text:
# Rice’s theorem and recursion theorem

Prove Rice’s theorem using recursion theorem.

I need some hints as to what must be done about it. Please use Davis' book notation: Computability, Complexity, and Languages, Second Edition: Fundamentals of Theoretical Computer Science.

-
cross-posted: cs.stackexchange.com/questions/11947/… –  A.Schulz May 11 '13 at 8:14
I thought that was another question. The other one is deleted now, sorry everyone. –  Gigili May 11 '13 at 8:37
This is a relatively standard construction - basically any proof of Rice's theorem can be reworked into a proof that uses the recursion theorem. –  Carl Mummert May 15 '13 at 13:30
Hint: The recursion theorem says that if you have any program $P$ that transforms programs into programs (i.e. P(Q) is a program), there must be a program $Q$ such that $P(Q)$ behaves exactly like $Q$. From a failure of Rice's theorem, you can build a program that violates this.

format: plain


text:
# How to understand the square of an AR(1) process?

I generate an AR(1) process as follows:

x=arima.sim(list(order = c(1,0,0),ar=0.67),n=1000,sd=sqrt(0.55))

When I square it, and fit AR(1) to the squared process, it seems still to be fitted well using AR(1) model:

y=x^2
fit=ar(y,order.max=1, method="ols")
acf(fit$resid[2:1000]) Box.test(fit$resid,lag=round(log(length(y))),type="Ljung-Box",fitdf=1)$p.value  The ACF plot of the model (for the squared process) residuals and the large residual test p value (larger than .05) both show that the residuals are independently distributed. What does this indicate? One simulation of time series$x$and$y$are as follows: - The data are still autocorrelated at lag 1 only. This wouldn't be the case if your data crossed zero though, would it? – Olivia Grigg Oct 18 '12 at 13:41 I think you need to take a piece of paper and a pencil, and derive the square of your AR(1) process and its ACF. Doing things first by simulation will likely be misleading, and trying to generalize from one simulated series is just a wrong approach. – StasK Oct 18 '12 at 14:19 @StasK Thanks for the comment. I did 1000 simulations, out of which over 50% were like what I showed in the question. – Elaine Oct 18 '12 at 22:38 The data definitely crossed zero. One plot of time series$x$and$y$has been added in the question. – Elaine Oct 18 '12 at 22:42 ## 2 Answers Quick and dirty - my thoughts on this. This might give you something to go on though I dunno if it is helpful. All usual assumptions $$x_i=\rho x_{i-1} + \epsilon_t \\ y_i=x^2\\ y_i=(\rho x_{i-1} + \epsilon_t)^2\\ y_i=\rho^2 x_{i-1}^2 + 2\rho x_{i-1} \epsilon_t + \epsilon_t ^ 2\\ y_i=\rho^2 x_{i-1}^2 + 2\rho x_{i-1} \epsilon_t + \delta\\ E[\delta] = E[\epsilon_t^2] = \sigma^2$$ Okay so then you get for some estimator which uncorrectly assuming AR(1) without including the quadratic thing $$y_i^*=\phi^*y_{i-1}\\ \delta^* = y_i-y_i^*\\ E[\delta^*]=E[ y_i-y_i^*]=E[\phi x_{i-1}^2 + 2\rho x_{i-1} \epsilon_t + \delta - \phi^*y_{i-1}]\\ = (\phi-E[\phi^*]) y_{i-1} + \sigma^2$$ So assuming the estimator is somehow unbiased we have$E[\delta^*] = \sigma^2$$$Cov[\delta^*_t, \delta^*_{t-1}] = E[(\delta^*_t-E[\delta^*_t])(\delta^*_{t-1}-E[\delta^*_{t-1}])]\\ = E[(\delta^*_t-\sigma^2)(\delta^*_{t-1}-\sigma^2)] \\ = E[\delta^*_t \delta^*_{t-1} - \sigma^2 \delta^*_{t-1} - \sigma^2 \delta^*_{t} + \sigma^4]\\ = E[\delta^*_t \delta^*_{t-1}] = \sigma^4$$ I can see where you are coming from, but maybe this is just too small a number? 0,55^4 is around 0,09. What is the result for a much larger simulated standard deviation? Maybe this also breaks down because the estimator is no longer unbiased. I will continue this for the biased case later but maybe you can go with this and check my thoughts there's probably some type of error in there. - This is pretty similar to the result of Flounderer posted above later, btw. We both kinda assume the estimator for the second AR to be unbiased, though. This could be the caveat. – IMA Jan 18 '13 at 8:14 I like the question. I think I have convinced myself of why you are seeing this, but it may be wishful thinking. The acf of the residuals shows that the residuals have low autocorrelations at lags 1 and above. It does not mean that they are independent. I think I can see why these autocorrelations are likely to be small. You can write the original process as $$x_i = \alpha x_i + \varepsilon_i$$ and squaring gives $$x_i^2 = \alpha^2 x_i + 2\alpha \varepsilon_i x_i + \varepsilon_i^2$$ Assuming that the ar() function gives the right value of$\alpha^2$, the residuals are $$r_i = 2\alpha\varepsilon_i x_i + \varepsilon_i^2$$ but$x_i = \sum_{k=0}^\infty \alpha^k \varepsilon_{i-k}$because this is an AR(1) process, so $$r_i = 2\alpha \varepsilon_i^2 + 2\alpha^2 \varepsilon_{i-1} \varepsilon_i + 2\alpha^3 \varepsilon_{i-2}\varepsilon_i + \cdots + \varepsilon_i^2$$ Now look at lag 1 (it's similar for higher lags). If we try to calculate the covariance of$r_i$with$r_{i-1}$and use the linearity of covariance, we end up with possibly nonzero terms of the form $$cov(\varepsilon_{i-1}^2, \varepsilon_i \varepsilon_{i-1})$$ and $$cov(\varepsilon_{i-1} \varepsilon_j, \varepsilon_{i} \varepsilon_j)$$ with other terms dropping out due to independence. But both of these are likely to be small (as can be checked by simulation) because if you imagine plotting$\varepsilon_{i-1}^2$versus$\varepsilon_{i-1}$you will get a parabola, and multiplying the points by an independent random amount in the y-direction will give you something which is likely to have a roughly horizontal line of best fit, hence small correlation, hence the covariance should be close to zero. Similarly, plotting$\varepsilon_{j}$versus$\varepsilon_{j}\$ and then multiplying all the points by independent random amounts in the x and y-directions will give something that looks very much like a random scatter of points, so the second covariance should be small. Again, it can be checked via simulation.

Notice that this explanation indicates that you should see the same behaviour regardless of the value of sd, which does indeed seem to be the case (I tried it with sd=1 and sd=5).

I do not think the same explanation works for exponential white noise. I tried doing it with exponential white noise and sometimes the software fits and AR(1), sometimes an AR(4).

-


text:
# New Brain Maps With Unmatched Detail May Change Neuroscience

A technique based on genetic bar codes can easily map the connections of individual brain cells in unprecedented numbers. Unexpected complexity in the visual system is only the first secret it has revealed.

Sitting at the desk in his lower-campus office at Cold Spring Harbor Laboratory, the neuroscientist Tony Zador turned his computer monitor toward me to show off a complicated matrix-style graph. Imagine something that looks like a spreadsheet but instead of numbers it’s filled with colors of varying hues and gradations. Casually, he said: “When I tell people I figured out the connectivity of tens of thousands of neurons and show them this, they just go ‘huh?’ But when I show this to people …” He clicked a button onscreen and a transparent 3-D model of the brain popped up, spinning on its axis, filled with nodes and lines too numerous to count. “They go ‘What the _____!’”

What Zador showed me was a map of 50,000 neurons in the cerebral cortex of a mouse. It indicated where the cell bodies of every neuron sat and where they sent their long axon branches. A neural map of this size and detail has never been made before. Forgoing the traditional method of brain mapping that involves marking neurons with fluorescence, Zador had taken an unusual approach that drew on the long tradition of molecular biology research at Cold Spring Harbor, on Long Island. He used bits of genomic information to imbue a unique RNA sequence or “bar code” into each individual neuron. He then dissected the brain into cubes like a sheet cake and fed the pieces into a DNA sequencer. The result: a 3-D rendering of 50,000 neurons in the mouse cortex (with as many more to be added soon) mapped with single cell resolution.

This work, Zador’s magnum opus, is still being refined for publication. But in a paper recently published by Nature, he and his colleagues showed that the technique, called MAPseq (Multiplexed Analysis of Projections by Sequencing), can be used to find new cell types and projection patterns never before observed. The paper also demonstrated that this new high-throughput mapping method is strongly competitive in accuracy with the fluorescent technique, which is the current gold standard but works best with small numbers of neurons.

The project was born from Zador’s frustration during his “day job” as a neurophysiologist, as he wryly referred to it. He studies auditory decision-making in rodents: how their brain hears sounds, processes the audio information and determines a behavioral output or action. Electrophysiological recordings and the other traditional tools for addressing such questions left the mathematically inclined scientist unsatisfied. The problem, according to Zador, is that we don’t understand enough about the circuitry of the neurons, which is the reason he pursues his “second job” creating tools for imaging the brain.

