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# Income, Wealth and Statistics

Tags: income, statistics, wealth
 Mentor P: 15,576 I've read a lot of posts, and think it might be helpful to point out some facts that I think would help clarify people's arguments. 1. Income is not wealth. Using one as a proxy for the other is like using velocity as a proxy for position. A small disparity in income, acting over time, becomes a much larger disparity in wealth. 2. Demographics matters. Over the course of a lifetime, people's income and wealth changes. Typically, income rises slowly and wealth less slowly over one's career, peaking just before retirement. At that point, income drops substantially and wealth decreases more slowly. Statistics that are not age-corrected can be highly misleading. Because of this, a plot of an income percentile (or quartile etc.) does not track a given cohort of people. People move into and out of that percentile. This is even more true for wealth than income. Statements like "such and such percentile gained/lost such and such" do not tell you anything at all about what is happening to individuals. This is even more true for wealth than income. Additionally, there has been an increase in the number of illegal immigrants to the US. Illegal immigrants make up about 7% of the population (according to the Bear Stearns remittances study), up from about 1.5% fifteen years ago. Today this group makes up a large chunk of the bottom decile in both wealth and income; that was much less true in the past. Any study needs to clearly state how this was treated for it to be interpreted. 3. Accounting is important. Perhaps this is best illustrated by example. Bob taught in the public school system and has just retired. He has a $40,000 a year pension. Joe taught at a private school system and has just retired. His 401(K) is returning$40,000 a year to him. Otherwise they have the same assets. Who is richer? In most studies, Joe would be considered about a million dollars richer than Bob, even though their standards of living are identical. This is solely due to how we usually calculate wealth - we include defined contribution plans and exclude defined benefit plans. As the fraction of people with defined contributions plans instead of defined benefits plans rises, the calculated wealth disparity will rise, even if the standard of living disparity stays the same. I hope people will take this into account, and will use this to make arguments like scientists, and not like cheerleaders for their favorite team.
P: 1,123
 Quote by Vanadium 50 3. Accounting is important. Perhaps this is best illustrated by example. Bob taught in the public school system and has just retired. He has a $40,000 a year pension. Joe taught at a private school system and has just retired. His 401(K) is returning$40,000 a year to him. Otherwise they have the same assets. Who is richer? In most studies, Joe would be considered about a million dollars richer than Bob, even though their standards of living are identical. This is solely due to how we usually calculate wealth - we include defined contribution plans and exclude defined benefit plans. As the fraction of people with defined contributions plans instead of defined benefits plans rises, the calculated wealth disparity will rise, even if the standard of living disparity stays the same. I hope people will take this into account, and will use this to make arguments like scientists, and not like cheerleaders for their favorite team.
I'm glad you posted this illustration. Please consider the person who works for minimum wage and progresses to an annual income of $40,000 - but will rely on SS for retirement. Next consider the person who works just a few hours per week, has a lifelong housing subsidy under Section 8, along with food stamps, Medicaid, EITC, Make Work Pay, other child credits (possibly SSDI) - then retires on Social Security (albeit based on the min income contributions) but retains other benefits. If paid from private investments, it would take a tremendous amount of assets to provide the benefits received by the welfare person over a lifetime. The cradle to grave welfare person is comparable to a lottery winner. The person that works at minimum wage from the ground up and can't afford to save - retires on funds contributed through payroll deductions over time. Given the drain on the SS system due to expansion and an aging population - those funds may not be available at the levels the worker had hoped. I say the welfare recipient is richer than the working person - and there's greater job security.  Mentor P: 21,999 Those are issues with the stats themselves. Of equal concern to me is the common implication that wealth inequality is a measure of/proxy for poverty. This comes largely from the fallacy that wealth is a zero sum game: "the rich get richer while the poor get poorer." P: 813 ## Income, Wealth and Statistics  Quote by Vanadium 50 I've read a lot of posts, and think it might be helpful to point out some facts that I think would help clarify people's arguments. 1. Income is not wealth. Using one as a proxy for the other is like using velocity as a proxy for position. A small disparity in income, acting over time, becomes a much larger disparity in wealth. True but surely some people in that 99% have the resources to save. Consequently wealth disparity would still give some measure of inequality. Of course both income and wealth statistics should be use to get a complete picture.  2. Demographics matters. Over the course of a lifetime, people's income and wealth changes. Typically, income rises slowly and wealth less slowly over one's career, peaking just before retirement. At that point, income drops substantially and wealth decreases more slowly. Statistics that are not age-corrected can be highly misleading. I agree. Perhaps the statistics should be divided up into age groups of 5-10 years. For income, this could be 20-30,30-35,35-40,40-50,50-60,60+. For wealth the 20-30 group is largely irrelevant. Still, it is convenient to be able to communicate the overall picture with just a few numbers. We could average the statics for each age bin and then this should give a number which should be independent of age demographics.  Because of this, a plot of an income percentile (or quartile etc.) does not track a given cohort of people. People move into and out of that percentile. This is even more true for wealth than income. Statements like "such and such percentile gained/lost such and such" do not tell you anything at all about what is happening to individuals. This is even more true for wealth than income. Please explain more.  Additionally, there has been an increase in the number of illegal immigrants to the US. Illegal immigrants make up about 7% of the population (according to the Bear Stearns remittances study), up from about 1.5% fifteen years ago. Today this group makes up a large chunk of the bottom decile in both wealth and income; that was much less true in the past. Any study needs to clearly state how this was treated for it to be interpreted. So, If we remove the bottom decile from the statistics does this significantly change the results?  3. Accounting is important. Perhaps this is best illustrated by example. Bob taught in the public school system and has just retired. He has a$40,000 a year pension. Joe taught at a private school system and has just retired. His 401(K) is returning $40,000 a year to him. Otherwise they have the same assets. Who is richer? I agree. Of course we can't count all social services as income someone receives. Someone young benefits little from Medicare. Someone on welfare might not consider it much benefit to have to see a social worker or do a drug test to get their welfare dollars. Housing certainly should be counted as income. If Medicare is counted as income it should be age adjusted. Social security is paid for via pay roll taxes, so only what is in excess of these fees should be counted as additional income. I'm curious what studies you think do a good job adjusting for these accounting concerns and how different the results are.  I hope people will take this into account, and will use this to make arguments like scientists, and not like cheerleaders for their favorite team. If you want people here to do this then you should show them where they could find the raw data and evaluate their code to analyze the data. My understanding is that these forms do not let the posters do their own interpretation/analyses of the raw data. PF Gold P: 3,021  Quote by Vanadium 50 2. Demographics matters. ... People move into and out of that percentile. ... We commonly see PF posts stating the opposite is mostly the case, that except for the higher income groups most people are stuck where they start. Long term data demonstrates instead what you state here, individual incomes are mobile in all groups, especially at the bottom. Caveat: for those that are unemployed, and at the moment in the US there are many, the usual methods for increasing income are blocked.  Mentor P: 15,576 In 2006, these are the figures: Age Median Income 15-24$31K 25-34 $50K 35-44$60K 45-54 $65K 55-64$55K 65+ $28K Age Median Net Worth 20-29$8K 30-39 $44K 40-49$118K 50-59 $182K 60-69$209K
P: 813
 Quote by mheslep We commonly see PF posts stating the opposite is mostly the case, that except for the higher income groups most people are stuck where they start. Long term data demonstrates instead what you state here, individual incomes are mobile in all groups, especially at the bottom. Caveat: for those that are unemployed, and at the moment in the US there are many, the usual methods for increasing income are blocked.
I think people are saying their is less social mobility. For instance while a University Graduate may get a higher salary they will have to pay off their student debt, wait until their are older to start earning money and will be later owning a house.

