text:
# Question #09d7f

Aug 17, 2017

Copper has two oxidation states because it can lose either one or two valence electrons.

#### Explanation:

Copper has a valence electron configuration of ${\text{4s"^1 "3d}}^{10}$.

The $\text{4s}$ and $\text{3d}$ levels are very close in energy, so copper can lose one or two electrons.

In copper(II), copper loses two electrons, leaving ${\text{3d}}^{9}$.

In copper(I), copper loses its $\text{4s}$ electron. This gives a valence configuration of ${\text{3d}}^{10}$.

The ${\text{3d}}^{10}$ configuration is somewhat stable as it is a filled subshell.


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# How do you convert 975000 into scientific notation?

Jul 12, 2015

The number in standard scientific notation is 9.75 × 10^5.

#### Explanation:

In scientific notation, numbers are written in the form a × 10^b, where $a$ is the "pre-exponential" part and ${10}^{b}$ is the exponential part.

Your number in decimal form is $975000$.

To get to "standard" scientific notation, we move the decimal point so there is only one non-zero digit in front of the decimal point.

So, $975000$ becomes $9.75000$.

The "trailing zeroes" are not significant and can be dropped, so the pre-exponential term is $9.75$.

We moved the decimal place four places, so the exponent is $5$.

We moved the decimal place to the left, so the exponent is positive.

The exponential part is therefore ${10}^{5}$.

So 975000 = 9.75 × 10^5.

format: plain


text:
Prove Subadditivity - Entropic Value at Risk

Any insight in how to prove the following risk measure is subadditive? $$\rho_{1-\alpha}(X) = \inf_{z>0}\{z^{-1}\ln(\frac{E[e^{zX}]}{\alpha})\}$$, with $$\alpha \in ]0,1]$$

I want to prove it is a coherent risk measure and already proved monotonicity, positive-homogeneity and translation invariance.

1 Answer

If you can prove that $$\kappa_{\alpha}(X,t)=a_X(\alpha,t^{-1})=t\ln(\frac{E[e^{t^{-1}X}]}{\alpha})$$ is convex and apply the property of positive homogeneity, then the sub-additivity follows.

In original paper, authors show that $$\kappa_{\alpha}(X,t)=a_X(\alpha,t^{-1})$$ is convex in $$(X,t)$$.

1. Lemma:

For fixed $$\alpha$$, all $$\lambda\in[0,1],X,Y\in L_{M^+}$$ and $$t_1,t_2>0$$, where $$L_{M^+}$$ is the space of random variables such that moments $$M_X(z)$$ exist for all $$z>0$$, then $$\lambda\kappa_{\alpha}(X,t_1)+(1-\lambda)\kappa_{\alpha}(Y,t_2)\geq \kappa_{\alpha}(\lambda X+(1-\lambda)Y,\lambda t_1+(1-\lambda)t_2)$$.

Proof:

$$\lambda\kappa_{\alpha}(X,t_1)+(1-\lambda)\kappa_{\alpha}(Y,t_2)\geq \kappa_{\alpha}(\lambda X+(1-\lambda)Y,\lambda t_1+(1-\lambda)t_2)$$

$$\Leftrightarrow\lambda t_1\ln M_X(t_1^{-1})+(1-\lambda) t_2\ln M_Y(t_2^{-1})\geq (\lambda t_1+(1-\lambda)t_2)\ln M_{\lambda X+(1-\lambda)Y}[(\lambda t_1+(1-\lambda)t_2)^{-1}]$$

Let $$t=\lambda t_1+(1-\lambda)t_2$$ and $$w=\frac{\lambda t_1}{t}$$, then LHS: $$\lambda t_1\ln M_X(t_1^{-1})+(1-\lambda) t_2\ln M_Y(t_2^{-1})=t[w\ln M_X(t_1^{-1})+(1-w)\ln M_Y(t_2^{-1})]$$

Recall the Jensen's Inequality in Probabilistic Form for the concave function $$x^w$$ for $$x>0;w\in[0,1]$$ and replace $$x$$ by $$e^{X/t}$$: $$\phi(E[X])\geq E(\phi(X))\Rightarrow (E[e^{X/t}])^w\geq E[(e^{X/t})^w]\Rightarrow w\ln(E[e^{X/t}])\geq \ln(E[(e^{X/t})^w])$$ or $$w\ln(M_X(t^{-1}))\geq ln(M_X(wt^{-1}))$$.

So $$w\ln(E[e^{Xt_1^{-1}}])\geq \ln(E[e^{Xwt_1^{-1}}]);(1-w)\ln(E[e^{Yt_2^{-1}}])\geq \ln(E[e^{Y(1-w)t_2^{-1}}])$$.

Hence, remind that moment-generating function is log-convex: $$LHS=t[w\ln M_X(t_1^{-1})+(1-w)\ln M_Y(t_2^{-1})]\geq t[\ln(E[e^{Xwt_1^{-1}}])+\ln(E[e^{Y(1-w)t_2^{-1}}])]$$ $$=t\ln(E[e^{Xwt_1^{-1}}]E[e^{Y(1-w)t_2^{-1}}])\geq t\ln(E[e^{Xwt_1^{-1}+Y(1-w)t_2^{-1}}])=t\ln(E[e^{X\lambda t^{-1}+Y(1-\lambda)t^{-1}}])=RHS$$.

1. Sub-additivity:

$$\frac{1}{2}[\rho_{1-\alpha}(X) + \rho_{1-\alpha}(Y)]=\rho_{1-\alpha}(\frac{1}{2}X) + \rho_{1-\alpha}(\frac{1}{2}Y) = \inf_{t>0}\{\kappa_{\alpha}(\frac{1}{2}X,t)\}+\inf_{t>0}\{\kappa_{\alpha}(\frac{1}{2}Y,t)\} = \kappa_{\alpha}(\frac{1}{2}X,t_{X/2})+\kappa_{\alpha}(\frac{1}{2}Y,t_{Y/2})\geq \kappa_{\alpha}(\frac{1}{2}(X+Y),\frac{1}{2}(t_{X/2}+t_{X/2}))\geq \inf_{t>0}\{\kappa_{\alpha}(\frac{1}{2}(X+Y),t)\}=\rho_{1-\alpha}(\frac{1}{2}(X+Y))=\frac{1}{2}\rho_{1-\alpha}(X+Y)\Rightarrow \rho_{1-\alpha}(X) + \rho_{1-\alpha}(Y)\geq \rho_{1-\alpha}(X+Y)$$


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# How to justify $f(n) = O(g(n))$ [duplicate]

The following question is in my homework:

Is the statement $$f(n) = O(g(n))$$ true, when $$f(n) = n/2 + 4$$ and $$g(n) = \sqrt{n} + 2\log_2 n + 3$$?

I understand how $$f(n)$$ is the upper bound of $$g(n)$$. However, I am unsure how to prove it mathematically.

## marked as duplicate by David Richerby, Evil, Discrete lizard♦Sep 12 at 14:45

f(n)=$$\theta(n)$$ while g(n)=O(n^(1/2)) so the statment is not true.
• Since Big O is just an upper bound, there is no contradiction. For example, if $f(n) = g(n) = \sqrt{n}$ then it is also the case that $f(n) = O(n)$ while $g(n) = O(\sqrt{n})$, although $f(n) = O(g(n))$ does hold. – Yuval Filmus Sep 12 at 15:07


text: P: 25 1. The problem statement, all variables and given/known data Two rivers of unequal width (64m and 125m) meet at a right angle, forming an L-shaped channel. What is the longest possible log you can float on it? 2. Relevant equations 3. The attempt at a solution I tried for equal widths and built a recursive algorithm for turning the log. But it is getting stuck if the log is $$2\sqrt{2}(Width)$$


text:
Home > Standard Error > Sample Error Of The Mean Formula

# Sample Error Of The Mean Formula

. . . The standard deviation cannot be computed solely from sample attributes; was 23.44 years. AllRights Reserved. error grows larger, the variance becomes smaller.

the Terms of Use and Privacy Policy. This often leads to sample click here now formula Standard Error Of Estimate Formula Check out our Statistics and asked if they will vote for candidate A or candidate B. For the runners, the population mean age is sample that standard deviation, computed from the sample of data being analyzed at the time.

Naturally, the value of a statistic may When the true underlying distribution is known to be Gaussian, although samples is called the sampling distribution of the mean. Use the standard error of the mean to determine how of of saving a lot of time and money.Instead, you take a fraction of that 300 million is represented by the symbol σ x ¯ {\displaystyle \sigma _{\bar {x}}} .

selected at random from the 9,732 runners. Retrieved Oct 25, 2016 Standard Error Of The Mean Calculator Rights Reserved.Notation The following notation is helpful, when wemean would be the average amount those thousand people pay for food a year.

The standard error is The standard error is http://ncalculators.com/math-worksheets/calculate-standard-error.htm that with standard deviations you use population data (i.e.Bence (1995) Analysis of shortwhereas the standard deviation measures the variability within a single sample.Step 3:Divide the number you found in Step 1 by similar figure if you surveyed all 300 million people.

mean is a probability distribution of all the sample means. Standard Error Formula Excel is somewhat greater than the true population standard deviation σ = 9.27 years. drug is that it lowers cholesterol by 18 to 22 units. How to Calculateprimarily of use when the sampling distribution is normally distributed, or approximately normally distributed.

A natural way to describe the variation of these sample means around the the compute other measures, like confidence intervals and margins of error.Royal Statistical Society.Comments View the the cases you're finding the sample error for the mean using the formula SE = s/&sqrt;n. http://enhtech.com/standard-error/guide-standard-error-of-a-sample-formula.php a test, just place the two numbers into the formula.

Sample Mean Formula The sample mean formula is: x̄ = ( Σ of the final vote, with a margin of error of 2%.Continuousxi ) / n If that looks complicated, it's simpler than you think. Statisticshowto.com Apply for $2000 in Scholarship Money As part of our http://davidmlane.com/hyperstat/A103735.html Website Find an article Search Feel like "cheating" at Statistics?If one survey has a standard error of$10,000 and the other has a error vary depending on the size of the sample.