The current state of the art for brain mapping is embodied by the Allen Brain Atlas, which was compiled from work in many laboratories over several years at a cost upward of $25 million. The Allen Atlas is what’s known as a bulk connectivity atlas because it traces known subpopulations of neurons and their projections as groups. It has been highly useful for researchers, but it cannot distinguish subtle differences within the groups or neuron subpopulations. If we ever want to know how a mouse hears a high-pitched trill, processes that the sound means a refreshing drink reward is available and lays down new memories to recall the treat later, we will need to start with a map or wiring diagram for the brain. In Zador’s view, lack of knowledge about that kind of neural circuitry is partly to blame for why more progress has not been made in the treatment of psychiatric disorders, and why artificial intelligence is still not all that intelligent. Justus Kebschull, a Stanford University neuroscientist, an author of the new Nature paper and a former graduate student in Zador’s lab, remarked that doing neuroscience without knowing about the circuitry is like “trying to understand how a computer works by looking at it from the outside, sticking an electrode in and probing what we can find. … Without ever knowing the hard drive is connected to the processor and the USB pod provides input to the whole system, it’s difficult to understand what’s happening.” Inspiration for MAPseq struck Zador when he learned of another brain mapping technique called Brainbow. Hailing from the lab of Jeff Lichtman at Harvard University, this method was remarkable in that it genetically labeled up to 200 individual neurons simultaneously using different combinations of fluorescent dyes. The results were a tantalizing, multicolored tableau of neon-colored neurons that displayed, in detail, the complex intermingling of axons and neuron cell bodies. The groundbreaking work gave hope that mapping the connectome — the complete plan of neural connections in the brain — was soon to be a reality. Unfortunately, a limitation of the technique in practice is that through a microscope, experimenters could resolve only about five to 10 distinct colors, which was not enough to penetrate the tangle of neurons in the cortex and map many neurons at once. That’s when the lightbulb went on in Zador’s head. He realized that the challenge of the connectome’s huge complexity might be tamed if researchers could harness the increasing speed and dwindling costs of high-throughput genomic sequencing techniques. “It’s what mathematicians call reducing it to a previously solved problem,” he explained. In MAPseq, researchers inject an animal with genetically modified viruses that carry a variety of known RNA sequences, or “bar codes.” For a week or more, the viruses multiply inside the animal, filling each neuron with some distinctive combination of those bar codes. When the researchers then cut the brain into sections, the RNA bar codes can help them track individual neurons from slide to slide. Zador’s insight led to the new Nature paper, in which his lab and a team at University College London led by the neuroscientist Thomas Mrsic-Flogel used MAPseq to trace the projections of almost 600 neurons in the mouse visual system. (Editor’s note: Zador and Mrsic-Flogel both receive funding from the Simons Foundation, which publishes Quanta.) Six hundred neurons is a modest start compared with the tens of millions in the brain of a mouse. But it was ample for the specific purpose the researchers had in mind: They were looking to discern whether there is a structure to the brain’s wiring pattern that might be informative about its function. A currently popular theory is that in the visual cortex, an individual neuron gathers a specific bit of information from the eye — about the edge of an object in the field of view, or a type of movement or spatial orientation, for example. The neuron then sends a signal to a single corresponding area in the brain that specializes in processing that type of information. To test this theory, the team first mapped a handful of neurons in mice in the traditional way by inserting a genetically encoded fluorescent dye into the individual cells. Then, with a microscope, they traced how the cells stretched from the primary visual cortex (the brain area that receives input from the eyes) to their endpoints elsewhere in the brain. They found that the neurons’ axons branched out and sent information to many areas simultaneously, overturning the one-to-one mapping theory. Next, they asked if there were any patterns to these projections. They used MAPseq to trace the projections of 591 neurons as they branched out and innervated multiple targets. What the team observed was that the distribution of axons was structured: Some neurons always sent axons to areas A, B and C but never to D and E, for example. These results suggest the visual system contains a dizzying level of cross-connectivity and that the pattern of those connections is more complicated than a one-to-one mapping. “Higher visual areas don’t just get information that is specifically tailored to them,” Kebschull said. Instead, they share many of the same inputs, “so their computations might be tied to each other.” Nevertheless, the fact that certain cells do project to specific areas also means that within the visual cortex there are specialized cells that have not yet been identified. Kebschull said this map is like a blueprint that will enable later researchers to understand what these cells are doing. “MAPseq allows you to map out the hardware. … Once we know the hardware we can start to look at the software, or how the computations happen,” he said. MAPseq’s competitive edge in speed and cost for such investigations is considerable: According to Zador, the technique should be able to scale up to handle 100,000 neurons within a week or two for only$10,000 — far faster than traditional mapping would be, at a fraction of the cost.

Such advantages will make it more feasible to map and compare the neural pathways of large numbers of brains. Studies of conditions such as schizophrenia and autism that are thought to arise from differences in brain wiring have often frustrated researchers because the available tools don’t capture enough details of the neural interconnections. It’s conceivable that researchers will be able to map mouse models of these conditions and compare them with more typical brains, sparking new rounds of research. “A lot of psychiatric disorders are caused by problems at the circuit level,” said Hongkui Zeng, executive director of the structured science division at the Allen Institute for Brain Science. “Connectivity information will tell you where to look.”

High-throughput mapping also allows scientists to gather lots of neurological data and look for patterns that reflect general principles of how the brain works. “What Tony is doing is looking at the brain in an unbiased way,” said Sreekanth Chalasani, a molecular neurobiologist at the Salk Institute. “Just as the human genome map has provided a scaffolding to test hypotheses and look for patterns in [gene] sequence and function, Tony’s method could do the same” for brain architecture.

The detailed map of the human genome didn’t immediately explain all the mysteries of how biology works, but it did provide a biomolecular parts list and open the way for a flood of transformative research. Similarly, in its present state of development, MAPseq cannot provide any information about the function or location of the cells it is tagging or show which cells are talking to one another. Yet Zador plans to add this functionality soon. He is also collaborating with scientists studying various parts the brain, such as the neural circuits that underlie fear conditioning.

“I think there are insights to be derived from connectivity. But just like genomes themselves aren’t interesting, it’s what they enable that is transformative. And that’s why I’m excited,” Zador said. “I’m hopeful it’s going to provide the scaffolding for the next generation of work in the field.”


text:
# Posts tagged ‘QuTiP’

## Some Small Useful Features of GitHub Actions

Due to recent problems free CI servers have had with abuse from cryptocurrency miners, Travis CI are scaling back what services are available to open source projects. We used to use Travis for QuTiP, but the new pricing model has meant that we won’t be able to afford it once the old free tier travis-ci.org completely shuts down. We trialled GitHub Actions to build the 28 different wheels we now distribute on PyPI, and then moved all our testing there as well.

Here are few nice features that I’ve been using since we started. These are mostly not unique to GitHub Actions, but all my example configuration code will be for it.

## QuTiP data layer and the end of Google Summer of Code 2020

This post is the final permalink for the work I’ve done for QuTiP in the Google Summer of Code 2020. My mentors have been Alex Pitchford, Eric Giguère and Nathan Shammah, and QuTiP is under the numFOCUS umbrella.

The main aim of the project was to make Qobj, the primary data type in QuTiP, able to use both sparse and dense representations of matrices and have them interoperate seamlessly. This was a huge undertaking that had far-reaching implications all across the library, but we have now succeeded. There is still plenty of work to be done in additional development documentation and on sanding out the edges to improve the UX, but we are moving towards a public beta of a major version update next year.

## Overview of the new QuTiP data layer

This post is partial documentation for the implementation of the data-layer that I wrote in the last week or so as part of Google Summer of Code with QuTiP. I may return to talk a bit more about how various algorithms are achieved internally, but for now, this is some indication of what I’ve been working on.

I had previously replaced the old fast_csr_matrix type with the new, custom CSR type as the data backing for Qobj, and all internal QuTiP data representations. This produced some speed-ups in some places due to improved algorithms and better cache usage in places, but its principle advantage was the massive reduction in overhead for function calls between Python and C space, which largely affected small objects.

The full aim, however, is to have QuTiP 5 support many different data representations as the backing of Qobj, and use the most suitable representation for the given data. This will not require every single QuTiP function to have an exponential number of versions for every possible combination of inputs, but only to have specialisations for the most common data combinations. This concept is the “data layer”.

All code examples in this post are prefixed with

1>>> from qutip.core import data

## Design draft: improving tensor-product dimensions in QuTiP

Qobj instantiation and mathematical operations have a large overhead, mostly because of handling the dims parameter in tensor-product spaces. I’m proposing one possible way to speed this up, while also gaining some additional safety and knowledge about mathematical operations on tensor-product spaces.

The steps:

1. rigourously define the “grammar” of dims, and allow all of dimensions.py to assume that this grammar is followed to speed up parsing
2. maintain a private data structure type dimensions._Parsed inside Qobj which is constructed once, and keeps all details of the parsing so they need not be repeated. Determine Qobj.type from this data structure
3. maintain knowledge of the individual type of every subspace in the full Hilbert space (e.g. with a list). There is still a “global” Qobj.type, but this can now be one in the set {'bra', 'ket', 'oper', 'scalar', 'super', 'other'}. 'other' is for when the individual elements do not all match each other. Individual elements cannot be 'other'. 'scalar' is added to operations can keep track of tensor elements which have been contracted, say by a bra-ket product—operations will then broadcast scalar up to the correct dimensions on certain operations.
4. dimension parsing is now sped up by using the operation-specific type knowledge. For example, bra + bra -> bra, and ket.dag() -> bra. Step 3 is necessary to allow matrix multiplication to work. These lookups could be done with enum values instead of string hashing.

Note: this is part of a design discussion for the next major release of QuTiP. I originally wrote this on 2020-07-13, and any further discussion may be found at the corresponding GitHub issue.

## Sphinx and the QuTiP Developers' Guide

Overhauling the internals of a mathematical library is no good if no other developers on the team know how to use the new systems you’ve put in place, and don’t know why you’ve made the choices you’ve made. In the last week I’ve been writing a new QuTiP developers’ guide to the new data layer that I’m creating as part of my Google Summer of Code project, which has involved learning a lot more about the Sphinx documentation tool, and a little bit of GitHub esoterica.

Currently we don’t have a complete plan for how this guide will be merged into the QuTiP documentation, and where exactly it will go, so for now it is hosted on my own GitHub repository. I have also put up a properly rendered version on a GitHub pages site linked to the repository.

My Sphinx conf.py file for this repository is not (at the time of commit 0edf49e) very exciting. Fortunately, Sphinx largely just works out-of-the-box as one would expect from a mature Python project. Perhaps the boldest part of that file is the intersphinx_mapping dictionary, which uses the intersphinx built-in to link to other projects’ documentation also built with Sphinx.

Right now, the intersphinx documentation is perhaps a little lacking, and sometimes seems to just involve some hope (and some disappointment). In particular, I have several external references set up as

1
2
3
4
5
6
7intersphinx_mapping = {
'qutip': ('http://qutip.org/docs/latest/', None),
'python': ('https://docs.python.org/3', None),
'numpy': ('https://numpy.org/doc/stable/', None),
'scipy': ('https://docs.scipy.org/doc/scipy/reference/', None),
}

## Design draft: data structure specifications

These are some of the discussion topics around how new linear algebra data structures will be defined in QuTiP as part of my Google Summer of Code project.

After some discussion, including some further designs of how the new dispatcher methods would function, we are trying to pursue a “light” data structure strategy. This will hopefully have very lightweight instances of the data, and dispatch methods are simplified; mulitple dispatch is a difficult concept to fit into a true object-oriented style, and we believe that dispatcher methods will make adding new data types and dispatched functions significantly easier.

Originally written on the 8th of June, 2020.

## Design draft: data layer separation

This is an early design document about the separation of data layer in QuTiP as part of my Google Summer of Code project. This is a very early-stage document, which is significantly liable to change, but is indicative of the direction we were planning to go.

Originally written on the 1st of June, 2020.

## Welcome to Google Summer of Code 2020

For this summer, I will be working full-time on the open-source project QuTiP with a stipend provided by Google as part of the Summer of Code 2020 project to the umbrella organisation numFOCUS. One requirement of the stipend is that I blog about what I am working on throughout the project, and on any interesting parts of team programming I encounter, so that future applicants to the programme have an idea of what to expect. I will be doing that here, under the “GSoC” tag.

QuTiP is a Python library for dynamics simulations of open quantum systems, something that I have made heavy use of during my PhD studies. At its core is the Qobj class, which represents all quantum objects. Currently, the underlying data storage format is always a compressed-sparse-row matrix, which allows efficient simulation of very large tensor-product spaces, but introduces significant overhead when handling smaller, few-qubit systems. The aim of the project is to abstract out the data layer so that higher-level components can function seamlessly without worrying about the representation format, but the advanced user will be able to access accelerated functions by using the right tool for the job. You can also read my full project proposal in PDF format hosted here.


text:
# Simpe Force Question

[SOLVED] Simpe Force Question

## Homework Statement

A 2.00 kg object placed on a frictionless, horizontal table is connected to a string that passes over a pulley and then is fastened to a hanging 7.00 kg object, as in Figure P5.24. Find the magnitude of the acceleration of the two objects and the tension in the string.

