text:
I have the blueprints. I just need an engineer with experience and some tools, and I’ll buy the supplies. [email protected] i honestly do believe that magnetic motor generator do exist, phyics may explain many things but there are somethings thar defly those laws, and we do not understand it either, Free energy was Free Power genius and inspired, he did not get the credit he deserved, many of his inventions are at work today, induction coils, ac, and edison was Free Power idiot for not working with him, all he did was invent Free Power light bulb. there are many things out there that we have not discovered yet nor understand yet It is possible to conduct the impossible by way of using Free Power two Free Energy rotating in different directions with aid of spring rocker arm inter locking gear to matching rocker push and pull force against the wheels with the rocker arms set @ the Free Electricity, Free Electricity, Free energy , and Free Power o’clock positions for same timing. No further information allowed that this point. It will cause Free Power hell lot of more loss jobs if its brought out. So its best leaving it shelved until the right time. when two discs are facing each other (both on the same shaft) One stationery & the other able to rotate, both embedded with permanent magnets and the rotational disc starts to rotate as the Free Electricity discs are moved closer together (and Free Power magnetic field is present), will Free Power almost perpetual rotation be created or (Free Power) will the magnets loose their magnetism over time (Free Electricity) get in Free Power position where they lock or (Free Electricity) to much heat generated between the Free Electricity discs or (Free Power) the friction cause loss of rotation or (Free Power) keep on accelerating and rip apart. We can have powerful magnets producing energy easily.
During the early 19th century, the concept of perceptible or free caloric began to be referred to as “free heat” or heat set free. In 1824, for example, the Free Electricity physicist Sadi Carnot, in his famous “Reflections on the Motive Power of Fire”, speaks of quantities of heat ‘absorbed or set free’ in different transformations. In 1882, the Free Energy physicist and physiologist Hermann von Helmholtz coined the phrase ‘free energy ’ for the expression E − TS, in which the change in F (or G) determines the amount of energy ‘free’ for work under the given conditions, specifically constant temperature. [Free Electricity]:Free Power.

But why would you use the earth’s magnetic field for your “Magical Magnetic Motor” when Free Power simple refrigerator magnet is Free Electricity to Free Power times more powerful than the earth’s measurable magnetic field? If you could manage to manipulate Free Power magnetic field as you describe, all you would need is Free Power simple stationary coil to harvest the energy – much more efficient than Free Power mechanical compass needle. Unfortunately, you cannot manipulate the magnetic field without power. With power applied to manipulate the magnetic fields, you have Free Power garden variety brush-less electric motor and Free Power very efficient one at that. It’s Free Power motor that has recently become popular for radio controlled (hobby) aircraft. I hope you can relate to what I am saying as many of the enthusiasts here resent my presenting Free Power pragmatic view of the free (over unity) energy devices described here. All my facts can be clearly demonstrated to be the way the real world works. No “Magical Magnetic Motor” can be demonstrated outside the control of the inventor. Videos are never proof of anything as they can be easily faked. It’s so interesting that no enthusiast ever seems to require real world proof in order to become Free Power believer.
Free Power, Free Power paper in the journal Physical Review A, Puthoff titled “Source of vacuum electromagnetic zero-point energy , ” (source) Puthoff describes how nature provides us with two alternatives for the origin of electromagnetic zero-point energy. One of them is generation by the quantum fluctuation motion of charged particles that constitute matter. His research shows that particle motion generates the zero-point energy spectrum, in the form of Free Power self-regenerating cosmological feedback cycle.

#### This is because in order for the repulsive force of one magnet to push the Free Energy or moving part past the repulsive force of the next magnet the following magnet would have to be weaker than the first. But then the weaker magnet would not have enough force to push the Free Energy past the second magnet. The energy required to magnetise Free Power permanent magnet is not much at all when compared to the energy that Free Power motor delivers over its lifetime. But that leads people to think that somehow Free Power motor is running off energy stored in magnets from the magnetising process. Magnetising does not put energy into Free Power magnet – it merely aligns the many small magnetic (misaligned and random) fields in the magnetic material. Dear friends, I’m very new to the free energy paradigm & debate. Have just started following it. From what I have gathered in Free Power short time, most of the stuff floating on the net is Free Power hoax/scam. Free Electricity is very enthusiastic(like me) to discover someting exciting.

And solar panels are extremely inefficient. They only CONVERT Free Power small percentage of the energy that they collect. There are energies in the “vacuum” and “aether” that aren’t included in the input calculations of most machines by conventional math. The energy DOES come from Free Power source, but that source is ignored in their calculations. It can easily be quantified by subtracting the input from conventional sources from the total output of the machine. The difference is the ZPE taken in. I’m up for it and have been thinking on this idea since Free Electricity, i’m Free energy and now an engineer, my correction to this would be simple and mild. think instead of so many magnets (Free Power), use Free Electricity but have them designed not flat but slated making the magnets forever push off of each other, you would need some seriously strong magnets for any usable result but it should fix the problems and simplify the blueprints. Free Power. S. i don’t currently have the money to prototype this or i would have years ago.
I don’t know what to do. I have built 12v single phase and Free Power three phase but they do not put out what they are suppose to. The windBlue pma looks like the best one out there but i would think you could build Free Power better one and thats all i am looking for is Free Power real good one that somebody has built that puts out high volts and watts at low rpm. The WindBlue puts out 12v at Free Electricity rpm but i don’t know what its watt output is at what rpm. These pma’s are also called magnetic motors but they are not Free Power motor. They are Free Power generator. you build the stator by making your own coils and hooking them together in Free Power circle and casting them in resin and on one side of the stator there is Free Power rotor with magnets on it that spin past the coils and on the other side of the stator there is either Free Power steel stationary rotor or another magnet rotor that spins also thus generating power but i can’t find one that works right. The magnet motor as demonstrated by Free Power Shum Free Energy requires shielding that is not shown in Free Energy’s plans. Free Energy’s shielding is simple, apparently on the stator. The Perendev shows each magnet in the Free Energy shielded. Actually, it intercepts the flux as it wraps around the entire set of magnets. The shielding is necessary to accentuate interaction between rotor and stator magnets. Without shielding, the device does not work. Hey Gilgamesh, thanks and i hope you get to build the motor. I did forget to ask one thing on the motor. Are the small wheels made of steel or are they magnets? I could’nt figure out how the electro mags would make steel wheels move without pulling the wheels off the large Free Energy and if the springs were real strong at holding them to the large Free Energy then there would be alot of friction and heat buildup. Ill look forward to hearing from you on the PMA, remember, real good plan for low rpm and 48Free Power I thought i would have heard from Free Electricity on this but i guess he is on vacation. Hey Free Power. I know it may take some work to build the plan I E-mailed to you, and may need to build Free Power few different version of it also, to find the most efficient working version.
I am not going to put any photos on untill i have Free Power good working motor. Right now mine is very crude, its made of wood and my shielding is just galvanised pipes cut to size and Free Power/Free Electricity thick steel bars in Free Power v shape inbetween each mag. Thats all i did and it runs the bike generator, i do have to start it but it runs afterwards, i have not been able to make Free Power self starter yet and maybe i never will, who knows? I will just keep collecting all the info i can and keep tinkering. Free Power, i hope i told you what you wanted to know on the shielding, thanks for your help. Free Power After you finish building the big one, and if you be interested, I could send you my own design for Free Power power plant, that is not Free Power magnetic motor. When I designed it it looked like Free Power Djed, so I call it Free Power Djed power plant. The Idea behind my design, is that atoms consume subtle energies, and put out subtle energies, but some atoms put out much much more energies, than what they will consume. A few alchemists would know, what I m talking about. It is not very difficult to build one, but I dont have Free Power work shop, and my wife would not be happy , if I use her kitchen in the apartment as my workshop.
If there are no buyers in LA, then you could take your show on the road. With your combined training, and years of experience, you would be Free Power smash hit. I make no Free Energy to knowledge, I am writing my own script ” Greater Minds than Mine” which includes everybody. My greatest feat in life is find Free Power warm commode, on time….. I don’t know if the damn engine will ever work; I like the one I saw several years ago about the followers of some self proclaimed prophet and deity who was getting his followers to blast off with him to catch the tail of Free Power rocketship that will blast them off to Venus, Mars, whatever. I think you’re being argumentative. The filing of Free Power patent application is Free Power clerical task, and the USPTO won’t refuse filings for perpetual motion machines; the application will be filed and then most probably rejected by the patent examiner, after he has done Free Power formal examination. Model or no model the outcome is the same. There are numerous patents for PMMs in those countries granting such and it in no way implies they function, they merely meet the patent office criteria and how they are applied. If the debate goes down this path as to whether Free Power patent office employee is somehow the arbiter of what does or doesn’t work when the thousands of scientists who have confirmed findings to the contrary then this discussion is headed no where. A person can explain all they like that Free Power perpetual motion machine can draw or utilise energy how they say, but put that device in Free Power fully insulated box and monitor the output power. Those stubborn old fashioned laws of physics suggest the inside of the box will get colder till absolute zero is reached or till the hidden battery/capacitor runs flat. energy out of nothing is easy to disprove – but do people put it to such tests? Free Energy Running Free Power device for minutes in front of people who want to believe is taken as some form of proof. It’s no wonder people believe in miracles. Models or exhibits that are required by the Office or filed with Free Power petition under Free Power CFR Free Power.
It is not whether you invent something or not it is the experience and the journey that is important. To sit on your hands and do nothing is Free Power waste of life. My electrical engineer friend is saying to mine, that it can not be done. Those with closed minds have no imagination. This and persistance is what it takes to succeed. The hell with the laws of physics. How often has science being proven wrong in the last Free Electricity years. Dont let them say you are Free Power fool. That is what keeps our breed going. Dont ever give up. I’ll ignore your attempt at sarcasm. That is an old video. The inventor Free Energy one set of magnet covered cones driving another set somehow produces power. No explanation, no test results, no published information.

