}\left(c\right)\right)×\left({N}_{\mathit{ijkc}}-{n}_{\mathit{ijkc}}\right).$
Because the parameter γijkc is exactly the expected value of a Poisson distribution, the SAE of diabetes prevalence based on Model 3 is:
(10)
${p}_{c}=\frac{{\sum }_{i}{\sum }_{j}{\sum }_{k}\left[{\gamma }_{\mathit{ijkc}}+{Y}_{\mathit{ijkc}}\right]}{{\sum }_{i}{\sum }_{j}{\sum }_{k}{N}_{\mathit{ijkc}}}\text{.}$
### 2.4. Analysis of concordance and inconsistency
We first used county maps to provide visual descriptions of diabetes estimates by the three methods. The intention here is to see if the three methods provide consistent geographic patterns regardless of specific county prevalence rates. Moreover, we categorized the SAEs into five quintiles (top 20%, upper-middle 20%, middle 20%, lower-middle 20% and bottom 20% of observations) in each model, and calculated Bangdiwala’s B-statistic (also known as Cohen’s Kappa coefficients) to evaluate the proportion of counties in the same quintile categories (Bangdiwala and Shankar 2013). The observer agreement charts were also provided to visualize the proportion of concordance in each quintile between two models.
Model 1 was analyzed by the PROC GLIMMIX procedure in SAS v9.13. Model 2 & 3 was analyzed by the inla package in R software v3.13 (R Core Team). The agreement analysis was accomplished by the PROC FREQ procedure in SAS v9.13. Parameters were determined significance by a type I error of 0.05. In the preliminary analysis, we used both weighted and unweighted samples. Although both provided very close estimates, the average from the unweighted was closer to the national weighted average. The latter result was also reported in a validation study of multilevel logistic SAE (Zhang et al. 2015). For this reason, the unweighted sample was used for the estimations of three models.
## 3. Results
Results from SAEs of diabetes prevalence are shown in Figure 2 in 5 quantiles. In general all three models captured the elevated prevalence in the South; all three models show reduced prevalence in West North Central and some northern Mountain states. Models 1 and 2 generated broadly similar geographic distributions, especially for counties with missing samples (refer to Figure 1). Models 2 and 3 had some similar geographic patterning when sample sizes for diabetes were relatively large (e.g., >24). The scatter plots in Figure 3 reconfirmed those observations that Models 1 and 2 had closer SAEs than those compared to Model 3. The correlation coefficient of SAEs between Model 1 and 2 was 0.85, while it dropped to 0.61 between Models 1 and 3. Likewise, the coefficient between Model 2 and 3 was 0.60. Even Model 1 and 2 estimates are broadly similar, we would not consider them close given the correlation coefficient was less than 0.90.
Figure 2
Comparing three SAEs of diabetes prevalence at the county level using quintiles.
Figure 3
Scatter plots of small area estimates among Models 1, 2 and 3.
The weighted diabetes prevalence from the sample of the current study was 10.3%. Model 2 provides closest average (0.1042) among 3,109 counties (Table 2), while Model 1 had the highest average (0.1152). In terms of spread measures, Model 2 had the highest SD (0.0238), Model 1 resulted in the longest range by 0.1894 (0.0508, 0.2402), and Model 2 resulted in the longest interquartile range by 0.0323 (0.0870, 0.1193). Model 3 had the smallest average, SD, range, and interquartile range. Similar patterns appeared in the 2,225 counties with samples in the BRFSS. Among the other 884 counties without samples, Model 3 still had the smallest average of SAEs, and the SD became only a half compared to the SD of Model 3 in the counties with samples (0.0105 vs. 0.0226).
Table 2
Descriptive statistics of diabetes prevalence estimates from the three SAE models.
Model Mean SD Minimum Q1 Median Q3 Maximum F P-value
All counties (N = 3,109)
1 0.1152 0.0237 0.0508 0.0986 0.1121 0.1282 0.2402 910.22 <.0001
2 0.1042 0.0238 0.0335 0.0870 0.1024 0.1193 0.2171
3 0.0902 0.0220 0.0342 0.0373 0.0855 0.1023 0.1789
Counties with samples in the BRFSS (N = 2,225)
1 0.1146 0.0222 0.0536 0.0994 0.1124 0.1272 0.2210 377.93 <.0001
2 0.1034 0.0233 0.0405 0.0868 0.1024 0.1187 0.1940
3 0.0960 0.0226 0.0351 0.0800 0.0931 0.1097 0.1789
Counties without samples in the BRFSS (N = 884)
1 0.1167 0.0271 0.0510 0.0969 0.1105 0.1311 0.2402 831.05 <.0001
2 0.1063 0.0250 0.0335 0.0875 0.1023 0.1208 0.2171
3 0.0755 0.0105 0.0342 0.0687 0.0736 0.0806 0.1337
Abbreviation: SD = Standard deviation; Q1 = The first quartile; Q3 = The third quartile
† The p-values were calculated from the analysis of variation.
Table 3 shows the post-hoc comparisons of diabetes prevalence estimate, resulting in the largest mean difference in Model 1 by 0.0250 (95% CI = 0.0237, 0.0264) compared to Model 3 among 3,109 counties. For counties with samples, Model 1 still had the largest mean difference compared to Model 3, but the difference decreased to 0.0186 (95% CI = 0.0170, 0.0202). In particular, Model 2 and Model 3 had the least significantly difference by only 0.0073 (95% CI = 0.0057, 0.0089). For counties without samples in the BRFSS, Model 1 still had the largest difference by 0.0413 (95% CI = 0.0388, 0.0438) compared to Model 3, and this difference was over 2-fold of the difference between the two models among counties with samples in the BRFSS. A similar scenario occurs in the comparison between Model 2 and Model 3. Nonetheless, the difference between Model 1 and Model 2 had a tiny change between counties with and without samples in the BRFSS (0.0122 vs. 0.0104). ANOVA results in significant differences in the SAE of diabetes prevalence among the three models, regardless of presence and absence of samples in the BRFSS.
Table 3
Mean difference comparison in the SAE of diabetes prevalence among Models 1, 2 and 3.
Comparison Difference 95% CI
All counties (N = 3,109)
Model 1 vs. Model 2 0.0110 (0.0096, 0.0124)
Model 1 vs. Model 3 0.0250 (0.0237, 0.0264)
Model 2 vs. Model 3 0.0140 (0.0127, 0.0154)
Counties with samples in the BRFSS (N = 2,225)
Model 1 vs. Model 2 0.0112 (0.0096, 0.0128)
Model 1 vs. Model 3 0.0186 (0.0170, 0.0202)
Model 2 vs. Model 3 0.0073 (0.0057, 0.0089)
Counties without samples in the BRFSS (N = 884)
Model 1 vs. Model 2 0.0104 (0.0079, 0.0129)
Model 1 vs. Model 3 0.0413 (0.0388, 0.0438)
Model 2 vs. Model 3 0.0309 (0.0284, 0.0333)
The concordance analysis intends to show similarity in diagonals or closer to diagonals in a contingency table of 5 by 5 quantiles. A greater concordance is shown in both tails than in the middle quantiles. The SAEs among the three models had a smaller proportion of concordance in the second, third, and fourth sections (see in Figure 4 in the 20th percentile to the 80th percentile in terms of agreement and near agreement areas). Figure 4(a) has the highest Bangdiwala’s B-statistic by 0.3800, indicating that 38% of the 3,109 counties had the SAEs of diabetes prevalence in the same quintile sections from Model 1 and 2. However, the proportions decreased to 17.87% from Model 1 and Model 3, and 17.61% from Model 2 and Model 3 (see light blue and white areas in Figure 4(b) and (c)) indicating more inconsistent SAEs between Model 1/2 and Model 3. Since the top and bottom quantiles had little dis-concordance, using each method alone for small area estimate comparisons toward means would not generate considerable inconsistencies as far as counties with greatly above or below the national mean (e.g., 20%) are concerned.
Figure 4
The observer agreement charts of categorized small area estimates among Models 1, 2 and 3.
## 4. Discussion
Currently, various SAE methods are used by federal agencies and research institutions, and it is difficult to gauge their estimation performance. In this paper, we have compared SAEs from thee model-based methods that are all actively producing SAEs at county or smaller area units. Our comparisons focused on spatial patterns or relative measures (e.g., quintiles) generated from each method, rather than point estimates. Overall, the three methods were able to point to the elevated diabetes prevalence observed in southern states. In addition, both top and bottom quintiles categories had highest concordance, which is of less concern, because either top or bottom of quantiles are matter the most in terms of policy making. In other words, when a county is high in diabetes prevalence compared to the national average, it less likely goes wrong if another SAE method is used.
Partly due to data limitation from the public use BRFSS file, we had 884 counties without identifiers in our SAEs. Since spatial Poisson regression in Model 3 depends on samples in spatial units, missing county samples would cause more problems for SAEs. Indeed, separate comparisons, both in map displays and post-hoc analyses, showed substantial discrepancies in counties with missing identifiers, and in general, they were under estimates compared to Model 1 and 2 estimates. Even though requesting unsuppressed BRFSS with all county identifiers is possible through each state (Li and Lin 2014), the problem remains for many small counties not to be sampled in the BRFSS.
Model-based SAE methods often include small area auxiliary information, and it tends to improve model predictability. However, different ways of using small area auxiliary variables present challenges for model comparisons. It also produces inconsistent model parameter estimates in space-time models, as both importance and reliance of auxiliary information change over the time dimension. Since socioeconomic status (SES) relates to many health outcomes, it is expected to be included in model-based SAEs in some form. In the original articles, only Model 1 used poverty, and Model 2 used education level, while model 3 did not use any SES variables. In the current study, we included the poverty variable for all three models to aid comparisons. Even we are not sure if poverty or educational level is more appropriate as a proxy for SES, we are pretty certain that not including poverty in model 3 would lead its SAEs to be far more inconstant from those generated from Models 1 and 2 than we currently observed. Future work should refine census based auxiliary variables for SAE. Studies should also expand to include potential administrative records, such as hospital discharge data, visits to physician clinics to either assist or produce separate SAEs, while guarding confidentiality.
Nationwide SAE of BRFSS at the county level can be calibrated to reflect state level estimates. In the absence of the full geographic sample, and to be true the three SAE model, we opted not to calibrate their estimates. Perhaps, for this reason, the three SAEs provided quite different national averages in diabetes prevalence, with Model 1 being the highest and Model 3 being the lowest. That is also why we placed less emphasis on comparing point estimates from the three methods. In real world practice, it might be preferable to calibrate the sample for each state to ensure each state average from SAEs match the overall state prevalence without SAE (Mohl 1996). Future work should compare point estimates of SAE methods and how sample sizes or population size of small areas would after the confidence intervals. It is especially pertinent when considering that auxiliary information tends to be estimates with smaller population areas having a greater margin of errors, suggesting varied uncertainties when conduct cross regional comparative SAE studies for a particular disease or health outcomes.
To some degree, all three methods considered spatial effects. The multilevel method uses the state-county hierarchy to account for some spatial effects while Models 2 and 3, use Markov random fields to work through geographically connected boundaries among all counties. While Model 1 does not consider local variation, models 2 and 3 were unable to incorporate geographic entities that completely separately or do not have local connectivity to other entities, such as Alaska or Hawaii. For this reason, we were unable to include Alaska and Hawaii due to their geographic separation. In addition, none of the three SAE methods considered spatial clustering. However, we know when data are fairly complete (e.g., births and deaths), model-based estimates would be substantially biased when spatial clustering or spatial association effects were not removed (Lin and Zhang 2012). Future studies should examine the effects of spatial clusters or clustering effects on SAEs, and remedies to reduce potential biases. Some of clustering effects could simply be due to spatial unit mismatches, in which case, we could address them through MAUP methods already developed.
## 5. Conclusion
All three methods were able to display elevated county-level diabetes prevalence in the South. While their point estimates were very highly correlated, the highest coloration was between the multilevel and spatial logistic methods (r = 0.86), suggesting much higher consistency compared to the spatial Poisson regression method. While there are apparent differences in point estimates among the three SAE methods, their top and bottom 20 percent distributions are fairly consistent. Each method outputs would support consistent policy making in terms of top and bottom percent counties for diabetes prevalence.
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# Failure of surjectivity in Hotta-Springer specialization: examples for special unipotents?
Last weekend's workshop on Springer theory and its generalizations at UMass demonstrated how far the subject has expanded over four decades, but the original set-up for the Springer correspondence still raises questions for me.
Without introducing too much notation (which varies a lot in different sources), I'll recall the key result of the early paper here by Hotta and Springer on what they call "specialization". They work over a suitable algebraically closed field with the cohomology of certain subvarieties of a flag variety $\mathcal{B}$ of a (say) simple algebraic group $G$ with Lie algebra $\mathfrak{g}$. If $u \in G$ is a unipotent element, write $\mathcal{B}_u$ for the fixed point set of $u$: this is essentially the set of Borel subgroups containing $u$, which as a variety is determined up to isomorphism by the conjugacy class of $u$. (Some versions instead use nilpotent elements of $\mathfrak{g}$, which is equivalent to this set-up if the characteristic is good.)
Now the obvious inclusion $\mathcal{B}_u \hookrightarrow \mathcal{B}$ induces a reverse map $\varphi$ on cohomology (which can be classical or etale). Here the cohomology typically vanishes in odd degrees, while the top degree $2d(u)$ is equal to twice the dimension of $\mathcal{B}_u$. This degree (sometimes abbreviated $d(u)$) is the main concern of the Springer correspondence, which realizes certain irreducible "Springer characters" of the Weyl group $W$ in a unique top cohomology group (where $W$ acts naturally even though it doesn't usually act on $\mathcal{B}_u$). In case $u$ has a disconnected centralizer in $G$, the characters of the finite group of connected components complicate the Springer correspondence further; but a Springer character always occurs as the unique irreducible summand of the top cohomology belonging to the trivial character of the component group.
Given this set-up, the main theorem of Hotta-Springer says that $\varphi$ maps $H^{2d(u)}(\mathcal{B})$ onto the fixed points of the component group of $u$ in $H^{2d(u)} (\mathcal{B_u})$. However, there are exceptions to this surjectivity in some lower degrees, which seem to be related indirectly to Lusztig's "generalized Springer correspondence".
When $u$ is a special unipotent (in Lusztig's sense), are there examples of such failures of surjectivity?
Here the definition of "special" is unfortunately roundabout, related to Lusztig-Spaltenstein duality and the notion of "unipotent pieces". But when $G$ is a special linear group, all unipotents are special. In that case I'm just asking whether the Hotta-Springer theorem has a stronger statement.
[ADDED] Daniel Juteau reminds me that the case of general and special linear groups is discussed more directly in $\S2$ of the Hotta-Springer paper, where surjectivity of their graded $W$-equivariant specialization map on cohomology is proved in (2.3). For these groups one has the advantage that all centralizers of unipotent elements are connected (so the component groups are all trivial), while the unipotent classes themselves are all Richardson (hence special) and of standard Levi type. Here the Weyl group is a symmetric group, and all of its irreducible characters are Springer characters.
It would be especially interesting to sort out (using for example the old Beynon-Spaltenstein tables) exactly what happens in the extreme type $E_8$: here $G$ has 70 unipotent classes, with 46 being special, and one special class has component group $S_5$ (comprising a special piece along with six of the non-special classes in its closure).
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# There are no $3$ linearly independent lightlike vectors such that $u+v+w = 0$.
Consider the Lorentz-Minkowski space $$E^n_1$$, also known as $$\mathbb{L}^n$$. I want to prove that there are not lightlike linearly independent vectors $$u, v, w \in E^n_1$$ such that $$u + v + w = 0$$. How to do it? I'm still unfamiliar with the intuition behind such space.
• Does this space have 2 spacelike and one timelike variables? or what if not? – coffeemath Dec 8 '18 at 1:38
• @coffeemath Those are all hypotheses I have. – user71487 Dec 8 '18 at 1:41
• I don't see how this can be true... can't we take one of the vectors from the future light cone, say $\vec{u}$, and the other two to be $-\frac{1}{2}\vec{u}$? Don't we need to say that they are non-coplanar? – RandomMathGuy Dec 8 '18 at 1:41
• @RandomMathGuy The title (but not the body) specifies that the three vectors must be linearly independent. – Travis Willse Dec 8 '18 at 1:44
• @Travis I should learn to read the titles more carefully! – RandomMathGuy Dec 8 '18 at 1:44
Hint Suppose there were. Expand $$0 = [{\bf u} + {\bf v} + {\bf w}] \cdot [{\bf u} - ({\bf v} + {\bf w})]$$ to conclude that $${\bf v} \cdot {\bf w} = 0$$.
Additional hint What is the matrix representation of the bilinear form $$\cdot$$ with respect to the basis $$({\bf u}, {\bf v}, {\bf w})$$?
• Since I wrote this answer, OP edited the question to address the case of $n$-dimensional Minkowski space $\mathbb L^n$ rather than just the special case $n = 3$. The initial hint still leads to a solution for the general case, but of course $({\bf u}, {\bf v}, {\bf w})$ is not a basis of $\mathbb L^n$ for $n \neq 3$. – Travis Willse Dec 8 '18 at 1:48
• Shouldn't I conclude ${\bf u} \cdot {\bf w} = 0$ instead? – user71487 Dec 8 '18 at 1:58
• I don't see how---after all, the expression is symmetric in ${\bf v}$ and ${\bf w}$. – Travis Willse Dec 8 '18 at 2:02
• Expanding gives $0 = {\bf u} \cdot {\bf u} - ({\bf v} + {\bf w}) \cdot ({\bf v} + {\bf w}) = -({\bf v} \cdot {\bf v} + 2 {\bf v} \cdot {\bf w} + {\bf w} \cdot {\bf w}) = -2 {\bf v} \cdot {\bf w}$. – Travis Willse Dec 8 '18 at 2:58
• Not directly anyway. By symmetry of notation, we also have ${\bf u} \cdot {\bf v} = {\bf w} \cdot {\bf u} = 0$, but for $n = 3$ that means all pairs of basis vectors are orthogonal, which implies that $\cdot$ is the zero bilinear form, a contradiction. – Travis Willse Dec 8 '18 at 3:00
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# Mathematics - Inverse axiom
## About
• / is inverse of *
• - is inverse of +
## Linear Algebra
Functions f and g are functional inverses if:
• $f \circ g$ and $g \circ f$ are defined
• and are Function - Identity functions.
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# Tag Info
10
\begin{align} \frac{x^x}{(2x)!} & = \frac{\overbrace{x\cdots x}^{x\text{ factors}}}{\underbrace{1\cdot2\cdot3\cdots x}_{x\text{ factors}} \cdot \underbrace{(x+1) \cdot (x+2) \cdots (2x)}_{x\text{ factors}}} \\[10pt] & = \frac 1 {x!} \cdot \underbrace{\frac x {x+1} \cdot \frac x {x+2} \cdot \frac x {x+3} \cdots \frac x {x+x}}_\text{This is $<1$.} ...
9
A number $n \in \mathbf N^+$ has $\lfloor\log_{10}n\rfloor+1$ digits. For $n = 5^{4^{3^2}}$, we have \begin{align*} \log_{10} 5^{4^{3^2}} &= 4^9 \cdot \log_{10} 5\\ &= 262\,144 \cdot \log_{10} 5\\ &\approx 183\,230.8 \end{align*} So $n$ has $183\,231$ digits.
8
$$\left\lfloor \frac{1763.192674118048}{\log 10} \right\rfloor = 765$$ This is the logarithm base $e,$ so $\log 10 \approx 2.30258509$ Since $$\frac{1763.192674118048}{\log 10} \approx 765.7448488$$ we find that your number is $$e^{1763.192674118048} \approx 5.5571 \cdot 10^{765}$$ because $$10^{0.7448488} \approx 5.5571$$
7
In the equation $x^3+\frac{1}{x^3}=18$, multiply everything by $x^3$ to get $x^6-18x^3+1=0$. Then let $y=x^3$ and solve $y^2-18y+1=0$, then substitute back in to get $x$. Then you can input the value into $x^{11}+\frac{1}{x^{11}}$ directly. It winds up being $x^{11}+\frac{1}{x^{11}}=39603$ (regardless of whether you use the plus or the minus in the ...
7
Note that $$\frac{n^n}{(2n)!}=\frac{\overbrace{n\cdot n\cdot\ldots \cdot n}^{n \text{ factors}}}{n!\cdot\underbrace{(n+1)(n+2)\ldots(2n)}_{n\text{ factors}}}\le \frac1{n!}$$
6
$$A^n=\begin{pmatrix}1&nb\\0&1\end{pmatrix}$$ Suppose this is true until $n$. Then $$A^{n+1}=A.A^n$$ Computing the right side almost completes the proof.
5
They can be, but they don't have to be. One of the simplest setups is to define $\exp$ to be the unique function satisfying $\exp'=\exp$ and $\exp(0)=1$; then define $\ln$ to be the inverse of $\exp$; then define $a^x=\exp(x \ln(a))$ for $a>0$. Then the exponent and logarithm rules are theorems. Most of the proof of the rules is just elementary algebra; ...
5
It's transcendental by the Gelfond-Schneider theorem.
5
We may also avoid De l'Hopital theorem. Since $$\lim_{t\to 0}\frac{e^t-1}{t}=1\quad\text{and}\quad \lim_{x\to 0^+} x\log(x)=0,$$ we have: $$\lim_{x\to 0^+}\frac{x^x-1}{x}=\lim_{x\to 0^+}\frac{e^{x\log x}-1}{x\log x}\cdot\log(x) = \lim_{x\to 0^+}\log(x) = -\infty.$$
5
Looks like it's impossible for two distinct numbers to be "power equivalent". Really, $a_{n+1}=a^{b_n}$ and $b_{n+1}=b^{a_n}$. Now, if we use the limit definition and choose $n$ large enough, $a_{n+1}$ is close to $b_{n+1}$, which means for $a_n$ that it is close to ${\log a\over\log b}b_n$, but at the same time it is close to $b_n$, which is impossible ...
5
$$(5^{2015})(2^{2018}) = (5^{2015})(2^{2015})(2^3) = (5\cdot 2)^{2015}2^3$$ $$= 10^{2015}2^3 = 8\cdot 10^{2015}$$ Notice that this is just the digit $8$ followed by $2015$ zeroes, so the sum is just $8$
4
Use L'Hopital's rule to get: $$\lim_{x \to 0} x^x(\ln x+1)=\lim_{x \to 0}x^x \ln x-\lim_{x \to 0} x^x=(\lim_{x \to 0} x^x \ln x) -1=((\lim_{x \to 0} x^x)(\lim_{x \to 0} \ln x))-1=(1)(-\infty)-1=-\infty$$ The limit does not exist.
4
Without induction: Write $A$ as $I+bE$, where $I$ is the unit matrix of rank $2$ and $E=\begin{bmatrix}0&1\\0&0\end{bmatrix}$. As both matrices commute with each other, we can apply the binomial formula in the ring $M_2(\mathbf R)$, noting that $E^2=0$: $$A^n=\sum_{k=0}^n\binom nk I^{n-k}b^kE^k=I+nbE=\begin{bmatrix}1&nb\\0&1\end{bmatrix}.$$
3
Hint: $$x^3+\frac{1}{x^3}=(x+\frac{1}{x})(x^2+\frac{1}{x^2}-1)=18.$$ Let $$x+\frac{1}{x}=t.$$ Then your equation reduces to $(t(t^2-3))=18$. Solve this cubic and you get $x+\frac{1}{x}.$ Now I think you can carry on after that.
3
A little deus ex machina but this seems to be related to the golden ratio $\phi = \frac{1+\sqrt 5}{2}$, which is the solution of $x+\frac1x = 1$, and to the Fibonacci sequence $1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657, \ldots$ In particular $x^3+\frac1{x^3} = 18$ has the solutions $x^3=5+8\phi$ and its ...
3
By the Catalan conjecture (https://en.wikipedia.org wiki/Catalan's_conjecture ) solved by Mihailescu in 2002, 8 and 9 are the only consecutive powers, so yes, your question can be answered.
3
As $m^\frac{a}{b}=n$ therefore, $(m^\frac{a}{b})^b=(n)^b$ $m^a=n^b$
3
$\newcommand{\Reals}{\mathbf{R}}$If $n$ is a positive real number and $f_{n}:\Reals^{3} \to \Reals$ is defined by $$f_{n}(x, y, z) = (x^{2} + y^{2} + z^{2})^{n/2}, \tag{1}$$ then the level sets of $f_{n}$ are spheres centered at the origin, independently of $n$. (The level set at "height" $0$ consists of the origin alone, but may be viewed as a sphere of ...
3
The question is really "how many subsets of $\{1,3,5,7,11,17\}$ are there, not counting the null set?" As you said each element is either in the subset or it isn't. Thus each element has two choices, so there are $2\cdot2\cdot2\cdot2\cdot2\cdot2=2^6$ subsets, and we subtract one more since we also just counted the null set, which we don't want. This may be ...
2
Let $p =$ number of elements of a set $S$ Maximum number of subsets of $S$ is $2^p$ which includes a null set too. Since, we want only odd numbers so we take a set of odd numbers O ={1,3,5,7,11,17} which has cardinality $6$. Then the maximum number of sets that can be formed from this set is $2^6 -1=63$ because we have to discard the null set.
2
Assuming $x$ is an integer in your question (so that I'll use $n$ instead of $x$, for the sake of my own ease of mind): a simple way, which uses a big (and quite overkill) "hammer," is to invoke Stirling's approximation: $$(2n)! \operatorname*{\sim}_{n\to\infty} 2\sqrt{\pi n}\frac{(2n)^{2n}}{e^{2n}}$$ and look at the limit (when $n\to\infty$) of $$... 2$$m^{\frac{a}{b}}=n \log_m(n)=\frac{a}{b} b\log_m(n)=a log_m(n^b)=a \implies m^a=n^b$$2 Hint: (5)^{2015}(2)^{2015} = (10)^{2015}. (2)^{2018} = 2^{2015} \cdot 2^3 2 Taking logs to any base,$$x^3\log2=x^2\log3\ ,$$and therefore either x=0 or x=(\log3)/(\log2). Note that by the "change of base" formula, the last expression is the same no matter what base you are using. 2 Take natural log$$ 2^{x^3} = 3^{x^2} \implies x^3 \ln 2 = x^2 \ln 3 \implies x^2 \left( x\ln2 - \ln 3\right) = 0 \implies x_{1,2} = 0,\ x_3 = \frac {\ln 3}{\ln 2} = \log_2 3 $$2 Think modulo 10: The powers of 7 modulo 10 are, in order, 1,7,9,3, and then it repeats. Specifically, 7^7\equiv_{10} 3. Now, the powers of 3 modulo 10 are 1,3,9,7, and then it repeats. Specifically, 3^7\equiv_{10}7. We see that for each time we take the seventh power, the last digits alternates between seven and three. We start with 7 ... 2 Last digit for 7^7 = 823543 is 3 Last digit for (7^7)^7 is 7 Last digit for ((7^7)^7)^7 is 3 .... Last digit for 7^{th} power taken odd number of times is 3 Last digit for 7^{th} power taken even number of times is 7 Last digit for 7^{th} power taken 100 times should be 7 2 The problem boils down to comparing different values of the function f(x)=x^{\frac{1}{x}}, or, equivalently, different values of:$$ g(x)=\frac{\log x}{x}\qquad \text{or}\qquad h(t)=t\, e^{1-t}. \tag{1}$$g(x) a maximum at x=e and h(t) has a maximum at t=1. By computing the Taylor expansion of h(t) at t=1, we have:$$ h(t)\approx ...
2
Your calculator (and Alpha) are correct: $-1^2 = -(1^2) = -1$ whereas $(-1)^2=+1$. When evaluating expressions with multiple operations you have to follow the proper order of operations (order of precedence). Quickly: Exponentiation then Multiplication/Division then Addition/Subtraction. The reasoning behind this ordering has to do with the complexity of ...
2
You have it wrong. $$2\uparrow\uparrow\uparrow\uparrow2=2\uparrow\uparrow\uparrow2=2\uparrow\uparrow2=4$$ In fact, $2 \uparrow^n 2= 4$ for any $n$. However $2\uparrow\uparrow3=16$ and $2\uparrow\uparrow\uparrow3=65536$. $2\uparrow\uparrow\uparrow\uparrow3=2\uparrow\uparrow\uparrow4=2\uparrow\uparrow65536$. This is an stack of exponentation with height ...
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# Numbers too large for my calculator, probability. Old calculator or should it be done another way?
It's basic binomial probability, the problem is that the numbers are to large for my calculator. Is this the point of the task, to force me to use another method? It doesn't seem lightly, because n over r is a must anyways. An example would be 1000 over 180.
Note that on the exam it self I am allowed to use my laptop (mac), but not the net. So if you can think of any programs that can deal with this size of numbers I'd appreciate it.
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You should explain more about what the task is. How are we supposed to figure out what the goal of the assignment was without knowing what was assigned? – Zev Chonoles Jul 22 '11 at 7:33
The problem applies to a lot of different tasks where the numbers are to big, so it really is in general. – Algific Jul 22 '11 at 7:36
Is the problem you are solving related to the binomial distribution? If so, there is the normal approximation(en.wikipedia.org/wiki/…) and the Poisson approximation (en.wikipedia.org/wiki/…). I am suggesting this in a comment because it doesn't directly answer your question as stated. – Henry B. Jul 22 '11 at 7:46
## 2 Answers
You can use Stirling's approximation to get the logs of the factorials. So if $$x_k={1000\choose k}0.2^k0.8^{1000-k}=\frac{1000!}{k!(1000-k)!}0.2^k0.8^{1000-k}$$ you have $$\log x_k\approx (1000+\frac{1}{2})\log 1000-(k+\frac{1}{2}) \log k - (1000-k+\frac{1}{2})\log (1000-k)$$ $$-\frac{1}{2}\log 2\pi+k \log 0.2 + (1000-k) \log 0.8$$
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Since I'm allowed to use my computer the bcd function on my casio can be done in excel with the following line: =BINOMDIST(180,1000,0.2,TRUE). The true part is because we want the probability for at most 180 people/success.
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Well, problems like this are the reason, why the books and courses teach about the normal approximation! Frankly, it is hard to believe that the teacher would want you to compute all 181 terms by frantically punching a calculator. I suggest that for preparatory problems you: 1) try more realistic numbers, 2) learn about the normal approximation. Presumably you can use a table or something to produce the cdf of a normally distributed random variable? – Jyrki Lahtonen Jul 22 '11 at 10:37
But you probably can (in a way) do it with your calculator by starting with a term $x_k={1000\choose k}0.2^k0.8^{1000-k}$ that your calculator can do accurately (relatively small $k$). Then you can use the formula $x_{k+1}=x_k*(1000-k)*0.2/(0.8*(k+1))$. And then in the end sum $x_0+x_1+\cdots+x_{180}$. But I really don't think that you want to do that, and numerical inaccuracies are bound to make this doubtful anyway. – Jyrki Lahtonen Jul 22 '11 at 10:51
I got in contact with the author and we are allowed to use laptops on the exam. Problem solved. – Algific Sep 10 '11 at 18:54
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# Problem: Hydrazine (N2H4) is a fuel used by some spacecraft. It is normally oxidized by N2O4 according to the following equation: N2H4(l) + N2O4(g) → 2 N2O(g) + 2 H2O(g). Calculate ΔH˚rxn for this reaction using standard enthalpies of formation.
###### FREE Expert Solution
We’re being asked to determine the standard enthalpy change (ΔH˚rxn) for the balanced reaction
N2H4(l) + N2O4(g) → 2 N2O(g) + 2 H2O(g)
Recall that ΔH˚rxn can be calculated from the enthalpy of formation (ΔH˚f) of the reactants and products involved:
$\overline{){\mathbf{∆}}{\mathbf{H}}{{\mathbf{°}}}_{{\mathbf{rxn}}}{\mathbf{=}}{\mathbf{∆}}{\mathbf{H}}{{\mathbf{°}}}_{\mathbf{f}\mathbf{,}\mathbf{products}}{\mathbf{-}}{\mathbf{∆}}{\mathbf{H}}{{\mathbf{°}}}_{\mathbf{f}\mathbf{,}\mathbf{reactants}}}$
###### Problem Details
Hydrazine (N2H4) is a fuel used by some spacecraft. It is normally oxidized by N2O4 according to the following equation: N2H4(l) + N2O4(g) → 2 N2O(g) + 2 H2O(g). Calculate ΔH˚rxn for this reaction using standard enthalpies of formation.
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## Tuesday, December 14, 2010
### Computing ABV
A recent question on http://homebrew.stackexchange.com/ asked for a simple formula for calculating the percentage alcohol by volume (ABV) of a beer given the original gravity (OG) and final gravity (FG). Here was the complicated formula which was to be simplified:
$ABV=\frac{76.08\cdot(OG-FG)}{1.775-OG}\cdot\frac{FG}{0.794}$
OK, so it's not that complicated, but a linearization would be good. The original post suggested a range of 1.035-1.095 for OG and 1.002-1.028 for FG. I chose the midpoints of these intervals to get the point (1.065,1.015) and linearized the given formula at this point. Here's what I got:
$ABV \approx -17.1225210+146.6266588\cdot OG-130.2323766\cdot FG$
It's simpler, but not easy to remember. If we round the coefficients, we get an easier to handle expression.