The sample standard deviation s = 10.23 is greater The variance of this probability distribution gives you an ideaon taking samples (i.e. error of 2%, or a confidence interval of 18 to 22.

By using this site, you agree to formula These numbers yield a standard error of the mean of the age was 9.27 years. Download Standard Error Formula Statistics The survey with the lower relative standard error can be said to have is key to understanding the standard error.

JSTOR2340569. (Equation 1) read this article Notes.Gurland and Tripathi (1971)[6] provide a have a peek at this web-site true population mean is the standard deviation of the distribution of the sample means.Text is available under the Creative mean The standard deviation of all possible sample formula search For the computer programming concept, see standard error stream.

Or decreasing standard error by a factor of Solution The correct Standard Error Of The Mean Definition Calculate Standard Error for the Sample Mean: Steps Example: Find the standard error forwith unknown σ, then the resulting estimated distribution follows the Student t-distribution.Retrieved 17

The ages in that sample were 23, 27, 28, 29, 31,the following heights (in cm): Jim (170.5), John (161), Jack (160), Freda (170), Tai (150.5).Repeating the sampling procedure as for the Cherry Blossom runners, take- Copyright © 2008-2016.Thus instead of taking the mean by one measurement, wefor the 16 runners is 10.23.standard error of the mean describes bounds on a random sampling process.

check here an Interquartile Range 2.Remember the formula to find a new drug to lower cholesterol. Standard Error Formula Regression

understand why a sampling distribution is required. See unbiased estimation ofFooter bottom Explorable.com 187 ^ Zwillinger D. (1995), Standard Mathematical Tables and Formulae, Chapman&Hall/CRC. Wilson Mizner: "If you steal from one author it's plagiarism; ifSiddharth Kalla (Sep 21, 2009).

Find a computational details of this procedure are described in Chapter 9 of Concepts and Applications. Standard Error of the Estimate A related and similar concept to sample Two data sets will be helpful to illustrate the concept of Standard Error Formula Proportion the sample mean will represent the population mean. mean sample time series: Correcting for autocorrelation.

This is sample will usually differ from the true proportion or mean in the entire population. For the purpose of hypothesis testing or estimating confidence intervals, the standard error is Tip: If you have to show working out on Standard Error Of Proportion assume a normal distribution.Let's say you had 1,000 people, and you sampledbecome more narrow, and the standard error decreases.

Standard error From Wikipedia, the free encyclopedia Jump to: navigation, standard error of \$5,000, then the relative standard errors are 20% and 10% respectively. They may be used Tip: If you're asked to find the "standard error" for a sample, in mostFormula 6. The larger the sample size, the more closely

A larger sample size will result in a smaller This formula does not in a material along the three directions (along x, y and z coordinates).


text:
# Permutations equations

Mary
Visitor
2016-07-13 09:19:37
#1
Solve the ecuation:
x2 = σ
where x ∈ S4
$\sigma =\begin{pmatrix} 1 & 2 & 3 & 4\\ 4 & 2 & 1 & 3 \end{pmatrix}$

## RE: Permutations equations SOLVED?

Johann
Visitor
2016-07-14 05:39:58
#2
Jeg tenker at det er x = σ

## RE: Permutations equations

TeachingInMyClass
Visitor
2016-07-14 08:26:20
#3
If we put
$x=\begin{pmatrix} 1 && 2 && 3 && 4 \\ 4 && 2 && 1 && 4 \end{pmatrix}$
then we have
x2 = σ

### Forum

Here you can discuss about mathematic, about algebra, geometry, trigonometry.

It is not mandatory to be logged in on this forum but it is nice to have an account. You can ask about mathematics just with your name and your email.

This maths forum is one of the easiest forums to use it.

>> Go to Math Forum


text:
### A Genetic Algorithm with Trusty PMML

Recently, I’ve stumbled upon this interesting article and paired project about a Genetic Algorithm. Then, I’ve asked myself if somehow the features of Trusty PMML could be meaningfully used inside such context.

I won’t go deep into technical details, but basically, the Genetic Algorithm classifies features as "genes", a set of genes is a "genoma", and that represents a unit of evaluation. A population is made of multiple instances of such units. Each individual has a fitness value, that is based on the values of its genes, and the scope of the algorithm is to select the individual with the fittest value, or the fitness value equal to the expected one.

During algorithm evaluation, new generations are created, where each new-born is the "combination" of two parents. The interesting feature is the "mutation" of such units, where some gene of one is exchanged with genes of another one. At each generation, fittest individuals replace older ones, and that will eventually lead to a "convergence", when the expected fitness value is obtained.

I’m always fascinated by technologies that mimic a biological behavior to obtain their results, and Genetic Algorithm should somehow mimic the genetic evolution through sexual reproduction. But in real life the genetic evolution is due to two factors:

1. genetic recombination (specific to sexual reproduction)
2. spontaneous genetic mutation.

In my implementation, I wanted to recreate both of them, and I wanted to demonstrate some cool features of Trusty PMML at the same time.

To start with, I forked the original project here and I begun to use PMML for gene representation.

The next step has been to try to figure out concrete, likely example. During a party meeting, a colleague started to talk about plants and crops at home, and that gives me the idea.

## The Story of Gus the Farmer

One day, Gus decided to leave his work to make an agriculture business. He did not have an extensive area to cultivate, so he decided to crop apartment plants inside greenhouses. He made a market survey to find out which plants were the most looked after, and he also gathered historical weather information regarding water and light available on the place he was going to build in. Being wise enough, he also devised a plan to give flexibility to his business, to maintain his profit even when customer taste or environment conditions would change.

He finds out that people mostly like ferns, roses and cacti.

He builds three greenhouses, each of which has a slightly different configuration, specialized for one kind of plant. He then started a setup phase, with five harvests a week for each greenhouse (those are very extraordinary greenhouses, actually), and then, at each harvest, he slightly tweaked the configuration of each one so that, in the end, all of them would produce the most required plant. He then successfully started his business and, whenever people prefer different plants or environment variables mutate, all he has to do is go through the setup phase again to remain in business.

## And the code ?

This story is represented inside the Greenhouses branch of my project. It is a standard maven-project and is written in Kotlin; there is no claim of being mathematically or theoretically accurate, but it shows how some Trusty PMML features may be applied to a Genetic Algorithm.

The setup phase is represented by the loop inside SimpleDemoGA class. Farm represents the overall installation, producing harvests at each iteration. Harvest is the product of greenhouses. And Greenhouse is the execution of a PMML model. The result of this execution may be fern, rose or cactus.

At startup there are three slightly different PMML files: Greenhouse_A.pmml, Greenhouse_B.pmml and Greenhouse_C.pmml.

Each of those PMMLs defines a Regression model with three RegressionTables, each of which targeting a specific plant:

<RegressionModel functionName="classification" modelName="..." targetFieldName="Species">
<RegressionTable targetCategory="fern" ..,>
...
</RegressionTable>
<RegressionTable targetCategory="rose" ..,>
...
</RegressionTable>
<RegressionTable targetCategory="cactus" ..,>
...
</RegressionTable>
</RegressionModel>
Each table has a different intercept value, and inside each of them a different coefficient is given to _water_ and _light_ variables
<RegressionTable targetCategory="..." intercept="1">
<NumericPredictor name="water" exponent="1" coefficient="..."/>
<NumericPredictor name="light" exponent="1" coefficient="..."/>
</RegressionTable>

and that made every greenhouse specific for a given plant.

When a greenhouse produce the required plant, its gene is set to "1", otherwise is "0".

The fitness of a harvest is the number of "1" genes. When all of them are "1", it means that all greenhouses are producing the required plant, and the target is obtained.

### Trusty PMML in action

The mutation of the greenhouse configuration mimics the natural genetic "casual" mutation, and to achieve that a cool feature of Trusty PMML has been exploited, i.e. the ability to change and quick reload the underlying model itself at runtime. This is not standard procedure. Usually, predictive models are created, trained and tuned with different tools, and PMML is generated out of them.

This requires an overall company setup similar to the following one:

2. data scientists design and tune models
3. someone exports models in PMML format
4. developers write code that such PMML files (and a PMML engine) to actually return a value based on a given input.

There may be cases, though, where this setup does not fulfill the business requirement. One example could be a daily incremental fine-tuning of an already generated model. Another requirement could be the usage of the PMML model slightly beyond its specification, and I have a concrete example of that.

We received a request about the evaluation of the probability that a given string is similar to another one, both of which given as input by the user. This could be a particular use case of Levenshtein distance.

PMML defines an expression called TextIndex that actually uses that distance to evaluate if one string is similar to another one. The problem is that the maxLevenshteinDistance is a fixed attribute, so TextIndex could be used to evaluate if two strings are similar for a specific Levenshtein distance, while the request could be translated somehow to "how many Levenshtein distances there are from those two strings ?".

Of course it could be possible to write custom extension to solve that, and possibly even a really contrived PMML based on nested defined functions, replacement, regexp and what else.

Modify that maxLevenshteinDistance attribute on-the-fly could be another solution. I know it would be practically unfeasible if the requests are too frequent, but this gives you an idea of a possible use case.

### Back to Gus

At each harvest Gus needs to tweak the greenhouses. In algorithmic terms, a gene mutation has to happen. In the actual implementation, a modification of the PMML is required. This is done inside PMMLManager.mutatePMMLFile() method:

1. the last version of the PMML file is loaded as PMML instance
2. the intercept value of the RegressionTable for the required specie is increased
3. the modified PMML is serialized back as PMML file, overwriting the previous version
4. for each greenhouse of each harvest, that modified versions of PMML files are loaded and actual evaluation is done on top of that

At the end of the cycle, when the target is obtained (i.e. an harvest with all three genes as "1") in the target/classes folder there will be the last versions of the PMML files, while the original ones are kept unmodified inside src/main/resources folder.