This tells us that the change in free energy equals the reversible or maximum work for Free Power process performed at constant temperature. Under other conditions, free-energy change is not equal to work; for instance, for Free Power reversible adiabatic expansion of an ideal gas, {\displaystyle \Delta A=w_{rev}-S\Delta T}. Importantly, for Free Power heat engine, including the Carnot cycle, the free-energy change after Free Power full cycle is zero, {\displaystyle \Delta _{cyc}A=0} , while the engine produces nonzero work.
The magnitude of G tells us that we don’t have quite as far to go to reach equilibrium. The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to Free Power. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, Go. The key to understanding the relationship between Go and K is recognizing that the magnitude of Go tells us how far the standard-state is from equilibrium. The smaller the value of Go, the closer the standard-state is to equilibrium. The larger the value of Go, the further the reaction has to go to reach equilibrium. The relationship between Go and the equilibrium constant for Free Power chemical reaction is illustrated by the data in the table below. As the tube is cooled, and the entropy term becomes less important, the net effect is Free Power shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when Free Power sealed tube containing NO2 gas is immersed in liquid nitrogen. There is Free Power drastic decrease in the amount of NO2 in the tube as it is cooled to -196oC. Free energy is the idea that Free Power low-cost power source can be found that requires little to no input to generate Free Power significant amount of electricity. Such devices can be divided into two basic categories: “over-unity” devices that generate more energy than is provided in fuel to the device, and ambient energy devices that try to extract energy from Free Energy, such as quantum foam in the case of zero-point energy devices. Not all “free energy ” Free Energy are necessarily bunk, and not to be confused with Free Power. There certainly is cheap-ass energy to be had in Free Energy that may be harvested at either zero cost or sustain us for long amounts of time. Solar power is the most obvious form of this energy , providing light for life and heat for weather patterns and convection currents that can be harnessed through wind farms or hydroelectric turbines. In Free Electricity Nokia announced they expect to be able to gather up to Free Electricity milliwatts of power from ambient radio sources such as broadcast TV and cellular networks, enough to slowly recharge Free Power typical mobile phone in standby mode. [Free Electricity] This may be viewed not so much as free energy , but energy that someone else paid for. Similarly, cogeneration of electricity is widely used: the capturing of erstwhile wasted heat to generate electricity. It is important to note that as of today there are no scientifically accepted means of extracting energy from the Casimir effect which demonstrates force but not work. Most such devices are generally found to be unworkable. Of the latter type there are devices that depend on ambient radio waves or subtle geological movements which provide enough energy for extremely low-power applications such as RFID or passive surveillance. [Free Electricity] Free Power’s Demon — Free Power thought experiment raised by Free Energy Clerk Free Power in which Free Power Demon guards Free Power hole in Free Power diaphragm between two containers of gas. Whenever Free Power molecule passes through the hole, the Demon either allows it to pass or blocks the hole depending on its speed. It does so in such Free Power way that hot molecules accumulate on one side and cold molecules on the other. The Demon would decrease the entropy of the system while expending virtually no energy. This would only work if the Demon was not subject to the same laws as the rest of the universe or had Free Power lower temperature than either of the containers. Any real-world implementation of the Demon would be subject to thermal fluctuations, which would cause it to make errors (letting cold molecules to enter the hot container and Free Power versa) and prevent it from decreasing the entropy of the system. In chemistry, Free Power spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of Free Power diamond turning into graphite, which can be written as the following reaction: Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be Free Power chemical reaction in Free Power beaker. Free Power we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?) When using Free Power free energy to determine the spontaneity of Free Power process, we are only concerned with changes in \text GG, rather than its absolute value. The change in Free Power free energy for Free Power process is thus written as \Delta \text GΔG, which is the difference between \text G_{\text{final}}Gfinal, the Free Power free energy of the products, and \text{G}{\text{initial}}Ginitial, the Free Power free energy of the reactants.

Of all the posters here, I’m certain kimseymd1 will miss me the most :). Have I convinced anyone of my point of view? I’m afraid not, but I do wish all of you well on your journey. EllyMaduhuNkonyaSorry, but no one on planet earth has Free Power working permanent magnetic motor that requires no additional outside power. Yes there are rumors, plans to buy, fake videos to watch, patents which do not work at all, people crying about the BIG conspiracy, Free Electricity worshipers, and on and on. Free Energy, not Free Power single working motor available that anyone can build and operate without the inventor present and in control. We all would LIKE one to be available, but that does not make it true. Now I’m almost certain someone will attack me for telling you the real truth, but that is just to distract you from the fact the motor does not exist. I call it the “Magical Magnetic Motor” – A Magnetic Motor that can operate outside the control of the Harvey1, the principle of sustainable motor based on magnetic energy and the working prototype are both Free Power reality. When the time is appropriate, I shall disclose it. Be of good cheer.
I do not fear any conspiracy from any nook & corner. I am simply taking my time and my space to stage the inevitable confrontation in the frozen face of the industry and geopolitics tycoons. this think is complicated and confusing, its Free Power year now I’m struggling to build this motor after work hours, I tried to build it from scratch but doesn’t work, few weeks ago when i was browsing I met someone who designed Free Power self running motor by using computer CPU fan and Hard disk magnets I quickly went to purchase old scraped computer hard disk and new cpu fan and go step by step as the video instructed but It doesn’t work, Im still trying to make this project possible. Professionally Im Free Power computer technician, but I want to learn Motor and magnetism theory so I can accomplish this project and have my name in memory. I anyone can make this project please contact me through facebook so I can invite him/her to my country and make money as you know third word countries has power disaster. My facebook Id is Elly Maduhu Nkonya, or use my E-mail. [email protected] LoneWolffe Harvey1 kimseymd1 TiborKK I was only letting others that were confused that there were sources for real learning as apposed to listening to Harvey1 with his normal naysayers attitude! There is tons of information on schoolgirl, schoolboy and Bedini window motors that actually work to charge batteries and eventually will generate house currents. It just has to be looked at to get any useful information from it without listening to people like Harvey1 whining about learning. Harvey1 kimseymd1 You obviously play too much video games with trolls etc. in them. Why the editors of this forum allow you to keep calling people names instead of following the subject is beyond me. This must be the last site to allow you on it. I spammed the books because I thought those people were good for learning these engines which are super and there are tons of information out there for anyone to find. You seem to only want to learn to be rude instead of electronics.
Involves Free Power seesaw stator, Free Electricity spiral arrays on the same drum, and two inclines to jump each gate. Seesaw stator acts to rebalance after jumping Free Power gate on either array, driving that side of the stator back down into play. Harvey1 is correct so far. Many, many have tryed and failed. Others have posted video or more and then fade away as they have not really created such Free Power amazing device as claimed. I still try every few weeks. My designs or trying to replicated others. SO far, non are working and those on the web havent been found to to real either. Perhaps someday, My project will work. I have been close Free Power few times, but it still didint work. Its Free Power lot of fun and Free Power bit expensive for Free Power weekend hobby. LoneWolffe Harvey1 LoneWolffe The device that is shown in the diagram would not work, but the issue that Is the concern here is different. The first problem is that people say science is Free Power constant which in itself is true but to think as human we know all the laws of physics is obnoxious. As our laws of physics have change constantly, through history. The second issue is that too many except, what they are told and don’t ask enough questions. Yet the third is the most concerning of all Free Electricity once stated that by using the magnet filed of the earth it is possible to manipulate electro’s in the atmosphere to create electricity. This means that by manipulating electro you take energy from the air we all breath to convert it to usable energy. Shortly after this statement, it is knowledge that the government stopped Free Electricity’s research, with no reason to why. Its all well and good reading books but you still question them. Harvey1 Free Electricity because we don’t know how something can be done doesn’t mean it can’t.
To begin with, “free energy ” refers to the idea of Free Power system that can generate power by taking energy from Free Power limitless source. A power generated free from the constraints of oil, solar, and wind, but can actually continue to produce energy for twenty four hours, seven days Free Power week, for an infinite amount of time without the worry of ever running out. “Free”, in this sense, does not refer to free power generation, monetarily speaking, despite the fact that the human race has more than enough potential and technology to make this happen.
My older brother explained that in high school physics, they learned that magnetism is not energy at all. Never was, never will be. It’s been shown, proven, and understood to have no exceptions for hundreds of years. Something that O. U. should learn but refuses to. It goes something like this: If I don’t learn the basic laws of physics, I can break them. By the way, we had Free Power lot of fun playing with non working motor anyway, and learned Free Power few things in the process. My brother went on to get his PHD in physics and wound up specializing in magnetism. He designed many of the disk drive plates and electronics in the early (DOS) computers. bnjroo Harvey1 Thanks for the reply! I’m afraid there is an endless list of swindlers and suckers out there. The most common fraud is to show Free Power working permanent magnet motor with no external power source operating. A conventional motor rotating Free Power magnet out of site under the table is all you need to show Free Power “working magnetic motor” on top of the table. How could I know this? Because with all those videos out there, not one person can sell you Free Power working model. Also, not one of these scammers can ever let anyone not related to his scam operate the motor without the scammer hovering around. The believers are victims of something called “Confirmation Bias”. Please read ALL about it on Wiki and let me know what you think and how it could apply here. This trap has ensnared some very smart people. Harvey1 bnjroo Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been created! NOTHING IS IMPOSSIBLE! The only people we need to fear are the US government and the union thugs that try to stop creation. Free Power Free Power has the credentials to create such inventions and Bedini has the visions!


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## Algebra 1: Common Core (15th Edition)

The answer is $10^{-2}$
To write $\frac{1}{100}$ in the form with the power of 10 and negative exponent, we can change the equation and bring the 100 to the numerator. We can write 100 with respect to 10 as $10^{2}$=100. Since 10 has an exponent of 2, the exponent changes from positive 2 to negative 2 when we bring it from the denominator to numerator. $\frac{1}{100}$ = $\frac{1}{10^{2}}$= $10^{-2}$

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# How to solve x^2 - 6x + 9 = 20 using the Square Root Property

How to solve ${x}^{2}-6x+9=20$ using the Square Root Property
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${x}^{2}-6x+9=20$ factor the left side ${\left(x-3\right)}^{2}=20$ now take the square root of each side $x-3=+\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}-\sqrt{20}$ $x=\sqrt{20}+3\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=-\sqrt{20}-3$ $x=2\sqrt{5}+3\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=-2\sqrt{5}-3$


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Introduction
2. Table project
Quiz
Summary

## Instruction

Good. Before we start, let us discuss the tables you're going to work with. They are used by a company to track their projects and the time employees spend on a given project.

## Exercise

Select all information from the table project.

The table is quite simple: each project has an id, a client_id and a name.


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# A nice generating set for the symmetric power of an algebra

I'm looking for a reference for the following fact.

Suppose $A$ is a finitely generated associative commutative unital algebra over an algebraically closed field of characteristic zero. Let $S^n(A)$ be the $n$-th symmetric power of $A$. That is, $S^n(A)$ is the subalgebra of $A^{\otimes n}$ fixed by the natural action of $S_n$ on this tensor product by permuting the factors. I would like to know a reference for the fact that $S^n(A)$ is generated by elements of the form

$(a \otimes 1 \otimes \cdots \otimes 1) + (1 \otimes a \otimes 1 \otimes \cdots \otimes 1) + \cdots + (1 \otimes \cdots \otimes 1 \otimes a)$

for $a$ in $A$.

Note that if $A = \mathbb{C}[t]$, then $S^n(A)$ is the ring of symmetric polynomials in $n$ variables and then the result follows from the fact that the power sum polynomials generate this ring.

I call the above a "fact" since I've seen it used in papers (without proof -- and I've contacted the authors and they don't know of a reference) and a colleague of mine says he could write out a proof. However, I can't help but think that there must be a reference for this in some text. But I'm unable to find one...