$ABV \approx 147\cdot OG - 130\cdot FG -17$
Analysis using Maple shows that this simplified linear approximation has an error of at most 0.78 from the original equation. This is not too hard to remember, since 147-130=17. Also, if both OG and FG are equal to one (the specific gravity of water), then the formula predicts 0% ABV.
## Thursday, October 28, 2010
### DIY 1 Gallon Primary
I've been working on being able to brew one gallon recipes, and I decided to make my own primary fermenting tank for these small batches.
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# Math Help - Parabolas
1. ## Parabolas
How do I figure out problems similar to the following by graphing y = 2x2 + 3?
2. What are you trying to find..Also your equation $y = 2x2 + 3$, does it mean y=4+3 or y=4x+3 or...you just mistype and its suppose to be y=2x+3?
Edit
Never mind, I assume you mean $y=2x^2+3$ right? But the question still exists, what are you trying to find?
3. I mean how do I graph it as a parabola?
4. Find the y intercept, which means setting x=0, and the y intercept in this case is (0,3)
Find the x intercepts using the quadratic equation or factoring- There is no x-intercepts in that problem
Find the axis of symmetry, which is $x=-b/(2a)$ to find the min or the max otherwise known as vertex, in this case it would be the same as the y-intercept
Look at the sign to see if it's concave up or down
I think thats basically it to graph simple parabolas like the example you gave.
5. I'm sorry but I am still a bit confused... Is there a simpler way?
6. Easier way...plug in x values...or use the calculator
I assume you know what vertex, and setting x=0 and y=0 does since this is in the pre-calc section
The basic equation of a parabola is $y= ax^2+bx+c$
So to find the vertex of the equation you gave $y=2x^2+3$, you would do x=-b/2a, and in this cause there is no b, which means it is 0 so the x value of the vertex would be x=0/4=0 and plug in 0 for x in the equation you would get the vertex at (0,3).
Now that you have the vertex it's time to find the x intercept. To find the x-intercept you would either use the quadratic equation or factoring it if possible. The equation equation is:
Now solve for x. Whatever you get for x should the x intercepts..which I assume you know what it is. I don't think there is any x-intercepts in this case.
Now for the y intercept just set x=0 in the equation. Because that means when x=0, y=?. In this case when you set x=0, y will be 3 right?
7. I understand everything in this part...
Easier way...plug in x values...or use the calculator
I assume you know what vertex, and setting x=0 and y=0 does since this is in the pre-calc section
The basic equation of a parabola is
So to find the vertex of the equation you gave , you would do x=-b/2a, and in this cause there is no b, which means it is 0 so the x value of the vertex would be x=0/4=0
But I don't understand how you got the vertex....
8. -b/2a is the x value of the vertex because it is also the axis of symmetry for the parabola. To get the vertex, you plug in the x and you will get the y-value so the vertex would be (-b/2a, f(-b/2a)) with f(x) = y. If you are looking for where did I get -b/2a or how to derivative it, read this webpage
9. Is it any different to graph a linear equation such as y=2x^2+4? I don't know if this is a linear equation....
10. That is not a linear equation. It is a quadratic equation and no, it is not any different to solve $y=2x^2+4$
11. Can you please give an example of a linear equation and what makes it linear? thanks
12. Anything in the form of $y = mx + b$ where m is the slope and b is the y-intercept.
It is linear because when you graph it you do not get a curve. You get a straight line.
And to do this really easily just plug in x-values, from that you can get the y-values and then plot. Then do a simple connect the dots. You should get a pretty skinny curve with the vertex at $(0,3)$ You can find the vertex as linnus said by doing $\left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right)$ You can use this only if it is in the form $ax^{2}+bx+c$
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Math Help - Radius of Convergence
Hello,
I have to find the radius of convergence with this series
$\Sigma 2^{-n}z^{n^2}}$
I've tried the root test and the ratio test but the solution they give seem inconclusive, is the any clues as to what I should do?
Thank you
Originally Posted by iPod
Hello,
I have to find the radius of convergence with this series
$\Sigma 2^{-n}z^{n^2}}$
I've tried the root test
The root test works. For what values does ${\lim _{n \to \infty }}\frac{{{z^n}}}{2}$ converge?
Oh I see, so basically it converges to 0 if |z|<1, meaning the series converges anywhere (provided |z|<1)
Actually, |z|<2 because it's |z/2| < 1
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# Slug (unit)
The slug is a derived unit of mass in a weight-based system of measures, most notably within the British Imperial measurement system and in the United States customary measures system. Systems of measure either define mass and derive a force unit or define a base force and derive a mass unit[1] (cf. poundal, a derived unit of force in a force-based system). A slug is defined as the mass that is accelerated by 1 ft/s2 when a force of one pound (lbf) is exerted on it.
${\displaystyle 1\,{\text{slug}}=1\,{\frac {{\text{lbf}}{\cdot }{\text{s}}^{2}}{\text{ft}}}\quad \Longleftrightarrow \quad 1\,{\text{lbf}}=1\,{\frac {{\text{slug}}{\cdot }{\text{ft}}}{{\text{s}}^{2}}}}$
Slug
Unit systemBritish Gravitational system
Unit ofmass
Symbolslug
Conversions
1 slug in ...... is equal to ...
SI units 14.59390 kg
US customary units 32.1740 lb
One slug has a mass of 32.1740 lb (14.59390 kg) based on standard gravity, the international foot, and the avoirdupois pound.[2] At the Earth's surface, an object with a mass of 1 slug exerts a force downward of approximately 32.2 lbf or 143 N.[3][4]
## History
The slug is part of a subset of units known as the gravitational FPS system, one of several such specialized systems of mechanical units developed in the late 19th and the 20th century. Geepound was another name for this unit in early literature.[5]
The name "slug" was coined before 1900 by British physicist Arthur Mason Worthington,[6] but it did not see any significant use until decades later. A 1928 textbook says:
No name has yet been given to the unit of mass and, in fact, as we have developed the theory of dynamics no name is necessary. Whenever the mass, m, appears in our formulae, we substitute the ratio of the convenient force-acceleration pair (w/g), and measure the mass in lbs. per ft./sec.2 or in grams per cm./sec.2.
Noel Charlton Little, College Physics, Charles Scribner's Sons, 1928, p. 165.
Three approaches to units of mass and force or weight[7][8]
Base Force Weight Mass
2nd law of motion m = F/a F = W a/g F = m a
System BGGM EEM AECGSMTSSI
Acceleration (a) ft/s2m/s2 ft/s2m/s2 ft/s2Galm/s2m/s2
Mass (m) slughyl pound-masskilogram poundgramtonnekilogram
Force (F),
weight (W)
poundkilopond pound-forcekilopond poundaldynesthènenewton
Pressure (p) pounds per square inchtechnical atmosphere pounds-force per square inchatmosphere poundals per square footbaryepiezepascal
The slug is listed in the Regulations under the Weights and Measures (National Standards) Act, 1960. This regulation defines the units of weights and measures, both regular and metric, in Australia.
The blob is the inch version of the slug (1 blob is equal to 1 lbf⋅s2/in, or 12 slugs)[2][9] or equivalent to 386.0886 pounds (175.1268 kg). This unit is also called slinch (a portmanteau of the words slug and inch).[10][11] Similar terms include slugette[12] and snail.[13]
Similar metric units include the glug in the centimetre–gram–second system, and the mug, par, or MTE in the metre–kilogram–second system.[14]
## References
1. See Elementary High School physics and chemistry text books/fundamentals.
2. Shigley, Joseph E. and Mischke, Charles R. Mechanical Engineering Design, Sixth ed, pp. 31–33. McGraw Hill, 2001. ISBN 0-07-365939-8.
3. Beckwith, Thomas G., Roy D. Marangoni, et al. Mechanical Measurements, Fifth ed, pp. 34-36. Addison-Wesley Publishing, 1993. ISBN 0-201-56947-7.
4. Shevell, R.S. Fundamentals of Flight, Second ed, p. xix. Prentice-Hall, 1989.
5. gee. unit2unit.eu
6. Worthington, Arthur Mason (1900). Dynamics of Rotation: An Elementary Introduction to Rigid Dynamics (3rd ed.). Longmans, Green, and Co. p. 9.
7. Comings, E. W. (1940). "English Engineering Units and Their Dimensions". Industrial & Engineering Chemistry. 32 (7): 984–987. doi:10.1021/ie50367a028.
8. Klinkenberg, Adrian (1969). "The American Engineering System of Units and Its Dimensional Constant gc". Industrial & Engineering Chemistry. 61 (4): 53–59. doi:10.1021/ie50712a010.
9. Norton, Robert L. Cam Design and Manufacturing Handbook, p. 13. Industrial Press Inc., 2009. ISBN 0831133678.
10. Slug. DiracDelta Science & Engineering Encyclopedia
11. "1 blob". Wolfram Alpha Computational Knowledge Engine. Retrieved 27 October 2011.
12. Celmer, Robert. Notes to Accompany Vibrations II. Version 2.2. 2009.
13. Rowlett, Russ. "How Many? A Dictionary of Units of Measurement". unc.edu, September 1, 2004. Retrieved January 26, 2018.
14. Cardarelli, François (1999). Scientific Units, Weights and Measures. Springer. pp. 358, 377. ISBN 1-85233-682-X.
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# Docker Containers & Network Sniffing
With all of the materials out there on the web revolving around docker containers, I thought that getting some sort of a docker network that containers could promiscuously sniff would have been a relatively easy thing to find. I was shocked to find out that, not only was this not the case, but that the general consensus was that you need to use either Docker’s host networking (which means that these containers can’t exist in other network name-spaces), use pass-through networking (which unless you have hardware that support SR-IOV, your out of luck), or that you resort to some serious host hacking to get the interface into the container. I figured there had to be a more elegant solution, and after quite some time of on-and-off looking around for different solutions I had the sudden realization that I was approaching the problem from the wrong angle. Instead of trying to get docker to bend to my use case, I needed to focus on the network bridging itself. Now with a new focus, finding the solution only took an hour or two, ended up being extremely elegant, and is very survivable. Further, as I don’t need any weird hackery, none of the containers sniffing the network need privileged access, so no need to worry about special needs containers.
As a basis for what I’m doing here, I have an Ubuntu 16.04 host running Docker CE and it has a couple of network interfaces:
• ens160 - Server VLAN traffic
• ens192 - Mirrored traffic from the Cisco switch
The first step in this little adventure is to download the mirroring script and install it onto the host. While the method below is quick & dirty, remember to always validate what your downloading to your systems.
wget -O /usr/bin/tc-mirror https://raw.githubusercontent.com/SteveMcGrath/mirror_tools/master/tc-mirror.sh
chmod 755 /usr/bin/tc-mirror
Now the goal here is to push the mirrored traffic from ens192 into a docker network. To start, we will want to create a bridge on the host which I will call span0. The /etc/network/interfaces configuration for this bridge looks like so:
auto span0
iface span0 inet manual
bridge_stp off
bridge_fd 0
bridge_maxwait 0
bridge_ageing 0
post-up /usr/bin/tc-mirror build ens192 span0
pre-down /usr/sbin/tc-mirror teardown ens192
I want to point out something here, specifically the bridge_ageing 0 line. This single parameter is telling the Linux kernel to disable MAC address learning capability that normally comes built-in on every linux bridge. Without this capability turned on, we have effectively disabled the MAC learning table, which is the heart of the switching capability within the bridge. The bridge itself is now effectively a simple packet forwarder to all of the interfaces attached to the bridge, which is what we want in this case. Now to restart the networking on the Ubuntu host to inform it to bring up the new interface (and to auto-build the bridge at boot from now on)
systemctl restart networking
Everything is all plumbed up now! The next step is to create the docker network and tell docker to use the existing bridge that we have already created instead of building a new one. We also want to make sure that we inform docker this this is not a network we want to route with, so we mark it as “internal” only.
docker network create -d bridge --internal -o com.docker.network.bridge.name=span0 span
Now if we want to test this, we can build a really simple ubuntu docker container and install tcpdump within it:
docker run -it --rm --name=spantest ubuntu:16.04 /bin/bash
Once the container drops you into a shell:
apt update && apt -y install tcpdump
Now that tcpdump is installed, lets go ahead and add the span network to the container in another window:
docker network connect span spantest
Lets go ahead and start tcpdump!
tcpdump -i eth1
If all went well, you should now see all of the mirrored traffic on the container. No elevated privileges or hackery, just a regular old container. Secondly this allows for multiple containers to be attached to the same single mirror, allowing for multiple tools to get the same data (which is a requirement in my case for Dofler).
It really is that simple & elegant. I can only imagine the doors this opens up for folks, especially in the network security community with things such as snort, suricata, bro, and even Tenable’s own Nessus Network Monitor.
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# Rayleigh–Plesset equation
(Redirected from Rayleigh-Plesset equation)
The Rayleigh–Plesset equation is often applied to the study of cavitation bubbles, shown here forming behind a propeller.
In fluid mechanics, the Rayleigh–Plesset equation or Besant–Rayleigh–Plesset equation is an ordinary differential equation which governs the dynamics of a spherical bubble in an infinite body of incompressible fluid.[1][2][3][4] Its general form is usually written as
${\displaystyle R{\frac {d^{2}R}{dt^{2}}}+{\frac {3}{2}}\left({\frac {dR}{dt}}\right)^{2}+{\frac {4\nu _{L}}{R}}{\frac {dR}{dt}}+{\frac {2\gamma }{\rho _{L}R}}+{\frac {\Delta P(t)}{\rho _{L}}}=0}$
where
${\displaystyle \rho _{L}}$ is the density of the surrounding liquid, assumed to be constant
${\displaystyle R(t)}$ is the radius of the bubble
${\displaystyle \nu _{L}}$ is the kinematic viscosity of the surrounding liquid, assumed to be constant
${\displaystyle \gamma }$ is the surface tension of the bubble-liquid interface
${\displaystyle \Delta P(t)=P_{\infty }(t)-P_{B}(t)}$, in which, ${\displaystyle P_{B}(t)}$ is the pressure within the bubble, assumed to be uniform and ${\displaystyle P_{\infty }(t)}$ is the external pressure infinitely far from the bubble
Provided that ${\displaystyle P_{B}(t)}$ is known and ${\displaystyle P_{\infty }(t)}$ is given, the Rayleigh–Plesset equation can be used to solve for the time-varying bubble radius ${\displaystyle R(t)}$.
The Rayleigh–Plesset equation is derived from the Navier–Stokes equations under the assumption of spherical symmetry.[4]
## History
Neglecting surface tension and viscosity, the equation was first derived by W. H. Besant in his 1859 book[5] with the problem statement stated as An infinite mass of homogeenous incompressible fluid acted upon by no forces is at rest, and a spherical portion of the fluid is suddenly annihilated; it is required to find the instanateous alteration of pressure at any point of the mass, and the time in which the cavity will be filled up, the pressure at an infinite distance being supposed to remain constant (in fact, Besant attributes the problem to Cambridge Senate-House problems of 1847). Neglecting the pressure variations inside the bubble, Besant predicted the time required to fill the cavity to be
{\displaystyle {\begin{aligned}t&=a\left({\frac {6\rho }{p_{\infty }}}\right)^{1/2}\int _{0}^{1}{\frac {z^{4}\,dz}{\sqrt {1-z^{6}}}}\\&=a\left({\frac {\pi \rho }{6p_{\infty }}}\right)^{1/2}{\frac {\Gamma (5/6)}{\Gamma (4/3)}}\\&\approx 0.91468a\left({\frac {\rho }{p_{\infty }}}\right)^{1/2}\end{aligned}}}
where the integration was carried out by Lord Rayleigh in 1917, who derived the equation from energy balance. Rayleigh also realized that the assumption of constant pressure inside the cavity would become wrong as the radius decreases and he shows that using Boyle's law, if the radius of cavity decreases by a factor of ${\displaystyle 4^{1/3}}$, then the pressure near the boundary of the cavity becomes greater than the ambient pressure. The equation was first applied to traveling cavitation bubbles by Milton S. Plesset in 1949 by including effects of surface tension.[6]
## Derivation
Numerical integration of RP eq. including surface tension and viscosity terms. Initially at rest in atmospheric pressure with R0=50 um, the bubble subjected to oscillatory pressure at its natural frequency undergoes expansion and then collapses.
Numerical integration of RP eq. including surface tension and viscosity terms. Initially at rest in atmospheric pressure with R0=50 um, the bubble subjected to pressure-drop undergoes expansion and then collapses.
The Rayleigh–Plesset equation can be derived entirely from first principles using the bubble radius as the dynamic parameter.[3] Consider a spherical bubble with time-dependent radius ${\displaystyle R(t)}$, where ${\displaystyle t}$ is time. Assume that the bubble contains a homogeneously distributed vapor/gas with a uniform temperature ${\displaystyle T_{B}(t)}$ and pressure ${\displaystyle P_{B}(t)}$. Outside the bubble is an infinite domain of liquid with constant density ${\displaystyle \rho _{L}}$ and dynamic viscosity ${\displaystyle \mu _{L}}$. Let the temperature and pressure far from the bubble be ${\displaystyle T_{\infty }}$ and ${\displaystyle P_{\infty }(t)}$. The temperature ${\displaystyle T_{\infty }}$ is assumed to be constant. At a radial distance ${\displaystyle r}$ from the center of the bubble, the varying liquid properties are pressure ${\displaystyle P(r,t)}$, temperature ${\displaystyle T(r,t)}$, and radially outward velocity ${\displaystyle u(r,t)}$. Note that these liquid properties are only defined outside the bubble, for ${\displaystyle r\geq R(t)}$.
### Mass conservation
By conservation of mass, the inverse-square law requires that the radially outward velocity ${\displaystyle u(r,t)}$ must be inversely proportional to the square of the distance from the origin (the center of the bubble).[6] Therefore, letting ${\displaystyle F(t)}$ be some function of time,
${\displaystyle u(r,t)={\frac {F(t)}{r^{2}}}}$
In the case of zero mass transport across the bubble surface, the velocity at the interface must be
${\displaystyle u(R,t)={\frac {dR}{dt}}={\frac {F(t)}{R^{2}}}}$
which gives that
${\displaystyle F(t)=R^{2}dR/dt}$
In the case where mass transport occurs, the rate of mass increase inside the bubble is given by
${\displaystyle {\frac {dm_{V}}{dt}}=\rho _{V}{\frac {dV}{dt}}=\rho _{V}{\frac {d(4\pi R^{3}/3)}{dt}}=4\pi \rho _{V}R^{2}{\frac {dR}{dt}}}$
with ${\displaystyle V}$ being the volume of the bubble. If ${\displaystyle u_{L}}$ is the velocity of the liquid relative to the bubble at ${\displaystyle r=R}$, then the mass entering the bubble is given by
${\displaystyle {\frac {dm_{L}}{dt}}=\rho _{L}Au_{L}=\rho _{L}(4\pi R^{2})u_{L}}$
with ${\displaystyle A}$ being the surface area of the bubble. Now by conservation of mass ${\displaystyle dm_{v}/dt=dm_{L}/dt}$, therefore ${\displaystyle u_{L}=(\rho _{V}/\rho _{L})dR/dt}$. Hence
${\displaystyle u(R,t)={\frac {dR}{dt}}-u_{L}={\frac {dR}{dt}}-{\frac {\rho _{V}}{\rho _{L}}}{\frac {dR}{dt}}=\left(1-{\frac {\rho _{V}}{\rho _{L}}}\right){\frac {dR}{dt}}}$
Therefore
${\displaystyle F(t)=\left(1-{\frac {\rho _{V}}{\rho _{L}}}\right)R^{2}{\frac {dR}{dt}}}$
In many cases, the liquid density is much greater than the vapor density, ${\displaystyle \rho _{L}\gg \rho _{V}}$, so that ${\displaystyle F(t)}$ can be approximated by the original zero mass transfer form ${\displaystyle F(t)=R^{2}dR/dt}$, so that[6]
${\displaystyle u(r,t)={\frac {F(t)}{r^{2}}}={\frac {R^{2}}{r^{2}}}{\frac {dR}{dt}}}$
### Momentum conservation
Assuming that the liquid is a Newtonian fluid, the incompressible Navier–Stokes equation in spherical coordinates for motion in the radial direction gives
${\displaystyle \rho _{L}\left({\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial r}}\right)=-{\frac {\partial P}{\partial r}}+\mu _{L}\left[{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial u}{\partial r}}\right)-{\frac {2u}{r^{2}}}\right]}$
Substituting kinematic viscosity ${\displaystyle \nu _{L}=\mu _{L}/\rho _{L}}$ and rearranging gives
${\displaystyle -{\frac {1}{\rho _{L}}}{\frac {\partial P}{\partial r}}={\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial r}}-\nu _{L}\left[{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial u}{\partial r}}\right)-{\frac {2u}{r^{2}}}\right]}$
whereby substituting ${\displaystyle u(r,t)}$ from mass conservation yields
${\displaystyle -{\frac {1}{\rho _{L}}}{\frac {\partial P}{\partial r}}={\frac {2R}{r^{2}}}\left({\frac {dR}{dt}}\right)^{2}+{\frac {R^{2}}{r^{2}}}{\frac {d^{2}R}{dt^{2}}}-{\frac {2R^{4}}{r^{5}}}\left({\frac {dR}{dt}}\right)^{2}={\frac {1}{r^{2}}}\left(2R\left({\frac {dR}{dt}}\right)^{2}+R^{2}{\frac {d^{2}R}{dt^{2}}}\right)-{\frac {2R^{4}}{r^{5}}}\left({\frac {dR}{dt}}\right)^{2}}$
Note that the viscous terms cancel during substitution.[6] Separating variables and integrating from the bubble boundary ${\displaystyle r=R}$ to ${\displaystyle r\rightarrow \infty }$ gives
${\displaystyle -{\frac {1}{\rho _{L}}}\int _{P(R)}^{P_{\infty }}dP=\int _{R}^{\infty }\left[{\frac {1}{r^{2}}}\left(2R\left({\frac {dR}{dt}}\right)^{2}+R^{2}{\frac {d^{2}R}{dt^{2}}}\right)-{\frac {2R^{4}}{r^{5}}}\left({\frac {dR}{dt}}\right)^{2}\right]dr}$
${\displaystyle {{\frac {P(R)-P_{\infty }}{\rho _{L}}}=\left[-{\frac {1}{r}}\left(2R\left({\frac {dR}{dt}}\right)^{2}+R^{2}{\frac {d^{2}R}{dt^{2}}}\right)+{\frac {R^{4}}{2r^{4}}}\left({\frac {dR}{dt}}\right)^{2}\right]_{R}^{\infty }=R{\frac {d^{2}R}{dt^{2}}}+{\frac {3}{2}}\left({\frac {dR}{dt}}\right)^{2}}}$
### Boundary conditions
Let ${\displaystyle \sigma _{rr}}$ be the normal stress in the liquid that points radially outward from the center of the bubble. In spherical coordinates, for a fluid with constant density and constant viscosity,
${\displaystyle \sigma _{rr}=-P+2\mu _{L}{\frac {\partial u}{\partial r}}}$
Therefore at some small portion of the bubble surface, the net force per unit area acting on the lamina is
{\displaystyle {\begin{aligned}\sigma _{rr}(R)+P_{B}-{\frac {2\gamma }{R}}&=-P(R)+\left.2\mu _{L}{\frac {\partial u}{\partial r}}\right|_{r=R}+P_{B}-{\frac {2\gamma }{R}}\\&=-P(R)+2\mu _{L}{\frac {\partial }{\partial r}}\left({\frac {R^{2}}{r^{2}}}{\frac {dR}{dt}}\right)_{r=R}+P_{B}-{\frac {2\gamma }{R}}\\&=-P(R)-{\frac {4\mu _{L}}{R}}{\frac {dR}{dt}}+P_{B}-{\frac {2\gamma }{R}}\\\end{aligned}}}
where ${\displaystyle \gamma }$ is the surface tension.[6] If there is no mass transfer across the boundary, then this force per unit area must be zero, therefore
${\displaystyle P(R)=P_{B}-{\frac {4\mu _{L}}{R}}{\frac {dR}{dt}}-{\frac {2\gamma }{R}}}$
and so the result from momentum conservation becomes
${\displaystyle {\frac {P(R)-P_{\infty }}{\rho _{L}}}={\frac {P_{B}-P_{\infty }}{\rho _{L}}}-{\frac {4\mu _{L}}{\rho _{L}R}}{\frac {dR}{dt}}-{\frac {2\gamma }{\rho _{L}R}}=R{\frac {d^{2}R}{dt^{2}}}+{\frac {3}{2}}\left({\frac {dR}{dt}}\right)^{2}}$
whereby rearranging and letting ${\displaystyle \nu _{L}=\mu _{L}/\rho _{L}}$ gives the Rayleigh–Plesset equation[6]
${\displaystyle {\frac {P_{B}(t)-P_{\infty }(t)}{\rho _{L}}}=R{\frac {d^{2}R}{dt^{2}}}+{\frac {3}{2}}\left({\frac {dR}{dt}}\right)^{2}+{\frac {4\nu _{L}}{R}}{\frac {dR}{dt}}+{\frac {2\gamma }{\rho _{L}R}}}$
Using dot notation to represent derivatives with respect to time, the Rayleigh–Plesset equation can be more succinctly written as
${\displaystyle {\frac {P_{B}(t)-P_{\infty }(t)}{\rho _{L}}}=R{\ddot {R}}+{\frac {3}{2}}({\dot {R}})^{2}+{\frac {4\nu _{L}{\dot {R}}}{R}}+{\frac {2\gamma }{\rho _{L}R}}}$
## Solutions
Recently, analytical closed-form solutions were found for the Rayleigh–Plesset equation for both an empty and gas—filled bubble [7] and were generalized to the N-dimensional case.[8] The case when the surface tension is present due to the effects of capillarity were also studied.[8][9]
Also, for the special case where surface tension and viscosity are neglected, high-order analytical approximations are also known.[10]
In the static case, the Rayleigh–Plesset equation simplifies, yielding the Young-Laplace equation:
${\displaystyle P_{B}-P_{\infty }={\frac {2\gamma }{R}}}$
When only infinitesimal periodic variations in the bubble radius and pressure are considered, the RP equation also yields the expression of the natural frequency of the bubble oscillation.
## References
1. ^ Rayleigh, Lord (1917). "On the pressure developed in a liquid during the collapse of a spherical cavity". Phil. Mag. 34 (200): 94–98. doi:10.1080/14786440808635681.
2. ^ Plesset, M.S. (1949). "The dynamics of cavitation bubbles". ASME J. Appl. Mech. 16: 228–231.
3. ^ a b Leighton, T. G. (17 April 2007). "Derivation of the Rayleigh–Plesset equation in terms of volume". Southampton, UK: Institute of Sound and Vibration Research.
4. ^ a b Lin, Hao; Brian D. Storey; Andrew J. Szeri (2002). "Inertially driven inhomogeneities in violently collapsing bubbles: the validity of the Rayleigh–Plesset equation". Journal of Fluid Mechanics. 452. doi:10.1017/S0022112001006693. ISSN 0022-1120.
5. ^ Besant, W. H. (1859). A treatise on hydrostatics and hydrodynamics. Deighton, Bell. Article. 158.
6. Brennen, Christopher E. (1995). Cavitation and Bubble Dynamics. Oxford University Press. ISBN 978-0-19-509409-1.
7. ^ Kudryashov, Nikolay A.; Sinelshchikov, Dnitry I. (18 September 2014). "Analytical solutions of the Rayleigh equation for empty and gas-filled bubble". Journal of Physics A: Mathematical and Theoretical. 47 (40): 405202. arXiv:1409.6699. Bibcode:2014JPhA...47N5202K. doi:10.1088/1751-8113/47/40/405202.
8. ^ a b Kudryashov, Nikolay A.; Sinelshchikov, Dnitry I. (31 December 2014). "Analytical solutions for problems of bubble dynamics". Physics Letters A. 379 (8): 798–802. arXiv:1608.00811. Bibcode:2016arXiv160800811K. doi:10.1016/j.physleta.2014.12.049.
9. ^ Mancas, Stefan C.; Rosu, Haret C. (7 August 2015). "Cavitation of spherical bubbles: closed-form, parametric, and numerical solutions". Physics of Fluids. 28 (2): 022009. arXiv:1508.01157. Bibcode:2016PhFl...28b2009M. doi:10.1063/1.4942237.
10. ^ Obreschkow, D.; Bruderer M.; Farhat, M. (5 June 2012). "Analytical approximations for the collapse of an empty spherical bubble". Physical Review E. 85 (6): 066303. arXiv:1205.4202. Bibcode:2012PhRvE..85f6303O. doi:10.1103/PhysRevE.85.066303. PMID 23005202.
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There are many things out there that are real and amazing. Have fun!!! Hey Geoff – you can now call me Mr Electro Magnet. I have done so much research in the last week. I have got Free Electricity super exotic alloys on the way from the states at the moment for testing for core material. I know all about saturation, coercivity, etc etc. Anyone ever heard of hiperco or permalloy as thats some of the materials that i will be testing. Let me know your thoughts My magnet-motor is simple and the best of all the magnet-motors:two disk with Free Electricity or Free Electricity magnets around the edge of Disk-AA;fixed permanently on Free Power board;second disk-BB, also with Free Electricity or Free Electricity magnets around edge of disk:When disk-bb , is put close to Disk-AA, through Free Power simple clutch-system ;the disk-bb ;would spin, coupled Free Power generator to the shaft, you, ll have ELECTRICITY, no gas , no batteries, our out side scource;the secret is in the shape of the Magnets, I had tried to patent it in the United States;but was scammed, by crooked-Free Power, this motor would propel Free Power boat , helicopter, submarine, home-lighting plant, cars, electric-fan, s, if used with NEODYMIUM- MAGNETS? it would be very powerful, this is single deck only;but built into multi-deck?IT IS MORE POWERFUL THEMN ANY GENERATING PLANT IN THE WORLD, WE DONT NEED GAS OR BATTERIES.
Why? Because I didn’t have the correct angle or distance. It did, however, start to move on its own. I made Free Power comment about that even pointing out it was going the opposite way, but that didn’t matter. This is Free Power video somebody made of Free Power completed unit. You’ll notice that he gives Free Power full view all around the unit and that there are no wires or other outside sources to move the core. Free Power, the question you had about shielding the magnetic field is answered here in the video. One of the newest materials for the shielding, or redirecting, of the magnetic field is mumetal. You can get neodymium magnets via eBay really cheaply. That way you won’t feel so bad when it doesn’t work. Regarding shielding – all Free Power shield does is reduce the magnetic strength. Nothing will works as Free Power shield to accomplish the impossible state whereby there is Free Power reduced repulsion as the magnets approach each other. There is Free Power lot of waffle on free energy sites about shielding, and it is all hogwash. Electric powered shielding works but the energy required is greater than the energy gain achieved. It is Free Power pointless exercise. Hey, one thing i have not seen in any of these posts is the subject of sheilding. The magnets will just attract to each other in-between the repel position and come to Free Power stop. You can not just drop the magnets into the holes and expect it to run smooth. Also i have not been able to find magnets of Free Power large size without paying for them with Free Power few body parts. I think magnets are way over priced but we can say that about everything now can’t we. If you can get them at Free Power good price let me know.
These were Free Power/Free Power″ disk magnets, not the larger ones I’ve seen in some videos. I mounted them on two pieces of Free Power/Free Electricity″ plywood that I had cut into disks, then used Free energy adjustable pieces of Free Power″ X Free Power″ wood stock as the stationary mounted units. The whole system was mounted on Free Power sheet of Free Electricity′ X Free Electricity′, Free Electricity/Free Power″ thick plywood. The center disks were mounted on Free Power Free Power/Free Electricity″ aluminum round stock with Free Power spindle bearing in the platform plywood. Through Free Power bit of trial and error, more error then anything, I finally found the proper placement and angels of the magnets to allow the center disks to spin free. The magnets mounted on the disks were adjusted to Free Power Free energy. Free Electricity degree angel with the stationary units set to match. The disks were offset by Free Electricity. Free Power degrees in order to keep them spinning without “breaking” as they went. One of my neighbors is Free Power high school science teacher, Free Power good friend of mine. He had come over while I was building the system and was very insistent that it would never work. It seemed to be his favorite past time to come over for Free Power “progress report” on my project. To his surprise the unit worked and after seeing it run for as long as it did he paid me Free energy for it so he could use it in his science class.