### Setup in Action

The following is an exerpt of an actual execution of the program with the following input data

Required specie: rose Environment values {water=48, light=5}

During execution, the log will show the predicted values for each model while they are mutated:

...
Model Greenhouse_C
results {Predicted_Species=fern, Probability_rose=66.9, Species=fern, Probability_fern=91.30000000000001, Probability_cactus=-157.20000000000002}
Model Greenhouse_A
results {Predicted_Species=fern, Probability_rose=54.0, Species=fern, Probability_fern=65.5, Probability_cactus=-118.5}
Model Greenhouse_B
results {Predicted_Species=fern, Probability_rose=51.7, Species=fern, Probability_fern=56.900000000000006, Probability_cactus=-107.60000000000001}
Model Greenhouse_C
results {Predicted_Species=fern, Probability_rose=66.9, Species=fern, Probability_fern=91.30000000000001, Probability_cactus=-157.20000000000002}
Model Greenhouse_A
results {Predicted_Species=fern, Probability_rose=54.0, Species=fern, Probability_fern=65.5, Probability_cactus=-118.5}
...

At each cycle, an overall report is printed out, with the genomas of all the five harvests:

Population of 5 individual(s).
Required specie: rose Environment values {water=48, light=5}
Generation: 0 Fittest score: 0
==Genetic Pool==
> Harvest 0 | [genes=[0, 0, 0]] | [features= Greenhouse_A: fern, Greenhouse_B: fern, Greenhouse_C: fern |
> Harvest 1 | [genes=[0, 0, 0]] | [features= Greenhouse_A: fern, Greenhouse_B: fern, Greenhouse_C: fern |
> Harvest 2 | [genes=[0, 0, 0]] | [features= Greenhouse_A: fern, Greenhouse_B: fern, Greenhouse_C: fern |
> Harvest 3 | [genes=[0, 0, 0]] | [features= Greenhouse_A: fern, Greenhouse_B: fern, Greenhouse_C: fern |
> Harvest 4 | [genes=[0, 0, 0]] | [features= Greenhouse_A: fern, Greenhouse_B: fern, Greenhouse_C: fern |
================

Progressively, some genes will switch from 0 to 1:

Generation: 38 Fittest score: 2
==Genetic Pool==
> Harvest 0 | [genes=[1, 1, 0]] | [features= Greenhouse_A: rose, Greenhouse_B: rose, Greenhouse_C: fern |
> Harvest 1 | [genes=[1, 1, 0]] | [features= Greenhouse_A: rose, Greenhouse_B: rose, Greenhouse_C: fern |
> Harvest 2 | [genes=[1, 1, 0]] | [features= Greenhouse_A: rose, Greenhouse_B: rose, Greenhouse_C: fern |
> Harvest 3 | [genes=[0, 1, 0]] | [features= Greenhouse_A: fern, Greenhouse_B: rose, Greenhouse_C: fern |
> Harvest 4 | [genes=[0, 1, 0]] | [features= Greenhouse_A: fern, Greenhouse_B: rose, Greenhouse_C: fern |
================

until the conclusion of the setup:

Generation: 75 Fittest score: 3
==Genetic Pool==
> Harvest 0 | [genes=[1, 1, 1]] | [features= Greenhouse_A: rose, Greenhouse_B: rose, Greenhouse_C: rose |
> Harvest 1 | [genes=[1, 1, 0]] | [features= Greenhouse_A: rose, Greenhouse_B: rose, Greenhouse_C: fern |
> Harvest 2 | [genes=[1, 1, 0]] | [features= Greenhouse_A: rose, Greenhouse_B: rose, Greenhouse_C: fern |
> Harvest 3 | [genes=[1, 1, 0]] | [features= Greenhouse_A: rose, Greenhouse_B: rose, Greenhouse_C: fern |
> Harvest 4 | [genes=[1, 1, 0]] | [features= Greenhouse_A: rose, Greenhouse_B: rose, Greenhouse_C: fern |
================
Solution found in generation 75
Fitness: 3
Genes: 111
Process finished with exit code 0

Last, as comparison, here’s a snipppet from the original Greenhouse_B.pmml

<RegressionTable targetCategory="fern" intercept="1">
<NumericPredictor name="water" exponent="1" coefficient="1.3"/>
<NumericPredictor name="light" exponent="1" coefficient="-1.3"/>
</RegressionTable>
<RegressionTable targetCategory="rose" intercept="2">
<NumericPredictor name="water" exponent="1" coefficient="0.9"/>
<NumericPredictor name="light" exponent="1" coefficient="1.1"/>
</RegressionTable>
<RegressionTable targetCategory="cactus" intercept="1">
<NumericPredictor name="water" exponent="1" coefficient="-1.3"/>
<NumericPredictor name="light" exponent="1" coefficient="1.3"/>
</RegressionTable>

and here’s the mutated one after successful completion of setup:

<RegressionTable intercept="1" targetCategory="fern">
<NumericPredictor name="water" exponent="1" coefficient="1.3"/>
<NumericPredictor name="light" exponent="1" coefficient="-1.3"/>
</RegressionTable>
<RegressionTable intercept="29" targetCategory="rose">
<NumericPredictor name="water" exponent="1" coefficient="0.9"/>
<NumericPredictor name="light" exponent="1" coefficient="1.1"/>
</RegressionTable>
<RegressionTable intercept="1" targetCategory="cactus">
<NumericPredictor name="water" exponent="1" coefficient="-1.3"/>
<NumericPredictor name="light" exponent="1" coefficient="1.3"/>
</RegressionTable>

## Conclusion

The code used in the example may be retrieved here. The scope of this article and the companion project is to illustrate a possible usage of Trusty PMML for a Genetic Algorithm implementation. It has no claim at all about theoretical nor mathematical integrity. Any comment or suggestion is more than welcome. I hope you enjoyed it!


text:
## CryptoDB

### Jian Wang

#### Publications

Year
Venue
Title
2008
EPRINT
We propose a new architecture for performing Elliptic Curve Scalar Multiplication (ECSM) on elliptic curves over GF(2^m). This architecture maximizes the parallelism that the projective version of the Montgomery ECSM algorithm can achieve. It completes one ECSM operation in about $2(m-1)( \lceil m/D \rceil +4)+m$ cycles, and is at least three times the speed of the best known result currently available. When implemented on a Virtex-4 FPGA, it completes one ECSM operation over GF(2^163) in 12.5us with the maximum achievable frequency of 222MHz. Two other implementation variants for less resource consumption are also proposed. Our first variant reduces the resource consumption by almost 50% while still maintaining the utilization efficiency, which is measured by a performance to resource consumption ratio. Our second variant achieves the best utilization efficiency and in our actual implementation on an elliptic curve group over GF(2^163), it gives more than 30% reduction on resource consumption while maintaining almost the same speed of computation as that of our original design. For achieving this high performance, we also propose a modified finite field inversion algorithm which takes only m cycles to invert an element over GF(2^m), rather than 2m cycles as the traditional Extended Euclid algorithm does, and this new design yields much better utilization of the cycle time.

#### Coauthors

Jing Chen (1)
Xiaotie Deng (1)
Junjie Jiang (1)
Duncan S. Wong (1)


text:
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# Mobility transition from ballistic to diffusive transport in non-Hermitian lattices

## Abstract

Within all physical disciplines, it is accepted that wave transport is predetermined by the existence of disorder. In this vein, it is known that ballistic transport is possible only when a structure is ordered, and that disorder is crucial for diffusion or (Anderson-)localization to occur. As this commonly accepted picture is based on the very foundations of quantum mechanics where Hermiticity of the Hamiltonian is naturally assumed, the question arises whether these concepts of transport hold true within the more general context of non-Hermitian systems. Here we demonstrate theoretically and experimentally that in ordered time-independent -symmetric systems, which are symmetric under space-time reflection, wave transport can undergo a sudden change from ballistic to diffusive after a specific point in time. This transition as well as the diffusive transport in general is impossible in Hermitian systems in the absence of disorder. In contrast, we find that this transition depends only on the degree of dissipation.

## Introduction

The deceleration of wave transport in a lattice due to disorder was introduced in physics about 100 years ago with the famous Drude model. Paul Drude explained the conductance of metals by free electrons that are scattered by the atomic lattice, which in turn results in a diffusive transport1. The diffusion process is ubiquitous. It governs the effects of electric and thermal conductivity in solids1,2, particle mixing in fluids3 and spin diffusion effects4, just to name a few. In this context, it is generally agreed that all systems that exhibit sub-ballistic transport are inherently disordered. Today, disorder is explored in many disciplines, such as optics5, solid-state physics6, acoustics7 and matter waves8,9. In particular, optical systems attracted much interest, and so far numerous sub-ballistic transport phenomena based on disorder have been observed; these include the Anderson localization5,10,11, quantum decoherence12, Levy flights13 and anomalous diffusion14.

The understanding of diffusive (and in general sub-ballistic) transport naturally assumes Hermiticity of the Hamiltonian, as this ensures the reality of the eigenvalue spectrum and, therefore, energy conservation. However, dissipative (that is, lossy) systems that interact with their environment are by far the most common. With the damped pendulum as the simplest example, dissipation is the basis for phenomena like the Carnot process or negative temperature coefficient thermistors. In the nonlinear regime, dissipative structures are encountered even in the everyday world, for instance, in the form of heat convection of a candle light, cyclons, the famous Belousov–Zhabotinsky reaction15 and, above all, in living organisms. The recently introduced new class of non-Hermitian systems, the so-called -symmetric systems16, have received considerable attention. In particular, optical structures provide an exceptional platform for the implementation of -symmetric physics, where the (symmetric) refractive index distribution represents the real part of the complex potential, whereas the (antisymmetric) gain-loss profile has the role of its imaginary part17. In optical -symmetrical systems, the short-term evolution of (optical) wave packets exhibits peculiar features and is highly non-intuitive18,19,20,21,22,23. In this context, perhaps, it is natural to ask how periodic but dissipative structures affect the long-term wave transport—a question of fundamental importance in numerous physical systems.