## The Attempt at a Solution

I have a specific question about this problem that I came upon at the end. Here is how I solved it.

$$T = m_1a$$
$$-T + m_2g = m_2a$$
So two equations to uknowns, and here is what I got.
$$-m_1a + m_2g = m_2a$$
$$a = \frac{m_2g}{(m_1+m_2)}$$

So solving for a I got -7.62. The thing I have a question about is the negative on the acceleration. At the beginning of the problem I defined rightward and downward direction as positive and upward direction as negative. So if acceleration is a negative value does that not mean it is traveling up, since I defined up as negative? I know that it doesn't not, but I am confused it can be this way.

Related Introductory Physics Homework Help News on Phys.org
alphysicist
Homework Helper
Hi Sheneron,

How did you get a negative answer? All the values (masses and g) are positive numbers.

Did you use g=-9.8? The value of g is a magnitude and therefore always a positive number; if we were talking about an object in free fall we would say that its acceleration is

$$a_y = -g$$

just to keep g a positive number.

Ah ok I see. thanks


text:
## Hokkaido Mathematical Journal

### On the stokes operator in general unbounded domains

#### Abstract

It is known that the Stokes operator is not well-defined in $L^q$-spaces for certain unbounded smooth domains unless $q=2$. In this paper, we generalize a new approach to the Stokes resolvent problem and to maximal regularity in general unbounded smooth domains from the three-dimensional case, see \cite{FKS1}, to the $n$-dimensional one, $n\geq 2$, replacing the space $L^q, 1\ltq\lt\infty$, by $\s{L}^q$ where $\s{L}^q = L^q\cap L^2$ for $q\geq 2$ and $\s{L}^q = L^q+L^2$ for $1\ltq\lt2$. In particular, we show that the Stokes operator is well-defined in $\s{L}^q$ for every unbounded domain of uniform $C^{1,1}$-type in $\R^n$, $n\geq 2$, satisfies the classical resolvent estimate, generates an analytic semigroup and has maximal regularity.

#### Article information

Source
Hokkaido Math. J. Volume 38, Number 1 (2009), 111-136.

Dates
First available in Project Euclid: 28 July 2009

https://projecteuclid.org/euclid.hokmj/1248787007

Digital Object Identifier
doi:10.14492/hokmj/1248787007

Mathematical Reviews number (MathSciNet)
MR2501897

Zentralblatt MATH identifier
1170.76011

#### Citation

FARWIG, Reinhard; KOZONO, Hideo; SOHR, Hermann. On the stokes operator in general unbounded domains. Hokkaido Math. J. 38 (2009), no. 1, 111--136. doi:10.14492/hokmj/1248787007. https://projecteuclid.org/euclid.hokmj/1248787007.


text:
# How do you turn 4/5 into a decimal and percentage?

How do you turn 4/5 into a decimal and percentage?

          Fraction %= 4/5%Decimal %= 0. 8 %Percentage %= 80%%
Explanation:
                                      To convert a fraction into a decimal, divide the top of the fraction (numerator) by the bottom of the fraction (denominator).  In this case, the fraction is <mathjax>%4/5%</mathjax>, so divide <mathjax>%4%</mathjax> by <mathjax>%5%</mathjax>.  This equals <mathjax>%0. 8%</mathjax>.  Then, to turn the decimal <mathjax>%0. 8%</mathjax> into a percentage, simply move the decimal point two places to the right, turning <mathjax>%0. 8%</mathjax> into <mathjax>%80%%</mathjax>.


text:
zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Remarks on the range properties of certain semilinear problems of Landesman-Lazer type. (English) Zbl 0993.34012
The authors consider the Dirichlet problem $$u''(t)+ u(t)+ g(u'(t))= f(t),\quad u(0)= u(\pi)= 0.$$ It is assumed that $g$ is continuous, has finite limits $g(+\infty)$, $g(-\infty)$, and $g(-\infty)< g(s)< g(+\infty)$ for all $s$. The case of odd and increasing $g$ with some asymptotic conditions is dealt with in more detail. The structure of the set of continuous functions $f$ is studied for which the problem is solvable.

MSC:
 34B15 Nonlinear boundary value problems for ODE
Full Text:
References:
 [1] Amann, H.; Ambrosetti, A.; Mancini, G.: Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities. Math. Z. 158, 179-194 (1978) · Zbl 0368.35032 [2] Cañada, A.; Drábek, P.: On semilinear problems with nonlinearities depending only on derivatives. SIAM J. Math. anal. 27, 543-557 (1996) · Zbl 0852.34018 [3] Cañada, A.; Roca, F.: Existence and multiplicity of solutions of some conservative pendulum-type equations with homogeneous Dirichlet conditions. Differential integral equations 10, 1113-1122 (1997) · Zbl 0938.34013 [4] Fučı\acute{}k, S.: Solvability of nonlinear equations and boundary value problems. (1980) [5] Habets, P.; Sanchez, L.: A two-point problem with nonlinearity depending only on the derivative. SIAM J. Math. anal. 28, 1205-1211 (1997) · Zbl 0886.34015 [6] Kannan, R.; Nagle, R. K.; Pothoven, K. L.: Remarks on the existence of solutions of x”+x+arctan$(x')=p(t); x(0)=$x({$\pi$})=0. Nonlinear anal. 22, 793-796 (1994) · Zbl 0802.34021 [7] Landesman, E. M.; Lazer, A. C.: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. mat. Mech. 19, 609-623 (1970) · Zbl 0193.39203


text:
# Smoothness of isotropy representation

In Introduction to Riemannian Manifolds John Lee states:

Corollary 3.18. If a Lie group G acts smoothly and transitively on a smooth manifold M with compact isotropy groups, then there exists a G-invariant Riemannian metric on M.

From what I understand the proof of this corollary follows like this:

There exist a point $$p\in M$$ such that $$G_p$$ is compact, therefore because the isotropy representation $$I_p:G_p \rightarrow GL(T_pM)$$ is smooth then $$I_p(G_p)$$ is compact and because of theorem 3.17 there exists a G-invariant Riemannian metric on M.

Is this proof correct? And if so why is the isotropy representation smooth or even the isotropy group a submanifold with differential structure?

The proof is correct. By construction (as a stabilizer of a point), the isotropy group $$G_p$$ is closed in $$G$$ and hence a Lie subgroup by the closed subgroup theorem. Hence you can restrict the action of $$G$$ to $$G_p$$, which defines a smooth map $$G_p\times M\to M$$. Smoothness of the tangent map of this easily implies that via a "partial derivative" you get a smooth map $$G_p\times TM\to TM$$. Restricting this to $$G_p\times T_pM$$, the values lie in $$T_pM$$ and what you get is exactly $$(g,X)\mapsto I_p(g)(X)$$, so the latter map is smooth by construction. Applying this to a basis of $$T_pM$$ and expanding the result in this basis, you see that the matrix describing $$I_p(g)$$ depends smoothly on $$G$$ and since these matrices are all invertible, $$I_p$$ is smooth as a map to $$GL(T_pM)$$.


text:
## FANDOM

162 Pages

The dot product is a vector operation, which, along with the cross product and scalar multiplication, comprise the three types of "multiplication" operations that can be performed on vectors. Vector division is not defined.

The fundamental definition of a dot product is the product of the scalar magnitudes (lengths) of each vector and the cosine of the angle in between them. $\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos(\theta)$

This definition can be equivalently written as a scalar projection, or component, of vector u onto vector v, times the magnitude of vector v: $\vec{u} \cdot \vec{v} = |\vec{v}| \mathrm{comp}_{\vec{v}} \vec{u}$

By consequence of the definition, a dot product is the sum of the products of corresponding elements within the vector. This is often regarded as an alternative definition. Given the two (3-dimensional) vectors u and v:

• $\vec{u} = \langle u_1,u_2,u_3 \rangle = u_1 \mathbf{\hat{i}} + u_2 \mathbf{\hat{j}} + u_3 \mathbf{\hat{k}}$
• $\vec{v} = \langle v_1,v_2,v_3 \rangle = v_1 \mathbf{\hat{i}} + v_2 \mathbf{\hat{j}} + v_3 \mathbf{\hat{k}}$

The dot product is:

$\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$

Community content is available under CC-BY-SA unless otherwise noted.


text:
# GATE EE 2015 SET 1

 Question 1
A random variable X has probability density function f(x) as given below:

$f(x)=\left\{\begin{matrix} a+bx & for \; 0 \lt x \lt 1\\ 0& otherwise \end{matrix}\right.$