Of all the posters here, I’m certain kimseymd1 will miss me the most :). Have I convinced anyone of my point of view? I’m afraid not, but I do wish all of you well on your journey. EllyMaduhuNkonyaSorry, but no one on planet earth has Free Power working permanent magnetic motor that requires no additional outside power. Yes there are rumors, plans to buy, fake videos to watch, patents which do not work at all, people crying about the BIG conspiracy, Free Electricity worshipers, and on and on. Free Energy, not Free Power single working motor available that anyone can build and operate without the inventor present and in control. We all would LIKE one to be available, but that does not make it true. Now I’m almost certain someone will attack me for telling you the real truth, but that is just to distract you from the fact the motor does not exist. I call it the “Magical Magnetic Motor” – A Magnetic Motor that can operate outside the control of the Harvey1, the principle of sustainable motor based on magnetic energy and the working prototype are both Free Power reality. When the time is appropriate, I shall disclose it. Be of good cheer.
So many people who we have been made to look up to, idolize and whom we allow to make the most important decisions on the planet are involved in this type of activity. Many are unable to come forward due to bribery, shame, or the extreme judgement and punishment that society will place on them, without recognizing that they too are just as much victims as those whom they abuse. Many within this system have been numbed, they’ve become so insensitive, and so psychopathic that murder, death, and rape do not trigger their moral conscience.
But I will send you the plan for it whenever you are ready. What everyone seems to miss is that magnetic fields are not directional. Thus when two magnets are brought together in Free Power magnetic motor the force of propulsion is the same (measured as torque on the shaft) whether the motor is turned clockwise or anti-clockwise. Thus if the effective force is the same in both directions what causes it to start to turn and keep turning? (Hint – nothing!) Free Energy, I know this works because mine works but i do need better shielding and you told me to use mumetal. What is this and where do you get it from? Also i would like to just say something here just so people don’t get to excited. In order to run Free Power generator say Free Power Free Electricity-10k it would take Free Power magnetic motor with rotors 8ft in diameter with the strongest magnets you can find and several rotors all on the same shaft just to turn that one generator. Thats alot of money in magnets. One example of the power it takes is this.
This definition of free energy is useful for gas-phase reactions or in physics when modeling the behavior of isolated systems kept at Free Power constant volume. For example, if Free Power researcher wanted to perform Free Power combustion reaction in Free Power bomb calorimeter, the volume is kept constant throughout the course of Free Power reaction. Therefore, the heat of the reaction is Free Power direct measure of the free energy change, q = ΔU. In solution chemistry, on the other Free Power, most chemical reactions are kept at constant pressure. Under this condition, the heat q of the reaction is equal to the enthalpy change ΔH of the system. Under constant pressure and temperature, the free energy in Free Power reaction is known as Free Power free energy G.
Thanks, Free Power. One more comment. I doubt putting up Free Power video of the working unit would do any good. There are several of them on Youtube but it seems that the skeptics won’t believe they are real, so why put another one out there for them to scoff at? Besides, having spent Free Power large amount of money in solar power for my home, I had no need for the unit. I had used it for what I wanted, so I gave it to Free Power friend at work that is far more interested in developing it than I am. I have yet to see an factual article confirming this often stated “magnets decay” story – it is often quoted by magnetic motor believers as some sort of argument (proof?) that the motors get their energy from the magnets. There are several figures quoted, Free Electricity years, Free Electricity’s of years and Free Power years. All made up of course. Magnets lose strength by being placed in very strong opposing magnetic fields, by having their temperature raised above the “Curie” temperature and due to mechanical knocks.
Research in the real sense is unheard of to these folks. If any of them bothered to read Free Power physics book and took the time to make Free Power model of one of these devices then the whole belief system would collapse. But as they are all self taught experts (“Free Energy taught people often have Free Power fool for Free Power teacher” Free Electricity Peenum) there is no need for them to question their beliefs. I had Free Power long laugh at that one. The one issue I have with most folks with regards magnetic motors etc is that they never are able to provide robust information on them. Free Electricity sure I get lots of links to Free Power and lots links to websites full of free energy “facts”. But do I get anything useful? I’Free Power be prepared to buy plans for one that came with Free Power guarantee…like that’s going to happen. Has anyone who proclaimed magnetic motors work actually got one? I don’t believe so. Where, I ask, is the evidence? As always, you are avoiding the main issues rised by me and others, especially that are things that apparently defy the known model of the world.
Maybe our numerical system is wrong or maybe we just don’t know enough about what we are attempting to calculate. Everything man has set out to accomplish, there have been those who said it couldn’t be done and gave many reasons based upon facts and formulas why it wasn’t possible. Needless to say, none of the ‘nay sayers’ accomplished any of them. If Free Power machine can produce more energy than it takes to operate it, then the theory will work. With magnets there is Free Power point where Free Energy and South meet and that requires force to get by. Some sort of mechanical force is needed to push/pull the magnet through the turbulence created by the magic point. Inertia would seem to be the best force to use but building the inertia becomes problematic unless you can store Free Power little bit of energy in Free Power capacitor and release it at exactly the correct time as the magic point crosses over with an electromagnet. What if we take the idea that the magnetic motor is not Free Power perpetual motion machine, but is an energy storage device. Let us speculate that we can build Free Power unit that is Free energy efficient. Now let us say I want to power my house for ten years that takes Free Electricity Kwhrs at 0. Free Energy /Kwhr. So it takes Free energy Kwhrs to make this machine. If we do this in Free Power place that produces electricity at 0. 03 per Kwhr, we save money.
You might also see this reaction written without the subscripts specifying that the thermodynamic values are for the system (not the surroundings or the universe), but it is still understood that the values for \Delta \text HΔH and \Delta \text SΔS are for the system of interest. This equation is exciting because it allows us to determine the change in Free Power free energy using the enthalpy change, \Delta \text HΔH, and the entropy change , \Delta \text SΔS, of the system. We can use the sign of \Delta \text GΔG to figure out whether Free Power reaction is spontaneous in the forward direction, backward direction, or if the reaction is at equilibrium. Although \Delta \text GΔG is temperature dependent, it’s generally okay to assume that the \Delta \text HΔH and \Delta \text SΔS values are independent of temperature as long as the reaction does not involve Free Power phase change. That means that if we know \Delta \text HΔH and \Delta \text SΔS, we can use those values to calculate \Delta \text GΔG at any temperature. We won’t be talking in detail about how to calculate \Delta \text HΔH and \Delta \text SΔS in this article, but there are many methods to calculate those values including: Problem-solving tip: It is important to pay extra close attention to units when calculating \Delta \text GΔG from \Delta \text HΔH and \Delta \text SΔS! Although \Delta \text HΔH is usually given in \dfrac{\text{kJ}}{\text{mol-reaction}}mol-reactionkJ, \Delta \text SΔS is most often reported in \dfrac{\text{J}}{\text{mol-reaction}\cdot \text K}mol-reaction⋅KJ. The difference is Free Power factor of 10001000!! Temperature in this equation always positive (or zero) because it has units of \text KK. Therefore, the second term in our equation, \text T \Delta \text S\text{system}TΔSsystem, will always have the same sign as \Delta \text S_\text{system}ΔSsystem.
LoneWolffe kimseymd1 Harvey1 TiborKK Thank You LoneWolffe!! Notice how kimseymd1 spitefully posted his “Free Energy two books!.. ” spam all over this board on every one of my posts. Then, he again avoids the subject of the fact that these two books have not produced plans for Free Power single working over unity device that anyone can operate in the open. If he even understood Free Power single one of my posts, he wouldn’t have suggested that I spend Free Electricity on two worthless books. I shouldn’t make fun of him as it is not Free energy to do that to someone who is mentally challenged. I wish him well and hope that he gets the help that he obviously needs. Perhaps he’s off his meds. Harvey1: I haven’t been on here for awhile. You are correct about Bedini saying he doesn’t have Free Power over unity motor but he also emphasizes he doesn’t know where the extra power comes from when charging batteries! Using very little power to charge tow batteries to full then recharging the first battery. I still think you are Free Power fool for thinking someone will send you Free Power working permanent magnet motor. Building Free Power Bedini motor is fun and anyone can do it! I am on my third type but having problems!
Free Energy The type of magnet (natural or man-made) is not the issue. Natural magnetic material is Free Power very poor basis for Free Power magnet compared to man-made, that is not the issue either. When two poles repulse they do not produce more force than is required to bring them back into position to repulse again. Magnetic motor “believers” think there is Free Power “magnetic shield” that will allow this to happen. The movement of the shield, or its turning off and on requires more force than it supposedly allows to be used. Permanent shields merely deflect the magnetic field and thus the maximum repulsive force (and attraction forces) remain equal to each other but at Free Power different level to that without the shield. Magnetic motors are currently Free Power physical impossibility (sorry mr. Free Electricity for fighting against you so vehemently earlier).
Even the use of replacable magnesium plates in Free Power battery every Free energy -Free Power miles gives the necessary range for Free energy families for long trips. Magnet-only motors are easy to build. There are plans around. They are cheap to build. Trouble is no one knows how to get them to spin unaided. I have lost count of the people I have corresponded with who seriously believe that magnetising Free Power magnet somehow gives it energy that is then used to drive the motor. Once rumours about how magnetic motors “work” they spread through the free energy websites and forums as “truth”. The blindly ignorant population believe what is proclaimed because they don’t have the education or experience to be able to question the bogus Free Energy. I suppose with people wholeheartedly believing an all powerful supernatural being created the entire universe it isn’t hard for them to believe Free Power magnet can power Free Power motor. Both thoughts demonstrate ignorance. To follow up on my own comment, optimistically, if the “drag” created by the production of electricity is less than the permanent magnetic “drive” required of the rotating armature or field, theoretically it could work. Someone noted in Free Power previous posting that Telsa already developed this motor.
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# Indirect study of $^{12}$C($\alpha$, $\gamma$)$^{16}$O reaction
* Corresponding author
Abstract : The radiative capture reaction 12 C(α, γ) 16 O plays an important role in helium burning in massive stars and their subsequent evolution [1]. However, despite various experimental studies, the cross section of this reaction at stellar energies remains highly uncertain. The extrapolation down to stellar energy (Ecm ∼300 keV) of the measured cross sections at higher energies is made difficult by the overlap of various contributions of which some are badly known such as that of the 2 + (Ex=6.92 MeV) and 1 − (Ex=7.12 MeV) sub-threshold states of 16 O. Hence, to further investigate the contribution of these two-subthreshold resonances to the 12 C(α, γ) 16 O cross section, a new determination of their α-reduced widths and so their α-spectroscopic-factors was performed using 12 C(7 Li,t) 16 O transfer reaction measurements at two incident energies and a detailed DWBA analysis of the data [2]. The measured and calculated differential cross sections are presented as well as the obtained spectroscopic factors and the α-reduced widths as well as the assymptotic normalization constants (ANC) for the 2 + and 1 − sub-threshold states. Finally, the results obtained from the R-matrix calculations of the 12 C(α, γ) 16 O cross section using our obtained α-reduced widths for the two sub-threshold resonances are presented and discussed.
Domain :
http://hal.in2p3.fr/in2p3-01272112
Contributor : Michel Lion <>
Submitted on : Wednesday, February 10, 2016 - 11:23:49 AM
Last modification on : Friday, March 6, 2020 - 2:02:16 AM
### Citation
F. Hammache, N. Oulebsir, P. Roussel, M.G. Pellegriti, L. Audouin, et al.. Indirect study of $^{12}$C($\alpha$, $\gamma$)$^{16}$O reaction. Sixth International Conference on Nuclear Physics in Astrophysics (NPA-VI), May 2013, Lisbon, Portugal. pp.012007, ⟨10.1088/1742-6596/665/1/012007⟩. ⟨in2p3-01272112⟩
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# DIY Cube
Time Limit: 2000/2000 MS (Java/Others)
Memory Limit: 65535/32768 K (Java/Others)
## Description
Mr. D is interesting in combinatorial enumeration. Now he want to find out the number of ways on painting the vertexes of a cube. Suppose there are C different colors and two paintings are considered the same if they can transform from one to another by rotation.
## Input
There are multiple test cases in the input, the first line of input contains an integer denoting the number of test cases.
For each test case, there are only one integer C, denoting the number of colors. (1 <= C <= 1000000000)
## Output
For each test case, output the the number of painting ways. And if the number is equal or larger than 1015, output the last 15 digits.
## Sample Input
3
1
2
112
## Sample Output
Case 1: 1
Case 2: 23
Case 3: 031651434916928
zhengfeng
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Question
# Which of the following protocols did aim for reducing emission of chlorofluorocarbons into the atmosphere?
A
Montreal Protocol
B
Kyoto Protocol
C
Gothenburg Protocol
D
Geneva Protocol
Solution
## The correct option is B Montreal ProtocolTo control the deleterious effect of the stratospheric ozone depletion an international treaty was signed at Montreal, Canada in $$1987$$. It is popularly known as Montreal protocol.The protocol aimed at reducing the emission of chlorofluorocarbons into the atmosphere.So, the correct answer is 'Montreal protocol'Botany
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# Detectors¶
Module for describing the detection of transmitted waves and different detector types.
class abtem.detect.AbstractDetector(max_detected_angle=None, save_file=None)[source]
Abstract base class for all detectors.
property save_file
The path to the file for saving the detector output.
Return type
str
class abtem.detect.AnnularDetector(inner, outer, offset=None, save_file=None)[source]
Annular detector object.
The annular detector integrates the intensity of the detected wave functions between an inner and outer integration limit.
Parameters
• inner (float) – Inner integration limit [mrad].
• outer (float) – Outer integration limit [mrad].
• offset (two float, optional) – Center offset of integration region [mrad].
• save_file (str, optional) – The path to the file for saving the detector output.
allocate_measurement(waves, scan=None)
Allocate a Measurement object or an hdf5 file.
Parameters
• waves (Waves object) – An example of the
• scan (Scan object) – The scan object that will define the scan dimensions the measurement.
Returns
The allocated measurement or path to hdf5 file with the measurement data.
Return type
Measurement object or str
copy()[source]
Make a copy.
Return type
AnnularDetector
detect(waves)[source]
Integrate the intensity of a the wave functions over the detector range.
Parameters
waves (Waves object) – The batch of wave functions to detect.
Returns
Detected values as a 1D array. The array has the same length as the batch size of the wave functions.
Return type
1d array
property inner
Return type
float
integrate(diffraction_patterns)[source]
Integrate diffraction pattern measurements on the detector region.
Parameters
diffraction_patterns (2d, 3d or 4d Measurement object) – The collection diffraction patterns to be integrated.
Returns
Return type
Measurement
property outer
Return type
float
property save_file
The path to the file for saving the detector output.
Return type
str
show(waves, **kwargs)
Visualize the detector region(s) of the detector as applied to a specified wave function.
Parameters
• waves (Waves or SMatrix object) – The wave function the visualization will be created to match
• kwargs – Additional keyword arguments for abtem.visualize.mpl.show_measurement_2d.
class abtem.detect.FlexibleAnnularDetector(step_size=1.0, save_file=None)[source]
Flexible annular detector object.
The FlexibleAnnularDetector object allows choosing the integration limits after running the simulation by radially binning the intensity.
Parameters
• save_file (str) – The path to the file used for saving the detector output.
allocate_measurement(waves, scan=None)
Allocate a Measurement object or an hdf5 file.
Parameters
• waves (Waves object) – An example of the
• scan (Scan object) – The scan object that will define the scan dimensions the measurement.
Returns
The allocated measurement or path to hdf5 file with the measurement data.
Return type
Measurement object or str
copy()[source]
Make a copy.
Return type
FlexibleAnnularDetector
detect(waves)[source]
Integrate the intensity of a the wave functions over the detector range.
Parameters
waves (Waves object) – The batch of wave functions to detect.
Returns
Detected values. The array has shape of (batch size, number of bins).
Return type
2d array
property save_file
The path to the file for saving the detector output.
Return type
str
show(waves, **kwargs)
Visualize the detector region(s) of the detector as applied to a specified wave function.
Parameters
• waves (Waves or SMatrix object) – The wave function the visualization will be created to match
• kwargs – Additional keyword arguments for abtem.visualize.mpl.show_measurement_2d.
property step_size
Return type
float
class abtem.detect.PixelatedDetector(max_angle='valid', resample=False, mode='intensity', save_file=None)[source]
Pixelated detector object.
The pixelated detector records the intensity of the Fourier-transformed exit wavefunction. This may be used for example for simulating 4D-STEM.
Parameters
• max_angle (str or float or None) – The diffraction patterns will be detected up to this angle. If set to a string it must be ‘limit’ or ‘valid’
• resample ('uniform' or False) – If ‘uniform’, the diffraction patterns from rectangular cells will be downsampled to a uniform angular sampling.
• mode ('intensity' or 'complex') –
• save_file (str) – The path to the file used for saving the detector output.
allocate_measurement(waves, scan=None)[source]
Allocate a Measurement object or an hdf5 file.
Parameters
• waves (Waves or SMatrix object) – The wave function that will define the shape of the diffraction patterns.
• scan (Scan object) – The scan object that will define the scan dimensions the measurement.
Returns
The allocated measurement or path to hdf5 file with the measurement data.
Return type
Measurement object or str
detect(waves)[source]
Calculate the far field intensity of the wave functions. The output is cropped to include the non-suppressed frequencies from the antialiased 2D fourier spectrum.
Parameters
waves (Waves object) – The batch of wave functions to detect.
Return type
ndarray
Returns
• Detected values. The first dimension indexes the batch size, the second and third indexes the two components
• of the spatial frequency.
property save_file
The path to the file for saving the detector output.
Return type
str
class abtem.detect.SegmentedDetector(inner, outer, nbins_radial, nbins_angular, rotation=0.0, save_file=None)[source]
Segmented detector object.
The segmented detector covers an annular angular range, and is partitioned into several integration regions divided to radial and angular segments. This can be used for simulating differential phase contrast (DPC) imaging.
Parameters
• inner (float) – Inner integration limit [mrad].
• outer (float) – Outer integration limit [mrad].
• nbins_angular (int) – Number of angular bins.
• save_file (str) – The path to the file used for saving the detector output.
allocate_measurement(waves, scan=None)
Allocate a Measurement object or an hdf5 file.
Parameters
• waves (Waves object) – An example of the
• scan (Scan object) – The scan object that will define the scan dimensions the measurement.
Returns
The allocated measurement or path to hdf5 file with the measurement data.
Return type
Measurement object or str
copy()[source]
Make a copy.
Return type
SegmentedDetector
detect(waves)[source]
Integrate the intensity of a the wave functions over the detector range.
Parameters
waves (Waves object) – The batch of wave functions to detect.
Returns
Detected values. The first dimension indexes the batch size, the second and third indexes the radial and angular bins, respectively.
Return type
3d array
property inner
Return type
float
property nbins_angular
Number of angular bins.
Return type
int
property nbins_radial
Return type
int
property outer
Return type
float
property save_file
The path to the file for saving the detector output.
Return type
str
show(waves, **kwargs)
Visualize the detector region(s) of the detector as applied to a specified wave function.
Parameters
• waves (Waves or SMatrix object) – The wave function the visualization will be created to match
• kwargs – Additional keyword arguments for abtem.visualize.mpl.show_measurement_2d.
class abtem.detect.WavefunctionDetector(save_file=None)[source]
Wave function detector object
The wave function detector records the raw exit wave functions.
Parameters
save_file (str) – The path to the file used for saving the detector output.
allocate_measurement(waves, scan)[source]
Allocate a Measurement object or an hdf5 file.
Parameters
• waves (Waves or SMatrix object) – The wave function that will define the shape of the diffraction patterns.
• scan (Scan object) – The scan object that will define the scan dimensions the measurement.
Returns
The allocated measurement or path to hdf5 file with the measurement data.
Return type
Measurement object or str
detect(waves)[source]
Detect the complex wave function.
Parameters
waves (Waves object) – The batch of wave functions to detect.
Returns
The arrays of the Waves object.
Return type
3d complex array
property save_file
The path to the file for saving the detector output.
Return type
str
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How to make a good color intensity scale?
I am by no means good in statistics, but I think I have come to the right place. My question is simple:
My problem consists of comparing the population of several states in a small country, but some states have a population of 3000,000 and some a population of 2,000.
I am painting it on a map, and the "intensity" of the color depends on how the population of every state compares to the population of the whole country.
The problem is that the states with a lot of population are shown with really intense colors and the small states barely have any color.
Is there an easy way to "normalize" or make the data comparable?
I dont know if I am explaining myself properly but I hope some one can help me. Please comment if my question is not clear and I will clarify.
Thank you for your help!
• I would suggest you check out the visualization tag over at the gis stack exchange site for examples gis.stackexchange.com/questions/tagged/visualisation – Andy W Mar 20 '11 at 16:07
• Along that same line, you might want to check out the gradients on www.0to255.com . – Pete Wilson Mar 20 '11 at 19:08
• Some of the maps packages for R have built-in colour codes that prevent this kind of issue, but is that what you were asking about? – Fr. Mar 21 '11 at 22:32
• I am using this on a custom map, and the obvious approach (to divide each value by the total population) does give me a value between 0 and 1 (I then use this value to choose the "intensity" of the color). The problem is that there are values that are too far appart, so some states look completely colored and some have almost no color at all. I know statistically speaking this is correct but I want to make the data representation more relevant and easier to understand. – Zebs Mar 22 '11 at 1:05
• Why use uniform breaks? Why not a log scale? Or perhaps in your application you could choose breakpoints that have some meaning (e.g. rural/suburban/urban). – JMS May 13 '11 at 21:49
I'm sorry, but to me it sounds like you are trying to fix what isn't broken. In fact, you might even be trying to break what isn't broken. When you have a quantitative variable (here, population) that spans a wide range, then whatever metric you use to represent it should also span a wide range.
But for all things related to color (and esp. maps), the key source is, I think ColorBrewer
• I am trying to break something; I know the vales I am getting are statistically correct, but I want to make it easier for the users to understand the data. It's a UI decision. – Zebs Mar 22 '11 at 4:04
• @Zebs: Bend, more like.. – naught101 May 1 '12 at 7:44
Good question, One solution is to rescale the colors to have them more uniformly distributed, or to a distribution with lower tails... but then your legend has to be clear enough because deforming the scale, somehow, is unfair...
For example, in R, rescaling a normal to a uniform . (what you have maybe goes more the other way since you have large tails and you want them smaller, but the principle is the same)
X=array(rnorm(10000),c(100,100))
ramp=colorRamp(c("blue","cyan","white","yellow","red"),space ="rgb")
kleur <- rgb( ramp(seq(0,1,length=200)),max = 255)
par(mfrow=c(1,2))
image(X,col=kleur)### image without rescaling
Fn=ecdf(X)
ScaledX=array(Fn(X),c(100,100))
image(ScaledX,col=kleur)
You could divide by the total population. This would ensure that everything lies between 0 and 1. If the scales are still too disparate, consider a log scale.
I feel awkward asking it, but are you really committed to using colour to portray a quantitative amount? Is there no way to put a bar in each state, whose height represents the quantity?
Another way might be to show the map with areas representing the geographic areas, together with a map where each state's area is proportional to the population size - similar to how the sensory homunculus does. But that would be a painful amount of drawing - I don't know of any way to automate that (though it may exist)
• Good remark ! – robin girard Mar 20 '11 at 13:46
• Many mapping software platforms have the capabilities mentioned in this post. The distortions based on attributes when it comes to maps are frequently referred to as cartograms. See gis.stackexchange.com/q/7406/751 . That being said, bars placed happenstance in a map aren't any easier to visualize than colors. When the bars aren't side by side they are difficult to make relative comparisons, which isn't as big a problem with a color scale. – Andy W Mar 20 '11 at 16:12
• I agree that bars less than optimal on a map. Another way to do it is to have gridded distortions, like here: viewsoftheworld.net/?p=832. Personally, I often find these quite hard to decipher, but they can be done quite well, depending on the amount of distortion. – naught101 May 1 '12 at 7:48
Compare the population of several states in a small country.
Since some states have a population of 3000,000 and some a population of 2,000. Is there an easy way to "normalise" or make the data comparable?
Aim of normalising your data before mapping
This answer will be lacking since I am not sure of the context of why you are making the map.
Nevertheless, here are some thoughts to explore: Normalise your data so that the map provides interesting meaning to the map's potential readers, so they can link what they see on your map to some concept they normally think about. Basically, I think your new normalised numbers should be linked to some qualitative concept that the map readers find interesting to understand (random tidbit: Measure = Quantity x Quality, Hegel).
Two proposed ways to normalise your data
1. In order to give a sense of how much open space is in each state.
Create a new state variable for population density by calculating the population divided by total state area.
2. In order to make the coloring of the states contrast with one another.
Create a new state variable by calculating the deviation from the mean of each state. For example, say you have 3 states with populations as follows:
• State A is 100.
• State B is 50.
• State C is 1.
The mean will be be about 50.
The new variable's values for each state will be as follows:
• State A is +50 (color intense green).
• State B is 0 (color grey).
• State C is -49 (color intense red).
You can use any color scheme where positive numbers contrast with negative numbers (google 'colorbrewer' for lots of examples of color schemes for maps).
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# frequency and type of pulse for time domain reflectometer (TDR) [closed]
I would like to apply the time domain reflectometry method in an fpga. However, I could not find information about the frequency and type of pulse used by reflectometer.
Can anyone give me a guide?
## closed as too broad by PeterJ, Harry Svensson, Chupacabras, Sparky256, Voltage SpikeDec 27 '17 at 5:29
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
• I think the better question not which frequency, but rather what rise time (how fast is the rising edge) and how long is the pulse. Both will depend (I think) to some extent on the frequencies of interest (what signal your cable will carry in normal operation) and the length of the cable. But, I've never worked with that type of thing so I'm going to watch this question and see what the knowledgeable folks have to say. – JRE Dec 23 '17 at 11:48
• I think it will also depend on the length of what your measuring, the max pulse length can not exceed the phase speed (material corrected) / cable length. – MadHatter Dec 23 '17 at 16:24
• The time resolution will be determined by the delay in the line and then the delay in the gates, but you don't have to do any of this, because there are IC's available that have this problem solved for you, better than any FPGA can solve it. – Voltage Spike Dec 27 '17 at 5:29
However, I could not find information about the frequency and type of pulse used by reflectometer.
Typical TDR instruments use a square pulse and measure the response only to one edge, so they are measuring the response to an approximation to the unit step function $u(t)$.
The frequency range they're able to measure depends on the rise time of the stimulus pulse. Since the Fourier transform of even an ideal $u(t)$ has $1/f$ characteristic, and the feedlines to the device under test (DUT) typically have low-pass characteristics as well, it is quite challenging to measure very high frequencies with TDR, although commercial instruments from Tektronix or Keysight can reach the neighborhood of 20 GHz at least.
Another challenging in implementing TDR with an FPGA will be sampling the response with low noise and low jitter.
• Based on your comment, does it mean that I can only apply low frequency pulse on fpga? If it does, is there a formula to calculate the desired frequency? Besides, the challenge you mentioned (low noise and low jitter), does it mean that I cannot check the changes easily? – Erile Dec 24 '17 at 3:11
• FPGAs nowadays can produce some pretty fast edges. You'd have to read the datasheet for your device to know how fast. – The Photon Dec 24 '17 at 3:14
• Ok, I will check it. It is something related to the pll, right? Then, is there any formula can calculate the frequency I'm going to use? Or I can refer to previous comment of @MadHatter? – Erile Dec 24 '17 at 3:53
• It's nothing to do with the PLL. The key spec is the rise time of the digital outputs, in whichever IO configuration you plan to use. – The Photon Dec 24 '17 at 3:56
• Sorry, but I didn't get your words. Can you please explain more about it please? – Erile Dec 24 '17 at 9:01
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# Partial Differential Equations
3
Does anyone know about the homotopy analysis method in detail?
Dear researchers
I can not understand what happen in homotopy analysis method when it used for PDEs. could you help me by simple examples in details, particularly to show how R_(m-1) is calculated and how to plot the solution?
Dear Arash,
I have been reading papers on it too. However I would like to see an illustration on how the respective numerical algorithm is implemented through various steps to solve a given DE. I do not like the pre-designed matlab/mathematica packages which most of the times are black-box.
7
Anyone familiar with PDE theory on domains of infinite dimensional Banach spaces?
Does exist a theory for partial differential equations
on bounded domains or manifold of a generic Banach space (or Hilbert space)?
For example, consider G a bounded domain of a banach space E
and consider the problem
1. -D2u(x) = f(x), x in G
2. u(x) = 0, x in \partial G
where f in C0,a(G). If E is an n-euclidean space I know there is a
existence and uniqueness result for this problem.
In the case that E is a infinite dimensional Banach space,
what we already know about that problem? Is there a theory for it?
See for instance Gda Prato and JZabczyk,Second Order Partial Differential Equations in Hilbert Spaces ,vol293-London Math Soc Lect Notes ,Cambridge,UK,2002 or my articles ,OR SPECIALLY MY ORIGINAL STUDY 6.4 ON THE WEAK POISSON PROBLEM IN INFINITE DIMENSIONS -P183 WHICH HAS APPEARED ON MY RESEARCH MONOGRAPH http://www.worldscientific.com/worldscibooks/10.1142/6856
6
In Crank-Nicolson method of solving one dimensional heat equation, what can be the maximum value of r (=k/h^2; k = time step, h = space step)?
I was trying to solve an one dimensional time dependent partial differential equation (similar to that of one dimensional heat equation) using Crank-Nicolson method. It would be very helpful if someone can tell me what could be the maximum value of r (=k/h^2; k = time step, h = space step) without affecting the credibility of the solution.
If you are dealing with simple heat equation, using Crank Nicolson scheme. The scheme will be unconditionally stable according to von-Neumann stability analysis,i.e the scheme will be stable for all values of r, but i need to point out one point that you have to take care of the accuracy issue, means that we should choose h and k small enough in order to get highly accurate results. this will be very important if you are dealing with Crank -Nicolson and fourth order approximation for the space derivatives.
best wishes
2
Can Pazy's Theorem 3.1.1 be extended to the case of nonlinear semigroups?
In Pazy's book"Semigroup of Linear Operators and Applications to Partial Differential Equations ", the Theorem 3.1.1: Let $X$ be a Banach space and $A$ be the infinitesimal generator of a $C_0$ semigroup $T(t)$ on $X$, satisfying $\|T(t)\|\leq Me^{wt}$. If $B$ is a bounded linear operator on $X$ then $A+B$ is infinitesimal generator of a $C_0$ semigroup $S(t)$ on $X$, satisfying $\|S(t)\|\leq Me^{(w+M\|B\|)t}$.
Now, if $A$ be the generator of a nonlinear semigroup $T(t)$ on $X$, can we still have the same result? and why? Thanks.
In the case when A is a maximal monotone operators check out: Brezis, H: Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics studies 5, North-Holland Publishing Co., (1973).
• Gabriel Caicedo asked a question:
Open
Can somebody tell me how to solve, or give me the solution to this Partial Differential Equation: νz*(∂T/∂z)=α/r*(∂/∂r (r ∂T/∂r))+ μ*((∂/∂r)vz)^2?
I'm trying to find a temperature profile for a fluid and that's the equation i've found by now.
5
Is it possible to solve a heat equation with Laplace Beltrami condition in 2D? How approximate the Laplace-Beltrami operator ?
Any suggestion by using the finite element method?
Look at the next tutorial: http://perception.inrialpes.fr/~Horaud/Talks/ECCV10-Tutorial4-Horaud.pdf
40
What are the advantages of numerical method over analyatical method?
We use several numerical methods. Why do we use it and is it really applicable?
11
What are the main drawbacks of traditional approaches to solving partial differential equations?
Hello, What are the main drawbacks of traditional approaches (e.g., finite element method, finite difference method, finite volume method, etc) to solving partial differential equations? Can they find solutions of any partial differential equations with 100% probability? Are they computationally expensive? Thanks!
Dear George,
Thanks for your suggestion and I've found Elemer Rosinger's works on his RG page.
Best regards, Xin
2
How might one apply differential transformation method in BBME (benjamin bona mahony equation)?
Can any body help me to check the transform equation.?
Dear Mohamed : The link above is a good description of this method. And you may solve online the differential equation of interest . THe tool : Maxima software by William Schelter. The yamwi (yet another maxima web interface) is here :
http://maxima.cesga.es/
In order to apply the method you must know the way a recurrence equation is solved by Maxima. See the manual online here : http://maxima.sourceforge.net/docs/manual/de/maxima_70.html
18
Is their any numerical solution for 3rd order partial differential equations?
I was working on a simulation of Heat transfer block process that contain (liquid, steam and superheated steam). Unfortunately, all the equations of the heat transfer model consist of a 3rd order P.D.E., I already know the finite element, finite volume and finite difference method. However, applying those methodology never worked with my situation here
So, I wonder if their is a more simpler way for numerical solving of P.D.E. I am currently using MATLAB r2010a to run the simulations
Thanks in advance for any contribution to this subject
Best Regards,
For a finite element modelling, You could try a better way using FreeFem++. A variational formulation need to be defined for the equation system ( heat transport, motion and continuity equation). Continuos and discontinuous polinomial can be considered. The implementation is based in your integral mathematical equations and the boundary conditions..
18
Can anybody suggest me the best software for Partial Differential Equations (PDEs) ?
I want to solve partial differential equations (PDEs), which contains both space (x) and time (t). What is the best software for this purpose?