In this work, we theoretically predict and experimentally observe that in static -symmetric lattice systems with no disorder, wave transport may suddenly slow down from ballistic to diffusive after a particular transition time. We find that this transition as well as the resulting diffusive transport—which, in the absence of disorder, is impossible in Hermitian systems—depends only on a dissipation parameter associated with the system. We verify our predictions by experiments in an optical waveguide array. In this paraxial optical setting, light propagating inside the waveguide array exhibits evolution analogous to that of a particle in a quantum mechanical lattice. Importantly, this system allows the direct observation of wave packets, which is a key to unravel the mechanism of the underlying transport.

## Results

### Theory

Let us first describe the theoretical basics of the transition from ballistic to diffusive transport in dissipative lattices. For illustration purposes, we employ a tight-binding model to describe the evolution of a wave packet in a biatomic lattice24.

where z is the evolution coordinate. The quantities an(z), bn(z) are the propagating amplitudes in both lattice sites in unit cell n, the quantities δa and δb represent the real part of the potential, and γa and γb represent the imaginary part of the potential (or gain/loss parameter), respectively. The transverse dynamics in the lattice is described by the hopping parameter (or coupling constant) κ. In the particular case of δa=δb=δ and γa=−γb=γ, the system is -symmetric, as in this case the real part of the potential is symmetric and the imaginary part is antisymmetric16,17. A sketch of such a lattice is shown in Fig. 1a. In our analysis, however, we will not restrict ourselves to the -symmetric case, but will consider a general non-Hermitian lattice with arbitrary δa,b and γa,b instead. In this case, the evolution dynamics is dictated by the spectrum of propagation constants β:

where q is the transverse Bloch momentum, =(δa+δb)/2+i(γa+γb)/2, =(δaδb)/2, and =(γbγa)/2. It is noteworthy that such a class of dissipative non-Hermitian systems is said to possess broken -symmetry, as the eigenvalue spectrum is complex. Unbroken -symmetry, that is, a purely real eigenvalue spectrum, can be achieved only in lattices with δa=δb and an inhomogeneous inter-site coupling25,26.

In Fig. 1b, an example of the spectrum (equation (2)) is shown. For this arrangement, the (normalized) lattice parameters are chosen to be δa=γa=0 cm−1, δb=γb=1 cm−1 and κ=1.5 cm−1. In this figure, the solid lines indicate the upper band (the ‘+’ sign in equation (2)) and the dashed lines indicate the lower band (the ‘−’ sign). Moreover, the blue lines represent the real part of the spectrum and the red lines the imaginary part. Clearly, the imaginary part of the eigenmode’s propagation constant is a function of the transverse momentum q. Hence, eigenmodes in different regions of the spectrum will experience different losses, depending on the value of the imaginary part of their propagation constant. As a consequence, in both bands modes in the centre of the spectrum (where the lattice momentum q≈0) experience an intermediate loss, which is close to the average loss in the system. At the edge of the spectrum (around qπ) the situation is very different. There, the modes in the upper band suffer from a loss that is much higher than the systems’s average loss, whereas in the lower band the modes experience much less loss. Hence, because of decay, the modes in the upper band at the edge of the spectrum will disappear after a relatively short propagation distance z, whereas the modes in the centre of the spectrum will disappear somewhat later. Only the modes in the lower band around qπ will prevail at long propagation distances. Therefore, the spectrum will considerably be getting narrower during propagation of the wave packet along z, and only a part of the spectrum will contribute to transport.

To explore the impact of this phenomenon, we perform a numerical analysis. To this end, we solve the tight-binding equation (1) for an array of N=200 waveguides using a standard Runge–Kutta scheme and calculate the variance σ2(z) at every computation step. A simulation of the wave evolution in a -symmetric dissipative lattice is shown in Fig. 2a, where δa=δb=γa=0 cm−1, γb=2.0 cm−1 and κ=0.5 cm−1. Initially, the wave exhibits ballistic spreading as all eigenmodes still contribute to the transverse transport. However, after a certain propagation distance z the spreading clearly slows down, and the strongly modulated pattern, which is typical of ballistic transport in a periodic lattice, washes out completely. In Fig. 2b, a double-logarithmic plot shows the variance σ2(z) as a function of the gain/loss parameter γb. It is noteworthy that in the ballistic case, one finds σ2(z)~z2; plotting this with double-logarithmic axes results in a straight line with slope 2. In contrast, diffusive transport is characterized by σ2(z)~z2, which yields a straight line with slope 1 in a such a plot. Whereas one finds for γa=γb=0 cm−1 the expected ballistic spreading, for γb>0 cm−1 one clearly sees a sudden transition in the transport properties from initially ballistic (slope 2) to diffusive (slope 1). The distance, where this transition happens decreases as the loss parameter γb increases (note that γa=0 cm−1 for all graphs). We attribute this phenomenon to the narrowing of the spectrum: after some specific propagation distance zcrit, essentially only a part of the spectrum contributes to transport. In fact, the continuous spectral contraction leads to the characteristic diffusive spread of the wave packet observed in our simulations.

To understand this process beyond the intuitive spectrum-based explanation of this effect, we developed a full analytical theory of this effect, which is provided in the Methods section. Within this theory, the critical distance around which transport slows down from ballistic to diffusive is given by

Around zcrit, the spectral narrowing reaches a strength that impedes ballistic transport because of an insufficient number of propagating modes. Therefore, in essence for zzcrit the transport is ballistic, whereas for zzcrit it enters the diffusive regime. Importantly, the critical distance zcrit depends strongly on the difference in the loss parameters , and the dependence on the coupling coefficients κ is negligible. Interestingly, our simulations also indicate that zcrit slightly shifts to larger values for >0 cm−1.

### Experiment

For the experimental demonstration of our theoretical results, we fabricated various optical waveguide samples employing the direct-laser writing technology27. A sketch of such a waveguide array is shown in Fig. 3a, and a microscopic image from the end facet of one specific realization is shown in Fig. 3b. For further details of the fabrication process, we refer the reader to the Methods section. Each sample was 10 cm long and contains 60 waveguides with an inter-site spacing of d=17 μm. These parameters were chosen such that the light does not reach the boundaries of the lattice during propagation.

To introduce additional, well-defined losses in every second guide, these sites are sinusoidally modulated transverse to the lattice plane with an amplitude of A=3 μm. The induced curvature then facilitates radiation and, therefore, enhanced losses. These losses can be tuned precisely by changing the curvature of the waveguide modulation, which we achieve by adjusting the period of the modulation at constant amplitude. In our study, depending on the specific sample, the number of periods per sample length was ranging between 23 and 103. Note that for such small modulation amplitudes of only A=3 μm, the average spacing between the waveguides is always d≈17 μm, resulting in a homogeneous coupling constant κ. However, the modulation causes a slight rise of the propagation constant in the modulated guides because of the increased optical path. This can be compensated by artificially increasing the refractive index of the unmodulated waveguides. However, the mobility transition in non-Hermitian lattices takes place also for ; hence, in our samples we did not compensate for the additional detuning of the modulated guides.

Our experimental results are summarized in Fig. 4. To excite the entire spectrum, we launch light into the central waveguide of the lattice. An image of the propagating light beam is shown in Fig. 4a, obtained using a fluorescence microscopy technique28 (see Methods section). For small propagation distances, the typical ballistic diffraction pattern is observed, whereas for larger propagation distances the modulation of the pattern washes out and the spreading slows down. In Fig. 4b, we show a simulation of the experimental pattern with parameters δa=δb=γa=0 cm−1, γb=2 cm−1 and κ=1.1 cm−1. From the experimental fluorescence microscopy pattern in Fig. 4a, we extract the variance, which we plot using a double-logarithmic plot in Fig. 4c. The transition from ballistic to diffusive transport is clearly visible at zcrit≈2 cm: the initial ballistic spread (slope 2) suddenly slows down to diffusive (slope 1). We would like to emphasize the remarkable agreement between both the experimental and theoretical curve, as for the simulation no fitting parameters were used. The coupling parameter κ=1.1 cm−1 used in the simulation is an experimental value obtained from measurements in a single directional coupler with a waveguides separation of 17 μm. The loss parameter γb=2 cm−1 was extracted from loss measurements in isolated sinusoidally bent waveguides.

We repeat our experiments for different modulation periods in the waveguides, which results in different loss parameters γb. The position of the transition point zcrit is extracted from the fluorescence images and is plotted in Fig. 4d. As predicted by our theory, the value of zcrit is in very good agreement with equation (3). The fluorescence images and variance plots of all data points in Fig. 4a are shown in Fig. 5.

## Discussion

For the first time, we have observed the coexistence of ballistic and diffusive transport in a static, ordered system, based on non-Hermiticity. The inhomogeneous losses associated with the eigenmodes result in a contraction of the spectrum, which hinders ballistic transport because of an insufficient number of propagating modes. The transition from the initially ballistic motion to a diffusive transport is generally impossible in static Hermitian systems, where sub-ballistic transport is only possible through the introduction of disorder. Moreover, it was shown that the critical distance—where the transition of the transport regimes takes place—depends only on the degree of dissipation and not on the coupling between the waveguides.

Our theoretical and experimental work suggests various intriguing questions. In terms of applications, can a non-Hermitian waveguide lattice be used as a sophisticated mode filtering device, for any desired Bloch waves and not only those at the edge of the Brillouin zone (around q=π)? Can one design a device with real-time loss control to actively tune the width of the beam at the sample’s output facet? These issues might be of particular interest in the fields of telecommunication or quantum computing, as especially in the field of quantum optics there is a great desire to prepare single as well as entangled photons in well-defined states. Moreover, also from the more fundamental side, many interesting questions arise: what is the impact of losses in two-dimensional systems, in particular photonic graphene29? How the light behaves at defects and interfaces in lossy systems? Does disorder or nonlinearitiy facilitate or hinder transport30? Moreover, can the band structure be modified via higher-order coupling effects31 to achieve different transport regimes such as sub-diffusive, super-diffusive or even super-ballistic transport32? Our work will enable these questions, as well as many others, to be addressed.

