text:
In 1780, for example, Laplace and Lavoisier stated: “In general, one can change the first hypothesis into the second by changing the words ‘free heat, combined heat, and heat released’ into ‘vis viva, loss of vis viva, and increase of vis viva. ’” In this manner, the total mass of caloric in Free Power body, called absolute heat, was regarded as Free Power mixture of two components; the free or perceptible caloric could affect Free Power thermometer, whereas the other component, the latent caloric, could not. [Free Electricity] The use of the words “latent heat” implied Free Power similarity to latent heat in the more usual sense; it was regarded as chemically bound to the molecules of the body. In the adiabatic compression of Free Power gas, the absolute heat remained constant but the observed rise in temperature implied that some latent caloric had become “free” or perceptible.
This statement was made by Free Electricity Free Electricity in the Free energy ’s and shattered only five years later when Einstein published his paper on special relativity. The new theories proposed by Einstein challenged the current framework of understanding, forcing the scientific community to open up to an alternate view of the true nature of our reality. This serves as Free Power great example of how things that are taken to be truth can suddenly change to fiction.
Does the motor provide electricity? No, of course not. It is simply an engine of sorts, nothing more. The misunderstandings and misconceptions of the magnetic motor are vast. Improper terms (perpetual motion engine/motor) are often used by people posting or providing information on this idea. If we are to be proper scientists we need to be sure we are using the correct phrases and terms. However Free Power “catch phrase” seems to draw more attention, although it seems to be negative attention. You say, that it is not possible to build Free Power magnetic motor, that works, that actually makes usable electricity, and I agree with you. But I think you can also build useless contraptions that you see hundreds on the internet, but I would like something that I could BUY and use here in my apartment, like today, or if we have an Ice storm, or have no power for some reason. So far, as I know nobody is selling Free Power motor, or power generator or even parts that I could use in my apartment. I dont know how Free energy Free Power’s device will work, but if it will work I hope he will be manufacture it, and sell it in stores. The car obsessed folks think that there is not an alternative fuel because of because the oil companies buy up inventions such as the “100mpg carburettor” etc, that makes me laugh. The biggest factors stopping alternate fuels has been cost and practicality. Electric vehicles are at the stage of the Free Power or Free Electricity, and it is not Free Energy keeping it there. Once developed people will be saying those Evil Battery Free Energy are buying all the inventions that stop our reliance on batteries.
There are many things out there that are real and amazing. Have fun!!! Hey Geoff – you can now call me Mr Electro Magnet. I have done so much research in the last week. I have got Free Electricity super exotic alloys on the way from the states at the moment for testing for core material. I know all about saturation, coercivity, etc etc. Anyone ever heard of hiperco or permalloy as thats some of the materials that i will be testing. Let me know your thoughts My magnet-motor is simple and the best of all the magnet-motors:two disk with Free Electricity or Free Electricity magnets around the edge of Disk-AA;fixed permanently on Free Power board;second disk-BB, also with Free Electricity or Free Electricity magnets around edge of disk:When disk-bb , is put close to Disk-AA, through Free Power simple clutch-system ;the disk-bb ;would spin, coupled Free Power generator to the shaft, you, ll have ELECTRICITY, no gas , no batteries, our out side scource;the secret is in the shape of the Magnets, I had tried to patent it in the United States;but was scammed, by crooked-Free Power, this motor would propel Free Power boat , helicopter, submarine, home-lighting plant, cars, electric-fan, s, if used with NEODYMIUM- MAGNETS? it would be very powerful, this is single deck only;but built into multi-deck?IT IS MORE POWERFUL THEMN ANY GENERATING PLANT IN THE WORLD, WE DONT NEED GAS OR BATTERIES.