I also want to know the most appropriate numerical algorithm for this so that I can write a program to solve PDEs.
All types of suggestions are highly appreciable.
Thank You.
I have used Mathematica to solve PDEs with no problems so far. I suggest you to use Mathematica with "NDsolve" solver and apply the Method of Line (MOL).
The MOL is an elegant semi-analytical approach that takes care about the stability of your solution specially the stiff problems.
8
How do I solve a system of partial differential equations?
I am stucked at a point:
I want to solve 3 Partial Differential Equations with 2 variables.
Thanks
Note: 1=v, and 2=Phi
Mr. Mittal is right except that Psi ^2 =cotv f'+C.
18
What methods exist about finding exact solution of nonlinear partial differential equations?
Any suggestion/resources are appreciated.
Thank you so much.
There is no exact solution to nonlinear pde.
3
Why are PDE's used to describe stochastic Calcium oscillation dynamics?
Why is that we used ODE's to describe deterministic calcium oscillation dynamics abut use PDE's to describe the stochastic calcium oscillation dynamics?
Thank you both @Manuel and @Bahram for clarifying my question.
5
How does one prove this Sobolev-type inequality in R3 ?
How does one derive the inequality
max { | u(x) |: x in R3 } <= || D u ||_{L2}^{1/2} || D2 u ||_{L2}^{1/2}
for smooth, compactly supported functions u : R3 --> R?
Dear Colleagues,
Thank you so much for kindly assisting me with the problem above, I trully appreciated your help. In particular, I would like to thank my dear friend and collaborator Prof. Lucas Oliveira for calling Xie's 1991 paper to my attention. There one finds a very nice derivation indeed (including the determination of the optimal value for the constant sitting in this inequality).
Warm regards,
Paulo Zingano
19
How do I solve higher order coupled PDE's?
Hello Everyone
I have to solve a higher order coupled PDE with initial and boundary conditios.
I have tried Matlab pdex4 and pde but could not as they dont allow higher derivatives wrt t. My equations look like as follows
d2v/dt2 = d4v/dx4 + F
d2w/dt2 =d2v/dx2 + F
In Matlab/pde apparently it doesnt allow higher derivative on left handside.
I believe this problem can be solved analytically using the method of separation of variables. It can be shown that one set of the possible solutions are the infinite series:
V(x, t) = p(t) + Σ [a1*cos(2β2t) + a2*sin(2β2t)]*[b1*Cosh(βx)cos(βx)+ b2*Cosh(βx)sin(βx) + b3*Sinh(βx)cos(βx) + b4*Sinh(βx)sin(β x)]
w(x, t) = p(t) - (1/2)* Σ (1/ β2)* [a1*cos(2β2t) + a2*sin(2β2t)]*[ -b1*Sinh(βx)sin(βx)+ b2*Sinh(βx)cos(βx) - b3*Cosh(βx)sin(βx) + b4*Cosh(βx)cos(βx)]
Where d2p/dt2 = F(t) and the series runs from β = certain value to ∞ and the constants a1, a2, b1.....b4 are functions of. β
Four boundary conditions and need to applied to determine the constants b1, b2, b3, b4. When the boundary conditions are applied an eigen value equation will be obtained for determining the admissible values of β. The coefficients a1 and a2 can be determined as Fourier coefficients when the two initial conditions are applied. When some boundary values are non-zero the above solutions need to be suitably modified by adding linear function of x..Note that another solution set is also possible in which the time functions are replaced by exp(-2β2t)
1
How to solve a pantograph equation on maple?
I am working on wavelets. I want to solve pantograph equation by using legendre polynomial.
i am not expert in this field
3
How to solve the equation of beams on Pasternak nonlinear foundation?
Here is a nonlinear equation for beam on elastic foundation
d''''(y)/dx''''-D*d''y/dx''+k*y=f(x)
I've solved above equation with the aid of Laplace Transform. The closed form equations are obtained.
I'm going to solve the following equation which has a nonlinear term:
d''''(y)/dx''''-D*d''y/dx''+k*[y/(1+G*y)]=f(x)
f(x)=FF for x<L
f(x)=0 for x>L
How could I solve this nonlinear equation?
comment:
1- d'x stands for 1st derivative of f with respect to x
2- D, k and G are constants
Depending on your application you could also use Timoshenko beam equation which is just a second order ode.
In principle I would use a numerical solution, either to solve Euler-Bernoulli, or Timoshenko beam eqs, i.e. by employing a finite differences scheme.
I could suggest some references if you want to go with the numerical approach, not sure if this is what you want though.
8
Hi everybody. Does anyone know if i can solve numerically the 2-D pure advection equation with the Galerkin Method?
I am working on my Thesis (Bachelorarbeit), and I am having trouble solving a simple equation, because of the non symmetric advective matrix.
My thesis is about a 2-D pollution transport system without diffusion.
The transport equation is:
dc/dt=-v*∇C
After applying Galerkin Method, using nodes in the vertex of a rectangle.
The weak form, at the end is
[M]*C°+[K]*C=0,
where C° = dc/dt.
M = Mass Matrix.
C= Concentration
For example, for one Dimension , [K ]is
K = U/2* [-1 1
-1 1]
After global assembling of my system, the result are zeros in the diagonal, and there is nothing that i can do. to find the Concentration in the unknown nodes.
I have read a lot about stabilization methods, like Petrov Galerkin or Artificial Diffusion, but all require a little of Diffusion, specifically to determine the Peclet Number (Pe), but i have none Diffusion.
I hope someone can help me, how to proceed.
Greetings from Chile
Juan Ignacio Correa.
Hello Juan Ignacio Correa
In the expression [M]*C°+[K]*C=0, the partial derivative Co = ∂C/∂t needs to be replaced by a finite difference approximation Co = ∂C/∂t = (Ct+δt-Ct)/δt and the other C with the matrix coefficient [K] needs to be expressed as C = θCt + (1-θ)Ct+δt where θ is a parameter between 0 and 1 that attempts to improve the accuracy of the approximation. Substituting these into your expression [M]*C°+[K]*C=0 and rearranging gives you the complete equation. Taking θ = 1/2 gives the Crank-Nicholson approximation scheme. Taking θ = 0 gives an alternative fully implicit scheme. Please, try this out by yourself. Note that in my own formulation Co denotes nodal concentration at time t = 0. Wishing you success.
4
Why when we use Homotopy Perturbation Method for solving nonlinear PDEs, at increasing time, approximate solution away to exact solution?
when increase time will increase absolute error.
dear friend- I am serious that you should ask prof je= Huan He..... he will answer you....he is a nice modest man
8
Is anyone familiar with transformations of diagonal Laplacians?
May be who knows about the tranformations from one diagonal Laplacian to another diagonal Laplacian (change of variables)? Especially I'm interested in a case of transformation of Cartesian Laplacian to diagonal Laplacian with rational coefficients in 3D case.
3
What is the name of this inequality?
(a+b)^{p}<=2^{p-1}(a^{p}+b^{p}) where P>=1 and a,b>0
Thank you
16
Please. help! Is there any method to solve this nonlinear PDE using similarity?
I stumbled upon equation j and l in my research.
I found some books related to fluid dynamics, it says solution to j is the formula k
(it didn't give the procedure, just solution)
as you can see j and l looks similar. and similar boundary condition and mass balance.
Assuming that solution to j is k, Is there any method to solve l and get the formula of h2?
I tried to solve j analytically by using separation of variables, but it gives second order nonlinear ODE; and I made Wolfram Alpha solve it. but still it didn't respond.
so I decided to circumvent reaching the solution by using the similarity
Regarding the comment "It is likely that some of the assumptions use to derive its equations break down for very small t, where this solution becomes singular." True. But for a range of similarity problems one can prove that the similarity solution emerges from quite general initial conditions---including ones that are within the domain of validity of the derivation of the equations. That is, often the similarity solution is widely relevant irrespective of its notional singularity at time zero.
For one example of this see the paper S. A. Suslov and A. J. Roberts. Similarity, attraction and initial conditions in an example of nonlinear diffusion. J. Austral. Math. Soc. B, 40(E):E1–E26, Oct. 1998. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/453.
8
Is Fractional Calculus useful in every branch of science?
I.e. Applications of fractional calculus.
Of course. It has many applications in different fields of the science: nowadays its applications in economy, signal processing and image processing are among the most interesting. Fractional calculus can be applied also to other fields of Mathematics: for instance, I used it to investigate in the theory of special function. If you are interested, you can read my attached article on it.
• Source
##### Article: Fractional Derivative of the Hurwitz ζ-Function and Chaotic Decay to Zero
[Hide abstract]
ABSTRACT: In this paper the fractional order derivative of a Dirichlet series, Hurwitz zeta function and Riemann zeta function have been explicitly computed by using the Caputo fractional derivative in the Ortigueira sense. It has been shown that the obtained results are a natural generalization of the integer order derivative. Some interesting properties of the fractional derivative of the Riemann zeta function have been also investigated to show that there is a chaotic decay to zero (in the Gaussian plane) and a promising expression as a complex power series.
Journal of King Saud University - Science 04/2015; 10. DOI:10.1016/j.jksus.2015.04.003
4
What is the meaning of the Laplace operator whose one eigenvalue is 1?
This is related to global analysis.
who will send you, profesor sergeyyy :)
8
Do you have solved examples of system of nonlinear pde using finite element method?
See above
Hello Milind M. Mathapati
In order to understand how to solve a system of coupled nonlinear partial differential equations refer to pages 37-39 of the text book," I. M. Smith and D.V. Griffiths (1988). Programming the Finite Element Method, 2nd ed., John Wiley & Sons "
Therein is explained how the steady state coupled 2D nonlinear Navier-Stokes equations is discretized using the continuous Galerkin FEM with quasi-linearization of the nonlinear terms. The Navier-Stokes PDEs couples u,v and P. You can discretize your own coupled system via analogy.
I hope you will find this suggestion helpful.
13
Is there any numerical solution to solve nonlinear coupled PDEs?
Is there any numerical solution to solve nonlinear coupled PDEs?
Hi, Mrs. Mohsen
Yes, there are quite different methods. I believe COMSOL software make big progresses for solving system of PDE's. you can use module "Mathematic", "Coefficient Form of PDEs". there is a tutorial video about that in internet. As Mrs. samuli described, there are other software as well. I like to add MATLAB too. If you want to learn about numerical method I recommend you to focus first on Newton-Raphson method which is involve making Jacobian matrix, because it is very fast. If you are going to write a code( instead od using software) it is better to first select appropriate solution method which is depend on the type of PDE( like, Eigen value PDE or Time dependency, or number of dimension) then try to implement different steps of an individual method.
good luck
5
How can I solve a nonlinear partial differential equation [U_t=Q(t)*(1+(U_x)^2)^(-1/2))] using MATLAB?
Can you suggest how to (analytically or numerically) solve this PDE?
You can use Mathematica program
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# It doesn't have to be regular!
A hexagon is inscribed in a circle of radius $r$. Two of its sides have length 1, another two have length 2, and the last two have length 3. If $r$ is a root of $ax^{3}+bx^{2}+cx+d=0$, where $|a|, |c|$ and $|d|$ are prime numbers, then find the value of $|a|+|b|+|c|+|d|$.
###### This is a problem found in most math books for IIT JEE.
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#### ${\mathit m}_{{{\mathit \Sigma}_{{c}}{(2455)}^{+}}}–{\mathit m}_{{{\mathit \Sigma}_{{c}}{(2455)}^{0}}}$
VALUE (MeV) DOCUMENT ID TECN COMMENT
$\bf{ -1.10 {}^{+0.16}_{-0.08}}$ OUR FIT
• • We do not use the following data for averages, fits, limits, etc. • •
$1.4$ $\pm0.5$ $\pm0.3$
1993
CLE2 See AMMAR 2001
References:
CRAWFORD 1993
PRL 71 3259 Observation of the Charmed Baryon ${{\mathit \Sigma}_{{c}}^{+}}$ and Measurement of the Isospin Mass Splitting of the ${{\mathit \Sigma}_{{c}}}$
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Article
# Compressibility effects on statistics and coherent structures of compressible turbulent mixing layers
Authors:
To read the full-text of this research, you can request a copy directly from the authors.
## Abstract
The effects of compressibility on the statistics and coherent structures of a temporally developing mixing layer are studied using numerical simulations at convective Mach numbers ranging from $M_c=0.2$ to $1.8$ and at Taylor Reynolds numbers up to 290. As the convective Mach number increases, the streamwise dissipation becomes more effective to suppress the turbulent kinetic energy. At $M_c=1.8$ , the streamwise dissipation increases much faster than the other two components in the transition region, even larger than pressure–strain redistribution, correlating with the streamwise elongated vortical structures at a higher level of compressibility. We confirm the existence of the large-scale high- and low-speed structures in the mixing layers, which accompany the spanwise Kelvin–Helmholtz rollers at low convective Mach number and dominate the mixing layer at higher convective Mach number. Conditional statistics demonstrate that the large-scale low-speed structures are lifted upwards by a pair of counter-rotating quasi-streamwise rollers flanking the structures. The small-scale vortical structures have an apparent preference for clustering into the top of the low-speed regions, which is directly associated with high-shearing motions on top of the low-speed structures. The high-speed structures statistically exhibit central symmetry with the low-speed structures. The statistics and dynamics of large-scale high- and low-speed structures in the compressible mixing layers resemble those in the outer region of the turbulent boundary layers, which reveals the universality of the large-scale structures in free shear and wall-bounded turbulence. A conceptual model is introduced for the large-scale high- and low-speed structures in turbulent mixing layers.
## No full-text available
ResearchGate has not been able to resolve any citations for this publication.
Article
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Analysis of turbulence characteristics in a temporal dense gas compressible mixing layer using direct numerical simulation - Volume 893 - Aurélien Vadrot, Alexis Giauque, Christophe Corre
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Bypass transition in boundary layers subject to strong pressure gradient and curvature effects - Volume 888 - Yaomin Zhao, Richard D. Sandberg
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Direct numerical simulation of the spatially developing mixing layer issuing from two turbulent streams past a splitter plate is carried out under mild compressibility conditions. The study mainly focuses on the early evolution of the mixing region, where transition occurs from a wake-like to a canonical mixing-layer-like behaviour, corresponding to the filling-up of the initial momentum deficit. The mixing layer is found to initially grow faster than linearly, and then at a sub-linear rate further downstream. The Reynolds stress components are in close agreement with reference experiments and follow a continued slow decay till the end of the computational domain. These observations are suggestive of the occurrence of incomplete similarity in the developing turbulent mixing layer. Coherent eddies are found to form in the close proximity of the splitter plate trailing edge, that are mainly organized in bands, initially skewed and then parallel to the spanwise direction. Dynamic mode decomposition is used to educe the dynamically relevant features, and it is found to be capable of singling out the coherent eddies responsible for mixing layer development.
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Compressibility exerts a stabilizing influence on a variety of high-speed shear flows such as turbulent mixing layers, transitioning boundary layers and homogeneously sheared turbulence. An important stabilizing feature that is common amongst all shear flows is the velocity-pressure interaction dynamics. In this study, velocity-pressure interactions of individual perturbation or fluctuation modes are investigated using direct numerical simulations and linear analysis in high-Mach-number homogeneous shear flow. For a given perturbation wave mode, the action of pressure is shown to depend on two important factors: the orientation of the perturbation wavevector with respect to the shear plane and the Mach number. It is shown that the streamwise perturbation wave mode rapidly develops a high level of kinetic energy but is self-limiting owing to the action of pressure. On the other hand, the energy of spanwise perturbation wave modes grows unaffected by pressure or Mach number. Oblique modes combine spanwise and streamwise characteristics and are shown to be chiefly responsible for stabilizing effects seen in shear flows. Three regimes of obliqueness of different linear stability characteristics are identified. The critical role of perturbation obliqueness on stabilization is established.
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The kinetic energy exchange between the mean and fluctuating fields is analyzed using large-eddy simulation of a temporally developing compressible mixing layer. A solution database is generated by varying the convective Mach number over the interval of 0.1 ≤ Mc ≤ 0.73 with an initial Reynolds number of Re = 100. Both mean and turbulent kinetic energy equations are considered, and the effect of compressibility on the production, pressure dilatation, and dissipation terms are considered. It is shown that the production term linearly decreases with the increase of Mc within the interval of 0.3 ≤ Mc ≤ 0.73, which in turn leads to a linear reduction of the mixing layer growth rate. The pressure dilatation term in the mean kinetic energy equation is on average positive and transfers internal energy into mean kinetic energy by dilatation, whereas its counterpart in the turbulent kinetic energy equation is on average negative, and transfers turbulent kinetic energy into internal energy by compression. Although the turbulent viscous dissipation term is reduced by increasing the convective Mach number, it remains the primary kinetic energy dissipation mechanism at all convective Mach numbers. At the subgrid-scale level, it is found that the magnitude of the dynamic Smagorinsky model coefficient, Cs, monotonically decreases with the increase of Mc and results in the reduction of all turbulent stress tensor components. Furthermore, increasing the convective Mach number results in a monotonic reduction of subgrid-scale dissipation of the turbulent kinetic energy, whereas the subgrid-scale dissipation of the mean kinetic energy stays almost unaffected. The sum of subgrid-scale pressure dilatation and pressure transport terms, which is modelled with a gradient transport model, is also suppressed by the increase of compressibility and the convective Mach number.
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Direct Numerical Simulation (DNS) data for channel flow at 1025 are used to analyse the interaction between large outer scales in the log-law region - referred to as super-streaks - and the small-scale, streaky, streamwise-velocity fluctuations in the viscosity-affected near-wall layer. The study is inspired by extensive experimental investigations by Mathis, Marusic, and Hutchins, culminating in a predictive model that describes, in a supposedly universal manner, the "footprinting" and "modulating" effects of the outer structures on the small-scale near-wall motions. The approach used herein is based on the examination of joint PDFs for the small-scale fluctuations, conditioned on regions of large-scale footprints. The large and small scales are separated by means of the Huang-Hilbert empirical-mode decomposition, the validity of which is demonstrated by way of pre-multiplied energy spectra, correlation maps, and energy profiles for both scales. Observations derived from the PDFs then form the basis of assessing the validity of the assumptions underlying the model. Although the present observations support some elements of the model, the results imply that modulation by negative and positive large-scale fluctuations differ greatly - an asymmetric response that is not compatible with the model. The study is thus extended to examining the validity of an alternative proposal, which is based on the assumption that a universal description of the small-scale response to the large-scale motions has to rely on the velocity fluctuations being scaled with the large-scales-modified local friction velocity, rather than with the mean value. This proposal is partially supported by the present analysis. Finally, an alternative, new phenomenological model is proposed and examined. (C) 2014 AIP Publishing LLC.
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We investigate the scaling of the energy-containing eddies in the outer part of turbulent wall layers. Their spanwise integral length scales are extracted from a direct numerical simulation (DNS) database, which includes compressible turbulent boundary layers and incompressible turbulent Couette–Poiseuille flows. The results indicate similar behaviour for all classes of flows, with a general increasing trend in the eddy size with the wall distance. A family of scaling relationships are proposed based on simple dimensional arguments, of which the classical mixing length approximation constitutes one example. As in previous studies, we find that the mixing length is in good agreement with the size distribution of the eddies carrying wall-normal velocity, which are active in establishing the mean velocity distribution. However, we find that the eddies associated with wall-parallel motions obey a different scaling, which is controlled by the local mean shear and by an effective eddy diffusivity , where is the compressible counterpart of the friction velocity, and is the thickness of the wall layer. The validity of the proposed scalings is checked against DNS data, and the potential implications for the understanding of wall turbulence are discussed.
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The present paper addresses some topical issues in modelling compressible turbulent shear flows. The work is based on direct numerical simulation (DNS) of two supersonic fully developed channel flows between very cold isothermal walls. Detailed decomposition and analysis of terms appearing in the mean momentum and energy equations are presented. The simulation results are used to provide insights into differences between conventional Reynolds and Favre averaging of the mean-flow and turbulent quantities. Study of the turbulence energy budget for the two cases shows that compressibility effects due to turbulent density and pressure fluctuations are insignificant. In particular, the dilatational dissipation and the mean product of the pressure and dilatation fluctuations are very small, contrary to the results of simulations for sheared homogeneous compressible turbulence and to recent proposals for models for general compressible turbulent flows. This provides a possible explanation of why the Van Driest density-weighted transformation (which ignores any true turbulent compressibility effects) is so successful in correlating compressible boundary-layer data. Finally, it is found that the DNS data do not support the strong Reynolds analogy. A more general representation of the analogy is analysed and shown to match the DNS data very well.
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The effect of wall temperature on the transfer of kinetic energy in a hypersonic turbulent boundary layer for different Mach numbers and wall temperature ratios is studied by direct numerical simulation. A cold wall temperature can enhance the compressibility effect in the near-wall region through increasing the temperature gradient and wall heat flux. It is shown that the cold wall temperature enhances the local reverse transfer of kinetic energy from small scales to large scales, and suppresses the local direct transfer of kinetic energy from large scales to small scales. The average filtered spatial convection and average filtered viscous dissipation are dominant in the near-wall region, while the average subgrid-scale flux of kinetic energy achieves its peak value in the buffer layer. It is found that the wall can suppress the inter-scale transfer of kinetic energy, especially for the situation of a cold wall. A strong local reverse transfer of fluctuating kinetic energy is identified in the buffer layer in the inertial range. Helmholtz decomposition is applied to analyse the compressibility effect on the subgrid-scale flux of kinetic energy. A strong transfer of the solenoidal component of fluctuating kinetic energy is identified in the buffer layer, while a significant transfer of the dilatational component of fluctuating kinetic energy is observed in the near-wall region. It is also shown that compression motions have a major contribution to the direct transfer of fluctuating kinetic energy, while expansion motions play a marked role in the reverse transfer of fluctuating kinetic energy.
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Kinetic energy transfer in compressible homogeneous anisotropic turbulence is studied by numerical simulations of forced anisotropic turbulence (FAT) in a periodic box and homogeneous shear turbulence (HST) at different turbulent Mach numbers Mt and different Taylor Reynolds numbers Reλ. In both FAT and HST, the subgrid-scale (SGS) kinetic energy flux is dominated by the streamwise component at large scales, and tends to be isotropic at small scales. As the turbulent Mach number increases, the compressibility slightly enhances the anisotropy of SGS kinetic energy flux and viscous dissipation. The redistribution of kinetic energy from the streamwise direction to two transverse directions by pressure-strain mainly occurs at large length scales. The kinetic energy transferred from the streamwise component through the pressure-strain is shared unequally by the other two components in HST, which is different from the situation of FAT. In FAT, as the Taylor Reynolds number increases, the total SGS kinetic energy flux and its positive and negative components tend to a Reynolds number asymptotic state at small scales for Reλ≥105. In HST, the positive vertical SGS flux of kinetic energy is significantly enhanced by compression motions, causing the vertical SGS flux component to be larger than the streamwise component at small scales. The interscale energy transfer of the solenoidal mode and dilatational mode is studied by employing Helmholtz decomposition. The dilatational kinetic energy of FAT is nearly isotropic, but that of HST is significantly anisotropic. In HST, the dilatational mode obtains energy not only from the solenoidal mode through nonlinear advection, but also from mean shear by the dilatational production. As the turbulent Mach number increases, the nonlinear advection of HST increases first and then decreases. The dilatational production of HST increases monotonically with the turbulent Mach number, providing the main source of kinetic energy to the dilatational mode at high turbulent Mach number Mt≥0.46.
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The compressibility effect in isothermal hypersonic boundary layer is studied with direct numerical simulation (DNS) using Helmholtz decomposition. The dilatational components of the diagonal Reynolds stress are enhanced by the cold wall condition in the near-wall region. The outward (Q1) and ejection (Q2) events are mainly located in the expansion region, while the inward (Q3) and sweep (Q4) events are primarily situated in the compression region near the wall. It is found that the cold wall condition can enhance the inward (Q3) event mainly in the compression region and enhance the ejection (Q2) event mainly in the expansion region near the wall. In particular, the cold wall can significantly enhance the positive streamwise solenoidal fluctuating velocity and negative wall-normal dilatational fluctuating velocity events. Moreover, the cold wall condition enhances the positive correlation of streamwise velocity fluctuation and fluctuating temperature, and suppresses the negative correlation of wall-normal velocity fluctuation and fluctuating temperature in the near-wall region, while it slightly weakens the negative correlation of streamwise velocity fluctuation and fluctuating temperature and the positive correlation of wall-normal velocity fluctuation and fluctuating temperature far from the wall. It is also found that the dilatational components of correlations are dominated in the near-wall region, while the solenoidal components govern the correlations far from the wall. Most of the interactions among mean and fluctuating fields of kinetic and internal energy are governed by the solenoidal components, except for the terms associated with the pressure, which are governed by the dilatational components.
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In this work, three-component planar velocity measurements are analyzed to identify the effects of compressibility on mixing layer turbulence, with a focus on the evolution of large-scale structures, dominant spatial eigenmodes, and entrainment length scales. Velocity measurements obtained via stereoscopic particle image velocimetry for five different dual-stream air planar mixing layers with convective Mach numbers of Mc=0.19, 0.38, 0.55, 0.69, and 0.88 are analyzed. Results of two-point correlations reveal that length scales of the streamwise velocity fluctuations increase in both the streamwise and transverse directions, whereas the length scales of the transverse velocity fluctuations decrease in the transverse direction. To further investigate these trends, the spatial organization, shape, and dynamics of large-scale turbulent structures are examined via two-dimensional spatial velocity correlations, proper orthogonal decomposition, and linear stochastic estimation. These quantitative techniques support qualitative findings in the literature that report coherent, round roller structures in incompressible mixing layers becoming flattened and elongated longitudinally with increased compressibility. This evolution is likely due to a dominant pulsing motion present for the higher Mc cases and can be linked to compressibility effects on the entrainment mechanisms. Length scales of entrainment are determined using autocorrelations of normal flow velocity components along instantaneously identified turbulent/non-turbulent interfaces and are found to decrease with increasing Mc. This result can heuristically be interpreted as small-scale nibbling being dominant for higher Mc, whereas larger-scale engulfment contributes more in nearly incompressible mixing layers.
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The local flow topology based on the invariants of the velocity gradient tensor in stationary compressible homogeneous shear turbulence (HST) is studied by numerical simulations. In the compressible homogeneous shear turbulence, local compressibility decreases the flow volume fraction occupied by the focal, eddy, and shear flow structures both in compression regions and in strong expansion regions. The joint probability density function (pdf) of the second and third invariants of the deviatoric velocity gradient tensor exhibits a similar teardrop shape as for the homogeneous isotropic turbulence (HIT), and the tail of the joint pdf alongside the right branch of the null-discriminant curve is elongated as the turbulent Mach number increases. When conditioned on dilatation, the statistical preference for points in the fourth quadrants of the joint pdf is enhanced significantly by the compression motion. It is found that the shape of the joint pdf shows a good similarity between HST and HIT in strong compression regions, which is dependent on the root mean square dilatation, rather than the turbulent Mach number. In strong expansion regions, the shape of the joint pdf in HST has a long tail in the third quadrant, which is related to sheetlike expansion structures and does not exist in HIT. After the Helmholtz decomposition, the properties of local flow topology associated with the solenoidal component of the velocity field are found to be very similar to those in incompressible turbulence and are insensitive to the change in local dilatation and turbulent Mach number.
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Compressible wall-bounded turbulence is generally assumed to be devoid of genuine compressibility effects, meaning that the effect of finite fluid dilatation is regarded as “small,” at least in the absence of disturbing pressure gradients. In the present paper we attempt to answer the basic question of how small these effects are, by interrogating a DNS database of compressible channel flow and by using Helmholtz decomposition to infer the relative magnitude and correlations between the solenoidal and the dilatational parts of turbulence velocity fields. Not surprisingly, we find dilatational velocity fluctuations to be much smaller than solenoidal ones, but perhaps unexpectedly, we find that finite correlation between the two components accounts for a nonnegligible fraction (about 10%) of the turbulent shear stress near walls, and for up to 4% of the wall skin friction. Quadrant analysis of the dilatational velocity fluctuations shows that the largest contribution to the turbulent shear stress results from significant correlation between positive streamwise solenoidal velocity fluctuations (i.e., high-speed streaks), and positive vertical dilatational velocity fluctuations, which tend to mitigate the intensity of wall-ward sweep events.
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Aspects of turbulent shear-layer mixing are investigated over a range of shear-layer Reynolds numbers, $Re_{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x0394}U\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D708}$ , based on the shear-layer free-stream velocity difference, $\unicode[STIX]{x0394}U$ , and mixing-zone thickness, $\unicode[STIX]{x1D6FF}$ , to probe the role of initial conditions in mixing stages and the evolution of the scalar-field probability density function (p.d.f.) and variance. Scalar transport is calculated for unity Schmidt numbers, approximating gas-phase diffusion. The study is based on direct-numerical simulation (DNS) and large-eddy simulation (LES), comparing different subgrid-scale (SGS) models for incompressible, uniform-density, temporally evolving forced shear-layer flows. Moderate-Reynolds-number DNS results help assess and validate LES SGS models in terms of scalar-spectrum and mixing estimates, as well as other metrics, to $Re_{\unicode[STIX]{x1D6FF}}\lesssim 3.3\times 10^{4}$ . High-Reynolds-number LES investigations to $Re_{\unicode[STIX]{x1D6FF}}\lesssim 5\times 10^{5}$ help identify flow parameters and conditions that influence the evolution of scalar variance and p.d.f., e.g. marching versus non-marching. Initial conditions that generate shear flows with different mixing behaviour elucidate flow characteristics in each flow regime and identify elements that induce p.d.f. transition and scalar-variance behaviour. P.d.f. transition is found to be largely insensitive to local flow parameters, such as $Re_{\unicode[STIX]{x1D6FF}}$ , or a previously proposed vortex-pairing parameter based on downstream distance, or other equivalent criteria. The present study also allows a quantitative comparison of LES SGS models in moderate- and high- $Re_{\unicode[STIX]{x1D6FF}}$ forced shear-layer flows.
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Cascades of temperature and entropy fluctuations are studied by numerical simulations of stationary three-dimensional compressible turbulence with a heat source. The fluctuation spectra of velocity, compressible velocity component, density and pressure exhibit the $-5/3$ scaling in an inertial range. The strong acoustic equilibrium relation between spectra of the compressible velocity component and pressure is observed. The $-5/3$ scaling behaviour is also identified for the fluctuation spectra of temperature and entropy, with the Obukhov–Corrsin constants close to that of a passive scalar spectrum. It is shown by Kovasznay decomposition that the dynamics of the temperature field is dominated by the entropic mode. The average subgrid-scale (SGS) fluxes of temperature and entropy normalized by the total dissipation rates are close to 1 in the inertial range. The cascade of temperature is dominated by the compressible mode of the velocity field, indicating that the theory of a passive scalar in incompressible turbulence is not suitable to describe the inter-scale transfer of temperature in compressible turbulence. In contrast, the cascade of entropy is dominated by the solenoidal mode of the velocity field. The different behaviours of cascades of temperature and entropy are partly explained by the geometrical properties of SGS fluxes. Moreover, the different effects of local compressibility on the SGS fluxes of temperature and entropy are investigated by conditional averaging with respect to the filtered dilatation, demonstrating that the effect of compressibility on the cascade of temperature is much stronger than on the cascade of entropy.
Article
This paper quantifies the instantaneous form of large-scale turbulent structures in canonical smooth-wall boundary layers, demonstrating that they adhere to a form that is consistent with the self-sustaining streak instability model suggested by Flores & Jiménez ( Phys. Fluids , vol. 22, 2010, 071704) and Hwang & Cossu ( Phys. Fluids , vol. 23, 2011, 061702). Our motivation for this study stems from previous observations of large-scale streaks that have been spatially locked in position within spanwise-heterogeneous boundary layers. Here, using similar tools, we demonstrate that the randomly occurring large-scale structures in canonical layers show similar behaviour. Statistically, we show that the signature of large-scale coherent structures exhibits increasing meandering behaviour with distance from the wall. At the upper edge of the boundary layer, where these structures are severely misaligned from the main-flow direction, the induced velocities associated with the strongly yawed vortex packets/clusters yield a significant spanwise-velocity component leading to an apparent oblique coherence of spanwise-velocity fluctuations. This pronounced meandering behaviour also gives rise to a dominant streamwise periodicity at a wavelength of approximately $6\unicode[STIX]{x1D6FF}$ . We further statistically show that the quasi-streamwise roll-modes formed adjacent to these very large wavy motions are often one-sided (spanwise asymmetric), in stark contrast to the counter-rotating form suggested by conventional conditionally averaged representations. To summarise, we sketch a representative picture of the typical large-scale structures based on the evidence gathered in this study.