Alternatively, if someone knows of a very short proof, that would be the next best thing to a reference. I want to use this fact in a paper I'm writing and I'd like to avoid having to digress from the main thread of the exposition by including a long proof of this fact.

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Pedantic remark: Your definition of $S^n(A)$ does not coincide with the usual definition of the symmetric algebra when $k$ has positive characteristic. But this definition is obviously useful here because it inherits $S^n(A)$ with an algebra structure. –  Martin Brandenburg Feb 18 '13 at 21:10
see my comment to MO 122287. –  Abdelmalek Abdesselam Feb 19 '13 at 14:15
@Martin: Thanks. I meant to say that the field has characteristic zero. I've updated the question to include this assumption. –  Alistair Savage Feb 19 '13 at 14:51
It's Lemma 4.56 (ii) in Etingof's "Introduction to representation theory" ( www-math.mit.edu/~etingof/replect.pdf ), though the proof of part (i) given there is not the easiest and relies on representation theory (unsurprisingly for a representation theory text). I believe this is one of those facts better proven than cited away for the sake of reader... –  darij grinberg Feb 19 '13 at 16:22
@darij: Thanks. Since the paper I'm writing is a representation theory paper, this is quite a good reference for me. –  Alistair Savage Feb 20 '13 at 14:02

Since your algebra is finitely generated, you only really need this result in the case when $A=\mathbb{C}[t_1,\ldots,t_n]$.

It seems that this result first appeared in

F. Junker, Über symmetrische Funktionen von mehreren Reihen von Veränderlichen, Math. Ann. 43 (1893), 225-270

(see here for a scanned version of it).

A much more modern reference (from which I know about Junker's paper) is

J. Dalbec, Multisymmetric functions, Beiträge zur Algebra und Geometrie, Vol. 40 (1999), No. 1, pp. 27-51.

(see here for an electronic version).

Edit: as Abdelmalek Abdesselam suggests, this result goes back to

L. Schläfli, Über die Resultante eines Systemes mehrerer algebraischer Gleichungen Ein Beitrag zur Theorie der Elimination. (A. d. IV Bd. der Denkschriften der math.-naturw. Classe der kais. Acad. d. Wiss. bes. abgedruckt), Wien 1852

(see here for a scanned version).

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it's even older than that, it is in the 1852 paper by Schlafli on resultants. –  Abdelmalek Abdesselam Feb 19 '13 at 19:34
Thanks! My command of German is not enough to conclude that Schlafli's paper contains a proof. I shall add a reference in the body of the answer. –  Vladimir Dotsenko Feb 19 '13 at 23:02
Thanks, Vladmir and Abdelmalek. Of course, I think one would need to include the proof that if the result is true for A then it is true for any homomorphic image of A, in order to deduce the result for finitely generated algebras. Luckily that isn't so hard (although not immediately obvious, I think). –  Alistair Savage Feb 20 '13 at 14:09


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# What does net mechanical efficiency mean?

I often see the term "net mechanical efficiency" used in literature, but I am not quite sure what it means, and what the difference is between it and "normal" efficiency. Take this sentence for example: ... increased the effectiveness, while reducing net mechanical efficiency. What does exactly does this mean?

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Mechanical efficiency is usually used as a metric to account for frictional losses in systems. For example, the transmission of a car transmits mechanical work from the engine to the wheels so the mechanical efficiency of the system will be $W_{transmitted}/W_{received/ideal}$. Even within an engine, there is friction between the piston and cylinder walls, bearings of crankshafts etc. Mechanical efficiency is: $W_{output}/W_{obtained\;from\; gas}$. The denominator is the work obtained from the work-fluid, the thermodynamic cycle. Therefor it is a way to quantify the frictional loses in the system.


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## G = Dic6⋊22D4order 192 = 26·3

### 10th semidirect product of Dic6 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic6⋊22D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×S3×Q8 — Dic6⋊22D4
 Lower central C3 — C2×C6 — Dic6⋊22D4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for Dic622D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=dad=a5, cbc-1=a6b, bd=db, dcd=c-1 >

Subgroups: 752 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×C6, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C4○D12, S3×Q8, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×Q8, Q85D4, Dic3⋊D4, C23.11D6, Dic6⋊C4, Dic35D4, C12⋊D4, D6⋊Q8, C4×C3⋊D4, C12.23D4, C3×C22⋊Q8, C2×C4○D12, C2×S3×Q8, Dic622D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2- 1+4, S3×D4, S3×C23, Q85D4, C2×S3×D4, Q8.15D6, S3×C4○D4, Dic622D4

Smallest permutation representation of Dic622D4
On 96 points
Generators in S96
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 69 7 63)(2 68 8 62)(3 67 9 61)(4 66 10 72)(5 65 11 71)(6 64 12 70)(13 35 19 29)(14 34 20 28)(15 33 21 27)(16 32 22 26)(17 31 23 25)(18 30 24 36)(37 74 43 80)(38 73 44 79)(39 84 45 78)(40 83 46 77)(41 82 47 76)(42 81 48 75)(49 96 55 90)(50 95 56 89)(51 94 57 88)(52 93 58 87)(53 92 59 86)(54 91 60 85)
(1 95 77 26)(2 88 78 31)(3 93 79 36)(4 86 80 29)(5 91 81 34)(6 96 82 27)(7 89 83 32)(8 94 84 25)(9 87 73 30)(10 92 74 35)(11 85 75 28)(12 90 76 33)(13 72 53 43)(14 65 54 48)(15 70 55 41)(16 63 56 46)(17 68 57 39)(18 61 58 44)(19 66 59 37)(20 71 60 42)(21 64 49 47)(22 69 50 40)(23 62 51 45)(24 67 52 38)
(1 22)(2 15)(3 20)(4 13)(5 18)(6 23)(7 16)(8 21)(9 14)(10 19)(11 24)(12 17)(25 64)(26 69)(27 62)(28 67)(29 72)(30 65)(31 70)(32 63)(33 68)(34 61)(35 66)(36 71)(37 92)(38 85)(39 90)(40 95)(41 88)(42 93)(43 86)(44 91)(45 96)(46 89)(47 94)(48 87)(49 84)(50 77)(51 82)(52 75)(53 80)(54 73)(55 78)(56 83)(57 76)(58 81)(59 74)(60 79)

G:=sub<Sym(96)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,69,7,63)(2,68,8,62)(3,67,9,61)(4,66,10,72)(5,65,11,71)(6,64,12,70)(13,35,19,29)(14,34,20,28)(15,33,21,27)(16,32,22,26)(17,31,23,25)(18,30,24,36)(37,74,43,80)(38,73,44,79)(39,84,45,78)(40,83,46,77)(41,82,47,76)(42,81,48,75)(49,96,55,90)(50,95,56,89)(51,94,57,88)(52,93,58,87)(53,92,59,86)(54,91,60,85), (1,95,77,26)(2,88,78,31)(3,93,79,36)(4,86,80,29)(5,91,81,34)(6,96,82,27)(7,89,83,32)(8,94,84,25)(9,87,73,30)(10,92,74,35)(11,85,75,28)(12,90,76,33)(13,72,53,43)(14,65,54,48)(15,70,55,41)(16,63,56,46)(17,68,57,39)(18,61,58,44)(19,66,59,37)(20,71,60,42)(21,64,49,47)(22,69,50,40)(23,62,51,45)(24,67,52,38), (1,22)(2,15)(3,20)(4,13)(5,18)(6,23)(7,16)(8,21)(9,14)(10,19)(11,24)(12,17)(25,64)(26,69)(27,62)(28,67)(29,72)(30,65)(31,70)(32,63)(33,68)(34,61)(35,66)(36,71)(37,92)(38,85)(39,90)(40,95)(41,88)(42,93)(43,86)(44,91)(45,96)(46,89)(47,94)(48,87)(49,84)(50,77)(51,82)(52,75)(53,80)(54,73)(55,78)(56,83)(57,76)(58,81)(59,74)(60,79)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,69,7,63)(2,68,8,62)(3,67,9,61)(4,66,10,72)(5,65,11,71)(6,64,12,70)(13,35,19,29)(14,34,20,28)(15,33,21,27)(16,32,22,26)(17,31,23,25)(18,30,24,36)(37,74,43,80)(38,73,44,79)(39,84,45,78)(40,83,46,77)(41,82,47,76)(42,81,48,75)(49,96,55,90)(50,95,56,89)(51,94,57,88)(52,93,58,87)(53,92,59,86)(54,91,60,85), (1,95,77,26)(2,88,78,31)(3,93,79,36)(4,86,80,29)(5,91,81,34)(6,96,82,27)(7,89,83,32)(8,94,84,25)(9,87,73,30)(10,92,74,35)(11,85,75,28)(12,90,76,33)(13,72,53,43)(14,65,54,48)(15,70,55,41)(16,63,56,46)(17,68,57,39)(18,61,58,44)(19,66,59,37)(20,71,60,42)(21,64,49,47)(22,69,50,40)(23,62,51,45)(24,67,52,38), (1,22)(2,15)(3,20)(4,13)(5,18)(6,23)(7,16)(8,21)(9,14)(10,19)(11,24)(12,17)(25,64)(26,69)(27,62)(28,67)(29,72)(30,65)(31,70)(32,63)(33,68)(34,61)(35,66)(36,71)(37,92)(38,85)(39,90)(40,95)(41,88)(42,93)(43,86)(44,91)(45,96)(46,89)(47,94)(48,87)(49,84)(50,77)(51,82)(52,75)(53,80)(54,73)(55,78)(56,83)(57,76)(58,81)(59,74)(60,79) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,69,7,63),(2,68,8,62),(3,67,9,61),(4,66,10,72),(5,65,11,71),(6,64,12,70),(13,35,19,29),(14,34,20,28),(15,33,21,27),(16,32,22,26),(17,31,23,25),(18,30,24,36),(37,74,43,80),(38,73,44,79),(39,84,45,78),(40,83,46,77),(41,82,47,76),(42,81,48,75),(49,96,55,90),(50,95,56,89),(51,94,57,88),(52,93,58,87),(53,92,59,86),(54,91,60,85)], [(1,95,77,26),(2,88,78,31),(3,93,79,36),(4,86,80,29),(5,91,81,34),(6,96,82,27),(7,89,83,32),(8,94,84,25),(9,87,73,30),(10,92,74,35),(11,85,75,28),(12,90,76,33),(13,72,53,43),(14,65,54,48),(15,70,55,41),(16,63,56,46),(17,68,57,39),(18,61,58,44),(19,66,59,37),(20,71,60,42),(21,64,49,47),(22,69,50,40),(23,62,51,45),(24,67,52,38)], [(1,22),(2,15),(3,20),(4,13),(5,18),(6,23),(7,16),(8,21),(9,14),(10,19),(11,24),(12,17),(25,64),(26,69),(27,62),(28,67),(29,72),(30,65),(31,70),(32,63),(33,68),(34,61),(35,66),(36,71),(37,92),(38,85),(39,90),(40,95),(41,88),(42,93),(43,86),(44,91),(45,96),(46,89),(47,94),(48,87),(49,84),(50,77),(51,82),(52,75),(53,80),(54,73),(55,78),(56,83),(57,76),(58,81),(59,74),(60,79)]])

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 4O 4P 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 4 4 6 6 6 6 6 12 12 12 12 12 12 12 12 size 1 1 1 1 4 6 6 12 12 2 2 2 2 2 4 4 4 4 6 ··· 6 12 12 2 2 2 4 4 4 4 4 4 8 8 8 8

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 C4○D4 2- 1+4 S3×D4 Q8.15D6 S3×C4○D4 kernel Dic6⋊22D4 Dic3⋊D4 C23.11D6 Dic6⋊C4 Dic3⋊5D4 C12⋊D4 D6⋊Q8 C4×C3⋊D4 C12.23D4 C3×C22⋊Q8 C2×C4○D12 C2×S3×Q8 C22⋊Q8 Dic6 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 D6 C6 C4 C2 C2 # reps 1 2 2 1 2 1 2 1 1 1 1 1 1 4 2 3 1 1 4 1 2 2 2

Matrix representation of Dic622D4 in GL6(𝔽13)

 0 8 0 0 0 0 8 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [0,8,0,0,0,0,8,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

Dic622D4 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes_{22}D_4
% in TeX

G:=Group("Dic6:22D4");
// GroupNames label

G:=SmallGroup(192,1192);
// by ID

G=gap.SmallGroup(192,1192);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,1571,297,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=d*a*d=a^5,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽


text:
# How do you simplify 8+24 div 4 using order of operations?