If the expected value E[X]=2/3, then $Pr[X \lt 0.5]$ is _____________.
 A 0.25 B 0.5 C 0.75 D 1
Engineering Mathematics   Probability and Statistics
Question 1 Explanation:
\begin{aligned} f(x)&=\left\{\begin{matrix} a+bx & \text{for }0 \lt x \lt 1\\ 0 & \text{otherwise} \end{matrix}\right.&\\ \text{Now given }E(X)&=2/3\\ \int_{0}^{1}xf(x)dx&=\frac{2}{3}\\ \int_{0}^{1}x(a+bx)dx&=\frac{2}{3}\\ a\left ( \frac{x^2}{2} \right )_0^1+b\left ( \frac{x^3}{3} \right )_0^1&=\frac{2}{3}\\ a\left ( \frac{1}{2} \right )+b\left ( \frac{1}{3} \right )&=\frac{2}{3}\\ 3a+2b&=4\;\;...(i)\\ \text{Now, }\int_{0}^{1}f(x)dx &=1\\ (\text{total probability} & \text{ is always equal to 1})\\ \int_{0}^{1}(a+bx)dx&=\left ( ax+\frac{bx^2}{2} \right )_0^1=1\\ a+\frac{b}{2}&=1\\ 2a+b&=2\;\;...(ii)\\ \text{Now solving } & \text{ (i) and (ii), we get}\\ a&=0,b=2\\ So,\; \;f(x)&=\left\{\begin{matrix} 2x & \text{for }0 \lt x lt 1\\ 0 & \text{otherwise} \end{matrix}\right.&\\ \text{Now we need}&\\ P(x \lt 0.5)&=\int_{0}^{0.5}2xdx=2\left ( \frac{x^2}{2} \right )_0^{0.5}\\ &=0.5^2-0^2=0.25 \end{aligned}
 Question 2
If a continuous function f(x) does not have a root in the interval [a,b], then which one of the following statements is TRUE?
 A $f(a)\cdot f(b)=0$ B $f(a)\cdot f(b) \lt 0$ C $f(a)\cdot f(b) \gt 0$ D $f(a)/ f(b)\leq 0$
Engineering Mathematics   Calculus
Question 2 Explanation:
Intermediate value theorem states that if a function is continious and $f(a) \cdot f(b) \lt 0$, then surely there is a root in (a,b). The contrapositive of this theorem is that if a function is continious and has no root in (a,b) then surely $f(a) \cdot f(b) \geq 0$. But since it is given that there is no root in the closed interval [a,b] it means $f(a) \cdot f(b) \neq 0$.
So surely $f(a) \cdot f(b) \gt 0$ which is choise(C).
 Question 3
If the sum of the diagonal elements of a 2x2 matrix is -6, then the maximum possible value of determinant of the matrix is ________.
 A 6 B 8 C 9 D 12
Engineering Mathematics   Linear Algebra
Question 3 Explanation:
Consider a symmetric matrix $A =\begin{bmatrix} a & b\\ b & d \end{bmatrix}$
Given $a+d=-6$
$|A|=ad-b^2$
Now since $b^2$ is always non-negative, maximum determinant will come when $b^2=0$.
So we need to maximize
\begin{aligned} |A|&=ad-0=ad=a \times -(6+a) \\ &= -a^2-6a\\ \frac{d|A|}{da} &=-2a-6=0 \\ \Rightarrow \;\; a &=-3 \text{ is the only stationary point} \end{aligned}
Since, $\left [\frac{d^2|A|}{da} \right ]_{a=-3}=-2 \lt 0$, we have a maximum at $a=-3$.
Since $a+d=-6$, corresponding value of $d=-3$.
$|A|=ad=-3 \times -3=9$
 Question 4
Consider a function $\vec{f}=\frac{1}{r^{2}}\hat{r}$, where r is the distance from the origin and $\hat{r}$ is the unit vector in the radial direction. The divergence of this function over a sphere of radius R, which includes the origin, is
 A 0 B 2$\pi$ C 4$\pi$ D R$\pi$
Electromagnetic Fields   Coordinate Systems and Vector Calculus
Question 4 Explanation:
$\bar{f}=\frac{1}{r^2}\hat{r}$
From divergence theorem as we know,
\begin{aligned} \int _{vol.} (\triangledown \cdot \bar{f})dV &=\oint \oint _S \bar{f}\cdot d\bar{S}\\ \oint \oint _S \bar{f}\cdot d\bar{S} &= \oint \oint _S \left ( \frac{1}{r^2} \cdot \hat{r}\right ) \times r^2 \sin \theta \cdot d\theta \cdot d\phi \cdot \hat{r} \\ &= \oint \oint _S \sin \theta \cdot d\theta \cdot d\phi=4\pi \end{aligned}
 Question 5
When the Wheatstone bridge shown in the figure is used to find the value of resistor $R_X$, the galvanometer G indicates zero current when $R_1=50\Omega ,R_2=65\Omega$ and $R_3=100\Omega$. If $R_3$ is known with $\pm 5$% tolerance on its nominal value of 100 $\Omega$ , what is the range of $R_X$ in Ohms?
 A [123.50, 136.50] B [125.89, 134.12] C [117.00, 143.00] D [120.25, 139.75]
Electrical and Electronic Measurements   Characteristics of Instruments and Measurement Systems
Question 5 Explanation:
$R_1=50\Omega$
$R_2=60\Omega$
$R_3=100\pm 5%\Omega$
The value of $R_3 \text{with} \pm 5$% of tollerance,
\begin{aligned} R_3&= 100\pm 5\% \\ &= 100+100 \times \frac{5}{100}=105 \Omega \\ &= 100-100 \times \frac{5}{100}=95 \Omega \end{aligned}
In both condition, the bridge is balance, so under balance condition,
\begin{aligned} \frac{R_1}{R_3}&=\frac{R_2}{R_x} \\ R_x&=\frac{R_2R_3}{R_1} \\ (i) \; \text{when}, \; R_3&=105\Omega \\ \therefore \;\;R_x &= \frac{105 \times 65}{50}=136.50\Omega \\ (ii)\; \text{when}, \; R_3&=95 \Omega \\ \therefore \;\;R_x &= \frac{95 \times 65}{50}=123.50\Omega \end{aligned}
 Question 6
A (0-50 A) moving coil ammeter has a voltage drop of 0.1 V across its terminals at full scale deflection. The external shunt resistance (in milliohms) needed to extend its range to (0-500 A) is _______.
 A 0.11 B 0.22 C 0.45 D 0.68
Electrical and Electronic Measurements   Galvanometers, Voltmeters and Ammeters
Question 6 Explanation:

$I_{fs}=50A,$
$V_m=0.1V,$
$R_m=\frac{0.1}{50}=2 \times 10^{-3}\Omega$
$\because \; m=10$
$R_{sh}=\frac{R_m}{(m-1)}$
$\;\;\;=\frac{2 \times 10^{-3}}{9}=0.22\Omega$
 Question 7
Of the four characteristics given below, which are the major requirements for an instrumentation amplifier?

P. High common mode rejection ratio
Q. High input impedance
R. High linearity
S. High output impedance
 A P, Q and R only B P and R only C P, Q and S only D Q, R and S only
Analog Electronics   Operational Amplifiers
 Question 8
In the following chopper, the duty ratio of switch S is 0.4. If the inductor and capacitor are sufficiently large to ensure continuous inductor current and ripple free capacitor voltage, the charging current (in Ampere) of the 5 V battery, under steady-state, is_____.
 A 1 B 1.5 C 2.5 D 3
Power Electronics   Choppers
Question 8 Explanation:

Step down chopper
\begin{aligned} \Rightarrow \; V_0&=\alpha V_s\\ V_0&=I_0R+E\\ I_0&=\frac{V_0-E}{R}\\ I_0&=\frac{\alpha V_s-E}{R}\\ &=\frac{(0.4)20-5}{3}=1A \end{aligned}
 Question 9
A moving average function is given by $y(t)=\frac{1}{T}\int_{t-T}^{t}u(\tau )d\tau$. If the input u is a sinusoidal signal of frequency $\frac{1}{2\tau }$ Hz, then in steady state, the output y will lag u (in degree) by ______ .
 A 30 B 60 C 90 D 120
Signals and Systems   Linear Time Invariant Systems
Question 9 Explanation:
System input: $\sin \omega _0 t$
\begin{aligned} f_0&=\frac{1}{2T}Hz\\ \text{Therefore,}\\ \omega _0&=2 \pi f_0=\frac{2 \pi}{2T}=\frac{\pi}{T}\; rad/sec\\ y(t)&=\frac{1}{T}\int_{t-T}^{t}\sin \omega _0\tau \; d\tau\\ &=\frac{1}{T}[-\cos\omega _0 t ]_{t-T}^t\\ &=-\frac{1}{T}[\cos\omega _0 t -\cos\omega _0 (t-T) ]\\ &\left [ \because \; \omega _0=\frac{t}{T} \right ]\\ &=-\frac{1}{T}[\cos\omega _0 t -\cos\left ( \frac{\pi t}{T}- \pi \right ) ]\\ \\ &=-\frac{1}{T}[\cos\omega _0 t +\cos\omega _0 t ]\\ &=-\frac{2}{T} \cos\omega _0 t \\ &=\frac{2}{T}\sin (\omega _0 t -90^{\circ}) \\ &=\text{Output is lagging by }90^{\circ} \text{ w.r.t. input.} \end{aligned}
 Question 10
The impulse response g(t) of a system, G , is as shown in Figure (a). What is the maximum value attained by the impulse response of two cascaded blocks of G as shown in Figure (b)?
 A $\frac{2}{3}$ B $\frac{3}{4}$ C $\frac{4}{5}$ D 1
Signals and Systems   Linear Time Invariant Systems
Question 10 Explanation:
\begin{aligned} g(t)&=u(t)-u(t-1)\\ G(s)&=\frac{1}{s}-\frac{e^{-s}}{s}\\ G(s) \times G(s)&=g(t)* g(t) \end{aligned}

Maximum value =1
There are 10 questions to complete.


text:
# Hawaiian Volcano Observatory

## Multimedia

Use the search options below to filter multimedia.

Examples:

• Webcams—Near-real-time images from webcams.
• Videos—Collection of videos recorded during field excursions or caught on our webcams.
• Image Galleries—Galleries of images and events with expanded descriptions.

The Kīlauea Photo and Video Chronology and Mauna Loa Photo and Video Chronology webpages also feature photos.

Filter Total Items: 1,728
November 19, 2009

### Lava within the Halemaumau vent cavity

movie shows lava within the Halemaumau vent cavity, at a depth of about 200 m below the vent rim. The lava surface, which is about 20 m wide, is extremely vigorous, with constant roiling and semi-continuous spattering. Spatter deposited on the walls around the lava surface creates a hot, unstable surface, which can be seen to disintegrate as large blocks break off and

October 3, 2009

### Disappearance of the lava pond within Halemaumau

movie shows the disappearance of the lava pond deep within the Halemaumau vent cavity during the early morning hours of October 3. The lava surface undergoes two filling and draining cycles, and then retreats to deeper levels in the conduit, completely out of view, around 2 am.

October 2, 2009

### Cycles of filling and draining within Halemaumau

movie shows lava pond activity within the Halemaumau vent cavity captured with the low-light camera situated at the Halemaumau Overlook. The entire night of Oct 1-2, 2009, is shown here in 26 seconds, so the speed of the video is increased considerably. You can see many cycles of filling and draining, with each cycle lasting about two hours. There is also alternating

October 1, 2009

### Fluid motion of pāhoehoe flow at Royal Gardens subdivision

This video shows the remarkable fluid motion of a finger of pāhoehoe flowing into an old skylight at the top of Royal Gardens subdivision. The movement of the viscous lava resembles that of soft-serve ice cream.

September 30, 2009

### Collapse of the unstable walls within Halemaumau vent

movie shows two separate rockfalls impacting and disrupting the active lava pond in Halemaumau. Rockfalls here result from collapse of the unstable walls of the vent cavity, and are a common occurrence. The first rockfall impacts the lava surface vertically, while the second features small vertical impacts followed by a slide of material that forces the lava surface to

September 30, 2009

### Vigorous spattering deep within Halemaumau vent

movie shows vigorous spattering at the northeast margin of the lava pond that is deep within the Halemaumau vent cavity. The lava surface, cut by incandescent cracks, is slowly migrating towards the southeast (down in this image). The lava pond has been visible the last several nights in the Halemaumau Overlook webcam (see 'webcams' link above).

September 29, 2009

### Spattering deep within Halemaumau vent

movie shows spattering on the northeast margin of a small lava pond deep within the Halemaumau vent cavity. The video was shot in "nightshot" mode, which provides improved views through the thick fume at night. Each spatter burst represents a gas bubble, or bubbles, breaking at the surface.

September 26, 2009

### Brown plume event within the Halemaumau vent cavity

movie shows a brown plume event resulting from a collapse within the Halemaumau vent cavity. Later views into the vent cavity with a thermal camera revealed that the floor of the vent cavity—at about 200 yards below the vent rim—had fallen away, deepening the vent cavity even further.