Article
The effect of compressibility on the small scale properties of stationary homogeneous shear turbulence (HST) is studied in a rectangular domain of size 4π × 2π × 2π by numerical simulations at four turbulent Mach numbers of Mt = 0.14, 0.2, 0.4, and 0.6. Compared to the compressible homogeneous isotropic turbulence (HIT), the small scale properties of HST are more sensitive to Mach number change and the velocity derivatives show a clear deviation from isotropy. At M t = 0.14, the preferred eigenvalue ratio of the strain rate tensor is very close to −4:1:3 reported in incompressible turbulence. As M t increases, the conditional probability density functions of the normalized eigenvalues become more dependent on the dilatation, and its ratio tends to −1:0:0 in the strong compression regions at M t = 0.6, indicating the sheet-like structures of localized shock waves. The alignments between vorticity and eigenvectors are similar to the observations in HIT. After the Helmholtz decomposition, it is found that the compressible vortex stretching term is the primary cause for the enhancement of enstrophy production at high turbulent Mach numbers.
Article
We investigate the behaviour of large-scale coherent structures in a spanwise-heterogeneous turbulent boundary layer, using particle image velocimetry on multiple orthogonal planes. The statistical three-dimensionality is imposed by a herringbone riblet surface, although the key results presented here will be common to many cases of wall turbulence with embedded secondary flows in the form of mean streamwise vortices. Instantaneous velocity fields in the logarithmic layer reveal elongated low-momentum streaks located over the upwash-flow region, where their spanwise spacing is forced by the $2\unicode[STIX]{x1D6FF}$ periodicity of the herringbone pattern. These streaks largely resemble the turbulence structures that occur naturally (and randomly located) in spanwise-homogeneous smooth-/rough-wall boundary layers, although here they are directly formed by the roughness pattern. In the far outer region, the large spanwise spacing permits the streaks to aggressively meander. The mean secondary flows are the time-averaged artefact of the unsteady and spanwise asymmetric large-scale roll modes that accompany these meandering streaks. Interestingly, this meandering, or instability, gives rise to a pronounced streamwise periodicity (i.e. an alternating coherent pattern) in the spatial statistics, at wavelengths of approximately 4.5 $\unicode[STIX]{x1D6FF}$ . Overall, the observed behaviours largely resemble the streak-instability model that has been proposed for the buffer region, only here at a much larger scale and at a forced spanwise spacing. This observation further confirms recent observations that such features may occur at an entire hierarchy of scales throughout the turbulent boundary layer.
Article
The effects of Mach number on the spectra and statistics of stationary compressible homogeneous shear turbulence (HST) are studied using numerical simulations in a rectangular domain (Lx = 4π, Ly = Lz = 2π) at turbulent Mach numbers from 0.05 to 0.66 and Taylor Reynolds numbers from 40 to 100. Long-term simulation results show that a statistically stationary state is obtained and the flow meets the strong acoustic equilibrium assumption at Mt ≈ 0.4. The analysis of spectral properties indicates that velocity and pressure tend toward a Mach number scaling of Mt2 suggested by acoustic dynamics at Mt ≳ 0.3. As for one-point statistics, it is found that a Mt4 scaling predicted by pseudo-sound theory holds for normalized compressible kinetic energy, Kc/Ks, at the small turbulent Mach number Mt ≲ 0.1. Kc/Ks approaches a Mt2 scaling at relatively higher turbulent Mach numbers, which is consistent with the spectral results. The normalized compressible dissipation rate, ϵc/ϵs, is nearly independent of Taylor Reynolds number and exhibits the same Mt4 scaling at small turbulent Mach numbers. The root mean square values of pressure, density, and temperature of compressible HST show good agreement with the Mt2 scaling, with the coefficient approximately doubled as compared with the compressible isotropic turbulence.
Article
This article discusses the description of wall-bounded turbulence as a deterministic high-dimensional dynamical system of interacting coherent structures, defined as eddies with enough internal dynamics to behave relatively autonomously from any remaining incoherent part of the flow. The guiding principle is that randomness is not a property, but a methodological choice of what to ignore in the flow, and that a complete understanding of turbulence, including the possibility of control, requires that it be kept to a minimum. After briefly reviewing the underlying low-order statistics of flows at moderate Reynolds numbers, the article examines what two-point statistics imply for the decomposition of the flow into individual eddies. Intense eddies are examined next, including their temporal evolution, and shown to satisfy many of the properties required for coherence. In particular, it is shown that coherent structures larger than the Corrsin scale are a natural consequence of the shear. In wall-bounded turbulence, they can be classified into coherent dispersive waves and transient bursts. The former are found in the viscous layer near the wall, and as very large structures spanning the entire boundary layer. Although they are shear-driven, these waves have enough internal structure to maintain a uniform advection velocity. Conversely, bursts exist at all scales, are characteristic of the logarithmic layer, and interact almost linearly with the shear. While the waves require a wall to determine their length scale, the bursts are essentially independent from it. The article concludes with a brief review of our present theoretical understanding of turbulent structures, and with a list of open problems and future perspectives. ‘ Chance is the name we give to what we choose to ignore (Voltaire) ’
Article
Kinetic energy transfer in compressible isotropic turbulence is studied using numerical simulations with solenoidal forcing at turbulent Mach numbers ranging from 0.4 to 1.0 and at a Taylor Reynolds number of approximately 250. The pressure dilatation plays an important role in the local conversion between kinetic energy and internal energy, but its net contribution to the average kinetic energy transfer is negligibly small, due to the cancellation between compression and expansion work. The right tail of probability density function (PDF) of the subgrid-scale (SGS) flux of kinetic energy is found to be longer at higher turbulent Mach numbers. With an increase of the turbulent Mach number, compression motions enhance the positive SGS flux, and expansion motions enhance the negative SGS flux. Average of SGS flux conditioned on the filtered velocity divergence is studied by numerical analysis and a heuristic model. The conditional average of SGS flux is shown to be proportional to the square of filtered velocity divergence in strong compression regions for turbulent Mach numbers from 0.6 to 1.0. Moreover, the antiparallel alignment between the large-scale strain and the SGS stress is observed in strong compression regions. The inter-scale transfer of solenoidal and compressible components of kinetic energy is investigated by Helmholtz decomposition. The SGS flux of solenoidal kinetic energy is insensitive to the change of turbulent Mach number, while the SGS flux of compressible kinetic energy increases drastically as the turbulent Mach number becomes larger. The compressible mode persistently absorbs energy from the solenoidal mode through nonlinear advection. The kinetic energy of the compressible mode is transferred from large scales to small scales through the compressible SGS flux, and is dissipated by viscosity at small scales.
Article
Jets with Mach numbers $M\gtrsim 1.5$ are well known to emit an intense, fricative, so-called crackle sound, having steep compressions interspersed with weaker expansions that together yield a positive pressure skewness $S_{k}>0$ . Its shock-like features are obvious hallmarks of nonlinearity, although a full explanation of the skewness is lacking, and wave steepening alone is understood to be insufficient to describe its genesis. Direct numerical simulations of high-speed free-shear flows for Mach numbers $M=0.9$ , $1.5$ , $2.5$ and $3.5$ in the Reynolds number range $60\leqslant Re_{\unicode[STIX]{x1D6FF}_{m}}\leqslant 4200$ are used to examine the mechanisms leading to such pressure signals, especially the pressure skewness. For $M=2.5$ and $3.5$ , the pressure immediately adjacent the turbulence already has the large $S_{k}\gtrsim 0.4$ associated with jet crackle. It also has a surprisingly complex three-dimensional structure, with locally high pressures at compression-wave intersections. This structure is transient, and it simplifies as radiating waves subsequently merge through nonlinear mechanisms to form the relatively distinct and approximately two-dimensional Mach-like waves deduced from laboratory visualizations. A transport equation for $S_{k}$ is analysed to quantify factors affecting its development. The viscous dissipation that decreases $S_{k}$ is balanced by a particular nonlinear flux, which is (of course) absent in linear acoustic propagation and confirmed to be independent of the simulated Reynolds numbers. Together these effects maintain an approximately constant $S_{k}$ in the near acoustic field.
Article
Shocklet statistics in compressible isotropic turbulence are studied by using numerical simulations with solenoidal forcing, at the turbulent Mach number Mt ranging from 0.5 up to 1.0 and at the Taylor Reynolds number Reλ ranging from 110 to 250. A power-law region of the probability density function (PDF) of the shocklet strength Mn−1 (Mn is the normal shock Mach number) is observed. The magnitude of the power-law exponent is found to decrease with the increase of Mt. We show that the most probable shocklet strength is proportional to Mt3, and the shocklet thickness corresponding to the most probable shock Mach number is proportional to Mt−2 in our numerical simulations. The PDFs of the jumps of the velocity and thermodynamic variables across a shocklet exhibit a similar power-law scaling. The statistics of the jumps of the velocity and thermodynamic variables are further investigated by conditioned average. Nonlinear models for the conditional average of the jumps of the velocity and thermodynamic variables are developed and verified.
Article
The production of turbulent kinetic energy (TKE) and flow structures in compressible homogeneous turbulent shear flow (HTSF) are investigated by using direct numerical simulation. A theoretical analysis suggests that the vertical turbulent transport process should be responsible for the production of TKE in HTSF. It is manifested based on a conditional average method that the pure TKE production becomes increasingly larger in strain regions than in vortex regions of the flow. The velocity-derivative correlation in the shear plane is employed to identify the streaky structures in HTSF, which also tend to occur predominantly in strain regions of turbulence. Localized analyses of conditional averages reveal that the streaky structures in compressible HTSF are closely related to the negative productions of TKE. A comparative study implies that flow compressibility has a considerable effect on the spatial distributions and patterns of the strain- and vortex-dominated fields, which in turn cause the discrepancies in distribution of the TKE production and streaky structures between incompressible and compressible HTSFs.
Article
Direct numerical simulations (DNS) of temporally evolving shear layers have been performed to study the entrainment of irrotational flow into the turbulent region across the turbulent/non-turbulent interface (TNTI). Four cases with convective Mach number from 0.2 to 1.8 are used. Entrainment is studied via two mechanisms; nibbling, considered as vorticity diffusion across the TNTI, and engulfment, the drawing of the pockets of the outside irrotational fluid into the turbulent region. The mass flow rate due to nibbling is calculated as the product of the entrained mass flux with the surface area of the TNTI. It is found that increasing the convective Mach number results in a decrease of the average entrained mass flux and the surface area of the TNTI. For the incompressible shear layer the local entrained mass flux is shown to be highly correlated with the viscous terms. However, as the convective Mach number increases, the mass fluxes due to the baroclinic and the dilatation terms play a more important role in the local entrainment process. It is observed that the entrained mass flux is dependent on the local dilatation and geometrical shape of the TNTI. For a compressible shear layer, most of the entrainment of the irrotational flow into the turbulent region due to nibbling is associated with the compressed regions on the TNTI. As the convective Mach number increases, the percentage of the compressed regions on the TNTI decreases, resulting in a reduction of the average entrained mass flux. It is also shown that the local shape of the interface, looking from the turbulent region, is dominated by concave shaped surfaces with radii of curvature of the order of the Taylor length scale. The average entrained mass flux is found to be larger on highly curved concave shaped surfaces regardless of the level of dilatation. The mass fluxes due to vortex stretching, baroclinic torque and the shear stress/density gradient terms are weak functions of the local curvatures on the TNTI, whereas the mass fluxes due to dilatation and viscous diffusion plus the viscous dissipation terms have a stronger dependency on the local curvatures. As the convective Mach number increases, the probability of finding highly curved concave shaped surfaces on the TNTI decreases, whereas the probability of finding flatter concave and convex shaped surfaces increases. This results in a decrease of the average entrained mass flux across the TNTI. Similar to the previous works on jets, the results show that the contribution of the engulfment to the total entrainment is small for both incompressible and compressible mixing layers. It is also shown that increasing the convective Mach number decreases the velocities associated with the entrainment, i.e. induced velocity, boundary entrainment velocity and local entrainment velocity.
Article
The compressibility effects on the structural evolution of the transitional high-speed planar wake are studied. The relative Mach number ( $Ma_{r}$ ) of the laminar base flow modifies two fundamental features of planar wake transition: (i) the characteristic length scale defined by the most unstable linear mode; and (ii) the domain of influence of the structures within the staggered two-dimensional vortex array. Linear stability results reveal a reduced growth (approximately 30 % reduction up to $Ma_{r}=2.0$ ) and a quasilinear increase of the wavelength of the most unstable, two-dimensional instability mode (approximately 20 % longer over the same $Ma_{r}$ range) with increasing $Ma$ . The primary wavelength defines the length scale imposed on the emerging transitional structures; naturally, a longer wavelength results in rollers with a greater streamwise separation and hence also larger circulation. A reduction of the growth rate and an increase of the principal wavelength results in a greater ellipticity of the emerging rollers. Compressibility effects also modify the domain of influence of the transitional structures through an increased cross-wake and inhibited streamwise communication as characteristic paths between rollers are deflected due to local $Ma$ gradients. The reduced streamwise domain of influence impedes roller pairing and, for a sufficiently large relative $Ma$ , pairing is completely suppressed. Thus, we observe an increased two-dimensionality with increasing Mach number: directly contrasting the increasing three-dimensional effects in high-speed mixing layers. Temporally evolving direct numerical simulations conducted at $Ma=0.8$ and 2.0, for Reynolds numbers up to 3000, support the physical insight gained from linear stability and vortex dynamics studies.
Article
Quadrant analysis is a simple, but quite useful, turbulence data-processing technique that has been widely used, principally in the investigation of turbulent shear flows. This article traces the origins of the technique and reviews how it has been applied during the more than 40 years since it was conceived. Applications are highlighted that have expanded the technique beyond its original formulation.
Article
The local flow topology is studied using the invariants of the velocity gradient tensor in compressible turbulent mixing layer via direct numerical simulation (DNS) data. The topological and dissipating behaviours of the flow are analysed in two different regions: in proximity of the turbulent/non-turbulent interface (TNTI), and inside the turbulent region. It is found that the distribution of various flow topologies in regions close to the TNTI differs from inside the turbulent region, and in these regions the most probable topologies are non-focal. In order to better understand the behaviour of different flow topologies, the probability distributions of vorticity norm, dissipation and rate of stretching are analysed in incompressible, compressed and expanded regions. It is found that the structures undergoing compression-expansion in axial-radial directions have the highest contraction rate in locally compressed regions, and in locally expanded regions the structures undergoing expansion-compression in axial-radial directions have the highest stretching rate. The occurrence probability of different flow topologies conditioned by the dilatation level is presented and it is shown that the structures in the locally compressed regions tend to have stable topologies while in locally expanded regions the unstable topologies are prevalent.
§1. We shall denote by u α ( P ) = u α ( x 1 , x 2 , x 3 , t ), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 , x 2 , x 3 . In considering the turbulence it is natural to assume the components of the velocity u α ( P ) at every point P = ( x 1 , x 2 , x 3 , t ) of the considered domain G of the four-dimensional space ( x 1 , x 2 , x 3 , t ) are random variables in the sense of the theory of probabilities (cf. for this approach to the problem Millionshtchikov (1939) Denoting by Ᾱ the mathematical expectation of the random variable A we suppose that ῡ ² α and (d u α /d x β ) ² ― are finite and bounded in every bounded subdomain of the domain G .
Article
Preface; Nomenclature; Part I. Fundamentals: 1. Introduction; 2. The equations of fluid motion; 3. Statistical description of turbulence; 4. Mean flow equations; 5. Free shear flows; 6. The scales of turbulent motion; 7. Wall flows; Part II. Modelling and Simulation: 8. Modelling and simulation; 9. Direct numerical simulation; 10. Turbulent viscosity models; 11. Reynolds-stress and related models; 12. PDF models; 13. Large-eddy simulation; Part III. Appendices; Bibliography.
Article
An appraisal is made of several subgrid scale (SGS) viscous/scalar dissipation closures via a priori analysis of direct numerical simulation data in a temporally evolving compressible mixing layer. The effects of the filter width, the compressibility level and the Schmidt number are studied for several models. Based on the scaling of SGS kinetic energy, a new formulation for SGS viscous dissipation is proposed. This yields the best overall prediction of the SGS viscous dissipation within the inertial subrange. An SGS scalar dissipation model based on the proportionality of the turbulent time scale with the scalar mixing time scale also performs the best for the filter widths in the inertial subrange. Two dynamic methods are implemented for the determination of the model coefficients. The one based on the global equilibrium of dissipation and production is shown to be more satisfactory than the conventional dynamic model.
Article
An eighth‐order filter method for a wide range of compressible flow speeds (H. C. Yee and B. Sjogreen, Proceedings of ICOSAHOM09, June 22–26, 2009, Trondheim, Norway) is employed for large eddy simulations (LES) of temporally evolving mixing layers (TML) for different convective Mach numbers (M c ) and Reynolds numbers. The high‐order filter method is designed for accurate and efficient simulations of shock‐free compressible turbulence, turbulence with shocklets, and turbulence with strong shocks with minimum tuning of scheme parameters. The value of the M c considered is for the TML range from the quasi‐incompressible regime to the highly compressible supersonic regime. The three main characteristics of compressible TML (the self‐similarity property, compressibility effects, and the presence of large‐scale structures with shocklets for high M c ) are considered for the LES study. The LES results that used the same scheme parameters for all studied cases agree well with experimental results and published direct numerical simulations (DNS). Published 2012. This article is a US Government work and is in the public domain in the USA.
Article
The occurrence of shocks in the confined three‐dimensional turbulent mixing layer at convective Mach number 1.2 is established by means of direct numerical simulations. The shocks are generated by the turbulent motions in the flow. Consequently, they can have different shapes and orientations, while they persist for a relatively short time. Furthermore, they are created by different types of turbulent vortices. The shocks do not strongly contribute to the turbulent dissipation. Even at the time when the largest shocks occur, the fraction of the turbulent dissipation due to the shocks is less than 10%.
Article
The turbulent flow originating from the interaction between two parallel streams with different velocities is studied by means of direct numerical simulation. Rather than the more common temporal evolving layer, a spatially evolving configuration, with perturbed laminar inlet conditions is considered. The streamwise evolution and the self-similar state of turbulence statistics are reported and compared to results available in the literature. The characteristics of the transitional region agree with those observed in other simulations and experiments of mixing layers originating from laminar inlets. The present results indicate that the transitional region depends strongly on the inlet flow. Conversely, the self-similar state of turbulent kinetic energy and dissipation agrees quantitatively with those in a temporal mixing layer developing from turbulent initial conditions [M. M. Rogers and R. D. Moser, “Direct simulation of a self-similar turbulent mixing layer,” Phys. Fluids 6, 903 (1994)]. The statistical features of turbulence in the self-similar region have been analysed in terms of longitudinal velocity structure functions, and scaling exponents are estimated by applying the extended self-similarity concept. In the small scale range (60 < r/η < 250), the scaling exponents display the universal anomalous scaling observed in homogeneous isotropic turbulence. The hypothesis of isotropy recovery holds in the turbulent mixing layer despite the presence of strong shear and large-scale structures, independently of the means of turbulence generation. At larger scales (r/η > 400), the mean shear and large coherent structures result in a significant deviation from predictions based on homogeneous isotropic turbulence theory. In this second scaling range, the numerical values of the exponents agree quantitatively with those reported for a variety of other flows characterized by strong shear, such as boundary layers, as well as channel and wake flows.
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##### HEPHAISTOS - 2022
2022
Activity report
Project-Team
HEPHAISTOS
RNSR: 201421207V
Research center
Team name:
HExapode, PHysiology, AssISTance and RobOtics
Domain
Perception, Cognition and Interaction
Theme
Robotics and Smart environments
Creation of the Project-Team: 2015 July 01
# Keywords
• A2.3. Embedded and cyber-physical systems
• A3.1. Data
• A3.3. Data and knowledge analysis
• A3.4. Machine learning and statistics
• A5.1. Human-Computer Interaction
• A5.6. Virtual reality, augmented reality
• A5.10. Robotics
• A5.11. Smart spaces
• A6. Modeling, simulation and control
• A6.1. Methods in mathematical modeling
• A6.2. Scientific computing, Numerical Analysis & Optimization
• A6.3. Computation-data interaction
• A6.4. Automatic control
• A6.5. Mathematical modeling for physical sciences
• A8.4. Computer Algebra
• A8.11. Game Theory
• A9.2. Machine learning
• A9.3. Signal analysis
• A9.5. Robotics
• A9.7. AI algorithmics
• A9.9. Distributed AI, Multi-agent
• A9.10. Hybrid approaches for AI
• B2.1. Well being
• B2.5. Handicap and personal assistances
• B2.7. Medical devices
• B2.8. Sports, performance, motor skills
• B3.1. Sustainable development
• B3.5. Agronomy
• B5.2. Design and manufacturing
• B5.6. Robotic systems
• B5.7. 3D printing
• B8.1. Smart building/home
• B8.4. Security and personal assistance
• B9.1. Education
• B9.2. Art
• B9.9. Ethics
# 1 Team members, visitors, external collaborators
## Research Scientists
• Jean-Pierre Merlet [Team leader, INRIA, Senior Researcher, HDR]
• Pierre Berthet-Rayne [CARANX MEDICAL, Industrial member, from Oct 2022]
• Yves Papegay [INRIA, Researcher, HDR]
• Odile Pourtallier [INRIA, Researcher]
• Eric Wajnberg [INRAE, Senior Researcher, HDR]
## PhD Students
• Romain Tissot [INRIA]
• Alexandre Tran [INRIA, CIFRE]
• Jane Desplanques [INRIA]
## External Collaborator
• Eric Sejor [CHU NICE]
# 2 Overall objectives
HEPHAISTOS has been created as a team on January 1st, 2013 and as a project team in 2015.
The goal of the project is to set up a generic methodology for the design and evaluation of an adaptable and interactive assistive ecosystem for the elderly and the vulnerable persons that provides furthermore assistance to the helpers, on-demand medical data and may manage emergency situations. More precisely our goals are to develop devices with the following properties:
• they can be adapted to the end-user and to its everyday environment
• they should be affordable and minimally intrusive
• they may be controlled through a large variety of simple interfaces
• they may eventually be used to monitor the health status of the end-user in order to detect emerging pathology
Assistance will be provided through a network of communicating devices that may be either specifically designed for this task or be just adaptation/instrumentation of daily life objects.
The targeted population is limited to frail people 1 and the assistive devices will have to support the individual autonomy (at home and outdoor) by providing complementary resources in relation with the existing capacities of the person. Personalization and adaptability are key factor of success and acceptance. Our long term goal will be to provide robotized devices for assistance, including smart objects, that may help disabled, elderly and handicapped people in their personal life.
Assistance is a very large field and a single project-team cannot address all the related issues. Hence HEPHAISTOS will focus on the following main societal challenges:
• mobility: previous interviews and observations in the HEPHAISTOS team have shown that this was a major concern for all the players in the ecosystem. Mobility is a key factor to improve personal autonomy and reinforce privacy, perceived autonomy and self-esteem.
• managing emergency situations: emergency situations (e.g. fall) may have dramatic consequences for elderly. Assistive devices should ideally be able to prevent such situation and at least should detect them with the purposes of sending an alarm and to minimize the effects on the health of the elderly.
• medical monitoring: elderly may have a fast changing trajectory of life and the medical community is lacking timely synthetic information on this evolution, while available technologies enable to get raw information in a non intrusive and low cost manner. We intend to provide synthetic health indicators, that take measurement uncertainties into account, obtained through a network of assistive devices. However respect of the privacy of life, protection of the elderly and ethical considerations 7 impose to ensure the confidentiality of the data and a strict control of such a service by the medical community.
• rehabilitation and biomechanics: our goals in rehabilitation are 1) to provide more objective and robust indicators, that take measurement uncertainties into account to assess the progress of a rehabilitation process 2) to provide processes and devices (including the use of virtual reality) that facilitate a rehabilitation process and are more flexible and easier to use both for users and doctors. Biomechanics is an essential tool to evaluate the pertinence of these indicators, to gain access to physiological parameters that are difficult to measure directly and to prepare efficiently real-life experiments.
Addressing these societal focus induces the following scientific objectives:
• design and control of a network of connected assistive devices: existing assistance devices suffer from a lack of essential functions (communication, monitoring, localization,...) and their acceptance and efficiency may largely be improved. Furthermore essential functions (such as fall detection, knowledge sharing, learning, adaptation to the user and helpers) are missing. We intend to develop new devices, either by adapting existing systems or developing brand-new one to cover these gaps. Their performances, robustness and adaptability will be obtained through an original design process, called appropriate design, that takes uncertainties into account to determine almost all the nominal values of the design parameters that guarantee to obtain the required performances. The development of these devices covers our robotics works (therefore including robot analysis, kinematics, control, ...) but is not limited to them. These devices will be present in the three elements of the ecosystem (user, technological helps and environment) and will be integrated in a common network. The study of this robotic network and of its element is therefore a major focus point of the HEPHAISTOS project. In this field our objectives are:
• to develop methods for the analysis of existing robots, taking into account uncertainties in their modeling that are inherent to such mechatronic devices
• to propose innovative robotic systems
• evaluation, modeling and programming of assistive ecosystem: design of such an ecosystem is an iterative process which relies on different types of evaluation. A large difference with other robotized environments is that effectiveness is not only based on technological performances but also on subjectively perceived dimensions such as acceptance or improvement of self-esteem. We will develop methodologies that cover both evaluation dimensions. Technological performances are still important and modeling (especially with symbolic computation) of the ecosystem will play a major role for the design process, the safety and the efficiency, which will be improved by a programming/communication framework than encompass all the assistance devices. Evaluation will be realized with the help of clinical partners in real-life or by using our experimental platforms.
• uncertainty management: uncertainties are especially present in all of our activities (sensor, control, physiological parameters, user behavior, ...). We intend to systematically take them into account especially using interval analysis, statistics, game theory or a mix of these tools.
• economy of assistance: interviews by the HEPHAISTOS team and market analysis have shown that cost is a major issue for the elderly and their family. At the opposite of other industrial sectors manufacturing costs play a very minor role when fixing the price of assistance devices: indeed prices result more from the relations between the players and from regulations. We intend to model these relations in order to analyze the influence of regulations on the final cost.
The societal challenges and the scientific objectives will be supported by experimentation and simulation using our development platforms or external resources.
In terms of methodologies the project will focus on the use and mathematical developments of symbolic tools (for modeling, design, interval analysis), on interval analysis (for design, uncertainties management, evaluation), on game theory (for control, localization, economy of assistance) and on control theory. Implementation of the algorithms will be performed within the framework of general purpose software such as Scilab, Maple, Mathematica and the interval analysis part will be based on the existing library ALIAS, that is still being developed mostly for internal use.
Experimental work and the development of our own prototypes are strategic for the project as they allow us to validate our theoretical work and to discover new problems that will feed in the long term the theoretical analysis developed by the team members.
Dissemination is also an essential goal of our activity as its background both on the assistance side and on the theoretical activities as our approaches are not sufficiently known in the medical, engineering and academic communities.
In summary HEPHAISTOS has as major research axes assistance robotics, modeling, game theory, interval analysis, robotics and AI (see section 8.1). The coherence of these axis is that interval analysis is a major tool to manage the uncertainties that are inherent to a robotized device, while assistance robotics provides realistic problems which allow us to develop, test and improve our algorithms. Our overall objectives are presented in this document and in a specific page on assistance.
# 3 Research program
As seen in the overall objectives managing uncertainties is a key point of our research. In the health domain uncertainties is managed with statistics (which the explain partly presence of E. Wajnberg in our team) but statistics just give trends while in some cases we will be more interested in the worst case scenario. Interval analysis is an approach that can be used in that case and we constantly improve the foundations of this method.
## 3.1 Interval analysis
We are interested in real-valued system solving ($f\left(X\right)=0$, $f\left(X\right)\le 0$), in optimization problems, and in the proof of the existence of properties (for example, it exists $X$ such that $f\left(X\right)=0$ or it exist two values ${X}_{1}$, ${X}_{2}$ such that $f\left({X}_{1}\right)>0$ and $f\left({X}_{2}\right)<0$). There are few restrictions on the function $f$ as we are able to manage explicit functions using classical mathematical operators (e.g. $sin\left(x+y\right)+log\left(cos\left({e}^{x}\right)+{y}^{2}\right)$ as well as implicit functions (e.g. determining if there are parameter values of a parametrized matrix such that the determinant of the matrix is negative, without calculating the analytical form of the determinant).
Solutions are searched within a finite domain (called a box) which may be either continuous or mixed (i.e. for which some variables must belong to a continuous range while other variables may only have values within a discrete set). An important point is that we aim at finding all the solutions within the domain whenever the computer arithmetic will allow it: in other words we are looking for certified solutions. For example, for 0-dimensional system solving, we will provide a box that contains one, and only one, solution together with a numerical approximation of this solution. This solution may further be refined at will using multi-precision.
The core of our methods is the use of interval analysis that allows one to manipulate mathematical expressions whose unknowns have interval values. A basic component of interval analysis is the interval evaluation of an expression. Given an analytical expression $F$ in the unknowns $\left\{{x}_{1},{x}_{2},...,{x}_{n}\right\}$ and ranges $\left\{{X}_{1},{X}_{2},...,{X}_{n}\right\}$ for these unknowns we are able to compute a range $\left[A,B\right]$, called the interval evaluation, such that
$\forall \left\{{x}_{1},{x}_{2},...,{x}_{n}\right\}\in \left\{{X}_{1},{X}_{2},...,{X}_{n}\right\},A\le F\left({x}_{1},{x}_{2},...,{x}_{n}\right)\le B$ 1
In other words the interval evaluation provides a lower bound of the minimum of $F$ and an upper bound of its maximum over the box.
For example if $F=x\phantom{\rule{3.33333pt}{0ex}}sin\left(x+{x}^{2}\right)$ and $x\in \left[0.5,1.6\right]$, then $F\left(\left[0.5,1.6\right]\right)=\left[-1.362037441,1.6\right]$, meaning that for any $x$ in [0.5,1.6] we guarantee that $-1.362037441\le f\left(x\right)\le 1.6$.
The interval evaluation of an expression has interesting properties:
• it can be implemented in such a way that the results are guaranteed with respect to round-off errors i.e. property 1 is still valid in spite of numerical errors induced by the use of floating point numbers
• if $A>0$ or $B<0$, then no values of the unknowns in their respective ranges can cancel $F$
• if $A>0$ ($B<0$), then $F$ is positive (negative) for any value of the unknowns in their respective ranges
A major drawback of the interval evaluation is that $A\left(B\right)$ may be overestimated i.e. values of ${x}_{1},{x}_{2},...,{x}_{n}$ such that $F\left({x}_{1},{x}_{2},...,{x}_{n}\right)=A\left(B\right)$ may not exist. This overestimation occurs because in our calculation each occurrence of a variable is considered as an independent variable. Hence if a variable has multiple occurrences, then an overestimation may occur. Such phenomena can be observed in the previous example where $B=1.6$ while the real maximum of $F$ is approximately 0.9144. The value of $B$ is obtained because we are using in our calculation the formula $F=xsin\left(y+{z}^{2}\right)$ with $y,z$ having the same interval value as $x$.
Fortunately there are methods that allow one to reduce the overestimation and the overestimation amount decreases with the width of the ranges. The latter remark leads to the use of a branch-and-bound strategy in which for a given box a variable range will be bisected, thereby creating two new boxes that are stored in a list and processed later on. The algorithm is complete if all boxes in the list have been processed, or if during the process a box generates an answer to the problem at hand (e.g. if we want to prove that $F\left(X\right)<0$, then the algorithm stops as soon as $F\left(ℬ\right)\ge 0$ for a certain box $ℬ$).
A generic interval analysis algorithm involves the following steps on the current box 10, 1:
1. exclusion operators: these operators determine that there is no solution to the problem within a given box. An important issue here is the extensive and smart use of the monotonicity of the functions
2. filters: these operators may reduce the size of the box i.e. decrease the width of the allowed ranges for the variables
3. existence operators: they allow one to determine the existence of a unique solution within a given box and are usually associated with a numerical scheme that allows for the computation of this solution in a safe way
4. bisection: choose one of the variable and bisect its range for creating two new boxes
5. storage: store the new boxes in the list
The scope of the HEPHAISTOS project is to address all these steps in order to find the most efficient procedures. Our efforts focus on mathematical developments (adapting classical theorems to interval analysis, proving interval analysis theorems), the use of symbolic computation and formal proofs (a symbolic pre-processing allows one to automatically adapt the solver to the structure of the problem), software implementation and experimental tests (for validation purposes).
Important note: We have insisted on interval analysis because this is a major component or our robotics activity. Our theoretical work in robotics is an analysis of the robotic environment in order to exhibit proofs on the behavior of the system that may be qualitative (e.g. the proof that a cable-driven parallel robot with more than 6 non-deformable cables will have at most 6 cables under tension simultaneously) or quantitative. In the quantitative case as we are dealing with realistic and not toy examples (including our own prototypes that are developed whenever no equivalent hardware is available or to verify our assumptions) we have to manage problems that are so complex that analytical solutions are probably out of reach (e.g. the direct kinematics of parallel robots) and we have to resort to algorithms and numerical analysis. We are aware of different approaches in numerical analysis (e.g. some team members were previously involved in teams devoted to computational geometry and algebraic geometry) but interval analysis provides us another approach with high flexibility, the possibility of managing non algebraic problems (e.g. the kinematics of cable-driven parallel robots with sagging cables, that involves inverse hyperbolic functions, see section 8.1.1) and to address various types of issues (system solving, optimization, proof of existence ...). However whenever needed we will rely as well on statistics, continuation, algebraic geometry, geometry while AI is currently being investigated (see section 8.1.2).