Now as for trying to be scientific about it, what statistics do you think best measure social mobility.
P: 813
 Quote by Vanadium 50 In 2006, these are the figures: Age Median Income 15-24 $31K 25-34$50K 35-44 $60K 45-54$65K 55-64 $55K 65+$28K Age Median Net Worth 20-29 $8K 30-39$44K 40-49 $118K 50-59$182K 60-69 $209K Do you have a source that gives this over several years? How would you suggest adjusting this for inflation. Shouldn't each demographic see a different rate of inflation? P: 193  Quote by John Creighto Do you have a source that gives this over several years? How would you suggest adjusting this for inflation. Shouldn't each demographic see a different rate of inflation? Why would each group see a different rate of inflation? I would like to think that$.79 buys a candy bar now (or $.59 15 years ago) if I'm 29 or 59. Age shouldn't matter. P: 4,570  Quote by Vanadium 50 I've read a lot of posts, and think it might be helpful to point out some facts that I think would help clarify people's arguments. 1. Income is not wealth. Using one as a proxy for the other is like using velocity as a proxy for position. A small disparity in income, acting over time, becomes a much larger disparity in wealth. Absolutely agree, but in saying this what is your definition of wealth? To me a definition of wealth means real ownership of resources. It could be a house, a factory, a business, an IP portfolio or some other resource (even land for example). Of course one would have to clearly define the kinds of resources explicitly, but I think a rule of thumb would be assets that generate some kind of income or something similar.  3. Accounting is important. Perhaps this is best illustrated by example. Bob taught in the public school system and has just retired. He has a$40,000 a year pension. Joe taught at a private school system and has just retired. His 401(K) is returning $40,000 a year to him. Otherwise they have the same assets. Who is richer? In most studies, Joe would be considered about a million dollars richer than Bob, even though their standards of living are identical. This is solely due to how we usually calculate wealth - we include defined contribution plans and exclude defined benefit plans. As the fraction of people with defined contributions plans instead of defined benefits plans rises, the calculated wealth disparity will rise, even if the standard of living disparity stays the same. I hope people will take this into account, and will use this to make arguments like scientists, and not like cheerleaders for their favorite team. Wealth does have a correlation to living standards but they aren't completely correlated. Also value is a very weird thing. It depends on who is doing the valuation since one group could overvalue (making some people happy) and some could undervalue. The idea of some equilbirium being met in a supply-demand free market situation can sound nice, but this isn't always the situation (i.e. there can be interference in the market that causes things to be undervalued or overvalued), so you need to be careful about using some valuation statistics. Again with the definition of wealth above, if you wanted to talk about standards of living you would need to incorporate purchasing power in addition to objects of wealth (i.e. resources that are owned and generate real income). But yeah I do agree with your sentiments about the arguments, but the whole point of these arguments (in my view anyway) should be for everyone to learn a thing or two about something that they don't really know anyway, and even people that do try to be scientific about it as much as they can, still have a thing to learn from people who may not consider themselves scientific, but none-the-less have some solid experience and good points to bring to the table. Not everything is a controlled experiment especially in something like economics and quite frankly I want to hear people with a lot of different backgrounds whether they are business people, employees, or otherwise because these people have a lot to bring to the table despite not considering themselves as a "scientist". P: 66  Quote by russ_watters Those are issues with the stats themselves. Of equal concern to me is the common implication that wealth inequality is a measure of/proxy for poverty." The attached international statistics are from a NYT article. Attached Thumbnails Mentor P: 21,999  Quote by John Creighto How would you suggest adjusting this for inflation. Shouldn't each demographic see a different rate of inflation? You wouldn't adjust it for inflation: it is a single snapshot in time. Mentor P: 21,999  Quote by LaurieAG The attached international statistics are from a NYT article. Ok....what is your reason for posting them?  Meh.... I'm going to go out on a limb and assume you posted that to show a positive corellation between income inequality and poverty, contradicting my post. I'm going to go further out on that limb and assume that you weren't aware that the OECD measures poverty using income inequality. So the data you posted is an example of, not a rebuttal of, the problem I was describing. The thing is, it has been my perception that gun to their heads, most people will acknowledge an acceptance of the dictionary definition of "poverty", which is a lack of basic needs. From the OECD website: "Poverty line: An income level that is considered minimally sufficient to sustain a family in terms of food, housing, clothing, medical needs, and so on." But (again, in my perception) this definition conflicts with the ideological desire for fairness through equality of outcome that we've discussed in multiple threads on PF lately and is common in Western society today: Because wealth and income are not zero sum games, you can have both high income inequality and low poverty (or vice versa). This causes some people to stray from the definition of "poverty" that nearly everyone accepts and instead utilize a measure of poverty that attempts to anchor the definition to their ideology. I've seen a lot from the OECD that bothers me, where it appears the organization allows an ideology to influence them to try to force the data to connect their favored ideology to their stated goal. Engineering Sci Advisor HW Helper Thanks P: 6,344  Quote by mege Why would each group see a different rate of inflation? I would like to think that$.79 buys a candy bar now (or $.59 15 years ago) if I'm 29 or 59. Age shouldn't matter. Sure, the candy bars cost the same for everybody, but your total purchases over a year will show a very different pattern for a 30 year old and a 60 year old. Obvious exaamples: Many 30 years olds will be spending a lot of money on raising their kids. Most 60 year olds will not (except perhaps buying the occasional candy bar for their grandkids). A 30 year old commuting long distances to work will spend much more on fuel and transport than 70 year old retirees. Etc, etc... P: 813  Quote by russ_watters You wouldn't adjust it for inflation: it is a single snapshot in time. You missed my point. I was saying that if we are going to break down wealth inequality per demographic then we should break down inflation per demographic so the inflation rate used for each demographic represents on average the inflation rate they actually see. I discussed the fact that not everyone sees the same rate of inflation in a previous thread: http://www.physicsforums.com/showthread.php?t=544676 Older people will already own a house so to them housing inflation is not as much of an issue as it is to younger people. Poor people spend a much greater percentage of their income on food and housing so changes in housing prices will have a much large impact on lower income earners then higher income earners.  Quote by russ_watters Ok....what is your reason for posting them? "Poverty line: An income level that is considered minimally sufficient to sustain a family in terms of food, housing, clothing, medical needs, and so on." Most people will agree on this but the question is should it be sufficient income to sustain a family or a single individual. If people are dependent on others to live, individuals will have considerably less freedom, dignity and there will be much greater rates of abuse. If only enough income is provided to sustain a family then what size of family should we try to make sustainable? To what extent should the government subsidize peoples choices to have families and how does this affect people that wait until they can afford to have a family. As for food, should they be happy with a high carb/fat diet, or should they be allowed meet which is better for you like say grass fed beef and be able to afford vegetables high in nutrients that is pesticide free? To what extent should they be able to buy pre packaged meals or should they have to make everything from scratch? Mentor P: 14,440  Quote by russ_watters You wouldn't adjust it for inflation: it is a single snapshot in time. Sure you could/would. That inflation is a part of the wealth picture (inflation is a cancer that eats into the wealth picture) is one of the justifications for taxing capital gains at a lesser rate than ordinary income. P: 669  I'm going to go further out on that limb and assume that you weren't aware that the OECD measures poverty using income inequality. The OECD measures poverty using INCOME, not income inequality. Its logically possible to have a high-income gini coefficient society with everyone is above a poverty line.  The thing is, it has been my perception that gun to their heads, most people will acknowledge an acceptance of the dictionary definition of "poverty", which is a lack of basic needs. From the OECD website: "Poverty line: An income level that is considered minimally sufficient to sustain a family in terms of food, housing, clothing, medical needs, and so on." I'm not sure the OECD definition is very different from your definition. Certainly I consider food, shelter, clothing and medical 'basic needs'? Obviously, using an income level is a proxy for counting those families who cannot afford food,housing,clothing,etc, but I don't see a cheap way to collect that sort of data without using income as a proxy.  But (again, in my perception) this definition conflicts with the ideological desire for fairness through equality of outcome.... This statement is the real reason I've responded to this post. I think the phrase "equality of outcome" (and its bastard brother "equality of opportunity") is a straw-man that no one thinks is desirable. I have no real intuition for this either way- but do the more conservative members of this forum TRULY believe that liberals/progressives/more leftish people truly want everyone to have the same outcome? Because if this is the case, we have serious and fundamental miscommunications. Mentor P: 21,999  Quote by D H Sure you could/would. That inflation is a part of the wealth picture (inflation is a cancer that eats into the wealth picture) is one of the justifications for taxing capital gains at a lesser rate than ordinary income. I'm aware that inflation is a problem in general, but I'm still not even seeing where one would put an inflation adjustment in a single-point-in-time snapshot. The income data, as it was posted, says something very important to me: Age Median Income 15-24$31K
25-34 $50K 35-44$60K
45-54 $65K 55-64$55K
65+ \$28K

This says that if I were a typical 22-year old, and the US economy held status quo (flat unemployment rate, just enough growth to counter population growth, etc.) I should expect to see annual 5% over inflation raises for the next 10 years or so, which will increase my annual before tax earnings by a net 61%.

Could you tell me, specifically, where you would apply an inflation adjustment and what it might tell us?

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# Magnetic Activity Cycles in the Exoplanet Host Star epsilon Eridani

@article{Metcalfe2012MagneticAC,
title={Magnetic Activity Cycles in the Exoplanet Host Star epsilon Eridani},
author={T. Metcalfe and A. Buccino and B. P. Brown and S. Mathur and D. Soderblom and T. Henry and P. Mauas and R. Petrucci and J. Hall and S. Basu},
journal={arXiv: Solar and Stellar Astrophysics},
year={2012}
}
• T. Metcalfe, +7 authors S. Basu
• Published 2012
• Physics
• arXiv: Solar and Stellar Astrophysics
• The active K2 dwarf epsilon Eri has been extensively characterized, both as a young solar analog and more recently as an exoplanet host star. As one of the nearest and brightest stars in the sky, it provides an unparalleled opportunity to constrain stellar dynamo theory beyond the Sun. We confirm and document the 3 year magnetic activity cycle in epsilon Eri originally reported by Hatzes and coworkers, and we examine the archival data from previous observations spanning 45 years. The data show… CONTINUE READING
51 Citations

format: plain


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## Main Question or Discussion Point

from http://en.wikipedia.org/wiki/Quantum_entanglement" [Broken]

A simple limit of Bell's inequality has the virtue of being completely intuitive. If the result of three different statistical coin-flips A, B, and C have the property that:
A and B are the same (both heads or both tails) 99% of the time
B and C are the same 99% of the time
then A and C are the same at least 98% of the time. The number of mismatches between A and B (1/100) plus the number of mismatches between B and C (1/100) are together the maximum possible number of mismatches between A and C.
so far so good. makes sense. the maximum number of mismatches (between A and C) can only be (1/100) + (1/100) = 2%

yet in Quantum Entanglement (QE) the number of mismatches is, say, = 4%

however if there were hidden variables (in QE) the mismatch rate would be only 2% as calculated above. thus the hidden variable hypothesis weakens.

In quantum mechanics, by letting A, B, and C be the values of the spin of two entangled particles measured relative to some axis at 0 degrees, θ degrees, and 2θ degrees
This is a chance for folks like me to understand it better. i thought the spin value is plus 1/2 or minus 1/2.

how is another value possible?

Let's assume x-axis. We measure spin at 0 degrees, 40 degrees and 80 degrees say.....we can only get -1/2 or +1/2 how are other values possible? what does the above mean?

Choosing the angle so that ε = 0.1, A and B are 99% correlated, B and C are 99% correlated and A and C are only 96% correlated.
between entangled photons: is the correlation not the same in all directions? does only one direction, generally, have 100% correlation?

what would be the actual (relative) value of the angle where A and B would be 99% correlated?

however can we entangle photons such that say along both x-axis and y-axis (separately) the correlation (between entangled photons) is 100%?

Imagine that two entangled particles in a spin singlet are shot out to two distant locations, and the spins of both are measured in the direction A. The spins are 100% correlated (actually, anti-correlated but for this argument that is equivalent). The same is true if both spins are measured in directions B or C. It is safe to conclude that any hidden variables that determine the A,B, and C measurements in the two particles are 100% correlated and can be used interchangeably.
If A is measured on one particle and B on the other, the correlation between them is 99%
.