September 17, 2009

### Thermal movie showing a view inside the Halemaumau vent cavity

movie shows a view into the Halemaumau vent cavity with a thermal camera. To the naked eye, the vent is entirely obscured by thick fume, but the thermal camera can 'see' through much of this and provide views of the cavity interior. These views show four puffing holes on the floor of the cavity. North is to the right. The 3:13am explosion originated from one of the

September 17, 2009

### Explosion from the vent in Halemaumau

An explosion at 3:13am from the vent in Halemaumau ejected hot particles about a hundred yards above the vent rim, and several tens of yards above the Halemaumau Overlook. This Quicktime movie (shown at x2 speed) shows the event captured by a low-light camera in the HVO observation tower.

September 13, 2009

### Two degassing holes on the floor of Halemaumau

movie shows two degassing holes on the floor of the Halemaumau cavity. Lava is just below the rim of the two holes, creating frequent spattering which falls around their rims. Within the larger of the two (on the right), lava can be seen vigorously sloshing. For scale, these openings are about 10 yards wide. The first half of the movie is shown in normal mode, with the

September 9, 2009

### Halemaumau changing of the typical white to dusty brown plume

movie shows a recent instance of the Halemaumau changing from its typical white to a dusty brown. The brown plume events are normally associated with collapses of the cavity walls.


text:
Thread: CURL library error View Single Post
 2021-03-01, 17:22 #1 LOBES   Mar 2019 USA 23·11 Posts CURL library error This isn't a problem with PrimeNet, but perhaps someone can help me out. I've got a laptop with no hard drive, it runs entirely from a USB stick using TAILS (https://tails.boum.org/). It runs 24x7 so I decided to put prime95 on it and let its cycles go towards the project, even if its not very speedy (its just doing P-1 tasks). Anyway, all is fine when I populate worktodo.txt manually, but when I set UsePrimenet=1, I get the following error: Code: CURL library error: Failed to connect to v5.mersenne.org port 80: Connection refused I'm almost certain this has to do with how TAILS handles connectivity with all its circuits and so forth. Anyway, if anyone has an idea I'll try it out, otherwise I'll just keep getting and submitting assignments manually... no big deal.


text:
Tag Info

A random variable $X$ is called continuous if its set of possible values is uncountable, and the chance that it takes any particular value is zero ($\text{P}(X = x) = 0$ for every real number $x$). A random variable is continuous if and only if its cumulative probability distribution function is a continuous function.

From Mood et al. (page 60, 1974):

"A random variable $X$ is called continuous if there exists a function $f_{X}(.)$ such that $F_{X}(.)=\int_{-\infty}^{x}f_{X}(u)du$ for every real number $x$. The cumulative distribution function $F_{X}(.)$ of a continuous random variable $X$ is called absolutely continuous".

Mood, A. M., Graybill, F. A., & Boes, D. C. (1974). Introduction to theory of statistics. (B. C. Harrinson & M. Eichberg, Eds.) (3rd ed., p. 564). McGraw-Hill, Inc.

Excerpt reference: Glossary of Statistical Terms from berkeley.edu

However, the term is also commonly used for variables that can take on a great many values, such as IQ.


text:
Volume 396 - The 38th International Symposium on Lattice Field Theory (LATTICE2021) - Oral presentation
Investigation of the Perturbative Expansion of Moments of Heavy Quark Correlators for $N_f=0$
L. Chimirri* and R. Sommer
Full text: pdf
Pre-published on: May 16, 2022
Published on:
Abstract
The QCD-coupling is a necessary input in the computation of many observables, and the parametric error on input parameters can be a dominant source of uncertainty. The coupling can be extracted by comparing high order perturbative computations and lattice evaluated moments of mesonic two-point functions with heavy quarks, which provide a high energy scale for perturbation theory. The truncation of the perturbative series is an important systematic uncertainty.\\
We report on our attempt to study this issue by measuring pseudo-scalar two-point functions in volumes of $L=2\, \text{fm}$ with twisted-mass Wilson fermions in the quenched approximation. We use full twist, the non-perturbative clover term and lattice spacings down to $a=0.015\,\text{fm}$ to tame the sizable discretization effects. Our preliminary results indicate that either higher order perturbative corrections
or the continuum limit are not under sufficient control despite our small lattice spacings and quark masses
extending beyond $2\,m_{\text{charm}}$.
DOI: https://doi.org/10.22323/1.396.0354
How to cite

Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete.

Open Access


text:
# Some questions regarding WIMPs as a DM candidate

1. Apr 20, 2006

### frazzle

WIMPs as a DM candidate - Some questions about detection.

Hello everyone, I am new here, so sorry if this has already been covered. I ran a search on 'WIMP' for the forum, and it seems I'm in the clear here..

So, I am writing a report on dark matter at the moment, but there are a few things that have me confused. At the moment, I am trying to figure out some aspects of detection of the WIMP.

1) I have been reading about indirect detection of WIMPs via their annihilation products. It suggests that WIMPs may cluster at the centre of bodies such as galaxies (due to their gravitational pull) and annihilate, as particles such as the Neutralino are their own anti-particle. However, I was previously under the impression that the majority of the dark matter present in galaxies was supposed to be located within the outer regions? i.e. a dark matter 'halo'? I am also confused as to why the WIMPs would be located in the halo at all. Would gravity not pull them into the centre, in much the same way the luminous matter is largely concentrated at the centre of the galaxy?

2) Another detection method involves the annual modulation of the WIMP flux signal in ground based detectors. As I understand it, this is due to the addition and subtraction of the Earth velocity to that of the WIMPs as the Earth orbits the sun, which is in turn orbiting the galaxy centre. My confusion here, is why would the WIMPs all move in one direction, like a 'wind'? Is this due to the fact that they too are in orbit around the galaxy centre?

3) Finally, I have been reading about possible indirect WIMP detection in the form of high energy gamma-rays. They suggest that the WIMPs that have annihilated to cause this signal might be TeV in magnitude. I was under the impression that this sort of mass is well outside the 'allowed' mass boundary of such a particle?

Sorry if I have rambled a bit here. Hope I'm not imposing at all. If you've read far enough to have reached this point, thanks very much

Last edited: Apr 20, 2006
2. Apr 21, 2006

### SpaceTiger

Staff Emeritus
The vast majority of the mass of dark matter is located in the halo, but it's mostly at a very low density. Only in the regions of the highest density could there be enough dark matter annihilation to detect a signal.

Unlike luminous matter, dark matter doesn't interact (at least, not very much), so it has no way of dissipating energy. As such, the dark matter particles will oscillate around the galaxy relatively unhindered, while the luminous matter will interact with its surroundings and radiate energy away.

On average, WIMPs shouldn't be rotating much with respect to the galactic center, so in that frame there would be no wind. However, the sun is in orbit about the GC, so there should be a net flow of WIMPs relative to the sun.

You can see in the figure you posted in the High-Energy physics section -- the mass scale goes up to 10^4 GeV (10 TeV).

3. Apr 21, 2006

### Chaos' lil bro Order

Are WIMPs neutrinos? Or are they other stuff.

4. Apr 21, 2006

### EL

WIMP stands for Weakly Interacting Massive Particles. Although neutrinos are weakly interacting, they are certainly not very massive, and hence does not belong to the class of particles we call WIMPs.
A typical WIMP is the (hypothetical) Neutralino, which exists in supersymmetric extensions of the Standard Model.

5. Apr 21, 2006

### frazzle

Thankyou so much for your reply Spacetiger, that's really cleared some things up for me.

Something that has further confused me however, is that I have read that dark matter is located largely in halos around galaxies, but largely in the centre of galaxy clusters. I don't understand why/how this could be so? i.e. why would the particles cluster differently in these two scenarios?

6. Apr 21, 2006

### SpaceTiger

Staff Emeritus
It's hard to say what the source meant by that, but dark matter halos all have similar shapes, whether they're in galaxy clusters or dwarf galaxies. In pretty much all such scenarios, the halo is the densest at the center and falls off as ~1/r^2 in the visible outskirts. The luminous matter in galaxies is more condensed than in clusters, so if that's what you're comparing to, one might say that the dark matter is more centrally concentrated in a cluster. I think the outer dark matter profiles are pretty poorly known, however.

7. Apr 21, 2006

### frazzle

in case you are curious, here is the quote:

An important difference between the distribution of dark matter in galaxies
and clusters needs to be emphasised: whereas dark matter appears to
increase with distance in galaxies, in clusters exactly the reverse is true, the dark matter distribution actually decreases with distance. Indeed, for certain dwarfs (such as DD0154) the rotation curve has been measured to almost 15 optical length scales indicating that the dark matter surrounding this object is extremely spread out (see also figure 1). A foreground cluster, on the other hand, acts as a gravitational lens which focuses the light from background objects such as galaxies and QSO’s thereby allowing us to determine the depth of the cluster potential well. Observations of strong lensing by clusters indicate that dark matter is strongly concentrated in central regions with a projected mass of 10^13 − 10^14[solar masses] being contained within 0.2 - 0.3 Mpc

from astro-ph/0403324, page 4.

i suppose it may be the case that the text is a series of statements without explainations because nobody really knows why the matter appears to be distributed this way?

8. Apr 22, 2006

### Chronos

Inflation nicely explains why DM is unevenly distributed in the universe. The pre-inflation universe was uniformly saturated with DM. Inflation smeared it out in clumpy, fractal patterns across what is now the entire universe. Baryonic matter was naturally attracted [gravitationally] to these overdense regions. The rest is history.

9. Apr 22, 2006

### Garth

ST, what matter (DM & bayonic) is thought to lie in the Inter Galactic Medium between galactic clusters? Although of very low density there may be significant contribution to $\Omega_m$ because of the vast volume.

Garth

10. Apr 22, 2006

### frazzle

Aha! Is this to say that the large amount of matter at the centre of the galaxy cluster is was there 'first', and the individual galaxies were attracted to it (and then to each other) which is what caused the cluster to form?

11. Apr 22, 2006

### SpaceTiger

Staff Emeritus
Outside of bound objects, dark matter is very difficult to observe. Gravitational lensing surveys (most of which are still in preparation) ought to give us a relatively direct measure of the overall distribution of matter in the universe on large scales. The theoretical prediction from $\Lambda CDM$ is that it will be a gaussian random field, with a scale-invariant power spectrum at the largest scales.

Baryonic matter in the IGM is a bit more complicated because it depends on the detailed history of the universe -- that is, how much matter was expelled or stripped from galaxies and galaxy clusters. This matter is generally observed in absorption along sightlines to distant quasars.

12. Apr 22, 2006

### SpaceTiger

Staff Emeritus
Galaxy clusters often have a great deal of mass at the center in the form of a giant elliptical galaxy, but the formation of the cluster wasn't due to an individual galaxy or small-scale concentration of matter. It was due to the combined gravitational effect of all galaxies within an "overdense" region of the universe.

13. Apr 22, 2006

### Garth

Thank you ST. Are there any upper limits? We have already looked elsewhere at WHIM, but what about cold hydrogen and the material causing the Lyman $\alpha$ forest?

Garth

14. Apr 22, 2006

### SpaceTiger

Staff Emeritus
We can estimate the amount of mass in the Ly$\alpha$ forest and I think it comes well short of what's needed for $\Omega_b$ from nucleosynthesis and the CMB. I don't know the numbers, but a search of arxiv might turn something up.