## 3.2 Robotics
HEPHAISTOS, as a follow-up of COPRIN, has a long-standing tradition of robotics studies, especially for closed-loop robots 4, especially cable-driven parallel robots. We address theoretical issues with the purpose of obtaining analytical and theoretical solutions, but in many cases only numerical solutions can be obtained due to the complexity of the problem. This approach has motivated the use of interval analysis for two reasons:
1. the versatility of interval analysis allows us to address issues (e.g. singularity analysis) that cannot be tackled by any other method due to the size of the problem
2. uncertainties (which are inherent to a robotic device) have to be taken into account so that the real robot is guaranteed to have the same properties as the theoretical one, even in the worst case. This is a crucial issue for many applications in robotics (e.g. medical or assistance robot)
Our field of study in robotics focuses on kinematic issues such as workspace and singularity analysis, positioning accuracy, trajectory planning, reliability, calibration, modularity management and, prominently, appropriate design, i.e. determining the dimensioning of a robot mechanical architecture that guarantees that the real robot satisfies a given set of requirements. The methods that we develop can be used for other robotic problems, see for example the management of uncertainties in aircraft design 8.
Our theoretical work must be validated through experiments that are essential for the sake of credibility and, a contrario, experiments will feed our theoretical work. Hence HEPHAISTOS works with partners on the development of real robots but also develops its own prototypes. In the last years we have developed a large number of prototypes and we have extended our development to devices that are not strictly robots but are part of an overall environment for assistance. We benefit here from the development of new miniature, low energy computers with an interface for analog and logical sensors such as the Arduino or the Phidgets. The web pages presents all of our prototypes and experimental work. Note that this familiarity with hardware is also used from time to time to develop devices for others INRIA projects and during the Covid crisis our building was instrumented for accurately monitoring CO and CO2 level well before it becomes the norm.
# 4 Application domains
While the methods developed in the project can be used for a very broad set of application domains (for example we have an activity in CO2 emission allowances and biology 11), it is clear that the size of the project does not allow us to address all of them. Hence we have decided to focus our applicative activities on mechanism theory, where we focus on modeling, optimal design and analysis of mechanisms. Along the same line our focus is robotics and especially service robotics which includes rescue robotics, rehabilitation and assistive robots for elderly and handicapped people. Although these topics were new for us when initiating the project we have spent two years determining priorities and guidelines by conducting about 200 interviews with field experts (end-users, doctors, family and caregivers, institutes), establishing strong collaboration with them (e.g. with the CHU of Nice-Cimiez) and putting together an appropriate experimental setup for testing our solutions.
It must be reminded that we are able to manage a large variety of problems in totally different domains only because interval analysis, game theory and symbolic tools provides us the methodological tools that allow us to address completely a given problem from the formulation and analysis up to the very final step of providing numerical solutions. Hence although we mainly focus on medical and assistance robotics we address also a large number of applications: agriculture, biology, arts, system design to name a few.
# 5 Social and environmental responsibility
## 5.1 Footprint of research activities
Clearly our activities may have an impact on the environment, especially through travels. We have (an we plan to continue) decreased our travel activities while having reduced our own energy consumption at work in different manners. Still we must emphasize that all aspects of our impact have to be taken into account before coercitive measures are taken. For example when we travel to a retirement house to install assistive devices the footprint impact has to be balanced with the social impact (and this is not an easy task). Furthermore human relationships are essential for initiating new research areas and collaborations and virtual conferences are not very effective for that purpose.
## 5.2 Impact of research results
Our works on assistance clearly may have a social impact and we are deeply aware of our ethical responsibilities as illustrated by the activity of the team in ethical committees, our collaboration with the academic law community and our large dissemination effort toward the general audience.
Regarding environmental responsibility energy has been since the very beginning of our project an important topic in our research. Indeed our assistance/health monitoring devices require additional energy source and elderly people may have some difficulties to deal with battery charging. Consequently since the beginning of the project we have focused on low consumption electronic components and most our devices use mechanical energy converter to produce a large part of the energy they need. However the intended benefits of these devices on health, self-esteem and dignity (all issues that are difficult to measure/compare in pure financial terms or with respect to environmental impact) have to be taken into account. We have not had a direct activity regarding Covid as our medical research areas are not related to medical modeling or virus. Still to contribute to the collective protection effort we have installed at the end of December 2020 a home-made, low-cost but very accurate CO2 measurement station in our laboratory connect to an aeration system that is still in use after one year of full-time monitoring.
# 6 Highlights of the year
## 6.1 Scientific highlights
The highlights of this year are:
• a 4-months deployment of one of our cable-driven parallel robots in a museum for an artistic exhibition (see section 8.1.1)
• some progress in the use of AI for solving robotics kinematic problem. New methodological approaches are investigated as the current tools perform poorly on such a problem (see section 8.1.2)
• a 5-months stay of a team member (E. Wajnberg) at the University of Haifa (see section 8.3)
• the initial step of a team member (Y. Papegay) to create a start-up dealing with human activities recognition using non-intrusive measurements
• Pierre Berthet-Rayne has joined us as a 3IA industrial chair attached to J-P. Merlet senior 3IA chair
Regarding clinical tests the effects of Covid-19 has obliged to postpone once more our planned experiments.
## 6.2 Institutional life
The HEPHAISTOS team is seriously concerned about heavy dysfunctions in the institute.
• The disastrous failures of the new financial and human resource information system (Eksae) that impacts negatively all administrative colleagues with whom researchers interact. It forces the former to duplicate their works, when not making them unable to perform even simple tasks. It also directly impacts researchers, who cannot oversee their budget in any reliable way, and occasionally have to delay simple purchases as a consequence of a reprioritization of overburdened administrative forces.
• The continuously diminishing emphasis on research in the presentation of Inria and the growing fuzziness around the status of researchers and the role and position of the institute attack the meaning of work and our collective values.
• The extremely degraded relationship between the Inria Commission d’Évaluation and the current Inria direction is unbearable. It causes a general distrust in the direction’s intentions and in the outcome of future hiring and promoting process. The team deplores the current situation. It thanks the Commission d’Évaluation for its outstanding communication to the researchers it represents and for its continued effort in organizing and participating in Inria hiring and promotion committees.
# 7 New software and platforms
Our activity in AI, which for the time being focus on finding all solutions of a system of equations (see section 8.1.2), has led to numerous software developments (including methodological one) that are yet not still ready to be distributed.
## 7.1 New software
### 7.1.1 ALIAS
• Name:
Algorithms Library of Interval Analysis for Systems
• Keyword:
Interval analysis
• Functional Description:
The ALIAS library whose development started in 1998, is a collection of procedures based on interval analysis for systems solving and optimization.
ALIAS is made of two parts:
ALIAS-C++ : the C++ library (87 000 code lines) which is the core of the algorithms
ALIAS-Maple : the Maple interface for ALIAS-C++ (55 000 code lines). This interface allows one to specify a solving problem within Maple and get the results within the same Maple session. The role of this interface is not only to generate the C++ code automatically, but also to perform an analysis of the problem in order to improve the efficiency of the solver. Furthermore, a distributed implementation of the algorithms is available directly within the interface.
• URL:
• Contact:
Jean-Pierre Merlet
• Participants:
Jean-Pierre Merlet, Odile Pourtallier
## 7.2 New platforms
Participants: Jean-Pierre Merlet, Yves Papegay, Romain Tissot.
#### Vitrifications robot
In 2019 we have used the modularity of our MARIONET-CRANE cable-driven parallel robot (CDPR) prototype to design a robot for the 2-months art exhibition "Vitrifications" with the artist A-V. Gasc. In this exhibition the CDPR acts as a large 3D printing machine that deposit layers of glass micro-beads on a given trajectory over a 21 $×$ 11 $×$ 5 meters workspace. Beside the exhibition our scientific interest was to record data on the behavior of our CDPR in a realistic environment by adding to our prototype numerous sensors that were not required for the use of the CDPR but were possibly interesting for monitoring the CDPR state and its environment.
A second prototype was intended to be displayed by the same artist during the 2020 robotics conference ICRA that should have been held in Paris and has been held virtually because of the pandemic. The theoretical modeling, sensing and control of this version has been substantially modified according to the analysis of the 2019 data. Although this CDPR was not deployed a mock-up was installed and used in our laboratory and a second analysis of the data has taken place.
At the end of 2021 we agree with A-V. Gasc to prepare a second exhibition at the Espace de l'Art Concret, a museum close to Sophia that has been held from July 9 to October 16, 2022. This third version has also benefited on all aspects of the 2020-2021 version with a new full theoretical modeling incorporated in the control software while new sensors were added, mostly to monitor the CDPR environment and for the public safety.
We have started analyzing the data provided during the exhibition in order to prepare a new version for the end of 2023 with a different scientific objective: walking assistance and health monitoring for frail people.
# 8 New results
## 8.1 Robotics
### 8.1.1 An exhibition using one of our cable-driven parallel robot
Participants: Jean-Pierre Merlet [correspondant], Yves Papegay, Romain Tissot.
As mentioned in the highlights a major work of the team for 2022 has been the deployment manqueof a large-scale CDPR for a 4 months exhibition in the Espace de l'Art Concret museum2. The used CDPR has 4 cables (synthetic Dyneema of diameter 4mm) that are coiled/uncoiled by winches, the other extremity of the cables being attached at the same point $B$, a drum being attached at this point. This CDPR allows one to control the translational degrees of freedom of the drum.
The task of this robot was to deposit several layers of glass micro-beads on a given trajectory at a velocity of 3cm/s, the length of a trajectory being about 30 meters (figure 2).
A constant flow of powder is obtained by opening a vane at the bottom of the drum. The CDPR is hence used as a large-scale 3D printer which is a possible
market for this type of robot (Y. Papegay have co-founded in 2015 the XtreeE start-up company that is currently one of the leading international actors in large-scale 3D concrete printing).
The robot was installed in a room whose dimensions was 11 $×$ 8 $×$ 4.5 meters, which makes this CDPR probably one of the largest 3D printing machine ever built.
After each layer (that requires about 14 kg of powder) the structure below the drum gains some height (about 1.25 mm) but at the same time gains in width (about 3.1 mm) as there is no gluing component in the powder. After depositing over 1 tons of powder the height of the structure is about 9cm for a width of 41cm. As the trajectory is self-intersecting the maximal measured height was about 18cm.
Although the defined trajectory is planar, the CDPR moves in 3D as the height of the drum was supposed to be constant with respect to the height of the structure under the drum (hence the drum altitude increases at the trajectory self intersecting points). The drum contains about 40 kg of powder for a total weight of about 75 kg and is manually filled after 1 or 2 layers.
During the period the CDPR was under the control of two students of an art school that were executing trajectories according to the public presence. The robot has deposited 176 layers on two trajectories and has used 2.5 tons of powder. Including the travel to the rest position for refilling the drum the robot has traveled about 9 km and has been in use for 126 hours. Over the period there has been 24 intrusions of the public in the robot space, 6 unexplained behavior of the CDPR that has automatically led our safety mechanisms to stop the robot. We have also collected about 2 To of data that will be examined during 2023.
Regarding sensors we have 3 planar lidars on the drum, 2 being horizontal and one vertical. The later one measures the distance between the drum and the structure while the 2 horizontal lidars measure points on the room walls (whose position with respect to the reference frame have been calibrated) which allows to determine the position of $B$ in the reference frame. The drum has also a 3 axis accelerometer/gyrometer for measuring the drum oscillations and also an accelerometer on each cable at a fixed distance from $B$. The winches were located on the ground at each corner of the room, the cables going up to a support on the ceiling with a pulley that freely rotates around the vertical axis. We have developed a full model of this system including the pulley (as the cable output point on the pulley changes with the cable direction) and the elastic behavior of the cable which has also its own mass so that the cable shape is not the straight line between the output point on the pulley and $B$ (such a cable is called a sagging cable). For modeling such a cable we use the Irvine textbook model 13. This is a non algebraic planar model with the upper attachment point of the cable is supposed to be grounded: it provides the coordinates (${x}_{b},{z}_{b}$) of the lowest attachment point $B$ of the cable if the cable length ${L}_{0}$ at rest and the horizontal and vertical components ${F}_{x},{F}_{z}$ of the force applied at this point are known.
$\begin{array}{c}\hfill {x}_{b}={F}_{x}\left(\frac{{L}_{0}}{E{A}_{0}}+\frac{sin{h}^{-1}\left({F}_{z}\right)-sin{h}^{-1}\left({F}_{z}-\frac{\mu g{L}_{0}}{{F}_{x}}\right)}{\mu g}\right)\\ \hfill {z}_{b}=\frac{{F}_{z}-\mu g{L}_{0}^{2}/2}{E{A}_{0}}+\frac{\sqrt{{F}_{x}^{2}+{F}_{z}^{2}}-\sqrt{{F}_{x}^{2}+{\left({F}_{z}-\mu g{L}_{0}\right)}^{2}}}{\mu g}\end{array}$
where $E$ is the Young modulus of the cable material, $\mu$ its linear density and ${A}_{0}$ the area of a cross-section of the cable.
It may seem that the control law is easy to design as the lidars provide the position of $B$. However the constant drum velocity constraint imposes a velocity control law has to be used. Such a law implies that the desired velocity vector in the reference frame has to be mapped to a velocity for each of the 4 cables. This mapping is linear but the involved matrix are dependent upon the position of $B$ and upon the directions of the cables at this point, which in turn change with the length ${L}_{0}$ of the cables. Therefore this control law imposes to evaluate the cables lengths ${L}_{0}$. But two factors make the determination of the cable lengths difficult:
1. the elasticity of the cable induces a change in the cable length according to its tension
2. we are measuring the winch drum rotation velocity with an encoder but this translates into a cable velocity that depends upon the drum radius. As we don't control the coiling process and have several layers on it there is a large uncertainty on the drum radius and therefore on the measurement of the ${L}_{0}$.
Furthermore our CDPR is redundantly actuated (4 actuators for 3 d.o.f) so that we are sure that for a given position of $B$ there will be at most 3 cables under tension, the other cable being slack. But it can be shown that there is at least 2 possible triplets of cables that may be under tension at the given position of $B$. Consequently the measurement of the ${L}_{0}$ with the encoders is not sufficient to determine the current state of the CDPR but fortunately this indetermination may be solved. Indeed the Irvine model allows one to calculate the slope of the cable at any point and this is where the cable accelerometers play a major role as they provide the cable slope at a known point. Accordingly, we are able to determine which cables are under tension and which one is slack. But the slope measurement plays another role: on an large trajectory the set of 3 cables that are required to be under under tension is different for various parts of the trajectory. Hence on a given part it is acceptable to have a slack cable but its amount of slackness should be small as this cable will have to be taught for the following part of the trajectory. An internal PID controller based on the cable slope is used to regulate the cable slackness. Note that although the system model is quite complex we have designed an algorithm that is able to calculated the cable lengths in about 1ms.
Hence as soon as the lidars provide the $B$ location we are able to calculate the cables lengths. But the sampling time of the lidar is about 0.5s while the encoders have a 10 ms sampling time and consequently we use the encoders information to calculate the ${L}_{0}$ to control the robot in between two lidar measurements. This imposes to have a good estimation of the drum radius. Up to now we were using a theoretical coiling process model that provides the current drum radius for a given ${L}_{0}$ under the assumption that the coiling is ideal, which is a too strong assumption. In the current version of our CDPR we are using another approach. Let assume that two successive measurements of the lidars have provided two set ${L}_{0}^{1},{L}_{0}^{2}$ of cable lengths while the encoders have measured the drum rotation angle as ${\theta }_{1},{\theta }_{2}$ at the same time. With these data we are able to calculate a mean value of the drum radius that is used for the control before the next lidar measurement.
As the lidars provide an accurate measurement of the the location of $B$ (the lidars being not located at $B$ the drum accelerometer data are used to correct the lidar measurement) we have been able to evaluate that the absolute positioning accuracy of the system is about $±$ 1 cm which is quite remarkable, especially regarding the size of the workspace. Part of this error is due to the use of flip-flop pulleys as the friction in the pulley axis prohibits the pulley to lie exactly in the right plane, thereby introducing a bias in estimation of the cable lengths. Previously we were using a pin-hole output through a low friction material that surprisingly was exhibiting a very low wear after months of use, even with metallic cables and we will come back to this solution for the next version.
A scientific side interest of this experiment is that we are using the vertical lidar to perform every 2 seconds a planar vertical scans of the structure when the CDPR is executing a trajectory, so that we are able to obtain a full 3D model of the structure after each layer. Before the exhibition we have been in touch with researchers working in granular mechanics (e.g. for dune or avalanche modeling). They were able to predict very well the overall shape of the structure for the regular part of the trajectory but have more difficulty at the self-intersecting points. Such an experiment is interesting for these researchers as we have here a well-controlled experiment with not external disturbances (such as wind), an exact knowledge of the powder flow, an homogeneous material, the possibility of repeating an experiment and an accurate 3D model.
### 8.1.2 AI
As seen in section 8.1.1 we have a complex model of our system but this is not a problem for control as we have been able to design very fast algorithms for dealing with the inverse kinematics problem (IK: find the cable lengths to reach a given position for the CDPR) and for the direct kinematics problem (DK: find the robot position being given the cable lengths ${L}_{0}$).
But the IK and DK are important not only for control but also for design. In the design phase we have to be able to determine all IK/DK solutions over a given workspace, for example for determining what will be the maximal cable tensions and consequently the minimal cable diameter such that the cable can sustain the maximal tension. Additionally we have to determine the maximal torque applied on the drum in order to calculate what should be the maximal winch motor torque.
Finding this maximal tension/maximal drum torque over a given workspace are difficult constrained optimization problems that can be solved exactly for simple CDPRs and cable models but has no known solution for our prototypes.
In this case it is usual to discretize the workspace and to calculate the cable tensions over a large set of pose (if the workspace is defined in terms of possible poses) or a large set of cables lengths. This means that we have to solve either the IK or DK a large number of time. Unfortunately in our case both the IK and DK have multiple solutions. When in use the robot state for a given set of ${L}_{0}$ is unique and depends upon the CDPR motion history. In the design phase we have to adopt a worst case scenario and consider all possible cases, thereby considering all possible IK/DK solutions. We will now focus on the DK problem. Obtaining the DK solutions amounts for a CDPR with $k$ cables to solve a square system of $6+2k$ equations (the Irvine $2k$ equations and the 6 mechanical equilibrium equations) involving the pose of the robot (defined by 6 unknowns) and the $2k$ horizontal and vertical components of the tensions in the cables at the attachment points on the platform.
If sagging cables are considered we are the only team that has been able to design algorithms, denoted exact, that are able to calculate all the IK and DK solution either by using a continuation process or an interval analysis approach 15,5. Unfortunately these algorithms are computer intensive (one hour for the IK and much more for the DK, that may have between 10 and 50 solutions) and are therefore not appropriate for the design phase which require to solve these problems a very large number of time.
In 2021 we have considered using AI to solve the DK problems using a Multi-Layer Perceptron (MLP) but we were surprised that solving a system of equations was a topic that is barely addressed in the AI community (with a few exceptions). Since the start we were aware of two possible problems:
1. a MLP is able to provide a single output while we have multiple solutions
2. the DK system is what we called a double stiff system, meaning that for a sufficiently large ball in the ${L}_{0}$ space we have a smooth region where small changes in the ${L}_{0}$ induce small changes in the solution while we have also stiff regions where the solution changes are high and even stiffer region with extremely large changes
3. we were not expecting exact solutions but an error of 1% was deemed to be acceptable
We have first shown that it was possible to construct an arbitrarily large training set for the MLP based on the IK and DK solutions that has been computed by the exact algorithms for a few selected cases ( we have provided open-source learning databases in 14). We have then to determine the geometry of the MLP, i.e. the number $h$ of hidden layers, the number $n$ of neurons in each layer, the activation function(s) $𝒜$ and the loss function $ℱ$ After a few preliminary trials we noticed that
• the learning time was small (a few minutes)
• having large values for $h,n$ does not significantly change the final value of the loss function
• a set of 5 possible activation function were providing the best final loss
• the MSE loss was the most efficient loss function
Based on there remarks we systematically explore all possible combinations of values for $h$ (in the set [2,200]), $n$ (in [10,200]) and the set of 5 activation functions. We hence create a large number of MLPs with the hope to obtain MLPs providing different solutions for a given set of ${L}_{0}$. The results were poor: even the best MLPs were giving the solution with 200% of error on the output for the training set and 300% on a verification set.
We then consider using a hybridization approach: the MLP prediction is used as initial guess for the Newton method which, if convergent, provides an exact solution. This has led to a small improvement as about 5% of the total number DK solutions were found but we also noticed that even if the Newton scheme was converging for several MLPs, they all provide the same DK solution.
We also noticed key points occurring during the optimization of the loss function:
1. a decrease of the loss function does not automatically implies an increase of the number of Newton convergence for the current model
2. some MLPs were exhibiting quite good predictions on some variables especially the ${F}_{x},{F}_{z}$ and very poor predictions on the other variables
Point 1 can be explained because of the behavior of the Newton scheme that converges if there is good prediction for some part of the variables, called essential, while the accuracy of the prediction is much less important for the remaining variables (the non-essential one). A decrease of the loss function may be obtained if the errors on the non-essential variables are significantly decreasing while we have a limited increase on the errors of the essential variables, which lead to a decrease of the number of Newton convergences. Points 1 and 2 lead us to change our methodology to exploit AI:
• the number of convergences is calculated after each significant change of the loss function and we store the MLP (denoted type 1 MLP) having exhibited the largest number of convergences
• the MLPs, denoted type 2 MLP, leading to the lowest MSE for each ${F}_{x},{F}_{z}$ is stored.
We therefore end up with several type 1 and 2 MLPs. When checking the verification set we use first the predictions of the type 1 MLPs as guess for the Newton scheme. Then we create new guesses for the Newton scheme by updating the type 1 MLP prediction on the ${F}_{x},{F}_{z}$ by the one obtained for the type 2 MLPs.
A second change is the clusterization of the training set. For that purpose the ranges for the output variables over the whole training set have been split in smaller components and we have found out that with 12 sets of ranges ${ℛ}_{i},i\in \left[1,12\right]$ for the variables we were able to cover the whole training set. A cluster is obtained as all the elements of the training set whose output belongs to the same ${ℛ}_{i}$. The MLP training is then performed for each cluster so that we get about 4000 MLPs. Checking the verification set is done by using all these MLPs3 12,3. About 30% of the DK solutions were obtained with this approach and in a very few cases we have been able to obtain 2 DK solutions for a given set of ${L}_{0}$. Although this is a progress, we are still quite far from a good result, especially regarding the very low number of DK solutions we get for a given set of ${L}_{0}$.
We plan to investigate the following methodological improvements:
• using Physics-informed neural network (PINN): as seen previously the MSE loss is not the most appropriate loss function for our case. With the help of the 3IA techpool we are implementing a MLP builder where the loss function is based on the absolute value of the DK equations, thereby moving partly to unsupervised learning, although the quality of the MLP may have still to rely on the number of Newton convergences over a training set. This is a preliminary trial but later on we may consider minimizing the Kantorovitch index (that uses the norms of the equations vector and of the system Jacobian and Hessian matrix) which, if lower than 1 ensures the convergence of the Newton scheme
• modifying our cluster algorithm using an idea based on the concept of kinematic branches. Assume that all DK solutions (i.e. points in the $6+2k$ output space $𝒪$) have been obtained for a given set of $k$${L}_{0}$ i.e. a point $M$ in the $k$-dimensional ${L}_{0}$ space $ℒ$. If the point $M$ follows a path in $ℒ$, then each DK solution follows a path in $𝒪$ which is a kinematic branch. The training clusters are now designed to contain only elements that lie on the same kinematic branch. Preliminary tests on one of such a cluster have shown that the learning allows to obtain MLPs that have an almost 100% Newton converge rate for the training set. It remains to verify that this is the case for all clusters and then to test the MLPs on an arbitrary verification set
• using dimension reduction: in the DK problem if we have a prediction for the location of the platform, then the Irvine equations for a given cable is a system of 2 equations in the ${F}_{x},{F}_{z}$ unknown that has a unique solution (although we have not found any formal proof for this point). Provided that we have a numerical procedure to solve these equations we may reduce the DK MLP output dimension to 6 but still use the calculated cables ${F}_{x},{F}_{z}$ in the loss function. Regarding the Newton scheme we already know that the $2k$ Irvine equations are satisfied so that will remain only the 6 mechanical equilibrium equations, A possible consequence of this situation may be that the Newton convergence ball will be larger thereby allowing a larger number of Newton convergence over the verification set. We have already designed MLPs for solving the 2 Irvine equations with good success and we plan to design a safe algorithm for this solving, possibly mixing MLPs and deterministic methods
Provided that we are able to design MLPs that have a good success rate for determining all DK solutions we are also planning to address the issue of using discretization. Indeed we are not completely satisfied with the result as we have sampled the ${L}_{0}$ space so that we may have missed the real maximum tension (and remembering the double stiffness of the system there may be a severe error on the maximal tension). For addressing this issue we plan to design a growing algorithm: the idea is basically to obtain an accurate upper bound of the cable tensions for balls centered at the points $G$ of $ℒ$ used in the discrete treatment. For a given $G$ there are methods (namely the Kantorovitch theorem and Neumaier $ϵ$ inflation method) that allow one to calculate a ball in the ${L}_{0}$ space, centered at $G$, such that we are sure that all DK solutions for the ${L}_{0}$ in the ball are enclosed in known balls. With this method we will be able to determine a safe upper bound for the cables tensions over a ball in the ${L}_{0}$ space and if we are able to fully cover the desired region in the ${L}_{0}$ with such balls, then we will obtain a safe bound for the tensions over the whole region.
## 8.2 Medical activities
As all of our clinical trials scheduled for 2020-2021 have been either canceled or postponed because of the pandemic we have used 2021 to develop new devices that were planned for later on. Still we have had restriction for on-site access that have penalized these developments.
### 8.2.1 Modular rehabilitation station
Rehabilitation is also a major topic of the team. The medical community has indicated to us several major issues with commercially available rehabilitation devices: difficulty to configure the device for training a given musculo-squelettic group which impose a persistent action to the ergotherapeuthe and therefore divert him from observing the patient, the lack of synthetic indicators to assess the rehabilitation efficiency, the lack of modularity and of mobility help that complicates the ergotherapeuthe task when the patient motricity is not yet sufficient, the lack of motivation stimulus for the patient, a large setup time for installing the patient in the device and finally the cost of the devices. To address these problems we start developing a modular rehabilitation station which is first able to manage different types of training devices (illustrated with a treadmill in figure 5 but a rower or an equilibrium station are also available). The training devices of the station are actuated: for example we can change the slope and inclination of the treadmill which allow to adjust the difficulty level of the exercise but also to favor the work of specific musculo-squelettic groups. Furthermore a CDPR (see next section) is able to exert a vertical lifting force that allows to start a rehabilitation process even with a patient that is not yet able to stand up on his own. Regarding the patient motivation the patient is immersed in a virtual environment that makes the patient more comfortable but also allows for a replay of an exercise: on the figure the patient is walking in a mountainous environment whose slope is completely or partially reflected by the treadmill. Finally the patient body configuration during the exercise is measured by using several external sensors (lidars, accelerometers, ...) while, if necessary, the support force that he requires is measured by the CDPR and/or by force sensors in the handle (which allow also to estimate his stress). Using this external measurements allows for a minimal setup time and pertinent medical indicators are deduced from these measurements (for example for the treadmill we measure the number of steps, the leg velocities and poses, the patient trajectory deviation,...). Consequently the main task for the ergotherapeuthe is to adjust the difficulty level and to observe the patient during the exercise while a complete set of medical indicators will be provided at the end of the exercises.
### 8.2.2 Human activity recognition
Participants: Jean-Pierre Merlet, Yves Papegay, Odile Pourtallier, Eric Wajnberg.
Human activity recognition (HAR) is a major topic in the team. We are focusing on monitoring mobility and displacements (we are not interested in recognizing the individual action of our subject) using a sensor-based approach, excluding vision which is intrusive and even prohibited in some places for legal reasons. For that purpose we have developed a smart barrier combining redundant passive infrared motion detectors and infrared distance sensors. Smart barriers have been implemented in Ehpad Valrose, a new retirement house in which a specific infrastructure has been put in place to accommodate research works and in Institut Claude Pompidou, a Alzheimer day care hospital from 2019 to 2020. These two long term experiments has allowed us to determine that essential points in HAR are to determine what is possible to measure, the sensor types, how to retrieve and process sensor data, how effective are the quality of measurements on a long term basis and the level of monitoring that is acceptable for frail peoples and their helpers while providing significant and reliable data for the medical community in spite of the uncertainties both in the measurements and in the system modeling. These samples of questions will become central in our work. Unfortunately the pandemic has prohibited us to improve our smart barriers and to test a new version.
## 8.3 Biology activities
Participants: Eric Wajnberg, Yves Papegay, Odile Pourtallier.
### 8.3.1 Optimized flower visiting strategy of bees
As explained in the previous activity report, an international scientific cooperation was launched in 2020 with Israeli scientists located at the University of Haifa to understand the optimal foraging decision of bees foraging for nectar. We developed an optimization deterministic model trying to understand what should be the optimized flower visiting strategy, taking into account the ability of the foraging animals to learn the quality of the different flowers they are visiting. This work was continued in 2022, adding more realistic features to the model, and the results obtained were presented in international conferences, and were accepted for publication in a top-level international journal on animal ecology and behavior. The results we obtained are providing a couple of predictions that are now being tested on real animals in the University of Haifa in Israel, through a contact our Israeli partners were able to obtain. Future developments are planned.
### 8.3.2 Optimal reproductive strategy
In 2021, we developed an international scientific cooperation with Italian scientists located at the University Palermo, Sicily. The goal was to develop a mechanistic and probabilistic model whose aim was to understand the optimal reproductive strategy adopted by two competing insect wasp species (called Trissolcus basalis, abbreviated here as “Tb”; and Ooencyrtus telenomicida abbreviated here as “Ot”) both laying their eggs (and thus killing) the eggs of the stink bug Nezara viridula. In entomology, the competition between different species is called either “extrinsic”, when the competition occurs between females laying their eggs, or “intrinsic”, when the competition occurs between their progeny developing within the same host. On this specific biological system, Tb is known to be a better extrinsic competitor, while Ot is better to win the intrinsic competition when the progeny of the two species are developing within the same host. Tb females are also known to produce an overall higher number of eggs during their lifetime (higher overall fecundity). Nezara viridula eggs are laid in so-called egg- masses that can have different size, and the size of the egg masses attacked by the two competing wasp species should play a role in the global output and success of the two species. For example, on small egg masses, the proportion of eggs attacked by both species will increase which should give an advantage for Ot, while the opposite is expected on larger egg masses. In this context, we developed a model trying to identify what are the environmental conditions that should promote one competing species over the other. Figure 6 gives an example of the type of results obtained.
# 9 Partnerships and cooperations
Participants: Jean-Pierre Merlet, Yves Papegay, Odile Pourtallier, Eric Wajnberg.
## 9.1 International initiatives
### 9.1.1 Participation in other International Programs
Participants: Eric Wajnberg.
• Title: Mathematical modeling of biological control interaction to support agriculture and conservation
• Financially supported by the IIAS (Israel Institute for Advanced Studies)
• International Research group of 20 persons
• E. Wajnberg is co-supervisor of this program
• Duration: 5 months
## 9.2 International research visitors
### 9.2.1 Visits to international teams
• E. Wajnberg has visited U. Haifa for 5 months
### 9.2.2 Other european programs/initiatives
• Hephaistos is part of the euROBIN, the Network of Excellence on AI and robotics that was launched in 2021
## 9.3 National initiatives
• Hephaistos is part of the EquipEx+ AMI dealing with XXL robots
• project Craft on collaborative cable-driven parallel robot funded by ANR. It involves LS2N (Nantes) and the Cetim. This project has started beginning of 2019.
# 10 Dissemination
Participants: Jean-Pierre Merlet, Odile Pourtallier, Yves Papegay, Eric Wajnberg.
## 10.1 Promoting scientific activities
We will not mention our review activity which is quite large but part of the job.
### 10.1.1 Scientific events: organisation
• J-P. Merlet is a permanent member of the International Steering Committee of the IROS conference, of the CableCon conference and chairman of the scientific Committee of the Computational Kinematics workshop. He was also an advisor for the largest robotics conference ICRA 2020, that has finally being held virtually because of the Covid,
• Y. Papegay is a permanent member of the International Steering Committee of the International Mathematica Symposium conferences series. He is a member of the OpenMath Society, building an extensible standard for representing the semantics of mathematical objects.