Why does the correlation drop from 100% to 99%?

so say at some angle (and we call it 0 degrees) along some axis there is 100% correlation but as we change the angle the correlation drops? why? anything to do with Malus law?

also here we have considered only one axis. if the correlation between two photons along, say x-axis, was 100%, what correlation can we expect along y and z axis?

If B is measured on one and C on the other, the correlation is 99%. This allows us to conclude that the hidden variables determining A and B are 99% correlated and B and C are 99% correlated. But if A is measured in one particle and C in the other, the results are only 96% correlated, which is a contradiction. The intuitive formulation is due to David Mermin, while the small-angle limit is emphasized in Bell's original article.
this is conceptually understandable from the statistical coin-flips example above.

Last edited by a moderator:

Related Quantum Physics News on Phys.org
xts
What text do you cite?

" This is a chance for folks like me to understand it better. i thought the spin value is plus 1/2 or minus 1/2. How is another value possible? "
if we speak about electrons (other fermions) you are right. If about photons: +1/-1. But that doesn't matter for the problem. A,B,C just denote the results of spin measurments if the axis of detector (polarizator, Stern-Gerlach magnet...) is such and such. A,B,C may take only two values {1,-1} for photons, {1/2, -1/2} for electrons.

Let's assume x-axis. We measure spin at 0 degrees, 40 degrees and 80 degrees say.....we can only get -1/2 or +1/2 how are other values possible? what does the above mean?

" is the correlation not the same in all directions? does only one direction, generally, have 100% correlation? "
We put one detector at some axis, and other rotated by some angle. If they are parallel - output is fully correlated. If perpendicular - there is no correlation.

" what would be the actual (relative) value of the angle where A and B would be 99% correlated? "
$corr=\cos^2(\theta)\, \rightarrow\, \theta\approx 6^{\circ}$

" however can we entangle photons such that say along both x-axis and y-axis (separately) the correlation (between entangled photons) is 100%? "
Sure. For entangled photons correlation for every possible axis is 100% if both detectors are in parallel.

" Why does the correlation drop from 100% to 99%? ... anything to do with Malus law? "
Exactly! That is Malus law.

" if the correlation between two photons along, say x-axis, was 100%, what correlation can we expect along y and z axis? "
Two would be enough, you can't measure polarisation parallel to the beam
Of course, correlation along any axis is 100% if both detectors are in parallel.

What text do you cite?

" This is a chance for folks like me to understand it better. i thought the spin value is plus 1/2 or minus 1/2. How is another value possible? "
if we speak about electrons (other fermions) you are right. If about photons: +1/-1. But that doesn't matter for the problem. A,B,C just denote the results of spin measurments if the axis of detector (polarizator, Stern-Gerlach magnet...) is such and such. A,B,C may take only two values {1,-1} for photons, {1/2, -1/2} for electrons.

Let's assume x-axis. We measure spin at 0 degrees, 40 degrees and 80 degrees say.....we can only get -1/2 or +1/2 how are other values possible? what does the above mean?

" is the correlation not the same in all directions? does only one direction, generally, have 100% correlation? "
We put one detector at some axis, and other rotated by some angle. If they are parallel - output is fully correlated. If perpendicular - there is no correlation.

" what would be the actual (relative) value of the angle where A and B would be 99% correlated? "
$corr=\cos^2(\theta)\, \rightarrow\, \theta\approx 6^{\circ}$

" however can we entangle photons such that say along both x-axis and y-axis (separately) the correlation (between entangled photons) is 100%? "
Sure. For entangled photons correlation for every possible axis is 100% if both detectors are in parallel.

" Why does the correlation drop from 100% to 99%? ... anything to do with Malus law? "
Exactly! That is Malus law.

" if the correlation between two photons along, say x-axis, was 100%, what correlation can we expect along y and z axis? "
Two would be enough, you can't measure polarisation parallel to the beam
Of course, correlation along any axis is 100% if both detectors are in parallel.
thanks xts. the link is from Wikipedia and i have inserted it into the original post now. i think you have answered 90% or more of my queries.

let me digest it and I will come back if i have anymore.

Sure. For entangled photons correlation for every possible axis is 100% if both detectors are in parallel.
i don't get this fully. the detectors are parallel to each other or to the polarization direction of the photons?

since the photons are separated, how do we figure out the x-axis of photon A corresponds to what axis of photon B?

i think i got it --- we can use the polarization direction as the base direction and thus the photons are now "direction synchronized" for experimental purpose/calculations

on a separate note:

since the photons are perfectly correlated (spin wise in same direction).

Isn't placing detectors on each of the photons at an angle is (in a sense), equivalent to

putting just one detector on just any one of the photons..........at the same angles to the polarization/chosen direction?

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xts
"the detectors are parallel to each other or to the polarization direction of the photons?"
If the detectors are parallel to each other the correlation is 100%.

"since the photons are separated, how do we figure out the x-axis of photon A corresponds to what axis of photon B?"
We measure that with our detectors. If both detectors are parallel (let's say vertical) than if both detectors click we know both photons were vertical.

"since the photons are perfectly correlated ..."


text:
Asymptotically almost periodic and asymptotically almost automorphic vector-valued generalized functions. (arXiv:1808.03163v1 [math.FA])

The main purpose of this paper is to introduce the notion of an asymptotically almost periodic ultradistribution and asymptotically almost automorphic ultradistribution with values in a Banach space, as well as to further analyze the classes of asymptotically almost periodic and asymptotically almost automorphic distributions with values in a Banach space. We provide some applications of the introduced concepts in the analysis of systems of ordinary differential equations.查看全文

Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 The main purpose of this paper is to introduce the notion of an asymptotically almost periodic ultradistribution and asymptotically almost automorphic ultradistribution with values in a Banach space, as well as to further analyze the classes of asymptotically almost periodic and asymptotically almost automorphic distributions with values in a Banach space. We provide some applications of the introduced concepts in the analysis of systems of ordinary differential equations.


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# K-omega models

(Difference between revisions)
 Revision as of 09:16, 15 May 2008 (view source) (→Introduction)← Older edit Latest revision as of 12:43, 12 October 2011 (view source) Line 2: Line 2: == Introduction == == Introduction == - The K-omega model is one of the most common [[Turbulence modeling|turbulence models]]. It is a [[Two equation models|two equation model]], that means, it includes two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. + The K-omega model is one of the most commonly used [[Turbulence modeling|turbulence models]]. It is a [[Two equation models|two equation model]], that means, it includes two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. The first transported variable is turbulent kinetic energy, $k$.  The second transported variable in this case is the specific dissipation, $\omega$. It is the variable that determines the scale of the turbulence, whereas the first variable, $k$, determines the energy in the turbulence. The first transported variable is turbulent kinetic energy, $k$.  The second transported variable in this case is the specific dissipation, $\omega$. It is the variable that determines the scale of the turbulence, whereas the first variable, $k$, determines the energy in the turbulence.

## Introduction

The K-omega model is one of the most commonly used turbulence models. It is a two equation model, that means, it includes two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy.

The first transported variable is turbulent kinetic energy, $k$. The second transported variable in this case is the specific dissipation, $\omega$. It is the variable that determines the scale of the turbulence, whereas the first variable, $k$, determines the energy in the turbulence.

To calculate boundary conditions for this model see turbulence free-stream boundary conditions.

## Miscellaneous

1. Near-wall treatment for k-omega models


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# zbMATH — the first resource for mathematics

## Merkin, John H.

Compute Distance To:
 Author ID: merkin.john-h Published as: Merkin, J. H.; Merkin, John H.; Merkin, John; Merkin, J. Homepage: https://www1.maths.leeds.ac.uk/applied/staff.dir/merkin/merkin.html External Links: Wikidata · ResearchGate · dblp
 Documents Indexed: 190 Publications since 1967
all top 5

#### Co-Authors

 33 single-authored 48 Scott, Stephen K. 30 Needham, David John 29 Pop, Ioan 16 Kalliadasis, Serafim 11 Simon, Peter L. 8 Ingham, Derek Binns 6 Brindley, John 6 Leach, Jannie A. 6 Ševčıková, Hana 5 Lazarovici, A. 5 Metcalf, Mark J. 5 Satnoianu, Razvan Alin 4 Kumaran, V. 4 Lok, Yian Yian 4 Sadiq, Muhammad Adil 4 Snita, D. 3 Finlayson, A. B. 3 Gray, Brian F. 3 Kiss, István Z. 3 Morris, Philip J. 3 Nazar, Roslinda Mohd 3 Pilling, Michael J. 3 Rennell, R. W. 3 Yannacopoulos, Athanasios N. 2 Collier, S. M. 2 Groşan, Teodor S. 2 Hasal, Pavel 2 Jivraj, N. A. 2 King, Andrew C. 2 Neufeld, Zoltán 2 Poole, A. J. 2 Sleeman, Brian D. 2 Smith, Frank T. 2 Tomlin, Alison S. 2 Trevelyan, Philip M. J. 2 Yang, Jianbo 1 Arifin, Norihan Md 1 Bachok, Norfifah 1 Dalman, M. T. 1 Godula, Tomas 1 Howe, Sally E. 1 Ishak, Anuar 1 Jaradat, Mohammad A. 1 Kaas-Petersen, Christian 1 Kay, Anthony 1 Kuiken, Hendrik K. 1 Liñán, Amable 1 McGreavy, C. 1 Mealey, L. 1 Merkin, M. R. 1 Nita, D. 1 Noszticzius, Zoltán 1 Revnic, Cornelia 1 Roberts, M. J. 1 Roşca, Alin V. 1 Roşca, Natalia C. 1 Salleh, Mohd Zuki 1 Smith, J. D. B. 1 Tamizharasi, R. 1 Taylor, Annette F. 1 Thompson, Barnaby W. 1 Vajravelu, Kuppalapalle 1 Volpert, Vitaly A. 1 Wake, Graeme Charles 1 Weidman, Patrick D.
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#### Serials