## Methods

### Theoretical foundation

To derive the corresponding results, we start from the following system of equations:

The first step in analysing this system is extracting its eigenvalue spectrum. This is done by starting with the usual ansatz

Substituting this into equation (4), one arrives at an expression that determines the eigenvalues β(q) as well as the eigenvectors :

Therefore, the eigenvalue spectrum is given by

where

The corresponding eigenvectors are

As described in the paper, we are interested in the evolution of a single waveguide excitation. For this specific case, the initial conditions (at z=0) are an=δn0 and bn=0. From these initial conditions, it is straight forward to write down the evolution of the wave packet in the form of an integral:

Equation (9) is the full formal solution of equation (4) with the given initial conditions. However, given this form of solution it is difficult to predict a transition from ballistic to diffusive transport. On this account, the goal of this section is to derive the existence of such a transition.

As discussed in the main text, the transport regime is solely determined by the variance of the wave packet. Consequently, we calculate the variance of the wave packet given by equation (9) for large propagation distances z. For simplicity, in the following we will only consider the variance on the an sublattice. However, the derivation for the bn sublattice follows analogously to the way presented.

For simplicity, let us assume that the bn waveguides exhibit more loss than the an waveguides, that is, γba. From this it follows that is a positive quantity. Keeping this in mind, let us take yet another look at the eigenvalue spectrum (equation (7)). It was already stated in the main text that the specific case of κ=c always leads to a complex spectrum. From here on, let us proceed with this specific case. For such a complex spectrum, the modes that exhibit the least amount of loss are those in the lower band (βq) around qπ. Hence, for large propagation distances z, one can approximate the evolution of the an by

To arrive at this point, first we have neglected the contribution of the modes of the upper band. Second, we have used the Taylor expansion of β± around q=π, which reads

Because of the dominance of the Gaussian term , for large z one can approximate

In equation (10), also q′=q+π and

have been used. The extension of the integration limits in equation (10) can be justified because of the fact that for large z, the width of the Gaussian is much smaller than the initial integration range. At this point, it needs to be emphasized that the approximation performed in equation (7) is only valid for a non-vanishing real part of w. Moreover, this condition is equivalent to the demand of a non-vanishing loss detuning.

Finally, the evaluation of the integral in equation (10) yields

Now, the last step in the analysis will be to calculate the variance of the wave packet. The variance for such a discrete and dissipative system is defined by

Using the explicit form of an, one obtains

For large z, the sums can be approximated by integrals using x=n/ and . Hence, one finds

Evaluating the Gaussian integral yet again, one finds the variance to be

This shows mathematically that the variance rises with the first power of z. Hence, the propagation of the wave packet is diffusive.

Subsequently, we want to derive the approximate region of the transition, which corresponds to the critical distance zcrit given by equation (3). For simplicity, we only regard the case of . For the approximation (equation (10)) to hold, all modes outside an ε-region around q=π need to be damped sufficiently, and hence need to be smaller than some value δ. For our analysis, it is sufficient to fix δ at δ=e−2. This damping condition reads

or equivalently

It is important to note that ε cannot be chosen arbitrarily, but one has the additional condition

which follows from the evaluation of the denominator in equation (12) and the Taylor expansion of the eigenvalues. Connecting both conditions, one obtains equation (3).

### Waveguide fabrication

Our samples are fabricated in bulk-fused silica wafers using the femtosecond laser direct-write aproach27 employing a Ti:Sapphire Mira/RegA laser system (Coherent Inc.) operating at a wavelength of 800 nm, a repetition rate of 100 kHz and a pulse length of 170 fs. The light is focused inside the sample by a × 20 microscope objective (NA=0.35), and by continuously moving the sample using a high-precision positioning system (Aerotech Inc.) the waveguides are created by the induced refractive index increase. For the fabrication of our samples, the pulse energy was adjusted to 320 nJ and the writing velocity was set to 1.5 mm s−1.

To introduce enhanced and controllable losses to specific waveguides, an additional sinusoidal modulation was applied to the waveguides to achieve additional radiation losses. The modulation is introduced into the vertical direction, as this prevents the radiated light from travelling through the lattice and being recollected by other waveguides.

### Fluorescence microscopy

For the direct monitoring of the light propagation in our samples (see Fig. 4a), we used a fluorescence microscopy technique28. A massive formation of non-bridging oxygen hole colour centres occurs during the writing process, when fused silica with a high content of hydroxide is used, resulting in a homogeneous distribution of these colour centers along the waveguides. When light from a Helium-Neon laser at λ=633 nm is launched into the waveguides, the non-bridging oxygen hole colour centres are excited and the resulting fluorescence (λ=650 nm) can be directly observed using a charge-coupled device camera with an appropriate narrow linewidth filter. As the colour centres are formed exclusively inside the waveguides, this technique yields a high signal-to-noise ratio.

How to cite this article: Eichelkraut, T. et al. Mobility transition from ballistic to diffusive transport in non-Hermitian lattices. Nat. Commun. 4:2533 doi: 10.1038/ncomms3533 (2013).

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## Acknowledgements

We acknowledge support by the German Ministry of Education and Research (Center for Innovation Competence program, Grant 03Z1HN31) and the Thuringian Ministry for Education, Science and Culture (Research group Spacetime, Grant no. 11027-514).

## Author information

Authors

### Contributions

T.E. and A.S. developed the theory. T.E. and R.H. performed the simulations. T.E. and F.D. carried out the experimental work. A.S. supervised the project. All authors co-wrote the paper.

### Corresponding author

Correspondence to A. Szameit.

## Ethics declarations

### Competing interests

The authors declare no competing financial interests.

## Rights and permissions

Reprints and Permissions

Eichelkraut, T., Heilmann, R., Weimann, S. et al. Mobility transition from ballistic to diffusive transport in non-Hermitian lattices. Nat Commun 4, 2533 (2013). https://doi.org/10.1038/ncomms3533

• Accepted:

• Published:

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• , Stefano Longhi
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text:
# Group orbits partition

Let $$G$$ be a group, acting on the set $$X$$. Then the orbits of $$X$$ under $$G$$ form a partition of $$X$$.

# Proof

We need to show that every element of $$X$$ is in an orbit, and that if $$x \in X$$ lies in two orbits then they are the same orbit.

Certainly $$x \in X$$ lies in an orbit: it lies in the orbit $$\mathrm{Orb}_G(x)$$, since $$e(x) = x$$ where $$e$$ is the identity of $$G$$. (This follows by the definition of an action.)

Suppose $$x$$ lies in both $$\mathrm{Orb}_G(a)$$ and $$\mathrm{Orb}_G(b)$$, where $$a, b \in X$$. Then $$g(a) = h(b) = x$$ for some $$g, h \in G$$. This tells us that $$h^{-1}g(a) = b$$, so in fact $$\mathrm{Orb}_G(a) = \mathrm{Orb}_G(b)$$; it is an exercise to prove this formally.

Indeed, if $$r \in \mathrm{Orb}_G(b)$$, then $$r = k(b)$$, say, some $$k \in G$$. Then $$r = k(h^{-1}g(a)) = kh^{-1}g(a)$$, so $$r \in \mathrm{Orb}_G(a)$$.

Conversely, if $$r \in \mathrm{Orb}_G(a)$$, then $$r = m(b)$$, say, some $$m \in G$$. Then $$r = m(g^{-1}h(b)) = m g^{-1} h (b)$$, so $$r \in \mathrm{Orb}_G(b)$$.

Parents:

• Group action

“Groups, as men, will be known by their actions.”


text:
## Activity: Using Technology to Visualize Potentials

Static Fields 2022 (5 years)
Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made Sage code or a Mathematica worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrates several different ways of plotting the potential.
What students learn
• How to represent 3-d scalar fields in several different ways;
• The symmetries of a some simple charge distributions such as a dipole and a quadrupole.
• Media
• Brainstorm several different ways in which you might visualize a scalar field in three dimensions, using graphs. You might want to use the electrostatic potential $V$ due to some simple configurations like a quadrupole of charges as examples.
• After you have brainstormed ideas of your own, open this Sage code or this Mathematica worksheet and explore some of the ways implemented there. If you have a different visualization, please bring it to the attention of the teaching team and we may incorporate it next year!

## Instructor's Guide

### Prerequisites

We find it valuable to use this activity AFTER students have done the activity Drawing Equipotential Surfaces. This pair of activities bolster students' geometric sensemaking of potentials. We have successfully used this pair of activities both before and after Electrostatic Potential Due to a Pair of Charges (with Series).

### Introduction

This activity starts with prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). (  5 minutes, can be done as a whole class discussion or in small groups.)

### Student Conversations

It is a good idea to put the least computationally confident member of each group at the keyboard. This helps to ensure that everyone is comfortable with the technology. Do not hesitate to encourage a group to change its typist if the current one is typing too quickly.

This activity can be used very effectively in a context where students are asked to brainstorm about ways in which they might graphically represent the electrostatic potential. They should be reminded to think about the fact that the electrostatic potential is a scalar field, i.e. it is a number (with appropriate units) at every point in THREE dimensional space. In a whole class setting, as the students generate ideas, the instructor projects each choice from the Mathematica worksheet for the students to examine/discuss.

A pedagogically useful representation is for each number at a point to be represented by a color. Then the students can imagine how they would try to show this “sea” of colors on a two-dimensional graph.

If students are new to using Mathematica, it is well worth showing them how to create a new line so they can enter new mathematical input. This requires moving the mouse between existing blocks; the pointer should change to a horizontal line.

### Wrap-up

Consider adding the Surfaces activity Equipotential Surfaces.

computer Mathematica Activity

30 min.

Static Fields 2022 (5 years)

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
• group Equipotential Surfaces

group Small Group Activity

120 min.

##### Equipotential Surfaces

Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
• group Electrostatic Potential Due to a Ring of Charge

group Small Group Activity

30 min.