Mar 29, 2018

$14$

#### Explanation:

Order of Operations

P arentheses

E xponent

M ultiplication

D ivision

A ddition

S ubtraction

$8 + 24 \div 4$

We need to simplify $24 \div 4$ to $6$ first, because it comes before addition in the Order of Operations

So our problem is $8 + 6$ which we simplify to $14$


text:
# Unnamed Problem -2 (Combinatorics)

Given a set of numbers $N = \{1,2,\ldots,n\}$ and numbers $a, b, c |\; a + b = c$ and $a \leq b \leq c \leq n$

Let $A = \{a_1, a_2,.\ldots,a_{n \choose a}\;|\; a_1 \neq a_2 \neq\ldots\neq a_{n \choose a}\;;\;|a_x| = a\}$; $|A|$ = $n \choose a$

Similarly,

$B = \{b_1, b_2,.\ldots,b_{n \choose b}\;|\; b_1 \neq b_2 \neq\ldots\neq b_{n \choose b}\;;\;|b_x| = b\}$; $|B|$ = $n \choose b$ and

$C = \{c_1, c_2,.\ldots,c_{n \choose c}\;|\; c_1 \neq c_2 \neq\ldots\neq c_{n \choose c}\;;\;|c_x| = c\}$; $|C|$ = $n \choose c$

Now, consider a subset of C, $C^1 = \{c^x | c^x = a^x \cup b^x;\;c^x \in C, a^x \in A; b^x \in B\}$

The question is to find the minimal cardinality of the subset of A, such that $C^1 = C$


text:
# Section 2.3: The Arithmetic of $latex\mathbb{Z}_p$ (p Prime) and $latex\mathbb{Z}_n$$In my analysis of this section I noted that Hungerford often switches between the conventional modular arithmetic notation (with the brackets []) and then uses the number by itself. While both are completely valid certain uses are better than others. That is the notation $[a]$ is much better than a if there are integers in being used at the same time. That is if $b\in\mathbb{Z}$ and $a\in\mathbb{Z}_n$ then $ab$ and $b\cdot [a]$ mean the same thing but the second notation indicates that b is not a congruence class. Now to continue with Section 2.3: ###### Question: Does this work for $\mathbb{Z}_13$? $[a]\odot [b]=[ab]=[0]$ means $13\mid ab$ implies $13\mid a$ or $13\mid b$. We’ll later cover this but note that if we try to solve $ax=1$ in $\mathbb{Z}$ then the only solutions for a are $a=\pm 1$. But the same equation in $\mathbb{Z}_5$ has multiple solutions: $[2]\odot [3]=[1]$ and $[4]\odot [4]=[1]$. #### Theorem 2.8: For a positive integer p, the following are equivalent: 1. p is prime 2. For non zero $a\in\mathbb{Z}_p$, the equation $ax=1$ has a solution 3. Whenever $bc=0$ in latex \mathbb{Z}_p then$$b=0$$or $c=0$ This theorem explains a lot but fails to explain when there are solutions to the equation $ax=1$. ### Theorem 2.9 Let$latex a,n\in\mathbb{Z}$, $n>1$. The equation $[a]\odot x=[1]$ has a solution$latex \mathbb{Z}_n\$ if and only if $(a,n)=1$.

To avoid repeating the equation $[a]\odot x=1$ let’s define something to indicate when we reference this:

Definition:

A non zero $[a]$ in $\mathbb{Z}_n$ is called a unit if $[a]\odot x=[1]$ has a solution $x\in\mathbb{Z}_n$.

A solution $x=[0]$ of $[a]\odot x=[1]$ is called a multiplicative inverse of a.

Using this new definition let’s re-write Theorem 2.9: $[a]$ is a unit if and only if $(a,n)=1$.

Definition:

A non zero element of $\mathbb{Z}_n$ is called a zero-divisor if $[a]\odot x=[0]$ has a non  zero solution $x=[b]$ in $\mathbb{Z}_n$.

Something not mentioned in the third edition of Hungerford’s Abstract Algebra but useful nevertheless are the following facts:

For any non-zero $a\in\mathbb{Z}_p$ (p is prime) $ax=b$ has a unique solution.

If $(a,n)=1$, then $ax=1$ has a unique solution.

If $(a,n)=d>1$, then $ax=b$ has solutions if and only if $d\mid b$, and in this case there are exactly d solutions.


text:
# Effect of adding a Pole and Zero to PID

I am confused about how adding a D (which adds a zero to the complete system) decreases the speed of the system. But when we normally add a zero to the system, it causes the system to overshoot.

The same goes for the I part of the PID. Normally when we add a pole to the system, it has less overshoot, but at the same time the integrator increases the overshoot!

How can I make sense out of this inverse relation?


text:
All the following studies have been made mainly thanks to private funding. Please, consider a donation to the Open Medicine Foundation, in order to speed up the research. See this page for how to donate to OMF.

1. Introduction

In this article, I report on the results from two research groups in which different experimental settings were used to measure electric impedance in blood samples from ME/CFS patients vs healthy controls. One of these studies comes from Stanford University and has been just published in PNAS: it is freely available here. The other one has been presented by Alan Moreau during the NIH conference on ME/CFS, and it is unpublished (R). In paragraph 2 I introduce the definition of impedance, in paragraph 3 you will learn something about the electric behaviour of cells, in paragraph 4 there is a description of the device used by the Stanford University group, in paragraph 5 there are the results of the experiment from Stanford University, in paragraph 6 there is a discussion of these results, in paragraph 7 the results from the other group are reported, and these two studies are compared in paragraph 8. In paragraph 9 I reported on two drugs that have shown the promise to be of therapeutic use in ME/CFS. Other notes follow in the last two paragraphs. If you are not interested in technical details on impedance (or if you don’t need them), go directly to paragraph 5.

2. Impedance

In this paragraph, I try to give a very simple and short introduction to circuits in a sinusoidal regime in general, and to impedance in particular. The main definition that we need, for that purpose, is the so-called Steinmetz transform for a sinusoidal function. Let’s consider the sinusoid

where A is called amplitude and is the maximal value that the function can reach, ω is the angular frequency (also called pulsatance) which is an indication of how fast the value of the function changes in time, α is the phase and it gives the indication of what the value of the function a(t) was for t = 0. The Steinmetz transform consists of the univocal association of the sinusoid a(t) with the complex number

also called phasor (which stands for phase vector), where j=√(-1) is the imaginary unity. A complex number can then be easily represented as a vector in the complex plane (see figure 1).

Let’s now consider the elementary circuit in figure 2 (which is also a simplified model of the device in the study by Ron Davis), where a generator of electrical potential is linked to another circuit (depicted as a box in the figure, on the left) that in our case is represented by the sample of peripheral blood white cells incubated in plasma. But it could be an arbitrarily complex net made up of conductors and what follows would still hold. Let’s assume that the electric current and the voltage of the generator are given respectively by

We can associate to these sinusoids their respective phasors with the Steinmetz transform, which gives

That said, we define impedance of the sample, the complex number that we obtain dividing the phasor of u(t) by the phasor of i(t):

Impedance describes several physical properties of the box in figure 2. Without going into details (this is beyond the scope of this article) just consider what follows.

• The real part of impedance represents the resistance of whatever is inside the box of figure 2, which can be seen as its ability to transform electric energy into heat, i.e. kinetic energy at a molecular level. The higher the value of the resistance, the more the ability to generate heat.
• The imaginary part of the impedance (called reactance) can be positive or negative. When it is positive it indicates the ability of whatever is inside the box to translate a magnetic field into voltage. The higher the positive reactance, the more its ability to generate a voltage from a magnetic field. A positive reactance is also called inductive reactance.
• When reactance is negative, it means that whatever is inside the box, it has the ability to store energy in an electric field: the higher the absolute value of the reactance, the more the energy stored in an electric field within the box. A negative reactance is also called capacitive reactance.

No matter how complex the system in the box is, its external electrical behaviour is completely characterized by its impedance, which means that the system can also be simplified in a series of an electrical component whose only relevant property is a resistance equal to the real component of impedance, and a second component completely characterized by a reactance with a value equal to the imaginary component of impedance (figure 2, on the right).

3. Impedance of cells

The study of the impedance of cellular cultures is a field that started probably in the early nineties. In a paper from the Rensselaer Polytechnic Insititute (NY), it was demonstrated that the measure of electrical impedance of a single cell layer was more sensitive than optical microscopy for the measure of changes of nanometers in the cell diameter or subnanometer changes in the distance between the cell layer and the electrodes (Giaever I. & Keese CR. 1991). In that pivotal paper, a mathematical model for the impedance of a layer of cells was also proposed and solved, but it is beyond the scope of this article. A simplified electrical model of a cell layer is provided by a parallel of a capacitance due to dielectric properties of the cell membrane, and a resistance due to the cell membrane, to the cytoplasm and to the layer between cells (Voiculescu I. et al. 2018). We can add a resistance for the solution in which cells are incubated and we obtain the circuit in figure 3.

Remember now that the only electrical property that we can directly measure is the total impedance (both the real component and the imaginary one). So we have to find the relationships between these two components and the physical parameters introduced in figure 3. For the equivalent impedance of the sample (see the last paragraph for the mathematical passages) we have:

The dependence of the real part of $Z_{cl}$ and of its imaginary component to $R_{cl}$ and $C_{cl}$ can be got from figure 4. The absolute value of $Z_{cl}$ is represented in figure 5.