 31 Journal of Engineering Mathematics 19 IMA Journal of Applied Mathematics 15 Journal of Mathematical Chemistry 13 Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 11 Acta Mechanica 11 ZAMP. Zeitschrift für angewandte Mathematik und Physik 10 Dynamics and Stability of Systems 9 Quarterly Journal of Mechanics and Applied Mathematics 7 International Journal of Heat and Mass Transfer 7 Journal of Fluid Mechanics 7 Physica D 6 Mathematical and Computer Modelling 5 Meccanica 5 European Journal of Mechanics. B. Fluids 4 Fluid Dynamics Research 4 Philosophical Transactions of the Royal Society of London. Series A 4 Mathematical Engineering in Industry 2 Nonlinearity 2 International Journal of Mathematics and Mathematical Sciences 2 SIAM Journal on Applied Mathematics 2 Chaos 1 Computers and Fluids 1 Journal of Mathematical Biology 1 Physics Letters. A 1 Applied Mathematics and Computation 1 Numerical Heat Transfer 1 European Journal of Mechanics. B. Fluids 1 Archive of Applied Mechanics 1 Physics of Fluids 1 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Communications in Nonlinear Science and Numerical Simulation 1 Combustion Theory and Modelling 1 Forma 1 Journal of Theoretical Biology
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#### Fields

 97 Fluid mechanics (76-XX) 82 Classical thermodynamics, heat transfer (80-XX) 69 Biology and other natural sciences (92-XX) 63 Partial differential equations (35-XX) 15 Ordinary differential equations (34-XX) 13 Numerical analysis (65-XX) 11 Dynamical systems and ergodic theory (37-XX) 7 Mechanics of deformable solids (74-XX) 3 General and overarching topics; collections (00-XX) 1 Special functions (33-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Optics, electromagnetic theory (78-XX) 1 Geophysics (86-XX) 1 Systems theory; control (93-XX)