15. Apr 23, 2006

### Garth

Yes thank you - I have two recent papers Rao's Damped Lyman alpha Surveys and Statistics: A Review and Wolfe et al's DAMPED Ly alpha SYSTEMS.

The latter paper gives the neutral gas $\Omega_g \sim 0.001$ and stellar $\Omega_* \sim 0.002$. However, the first paper makes the point:
but later makes the point
So there may be more out there than is thought.

Garth

Last edited: Apr 23, 2006
16. Apr 23, 2006

### SpaceTiger

Staff Emeritus
Damped systems are only a small fraction of the full Ly$\alpha$ forest. Specifically, it's only those absorbers with hydrogen column densities greater than about 1020 cm-2. Even though they may carry most of the neutral gas, the Ly$\alpha$ lines in the forest will often be from warmer, more heavily ionized gas with large hydrogen column densities.

Last edited: Apr 23, 2006
17. Apr 23, 2006

### Garth

Is it thought that this latter ionized gas is WHIM at 105 - 107oK or just warm gas?

Garth

18. Apr 23, 2006

### SpaceTiger

Staff Emeritus
The WHIM is very highly ionized, so it produces virtually no signature in the Ly$\alpha$ forest. This is a part of the reason that we have such difficulty constraining the mass of the WHIM. The Ly$\alpha$ forest is largely diffuse, ionized gas, presumably at temperatures nearer 104 K.

19. Apr 23, 2006

### SpaceTiger

Staff Emeritus
Last edited by a moderator: Apr 22, 2017
20. Apr 24, 2006

### Garth

Just to include a paper today on WHIM Warm-hot intergalactic medium contribution to baryonic matter which states
, which is even lower (by a factor of ~ 3) than the lower limit of Nicastro, Elvis, Fiore & Mathur's paper: Measured Cosmological Mass Density in the WHIM: the Solution to the 'Missing Baryons' Problem, which as I have posted before gives an upper limit of:
$\Omega_b$WHIM > 4.3 × 100.47 % = 12.6%
and a lower limit of:
$\Omega_b$WHIM > 1.3 × 100.32 % = 2.7%.

Garth

Last edited by a moderator: Apr 22, 2017


text:
Copied to
clipboard

## G = C2×C4×D20order 320 = 26·5

### Direct product of C2×C4 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C4×D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×D20 — C2×C4×D20
 Lower central C5 — C10 — C2×C4×D20
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C2×C4×D20
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1518 in 426 conjugacy classes, 183 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×6], C22, C22 [×6], C22 [×32], C5, C2×C4 [×14], C2×C4 [×26], D4 [×16], C23, C23 [×20], D5 [×8], C10 [×3], C10 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×18], C2×D4 [×12], C24 [×2], Dic5 [×4], C20 [×8], C20 [×2], D10 [×8], D10 [×24], C2×C10, C2×C10 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C4×D5 [×16], D20 [×16], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×14], C2×C20 [×2], C22×D5 [×12], C22×D5 [×8], C22×C10, C2×C4×D4, C4⋊Dic5 [×4], D10⋊C4 [×8], C4×C20 [×4], C2×C4×D5 [×8], C2×C4×D5 [×8], C2×D20 [×12], C22×Dic5 [×2], C22×C20 [×3], C23×D5 [×2], C4×D20 [×8], C2×C4⋊Dic5, C2×D10⋊C4 [×2], C2×C4×C20, D5×C22×C4 [×2], C22×D20, C2×C4×D20
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D5, C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C4×D5 [×4], D20 [×4], C22×D5 [×7], C2×C4×D4, C2×C4×D5 [×6], C2×D20 [×6], C4○D20 [×2], C23×D5, C4×D20 [×4], D5×C22×C4, C22×D20, C2×C4○D20, C2×C4×D20

Smallest permutation representation of C2×C4×D20
On 160 points
Generators in S160
(1 79)(2 80)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 100)(22 81)(23 82)(24 83)(25 84)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 99)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 141)(134 142)(135 143)(136 144)(137 145)(138 146)(139 147)(140 148)
(1 36 101 150)(2 37 102 151)(3 38 103 152)(4 39 104 153)(5 40 105 154)(6 21 106 155)(7 22 107 156)(8 23 108 157)(9 24 109 158)(10 25 110 159)(11 26 111 160)(12 27 112 141)(13 28 113 142)(14 29 114 143)(15 30 115 144)(16 31 116 145)(17 32 117 146)(18 33 118 147)(19 34 119 148)(20 35 120 149)(41 122 79 95)(42 123 80 96)(43 124 61 97)(44 125 62 98)(45 126 63 99)(46 127 64 100)(47 128 65 81)(48 129 66 82)(49 130 67 83)(50 131 68 84)(51 132 69 85)(52 133 70 86)(53 134 71 87)(54 135 72 88)(55 136 73 89)(56 137 74 90)(57 138 75 91)(58 139 76 92)(59 140 77 93)(60 121 78 94)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 80)(10 79)(11 78)(12 77)(13 76)(14 75)(15 74)(16 73)(17 72)(18 71)(19 70)(20 69)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 100)(41 110)(42 109)(43 108)(44 107)(45 106)(46 105)(47 104)(48 103)(49 102)(50 101)(51 120)(52 119)(53 118)(54 117)(55 116)(56 115)(57 114)(58 113)(59 112)(60 111)(121 160)(122 159)(123 158)(124 157)(125 156)(126 155)(127 154)(128 153)(129 152)(130 151)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 143)(139 142)(140 141)

G:=sub<Sym(160)| (1,79)(2,80)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,100)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148), (1,36,101,150)(2,37,102,151)(3,38,103,152)(4,39,104,153)(5,40,105,154)(6,21,106,155)(7,22,107,156)(8,23,108,157)(9,24,109,158)(10,25,110,159)(11,26,111,160)(12,27,112,141)(13,28,113,142)(14,29,114,143)(15,30,115,144)(16,31,116,145)(17,32,117,146)(18,33,118,147)(19,34,119,148)(20,35,120,149)(41,122,79,95)(42,123,80,96)(43,124,61,97)(44,125,62,98)(45,126,63,99)(46,127,64,100)(47,128,65,81)(48,129,66,82)(49,130,67,83)(50,131,68,84)(51,132,69,85)(52,133,70,86)(53,134,71,87)(54,135,72,88)(55,136,73,89)(56,137,74,90)(57,138,75,91)(58,139,76,92)(59,140,77,93)(60,121,78,94), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,80)(10,79)(11,78)(12,77)(13,76)(14,75)(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,100)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,153)(129,152)(130,151)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141)>;

G:=Group( (1,79)(2,80)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,100)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148), (1,36,101,150)(2,37,102,151)(3,38,103,152)(4,39,104,153)(5,40,105,154)(6,21,106,155)(7,22,107,156)(8,23,108,157)(9,24,109,158)(10,25,110,159)(11,26,111,160)(12,27,112,141)(13,28,113,142)(14,29,114,143)(15,30,115,144)(16,31,116,145)(17,32,117,146)(18,33,118,147)(19,34,119,148)(20,35,120,149)(41,122,79,95)(42,123,80,96)(43,124,61,97)(44,125,62,98)(45,126,63,99)(46,127,64,100)(47,128,65,81)(48,129,66,82)(49,130,67,83)(50,131,68,84)(51,132,69,85)(52,133,70,86)(53,134,71,87)(54,135,72,88)(55,136,73,89)(56,137,74,90)(57,138,75,91)(58,139,76,92)(59,140,77,93)(60,121,78,94), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,80)(10,79)(11,78)(12,77)(13,76)(14,75)(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,100)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,153)(129,152)(130,151)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141) );

G=PermutationGroup([(1,79),(2,80),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,100),(22,81),(23,82),(24,83),(25,84),(26,85),(27,86),(28,87),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,94),(36,95),(37,96),(38,97),(39,98),(40,99),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,141),(134,142),(135,143),(136,144),(137,145),(138,146),(139,147),(140,148)], [(1,36,101,150),(2,37,102,151),(3,38,103,152),(4,39,104,153),(5,40,105,154),(6,21,106,155),(7,22,107,156),(8,23,108,157),(9,24,109,158),(10,25,110,159),(11,26,111,160),(12,27,112,141),(13,28,113,142),(14,29,114,143),(15,30,115,144),(16,31,116,145),(17,32,117,146),(18,33,118,147),(19,34,119,148),(20,35,120,149),(41,122,79,95),(42,123,80,96),(43,124,61,97),(44,125,62,98),(45,126,63,99),(46,127,64,100),(47,128,65,81),(48,129,66,82),(49,130,67,83),(50,131,68,84),(51,132,69,85),(52,133,70,86),(53,134,71,87),(54,135,72,88),(55,136,73,89),(56,137,74,90),(57,138,75,91),(58,139,76,92),(59,140,77,93),(60,121,78,94)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,63),(7,62),(8,61),(9,80),(10,79),(11,78),(12,77),(13,76),(14,75),(15,74),(16,73),(17,72),(18,71),(19,70),(20,69),(21,99),(22,98),(23,97),(24,96),(25,95),(26,94),(27,93),(28,92),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,84),(37,83),(38,82),(39,81),(40,100),(41,110),(42,109),(43,108),(44,107),(45,106),(46,105),(47,104),(48,103),(49,102),(50,101),(51,120),(52,119),(53,118),(54,117),(55,116),(56,115),(57,114),(58,113),(59,112),(60,111),(121,160),(122,159),(123,158),(124,157),(125,156),(126,155),(127,154),(128,153),(129,152),(130,151),(131,150),(132,149),(133,148),(134,147),(135,146),(136,145),(137,144),(138,143),(139,142),(140,141)])

104 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4H 4I ··· 4P 4Q ··· 4X 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 ··· 10 1 ··· 1 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 D4 D5 C4○D4 D10 D10 C4×D5 D20 C4○D20 kernel C2×C4×D20 C4×D20 C2×C4⋊Dic5 C2×D10⋊C4 C2×C4×C20 D5×C22×C4 C22×D20 C2×D20 C2×C20 C2×C42 C2×C10 C42 C22×C4 C2×C4 C2×C4 C22 # reps 1 8 1 2 1 2 1 16 4 2 4 8 6 16 16 16

Matrix representation of C2×C4×D20 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 1 0 0 0 0 5 35 0 0 0 0 0 0 1 40 0 0 0 0 36 6 0 0 0 0 0 0 9 11 0 0 0 0 30 14
,
 1 0 0 0 0 0 36 40 0 0 0 0 0 0 40 0 0 0 0 0 5 1 0 0 0 0 0 0 9 11 0 0 0 0 30 32

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,36,0,0,0,0,40,6,0,0,0,0,0,0,9,30,0,0,0,0,11,14],[1,36,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,0,1,0,0,0,0,0,0,9,30,0,0,0,0,11,32] >;