The capacitance in this formula is due – as said – to the dielectric properties of the plasma membrane. We can see a cell as a spherical capacitor, where two conductive layers (one in the cytoplasm and the other one in the extracellular space) are separated by the outer membrane. The insulating portion of a phospholipid membrane is of about 4.5 nm and it has been found that the capacitance per square cm of the cell membrane is one μF (Matthews GG, 2002). Since the permittivity constant ε is known, we can calculate the dielectric constant κ of a lipid membrane quite easily (see the last paragraph), and we find κ=5.

4. The nanoneedle

The device used for the measurement of the impedance of blood samples from ME/CFS patients is an array of thousands of sensors. Each sensor is made up of two conductive layers, separated by a dielectric material (figure 6). Each sensor is a sinusoid circuit that operates at a frequency of 15 kHz and at a voltage with an amplitude of about 350 mV. In figure 6, I have added the electric scheme for the circuit made up by the sensor itself and the sample, according to what seen in the previous paragraph. I have added some resistances and capacitors for the electrodes, according to (Esfandyarpour R et al. 2014).

As you can see from the picture, one of the dimensions of the sensor is below one micron, while the other is of about 3 microns. Keep in mind that the diameter of the average white blood cell is of about 15 microns… To me, such a small size makes it difficult the application to this system of both the electrical model by Ivar Giaever and Charles Keese and of the simplified one presented in the previous paragraph, which have been designed to describe the behaviour of a layer of cells that grow above an electrode that can harbour many cells on its surface. And in fact, in their paper, Esfandyarpour R. and his colleagues have sketched a different model (R, B), even though they haven’t used it to draw any conclusion or interpretation from the experimental data, yet.

5. The experiment

The measurement of the impedance of samples from ME/CFS patients and controls has been made with an array of thousands of electrodes, each one like the one in figure 6. The system took 5 measures of impedance for second and the experiment on each sample lasted for about 3 hours. The researchers measured, for each point in time, both the real and the imaginary component of the impedance of the sample. They also measured the module of the impedance.

Each sample consisted of peripheral blood mononuclear cells (PBMC) incubated in patient’s own plasma (plasma is blood without erythrocytes, platelets and white blood cells), at a concentration of 200 cells per μL. It might be useful to remember that PBMCs are basically all the white blood cells that are present in peripheral blood but granulocytes, which have multi-lobed nuclei and, as such, are not “monuclear”.

The researchers drew blood from 5 severe patients, 15 moderate patients (diagnosed by a physician according to the Canadian Consensus Criteria) and 20 healthy controls, with 5 of them age- and gender-matched to 5 of the ME/CFS patients.

About 20 minutes were required for the impedance to reach a steady state (the baseline level, characterized by swings in impedance below 2% of its value). The measures for each sample have been divided by the value of impedance at the baseline. This is the reason why the baseline has a value of 1 in the diagrams. After the steady state was reached, the researchers added 6 μL of NaCl to the samples. After a transient reduction in impedance, the samples from healthy controls returned to the baseline value. In samples from patients, the initial reduction in impedance after NaCl introduction was followed by a dramatic change in both the real component and the imaginary component of impedance. The normalized real part, in particular, had an increase of 301.67% ± 3.55 (see figure 7 and R).

6. What does it mean?

In the experiment by Stanford University, they added NaCl to the samples and this likely led to the activation of the sodium-potassium pump that requires a molecule of ATP in order to transport 3 Na ions outside the cells (and two K ions inside). This would be the only way for these cells to maintain the correct intracellular concentration of sodium, pumping out those Na ions that found their way to the cytoplasm from the plasma. This is like putting a cell on a stationary bike. What the experiment says is that this effort made by the cells to maintain homeostasis leads to huge changes in the electrical properties of the samples from ME/CFS patients, while producing virtually no changes in the samples from healthy controls. But what is the origin of the change in impedance?

If we consider the electrical model that I have proposed in figures 3 and 6 and looking at figure 4 (left), we might hypothesise that the change comes from a reduction in the capacitance $C_{cl}$ which is due to the dielectric properties of cell membranes. A change in composition in these membranes could lead to a reduction in $C_{cl}$ and thus to the observed increase in the real component of the total impedance. This might perhaps be linked to the reduction in the metabolism of the main components of the plasma membrane (sphingolipid, phospholipid and glycosphingolipid) in patients vs controls previously reported in a metabolomic study (Naviaux R et al. 2016). A reduction in the dielectric properties of cell membranes could also explain the increase in the module of impedance observed in this study (see figure 5). But it is worth noting again that the model I used for the description of the electrical properties of the sample is a hugely simplified version of the one proposed in (Giaever I. & Keese CR. 1991) and it has been developed for electrodes that are many times larger than the one used by Esfandyarpour R and colleagues. As said elsewhere, the authors have proposed a different, more complex, electric circuit (R, B) and they wrote that the process of using it to interpret the experimental data is currently on-going. But they did note that a change in plasma membrane composition might be responsible for the observed change in impedance, in one point of the article, among other possible explanations.

A release of molecules (cytokines?) from the PBMCs into the plasma might also be the cause of the change in impedance, but if we assume that our model in figure 3 is reliable, these molecules would only change the value of $R_{su}$, so the imaginary component of the impedance would not be affected, while we know that there is a change in that component too. But again, our model is a very simplistic one.

A change in the shape or size of the cells would lead to a change in $C_{cl}$. But the authors observed the samples in standard live microscopy imaging and they were not able to record any significant cell size difference in samples from ME/CFS patients vs samples from healthy controls.

During the NIH conference on ME/CFS, the Canadian group led by Alan Moreau, presented, at the end of a speech about microRNAs, a measure of impedance on immortalized T cells incubated with plasma from healthy controls, plasma from ME/CFS patients, and plasma from patients with idiopathic scoliosis (figure 8) and, as you can see, there is an increase in impedance with the increase in plasma concentration only in the second group (R). This measure has been made with the CellKey system, after stimulation of cells with G-coupled protein receptors agonists (Garbison KE et al. 2012). It is also worth mentioning that this impedance is the one due to the flow of charges in the extracellular space and that it seems to be the module of impedance, rather than the real or the imaginary part.

Alan Moreau also noted that if we subgroup ME/CFS patients according to differences in circulating microRNA, we find that plasma from two of these groups leads to an increase in impedance while plasma from three other groups induces a decrease in impedance, if compared with T cells incubated with plasma from healthy controls (figure 9).

8. The X factor

Even though the Canadian experiment is not directly comparable to the one from the Stanford University group, nevertheless it is a partial confirmation of that result. Moreover, since in the Canadian experiment the cells are the same for all the groups (it is a line of immortalized T cells) and what changes is only the plasma they are incubated in, we can say that the origin of the electrical shift in these samples is something that is present in the plasma of patients (an X factor) and it might be due to the interaction between this X factor and cells. This interpretation is in agreement with a previous observation from a Norwegian group who incubated muscular cells in serum from 12 patients and from 12 healthy donors: they found an increase in oxygen consumption and in lactic acid production in cells incubated with sera from patients vs cells incubated with sera from healthy controls. This experiment was performed using the Seahorse instrument (Fluge et al. 2016). It is worth noting that in this case only serum was used, and serum is plasma without clotting factor.

The idea of an X factor present in plasma (or serum) of patients is even more suggestive if we take into account the unpublished result presented by Ron Davis during the NIH conference, called the “plasma swap experiment”, performed with the nanoneedle device (R). As you can see from figure 10, the increase in impedance happens only when cells are incubated with plasma from ME/CFS, no matter whether the cells are from healthy controls or from ME/CFS patients.

It is extremely important here to note that several filtrations of the plasma from patients have been made by the Stanford Group in order to discover what the X factor is: they have concluded that it is neither a metabolite nor a cytokine. Alan Moreau noted also that it is probably not an antibody. It turned out that it might be an exosome, a vesicle released by cells which contains – among other molecules – microRNA molecules. As Ron Davis said: “I guess that the signal is coming from damaged mitochondria, but it is only a guess” (R).

9. Drug testing

The authors of the study on the nanoneedle device are interested in using it for drug testing. Ron Davis reported during the last Emerge Australia conference (R) that two compounds are able to reduce the alteration in impedance seen in PBMCs incubated with plasma from patients: Copaxone, a peptide currently used in the treatment of multiple sclerosis, and SS31, a molecule not available yet, that can scavenge mitochondrial reactive oxygen species (ROS), thereby promoting mitochondrial function (Escribano-Lopez I. et al. 2018), (Thomas DA et al 2007).

10. Limitations of the study from Stanford University

Even though the differences observed in the electric properties of the samples from ME/CFS patients vs controls, after the addition of the osmotic stressor, are striking, there are some potential limitations that ought to be mentioned.

• Only 5 of the 20 healthy controls were age and gender-matched to 5 ME/CFS patients. So the difference observed might be due, at least in part, to age or gender.
• The difference in impedance might be due to some consequence of the disease, like deconditioning, since the healthy control was not a sedentary one.

I presented the content of this blog post after the screening of Unrest in Turin (Italy) in May 2019 (video in Italian).

11. Mathematical notes

The calculation of the impedance $Z_{cl}$ of the sample (figure 3) is as follows:

Then you have to add the resistance $R_{su}$ to the real part and you obtain $Z_{tot}$. In order to calculate the dielectric constant of the lipid membrane just follow these passages:

In order to choose the range of variation for $C_{cl}$ and $R_{cl}$ in the diagrams in figures 4 and 5, I calculated the capacitance of a cell, assuming a spheric shape, a radius of 5 μm, a capacitance for square cm of 1 μF, a thickness of the plasma membrane of 4.5 nm, and a dielectric constant κ=5. This gives for $C_{cl}$ a value of $\pi10^{-12}$ Farad. I then found the value of the imaginary component of the impedance of a culture of yeast cells measured by the nanoneedle, which is 800 kΩ and I set the angular frequency at 2π·15 kHz (which is the frequency of the generator of voltage of the nanoneedle). Then we have a reference value for resistance too: 1.9·$10^{6}\Omega$. The simple code (Matlab) that I used to plot the diagrams in figure 4 and 5 is the following one.