#### Citations contained in zbMATH Open

156 Publications have been cited 947 times in 574 Documents Cited by Year
On dual solutions occurring in mixed convection in a porous medium. Zbl 0597.76081
Merkin, J. H.
1986
On roll waves down an open inclined channel. Zbl 0553.76013
Needham, D. J.; Merkin, J. H.
1984
Mixed convection boundary layer flow on a vertical surface in a saturated porous medium. Zbl 0433.76078
Merkin, J. H.
1980
The effect of buoyancy forces on the boundary-layer flow over a semi- infinite vertical flat plate in a uniform free stream. Zbl 0164.56201
Merkin, J. H.
1969
Mixed convection boundary layer similarity solutions: Prescribed wall heat flux. Zbl 0667.76126
Merkin, J. H.; Mahmood, T.
1989
Triple-deck solutions for subsonic flow past humps, steps, concave or convex corners and wedged trailing edges. Zbl 0516.76043
Smith, F. T.; Merkin, J. H.
1982
A simple isothermal model for homogeneous-heterogeneous reactions in boundary-layer flow. I: Equal diffusivities. Zbl 1052.80505
Chaudhary, M. A.; Merkin, J. H.
1995
Propagating reaction-diffusion waves in a simple isothermal quadratic autocatalytic chemical system. Zbl 0707.76083
Merkin, J. H.; Needham, D. J.
1989
On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme. Zbl 0627.92026
Merkin, J. H.; Needham, D. J.; Scott, S. K.
1987
Similarity solutions in free convection boundary-layer flows adjacent to vertical permeable surfaces in porous media. I: Prescribed surface temperature. Zbl 0835.76100
Chaudhary, M. A.; Merkin, J. H.; Pop, I.
1995
The propagation of a voidage disturbance in a uniformly fluidized bed. Zbl 0543.76130
Needham, D. J.; Merkin, J. H.
1983
The development of travelling waves in a simple isothermal chemical system. II: Cubic autocatalysis with quadratic and linear decay. Zbl 0718.92029
Merkin, J. H.; Needham, D. J.
1990
Free convection with blowing and suction. Zbl 0236.76068
Merkin, J. H.
1972
The development of travelling waves in a simple isothermal chemical system with general orders of autocatalysis and decay. Zbl 0749.92025
Needham, D. J.; Merkin, J. H.
1991
Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis. Zbl 0809.92027
Metcalf, M. J.; Merkin, J. H.; Scott, S. K.
1994
The development of travelling waves in a simple isothermal chemical system. I: Quadratic autocatalysis with linear decay. Zbl 0682.92024
Merkin, J. H.; Needham, D. J.; Scott, S. K.
1989
A note on the similarity solutions for free convection on a vertical plate. Zbl 0577.76083
Merkin, J. H.
1985
The existence and stability of quasi-steady periodic voidage waves in a fluidized bed. Zbl 0609.76040
Needham, D. J.; Merkin, J. H.
1986
Spatio-temporal structures in a differential flow reactor with cubic autocatalator kinetics. Zbl 0952.76095
Satnoianu, Razvan A.; Merkin, John H.; Scott, Stephen K.
1998
Free convection boundary layers in a saturated porous medium with lateral mass flux. Zbl 0391.76061
Merkin, J. H.
1978
Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension. Zbl 0779.35057
Merkin, J. H.; Needham, D. J.
1993
A model for isothermal homogeneous-heterogeneous reactions in boundary-layer flow. Zbl 0884.76090
Merkin, J. H.
1996
On infinite period bifurcations with an application to roll waves. Zbl 0588.76024
Merkin, J. H.; Needham, D. J.
1986
Similarity solutions in axisymmetric mixed-convection boundary-layer flow. Zbl 0656.76073
Mahmood, T.; Merkin, J. H.
1988
A note on the solution of a differential equation arising in boundary- layer theory. Zbl 0532.76038
Merkin, J. H.
1984
Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system. Zbl 0780.35053
Merkin, J. H.; Needham, D. J.; Scott, S. K.
1993
A simple model for sustained oscillations in isothermal branched-chain or autocatalytic reactions in a well stirred open system. I: Stationary states and local stabilities. Zbl 0606.92029
Merkin, J. H.; Needham, D. J.; Scott, S. K.
1985
The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition. Zbl 1419.76591
Merkin, J. H.; Pop, I.
2011
Evans function analysis of the stability of non-adiabatic flames. Zbl 1068.80531
Simon, Peter L.; Kalliadasis, Serafim; Merkin, John H.; Scott, Stephen K.
2003
Free convection boundary layers near corners and sharp trailing edges. Zbl 0487.76045
Merkin, J. H.; Smith, F. T.
1982
Conjugate free convection on a vertical surface. Zbl 0963.76582
Merkin, J. H.; Pop, I.
1996
The effects of blowing and suction on free convection boundary layers. Zbl 0295.76019
Merkin, J. H.
1975
On solutions of the boundary-layer equations with algebraic decay. Zbl 0389.76022
Merkin, J. H.
1978
The effects of blowing and suction on free convection boundary layers on vertical surfaces with prescribed heat flux. Zbl 0781.76077
Chaudhary, M. A.; Merkin, J. H.
1993
Mixed convection similarity solutions on a horizontal surface. Zbl 0611.76092
Merkin, J. H.; Ingham, D. B.
1987
Combustion initiation and extinction in a 2D chaotic flow. Zbl 1032.80006
Kiss, I. Z.; Merkin, J. H.; Neufeld, Z.
2003
The effects of an electric field on an autocatalytic ionic reaction in a system with high ionic strength. Zbl 0903.35079
Merkin, J. H.; Ševčíková, H.; Šnita, D.; Marek, M.
1998
Mixed convection on a vertical circular cylinder. Zbl 0642.76100
Mahmood, T.; Merkin, J. H.
1988
Forced-convection heat transfer over a circular cylinder with Newtonian heating. Zbl 1325.76171
Salleh, M. Z.; Nazar, R.; Arifin, N. M.; Pop, I.; Merkin, J. H.
2011
The development of spatial structure in an ionic chemical system induced by applied electric fields. Zbl 0968.35119
Merkin, John; Satnoianu, Razvan; Scott, Stephen
2000
Free-convection boundary layers on vertical surfaces driven by an exothermic surface reaction. Zbl 0812.76089
Merkin, J. H.; Chaudhary, M. A.
1994
The use of algebraic sets in the approximation of inertial manifolds and lumping in chemical kinetic systems. Zbl 0885.92039
Yannacopoulos, A. N.; Tomlin, A. S.; Brindley, J.; Merkin, J. H.; Pilling, M. J.
1995
Stable pattern and standing wave formation in a simple isothermal cubic autocatalytic reaction scheme. Zbl 0847.35064
Hill, R.; Merkin, J. H.; Needham, D. J.
1995
Free-convection stagnation-point boundary layers driven by catalytic surface reactions. I: The steady states. Zbl 0801.76093
Chaudhary, M. A.; Merkin, J. H.
1994
Disjoint bifurcation diagrams in combustion systems. Zbl 0758.35083
Gray, B. F.; Merkin, J. H.; Wake, G. C.
1991
Steady states, oscillations and heat explosion in a combustion problem with convection. Zbl 1060.76127
Lazarovici, A.; Volpert, V.; Merkin, J. H.
2005
The development of nonlinear waves on the surface of a horizontally rotating thin liquid film. Zbl 0642.76047
Needham, D. J.; Merkin, J. H.
1987
Creation of spatial structure by an electric field applied to an ionic cubic autocatalator system. Zbl 0948.92026
Finlayson, A. B.; Merkin, J. H.
2000
The unsteady MHD boundary-layer flow on a shrinking sheet. Zbl 1196.76086
Merkin, J. H.; Kumaran, V.
2010
Mixed convection boundary layer flow over a permeable vertical surface with prescribed wall heat flux. Zbl 1133.76048
Ishak, A.; Merkin, J. H.; Nazar, R.; Pop, I.
2008
Forced convective structures in a differential flow reactor. Zbl 0935.35063
Satnoianu, Razvan A.; Merkin, John H.; Scott, Stephen K.
1999
Pattern formation in a coupled cubic autocatalator system. Zbl 0815.92003
Hill, R.; Merkin, J. H.
1994
Reaction-diffusion waves in coupled isothermal autocatalytic chemical systems. Zbl 0808.35164
Metcalf, M. J.; Merkin, J. H.; Scott, S. K.
1993
Mixed convection along a vertical surface: Similarity solutions for uniform flow. Zbl 1064.76610
Merkin, J. H.; Pop, I.
2002
Natural convection from a semi-infinite flat plate inclined at a small angle to the horizontal in a saturated porous medium. Zbl 0605.76103
Ingham, D. B.; Merkin, J. H.; Pop, I.
1985
Free convection on a heated vertical plate: The solution for small Prandtl number. Zbl 0702.76100
Merkin, J. H.
1989
The effect of an electric field on the local stoichiometry of front waves in an ionic chemical system. Zbl 0945.92029
Merkin, J. H.; Ševčíková, H.; Šnita, D.
2000
The collision of free-convection boundary layers on a horizontal cylinder embedded in a porous medium. Zbl 0524.76083
Ingham, D. B.; Merkin, J. H.; Pop, I.
1983
Free convection boundary layers over humps and indentations. Zbl 0537.76055
Merkin, J. H.
1983
Flow past a suddenly cooled vertical flat surface in a saturated porous medium. Zbl 0502.76106
Ingham, D. B.; Merkin, J. H.; Pop, I.
1982
A note on the similarity equations arising in free convection boundary layers with blowing and suction. Zbl 0803.76078
Merkin, J. H.
1994
Conjugate free convection on a vertical surface in a saturated porous medium. Zbl 1051.76611
Pop, I.; Merkin, J. H.
1995
An infinite period bifurcation arising in roll waves down an open inclined channel. Zbl 0593.76024
Merkin, J. H.; Needham, D. J.
1986
The effects of a complexing agent on travelling waves in autocatalytic systems with applied electric fields. Zbl 1084.35032
Merkin, J. H.; Ševčíková, H.
2005
Inhibition of flame propagation by an endothermic reaction. Zbl 1058.80006
Simon, Peter L.; Kalliadasis, Serafim; Merkin, John H.; Scott, Stephen K.
2003
On the free convection boundary layer on a vertical plate with prescribed surface heat flux. Zbl 0708.76119
Merkin, J. H.; Mahmood, T.
1990
Stationary-state solutions for coupled reaction-diffusion and temperature-conduction equations. I: Infinite slab and cylinder with general boundary conditions. Zbl 0713.35030
Brindley, J.; Jivraj, N. A.; Merkin, J. H.; Scott, S. K.
1990
Stationary-state solutions for coupled reaction-diffusion and temperature-conduction equations. II: Spherical geometry with Dirichlet boundary conditions. Zbl 0713.35031
Brindley, J.; Jivraj, N. A.; Merkin, J. H.; Scott, S. K.
1990
The development of travelling waves in a simple isothermal chemical system. III: Cubic and mixed autocatalysis. Zbl 0718.92030
Gray, P.; Merkin, J. H.; Needham, D. J.; Scott, S. K.
1990
Dynamics of a vertically falling film in the presence of a first-order chemical reaction. Zbl 1185.76372
Trevelyan, Philip M. J.; Kalliadasis, Serafim; Merkin, John H.; Scott, Stephen K.
2002
The effects of an outer flow on the unsteady free convection boundary layer near a stagnation point in a heat-generating porous medium. Zbl 1302.76063
Merkin, J. H.
2014
Quenching of flame propagation with heat loss. Zbl 1023.92049
Simon, Peter L.; Kalliadasis, Serafim; Merkin, John H.; Scott, Stephen K.
2002
Wave propagation in spatially distributed excitable media. Zbl 1048.92002
Yang, Jianbo; Kalliadasis, Serafim; Merkin, John H.; Scott, Stephen K.
2002
On the spread of morphogens. Zbl 1066.92003
Merkin, J. H.; Sleeman, B. D.
2005
Free convection boundary layers driven by exothermic surface reactions: Critical ambient temperatures. Zbl 0832.92029
Chaudhary, M. A.; Liñan, A.; Merkin, J. H.
1995
The effects of geometrical spreading in two and three dimensions on the formation of travelling wavefronts in a simple, isothermal, chemical system. Zbl 0800.35052
Needham, D. J.; Merkin, J. H.
1992
An analysis of a two-cell coupled nonlinear chemical oscillator. Zbl 0761.92048
Leach, J. A.; Merkin, J. H.; Scott, S. K.
1991
Stability of flames in an exothermic-endothermic system. Zbl 1149.80302
Simon, Peter L.; Kalliadasis, Serafim; Merkin, John H.; Scott, Stephen K.
2004
The structure of flame filaments in chaotic flows. Zbl 1006.80002
Kiss, I. Z.; Merkin, J. H.; Scott, S. K.; Simon, P. L.; Kalliadasis, S.; Neufeld, Z.
2003
The propagation of travelling waves in an open cubic autocatalytic chemical system. Zbl 0872.92022
Merkin, J. H.; Sadiq, M. A.
1996
Oscillations and waves in the Belousov-Zhabotinskii reaction in a finite medium. Zbl 0857.92023
Leach, J. A.; Merkin, J. H.; Scott, S. K.
1994
Mixed convection boundary-layer on a vertical cylinder embedded in a saturated porous medium. Zbl 0608.76071
Merkin, J. H.; Pop, I.
1987
A mathematical model for the spread of morphogens with density dependent chemosensitivity. Zbl 1078.92010
Merkin, J. H.; Needham, D. J.; Sleeman, B. D.
2005
Pattern formation through reaction and diffusion in a simple pooled- chemical system. Zbl 0702.58083
Needham, D. J.; Merkin, J. H.
1989
Travelling waves in an ionic autocatalytic chemical system with an imposed electric field. Zbl 1067.92504
Snita, D.; Sevcikova, H.; Marek, M.; Merkin, M.; Merkin, J. H.
1997
Reaction fronts in an ionic autocatalytic system with an applied electric field. Zbl 0948.92027
Merkin, J. H.; Ševčíková, H.
1999
On the structural stability of the cubic autocatalytic scheme in a closed vessel. Zbl 0673.93070
Gray, B. F.; Merkin, J. H.; Roberts, M. J.
1989
Free convection boundary layers on a vertical surface in a heat-generating porous medium. Zbl 1138.76060
Mealey, L.; Merkin, J. H.
2008
Natural convective boundary-layer flow in a heat generating porous medium with a prescribed wall heat flux. Zbl 1171.76051
Merkin, J. H.
2009
Unsteady free convective boundary-layer flow near a stagnation point in a heat-generating porous medium. Zbl 1294.76227
Merkin, J. H.
2013
Combustion in a porous material with reactant consumption: The role of the ambient temperature. Zbl 0805.34024
Sadiq, M. A.; Merkin, J. H.
1994
Multistability, oscillations and travelling waves in a product-feedback autocatalator model. II: The initiation and propagation of travelling waves. Zbl 0821.92026
Collier, S. M.; Merkin, J. H.; Scott, S. K.
1994
Two-cell coupled cubic autocatalator: The effect of the uncatalysed reaction. Zbl 0782.92023
Leach, J. A.; Merkin, J. H.; Scott, S. K.
1992
Reaction-diffusion model for combustion with fuel consumption. I: Dirichlet boundary conditions. Zbl 0775.76214
Zhang, G.; Merkin, J. H.; Scott, S. K.
1991
Mixed convection on a vertical surface with a prescribed heat flux: The solution for small and large Prandtl numbers. Zbl 0749.76064
Merkin, J. H.; Pop, I.; Mahmood, T.
1991
A simple isothermal model for homogeneous-heterogeneous reactions in boundary-layer flow. II: Different diffusivities for reactant and autocatalyst. Zbl 1052.80506
Chaudhary, M. A.; Merkin, J. H.
1995
The suppression of an exothermic branched-chain flame through endothermic reaction and radical scavenging. Zbl 1013.80006
Gray, B. F.; Kalliadasis, S.; Lazarovici, A.; Macaskill, C.; Merkin, J. H.; Scott, S. K.
2002
Free convection stagnation point boundary layers driven by catalytic surface reactions. II: Times to ignition. Zbl 0876.76081
Chaudhary, M. A.; Merkin, J. H.
1996
A simple model for sustained oscillations in isothermal branched-chain or autocatalytic reactions in a well stirred open system. II: Limit cycles and non-stationary states. Zbl 0606.92030
Merkin, J. H.; Needham, D. J.; Scott, S. K.
1985
Free convection from a vertical cylinder embedded in a saturated porous medium. Zbl 0608.76075
Merkin, J. H.
1986
An anomalous self-similar solution for a laminar wall jet flowing up a heated vertical wall. Zbl 07167122
Merkin, J. H.; Weidman, P. D.
2020
On an equation arising in the boundary-layer flow of stretching/shrinking permeable surfaces. Zbl 1443.76123
Merkin, J. H.; Pop, I.
2020
Stagnation point flow past a stretching/shrinking sheet driven by arrhenius kinetics. Zbl 1427.76059
Merkin, J. H.; Pop, I.
2018
Mixed convection boundary-layer flow near the lower stagnation point of a horizontal circular cylinder with a second-order wall velocity condition and a constant surface heat flux. Zbl 1328.35179
Roşca, Natalia C.; Roşca, Alin V.; Merkin, John H.; Pop, Ioan
2015
MHD oblique stagnation-point flow towards a stretching/shrinking surface. Zbl 1336.76045
Lok, Y. Y.; Merkin, J. H.; Pop, I.
2015
The effects of an outer flow on the unsteady free convection boundary layer near a stagnation point in a heat-generating porous medium. Zbl 1302.76063
Merkin, J. H.
2014
Unsteady free convective boundary-layer flow near a stagnation point in a heat-generating porous medium. Zbl 1294.76227
Merkin, J. H.
2013
A similarity solution for the flow and heat transfer over a moving permeable flat plate in an external free stream: case of strong injection. Zbl 1293.76119
Merkin, J. H.; Bachok, N.; Jaradat, M. A.; Pop, I.
2013
Mixed convection boundary-layer flow on a horizontal flat surface with a convective boundary condition. Zbl 1293.74374
Grosan, T.; Merkin, J. H.; Pop, I.
2013
The development of forced convection heat transfer near a forward stagnation point with Newtonian heating. Zbl 1268.76059
Merkin, J. H.; Nazar, R.; Pop, I.
2012
On thin film flow of a third-grade fluid down an inclined plane. Zbl 1293.76038
Kumaran, V.; Tamizharasi, R.; Merkin, J. H.; Vajravelu, K.
2012
Natural convective boundary-layer flow in a heat generating porous medium with a constant surface heat flux. Zbl 1291.76296
Merkin, J. H.
2012
The unsteady heat transfer due to a heat source in an MHD stretching sheet flow. Zbl 1293.76174
Merkin, John H.; Kumaran, V.
2012
The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition. Zbl 1419.76591
Merkin, J. H.; Pop, I.
2011
Forced-convection heat transfer over a circular cylinder with Newtonian heating. Zbl 1325.76171
Salleh, M. Z.; Nazar, R.; Arifin, N. M.; Pop, I.; Merkin, J. H.
2011
The unsteady MHD boundary-layer flow on a shrinking sheet. Zbl 1196.76086
Merkin, J. H.; Kumaran, V.
2010
Natural convective boundary-layer flow in a heat generating porous medium with a prescribed wall heat flux. Zbl 1171.76051
Merkin, J. H.
2009
Travelling waves in the Oregonator model for the BZ reaction. Zbl 1185.35299
Merkin, J. H.
2009
Mixed convection flow near an axisymmetric stagnation point on a vertical cylinder. Zbl 1160.76044
Revnic, Cornelia; Grosan, Teodor; Merkin, John; Pop, Ioan
2009
Mixed convection boundary layer flow over a permeable vertical surface with prescribed wall heat flux. Zbl 1133.76048
Ishak, A.; Merkin, J. H.; Nazar, R.; Pop, I.
2008
Free convection boundary layers on a vertical surface in a heat-generating porous medium. Zbl 1138.76060
Mealey, L.; Merkin, J. H.
2008
Free convective boundary-layer flow in a heat-generating porous medium: similarity solutions. Zbl 1138.76032
Merkin, J. H.
2008
Travelling waves in autocatalytic chemical systems with decay: bounds on existence. Zbl 1120.92050
Merkin, J. H.
2007
Steady states, oscillations and heat explosion in a combustion problem with convection. Zbl 1060.76127
Lazarovici, A.; Volpert, V.; Merkin, J. H.
2005
The effects of a complexing agent on travelling waves in autocatalytic systems with applied electric fields. Zbl 1084.35032
Merkin, J. H.; Ševčíková, H.
2005
On the spread of morphogens. Zbl 1066.92003
Merkin, J. H.; Sleeman, B. D.
2005
A mathematical model for the spread of morphogens with density dependent chemosensitivity. Zbl 1078.92010
Merkin, J. H.; Needham, D. J.; Sleeman, B. D.
2005
The effects of a complexation reaction on travelling wave-fronts in a quadratic autocatalytic system. Zbl 1083.92046
Needham, D. J.; Leach, J. A.; Merkin, J. H.
2005
The effect of an electric field on reaction fronts in autocatalytic systems. Zbl 1134.35372
Merkin, J. H.
2005
The effect of a radical scavenger on the propagation of flames in an exothermic-endothermic system. Zbl 1217.80136
Simon, Peter L.; Scott, Stephen K.; Kalliadasis, Serafim; Merkin, John H.
2005
Stability of flames in an exothermic-endothermic system. Zbl 1149.80302
Simon, Peter L.; Kalliadasis, Serafim; Merkin, John H.; Scott, Stephen K.
2004
Electric field effects on travelling waves in the Oregonator model for the Belousov-Zhabotinsky reaction. Zbl 1060.92008
Kiss, I.; Merkin, J. H.; Scott, S. K.; Simon, P. L.
2004
The propagation and inhibition of an exothermic branched-chain flame with an endothermic reaction and radical scavenging. Zbl 1125.80008
Lazarovici, A.; Kalliadasis, S.; Merkin, J. H.; Scott, S. K.; Simon, P. L.
2004
On the structure of the spectra for a class of combustion waves. Zbl 1170.80334
Simon, Peter L.; Scott, Stephen K.; Kalliadasis, Serafim; Merkin, John H.
2004
Evans function analysis of the stability of non-adiabatic flames. Zbl 1068.80531
Simon, Peter L.; Kalliadasis, Serafim; Merkin, John H.; Scott, Stephen K.
2003
Combustion initiation and extinction in a 2D chaotic flow. Zbl 1032.80006
Kiss, I. Z.; Merkin, J. H.; Neufeld, Z.
2003
Inhibition of flame propagation by an endothermic reaction. Zbl 1058.80006
Simon, Peter L.; Kalliadasis, Serafim; Merkin, John H.; Scott, Stephen K.
2003
The structure of flame filaments in chaotic flows. Zbl 1006.80002
Kiss, I. Z.; Merkin, J. H.; Scott, S. K.; Simon, P. L.; Kalliadasis, S.; Neufeld, Z.
2003
Mixed convection along a vertical surface: Similarity solutions for uniform flow. Zbl 1064.76610
Merkin, J. H.; Pop, I.
2002
Dynamics of a vertically falling film in the presence of a first-order chemical reaction. Zbl 1185.76372
Trevelyan, Philip M. J.; Kalliadasis, Serafim; Merkin, John H.; Scott, Stephen K.
2002
Quenching of flame propagation with heat loss. Zbl 1023.92049
Simon, Peter L.; Kalliadasis, Serafim; Merkin, John H.; Scott, Stephen K.
2002
Wave propagation in spatially distributed excitable media. Zbl 1048.92002
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2000
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2000
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1999
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1999
Travelling waves in an ionic quadratic autocatalytic chemical system. Zbl 0991.92042
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1999
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1998
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1998
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1998
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1997
A model for isothermal homogeneous-heterogeneous reactions in boundary-layer flow. Zbl 0884.76090
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1996
Conjugate free convection on a vertical surface. Zbl 0963.76582
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1996
The propagation of travelling waves in an open cubic autocatalytic chemical system. Zbl 0872.92022
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1996
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1996
A reaction-diffusion model for autocatalytic polymerization. II: The initial value problem. Zbl 0852.92030
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1996
Homogeneous-heterogeneous reactions in boundary layer flow: Effects of loss of reactant. Zbl 0859.76077
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1996
Analysis of the bromate-ferroin clock reaction. Zbl 0903.92038
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1996
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1996
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1995
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1995
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1995
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1995
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1994
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# Creo integration donot open the model after revised