C2×C4×D20 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_{20}
% in TeX

G:=Group("C2xC4xD20");
// GroupNames label

G:=SmallGroup(320,1145);
// by ID

G=gap.SmallGroup(320,1145);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,80,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽


text:
# Article

Full entry | PDF   (0.2 MB)
Keywords:
Higher order equations; nonlinear limit-point; nonlinear limit-circle
Summary:
We describe the nonlinear limit-point/limit-circle problem for the $n$-th order differential equation $y^{(n)} + r(t)f(y,y^{\prime }, \dots , y^{(n-1)}) = 0.$ The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated.
References:
[1] M. Bartušek, Z. Došlá: On the limit-point/limit-circle problem for nonlinear third order differential equations. Math. Nachr. 187 (1997), 5–18. MR 1471135
[2] M. Bartušek Z. Došlá, and J. R. Graef: On $L^2$ and limit-point type solutions of fourth order differential equations. Appl. Anal. 60 (1996), 175–187. MR 1623388
[3] M. Bartušek Z. Došlá, and J. R. Graef: Limit-point type results for nonlinear fourth order differential equations. Nonlinear Anal. 28 (1997), 779–792. MR 1422183
[4] M. Bartušek Z. Došlá, and J. R. Graef: Nonlinear limit-point type solutions of $n$th order differential equations. J. Math. Anal. Appl. 209 (1997), 122–139. MR 1444516
[5] N. Dunford, J. T. Schwartz: Linear Operators; Part II: Spectral Theory. Wiley, New York, (1963). MR 1009163 | Zbl 0128.34803
[6] M. S. P. Eastham: The limit-$2n$ case of symmetric differential operators of order $2n$. Proc. London Math. Soc. (3) 38 (1979), 272–294. MR 0531163 | Zbl 0398.34021
[7] M. S. P. Eastham, C. G. M. Grudniewicz: Asymptotic theory and deficiency indices for fourth and higher order self-adjoint equations: a simplified approach. in: Ordinary and Partial Differential Equations (W. N. Everitt and B. D. Sleeman, eds.), Lecture Notes in Math. Vol 846 Springer Verlag, New York, 1981, pp. 88–99. MR 0610637 | Zbl 0514.34047
[8] W. N. Everitt: On the limit-point classification of fourth-order differential equations. J. London Math. Soc. 44 (1969), 273–281. MR 0235187 | Zbl 0162.39201
[9] M. V. Fedorjuk: Asymptotic method in the theory of one-dimensional singular differential operators. Trudy Mosk. Matem. Obsch. 15 (1966), 296–345. MR 0208060
[10] J. R. Graef: Limit circle criteria and related properties for nonlinear equations. J. Differential Equations 35 (1980), 319–338. MR 0563385 | Zbl 0441.34024
[11] J. R. Graef: Limit circle type results for sublinear equations. Pacific J. Math. 104 (1983), 85–94. MR 0683730 | Zbl 0535.34024
[12] J. R. Graef: Some asymptotic properties of solutions of $(a(t)x^{\prime })^{\prime } - q(t)f(x) = r(t)$. in: Differential Equations: Qualitative Theory (Szeged, 1984), Colloquia Mathematica Societatis János Bolyai, Vol. 47, North-Holland, Amsterdam, 1987, pp. 347–359. MR 0890550
[13] J. R. Graef, P. W. Spikes: On the nonlinear limit-point/limit-circle problem. Nonlinear Anal. 7 (1983), 851–871. MR 0709039 | Zbl 0535.34023
[14] R. M. Kauffman T. T. Read, and A. Zettl: The Deficiency Index Problem for Powers of Ordinary Differential Expressions. Lecture Notes in Math. Vol. 621, Springer-Verlag, New York, 1977. MR 0481243
[15] R. M. Kauffman: On the limit-$n$ classification of ordinary differential operators with positive coefficients. Proc. London Math. Soc. 35 (1977), 496–526. MR 0460780 | Zbl 0382.47025
[16] M. A. Naimark: Linear Differential Operators, Part II. George Harrap & Co., London, 1968. Zbl 0227.34020
[17] R. B. Paris, A. D. Wood: On the $L^2$ nature of solutions of $n$-th order symmetric differential equations and McLeod’s conjecture. Proc. Roy. Soc. Edinburgh 90A (1981), 209–236. MR 0647603 | Zbl 0483.34014
[18] B. Schultze: On singular differential operators with positive coefficients. Proc. Roy. Soc. Edinburgh 10A (1992), 361–365. MR 1159190 | Zbl 0767.34058
[19] P. W. Spikes: On the integrability of solutions of perturbed nonlinear differential equations. Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 309–318. MR 0514125
[20] P. W. Spikes: Criteria of limit circle type for nonlinear differential equations. SIAM J. Math. Anal. 10 (1979), 456–462. MR 0529063 | Zbl 0413.34033
[21] H. Weyl: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörige Entwicklungen willkürlicher Funktionen. Math. Ann. 68 (1910), 220–269. MR 1511560

Partner of


text:
# Attractive Basins and Loops in Julia Sets

I recently learned about the Mandelbrot set for the first time from a presentation by some undergraduates in honor of Mandelbrot's death. The presentation was short and by non-experts so I left with a few questions.

When I heard about the fundamental dichotomy, it seemed odd to me that the attractive basin at infinity was a distinguished point, in that filled Julia sets are the complement of the attractive basin of infinite. But when I naively expected the set of bounded points to be the attractive basin of the origin I was caught by the lack of duality--the filled Julia set is either simply connected or a Cantor set, and the complement of a simply connected set is simply connected, while I have only the slightest idea what the complement of a Cantor set looks like; it certainly isn't simply connected.

Talking the subject over with a few graduate students I realized that there should be at least countably many bounded attractive basins or loops for any curve, and it seems unlikely that in the case that the basin of infinity is the complement of a Cantor set that the other attractive basins (including the loops of the roots of $f ^{n}(x) - x\$, where $f(x) = z^2+c$ ) are also complements of Cantor sets.

When $c=0$ the picture is very nice and symmetric, with two simply connected attractive basins, countably many finite loops at the $\pi$-rational angles around the unit circle, and uncountably many infinite loops. I'm curious what the picture looks like for other $c$.

So my questions are:

What distinguishes the point at infinity from the other points here?

What do the loops look like?

Are there other attractive basins? For which $c$? What do they look like?

Or what book(s) or paper(s) could I look through to get satisfying answers to these questions?

-
perhaps this question would benefit from a "complex dynamics" tag, or something similar –  Yemon Choi Nov 1 '10 at 4:14
Here is a naïve observation. The asymmetry is related to the fact that the family of functions of the form $z \mapsto z^2 + c$ is not invariant under any Möbius transformations. In particular, if we conjugate by a map that switches zero and infinity, e.g., $w = 1/z$, we end up with $w \mapsto w^2/(1+cw^2)$, which has qualitatively different behavior (as you would expect). –  S. Carnahan Nov 1 '10 at 4:54
Your questions appear quite vague. The math.stackexchange site might be more appropriate. But perhaps more appropriate would be to pick up a book like Devaney's "An Introduction To Chaotic Dynamical Systems." What is your background? Devaney has some undergraduate oriented textbooks as well. –  Ryan Budney Nov 1 '10 at 5:17
You might want to look at golem.ph.utexas.edu/category/2010/10/benot_mandelbrot.html for a discussion of the Mandelbrot and Julia sets. Also, $\infty$ is a superattracting fixed point for a polynomial (this is related to Scott's observation that polynomials aren't invariant under Mobius transformations), which is not the case for the other points. –  David Roberts Nov 1 '10 at 5:31
Loop -> orbit, I think... –  Sam Nead Nov 1 '10 at 11:22

## 1 Answer

It seems you are basically interested in an introduction to complex dynamical systems. The books by Beardon, Milnor and Steinmetz all give good introductions.

Regarding your specific questions:

a) The point at infinity is a superattracting fixed point, but more importantly its immediate basin of attraction - that is, the component of the basin containing the fixed point itself - is completely invariant (invariant under forward and backwards iteration). This is the case for all polynomials (of degree at least two), and is one of the reasons that studying polynomials is easier than studying general rational maps (where e.g. the Julia set - where the dynamics is chaotic - may in fact be the whole Riemann sphere). The basin of infinity supports foliations into "external rays" and "equipotentials", and this allows one to study the Julia set. This idea was introduced by Douady and Hubbard, and is the basis of the famous "Yoccoz puzzle".

b) There are all kinds of possible orbits in the Julia set. This isn't the place to discuss them. Stable orbits of polynomials may converge to a (super)-attracting fixed point, to a parabolic fixed point (where the multiplier is a root of unity), or belong to a rotation domain (a simply connected domain on which the dynamics is conjugate to a rotation).

c) See above. There may be at most one attracting cycle for a quadratic polynomial (apart from infinity), because there is at most one critical point. (This is a theorem of Fatou.) Conjecturally, systems with an attracting cycle are dense in the Mandelbrot set; this is perhaps the most celebrated conjecture in one-dimensional dynamics at the moment.

-


text:
## Paraphrasing

### Non-integral and integral orientation

There are two ways of incorporating paraphrases and quotes into your work: non-integral orientation and integral orientation.

Non-integral orientation, also known as information-prominent, is where you paraphrase or quote information from another source and only mention the author in your reference. This gives greater emphasis to the information you're providing rather than its origin:

Integral orientation, also known as author-prominent, is where you include the author's name as part of your sentence. This draws attention to the author's important role in developing the information you're using.

Though most academic papers contain a mixture of integral and non-integral orientation, in most disciplines there are usually more non-integral references than integral ones.


text:
# Preface

Hello！ buddy ！
Thank you very much for reading Haihong's article , If there are mistakes in the text , You are welcome to point out ～

Self introduction. ଘ(੭ˊᵕˋ)੭
nickname ： Sea boom
label ： Program the ape ｜C++ player ｜ Student
brief introduction ： because C Language and programming , Then I turned to computer science , I was lucky to win some national awards 、 Provincial award … It has been insured . Currently learning C++/Linux/Python
Learning experience ： Solid foundation + Take more notes + Knock more code + Think more + Learn English well ！

Machine learning white stage
The article is only for your own study notes For knowledge system establishment and review
Know what it is Know why ！

# The articles

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （1）： Second and third order determinants 、 Total permutation and its reverse order number

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （2）：n Step determinant 、 exchange

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （3）： The nature of determinants

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （4）： Determinant by line （ Column ） an

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （5）： Kramer's law

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （6）： Matrix operation

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （7）： Inverse matrix

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （8）： Elementary transformation of matrix

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （9）： The rank of a matrix 、 Solutions of linear equations

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （10）： Vector group and its linear combination

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （11）： Linear correlation of vector groups

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （12）： The rank of a vector group

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （13）： Structure of solutions of linear equations

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （14）： Vector space

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （15）： The inner product of a vector 、 Length and orthogonality

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （16）： Eigenvalues and eigenvectors of square matrices

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （17）： Similarity matrix

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （18）： Diagonalization of symmetric matrices

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （19）： Quadratic form and its canonical form

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （20）： The quadratic form is transformed into standard form by formula method

【 machine learning ｜ Mathematical basis 】Mathematics for Machine Learning Linear algebra of series （21）： Positive definite quadratic form