% file name = impedance
clear all
% we define the angular frequency
w = 2*pi*15*(10^3)
% we register the array of the capacitance axis (pico Farad)
c_span = 4.;
delta_c = c_span/30.;
n_c = c_span/delta_c;
% we register the array
c(1) = 0.;
for i = 2:30+1
c(i) = c(i-1) + delta_c;
end
% we define the array of resistance (mega Ohm)
r_span = 9.;
delta_r = r_span/30.;
n_r = r_span/delta_r;
r(1) = 0.;
for i = 2:30+1
r(i) = r(i-1) + delta_r;
end
% we register the array of the real part and of the imaginary part of impedance and its module
for i=1:n_c
for j=1:n_r
Rcl = r(j)*(10^6);
Ccl = c(i)*(10^(-12));
Z_r (i,j) = Rcl/( 1 + ( (Rcl^2)*(w^2)*(Ccl^2) ) );
Z_i (i,j) = (-1)*( w*Ccl )/( ( 1/(Rcl^2) ) + (w*Ccl)^2 );
Z_m (i,j) = sqrt( (Z_r (i,j)^2)+(Z_i (i,j)^2) );
endfor
endfor
% we plot the real part of the impedance
figure(1)
mesh(r(1:n_r), c(1:n_c), Z_r(1:n_c,1:n_r));
grid on
xlabel('resistance (Mega Ohm)');
zlabel('Ohm');
legend('Real part of Impedance',"location","NORTHEAST");
% we plot the imaginary part of the impedance
figure(2)
mesh(r(1:n_r), c(1:n_c), Z_i(1:n_c,1:n_r));
grid on
xlabel('resistance (Mega Ohm)');
zlabel('Ohm');
legend('Imaginary part of Impedance',"location","NORTHEAST");
figure(3)
mesh(r(1:n_r), c(1:n_c), Z_m(1:n_c,1:n_r));
grid on
xlabel('resistance (Mega Ohm)');
zlabel('Ohm');
legend('Module of Impedance',"location","NORTHEAST");

## 5 thoughts on “Impedance, the biomarker you wouldn’t expect”

1. Libby Boone says:

Do you feel that this impedance has some relevance to the paper produced by Naviaux regarding the cell-danger response in terms of mitochodrial damage?

https://www.sciencedirect.com/science/article/pii/S1567724913002390?via%3Dihub

I think the Naviaux paper was one that Open Medicine Foundation was looking into before the metabolic trap hypothesis by Robert Phair.

This electrical impedance you write about would indicate that immune modulating drugs would possibly be useful, such as capaxone for multiple sclerosis.

As you say, we cannot yet say if the differences in the impedance are related to factors such as aging, etc.

Naviaux was looking at similarities between ME and “sleeping sickness,” and at one point was trying to get a trail of the old medication for sleeping sickness called suramin.

From Wikipedia: “The mechanism of action for suramin is unclear, however, it is thought that parasites are able to selectively uptake suramin via receptor-mediated endocytosis of drug that is bound to low-density lipoproteins and to a lesser extent, other serum proteins.[12] Once inside parasites, suramin combines with proteins, especially trypanosomal glycolytic enzymes to inhibit energy metabolism.[14]”

The clinical presentation factor that seems to distinguish ME from MS is the post exertional collapse, or PEM.
Also, MS involves lesions on the myelin sheath, whereas the inflammation in the brain of ME patients seems to be more widespread and possibly related to the hypothalamic pituitary axis.

And the impingement on the spinal column in patients who have had the cranio-cervical instablity would seem to be primarily structural.

Thanks. This is very interesting.

Liked by 1 person

2. Suramin is an antagonist of the purinergic receptor P2X, and this is the reason why Naviaux is interested in this drug. He believes that this receptor is responsible, at least in part, for the cell danger response.

At present, it is not possible to figure out whether there is a link between the increase of blood impedance and the theory by Naviaux. They seem unrelated, at present.

Copaxone decreases the impedance in blood from ME/CFS patients, but the mechanism by which it is supposed to work in MS is completely different.

Like


text:
# Area of squares and rectangles

Find the area of rectangles and squares when given side lengths.  Find the side length of a square when given the area.

### Problem

The area of a square is 49 square meters. How long is each side?
meters
Get first 3 correct, or 5 in a row.


text:
# Covariant connections on bicovariant differential calculus

@article{Bhowmick2019CovariantCO,
title={Covariant connections on bicovariant differential calculus},
journal={arXiv: Quantum Algebra},
year={2019}
}
• Published 10 December 2019
• Mathematics
• arXiv: Quantum Algebra
3 Citations
We prove that the $4D_\pm$ calculi on the quantum group $SU_q(2)$ satisfy a metric-independent sufficient condition for the existence of a unique bicovariant Levi-Civita connection corresponding to
• Mathematics
• 2022
Arbitrary right connections on bicovariant bimodules of a quantum group (Hopf algebra) are shown to canonically extend to tensor products of bicovariant bimodules. In particular, a connection on a
• Mathematics
• 2019
We study pseudo-Riemannian invariant metrics on bicovariant bimodules over Hopf algebras. We clarify some properties of such metrics and prove that pseudo-Riemannian invariant metrics on a

## References

SHOWING 1-10 OF 37 REFERENCES

• Mathematics
• 2016
We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the

### Schmüdgen : Levi-Civita Connections on the Quantum Groups SLq(N) Oq(N) and Sp q (N)

• Comm. Math. Phys. 185,
• 1997
• Mathematics
• 2019
We study pseudo-Riemannian invariant metrics on bicovariant bimodules over Hopf algebras. We clarify some properties of such metrics and prove that pseudo-Riemannian invariant metrics on a
The paper deals with non-commutative differential geometry. The general theory of differential calculus on quantum groups is developed. Bicovariant bimodules as objects analogous to tensor bundles
Noncommutative Spaces It was noticed a long time ago that various properties of sets of points can be restated in terms of properties of certain commutative rings of functions over those sets. In
• Mathematics
• 2016
We prove the existence and uniqueness of Levi-Civita connections for a noncommutative pseudo-Riemannian metric on a class of centered bimodule of one forms. As an application, we compute the Ricci
• Mathematics
Reviews in Mathematical Physics
• 2020
We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul


text:
Last updated on February 29, 2016 by Jean-Philippe Bernardy

# Abstract

In this post I explain how to write Haskell code in a way that lets the programmer control when fusion happens. The key idea is to polarize the type to deforest.

# Explicit Fusion

From a sufficiently high-level point of view, the idea behind short-cut fusion is to:

1. Convert functions on Lists to a representation which does not involve explicit (general) recursion. Let’s call this representation List'.
2. Let the inliner simplify expressions. The absence of general recursion means that it can do a reliable job, only using well-known and predictable means. This simplification process will fuse some functions together.
3. Revert the eventual non-eliminated occurrences of List'.

The above modus operandi effectively means that the programmer works with a representation of lists which is not that being used by the optimizer. Consequently, it can be hard to predict when fusion occurs in any given program. One way to circumvent this issue is to work directly with a non-recursive representation.

# Mu and Nu Lists

But which representation should be chosen? There are two obvious choices: either the least fixpoint or the greatest fixpoint; called respectively Mu (μ) and Nu (ν).

newtype Mu f = Mu {fold :: forall x. (f x -> x) -> x}

data Nu f where
Unfold :: (x -> f x) -> x -> Nu f

The representation chosen by Gill et al. are Mu lists. This choice yields the shortcut fusion system used in GHC, also known as the ‘foldr/build’ rule. Nu lists are used by Svenningsson and yield the ‘destroy/unfoldr’ rule. Both these representations use the same, standard functor for lists:

data F a x = Stop | More a x
deriving Functor

(The ‘stream fusion’ of Coutts et al. use another functor; I’m not going to get into why here.)

The choice of either Mu or Nu lists influences the set of functions which can be fused. When the choice is automatic (as in all the user-facing shortcut fusion facilities currently available), the programmer is at the mercy of the choice made by the implementer. On the other hand, if the programmer controls the choice, they can pick the best representation for the job at hand. However, to make a sensible choice one must have a good grasp of which representation is good for what. Let me try to help by giving an intuitive description of Mu and Nu types.

Least fixpoints (Mu) are computations that can consume the structure, in one go. That is, they want to control when to consume each level of the structure. They are naturally masters. We say that they are negative types.

Greatest fixpoints (Nu) are computations that can generate the structure, on demand. That is, they can respond to requests to produce the next level of the structure. They are naturally slaves. We say that they are positive types.

# Good citizens

We can now characterize the functions which ‘behave well’ given a selection of polarity for their arguments and results. Intuitively, there must be a master which drives the order the computation, and some slaves which follow that order.

More precisely, a computation can fuse with its context if and only if the context contains exactly one master. This means that to be fusible with its context, any given function must have exactly one negative argument — and for this purpose, one should count results as arguments of the opposite polarity. (This rule is a generalization of the ‘good consumer/good producer’ rules of thumb explained in the GHC manual.)

Let’s take a look at a few simple examples. First, a map.

mapMu :: (a -> b) -> Mu (F a) -> Mu (F b)

In this case we have one negative argument (master) and one negative result (slave). Thus, we can write a fusible implementation, as follows:

mapMu :: (a -> b) -> Mu (F a) -> Mu (F b)
mapMu f (Mu phi0) = Mu $\phi1 -> phi0 (phi1 . mapF f) where mapF _ Stop = Stop mapF f (More a x) = More (f a) x (Note the lack of recursion.) Let’s consider a zip-like function: zipNu :: Nu (F a) -> Nu (F b) -> Nu (F (a,b)) We have two positive arguments (slaves) and one positive result (master). We can thus write: zipNu (Unfold psi0 s0) (Unfold psi1 s1) = Unfold psi (s0,s1) where psi (s,t) = case (psi0 s, psi1 t) of (More a s', More b t') -> More (a,b) (s',t') _ -> Stop (You may want to try your hand at writing other zip-like functions, mixing several Nu’s and Mu’s.) # Not so good citizens Not all programs have a single master. One way to proceed is to provide conversion functions from Mu to Nu and vice-versa. These functions won’t be fusible, but the programmer will know exactly when fusion does not happen, since they’ll call those functions explicitly. ## Nu to Mu What happens when we convert a slave to a master? In this case, we effectively have to conjure-up an evaluation order. That is, write a loop. The only way to do this in Haskell is to use recursion: loop :: Functor f => Nu f -> Mu f loop (Unfold psi s0) = Mu$ \phi -> let go = phi . fmap go . psi
in go s0

When loop is composed with a fusible function f, the transformations performed by f will be executed one bit at a time, as the lists are being consumed or produced by the loop.