Experimenter

Creo:4.0

TC:11.2.1

TC&Creo: 11.5.0

I can not open the 3D model after revised a item from "A" to "B";  a error  is reported; Refer to the attachments.

Found this question:

the two naming reference's names of the 3D model  is the same between "A" and "B" ; But the two datasets  of the 3D model is different ;

Changed  the deep copy rule "copyasobject",and all so .

I maybe still some configuration needed , but don't know.


text:
Assymmetric Risk Call Option

I'm reading in Radu Tunaru's book "Model Risk in Financial Markets" and he argues that two parties in the ﬁnancial contract do not have the same magnitude of exposure to model risk. The reason for that is the following:

"call option contracts have no downside and variation comes from the upside and also because the process used for modelling the underlying index cannot become non-positive"

Can someone clarify what he means by "call option contracts have no downside" and "variation comes from the upside"? What variation is he talking about?

And why is the fact that the underlying can't be negative an influence for the risk?

If you have bought an option contract, your payoff is $$(S-K)^+\geq0$$ where $$S$$ is your underlying and $$K$$ the strike. Therefore you can only receive a positive cash flow: there is "no downside". I guess he means that, because you can only get a positive payoff, it matters less whether you got it right when modelling the option price: anyway, you can't lose money once you've paid the premium.
By "variation comes from the upside", I guess he means that, the more in-the-money an option is (so for a call option, the more $$S$$ is above $$K$$), the more your delta is closer to 1. So the variation in (absolute) value from your option is greater the greater you are in-the-money. On the other hand, depending on how out-the-money your option is, then a change in the underlying won't really have much impact on the option value. So: when you're losing, modelling risk accurately is not necessary because anyway the variation in the option value is low; and when the variation is large, it doesn't matter that much either because anyway you're in a winning situation.
• It also seems to ignore mark-to-market accounting rules: even if you hold a call option which can only pay a positive cash flow at maturity, in the meantime you hold that option in your balance sheet. The underlying value $$S$$ can vary; this will make the value of your option vary; and this will be reflected in your balance sheet and income statement. I think it's pretty important to have an accurate model to represent those values.