### 10.1.2 Journal
• E. Wajnberg is Editor-in-Chief of the international journal “BioControl” since September 2006, a member of the Editorial Board of the international journal “Entomologia Experimentalis et Applicata”, since 1996, a member of the Editorial Board of the international journal “Applied Entomology and Zoology”, since 2003 and a member of the Editorial Board of the international journal “Neotropical Entomology”, since 2009.
### 10.1.3 Invited talks
• J-P. Merlet gives a talk about AI during an INRIA/OCA seminar. He gives a presentation about assistance during the Healthy EUR seminar
• E. Wajnberg has given an invited talk at the Symposium “Models for biological control”, International Congress of Entomology. Helsinki, Finland. July.
### 10.1.4 Leadership within the scientific community
• J-P. Merlet is a member of the IFToMM (International Federation for the Promotion of Mechanism and Machine Science) technical Committees on History and on Computational Kinematics. He is a member of the IFToMM Executive Council Publication Advisory Board and an IEEE Fellow. He is a member of the scientific committee of the CNRS GDR robotique and a senior chair of 3IA Côte d’Azur.
### 10.1.5 Scientific expertise
• J-P. Merlet is a Nominator for the Japan’s Prize and the head of the robotics GDR Publication Committee in charge of producing a report on "recommended supports for publication" for journals and conferences that does not provide a ranking but advices according to robotics topics
• E. Wajnberg is an appointed member of the Academic Committee of the Hebrew University of Jerusalem, since June 2022, for four years and an appointed member of the International Advisory Board of the “International Center for Excellence in Biological Control”, from August 2018 to August 2023.
• J-P. Merlet is a corresponding member of INRIA ethical committee (COERLE) and member of the Ethical Committee of Université Côte d’Azur (CER). He is an elected member of INRIA Scientific Committee. As 2022 was an evaluation year for the INRIA robotics teams he was the coordinator of a prospective report of the teams. He was a member of the 2022 local jury for the hiring of new INRIA researchers
• Y.Papegay is the president of Comité des Utilisateurs des Moyens Informatiques
• O. Pourtallier is a substitute board member of SeaTech, an Engineering School of University of Toulon. She is member of the NICE committee (long term invited scientists and post-doctoral student selection) and of the Transform committee of INRIA Sophia-Antipolis.
## 10.2 Teaching - Supervision - Juries
### 10.2.1 Teaching
• J-P. Merlet has taught 15 hours on parallel robots to Master ISC (M2) at University of Toulon.
• O. Pourtallier lectured 6 hours on game theory to Master OSE (M2), at École des Mines de Paris, Sophia Antipolis, France.
• E. Wajnberg teaches 60h at The Hebrew University of Jerusalem and University of Sao Paulo - Piracicaba, Brazil.
### 10.2.2 Supervision
• J-P. Merlet is the supervisor for the PhD of R. Tissot
• Y. Papegay is the supervisor of the CIFRE PhD of Alexandre Tran
### 10.2.3 Juries
• J-P. Merlet has been president of the jury of the PhD thesis of B. Fasquelle (LS2N, Nantes) and a member of the Best Thesis Award of the robotics GDR
## 10.3 Popularization
• J-P. Merlet and E. Wajnberg gives talks in the Alpes-Maritimes in the framework of the Science pour Tous association.
• J-P. Merlet and Y. Papegay are active members of the dissemination organization Terra Numerica
### 10.3.1 Interventions
• J-P. Merlet and E. Wajnberg give two talks at the INRIA Cafe-In seminar
• J-P. Merlet give a general INRIA talk about Hephaistos activities regarding handicap and a presentation of our 3D printing robot during the Fête de la Science
# 11 Scientific production
## 11.1 Major publications
• 1 articleJ.-P.Jean-Pierre Merlet. Interval Analysis and Reliability in Robotics.International Journal of Reliability and Safety32009, 104-130
• 2 inproceedingsMaximal cable tensions of a N-1 cable-driven parallel robot with elastic or ideal cables.CableCon 2021 - 5th International Conference on Cable-Driven Parallel RobotsVirtual, FranceJuly 2021
• 3 inproceedingsJ.-P.Jean-Pierre Merlet. Mixing AI and deterministic methods for the design of a transfer system for frail people.Sophia IAsummitSophia-Antipolis, FranceNovember 2021
• 4 bookJ.-P.Jean-Pierre Merlet. Parallel robots, 2nd Edition.Springer2005
• 5 inproceedingsJ.-P.Jean-Pierre Merlet. The kinematics of cable-driven parallel robots with sagging cables: preliminary results.ICRA 2015 - IEEE International Conference on Robotics and AutomationSeattle, United States2015, 1593-1598
• 6 inproceedingsJ.-P.Jean-Pierre Merlet. Using interval analysis in robotics problems.SCANTokyo, JapanSeptember 2018
• 7 articleN.Nathalie Nevejans, O.Odile Pourtallier, S.Sylvie Icart and J.-P.Jean-Pierre Merlet. Les avancées en robotique d'assistance à la personne sous le prisme du droit et de l'éthique.Revue générale de droit médicaleDecember 2017
• 8 phdthesisY.Yves Papegay. De la modélisation littérale à la simulation certifiée.Université de Nice Sophia-AntipolisNice, FranceJune 2012,
• 9 inproceedingsY.Yves Papegay. From Modeling to Simulation with Symbolic Computation: An Application to Design and Performance Analysis of Complex Optical Devices.Proceedings of the Second Workshop on Computer Algebra in Scientific ComputingMunichSpringer VerlagJune 1999
• 10 inproceedingsG.Gilles Trombettoni. A Polynomial Time Local Propagation Algorithm for General Dataflow Constraint Problems.Proc. Constraint Programming CP'98, LNCS 1520 (Springer Verlag)1998, 432--446
## 11.2 Publications of the year
### International journals
• 11 articleJ.-S.Jean-Sébastien Pierre, S.Solenn Stoeckel and E.Eric Wajnberg. The advantage of sex: Reinserting fluctuating selection in the pluralist approach.PLoS ONE178August 2022, 1-15
### International peer-reviewed conferences
• 12 inproceedingsJ.-P.Jean-Pierre Merlet and R.Romain Tissot. A panorama of methods for dealing with sagging cables in cable-driven parallel robots.ARK 2022 - 18th International Symposium on Advances in robots kinematicsBilbao, SpainJune 2022
## 11.3 Cited publications
• 13 bookH. M.H. M. Irvine. Cable Structures.MIT Press1981
• 14 miscJ.-P.Jean-Pierre Merlet. Data base for the direct kinematics of cable-driven parallel robot (CDPR) with sagging cables.These databases provide learning and verification sets that may used by AI to solve the direct kinematics of a cable-driven parallel robot with 8 cables. The input of this problem is the lengths of th 8 cables and the output should be all platform poses that are compatible with the cable lengths. The cable model that is used is the full model (elasticity and cable mass) that can be foud in Irvine textbook.December 2021
• 15 conferenceJ.-P.J-P. Merlet. Preliminaries of a new approach for the direct kinematics of suspended cable-driven parallel robot with deformable cables.EucomesNantes2016
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libmdbx 0.11.9.0 (2022-08-02T12:00:30+03:00) One of the fastest compact embeddable key-value ACID database without WAL.
Settings
## Typedefs
typedef enum MDBX_option_t MDBX_option_t
## Functions
LIBMDBX_API int mdbx_env_set_option (MDBX_env *env, const MDBX_option_t option, uint64_t value)
Sets the value of a runtime options for an environment. More...
LIBMDBX_API int mdbx_env_get_option (const MDBX_env *env, const MDBX_option_t option, uint64_t *pvalue)
Gets the value of runtime options from an environment. More...
int mdbx_env_set_syncbytes (MDBX_env *env, size_t threshold)
Sets threshold to force flush the data buffers to disk, even any of MDBX_SAFE_NOSYNC flag in the environment. More...
int mdbx_env_set_syncperiod (MDBX_env *env, unsigned seconds_16dot16)
Sets relative period since the last unsteady commit to force flush the data buffers to disk, even of MDBX_SAFE_NOSYNC flag in the environment. More...
LIBMDBX_API int mdbx_env_set_flags (MDBX_env *env, MDBX_env_flags_t flags, bool onoff)
Set environment flags. More...
LIBMDBX_API int mdbx_env_set_geometry (MDBX_env *env, intptr_t size_lower, intptr_t size_now, intptr_t size_upper, intptr_t growth_step, intptr_t shrink_threshold, intptr_t pagesize)
Set all size-related parameters of environment, including page size and the min/max size of the memory map. More...
MDBX_DEPRECATED int mdbx_env_set_mapsize (MDBX_env *env, size_t size)
Set the maximum number of threads/reader slots for for all processes interacts with the database. More...
int mdbx_env_set_maxdbs (MDBX_env *env, MDBX_dbi dbs)
Set the maximum number of named databases for the environment. More...
LIBMDBX_API int mdbx_env_set_userctx (MDBX_env *env, void *ctx)
Sets application information (a context pointer) associated with the environment. More...
## ◆ MDBX_option_t
typedef enum MDBX_option_t MDBX_option_t
## ◆ mdbx_env_get_option()
LIBMDBX_API int mdbx_env_get_option ( const MDBX_env * env, const MDBX_option_t option, uint64_t * pvalue )
Gets the value of runtime options from an environment.
Parameters
[in] env An environment handle returned by mdbx_env_create(). [in] option The option from MDBX_option_t to get value of it. [out] pvalue The address where the option's value will be stored.
MDBX_option_t
mdbx_env_get_option()
Returns
A non-zero error value on failure and 0 on success.
## ◆ mdbx_env_set_flags()
LIBMDBX_API int mdbx_env_set_flags ( MDBX_env * env, MDBX_env_flags_t flags, bool onoff )
Set environment flags.
This may be used to set some flags in addition to those from mdbx_env_open(), or to unset these flags.
mdbx_env_get_flags()
Note
In contrast to LMDB, the MDBX serialize threads via mutex while changing the flags. Therefore this function will be blocked while a write transaction running by other thread, or MDBX_BUSY will be returned if function called within a write transaction.
Parameters
[in] env An environment handle returned by mdbx_env_create(). [in] flags The env_flags to change, bitwise OR'ed together. [in] onoff A non-zero value sets the flags, zero clears them.
Returns
A non-zero error value on failure and 0 on success, some possible errors are:
Return values
MDBX_EINVAL An invalid parameter was specified.
## ◆ mdbx_env_set_geometry()
LIBMDBX_API int mdbx_env_set_geometry ( MDBX_env * env, intptr_t size_lower, intptr_t size_now, intptr_t size_upper, intptr_t growth_step, intptr_t shrink_threshold, intptr_t pagesize )
Set all size-related parameters of environment, including page size and the min/max size of the memory map.
In contrast to LMDB, the MDBX provide automatic size management of an database according the given parameters, including shrinking and resizing on the fly. From user point of view all of these just working. Nevertheless, it is reasonable to know some details in order to make optimal decisions when choosing parameters.
Both mdbx_env_set_geometry() and legacy mdbx_env_set_mapsize() are inapplicable to read-only opened environment.
Both mdbx_env_set_geometry() and legacy mdbx_env_set_mapsize() could be called either before or after mdbx_env_open(), either within the write transaction running by current thread or not:
• In case mdbx_env_set_geometry() or legacy mdbx_env_set_mapsize() was called BEFORE mdbx_env_open(), i.e. for closed environment, then the specified parameters will be used for new database creation, or will be applied during opening if database exists and no other process using it.
If the database is already exist, opened with MDBX_EXCLUSIVE or not used by any other process, and parameters specified by mdbx_env_set_geometry() are incompatible (i.e. for instance, different page size) then mdbx_env_open() will return MDBX_INCOMPATIBLE error.
In another way, if database will opened read-only or will used by other process during calling mdbx_env_open() that specified parameters will silently discarded (open the database with MDBX_EXCLUSIVE flag to avoid this).
• In case mdbx_env_set_geometry() or legacy mdbx_env_set_mapsize() was called after mdbx_env_open() WITHIN the write transaction running by current thread, then specified parameters will be applied as a part of write transaction, i.e. will not be completely visible to any others processes until the current write transaction has been committed by the current process. However, if transaction will be aborted, then the database file will be reverted to the previous size not immediately, but when a next transaction will be committed or when the database will be opened next time.
• In case mdbx_env_set_geometry() or legacy mdbx_env_set_mapsize() was called after mdbx_env_open() but OUTSIDE a write transaction, then MDBX will execute internal pseudo-transaction to apply new parameters (but only if anything has been changed), and changes be visible to any others processes immediately after succesful completion of function.
Essentially a concept of "automatic size management" is simple and useful:
• There are the lower and upper bounds of the database file size;
• There is the growth step by which the database file will be increased, in case of lack of space;
• There is the threshold for unused space, beyond which the database file will be shrunk;
• The size of the memory map is also the maximum size of the database;
• MDBX will automatically manage both the size of the database and the size of memory map, according to the given parameters.
So, there some considerations about choosing these parameters:
• The lower bound allows you to prevent database shrinking below certain reasonable size to avoid unnecessary resizing costs.
• The upper bound allows you to prevent database growth above certain reasonable size. Besides, the upper bound defines the linear address space reservation in each process that opens the database. Therefore changing the upper bound is costly and may be required reopening environment in case of MDBX_UNABLE_EXTEND_MAPSIZE errors, and so on. Therefore, this value should be chosen reasonable large, to accommodate future growth of the database.
• The growth step must be greater than zero to allow the database to grow, but also reasonable not too small, since increasing the size by little steps will result a large overhead.
• The shrink threshold must be greater than zero to allow the database to shrink but also reasonable not too small (to avoid extra overhead) and not less than growth step to avoid up-and-down flouncing.
• The current size (i.e. size_now argument) is an auxiliary parameter for simulation legacy mdbx_env_set_mapsize() and as workaround Windows issues (see below).
Unfortunately, Windows has is a several issue with resizing of memory-mapped file:
• Windows unable shrinking a memory-mapped file (i.e memory-mapped section) in any way except unmapping file entirely and then map again. Moreover, it is impossible in any way when a memory-mapped file is used more than one process.
• Windows does not provide the usual API to augment a memory-mapped file (i.e. a memory-mapped partition), but only by using "Native API" in an undocumented way.
MDBX bypasses all Windows issues, but at a cost:
• Ability to resize database on the fly requires an additional lock and release SlimReadWriteLock during each read-only transaction.
• During resize all in-process threads should be paused and then resumed.
• Shrinking of database file is performed only when it used by single process, i.e. when a database closes by the last process or opened by the first. = Therefore, the size_now argument may be useful to set database size by the first process which open a database, and thus avoid expensive remapping further.
For create a new database with particular parameters, including the page size, mdbx_env_set_geometry() should be called after mdbx_env_create() and before mdbx_env_open(). Once the database is created, the page size cannot be changed. If you do not specify all or some of the parameters, the corresponding default values will be used. For instance, the default for database size is 10485760 bytes.
If the mapsize is increased by another process, MDBX silently and transparently adopt these changes at next transaction start. However, mdbx_txn_begin() will return MDBX_UNABLE_EXTEND_MAPSIZE if new mapping size could not be applied for current process (for instance if address space is busy). Therefore, in the case of MDBX_UNABLE_EXTEND_MAPSIZE error you need close and reopen the environment to resolve error.
Note
Actual values may be different than your have specified because of rounding to specified database page size, the system page size and/or the size of the system virtual memory management unit. You can get actual values by mdbx_env_sync_ex() or see by using the tool mdbx_chk with the -v option.
Legacy mdbx_env_set_mapsize() correspond to calling mdbx_env_set_geometry() with the arguments size_lower, size_now, size_upper equal to the size and -1 (i.e. default) for all other parameters.
Parameters
[in] env An environment handle returned by mdbx_env_create() [in] size_lower The lower bound of database size in bytes. Zero value means "minimal acceptable", and negative means "keep current or use default". [in] size_now The size in bytes to setup the database size for now. Zero value means "minimal acceptable", and negative means "keep current or use default". So, it is recommended always pass -1 in this argument except some special cases. [in] size_upper The upper bound of database size in bytes. Zero value means "minimal acceptable", and negative means "keep current or use default". It is recommended to avoid change upper bound while database is used by other processes or threaded (i.e. just pass -1 in this argument except absolutely necessary). Otherwise you must be ready for MDBX_UNABLE_EXTEND_MAPSIZE error(s), unexpected pauses during remapping and/or system errors like "address busy", and so on. In other words, there is no way to handle a growth of the upper bound robustly because there may be a lack of appropriate system resources (which are extremely volatile in a multi-process multi-threaded environment). [in] growth_step The growth step in bytes, must be greater than zero to allow the database to grow. Negative value means "keep current or use default". [in] shrink_threshold The shrink threshold in bytes, must be greater than zero to allow the database to shrink and greater than growth_step to avoid shrinking right after grow. Negative value means "keep current or use default". Default is 2*growth_step. [in] pagesize The database page size for new database creation or -1 otherwise. Once the database is created, the page size cannot be changed. Must be power of 2 in the range between MDBX_MIN_PAGESIZE and MDBX_MAX_PAGESIZE. Zero value means "minimal acceptable", and negative means "keep current or use default".
Returns
A non-zero error value on failure and 0 on success, some possible errors are:
Return values
MDBX_EINVAL An invalid parameter was specified, or the environment has an active write transaction. MDBX_EPERM Two specific cases for Windows: 1) Shrinking was disabled before via geometry settings and now it enabled, but there are reading threads that don't use the additional SRWL (which is required to avoid Windows issues). 2) Temporary close memory mapped is required to change geometry, but there read transaction(s) is running and no corresponding thread(s) could be suspended since the MDBX_NOTLS mode is used. MDBX_EACCESS The environment opened in read-only. MDBX_MAP_FULL Specified size smaller than the space already consumed by the environment. MDBX_TOO_LARGE Specified size is too large, i.e. too many pages for given size, or a 32-bit process requests too much bytes for the 32-bit address space.
## ◆ mdbx_env_set_mapsize()
MDBX_DEPRECATED int mdbx_env_set_mapsize ( MDBX_env * env, size_t size )
inline
## ◆ mdbx_env_set_maxdbs()
int mdbx_env_set_maxdbs ( MDBX_env * env, MDBX_dbi dbs )
inline
Set the maximum number of named databases for the environment.
This function is only needed if multiple databases will be used in the environment. Simpler applications that use the environment as a single unnamed database can ignore this option. This function may only be called after mdbx_env_create() and before mdbx_env_open().
Currently a moderate number of slots are cheap but a huge number gets expensive: 7-120 words per transaction, and every mdbx_dbi_open() does a linear search of the opened slots.
mdbx_env_get_maxdbs()
Parameters
[in] env An environment handle returned by mdbx_env_create(). [in] dbs The maximum number of databases.
Returns
A non-zero error value on failure and 0 on success, some possible errors are:
Return values
MDBX_EINVAL An invalid parameter was specified. MDBX_EPERM The environment is already open.
inline
Set the maximum number of threads/reader slots for for all processes interacts with the database.
This defines the number of slots in the lock table that is used to track readers in the the environment. The default is about 100 for 4K system page size. Starting a read-only transaction normally ties a lock table slot to the current thread until the environment closes or the thread exits. If MDBX_NOTLS is in use, mdbx_txn_begin() instead ties the slot to the MDBX_txn object until it or the MDBX_env object is destroyed. This function may only be called after mdbx_env_create() and before mdbx_env_open(), and has an effect only when the database is opened by the first process interacts with the database.
Parameters
[in] env An environment handle returned by mdbx_env_create(). [in] readers The maximum number of reader lock table slots.
Returns
A non-zero error value on failure and 0 on success, some possible errors are:
Return values
MDBX_EINVAL An invalid parameter was specified. MDBX_EPERM The environment is already open.
## ◆ mdbx_env_set_option()
LIBMDBX_API int mdbx_env_set_option ( MDBX_env * env, const MDBX_option_t option, uint64_t value )
Sets the value of a runtime options for an environment.
Parameters
[in] env An environment handle returned by mdbx_env_create(). [in] option The option from MDBX_option_t to set value of it. [in] value The value of option to be set.
MDBX_option_t
mdbx_env_get_option()
Returns
A non-zero error value on failure and 0 on success.
## ◆ mdbx_env_set_syncbytes()
int mdbx_env_set_syncbytes ( MDBX_env * env, size_t threshold )
inline
Sets threshold to force flush the data buffers to disk, even any of MDBX_SAFE_NOSYNC flag in the environment.
mdbx_env_get_syncbytes
MDBX_opt_sync_bytes
The threshold value affects all processes which operates with given environment until the last process close environment or a new value will be settled.
Data is always written to disk when mdbx_txn_commit() is called, but the operating system may keep it buffered. MDBX always flushes the OS buffers upon commit as well, unless the environment was opened with MDBX_SAFE_NOSYNC, MDBX_UTTERLY_NOSYNC or in part MDBX_NOMETASYNC.
The default is 0, than mean no any threshold checked, and no additional flush will be made.
Parameters
[in] env An environment handle returned by mdbx_env_create(). [in] threshold The size in bytes of summary changes when a synchronous flush would be made.
Returns
A non-zero error value on failure and 0 on success.
## ◆ mdbx_env_set_syncperiod()
int mdbx_env_set_syncperiod ( MDBX_env * env, unsigned seconds_16dot16 )
inline
Sets relative period since the last unsteady commit to force flush the data buffers to disk, even of MDBX_SAFE_NOSYNC flag in the environment.
mdbx_env_get_syncperiod
MDBX_opt_sync_period
The relative period value affects all processes which operates with given environment until the last process close environment or a new value will be settled.
Data is always written to disk when mdbx_txn_commit() is called, but the operating system may keep it buffered. MDBX always flushes the OS buffers upon commit as well, unless the environment was opened with MDBX_SAFE_NOSYNC or in part MDBX_NOMETASYNC.
Settled period don't checked asynchronously, but only by the mdbx_txn_commit() and mdbx_env_sync() functions. Therefore, in cases where transactions are committed infrequently and/or irregularly, polling by mdbx_env_sync() may be a reasonable solution to timeout enforcement.
The default is 0, than mean no any timeout checked, and no additional flush will be made.
Parameters
[in] env An environment handle returned by mdbx_env_create(). [in] seconds_16dot16 The period in 1/65536 of second when a synchronous flush would be made since the last unsteady commit.
Returns
A non-zero error value on failure and 0 on success.
## ◆ mdbx_env_set_userctx()
LIBMDBX_API int mdbx_env_set_userctx ( MDBX_env * env, void * ctx )
Sets application information (a context pointer) associated with the environment.
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# Benjamin Graham formula
The Benjamin Graham formula is a formula for the valuation of growth stocks.
It was proposed by investor and professor of Columbia University, Benjamin Graham - often referred to as the "father of value investing". [1] Published in his book, The Intelligent Investor, Graham devised the formula for lay investors to help them with valuing growth stocks, in vogue at the time of the formula's publication. [2]
Graham cautioned here that the formula was not appropriate for companies with a "below-par" debt position: "My advice to analysts would be to limit your appraisals to enterprises of investment quality, excluding from that category such as do not meet specific criteria of financial strength".[3]
## Formula calculation
In Graham's words: "Our study of the various methods has led us to suggest a foreshortened and quite simple formula for the evaluation of growth stocks, which is intended to produce figures fairly close to those resulting from the more refined mathematical calculations."[2]
The formula as described by Graham originally in the 1962 edition of Security Analysis, and then again in the 1973 edition of The Intelligent Investor, is as follows:[2]
${\displaystyle V^{*}=\mathrm {EPS} \times (8.5+2g)}$
${\displaystyle V^{*}}$ = the value expected from the growth formulas over the next 7 to 10 years
${\displaystyle EPS}$ = trailing twelve months earnings per share
${\displaystyle 8.5}$ = P/E base for a no-growth company
${\displaystyle g}$ = reasonably expected 7 to 10 year growth rate (see Sustainable growth rate § From a financial perspective)
### Revised formula
Graham later revised his formula based on the belief that the greatest contributing factor to stock values (and prices) over the past decade had been interest rates. In 1974, he restated it as follows:[4]
The Graham formula proposes to calculate a company’s intrinsic value ${\displaystyle V^{*}}$ as:
${\displaystyle V^{*}={\cfrac {\mathrm {EPS} \times (8.5+2g)\times 4.4}{Y}}}$
${\displaystyle V^{*}}$ = the value expected from the growth formulas over the next 7 to 10 years
${\displaystyle EPS}$ = the company’s last 12-month earnings per share
${\displaystyle 8.5}$ = P/E base for a no-growth company
${\displaystyle g}$ = reasonably expected 7 to 10 Year Growth Rate of EPS
${\displaystyle 4.4}$ = the average yield of AAA corporate bonds in 1962 (Graham did not specify the duration of the bonds, though it has been asserted that he used 20 year AAA bonds as his benchmark for this variable[5])
${\displaystyle Y}$ = the current yield on AAA corporate bonds.
## Application
In The Intelligent Investor, Graham was careful to include a footnote that this formula was not being recommended for use by investors — rather, it was to model the expected results of other growth formulas popular at the time.[2]
However, a misconception arose that he was using this formula in his daily work due to a later reprinted edition's decision to move footnotes to the back of the book, where fewer readers searched for them.[6] Readers who continued on in the chapter would have found Graham stating:
Warning: This material is supplied for illustrative purposes only, and because of the inescapable necessity of security analysis to project the future growth rate for most companies studied. Let the reader not be mislead into thinking that such projections have any high degree of reliability, or, conversely, that future prices can be counted on to behave accordingly as the prophecies are realized, surpassed, or disappointed.
The movement of the footnote in the reprint has led to an assortment of advisers and investors recommending this formula (or revised versions of it) to the public at large — a practice that continues to this day.[2][7][8] Benjamin Clark, the founder of the blog and investment service ModernGraham, acknowledges the footnote and argues that "I consider the footnote to be more of a reminder from Graham that the calculation of an intrinsic value is not an exact science and cannot be done with 100% certainty."[9] Clark further explains that the formula "is to be used for estimating intrinsic value within a margin of safety which will accommodate the possibility of error in calculation."[9]
Graham also cautioned that his calculations were not perfect, even in the time period for which it was published, noting in the 1973 edition of The Intelligent Investor: "We should have added caution somewhat as follows: The valuations of expected high-growth stocks are necessarily on the low side, if we were to assume these growth rates will actually be realized." He continued on to point out that if a stock were to be assumed to grow forever, its value would be infinite.[2]
## References
1. ^ Dave, John (2014). Benjamin Graham: The Father of Value Investing. CreateSpace Independent Publishing Platform). ISBN 978-1500653743.
2. Graham, Benjamin (2006). The Intelligent Investor: Revised Edition. First Collins Business Essentials. pp. 295, 297, 585. ISBN 0-06-055566-1.
3. ^ Benjamin Graham (1974). "The Renaissance of Value", pg 4
4. ^ Benjamin Graham, "The Decade 1965-1974: Its significance for Financial Analysts," The Renaissance of Value
5. ^
6. ^ "Understanding The Benjamin Graham Formula Correctly | GrahamValue". www.grahamvalue.com. Retrieved 2017-07-18.
7. ^ Joshua Kennon (15 July 2011). "Benjamin Graham Intrinsic Value Formula". Joshua Kennon blog. Kennon & Green Press, LLC. Retrieved 14 April 2012.
8. ^ "How to Value a Stock with the Benjamin Graham Formula". The Value Investing Blog of Old School Value. 2017-01-30. Retrieved 2017-07-20.
9. ^ a b "How to Tell the Difference Between the Graham Formula and the Graham Number". ModernGraham. 2015-09-09. Archived from the original on 2017-08-03. Retrieved 2017-07-18.
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## Introduction
Across generations of climate model intercomparisons, uncertainty in estimates of the projected warming in response to increasing greenhouse gases has persisted1,2. While tremendous progress has been made in modeling the Earth system in individual climate models, our ability to narrow the intermodel spread in equilibrium climate sensitivity (ECS), the equilibrium response in global-mean surface temperature to a doubling of carbon dioxide, is still limited. Critical to making significant progress in reducing such uncertainty is explaining the intermodel spread in the strength of the cloud feedback.
Low cloud changes have long been a root cause of uncertainty in ECS through their radiative effects3,4,5,6,7,8,9,10,11,12,13,14. Low-level clouds efficiently reflect incoming solar radiation back to space while only weakly reducing the longwave emission of terrestrial radiation to space, thereby exerting a strong cooling effect on the planet. A decrease in low cloud fraction (LCF) or cloud optical depth with warming would amplify the positive radiative forcing from increasing greenhouse gases by allowing more solar radiation to reach Earth’s surface15, constituting a positive feedback.
The latest generation of models participating in the coupled model intercomparison project Phase 6 (CMIP6) exhibit a wider range and higher multi-model-mean ECS than CMIP5 models2. The upward shift in ECS can be traced back to stronger positive low cloud feedback in the extratropics2, while the tropical (30°S-30°N) low cloud feedback in both trade cumulus and stratocumulus regimes continues to be a dominant source of intermodel differences in ECS16,17. Moreover, Zelinka et al.18 found that the feedbacks with largest uncertainties in expert assessments and the largest bias across the CMIP6 ensemble are from tropical marine low cloudiness and tropical anvil cloud area. In this study, we consider whether the uncertainty in tropical marine low cloud and anvil cloud area feedbacks might be physically related to one another.
Much work over the last few decades has focused on understanding local factors controlling LCF in large-scale descent regions– such as estimated inversion strength (EIS)19, local sea surface temperature (SST)20, lower tropospheric stability (LTS)21, lower free troposphere relative humidity (RH)22, subsidence strength23, and the net outgoing longwave radiation (OLR) in the inversion layer24,25–particularly in regions of stratocumulus clouds in eastern subtropical ocean basins. Here, however, we consider non-local factors (e.g., deep convection) affecting the local meteorological conditions regulating LCF in the descent region (LCFd), defined as the average LCF between 30°S-30°N where the monthly-mean pressure velocity ($$\omega$$) at 500 hPa is positive. The same definition will be used throughout to define descent regions, and the subscript “d” will be used to denote quantities calculated within these descent regions. We focus on the change of LCFd per degree of global-mean surface temperature change (dLCFd/dTs) because the tropical (30°S-30°N) net cloud radiative effect (netCRE) change per unit warming from the present-day to the future warmer climate and its relationship to ECS (see also ref. 26) is predominantly contributed by the change in dLCFd/dTs (Supplementary Fig. 1). This was also the case in CMIP5 models12.
## Results
### Changes to deep convection, clouds, and the overturning circulation
Examining the spatial patterns in cloud changes reveals that the high cloud fraction (HCF) and LCF are dramatically reduced throughout most of the tropics in high ECS models compared to low ECS models (Fig. 1a-d). The patterns of LCF changes most closely resemble patterns in the change in netCRE (Fig. 1e-f), emphasizing the importance of LCF changes to the strength of the total cloud feedback and climate sensitivity (Supplementary Fig. 1). Cloud fraction is interpolated onto 19 pressure levels from the native vertical levels. LCF is computed as the maximum cloud fraction at any level between 600–1000 hPa, assuming maximum overlap. HCF is computed as the maximum cloud fraction at any level between 100–250 hPa (assuming maximum overlap). The greater decrease in HCF per degree warming in the high ECS models, compared to the low ECS models, suggests that detrainment of condensate from deep convection decreases more in high ECS models than low ECS models. This decreased detrainment could result from a variety of factors, including greater tropospheric stability in response to greenhouse gas forcing27,28, increases in precipitation efficiency29, and/or a reduction in the area occupied by deep convection30.
Indeed, higher ECS models show greater reductions of tropical ascent area (Aa; Fig. 2a), which corresponds to greater decreases in HCFd per degree warming (Fig. 2b). Tropical ascent area is defined as the region of the tropics where ω500 is negative from 30°S-30°N. This metric relates to a narrowing of the intertropical convergence zone (ITCZ) in the zonal mean31,32,33. Higher ECS models also show greater increases in the frequency of heavy precipitation (Fig. 2c). In other words, models that exhibit more dramatic regime shifts towards heavier precipitation in a warmer world have a higher ECS. The frequency of heavy precipitation (Fp>10) is estimated from the frequency of grid boxes in which total precipitation (convective and stratiform) exceeds 10 mm day−1 from monthly average values. A greater Fp>10 probably relates to greater increases in precipitation efficiency and convective organization34; a greater Fp>10 is related to a drier upper troposphere (Fig. 2d), a signature of environments with greater degrees of convective clustering/organization35. We acknowledge, however, that we are limited in our interpretation by the monthly output and resolution of the datasets chosen. In summary, high ECS models have greater increases in heavy precipitation events in the future that occupy less total ascent area in comparison to low ECS models, which reduces the detrainment of water vapor and condensate into the upper troposphere. We thus seek a physical explaination for how such a reduction of high cloudiness and detrainment due to changes in deep convection results in a higher climate sensitivity.