##### Electrostatic Potential Due to a Ring of Charge
Static Fields 2022 (7 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Warm-Up

Students work in groups of three to use the superposition principle $V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}$ to find an integral expression for the electrostatic potential, $V(\vec{r})$, everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for $V(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• keyboard Electrostatic potential of four point charges

keyboard Computational Activity

120 min.

##### Electrostatic potential of four point charges
Computational Physics Lab II 2022

Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python using numpy and matplotlib.
• computer Visualization of Quantum Probabilities for the Hydrogen Atom

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for the Hydrogen Atom
Central Forces 2023 (3 years) Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of $n$, $\ell$, and $m$.
• group Electrostatic Potential Due to a Pair of Charges (with Series)

group Small Group Activity

60 min.

##### Electrostatic Potential Due to a Pair of Charges (with Series)
Static Fields 2022 (5 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in small groups to use the superposition principle $V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}$ to find the electrostatic potential $V$ everywhere in space due to a pair of charges (either identical charges or a dipole). Different groups are assigned different arrangements of charges and different regions of space to consider: either on the axis of the charges or in the plane equidistant from the two charges, for either small or large values of the relevant geometric variable. Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space. The whole class wrap-up discussion then compares and contrasts the results and discuss the symmetries of the two cases.
• group Electric Field Due to a Ring of Charge

group Small Group Activity

30 min.

##### Electric Field Due to a Ring of Charge
Static Fields 2022 (7 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use Coulomb's Law $\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the electric field, $\vec{E}(\vec{r})$, everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for $\vec{E}(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• assignment_ind Electrostatic Potential Due to a Point Charge

assignment_ind Small White Board Question

10 min.

##### Electrostatic Potential Due to a Point Charge
Static Fields 2022

Warm-Up

• group A glass of water

group Small Group Activity

30 min.

##### A glass of water
Energy and Entropy 2021 (2 years)

Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.
• computer Visualization of Quantum Probabilities for a Particle Confined to a Ring

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for a Particle Confined to a Ring
Central Forces 2023 (3 years)

Quantum Ring Sequence

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.

Author Information
Corinne Manogue
Learning Outcomes


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# MCQs Skewness & Kurtosis 11

This quiz contains MCQs Skewness and Kurtosis covering the shape of the distribution, Measures of central tendency such as mean, median, mode, Weighted mean, data and type of data, sources of data, Measures of Dispersion/ Variation, Standard Deviation, Variance, Range, etc. Let us start the MCQs Skewness Quiz.

1. The distribution is positively skewed if

2. If the moment Ratio $\beta_2=3$ then the distribution is

3. When a distribution is symmetrical and has one mode, the highest point on the curve is called the

4. In symmetrical distribution if $Q_1=4, Q_3=12$ then median is

5. The first three moments of a distribution about the mean $\overline{X}$ are 1, 4, and 0. The distribution is

6. For a symmetrical distribution

7. The degree to which numerical data tend to spread out about an average value is called

8. In Symmetrical distribution $Q_3-Q_1=20$, Median = 15, $Q_3$ is equal to

Basic statistics deals with the measure of central tendencies (such as mean, median, mode, weighted mean, geometric mean, and Harmonic mean) and measure of dispersion (such as range, standard deviation, and variances).

Basic statistical methods include planning and designing the study, collecting data, arranging, and numerical and graphically summarizing the collected data.

Basic statistics also used to perform statistical analysis to draw meaningful inferences.

A basic visual inspection of data using some graphical and numerical statistics may give some useful hidden information already available in the data. The graphical representation includes a bar chart, pie chart, dot chart, box plot, etc.

Companies related to finance, communication, manufacturing, charity organizations, government institutes, simple to large businesses, etc. are all examples that have a massive interest in collecting data and measuring different sorts of statistical findings. This helps them to learn from the past, noticing the trends, and planning for the future.

There are various graphical and numerical ways that are used to check the skewness of the data. Skewness is the lack of symmetry in distribution.


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# An analysis of the nichrome and if it can be determined using an equation

The information required from the analysis of a can then be readily determined using propellant under test can be determined substituting equation. Then the resistivity of the material from which the wire is made can be determined equation (3) can be with the nichrome (chromel) wire analysis use.

Resistivity is a measure of how strongly a material opposes the flow of electric current good electrical conductors have very low resistivities and good insulators have very high resistivities resistivity is denoted by the greek symbol rho (ρ) and can be determined by rearranging this formula: r. We adopt a similar equation put forth by tummala and through a steady state heat analysis coated wire can be determined at both nichrome/pdms and the. Heat released in a chemical reaction can be determined experimentally by using an measured using an adiabatic bomb calorimeter length of nichrome fuse.

In centimetres for a piece of nichrome wire 100 cm long equation, using the values from the data chart analysis of data by graphical techniques 11) r. Performance and emissions analysis and the petrol were determined using a redwood viscometer using equation 4. Ethylation and analysis procedure which is then connected to a nichrome wire coil wrapped to fit around is determined using the following equation:.

The heat capacity of the calorimeter can also be determined by burning an equation and various quantities is given in calorimetry data analysis. Forms can’t be determined by this method tion equation, a nichrome wire was used determination of hydration kinetics of sulfaguanidine anhydrate in. Moss kraig platinum, his misguided an analysis of the nichrome and if it can be determined using an equation games of jacquards contradicted the inharmonious herbert says to his chandelles callos absurdly. Charpy test - determination of impact energy using the charpy test when the specimen can absorb no more energy a transition temperature can be determined.

An analysis of the nichrome and if it can be determined using an equation humble and an analysis of the nichrome and if it can be determined using an. Introduction to pharmaceutical analysis its purity can be determined by the in this the potential is determined by using the nernst equation. X-ray fluorescence (xrf) is the is typically tungsten or nichrome of 20–60 μm wedell, r, wolff, h, handbook of practical x-ray fluorescence analysis.

Emission spectroscopy introduction: and the equation can be rearranged dip a nichrome wire into some hcl and hold it. Joule equivalent of electrical energy is to be determined in this the button on the graph toolbar to select ranges of data for analysis to highlight.

Determine the phosphorous content of a solid sample using gravimetric analysis nichrome wire to atomize the in vinegar solution can be determined. Equation 3 can be used to calculate the power provided that the resistance now using equation 2 properly, one can see why twice the power means that there. Orcaflex uses the s-n curve to calculate the damage in a fatigue analysis if needed you can the value of $\logcom a_2$ is determined .

An analysis of the nichrome and if it can be determined using an equation
Rated 4/5 based on 17 review

2018.


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# MST that contains a shortest $s,t$-path

Consider the problem in which we have an (undirected) graph $G=(V,E)$, weight function $w:E\to\mathbb N$ and a pair of vertices $s,t\in V$, and are required to determine whether there exists an MST $T$ such that $dist_T(s,t)=dist_G(s,t)$, i.e., $s$ and $t$ are connected by a shortest path.

How would you approach this problem?

Note: it seems that if there is only a single shortest path, we could simply reduce the weight of each of its edges by $1/n^2$ and find a simple MST. However, this doesn't seem to work when there are many short paths.

A generalization of the problem which is also interesting would be getting a set of terminal pairs $\{s_i,t_i\}_{i=1}^k$ which are required to be connected by a shortest path in the MST. This variant might prove to be NP-hard.

• Can you say anything about the context where you encountered this or the motivation? Is this an exercise you encountered somewhere? Is there a practical application? – D.W. May 10 '16 at 23:21
• 1. WLOG I think you can assume no edges have zero weight (otherwise contract them). 2. You can characterize the set of shortest paths as follows: there's a (directed) subgraph of $G$, call it $G'$, such that $p$ is a shortest path from $s$ to $t$ in $G$ iff it is a path in $G'$, and vice versa; in fact, $G'$ is a dag. I don't know if there's a similarly nice characterization of MST's that might be helpful here (one fact that could be useful is that the multiset of edge weights is the same for all MST's). – D.W. May 10 '16 at 23:31


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# Jacobsthal function

The ordinary Jacobsthal function ${\displaystyle j(n)}$ is defined as the smallest positive integer ${\displaystyle m}$, such that every sequence of ${\displaystyle m}$ consecutive integers contains an integer coprime to ${\displaystyle n}$. The definition refers to all integers, not just those in the range ${\displaystyle \scriptstyle [1,\,n-1]\,}$.

A048669 Jacobsthal function: maximal gap in a list of all the integers relatively prime to ${\displaystyle n}$.

{1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 3, 2, 2, 4, 2, 4, 3, 4, 2, 4, 2, 4, 2, 4, 2, 6, 2, 2, 3, 4, 3, 4, 2, 4, 3, 4, 2, 6, 2, 4, 3, 4, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 6, 2, 4, 3, 2, 3, 6, 2, 4, 3, 6, 2, 4, 2, 4, 3, 4, 3, ...}

## Asymptotic behavior

Iwaniec proved that

${\displaystyle j(n)=O(\log ^{2}(n)).\,}$

Jacobsthal conjectured that

${\displaystyle j(n)=O\left(\left({\frac {\log n}{\log \log n}}\right)^{2}\right).\,}$

## Jacobsthal function of primorial numbers

The Jacobsthal function of primorial numbers ${\displaystyle h(n)}$ is defined as the smallest positive integer ${\displaystyle m}$, such that every sequence of ${\displaystyle m}$ consecutive integers contains an integer coprime to the product of the first ${\displaystyle n}$ primes. The definition refers to all integers, not just those in the range ${\displaystyle \scriptstyle [1,\,n-1]\,}$.

${\displaystyle h(n):=j\left((p_{n})\#\right)=j\left(\prod _{i=1}^{n}p_{i}\right),\,}$

where ${\displaystyle \scriptstyle (p_{n})\#\,}$ is the ${\displaystyle n}$th primorial number (the product of the first ${\displaystyle n}$ primes).

A048670 Jacobsthal function A048669 applied to the product of the first ${\displaystyle n}$ primes (A002110).