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## Answers to exercises

##### Exercise 1
$$\lim\limits_{x\to 0} \dfrac{\tan x}{x} = \lim\limits_{x\to 0}\Bigl(\dfrac{\sin x}{x}\times \dfrac{1}{\cos x}\Bigr) = 1\times 1 = 1$$.
##### Exercise 2
$$\lim\limits_{x\to 0} \dfrac{\sin 3x+\sin 7x}{5x} = \lim\limits_{x\to 0} \Bigl(\dfrac{3}{5}\times\dfrac{\sin 3x}{3x}\Bigr) + \lim\limits_{x\to 0} \Bigl(\dfrac{7}{5}\times\dfrac{\sin 7x}{7x}\Bigr) = \dfrac{3}{5}+\dfrac{7}{5} = 2$$.
##### Exercise 3
1. Multiplying top and bottom by $$1+\cos x$$ gives $\lim_{x\to 0} \dfrac{1-\cos x}{x^2} = \lim_{x\to 0} \dfrac{\sin^2x}{x^2(1+\cos x)} = \Bigl(\lim_{x\to 0}\dfrac{\sin x}{x}\Bigr)^2 \times \Bigl(\lim_{x\to 0} \dfrac{1}{1+\cos x}\Bigr) = \dfrac{1}{2}.$
2. Hence, for small $$x$$, we have $\dfrac{1-\cos x}{x^2} \approx \dfrac{1}{2},$ and so $$\cos x \approx 1-\dfrac{1}{2}x^2$$.
##### Exercise 4
1. We use two compound-angle formulas: \begin{align*} \cos(A-B) &= \cos A\,\cos B + \sin A\,\sin B \\ \cos(A+B) &= \cos A\,\cos B - \sin A\,\sin B. \end{align*} Let $$C=A+B$$ and $$D=A-B$$. Then $$A=\dfrac{1}{2}(C+D)$$ and $$B=\dfrac{1}{2}(C-D)$$. So it follows that $\cos C - \cos D = -2\,\sin A\,\sin B = -2\,\sin\Bigl(\dfrac{C+D}{2}\Bigr)\,\sin\Bigl(\dfrac{C-D}{2}\Bigr).$
2. \begin{align*} \dfrac{d}{dx} (\cos x) &= \lim_{h\to 0} \dfrac{\cos(x+h)-\cos x}{h} \\ &= \lim_{h\to 0} \dfrac{-2\,\sin(x+\dfrac{h}{2})\,\sin\dfrac{h}{2}}{h} \\ &= -\Bigl(\lim_{h\to 0} \,\sin\bigl(x+\dfrac{h}{2}\bigr)\Bigr) \times \Bigl(\lim_{h\to 0} \dfrac{\sin\dfrac{h}{2}}{\dfrac{h}{2}}\Bigr) = -\sin x. \end{align*}
##### Exercise 5
$$\dfrac{d}{dx}(\tan x) = \dfrac{d}{dx}\Bigl( \dfrac{\sin x}{\cos x} \Bigr) = \dfrac{\cos^2x +\sin^2x}{\cos^2x} = \dfrac{1}{\cos^2 x} = \sec^2 x$$.
##### Exercise 6
1. $$\dfrac{d}{dx} (\operatorname{cosec} x) = \dfrac{d}{dx}(\sin x)^{-1} = -1(\sin x)^{-2} \times \cos x = -\dfrac{\cos x}{\sin^2 x} = -\operatorname{cosec} x\,\cot x$$.
2. $$\dfrac{d}{dx} (\sec x) = \dfrac{d}{dx}(\cos x)^{-1} = -1(\cos x)^{-2}\times -\sin x = \dfrac{\sin x}{\cos^2 x} = \tan x\,\sec x$$.
3. $$\dfrac{d}{dx} (\cot x) = \dfrac{d}{dx}\Bigl(\dfrac{\cos x}{\sin x}\Bigr) = \dfrac{-\sin^2x-\cos^2x}{\sin^2x}= -\operatorname{cosec}^2 x$$.
##### Exercise 7
\begin{align*} \dfrac{d}{dx} \log_e\Bigl(\dfrac{1+\sin x}{\cos x}\Bigr) &= \dfrac{d}{dx}\bigl(\log_e(1+\sin x) - \log_e(\cos x)\bigr) \\ &= \dfrac{\cos x}{1+\sin x} + \dfrac{\sin x}{\cos x} = \dfrac{\cos^2x + \sin^2x + \sin x}{(1+\sin x)\,\cos x} \\ &= \dfrac{1+\sin x}{(1+\sin x)\,\cos x} = \dfrac{1}{\cos x} = \sec x. \end{align*}
##### Exercise 8

The derivative of the function is

$\dfrac{dy}{dx} = \dfrac{d}{dx} \Bigl(\dfrac{\sin x}{3+4\cos x}\Bigr) = \dfrac{3\cos x + 4}{(3+4\cos x)^2}.$

Since $$\cos x \geq -1$$, it follows that $$\dfrac{dy}{dx} > 0$$ whenever $$3 + 4\cos x \neq 0$$. Hence, $$y$$ is an increasing function wherever it is defined.

##### Exercise 9

The perimeter of the triangle is $$P = 2a + 2a\cos \theta = 2a(1+\cos\theta)$$, so

$a = \dfrac{P}{2(1+\cos\theta)}.$

The area of the triangle is

$A = \dfrac{1}{2}\,\bigl(2a\cos \theta\bigr)\,\bigl(a\,\sin \theta\bigr) = a^2\cos\theta\,\sin\theta = \dfrac{1}{2}a^2\sin 2\theta.$

Substituting for $$a$$ gives

$A = \dfrac{P^2\sin2\theta}{8(1+\cos\theta)^2}.$

We want to find $$\theta$$ in the range 0 to $$\dfrac{\pi}{2}$$ such that $$\dfrac{dA}{d\theta} = 0$$. By the quotient rule, if $$\dfrac{dA}{d\theta} = 0$$, then

$16P^2\cos2\theta\,(1+\cos\theta)^2 + 16P^2\sin2\theta\,(1 + \cos\theta)\,\sin\theta = 0.$

It follows after some calculation that $$(1 + \cos\theta)^2(2\cos\theta - 1) = 0$$. So $$\cos\theta = -1$$ or $$\cos\theta = \dfrac{1}{2}$$. Hence, for $$0 < \theta < \dfrac{\pi}{2}$$, the only solution is $$\theta = \dfrac{\pi}{3}$$. The triangle is equilateral.

##### Exercise 10
1. $$\displaystyle \int_{\frac{\pi}{6}}^\frac{\pi}{3} (\sin 2x + \cos 3x)\, dx = \Bigl[-\dfrac{1}{2}\cos 2x + \dfrac{1}{3} \sin 3x \Bigr]_{\frac{\pi}{6}}^\frac{\pi}{3} = \dfrac{1}{6}$$.
2. We have $$\dfrac{d}{dx} (x\,\sin x) = \sin x + x\,\cos x$$. Hence $\int_0^{\frac{\pi}{2}} x\,\cos x\, dx = \Bigl[x\,\sin x\Bigr]_{0}^\frac{\pi}{2} - \int_0^{\frac{\pi}{2}} \sin x\, dx = \dfrac{\pi}{2}-1.$
3. $$\displaystyle \int \tan^2 x\, dx = \int \bigl(\sec^2 x - 1\bigr)\, dx = \tan x - x + C$$.
##### Exercise 11

We have $$\sin^2\theta = \dfrac{1}{2}(1-\cos 2\theta)$$, and so

$\int \sin^2\theta\, d\theta = \dfrac{1}{2}\Bigl(\theta - \dfrac{1}{2}\sin 2\theta\Bigr) + C.$
##### Exercise 12

Since the period is $$\pi \sqrt{2}$$, we have $$\dfrac{2\pi}{n} = \pi\sqrt{2}$$, so $$n=\sqrt{2}$$. The amplitude is $$C=2$$. Hence, from $$v^2=n^2(C^2-x^2)$$, when $$x=0$$, $$v=\pm 2\sqrt{2}$$. Thus the speed is $$2\sqrt{2}$$ m/s.

##### Exercise 13

1. This restricted function has domain $$[0,\pi]$$ and range $$[-1,1]$$. So its inverse has domain $$[-1,1]$$ and range $$[0,\pi]$$.
2. Graphs of inverse cosine of minus x and minus inverse cosine of x showing one is a translation of the other.

3. Detailed description
Since $$\cos \dfrac{\pi}{3} = \dfrac{1}{2}$$, we have $$\cos^{-1}(\dfrac{1}{2}) = \dfrac{\pi}{3}$$ and so $$\cos^{-1}(-\dfrac{1}{2}) = \pi - \cos^{-1}(\dfrac{1}{2}) = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}$$.
4. Let $$y = \cos^{-1}x$$. Then $$x = \cos y$$, and so $\dfrac{dx}{dy} = -\sin y = -\sqrt{1 - \cos^2 y} = -\sqrt{1 - x^2}.$ (Note that $$0 \leq y \leq \pi$$ and so $$0 \leq \sin y \leq 1$$.) Hence, $\dfrac{dy}{dx} = -\dfrac{1}{\sqrt{1 - x^2}}.$
5. Let $$f(x) = \sin^{-1}x + \cos^{-1}x$$, for $$x \in [-1, 1]$$. Then $f'(x) = \dfrac{1}{\sqrt{1-x^2}}- \dfrac{1}{\sqrt{1-x^2}} = 0.$ So $$f(x) = C$$, for some constant $$C$$. Now $$f(0)=\dfrac{\pi}{2}$$ and so $$\sin^{-1} x + \cos^{-1} x = \dfrac{\pi}{2}$$.


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SEARCH HOME
 Math Central Quandaries & Queries
 Question from Hope, a student: cos (theta / 30 = 1 I am very confused as to how to solve it. Can you help?

Hi Hope,

I think you know that $\cos 0 = 1$ so one solution to $\cos \large \frac{\theta}{30} = 1$ is $\large \frac{\theta}{30} = 0.$ But if $\large \frac{\theta}{30} = 0$ then $\theta = 0$ and hence one solution is $\theta = 0.$

This is not the only solution however since $\cos(360^o)$ is also $1$ so $\large \frac{\theta}{30} = 360^o$ gives $\cos \large \frac{\theta}{30} = 1.$ What is the value of $\theta?$ Are there other solutions?