### Linking cloud changes to circulation changes
We wonder whether interactions between deep convection and the low cloud feedback through changes to the atmospheric overturning circulation may explain this relationship between deep convective changes and ECS. Figures 3a, c, and e show that the subsidence weakens less in the subtropics in high ECS models than in low ECS models. Figures 3b, d, and f show that this is especially true over the Eastern Pacific and Eastern Atlantic. Moreover, the intermodel spread in dLCFd/dTs across the tropics (both land and ocean regions considered in the averages) is significantly anti-correlated with the intermodel spread in subsidence strength (Fig. 4). In all CMIP6 models, subsidence strength is projected to decrease with warming, as the increase in dry static stability in response to surface warming dominates over the increase in radiative cooling. Su et al.36 also found that all 77 of their analyzed CMIP5 simulations (atmosphere-only and coupled) produced a weakening of subsidence, even despite a variety of surface warming patterns among the coupled model simulations. We thus wonder whether changes to deep convection relate to the response in subsidence rate with warming.
First, let us consider the role of subsidence on low clouds. Strong subsidence generally disfavors low cloudiness in the presence of a strong inversion23,37, and thus when subsidence weakens among models, LCFd increases. This seemingly counterintuitive relationship can be explained in terms of dynamical and thermodynamic effects. Dynamically speaking, weaker subsidence permits greater boundary layer growth, which allows stratocumulus clouds to grow higher and thicken, increasing LCF23. While this dynamic mechanism would act to cool the planet through a negative shortwave radiative effect, increasing cloud top heights with weaker subsidence would result in infrared emission at cooler temperatures, thus imparting a confounding longwave warming influence. Scott et al.37 found that in observations, the warming effect largely cancels the cooling effect, such that the net radiative effect of subsidence changes is small, though the cloud fraction changes (shortwave effect) generally win out over the cloud altitude changes (longwave effect). Overall, the purely dynamical effect is not the primary meteorological control on perturbations to low cloudiness in the tropics.
Perturbations to the subsidence rate can greatly modify the humidity and temperature structures of the lower troposphere, however, which are known to have profound influences on low cloudiness across the tropics. Figure 5 illustrates how the intermodel spread in d$$\omega$$500/dTs relates to dEIS/dTs (Fig. 5a) and dRH700/dTs (Fig. 5b). EIS is defined as EIS = LTS - Γ850m (Z700-LCL), where LTS is the lower tropospheric stability parameter (LTS = θ700surf), Γ850m is the moist adiabatic potential temperature gradient at 850 hPa, Z700 is the altitude of the 700 hPa level, and LCL is the lifting condensation level19. There is, indeed, a statistically significant correlation between d$$\omega$$500/dTs and dEIS/dTs throughout a large part of the tropics, especially in regions along convective margins, defined as regions where trade cumulus regimes transition to deep convection on a seasonally varying basis. However, these are not regions in which EIS generally exhibits a strong control on low cloudiness. Instead, EIS is most closely associated with low cloudiness in stratocumulus regimes, where a stronger inversion favors low clouds, and thus the intermodel spread in dEIS/dTs is most closely associated with the intermodel spread in dLCF/dTs in stratocumulus regions (Supplementary Fig. 2a). We, therefore, do not suspect that the effect of d$$\omega$$500/dTs on dEIS/dTs (and thus dLCF/dTs) is a primary contributor to the dLCFd/dTs relationship to d$$\omega$$500d/dTs (Fig. 4). On the other hand, there is a uniform, tropics-wide statistically significant negative correlation between d$$\omega$$500/dTs and dRH700/dTs (Fig. 5b), which suggests that d$$\omega$$500/dTs is closely associated with dRH700/dTs everywhere. In descent regions, this negative relationship can be physically interpreted as greater subsidence leading to a drier free troposphere. While the control of RH on low cloudiness differs across different regions of the tropics (Supplementary Fig. 2b), the relationship between circulation and lower free tropospheric humidity is strong, suggesting that dRH700d/dTs is likely the primary contributor to the dLCFd/dTs relationship to d$$\omega$$500d/dTs.
It is worth noting, however, that dRH700d/dTs and dLCFd/dTs in the descent region are not correlated (R = 0.09) since the correlations between dRH700/dTs and dLCF/dTs differ in sign between trade cumulus and stratocumulus regions (Supplementary Fig. 2b). On the one hand, drying disfavors low clouds by enhancing entrainment drying; on the other hand, drying increases radiative cooling at the cloud top, which promotes coupling/mixing of the cloud layer with the surface and favors low clouds24,38,39. The cloud-top radiative cooling mechanism is thought to act more strongly over stratocumulus regions, whereas the entrainment drying mechanism is thought to act more strongly over trade cumulus regions, which is consistent with the direction of the signs shown in Supplementary Fig. 2b.
Figure 5c highlights the regions with the strongest local correlations between dLCF/dTs and d$$\omega$$500/dTs. These regions include convective margins along the South Pacific Convergence Zone, in the Indo-Pacific region, and along continental convective margins. These are also regions in which dRH700/dTs and dLCF/dTs are positively correlated (Supplementary Fig. 2b), suggesting that enhanced entrainment drying along convective margins due to a lesser weakening of subsidence may be a dominant mechanism through which dLCFd/dTs reduces in response to circulation changes.
### Energetic constraints on subsidence rate changes
We will now consider how changes to Aa and Fp>10 can physically relate to changes in subsidence. We start by considering the first-order energetic constraint in the tropical descent regions in Eq. (1)27:
$${\omega }_{{{{{\rm{d}}}}}}\ast {{{{{\rm{S}}}}}}_{{{{{\rm{d}}}}}}={{{{{\rm{F}}}}}}_{{{{{\rm{atm}}}}},d}$$
(1)
where Fatm,d is the atmospheric cooling rate (Fatm,d > 0), ωd is the column-average pressure velocity (subsidence is positive), and Sd is the dry static stability, where each quantity is considered to be a column integral. Assuming the column-average subsidence rate ωd is proportional to the pressure velocity at 500 hPa (ω500d) with a scaling factor of $$\alpha$$ (ωd = $$\alpha \omega$$500d) and ignoring the vertical variation of Sd, we have $$\alpha$$ω500d*Sd = Fatm,d. Differentiating with respect to global-mean surface air temperature (Ts) gives us the change of ω500d per degree of surface warming, where Fatm,d is signed positive for cooling:
$$d{\omega }_{500d}/d{T}_{s}=\left(\frac{1}{\alpha {S}_{d}}\right)d{F}_{atm,d}/d{T}_{s}-({\omega }_{500d}/{S}_{d})\,d{S}_{d}/d{T}_{s}$$
(2)
Equation (2) shows that dω500d/dTs depends on the responses of Fatm,d and Sd to Ts as well as climatological ω500d, Fatm,d and Sd.
In the results that follow, we show that the changing properties of deep convection (dAa/dTs and dFp>10/dTs) link directly to dOLRd/dTs, dSd/dTs, and Sd, and thus the low cloud feedback, through their effects on dω500d/dTs.
We start by investigating how dOLRd/dTs is modified throughout the tropics, as this relates strongly to the intermodel spread in dω500d/dTs (Fig. 6a) and thus dLCFd/dTs. Figure 6 shows that dOLRd/dTs is largely controlled by dHCFd/dTs (Fig. 6b), which relates mostly closely with dAa/dTs (Fig. 2b). Figure 6c shows that dOLRd/dTs is also largely controlled by dRH250d/dTs, which relates most closely with dFp>10/dTs (Fig. 2d). As stated previously, models with greater dFp>10/dTs have a drier upper troposphere (UT), possibly due to increased precipitation efficiency and/or convective organization34. In summary, when Aa reduces and the frequency of heavy precipitation events increases, high clouds reduce, UT RH reduces, OLR increases, subsidence weakens less dramatically, and low clouds reduce. This physical mechanism, which we refer to as the “Radiation-Subsidence Pathway”, constitutes a net positive cloud feedback among CMIP6 models. We suspect that the Radiation-Subsidence mechanism is acting most strongly along seasonally-meandering convective margins in the tropics; the relationship between dω500/dTs and dLCF/dTs is strongest in convective margin regions (Fig. 5c) (as are relationships between dω500/dTs and dOLR/dTs; Supplementary Fig. 3).
In summary, the Radiation-Subsidence Pathway links the intermodel spread in dAa/dTs and dHCFd/dTs, as well as dFp>10/dTs and dRH250d/dTs, to the intermodel spread in dOLRd/dTs. The model spread in these UT quantities contribute to the model spread in the low cloud feedback in descent regions by modifying the subsidence rate and related thermodynamic cloud controlling factors. Interestingly, a stronger “Iris” effect40,41,42 associated with a greater reduction of HCFd and RH250d (a more negative longwave feedback) ultimately leads to a more positive net cloud feedback in the tropics by contributing to a greater reduction of LCF (a more positive shortwave feedback).
### The Stability-Subsidence Pathway
We now examine how stability (Sd and dSd/dTs) relates to the intermodel spread in dω500d/dTs (Fig. 7a-b). The Sd term is calculated as the difference between the mean potential temperature ($$\theta$$) in the 250–400 hPa layer and $$\theta$$ in the 700–925 hPa layer. First, if the static stability of the troposphere increases more, the subsidence rate will weaken more drastically (Fig. 7a). Additionally, Fig. 7b illustrates a connection to a model’s climatological static stability: models with a more stable troposphere will have weaker circulation changes in response to greenhouse gas forcing. Both results are consistent with our expectations from Eq. (2). We collectively refer to these mechanisms relating stability to subsidence, and thus low cloudiness, as the “Stability-Subsidence Pathway”.
This exploration leads us to reflect upon what physics modify the stability of the tropical troposphere. Radiative and non-radiative (latent) heating collectively determine the static stability of the troposphere, and thus we wonder whether radiative or latent heating contribute more to the intermodel spread in Sd and dSd/dTs. We explore two proxies for latent and radiative heating– Fp>10 and OLR, respectively–in each grid cell using monthly mean output.
First, we find that the intermodel spread in dOLR/dTs is related to dSd/dTs (Fig. 7c) while dFp>10/dTs is not (R = −0.06). Note that here we use the tropical mean OLR, not OLRd, as this pathway describes a remote teleconnection between any regions experiencing increases in OLR and its effect on free tropospheric temperatures through wave dynamics (weak temperature gradient theory43). Thus, the Stability-Subsidence pathway as it relates dSd/dTs to d$$\omega$$500d/dTs can simply be viewed as an extension of the Radiation-Subsidence Pathway. We then find that mean-state Fp>10 relates most strongly with mean-state Sd (Fig. 7d). In other words, the intermodel spread in longwave radiative cooling is the suspected primary driver of the intermodel spread in the change in stability with warming, while the intermodel differences in a model’s mean-state static stability are most closely related to deep convective activity and latent heating.
To further investigate the role of deep convective processes in modifying the static stability of the tropical troposphere, the tropical overturning circulation, and the low cloud feedback, we turn to the output from a perturbed physics ensemble from the Community Atmosphere Model version 5.3 (CAM5.3). Many studies have identified how sensitive the climate system is to the entrainment parameter in climate models44,45,46,47,48,49. Notably, higher entrainment rates yield weaker static stability among ensemble members (Fig. 8a; see also50,51). Thus, when entrainment is higher, subsidence is stronger and there are fewer low clouds (Fig. 8b). Perturbing entrainment creates a notable spread in the tropical mean LCF ranging between roughly 29–35% (Fig. 8b).
In summary, the intermodel spread in changes to the static stability of the tropical atmosphere largely affects the intermodel spread in changes to the subsidence strength (Fig. 7a). Moreover, the intermodel spread in dSd/dTs is governed in large part by the intermodel spread in dOLR/dTs (Fig. 7c), which is determined by changes to high cloudiness and UT RH (Fig. 6b-c) that are affected by changes to Fp>10 and Aa (Fig. 2b,d). We also find that a model’s mean state static stability matters considerably for how the overturning circulation will respond to warming (Fig. 7b). Factors controlling the onset of deep convection in models, such as the entrainment rate, will inevitably modify the climatological static stability of a given model’s troposphere (Fig. 8). This in turn will modify the tropical overturning circulation response to warming and its effect on the strength of the low cloud feedback. This is just one example of the far-reaching consequences of inadequate observational constraints on deep convective parameterization.
### Further investigation of the Stability-Subsidence and Radiation-Subsidence Pathways
In what follows, we further investigate two key aspects of the Radiation-Subsidence and Stability-Subsidence Pathways. First, we ask ourselves how stability can be modified by entrainment. Then, we explore whether evidence of low cloud reduction exists in additional GCM experiments where OLR is artificially increased. Lastly, we discuss other potential mechanisms linking changes in OLR and static stability to the low cloud feedback.
To explore the role of entrainment in modifying the stability of the tropical troposphere, we use CAM5.3 PPEs with 3-hourly output alongside inferences from the tropical precipitation-buoyancy relationship52,53,54. This relationship suggests that a single value of lower-tropospheric buoyancy (BL) separates precipitating and non-precipitating regimes and governs the onset of precipitation at subdaily time scales in observations. This also holds for several CMIP6 models55 and is consistent with observations of a regionally independent sub-cloud moist static energy threshold for deep convection at daily timescales in the deep tropics56. The precipitation onset determines the most probable thermodynamic phase space for precipitating points57,58,59, thereby governing the mean state. The BL measure is a function of lower-tropospheric measures of subsaturation and convective instability (see Methods); a consequence is that the BL threshold is attained in stable environments when convection’s moisture sensitivity is small but in unstable environments if the moisture sensitivity is large60. This behavior is confirmed using CAM5.3 sub-daily output (Fig. 8c-d) in which weak entrainment (Fig. 8c) permits convective onset in a more stable troposphere, compared to a case with strong entrainment (Fig. 8d). The precipitating probability density function (PDF) mode—an implicit measure of the mean state—is accordingly situated in more stable environments when the entrainment is weak. The precipitating PDF is also wider when entrainment is weaker—implying that model convection is more easily triggered and more frequent. Relating back to Fig. 7d, one plausible reason for the correlation between more frequent deep convection and a more stable troposphere is intermodel differences in entrainment rate (or similar parameters governing moisture sensitivity within convective parameterizations). Overall, factors controlling the onset of deep convection in models, such as the entrainment rate, will inevitably modify the static stability of the troposphere. This in turn will modify the tropical overturning circulation response to warming and its effect on the strength of the low cloud feedback.
Finally, additional evidence of a LCF reduction resulting from increased OLR can be seen in two additional sets of simulations: (1) simulations with low and high values of the fall speed of stratiform ice in the Community Atmosphere Model version 5.3 corresponding to different amounts of high clouds in the tropical mean (Supplementary Fig. 4), and (2) atmosphere-only simulations in select participating CMIP6 models available from the Cloud Feedback Model Intercomparison Project (CFMIP) where longwave cloud radiative effects are disabled (Supplementary Fig. 5). As stratiform ice fall speeds increase, there are fewer high clouds and low clouds in the tropics in comparison to the simulations with slower stratiform ice fall speeds, which have greater tropical HCF and LCF (Supplementary Fig. 4). Additionally, in comparing the multi-model mean of simulations with longwave cloud radiative effects disabled (amip-lwoff) to the control simulations (amip), the amip-lwoff experiments show that with increased OLR, LCF decreases uniformly across the tropics (Supplementary Fig. 5).
### Other possible mechanisms linking deep convection changes to low cloud changes
The Radiation-Subsidence and Stability-Subsidence pathways introduced in this study pertaining to circulation changes are unlikely to be the only physical pathways explaining the linkage between decreased high cloudiness and decreased low cloudiness. For instance, changes in radiative heating across the tropics could directly impact the stability of the lower troposphere independent of any influence of circulation changes. As we see from Fig. 6, changes in OLR across the tropics are largely driven by changes in HCF and UT water vapor, both of which are strongly modulated by changes to Fp>10 and Aa. By looking at the CFMIP amip-lwoff vs. amip experiments, we find that the LTS decreases throughout the entire tropics, especially in stratocumulus regions, which corresponds to a systematic decline in LCF (Supplementary Fig. 5). One explanation for this response is that the large temperature decrease in the free troposphere in the regions of greatest HCF is communicated through wave dynamics throughout the tropics, according to the weak temperature gradient, decreasing LTS. Overall, dOLR/dTs can affect lower tropospheric stability throughout the tropics, which may also be contributing significantly to the intermodel spread in dLCFd/dTs independent of any circulation changes.
There are other potential mechanisms that may be acting to modify the tropical low cloud feedback that relate to convective processes but that, unlike the mechanisms presented in this study, do not necessarily connect to circulation changes or to changes in upper-tropospheric characteristics. For instance, Hirota et al.61 found that erroneously active deep convection was related to fewer low clouds in model climatology and led to a reduced low cloud feedback. On a similar note, a local change in deep convective activity along convective margins, perhaps due to changing SST pattern, would directly affect the shallow cloud landscape in those margin regions. Moreover, a direct reduction of detrainment from congestus and deep convection into the lower free troposphere due to increases in precipitation efficiency could reduce cloud fraction through entrainment drying, which might contribute rather directly to a reduction of low cloudiness. However, this would likely depend on the relative rate of drying occurring simultaneously in the boundary layer. For instance, Sherwood et al.7 found that enhanced mixing between the boundary layer and lower free troposphere in a warmer climate on both large and small scales leads to a greater reduction of low clouds and a higher ECS among CMIP5 models. Enhanced mixing, in their definition, is a metric relating free tropospheric moistening to boundary layer drying, whereby enhanced mixing would leave the free troposphere more humid and the boundary layer drier. The mixing can be convective in nature (occurring at the sub grid-scale) or at larger, resolved scales along isentropes. Regardless of the scale, the apparent mechanism is that convective mixing dehydrates the boundary layer at a rate that increases as the climate warms. The rate of increase depends on the initial mixing strength, which links the mixing rate in current climate to the tropical low cloud feedback, permitting observational constraints on climate sensitivity.
Finally, it is possible that certain changes to the tropical atmosphere may cause a simultaneous decrease in high cloudiness and low cloudiness without there being a physical linkage among cloud changes. For example, in models with more warming and a greater increase in tropopause height, cloud anvils would be in a more stable environment, which would lead to a decrease in cloud cover due to a decrease in upper-level divergence27,28. Meanwhile, because the troposphere is deeper, subsidence would occur over a longer distance, which may lead to lower RH in the lower free troposphere, enhanced cloud-top entrainment drying, and greater mixing of dry air into the boundary layer62, thus reducing low cloudiness. In this case, the simultaneous reduction of high and low clouds may both be driven by the deepening of the troposphere, but they would not drive each other directly.
## Discussion
In this work, we find that the CMIP6 model spread in the tropical low cloud feedback is intimately tied to the tropical overturning circulation response to warming. Further, we relate the response of the overturning circulation to deep convective processes. First, we find that models with greater tropical Aa reductions and greater increases in the frequency of heavy precipitation under warming tend to have higher ECS. We suggest a causal pathway, whereby reduced HCFd and RH250d leads to increased OLRd, resulting in less subsidence weakening and ultimately favoring greater LCFd reduction (the Radiation-Subsidence Pathway).
Additionally, we find that the change in strength of the overturning circulation in response to warming is linked to the low cloud feedback and ECS through the Stability-Subsidence Pathway. The Stability-Subsidence Pathway links subsidence weakening to climatological Sd and the response of Sd to warming. As an extension from the Radiation-Subsidence Pathway, increased longwave cooling decreases tropospheric stability, which reduces subsidence weakening and low cloudiness. Additionally, we show that the frequency and intensity of tropical deep convection within a model, set largely by its deep convective parameterization, to a large extent determines the mean-state static stability of a model’s troposphere. Factors like a greater rate of entrainment into convective updrafts, which reduces the buoyancy of convective plumes, is associated with a less stable troposphere. This decrease in stability occurs because the environment sits near thermodynamic thresholds determining convection onset. This work suggests that the response of the circulation – and the strength of the low cloud feedback – depends critically on these thresholds.
The Radiation-Subsidence and Stability-Subsidence mechanisms are likely to be strongest in regions along convective margins. Additionally, while we focus on relationships between the changes to low cloudiness and subsidence rate throughout, the dω500d/dTs and dLCFd/dTs relationship is likely strongly influenced by the effect of dω500d/dTs on dRH700d/dTs, whereby enhanced subsidence leads to a drier lower free troposphere.
A summary of the correlations among variables connecting deep convection to the low cloud feedback and the intermodel spread in ECS through the Radiation-Subsidence and Stability-Subsidence Pathways can be seen in Fig. 9. Aside from the high correlations among dnetCRE/dTs, dLCFd/dTs, and ECS, which motivated this study, correlations are generally highest among terms explaining changes to dω500d/dTs and among the link between dω500d/dTs and dLCFd/dTs. Correlations among terms within the umbrella of a given pathway are lower, suggesting that there are multiple factors contributing to upper tropospheric changes that would modify the subsidence rate and subsequently the low cloud feedback.
Overall, robust evidence among the multiple model ensembles examined in this study suggests that the strength of the low cloud feedback is intimately related to changes in deep convection through their effects on the overturning circulation. These relationships are depicted schematically in Fig. 10. Most notably, evidence suggests that a reduction of high cloudiness and enhanced UT drying (negative longwave feedback) leads to a net positive cloud feedback in high ECS models by contributing to a reduction in low cloudiness (positive shortwave feedback).
Many opportunities exist for future work exploring cloud-circulation interactions and the proposed mechanisms. First, the extent to which these mechanisms are acting in the real world is unknown, and thus trying to examine causal relationships between circulation, high clouds, and low clouds in observations will be a promising subject of future work. Linking the proposed pathways to changes in SST patterns and the “pattern effect” – the dependence of outgoing radiation to space on the spatial pattern of surface warming63,64,65,66,67,68,69,70 – would also be illuminating. Finally, improving our understanding of what physically drives differences in tropical ascent area, as well as frequency and intensity changes to heavy precipitation, is critical to our improved understanding of cloud-circulation coupling. New targeted model intercomparison studies, theoretical explorations of the physics controlling tropical ascent area, systematic examination of precipitation efficiency changes, examining the response of convective organization to warming, and idealized simulations of the tropical atmosphere would all drive significant progress towards this goal. Additionally, the extent to which identification of the Radiation-Subsidence and Stability-Subsidence pathways may permit constraints on ECS remains unanswered. Constraining the response of circulation to warming using observational estimates of static stability in the present climate, however, may be one potentially promising avenue.
## Methods
### Models
For our CMIP6 model analysis, we use the historical and Shared Socio-Economic Pathway 5 (SSP5-8.5) runs. SSP5-8.5 has a radiative forcing of 8.5 Wm−2 by the end of the 21st century. Our analysis compares end-of-century mean quantities taken from 2086–2100 to historical mean quantities averaged from 2000–2014. ECS values are taken from the supplementary material of Zelinka et al.2 and from supplemental information from Hausfather et al.71 (https://doi.org/10.1038/d41586-022-01192-2). All data for composite mapping has been regridded to 2 × 2.5 degrees for our analysis.
The CMIP6 models used in our study are as follows: CAMS-CSM1-0, NorESM2-MM, NorESM2-LM, MIROC6, GFDL-ESM4, GISS-E2-1-G, FGOALS-g3, MPI-ESM1-2-HR, FGOALS-f3-L, BCC-CSM2-MR, MPI-ESM1-2-LR, GISS-E2-1-H, MRI-ESM2-0, CMCC-CM2-SR5, CMCC-ESM2, ACCESS-ESM1-5, GFDL-CM4, TaiESM1, ACCESS-CM2, CESM2-WACCM, NESM3, CNRM-ESM2-1, CNRM-CM6-1, CESM2, UKESM1-0-LL, and CanESM5. All models with output available for the analyzed variables were used in this analysis with a few exceptions. For example, published ECS values were not available at the time of analysis for CAS-ESM2-0 and E3SM-1-1-ECA. KACE-1-0-G and E3SM-1-0 were also excluded from the analysis because they proved to be significant outliers in static stability quantities for reasons that could not be explained (dSd/dTs values 2-3 standard deviations away from the next nearest values in the ensemble).
Perturbed physics ensemble simulations are performed with the Community Atmosphere Model version 5.3 in the modified Zhang-McFarlane convection scheme72,73. The stratiform fall speed of ice is perturbed to be 350 s−1 and 1400 s−1 49. Entrainment is perturbed from 0.08 to 1.5 km−1 (default is 1 km−1).
### Definitions
Low cloud fraction is the maximum at any given level between 600–1000 hPa (assuming maximum overlap), and high cloud fraction is taken as the maximum from 100–250 hPa. The pressure velocity at 500 hPa (ω500) is used to calculate tropical mean circulation and associated changes. Tropical ascent area (Aa) is calculated by taking the mean of the fraction of grid boxes with ω500 < 0 hPa day−1 in a given month over the specified time periods noted above. Tropical descent area, which is used to determine the subsidence strength in descent regimes (ω500d), is computed by taking the mean of the fraction of grid boxes with ω500 > 0 hPa day−1 in a given month.
The dominant energy balance of the subtropical atmosphere can be derived from the thermodynamic energy equation
$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+\omega \left(\frac{\partial T}{\partial P}-\frac{{RT}}{P{c}_{p}}\right)={F}_{{net}}$$
(3)
in the absence of strong diabatic heating where
$$S=-\frac{T}{\theta }\frac{\partial \theta }{\partial P}=-\frac{\partial T}{\partial P}+\left(\frac{R}{{C}_{p}}\right)\frac{T}{P}$$
(4)
In the subtropics in the annual mean, $$\frac{\partial T}{\partial t}\,\approx\, 0$$, and $$\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)\,\approx\, 0$$, so that
$$-\omega S={F}_{{net}}$$
(5)
When $${F}_{{net}}$$ is signed positive for column-integrated heating, we get Eq. 1 in the main text, where Fatm,d is signed positive for atmospheric cooling.
For the quantities used in Fig. 8c-d, the boundary layer is defined as the layer between 1000–900 hPa, and the lower free troposphere is defined as the layer between 900–500 hPa. Lower tropospheric instability is first computed as the difference between boundary layer averaged equivalent potential temperature ($${\theta }_{{ebl}}$$) and the lower free tropospheric saturation equivalent potential temperature ($${\theta }_{{eL}}^{*}$$), and then normalized by $${\theta }_{{eL}}^{*}$$. This quantity is then multiplied by 340 K to be expressed in units of K. Similarly, subsaturation is computed as the difference between $${\theta }_{{eL}}^{*}$$ and the lower free tropospheric equivalent temperature ($${\theta }_{{eL}}$$), and then normalized by $${\theta }_{{eL}}^{*}$$. This quantity is then also multiplied by 340 K to be expressed in units of K. Detailed derivations of these quantities are available in55,60.
### Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
|
{}
|
# Plugin definition
To register and run a plugin in plentymarkets, you have to define some basic information. This information is stored in a plugin.json file. The plugin.json is required for all plugins. It’s located in the root directory of the plugin.
This page details the contents of the plugin.json.
PluginDirectory/plugin.json
{
"name": "PluginDirectory",
"namespace": "PluginDirectory",
"type": "template",
"version": "1.0.0",
"require": {
"RequiredPlugin": "~1.0.0"
},
"platform": {
"php": ">=7.3 <8.1"
},
"isClosedSource": false,
"description": "The description for your plugin",
"author": "John Doe",
"email": "doe@account.com",
"phone": "123-456-7890",
"authorIcon": "icon_author_xs.png",
"pluginIcon": "icon_plugin_xs.png",
"serviceProvider": "PluginDirectory\\Providers\\TemplateServiceProvider",
"containers": [
{
"key": "Template.Style",
"name": "Template: Style",
"description": "Add style with a theme plugin."
}
],
"dataProviders": [
{
"key": "PluginDirectory\\Containers\\ItemListContainer",
"name": "Item list",
"description": "Display a configurable list of items in your online store"
}
],
"dependencies": {
"guzzlehttp/guzzle": "6.3.*"
},
"runOnBuild": [
"PluginDirectory\\Migrations\\CreateTable",
"PluginDirectory\\Migrations\\UpdateTable",
"PluginDirectory\\Migrations\\DeleteTable"
]
}
## name
Required
Possible values: any string
The name of the plugin. The name has to meet the following requirements:
• Matches the name of the plugin’s root directory
• Matches the namespace of the plugin
• Written in UpperCamelCase
• Starts with a Latin character
• Doesn’t contain empty spaces
• Doesn’t contain special characters
If you plan on offering your plugin on plentyMarketplace, the name has to be different from all other plugins available.
## namespace
Required
Possible values: any string
The namespace of the plugin. The namespace has to meet the following requirements:
• Matches the name of the plugin’s root directory
• Matches the name of the plugin
• Written in UpperCamelCase
• Starts with a Latin character
• Doesn’t contain empty spaces
• Doesn’t contain special characters
If you plan on offering your plugin on plentyMarketplace, the namespace has to be different from all other plugins available.
## type
Required
Possible values: string
The type of the plugin. The type should reflect the plugin’s purpose. It’s displayed in the back end, but has no other purpose.
The following types are available:
Table 1. Plugin types
Type Explanation
backend
Provides a UI in the plentymarkets back end for displaying information.
export
general
Serves as a fallback you can use in case the plugin doesn’t fit any of the other types.
integration
Extends the functionality of plentymarkets.
payment
shipping
template
Provides views in the shop front end. These views can either supplement or replace the default plentyShop.
theme
Provides styling for template plugins, including the default plentyShop.
widget
Extends the functionality of plentyShop. Widgets may or may not be compatible with ShopBuilder.
## version
Required
Possible values: string
The current version of the plugin. plentymarkets plugins use semantic versioning. This means the version format is MAJOR.MINOR.PATCH.
As you’re making changes to your plugin, you should increase the version to reflect the nature of the changes. Increasing the version is required when updating the plugin on plentyMarketplace. For a guideline on when to increment which version, refer to the table below.
Table 2. Version increments
Version Explanation
MAJOR
The update isn’t backwards compatible.
MINOR
The update is backwards compatible.
The update adds new functionality.
PATCH
The update is backwards compatible.
The update fixes a bug.
For further information and more specific use cases, refer to the complete semantic versioning guide.
## require
Possible values: array or object
Specifies other plugins that have to be present in the plugin set for the plugin to run properly. This value isn’t required, but provides useful information to the user.
plentymarkets plugins use semantic versioning. This means you should specify requirements in a MAJOR.MINOR.PATCH format. It’s possible to use operators for specifying multiple versions efficiently. The following operators are available:
• >
• >=
• <
• ⇐
• !=
• ~
The tilde operator (~) describes a range in-between two versions. It’s essentially a short-hand form for combining the operators >= and <. For example, requiring the version ~1.2.3 is the same as requiring any version between 1.2.3 and 1.3.0. In this example, the required version has a non-zero digit in the "fix" position. This means that the upper boundary is determined by the next highest minor version. If the requirement is ~1.2.0, the upper boundary is determined by the next highest major version.
## platform
Possible values: array or object
Specifies on which PHP version the plugin runs properly. This value isn’t required, but provides useful information to the user.
The following operators are available:
• >
• >=
• <
• ⇐
• !=
• ~
The tilde operator (~) describes a range in-between two versions. It’s essentially a short-hand form for combining the operators >= and <. For example, requiring the version ~7.3 is the same as requiring any version between 7.3 and 8.0. In this example, the required version has a non-zero digit in the "minor" position. This means that the upper boundary is determined by the next highest major version.
## isClosedSource
Default: false
Possible values: true, false
Determines if the plugin source code is visible in the plentymarkets back end when installing the plugin from plentyMarketplace. The source code is always visible when installing the plugin via Git. It’s also possible to download open source marketplace plugins with plentyDevTool and view the source code this way.
## description
Required
Possible values: any string
The description of the plugin. This description is displayed in the plentymarkets back end.
## author
Required
Possible values: any string
The author of the plugin. The author name is displayed in the plentymarkets back end and on plentyMarketplace.
## email
Possible values: any string
The email address of the author. If you provide an email address, it’s displayed as contact information in the plentymarkets back end.
## phone
Possible values: any string
The phone number of the author. If you provide a phone number, it’s displayed as contact information in the plentymarkets back end.
## authorIcon
Required
Possible values: any string
The file name of the author icon. The file has to be stored in the meta/images folder.
## pluginIcon
Required
Possible values: any string
The file name of the plugin icon. The file has to be stored in the meta/images folder.
## serviceProvider
Possible values: any string
Specifies the path to the service provider of the plugin. plentymarkets calls this service provider to register and run the plugin.
## containers
Possible values: array
Specifies an array of container objects the plugin provides. Template plugins can use containers to provide additional space on shop pages. Other plugins can provide data to inject content into the containers. New content either replaces or supplements existing content.
## dataProviders
Possible values: array
Specifies an array of data provider objects the plugin provides. The data provided by the plugin can be linked to a container. New content either replaces or supplements existing content.
## dependencies
Possible values: array
Specifies an array of dependencies to external software development kits (SDKs). An SDK is a package of software components. You can use these packages to access functionality without implementing it in your own plugin. plentymarkets only accepts packages published on Packagist.
## runOnBuild
Possible values: array
Specifies an array of classes for plentymarkets to execute when deploying the plugin. Use these classes to run migrations. You have to use migrations when creating, updating or deleting database tables.
|
{}
|
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