## Mu to Nu

What happens when we convert a master to a slave? Is such a function impossible to write? No! It just means that we have to allocate a buffer so that the master can proceed at its own pace. This in turn means that we have to allocate an intermediate data structure, and thus that fusion can’t happen.

alloc :: Mu f -> Nu f

The only means to do this in Haskell is also recursion; this time in the definition of a data type:

newtype Fix f = In {out :: f (Fix f)}

Allocating a structure of type Fix f from a Mu f is immediate:

muAlloc :: Mu f -> Fix f
muAlloc m = fold m In

Traversing the structure to get a Nu type is not harder:

nuWalk :: Fix f -> Nu f
nuWalk = Unfold out

We can thus define alloc as the composition of the above two functions:

alloc = nuWalk . muAlloc

For completeness, let’s write the two missing conversions between Fix and Mu/Nu. They correspond to folding and unfolding the Fix structure. Those are standard generic programming functions that have the following implementations:

fixFold :: Functor f => (f a -> a) -> Fix f -> a
fixFold f = f . fmap (fixFold f) . out

fixUnfold :: Functor f => (a -> f a) -> a -> Fix f
fixUnfold f = In . fmap (fixUnfold f) . f

And the conversions simply wrap those:

nuAlloc :: Functor f => Nu f -> Fix f
nuAlloc (Unfold psi s0) = fixUnfold psi s0

muWalk :: Functor f => Fix f -> Mu f
muWalk s = Mu $\phi -> fixFold phi s Because no Haskell blog post is complete without a little proof, we can finish by showing that loop = muWalk . nuAlloc (but fused). We start by unfolding the equality: loop (Unfold psi s0) = Mu$ \phi -> (fixFold phi . fixUnfold psi) s0

Then we give a name to fixFold phi . fixUnfold psi and calculate:

go = fixFold phi . fixUnfold psi
= phi . fmap (fixFold phi) . out . In . fmap (fixUnfold psi) . psi
= phi . fmap (fixFold phi) . fmap (fixUnfold psi) . psi
= phi . fmap (fixFold phi . fixUnfold psi) . psi
= phi . fmap go . psi

# Conclusion

In summary, my recommendations for getting reliable and flexible fusion are to:

1. Roll out you own non-recursive types, or use a library which internally use non-recursive types;

2. Select representations accordingly to the the behavior that they should have (master/slave). Library writer should expose both versions, for the user to select.

3. Add non-fusible conversion functions at the appropriate places. Library writer should provide such conversion functions.


text:
# How do you write an equation in standard form for a line passing through (2, -3), m= -3.6?

Apr 8, 2018

$18 x + 5 y = 21$

#### Explanation:

The standard form of a linear equation is $A x + B y = C$.

Since we are given a slope and a point, the slope-intercept form of a linear equation will quickly provide us with an answer.

Recall that any (non-vertical) line can be written in the form $y = m x + b$, where $m$ is our slope and $b$ is our initial value. We find $b$ by plugging in the provided point and slope:

$y = m x + b$
$y = - 3.6 x + b$
$- 3 = - 3.6 \left(2\right) + b$
$- 3 = - 7.2 + b$
$4.2 = b$

Thus, our linear equation in slope-intercept form is $y = - 3.6 x + 4.2$. By moving our term containing $x$ to the left-hand side of the equation, we obtain our equation in standard form:

$y = - 3.6 x + 4.2$
$3.6 x + y = 4.2$

We can clean up our answer by multiplying through by $10$ to remove our decimals, then simplifying.

$3.6 x + y = 4.2$
$36 x + 10 y = 42$
$18 x + 5 y = 21$


text:
# How to show that a leontief utility function is homothetic?

usually what i do to show that a utility function is homothetic is by either showing that the function is homogeneous or if the MRS is homogeneous of degree 0. However the MRS is not defined. So im not even sure if the function is homogeneous. What i have been thinking is that it has something to do with the expansión path, as it is always constant. But how about the homogeneous degree? The specific function that i have is U(X,Y)=min{2X,6Y} Thank you

• A function $U(x,y)$ is homothetic if $U(\alpha x, \alpha y)=\alpha U(x,y)$. What is $\min(2\alpha x,6\alpha y)$? – Angela Pretorius Sep 1 '19 at 17:01
• Hi Angela. I think that is homogeneous degree 1. But you are right by saying that homogeneous implies homothetic property. However the loentief utility function is a little different so i dont think that it applys the same analysis. – neto333 Sep 1 '19 at 18:43


text:
# Composition of Two Functions

• January 9th 2008, 04:47 PM
Macleef
Composition of Two Functions
Given $f(x) = -5x + 2$ and $g(x) = -x^2 + 3$

Find $(g \circ f)(x)$:

$= g(f(x))$

$= g(-5x + 2)$

$= -(-5x + 2)^2 + 3$

$= - (25x^2 + 4) + 3$

$= -25x^2 - 4 + 3$

$= -25x^2 - 1$

And textbook answer is $(g \circ f)(x) = -25x^2 + 20x - 1$

I don't know how to get the $"20x"$ ? Please help me !
• January 9th 2008, 04:53 PM
Plato
$- \left( { - 5x + 2} \right)^2 = - \left( {25x^2 - 20x + 4} \right)$


text:
100 LED floodlight schematic help

I need a floodlight for a large yard.

I want to build one using 100 white LEDs (e.g. 10x10).

The input voltage is not important for me (220v AC , 24v DC ,12v DC all are OK)

How the schematic should be?

-
Have you added up the cost of 100 white LEDs? That's going to be a lot of money. –  Olin Lathrop Aug 22 '12 at 21:05
@OlinLathrop: 100 5mm LEDs will not be expensive! about 6$in my country. – 4r1y4n Aug 22 '12 at 21:17 Not at ebay :) – Wilhelmsen Aug 22 '12 at 21:18 You might be able to get ordinary low efficiency green or red LEDs for$.06 each, but that's not going to be true for white LEDs, especially reasonably efficient ones. Also, where is "my country"? You haven't filled in some stuff in your profile. Remember, that information is a courtesy for us, it's not for you. If you can find reasonable white LEDs for $.06 each, please post a link. – Olin Lathrop Aug 22 '12 at 21:20 @OlinLathrop: I bought 50pcs of 5mm white LEDs (alibaba.com/product-gs/317721444/…) and I paid 6$! –  4r1y4n Aug 22 '12 at 21:27

http://ledcalc.com/ offer the perfect calculator for your question.

-
Is it possible to calculate the luminance? –  4r1y4n Aug 22 '12 at 21:20
That depends on the LEDs. They both have a luminance factor and angle of light, so the distance between each, the wattage and luminance has to be calculated after you choose the LEDs –  Wilhelmsen Aug 22 '12 at 21:24
My LEDs are these: sparkinter.com/pdf/5AW4UC_8000-10000MCD.pdf –  4r1y4n Aug 22 '12 at 21:29
The mcd (milli candela, a luminence factor) is explained in japanese or something :P But I think that with a hundred of these you will have good lighting in the dark. I had 40, lit up a room VERY well –  Wilhelmsen Aug 22 '12 at 21:33
;-) its mcd is 8000 to 10000 (i say from the file name) –  4r1y4n Aug 22 '12 at 21:36


text:
TITLE

# The scalar curvature problem on the four dimensional half sphere

AUTHOR(S)
Ben Ayed, M.; El Mehdi, K.; Ould Ahmedou, M.
PUB. DATE
April 2005
SOURCE
Calculus of Variations & Partial Differential Equations;Apr2005, Vol. 22 Issue 4, p465
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature to a metric conformal to the standard one. Our proof involves the study of critical points at infinity of the associated variational problem.
ACCESSION #
16176839

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A set of 16 scalar invariants is given of the Riemann tensor which is shown to contain complete minimal sets in the Einsteinâ€“Maxwell and perfect fluid cases. All previously known sets fail to be complete in the perfect fluid case.

• Phases of Soliton-like Solutions in the Scalar-tensor Theories of Gravity. Georgieva, D. A.; Stefanov, I. Zh.; Todorov, M. D.; Yazadjiev, S. S. // AIP Conference Proceedings;11/26/2010, Vol. 1301 Issue 1, p321

The aim of the present work is to investigate numerically phases of soliton-like solutions coupled to non-linear electrodynamics in the scalar-tensor theories of gravity. As a result of the numerical experiments new non-unique solutions of the corresponding boundary value problem and their...

• Deforming metrics with negative curvature by a fully nonlinear flow. Jiayu Li; Weimin Sheng // Calculus of Variations & Partial Differential Equations;May2005, Vol. 23 Issue 1, p33

By studying a fully nonlinear flow deforming conformal metrics on compact and connected manifold, we prove that for, any metricgwith its modified Schouten tensoralways can be deformed in a natural way to a conformal metric with constant-scalar curvature at exponential rate.

• On Einstein Hermitian Surfaces. Kim, J. // Siberian Mathematical Journal;Jan2006, Vol. 47 Issue 1, p64

We show that every compact Einstein Hermitian surface with constant conformal scalar curvature is a Kahler surface and that, in contrast to the compact case, there exits a noncompact Einstein Hermitian and non-Kahler surface with constant conformal scalar curvature.

• SOME ASPECTS CONCERNING COMPACT EINSTEIN WARPED PRODUCTS. Dumitru, Dan // Bulletin of the Transilvania University of Brasov, Series III: M;2008, Vol. 1 Issue 50, p123

We establish some properties of Einstein warped products. Between others we establish that, in some conditions, there does not exist compact Einstein warped products manifolds.

• Cosmological Model with Dynamical Cancellation of Vacuum Energy and Dark Energy. Dolgov, A. D.; Kawasaki, M. // Physics of Atomic Nuclei;May2005, Vol. 68 Issue 5, p828

We propose a model with a compensating scalar field whose back reaction to the cosmological curvature cancels possible vacuum energy density down to the terms of the order of the time-dependent critical energy density. Thus, the model simultaneously solves the mystery of the compensation of...

• Nonexistence of static conformally flat solutions in a new scalarâ€“tensor theory. Singh, Tarkeshwar // Journal of Mathematical Physics;Jun87, Vol. 28 Issue 6, p1383

For the special case when the scalar field is massless and conformally invariant, it is shown that there do not exist spherically symmetric conformally flat solutions in a new scalarâ€“tensor theory proposed by Schmidt et al. [Phys. Rev. D 24, 1484 (1981)] representing disordered radiation...

• Simulating Gradient Contour and Mesh of a Scalar Field. Khan, Usman Ali; Tariq, Bismah; Raza, Khalida; Malik, Saima; Muhammad, Aoun // World Academy of Science, Engineering & Technology;Aug2011, Issue 56, p617

This research paper is based upon the simulation of gradient of mathematical functions and scalar fields using MATLAB. Scalar fields, their gradient, contours and mesh/surfaces are simulated using different related MATLAB tools and commands for convenient presentation and understanding....

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text:
# Tangent bundle to Grassmannian

Let $G :=G(k,n)$ be the Grassmannian of $k$-planes in an $n$-dimensional vector space. We automatically have the exact sequence for the universal (tautological) bundle $\mathcal{S}$:

$$0 \to \mathcal{S} \to \mathcal{O}_{G}^{n} \to \mathcal{Q} \to 0.$$

Then we have the following description of the tangent sheaf for $G$:

$$\mathcal{T}_G \sim \mathcal{Hom} (\mathcal{S}, \mathcal{Q}).$$

The way I have seen this isomorphism is described is by basically showing that cocycles for those bundles coincide (e.g. in "3254 and all that" by Eisenbud and Harris).

I was wondering if there is a more conceptual way to show this isomorphism other than comparing cocycles?

Ideally, I'd like to know the geometric construction of this map in the algebraic category (so without germs of curves, etc.).