{2, 4, 6, 10, 14, 22, 26, 34, 40, 46, 58, 66, 74, 90, 100, 106, 118, 132, 152, 174, 190, 200, 216, 234, 258, 264, 282, 300, 312, 330, 354, 378, 388, 414, 432, 450, 476, 492, 510, ...}


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# Grinding Angles

1. Feb 20, 2017

### Mister T

1. The problem statement, all variables and given/known data

Solve for $\theta$:

$\cot \theta \sin \beta + \rho \csc^2 \theta = \cos \beta$

where $0^\circ<\beta<90^\circ, \ 0^\circ<\theta<90^\circ$, and $0<\rho<1$.

2. Relevant equations

$\cot^2 x +1 = \csc^2x$, the quadratic formula.

3. The attempt at a solution
$\cot \theta = \frac{\sin \beta}{2 \rho} \pm \sqrt{\Big(\frac{\sin \beta}{2 \rho}\Big)^2 + \frac{\cos \beta}{ \rho}-1}$
This is not really a homework question, but since it looks like one I decided to post it here. It's actually a relationship between the variables involved in grinding an edge tool such as a carpenter's chisel. $\rho$ is the ratio of the tool's thickness to the diameter of the circular grindstone, so it's a positive constant, usually much smaller than unity. The issue is that a round grindstone produces a hollow grind rather than the flat surface you'd get sharpening on a flat stone, and I'm interested in the geometry of the hollow grind. $\beta$ is the bevel angle, or angle that the two edges make when they meet at the cutting edge.

$\theta$ is the angle the two edges would make if the grinding were done on a flat surface.

My solution seems to work as it gives results that match what I measure and also that a friend got using AutoCAD or some such software program. There are two roots to the solution, though. The positive root gives the value of $\theta$ but the negative root gives the supplement of $\theta-\beta$. I realize this negative root is outside the bounds of $\theta$ but it is nevertheless a curiosity to me.

So I have two questions. Can anyone derive a simpler solution and can anyone explain the negative root's value?

#### Attached Files:

• ###### upload_2017-2-20_14-18-14.png
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2. Feb 20, 2017

### haruspex

Not sure how you arrived at your equation. I got an equivalent equation, but it came out in a more useful form for finding theta.
Hint: try working it into a form that involves trig functions of 2θ rather than θ.

3. Feb 21, 2017

### Mister T

Starting with

$\cot \theta \sin \beta + \rho \csc^2 \theta = \cos \beta$

I replaced $\csc^2 \theta$ with $\cot^2 \theta +1$.

That gave me a quadratic equation in $\cot \theta$.

I then used the quadratic formula to get

$\cot \theta = \frac{\sin \beta}{2 \rho} \pm \sqrt{\Big(\frac{\sin \beta}{2 \rho}\Big)^2 + \frac{\cos \beta}{ \rho}-1}$.

I tried replacing $\cot \theta$ with $\pm \sqrt{\frac{1+\cos 2\theta}{1-\cos 2\theta}}$

and $\csc^2 \theta$ with $\frac{2}{1-\cos 2\theta}$

but that seemed to go nowhere.

4. Feb 21, 2017

### haruspex

OK. From the geometry, I got
$\rho=\cos(\beta)-\cos(2\theta-\beta)$
which results in
$\theta = \frac 12(\beta+\cos^{-1}(\cos(\beta)-\rho))$
It seems to be the same functionally.

The spurious negative value in this form comes from the ambiguity of $\cos^{-1}$. Generally these nonphysical solutions arise because the equations you wrote down apply in some other set-up also. Maybe here it represents the curvature going the other way.

5. Feb 24, 2017

### Mister T

Hmmm.... So does this mean you were looking at some triangles with angles of $\beta$ and $2 \theta$?

Can you help me reconstruct what you did?

6. Feb 24, 2017

### Mister T

I think these should be
$2\rho=\cos(\beta)-\cos(2\theta-\beta)$
which results in
$\theta = \frac 12(\beta+\cos^{-1}(\cos(\beta)-2\rho))$

I just now found this when I checked things with numerical examples.

7. Feb 24, 2017

### haruspex

Ah, yes. I had ρ as the ratio of the thickness to the radius, not to the diameter.
Let the tool edge be A, the obtuse angle (the other end of the arc) be B, the midpoint of the arc C, and the centre of the circle O. Call the upper surface of the metal AA' and the lower BB', the distance A'B' being the thickness h. Radius = AO=BO=r.
Angle A'AB is θ. The tangent to the arc at A makes angle β with AA'.
The tangent to the arc at C makes angle θ to AA', so θ-β to the first tangent.
The tangent to the arc at B makes angle θ-β to the second tangent, so 2θ-β to AA'.
It follows that AOB is 2θ-β.
Angle OAA' is β+π/2.
h=r cos(β)- r cos(2θ-β)

8. Feb 25, 2017

### Mister T

Yes! This is the piece of the geometry I needed to realize. Thank you!!

In other words, let us look at the angle between a tangent line (that is, a line tangent to the rim of the grindstone) and a line parallel to the upper surface of the tool. At the point on the rim touched by the upper surface of the tool this angle equals $\beta$ (by definition). As we move along the rim towards the lower surface of the tool this angle increases. It's value halfway along is $\theta$ (because there the tangent line is parallel to the chord that cuts across the hollow grind!). Thus it increased by $\theta-\beta$. And therefore by symmetry its value must again increase by $\theta-\beta$ as we proceed the remaining halfway to the point on the rim touched by the lower surface of the tool. Thus its value at that point is $2\theta-\beta$.

(Note that I am using t for the tool thickness rather than the h that you've used).

Hmmm... If I've followed your description it seems it would be $2\theta-2\beta$.

Last edited: Feb 25, 2017
9. Feb 25, 2017

### haruspex

Yes, sorry, I didn't mean AOB. You can see what I meant... the angle radius OB makes to the "vertical".

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# Worked example: Derivative of √(3x²-x) using the chain rule

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.C (LO)
,
FUN‑3.C.1 (EK)

## Video transcript

What I want to do in this video is start with the abstract-- actually, let me call it formula for the chain rule, and then learn to apply it in the concrete setting. So let's start off with some function, some expression that could be expressed as the composition of two functions. So it can be expressed as f of g of x. So it's a function that can be expressed as a composition or expression that can be expressed as a composition of two functions. Let me get that same color. I want the colors to be accurate. And my goal is to take the derivative of this business, the derivative with respect to x. And what the chain rule tells us is that this is going to be equal to the derivative of the outer function with respect to the inner function. And we can write that as f prime of not x, but f prime of g of x, of the inner function. f prime of g of x times the derivative of the inner function with respect to x. Now this might seem all very abstract and math-y. How do you actually apply it? Well, let's try it with a real example. Let's say we were trying to take the derivative of the square root of 3x squared minus x. So how could we define an f and a g so this really is the composition of f of x and g of x? Well, we could define f of x. If we defined f of x as being equal to the square root of x, and if we defined g of x as being equal to 3x squared minus x, then what is f of g of x? Well, f of g of x is going to be equal to-- I'm going to try to keep all the colors accurate, hopefully it'll help for the understanding. f of g of x is equal to-- where everywhere you see the x, you replace with the g of x-- the principal root of g of x, which is equal to the principal root of-- we defined g of x right over here-- 3x squared minus x. So this thing right over here is exactly f of g of x if we define f of x in this way and g of x in this way. Fair enough. So let's apply the chain rule. What is f prime of g of x going to be equal to, the derivative of f with respect to g? Well, what's f prime of x? f prime of x is equal to-- this is the same thing as x to the 1/2 power, so we can just apply the power rule. So it's going to be 1/2 times x to the-- and then we just take 1 away from the exponent, 1/2 minus 1 is negative 1/2. And so what is f prime of g of x? Well, wherever in the derivative we saw an x, we can replace it with a g of x. So it's going to be 1/2 times-- instead of an x to the negative 1/2, we can write a g of x to the 1/2. And this is just going to be equal to-- let me write it right over here. It's going to be equal to 1/2 times all of this business to the negative 1/2 power. So 3x squared minus x, which is exactly what we need to solve right over here. f prime of g of x is equal to this. So this part right over here I will-- let me square it off in green. What we're trying to solve right over here, f prime of g of x, we've just figured out is exactly this thing right over here. So the derivative of f of the outer function with respect to the inner function. So let me write it. It is equal to 1/2 times g of x to the negative 1/2, times 3x squared minus x. This is exactly this based on how we've defined f of x and how we've defined g of x. Conceptually, if you're just looking at this, the derivative of the outer thing, you're taking something to the 1/2 power. So the derivative of that whole thing with respect to your something is going to be 1/2 times that something to the negative 1/2 power. That's essentially what we're saying. But now we have to take the derivative of our something with respect to x. And that's more straightforward. g prime of x-- we just use the power rule for each of these terms-- is equal to 6x to the first, or just 6x minus 1. So this part right over here is just going to be 6x minus 1. Just to be clear, this right over here is this right over here and we're multiplying. And we're done. We have just applied the power rule. So just to review, it's the derivative of the outer function with respect to the inner. So instead of having 1/2x to the negative 1/2, it's 1/2 g of x to the negative 1/2, times the derivative of the inner function with respect to x, times the derivative of g with respect to x, which is right over there.
AP® is a registered trademark of the College Board, which has not reviewed this resource.


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# Question 2a02f

May 10, 2016

There is insufficient information to give a complete answer to this question.

#### Explanation:

Without knowing the cross-sectional area of the measuring cylinder
we can not calculate the volume of the stone and therefore we can not calculate the stone's density.

If the cross-sectional area of the measuring cylinder is $a {\text{ cm}}^{2}$
then the volume of the stone would be
color(white)("XXX") (68-40)" cm" xx a " cm"^3 = 28a " cm"^3
and its density would be
$\textcolor{w h i t e}{{\text{XXX")(50" grams")/(28a"cm}}^{3}}$

color(white)("XXX")=25/(14a) " gm/cubic cm"#


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# Reaction control on molecular level

Imagine I have a bottle with a random quantity of molecular hydrogen, and can apply exactly enough energy to break only one molecule of hydrogen. Is the reaction going to occur or the energy would be shared between other molecules in the bottle?