# 6.1 Definition and properties of linear space

## Definition 1： Linear space

set up V V It's a Nonempty set , R \mathbb{R} by Real number field

If for any two elements α , β ∈ V \alpha,\beta \in V , There is always one element γ ∈ V \gamma \in V With the corresponding , be called α \alpha And β \beta And , Write it down as γ = α + β \gamma=\alpha+ \beta

For any number λ ∈ R , α ∈ V \lambda\in \mathbb{R},\alpha \in V , There is always one element δ ∈ V \delta\in V With the corresponding , be called λ \lambda And α \alpha Product of , Write it down as δ = λ α \delta=\lambda \alpha

And these two kinds of transportation meet Eight operation rules

1. α + β = β + α \alpha+\beta=\beta+\alpha
2. ( α + β ) + γ = α + ( β + γ ) (\alpha+\beta)+\gamma=\alpha+(\beta+\gamma)
3. stay V V There is a zero element in 0 \boldsymbol0 , To any α ∈ V \alpha\in V , There are α + 0 = α \alpha+\boldsymbol0=\alpha
4. To any α ∈ V \alpha \in V , There are α \alpha The negative element of β ∈ V \beta \in V , send α + β = 0 \alpha+\beta=\boldsymbol0
5. 1 α = α 1\alpha=\alpha
6. λ ( μ α ) = ( λ μ ) α \lambda(\mu\alpha)=(\lambda \mu)\alpha
7. ( λ + μ ) α = λ α + μ α (\lambda+\mu)\alpha=\lambda \alpha+\mu\alpha
8. λ ( α + β ) = λ α + λ β \lambda(\alpha+\beta)=\lambda \alpha+\lambda\beta

notes ： α , β , γ ∈ V ; λ , u ∈ R \alpha,\beta,\gamma \in V;\lambda,u\in \mathbb{R}

that V V It's called the real number field R \mathbb{R} Upper Vector space （ Or linear space ）

in short

• All addition and multiplication that satisfy the above eight laws operation , It's called Linear operation
• Any set that defines a linear operation , It's called vector space

## Properties of linear spaces

### nature 1

The zero element is unique

prove （ Reduction to absurdity ）

Suppose there are two zero elements 0 1 , 0 2 ∈ V 0_1,0_2 \in V

According to the definition of zero element , Yes

{ 0 1 + 0 2 = 0 1 ( 0 2 see become zero element plain ) 0 1 + 0 2 = 0 2 ( 0 1 see become zero element plain ) \begin{cases} 0_1 + 0_2=0_1(0_2 As a zero element )\\ 0_1 + 0_2=0_2(0_1 As a zero element )\\ \end{cases}

obtain

0 1 = 0 2 0_1=0_2

It shows that the zero element is unique

### nature 2

The negative element of any element is unique , α \alpha The negative element of is written as − α -\alpha

prove （ Reduction to absurdity ）

hypothesis α ∈ V \alpha \in V There are two negative elements , Write it down as β , γ \beta,\gamma

According to the definition of negative element , Yes

{ α + β = 0 α + γ = 0 \begin{cases} \alpha + \beta = 0\\ \alpha + \gamma = 0 \end{cases}

also

β = β + 0 = β + ( α + γ ) = ( β + α ) + γ = 0 + γ = γ \beta=\beta+0=\beta+(\alpha+\gamma)=(\beta+\alpha)+\gamma=0+\gamma=\gamma

namely

β = γ \beta=\gamma

Sum up , The negative element of any element is unique

### nature 3

（ 1 ） 0 α = 0 （1）0\alpha=\boldsymbol0
（ 2 ） ( − 1 ) α = − α （2）(-1)\alpha=-\alpha
（ 3 ） λ 0 = 0 （3）\lambda \boldsymbol0=\boldsymbol0

Prove （1）

α + 0 α = 1 α + 0 α = ( 1 + 0 ) α = α \alpha+0\alpha=1\alpha+0\alpha=(1+0)\alpha=\alpha

obtain

0 α = 0 0\alpha=\boldsymbol0

Prove （2）

α + ( − 1 ) α = ( 1 − 1 ) α = 0 \alpha+(-1)\alpha=(1-1)\alpha=\boldsymbol0

According to the definition of negative element , obtain

( − 1 ) α = − α (-1)\alpha=-\alpha

Prove （3）

λ 0 = λ ( α + ( − 1 ) α ) = λ α + ( − λ ) α = ( λ + ( − λ ) ) α = 0 α = 0 \lambda \boldsymbol0=\lambda(\alpha+(-1)\alpha)=\lambda\alpha+(-\lambda)\alpha=(\lambda+(-\lambda))\alpha=0\alpha=\boldsymbol0

namely

λ 0 = 0 \lambda \boldsymbol0=\boldsymbol0

### nature 4

If λ α = 0 \lambda \alpha=\boldsymbol0 , be λ = 0 \lambda=0 or α = 0 \alpha=\boldsymbol0

prove

When λ = 0 \lambda=0 when , λ α = 0 \lambda \alpha=\boldsymbol0

When λ ≠ 0 \lambda\neq0 when ,

equation λ α = 0 \lambda \alpha=\boldsymbol0 Ride on both sides 1 λ \frac{1}{\lambda} , have to

1 λ ( λ α ) = 1 λ 0 = 0 \frac{1}{\lambda}(\lambda \alpha)=\frac{1}{\lambda}\boldsymbol0=\boldsymbol0

Again because

1 λ ( λ α ) = ( 1 λ λ ) α = 1 α = α \frac{1}{\lambda}(\lambda \alpha)=(\frac{1}{\lambda}\lambda)\alpha=1\alpha=\alpha

Introduction

α = 0 \alpha=\boldsymbol0

（2） The necessity of proof ：

from λ = 0 \lambda=0 or α = 0 \alpha=\boldsymbol0

It's easy to get λ α = 0 \lambda \alpha=\boldsymbol0

Sum up , If λ α = 0 \lambda \alpha=\boldsymbol0 , be λ = 0 \lambda=0 or α = 0 \alpha=\boldsymbol0

## give an example

### example 1

explain P [ x ] n P[x]_n It's vector space , among P [ x ] n P[x]_n Indicates that the number of times does not exceed n All of the polynomials of

P [ x ] n = { a n x n + a n − 1 x n − 1 + . . . + a 1 x + a 0 ∣ a n , . . . , a 1 , a 0 ∈ R } P[x]_n=\{a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0|a_n,...,a_1,a_0 \in \mathbb{R} \}

prove

Prove the closure of addition operation ：

set up

α ∈ P [ x ] n , β ∈ P [ x ] n \alpha\in P[x]_n,\beta\in P[x]_n

Yes

α + β ∈ P [ x ] n \alpha + \beta \in P[x]_n

α + β \alpha + \beta The number of times of any item in the will not exceed n n , So the result also belongs to P [ x ] n P[x]_n

The closure of the multiplication of the number of certificates ：

set up

k ∈ R , α ∈ P [ x ] n k\in \mathbb{R},\alpha \in P[x]_n

Yes

k α ∈ P [ x ] n k\alpha \in P[x]_n

Multiplication does not make P [ x ] n P[x]_n The maximum number of times exceeded n, The result is also in P [ x ] n P[x]_n in

The addition of polynomials 、 The number multiplication operation satisfies the law of linear operation , That is, eight operation laws , No more details here

Sum up , P [ x ] n P[x]_n It's vector space

### example 2

explain Q [ x ] n Q[x]_n It's vector space , among Q [ x ] n Q[x]_n Expressed as

Q [ x ] n = { a n x n + a n − 1 x n − 1 + . . . + a 1 x + a 0 ∣ a n , . . . , a 1 , a 0 ∈ R , And a n ≠ 0 } Q[x]_n=\{a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0|a_n,...,a_1,a_0 \in \mathbb{R}, And a_n\neq0 \}

prove

Prove the closure of addition operation ：

set up

α ∈ Q [ x ] n , β ∈ Q [ x ] n \alpha \in Q[x]_n,\beta \in Q[x]_n

Yes

α + β ∈ Q [ x ] n \alpha + \beta \in Q[x]_n

The closure of the multiplication of the number of certificates ：

set up

k ∈ R , α ∈ Q [ x ] n k\in \mathbb{R},\alpha \in Q[x]_n

obtain k α k\alpha It doesn't necessarily belong to Q [ x ] n Q[x]_n

The special case is when k = 0 k=0 when

k α = 0 α = 0 k\alpha=0\alpha=0

Be careful Q [ x ] n Q[x]_n In the definition a n ≠ 0 a_n\neq0 , explain Q [ x ] n Q[x]_n It must be non-zero

So number multiplication is not closed

Sum up , Q [ x ] n Q[x]_n It's not vector space

### example 5

The totality of positive real numbers , Write it down as R + \mathbb{R}^+ , In which the addition and multiplication operations are defined as

{ a ⊕ b = a b ( a , b ∈ R + ) λ ⊙ a = a λ ( λ ∈ R , a ∈ R + ) \begin{cases} a\oplus b=ab(a,b\in \mathbb{R}^+)\\ \lambda \odot a=a^{\lambda}(\lambda\in \mathbb{R},a\in \mathbb{R^+} ) \end{cases}

Test certificate R + \mathbb{R}^+ The above addition and multiplication operations form a linear space

prove

Prove the closure of addition operation ：

To any a , b ∈ R + a,b\in\mathbb{R}^+ , There are

a ⊕ b = a b ∈ R + a\oplus b=ab \in \mathbb{R}^+

The closure of the multiplication of the number of certificates ：

To any λ ∈ R , a ∈ R + \lambda\in \mathbb{R},a\in \mathbb{R^+} , Yes

λ ⊙ a = a λ ∈ R + \lambda \odot a=a^{\lambda}\in \mathbb{R^+}

Prove eight operation laws ：

It's not proved one by one here , The result is that all eight operation laws meet

But it should be noted that

At this time The zero element is 1

That is, for any a ∈ R + a \in \mathbb{R^+} , Yes a ⊕ 1 = a a\oplus1=a

Just follow the negative element 、 Definition of zero element , Then we can solve it according to our custom operation rules

### Summary

（1） Prove whether a set constitutes a vector space , Definitely not only To verify the addition 、 Closure of number multiplication operation

（2） When the defined addition and multiplication operations are not the usual addition and multiplication operations between real numbers , It also needs to prove whether eight linear operation laws are satisfied

（3） When proving uniqueness , have access to Reduction to absurdity , Suppose there are multiple elements at the same time , Then prove that these elements are equal .

# Conclusion

explain ：

• Refer to textbook 《 linear algebra 》 The fifth edition Department of mathematics, Tongji University
• Cooperate with the concept explanation in the book Combined with some of my own understanding and thinking

The article is only for study notes , Record from 0 To 1 A process of

I hope it will be of some help to you , If you have any mistakes, you are welcome to correct them ～

I am a Sea boom ଘ(੭ˊᵕˋ)੭

If you think it's ok , Please like it

Thank you for your support

https://cdmana.com/2021/10/20211002145440543h.html

Scroll to Top