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.


text:
# Black building

Keith built building with a rectangular shape 6.5 m × 3.9 m. Calculate how much percent exceeded the limit 25 m2 for small building. Building not built in accordance with the law is called "black building". Calculate the angle that the walls were clenching each other to meet limit footprint limit 25 m2 with given lengths of the walls, if it stands angular.

p =  1.4 %
α =  80.5 °

### Step-by-step explanation:

$p=100\frac{6.5\cdot 3.9-25}{25}=1.4\mathrm{%}$
$\alpha =\mathrm{arcsin}\left(\frac{25}{6.5\cdot 3.9}\right)=80.{5}^{\circ }$

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How many CZK we pay for lining the perimeter walls of the bathroom with rectangular shape with dimensions 3.5 m and 4 m, high 1.5 m if 1 square m tile cost 300 CZK?
• The hall
The hall had a rectangular ground plan one dimension 20 m longer than the other. After rebuilding the length of the hall declined by 5 m and the width has increased by 10 m. Floor area increased by 300 m2. What were the original dimensions of the hall?
• Office
Office building was built in the shape of a regular hexagon inscribed in a circle with a radius of 12 m. The height of the walls is 7m. How much CZK cost plastering the walls of the building, if per 1 m square cost CZK 400?
• The mast
A 40 m high mast is secured in half by eight ropes of 25 m long. The ends of the ropes are equidistant from each other. Calculate this distance.
• Rectangular land
On a rectangular land with dimensions of 35 m and 18.5 m is a house with a square floor plan with a side of 14 m. What % of the land is not occupied?
• Orchard
Route passes trapezoidal orchard perpendicular to the parallel sides. It is 80 cm wide. The lengths of the bases are in the ratio 5:3 and the length of the longer base to the length of the path is in the ratio 5:6. How many square meters occupies the rout
• Rectangular garden
Rectangular garden has a length of 48.7 m, a width of 6.3 meters shorter than the length. How much mesh should be bought for its fencing if the gate is 2.9 m long and the gate 1.1 m? What is the area of the garden?
• Uphill garden
I have a garden uphill, increasing from 0 to 4.5 m for a length of 25 m, how much is the climb in percent?
• Flower bed
The park has a rectangular flower bed with dimensions of 3.2m and 1.5m. How many rose bushes will we plant in the flower bed if 20 dm2 are needed for one bush?
• The room
The room has a cuboid shape with dimensions: length 50m and width 60dm and height 300cm. Calculate how much this room will cost paint (a floor is not painted) if the window and door area is 15% of the total area and 1m2 cost 15 euro.
• Trapezoid MO
The rectangular trapezoid ABCD with the right angle at point B, |AC| = 12, |CD| = 8, diagonals are perpendicular to each other. Calculate the perimeter and area of the trapezoid.
• Roof cover
Above the pavilion with a square ground plan with a side length of a = 12 m is a pyramid-shaped roof with a height v = 4.5 m. Calculate how much m2 of sheet metal is needed to cover this roof if 5.5% of the sheet we must add for joints and waste.
• Building
The building I focused at an angle 30°. When I moved 5 m building I focused at an angle 45°. What is the height of the building?


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## Dimmable night light

I’m not a big fan of using technology just for the sake of it. We used to have a wireless battery-powered door bell, but it was unreliable: once, a heavy-handed delivery driver pushed in the rubber button so far that it got stuck under the case. Then every caller for a week after that just pressed it anyway, and assumed it was working and that the lack of sound must be because the buzzer couldn’t be heard from outside the house. So I replaced it with a ship bell, which has the advantage of providing instant ear-splitting feedback to the user as to whether it is working or not. It has had 100% uptime over several years of operation.

So it is with dismay that I observe the trend towards push-button or touch controls on everything. My wife needs a dimmable bedside lamp: bright enough to help her breastfeed in bed at night, but not so bright so as to interfere with sleeping. I went shopping and found a wide variety of inappropriate designs. For example, some have only a single button and need to be cycled through the brightest setting in order to turn them off. How can a designer make such a thing and still take pride in their work? I know potentiometers are expensive, costing $1 or so, but even at the top end of the market, with lamps costing$200, the best they can do is put half a dozen touch switches on them, giving you bidirectional control over brightness and cycling through colour temperature settings. But they are still harder to use than an old-fashioned knob, and with an inappropriate minimum brightness for my application.

Hazel threw an LED torch down the stairs and broke it. Angela asked me to fix it. Well, it only cost \$4 from the Reject Shop in the first place, so my repairs weren’t very careful. I opened up the aluminium case lengthwise with a Dremel cut-off wheel and found the problem — the circuit board had been soldered to the case, and that solder joint had broken. Oh well, every failure is an opportunity, right?

For days, as I walked around the house, I looked at every item for its potential as a lamp. Eventually I settled on the acrylic case for an old iPod Shuffle. I put the iPod itself in the bin.

It provides indirect lighting: the upward-facing LEDs put a spot on the ceiling, lighting up the room without excessive glare for people lying in bed.

Bill of materials:

• Salvaged LED array
• 10kΩ linear potentiometer
• 100Ω resistor
• BC548 NPN transistor
• 1N914 type power diode
• iPod shuffle case
• 2.1mm DC barrel jack
• Universal switch-mode power supply set to 9V

All items were from my stock or salvaged.

To dim an LED with a potentiometer, you need to control the current rather than the voltage. If you control the voltage, then you’ll get nothing at all until it reaches a certain threshold, and then the brightness will rise exponentially until something overheats. So I adapted a simple voltage-controlled current source from Horowitz & Hill (2nd edition) to provide roughly linear current control as you turn the knob. Biasing the lower end with a diode brings the zero current point to approximately the zero position of the potentiometer. In practice, at the lowest setting, there is a very slight glow from the LED array which is only visible in a very dark room. I’ll call that a feature, to help you find the knob at night.

## Battery power?

Update: A friend asked me about battery power. The torch the LED array came from used 3 AAA batteries, so about 4.5V, but with this inefficient current source the supply voltage needs to be doubled, since at full brightness, the voltage drop across the resistor R2 is as much as across the LED. And even if you used a current source that could go from rail to rail, you would still waste up to half the power.

A better solution is to power it from its original 4.5V, but with PWM. No doubt something could be cooked up with 555 timers, but they’re not really my style, I don’t have them in stock. I do have microcontrollers, and a microcontroller solution for this would have some nice advantages.

So I would use an ATtiny44, with a circuit very similar to my season clock (which, by the way, is still running on its original AA batteries, almost 3 years later). I would measure the potentiometer voltage with the microcontroller’s ADC, and when it drops below a certain voltage, go into sleep mode, waking say once every 100ms. I would power the potentiometer from a digital output, saving 450µA in sleep mode, just turning it on long enough to measure it. Maximum DC output for this chip is 40mA, so an outboard transistor may be needed, depending on choice of the LED array.


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# What will happen to a DC series motor if the series windings are shunted by a variable resistance?

This question was previously asked in
UPRVUNL TG 2 (Electrician) Shift 2 Official Paper (Held on 15 July 2021)
View all UPRVUNL TG2 Papers >
1. The speed of the motor will decrease
2. The speed of the motor will increase
3. The iron losses in the motor will increase
4. The leakage flux will be reduced to zero

Option 2 : The speed of the motor will increase
Free
General Knowledge Free Mock Test
3.5 K Users
20 Questions 20 Marks 15 Mins

## Detailed Solution

Speed control of DC series motor by field diverter method:

The speed of a DC series motor is given by:

E α kϕω

$$\omega \space \alpha \space {1\over \phi}$$

Therefore, the speed is inversely proportional to the flux.

• In this method, the series winding is shunted by variable resistance R. Due to diverting the field current through the variable resistance path, the flux Φ reduces.
• Any desired amount of current can be passed through the diverter by adjusting its resistance.
• This will result in an increase in the speed of the motor.
• Finally, we conclude that the speed of the DC series motor increases above its normal speed due to the field diverter.


text:
Need help with writing proofs for matrices that are row equivalent

I have two proofs I do not know how to start:

Q1: Prove that if A is row-equivalent to B and B is row-equivalent to C, then A is row-equivalent to C.

Q2: Let A be a nonsingular matrix. Prove that if B is row-equivalent to A, then B is also nonsingular.

"Getting started: To prove that A is row-equivalent to C, you have to find elementary matrices $$E_1E_2… E_k$$ such that A = $$E_k… E_2E_1C$$.

I. Begin by observing that A is row-equivalent to B and B is row equivalent to C.

II. This means that there exist elementary matrices $$F_1F_2… F_n$$ and $$G_1 G_2… G_m$$ such that A = $$F_nF_{n-1}… F_1B$$ and B = $$G_mG_{m-1}… G_1C$$.

III. Combine the matrix equations from step II."

I am unsure what row equivalence means and have no idea how A having an inverse makes B have an inverse if it is row equivalent; I've always froze up at proof questions for some reason so please do not be afraid to be verbose; I'd like to be able to explain and understand this stuff someday such that I will be able to write proofs properly.

• for the first one, Do I literally just make $$A = F_n...F_2F_1B$$ into $$A = F_n...F_2F_1 G_m...G_2G_1C$$? Wouldn't the elementary matrices being multiplied to themselves once more do something different from what we might predict to happen? Also row-equivalence just means that the matrices have the same numbers in the same places instead of meaning just one row matches right? Sep 18 '18 at 6:15
• Ok I figured out row equivalence means more that 2 matrices share the same values and does not necessary look the same but give the same information that could be shown in different ways. Still unsure on how step 3 wants me to figure out A = C if A = B and B = C by elementary operations. Sep 18 '18 at 6:38
• As you suggested in your first comment, you simply have $A=FGC$ which implies that $A$ is row equivalent to $C$
– user418131
Sep 18 '18 at 8:49
• It simply means that by performing elementary row operations, we can convert one to another
– user418131
Sep 18 '18 at 9:12
• That makes things much easier to understand; thank you @AnotherJohnDoe Sep 19 '18 at 23:58

$$A=F_n...F_1B$$

$$B=G_n...G_1C$$

$$A=F_n...F_1G_n...G_1C$$

This shows that matrix A is equal to matrix C as B is equal to C through elementary row operations.

As for proving that if A is nonsingular, B is also nonsingular if row-equivalent to A...

B ~ A

$$T_n...T_1B = A$$ $$T^{-1}(T_n...T_1B) = T^{-1}(A)$$ $$B = T_n^{-1}...T_1^{-1}A=E_1...E_nA$$

Should show that B is also row-equivalent.