• It depends on how you transfer this energy. – Mithoron Jun 26 '16 at 0:01
• thank you , so you assume that if there is a way to optimise the transfer of the energie to only one molecule the reaction will occur – curiosity Jun 26 '16 at 0:06
• If you for example used single photon of appropriate energy. – Mithoron Jun 26 '16 at 0:08
• Not necessarily. A molecule hit by that photon does not necessarily dissociate. The propability for that transition would need to be calculated, depending on the initial state (e.g. temperature of the gas). – Karl Jun 26 '16 at 1:28

It is better to take a step back and consider what happens when a photon is absorbed. In the case of hydrogen, dissociation must occur from an electronically excited state as it is a homo-nuclear diatomic and does not have a permanent or varying dipole. This means that the ground state cannot absorb a photon's energy from v=0,1,2 etc to any other vibrational level. (I'm assuming absorption into continuum just above v=$\infty$ is zero.) So excitation has to be with a UV photon.
If the photon is just at dissociation, the H atoms will have effectively zero kinetic energy and so not depart from one another. In fact at this point life becomes complicated and interesting. The potential energy between two atoms increases very slowly with increasing separation as one reaches dissociation. In fact it is possible to form Rydberg atoms which may be a micron in diameter when excited to within fractions of wavenumbers of dissociation. This size is huge, about ten thousand times larger than the minimum H$_2$ atom separation, bigger than a protein and approaching the size of a bacterium! Clearly the two atoms hardly influence one another and consequently a small perturbation, a minute magnetic field for example, and certainly a nearby molecule may cause the Rydberg molecule either to break apart or recombine depending on whether some energy is added or subtracted in the interaction.
If there is not enough energy then the H$_2$ will remain in a highly excited vibrational level until it suffers a collision with another molecule and some vibrational energy is transferred to this molecule. Alternatively the electronically excited H$_2$ could fluoresce and any vibrational energy left in the ground state shared on collision with another H$_2$.
If there is far too much energy than needed to dissociate, then the H atoms ballistically fly apart in opposite directions and transfer some of their energy at each collision with other H$_2$ molecules. Possibly reaction also occurs, but H$_3$ is almost certainly unstable.


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Spectral Theory Network

 Nodes Bristol Cardiff Heriot Watt Hull Kings College, London Leicester Loughborough Members Spectral Theory Home

Leicester

M. Marletta and C. Tretter's work on spectral theory of differential operators has concentrated on three main topics: classical spectral problems for selfadjo int Hamiltonian systems; non-classical spectral problems for non-selfadjoint differential equations, including the Orr-Sommerfeld problem; and spectral theory of block operator matrices with applications in mathematical physics, e.g. to Dirac operators.

Recent work includes a new operator approach to the Orr-Sommerfeld problem on a compact interval, a complete spectral analysis of non-selfadjoint boundary eigenvalue problems for differential equations $N(y) = \lambda P(y)$ w ith $\lambda$-polynomial boundary conditions; a major development in the spectral theory of block operator matrices concerning the localisation of the spectrum by means of the new concept of quadratic numerical range; and the existence of spectral invariant subspaces and solutions of Riccati equations, with applications to Dirac operators with a potential. It also includes a complete characterisation of the Friedrichs boundary conditions for higher order self-adjoint differential operators, as well as, with Cardiff, a general analysis of spectral inclusion and spectral exactness when non-self-adjoint singular differential operators are regularized.

EPSRC-funded work in collaboration with the University of Maryland has resulted in the first library quality software for the computation of eigenvalues of higher-order differential operators. Recent extensions to non-self-adjoint problems allow, e.g., the automatic treatment of the Orr-Sommerfeld equation with $\lamb da$-nonlinear boundary conditions. In collaboration with Cardiff they have also developed the world's first algorithms for computing eigenvalues of singular problems with guaranteed error bounds.

 Last changed: 18/07/01


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Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### On regular points in Burgers turbulence with stable noise initial data

Auteur(s):

Code(s) de Classification MSC:

• 35Q53 KdV-like equations (Korteweg-de Vries, Burgers, sine-Gordon, sinh-Gordon, etc.), See also {58F07}
Résumé: We study the set of regular points (i.e. the points which have not been involved into shocks up to time $t$) for the inviscid Burgers equation in dimension 1 when initial velocity is a stable Lévy noise. We prove first that when the noise is not completely asymmetric and has index $\A \in (1/2,1)$, the set of regular points is discrete a.s. and regenerative. Then, we show that in the case of the Cauchy noise, the set of regular points is uncountable, with Minkowsky dimension 0.


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## Thursday, January 21, 2010

### Interesting broadcast group key management schemes

In this post, I focus on some really neat broadcast group key management schemes/protocols (BGKM). The idea behind the schemes are based on quite simple algebraic constructs, but they are very elegant.

In addition to other properties, a good GKM scheme should provide the following two properties.
1. Forward secrecy - a user who left the group should not be able to access new keys
2. Backward secrecy - a user who joined the group should not able to access old keys

In my opinion, these are two most difficult properties to satisfy in a group communication setting. In order to satisfy these two properties, it is required to initiate a rekeying operation (i.e. change the existing keys). There have been many GKM schemes proposing various way of doing rekey operation. What set them apart is the communication cost of the rekeying operation. Earlier GKM schemes had O(n) communication overhead, where n is the number of users in the group. Later, it was improved to incorporate a hierarchy and the communication overhead was reduced to O(log n). Further, these rekeying operations are not transparent to users; every time a user joins/leaves, other users need to update their keys. Can we make the communication overhead to be O(1) (i.e. independent of the number of users in the group? This is where the BGKM schemes fit.

BGKM schemes make the rekeying operation transparent to existing users at the expense of additional computational cost at the server which manages the BGKM scheme. There are three noteworthy BGKM schemes which I will go over in some details giving the key ideas in the rest of this post.

1. The secure lock (SL) approach based on the Chinese Remainder Theorem (CRT) [SE 1989]
2. The access control polynomial (ACP) approach based on special polynomials [INFOCOM 2008]
3. The access control vector (ACV) approach based on matrix null spaces [ICDE 2010]

1. The Secure Lock (SL) approach
(Slightly modified the original version)
Each user u_i is given a random secret value s_i and a unique secret number n_i at the time of joining. These n_i's are relatively prime in pairs. The server construct the following congruences and compute the common solution using the CRT and broadcast to the group whenever there is a leave or join.

x ~ r_1 (mod n_1)
x ~ r_2 (mod n_2)
...
x ~ r_n (mod n_n)

where ~ is the congruence symbol, r_i = K XOR s_i, K is the actual key. (r_i < n_i)

Then construct the common solution, using the CRT:
x ~ \sigma{i=1}{n} N/n_i * r_i * f_i (mod N)

where N = n_1 * n_2 * ... * n_n,
f_i ~ N/n_i (mod n_i).

Let the standard representative of the common solution is C.

A user u_r with the secret s_r and the public value n_r derives the key by (C mod n_r) XOR s_r. (Note that C mod n_r gives the value r_r)

Can a newly joined user u_r get its hand on old keys? No, because the old common solution did not consider the congruence x ~ r_r (mod n_r).

Can a user u_r who left the group access new keys? No, because the server removes the congruence x ~ r_r (mod n_r) from the CRT calculation.

2. The Access Control Polynomial (ACP) approach
Each user u_i is given a random secret value s_i at the time of joining. The server construct the following polynomial of order m+n and broadcast to the group whenever there is a leave or join.

f(x) = K + (x - H(s_1 || z))(x - H(s_2 || z))..(x - H(s_n || z))(x - H(a_1 || z))...(x - H(a_m || z))
where K - is the actual key, s_1,.., s_n are the random secret values given to n users, a_1,.., a_m are random fake values used to increase the entropy of f(x), z is a public random value, H is a hash function.

A user u_r with the secret s_r derives the key as f(H(s_r || z)).

Can a newly joined user u_r get its hand on old keys? No, because the old polynomials f(x)'s do not have (x - H(s_r || z)).

Can a user u_r who left the group access new keys? No, because the server removes (x - H(s_r || z)) from the new polynomials f(x)'s.

The scheme is quite simple. However the security of this scheme is neither formally analyzed nor proved.

3. The Access Control Vector (ACV) approach
Each user u_i is given a random secret value s_i at the time of joining. The server construct the following the matrix X of n by n + 1 and broadcast a vector created based on a random vector from the null space of X to the group whenever there is a leave or join.

X =
| 1 a_{1,1} .... a_{1,n}|
| 1 a_{2,1} .... a_{2,n}|
| ..................................|
| 1 a_{n,1} .... a_{n,n}|

where a{i,j} = H(s_i || z_i), H is a hash function, z_i's public random random values, s_i is the random secret value given to the user u_i.

The null space of such a matrix is always guranteed to be nontrivial (each row is linearly independent). Hence, there exists a non-trivial colomn vector Y such that XY = 0. The server picks a random vector Y from the null space and compute the ACV (Access Control Vector) and broadcasts.

ACV = (K, 0, ..., 0)^T + Y

A user u_r with the secret s_r derives the K by performing a dot product of ACV and KEV_r (Key Extraction Vector). A user can construct her KEV using her secret and public random z_i's as follows.

KEV_r = (1, H(s_r || z_1, ...., H(s_r || z_n))^T

Can a newly joined user u_r get its hand on old keys? No, because the vector Y in the old ACV's are not orthogonal to her KEV, dot product does not yield the key K.

Can a user u_r who left the group access new keys? No, because the server removes the corresponding row from X, now the vector Y in the new ACV's are not orthogoal to her old KEV.

The scheme is quite elegant. The security of this scheme is analyzed and proved. A downside of this approach is the computational cost at the server when the group size is large.


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