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# PHP Authentication Security

Can someone look over this? This is the entire authentication file. It's referenced at the beginning of all my catalog editing files to make sure only the specified user is logged in. I want to make sure it's secure.

<?php
//      pageauth.php

if($_SESSION['logged_in'] != 1){ if(!isset($_SERVER['PHP_AUTH_USER'])){
echo '
Hey! You can\'t be here!

Try logging in first!
';
exit;
}elseif(md5($_SERVER['PHP_AUTH_USER']) != "04b2f0a4ad7772ca864aa569917b2d2d"){ echo '
Wrong Username!

Only the admin username and password are accepted.
'; exit; }elseif(md5($_SERVER['PHP_AUTH_PW']) != "ed972411dfcca5313ab151694af01da8"){

For obvious reasons, we need a correct password!
';
exit;
}else{
session_start();
$_SESSION['logged_in'] = 1; } } ?>  I just need to authenticate for one user. I don't need to go into databases or anything, so I thought PHP_AUTH would be a good solution. • If you're using Basic authentication, you shouldn't need to deal with sessions; the browser will re-send whatever credentials it last used. – icktoofay Jan 26 '14 at 2:17 • Thanks! Other than that, anything else? – ndm13 Jan 26 '14 at 2:21 ## 2 Answers Basic authentication is very basic. It is not really designed to allow responses, such as Wrong Username/Wrong Password. If a login attempt fails it will prompt you for the username/password again and again, (usually the browsers allow 3 attempts before failing) Once the login fails you get 1 error message. This could be caused by wrong username/password from cancel option. If you want to show bad username/password messages I suggest you implement your own authentication form (which is not difficult). It is also not a good idea to show wrong username, and wrong password explicitly as once I guess the correct username, I will be able to tell from your response saying bad password only. Then i can work on cracking your password knowing I have the username correct. I have altered your code substantially and tried to explain why in the comments If you are using apache websserver, all of what you have done can also be achieved using a a simple .htaccess and .htpasswd file Here is a website that can generate those files for you http://www.htaccesstools.com/htpasswd-generator/ <?php // pageauth.php // Asks for a username and a password and checks it. // lets define username/password first, so it is easier to change later without digging through the code.$auth_hash_algorithm = "sha256";
$user_hash = '8c6976e5b5410415bde908bd4dee15dfb167a9c873fc4bb8a81f6f2ab448a918'; // admin$pass_hash = '5e884898da28047151d0e56f8dc6292773603d0d6aabbdd62a11ef721d1542d8'; // password

// If i run your code up under E_STRICT then i get a warning
// Notice: Undefined variable: _SESSION in test.php
// We should really start the session before we use it, not in the conditional logic below

session_start();

// Next I get
// Notice: Undefined index: logged_in in test.php
// the first time through we should test the index logged_in exists before referencing it.
// Assuming logged_in could only ever be set to 1, then we could just test to see if logged_in is not set, rather then != 1

// if($_SESSION['logged_in'] != 1){ if(!isset($_SESSION['logged_in'])){

// First time through it prompts me for a user/name password
// $_SERVER['PHP_AUTH_USER'] is now set to blank and i can't even attempt to login again unless i restart the browser // I am assuming it was not the intention to only allow 1 login attempt?$user = isset($_SERVER['PHP_AUTH_USER']) ?$_SERVER['PHP_AUTH_USER'] : null;
$pass = isset($_SERVER['PHP_AUTH_PW']) ? $_SERVER['PHP_AUTH_PW'] : null; if (hash($auth_hash_algorithm, $user) ==$user_hash && hash($auth_hash_algorithm,$pass) == $pass_hash) {$_SESSION['logged_in'] = 1;
} else {
echo '
Hey! You can\'t be here!

Try logging in first!
';
exit;
}
}

echo "Success You must be logged in";

• Thank you so much! This is an amazing improvement over my code! – ndm13 Jan 26 '14 at 20:29

md5 usage may be okay on small projects, but it's generally considered weak and shouldn't be used. As of the MD5 wikipedia page:

The security of the MD5 hash function is severely compromised.

It is not recommended to use this function to secure passwords, due to the fast nature of this hashing algorithm.

So perhaps you should look into PHP's other hashing methods, say hash() itself. You can read more here: http://us1.php.net/manual/en/function.hash.php

### Important Edit:

I read a little bit more and I am incorrect! The hash method is okay. It's not great like bcrypt! I highly suggest you read both of these and understand them, I'm sure it will come in handy later on! https://security.stackexchange.com/questions/17421/how-to-store-salt and https://security.stackexchange.com/questions/211/how-to-securely-hash-passwords. And directly relating to PHP, http://us1.php.net/manual/en/function.password-hash.php

• Thanks for the feedback! I'll use hash() instead! – ndm13 Jan 26 '14 at 3:50
• I have updated my answer, please read the edit! – Alex L Jan 29 '14 at 6:26
• Thanks for the update! I feel comfortable using hash(SHA256,\$password), but if I think I need more security (for something like an entire user system in a database) I'll look into bcrypt! – ndm13 Jan 30 '14 at 23:55


text:
• ## ed762

There is no available content written by ed762

• ### Recent Forum Posts

#### Re: Permutations

Is the only difference that three have to be numbers and three have to be letters? Because your calculation "10*9*8*26*25*24" suggests that

joypav Today, 01:02

#### Permutations

Hi,

I'm not sure if this is the correct forum so if I need to post elsewhere please let me know.

I'm having trouble with

Mackenzie3 Yesterday, 23:13

#### Re: Vector calculations - Geogebra

Good!

There doesn't seem to be a command for a projection, but

(u*v)/v^2 * v
works for me.
It shows up as $w=\frac{u\,v}{v^2}v$.

Klaas van Aarsen Yesterday, 16:36

#### Re: Vector calculations - Geogebra

Ahh ok! Now I get only $\frac{5\pi}{6}$.

Now I got everything but I got stuck at the projection. Is there

mathmari Yesterday, 16:27

#### Re: Vector calculations - Geogebra

Hey [unm]mathmari[/unm]!!

Shouldn't we have 1 angle?
The angle between two vectors is usually defined as the unique shortest angle between

Klaas van Aarsen Yesterday, 15:58


text:
# Different ways to set figure size in RMarkdown

## November 02, 2016

Markdown is thought as a “lightweight” markup language, hence the name markdown. That’s why formatting options are scarce. However, there are some extensions, for instance brought by RMarkdown.

One point of particular interest is the sizing of figures. Let’s look at some ways how to size a figure with RMarkdown.

We take some data first:

data(mtcars)
names(mtcars)

##  [1] "mpg"  "cyl"  "disp" "hp"   "drat" "wt"   "qsec" "vs"   "am"   "gear"
## [11] "carb"

Not let’s plot.

## Define size in YAML header

We can define the size of figures globally in the YAML part, like this for example.

---
title: "My Document"
output: html_document:
fig_width: 6
fig_height: 4
---

# Defining figures size for R plots

## Define figure size as global chunk option

As a first R-chunk in your RMD document, define the general chunk settings like this:

knitr::opts_chunk\$set(fig.width=12, fig.height=8)

## Chunk options

We can set the chunk options for each chunk too. With figh.height and fig.width we can define the size. Note that the numbers default to inches as unit: {r fig1, fig.height = 3, fig.width = 5}.

plot(pressure)

For a plot of different size, change simple the numbers: {r fig2, fig.height = 3, fig.width = 3, fig.align = "center"}.

plot(pressure)

Alternatively, you may change the aspect ratio of the image: {r fig3, fig.width = 5, fig.asp = .62}.

plot(pressure)

Note that the aspect ratio is based on the fig.width specified by you. See here.

## Different options for different output formats

The options for figure sizing also depend on the output format (HTML vs. Latex, we do not mention Word here). For instance, in Latex percentage is allowed, as is specified on the options page: {r fig4, out.width = '40%'}.

plot(pressure)

But note that it appears to work in HTML too.

## Differnce between figure size and output size

We are allowed to specify the figure size, and secondly the size of the figure as to appear in the output. For example, if you set the size of a ggplot figure to large, then fonts etc. will appear tiny. Better do not scale up fig.height, but set out.width accordingly, eg., like this out.width = "70%".

# Using Pandoc’s Markdown for figure sizing

Alternatively, instead of using R for plotting, you can just load an image. Of course, it is possible to just use markdown for that: ![](path/to/figure/figure.png).

Change the figure size like this: ![](file.jpg){ width=50% }. Note that no space are allowed around the = (equal sign), and the curly brace { needs to come right after the ) brace; no space allowed.

Similarly, with path to local folder:

![](../../sebastiansauer.github.io/images/2016-10-17/unnamed-chunk-5-1.png.png)

{ width=20% }

Centering is not really part of markdown. But there are some workarounds. See:

![](https://sebastiansauer.github.io/images/2016-10-17/unnamed-chunk-5-1.png){ width=20% }

I used this code:

<center>
![](https://sebastiansauer.github.io/images/2016-10-17/unnamed-chunk-5-1.png){
width=20% }
</center>

# Using the knitr function include_graphics

We can use the knitr function include_graphics which is convenient, as it takes care for the different output formats and provides some more features (see here the help file).

Note that online sources are allowed. Don`t forget to load knitr previously.

# If all fails

Just resize the image with your favorite photo/image manager such as Gimp, Photoshop, Preview App etc.

here. The Book “R for Data Science” by Hadley Wickham and Garrett Grolemund (read here) is a great resource too. Read chapter 28 on diagrams here. Pandoc’s user guide has some helpful comments on figures sizing with Pandoc’s markdown .


text:
# Antiderivative

Definition: A function $F(x)$ is an antiderivative of a function $f(x)$ if $F'(x)=f(x)$.

Examples:
1. $F(x)=x^3+2x$ is an antiderivative of $f(x)=3x^2+2$, because $F'(x)=f(x)$;
2. $F(x)=4e^x$ is an antiderivative of $f(x)=4e^x$, because $F'(x)=f(x)$;
3. $F(x)=2\sqrt{x+1}$ is an antiderivative of $f(x)=\frac{1}{\sqrt{x+1}}$, because $F'(x)=f(x)$.

Note that in Example 1 not only $F(x)=x^3+2x$, but also $F(x)=x^3+2x+4$, $F(x)=x^3+2x-16$ and $F(x)=x^3+2x - 2\sqrt{5}$ are antiderivatives of $f(x)=3x^2+2$. This leads to the following result.

Theorem: If the function $F(x)$ is an antiderivative of the function $f(x)$, then for every constant $c$, also $F(x)+c$ is an antiderivative of the function $f(x)$.

Hence, in Example 2 and 3 it holds that $F(x)=4e^x+c$ and $F(x)=2\sqrt{x+1}+c$ describe all the antiderivatives of the functions $f(x)=4e^x$ and $f(x)=\frac{1}{\sqrt{x+1}}$, respectively.