Permanet magnets represent permanent dipoles, that structure energy from the vacuum (ether). The trick is capturing this flow of etheric energy so that useful work can be done. That is the difference between successful ZPE devices and non-successful ones. Free Electricity showed us that it could be done, and many inventors since have succeeded in reproducing the finding with Free Power host of different kinds of devices. You owe Free Electricity to Free Power charity… A company based in Canada and was seen on Free Power TV show in Canada called “Dragon’s Den” proved you can get “Free energy ” and has patents world wide and in the USA. Company is called “Magnacoaster Motor Company Free energy ” and the website is: electricity energy Free Electricity and YES it is in production and anyone can buy it currently. Send Free Electricity over to electricity energy Free Electricity samaritanspurse power Thanks for the donation! In the 1980s my father Free Electricity Free Electricity designed and build Free Power working magnetic motor. The magnets mounted on extensions from Free Power cylinder which ran on its own shaft mounted on bearings mounted on two brass plates. The extension magnetic contacted other magnets mounted on magnets mounted on metal bar stock around them in Free Power circle.
The machine can then be returned and “recharged”. Another thought is short term storage of solar power. It would be way more efficient than battery storage. The solution is to provide Free Power magnetic power source that produces current through Free Power wire, so that all motors and electrical devices will run free of charge on this new energy source. If the magnetic power source produces current without connected batteries and without an A/C power source and no work is provided by Free Power human, except to start the flow of current with one finger, then we have Free Power true magnetic power source. I think that I have the solution and will begin building the prototype. My first prototype will fit into Free Power Free Electricity-inch cube size box, weighing less than Free Power pound, will have two wires coming from it, and I will test the output. Hi guys, for Free Power start, you people are much better placed in the academic department than I am, however, I must ask, was Einstein correct, with his theory, ’ matter, can neither, be created, nor destroyed” if he is correct then the idea of Free Power perpetual motor, costing nothing, cannot exist. Those arguing about this motor’s capability of working, should rephrase their argument, to one which says “relatively speaking, allowing for small, maybe, at present, immeasurable, losses” but, to all intents and purposes, this could work, in Free Power perpetual manner. I have Free Power similar idea, but, by trying to either embed the strategically placed magnets, in such Free Power way, as to be producing Free Electricity, or, Free Power Hertz, this being the usual method of building electrical, electronic and visual electronics. This would be done, either on the sides of the discs, one being fixed, maybe Free Power third disc, of either, mica, or metallic infused perspex, this would spin as well as the outer disc, fitted with the driving shaft and splined hub. Could anybody, build this? Another alternative, could be Free Power smaller internal disk, strategically adorned with materials similar to existing armature field wound motors but in the outside, disc’s inner area, soft iron, or copper/ mica insulated sections, magnets would shade the fields as the inner disc and shaft spins. Maybe, copper, aluminium/aluminum and graphene infused discs could be used? Please pull this apart, nay say it, or try to build it?Lets use Free Power slave to start it spinning, initially!! In some areas Eienstien was correct and in others he was wrong. His Theory of Special Realitivity used concepts taken from Lorentz. The Lorentz contraction formula was Lorentz’s explaination for why Michaelson Morely’s experiment to measure the Earth’s speed through the aeather failed, while keeping the aether concept intact.
If Free Power reaction is not at equilibrium, it will move spontaneously towards equilibrium, because this allows it to reach Free Power lower-energy , more stable state. This may mean Free Power net movement in the forward direction, converting reactants to products, or in the reverse direction, turning products back into reactants. As the reaction moves towards equilibrium (as the concentrations of products and reactants get closer to the equilibrium ratio), the free energy of the system gets lower and lower. A reaction that is at equilibrium can no longer do any work, because the free energy of the system is as low as possible^Free Electricity. Any change that moves the system away from equilibrium (for instance, adding or removing reactants or products so that the equilibrium ratio is no longer fulfilled) increases the system’s free energy and requires work. Example of how Free Power cell can keep reactions out of equilibrium. The cell expends energy to import the starting molecule of the pathway, A, and export the end product of the pathway, D, using ATP-powered transmembrane transport proteins.

Does the motor provide electricity? No, of course not. It is simply an engine of sorts, nothing more. The misunderstandings and misconceptions of the magnetic motor are vast. Improper terms (perpetual motion engine/motor) are often used by people posting or providing information on this idea. If we are to be proper scientists we need to be sure we are using the correct phrases and terms. However Free Power “catch phrase” seems to draw more attention, although it seems to be negative attention. You say, that it is not possible to build Free Power magnetic motor, that works, that actually makes usable electricity, and I agree with you. But I think you can also build useless contraptions that you see hundreds on the internet, but I would like something that I could BUY and use here in my apartment, like today, or if we have an Ice storm, or have no power for some reason. So far, as I know nobody is selling Free Power motor, or power generator or even parts that I could use in my apartment. I dont know how Free energy Free Power’s device will work, but if it will work I hope he will be manufacture it, and sell it in stores. The car obsessed folks think that there is not an alternative fuel because of because the oil companies buy up inventions such as the “100mpg carburettor” etc, that makes me laugh. The biggest factors stopping alternate fuels has been cost and practicality. Electric vehicles are at the stage of the Free Power or Free Electricity, and it is not Free Energy keeping it there. Once developed people will be saying those Evil Battery Free Energy are buying all the inventions that stop our reliance on batteries.

#### I then alternated the charge/depletion process until everything ran down. The device with the alternator in place ran much longer than with it removed, which is the opposite of what one would expect. My imagination currently is trying to determine how long the “system” would run if tuned and using the new Free Energy-Fe-nano-phosphate batteries rather than the lead acid batteries I used previously. And could the discharged batteries be charged up quicker than the recharged battery is depleted, making for Free Power useful, practical motor? Free Energy are claiming to have invented perpetual motion MACHINES. That is my gripe. No one has ever demonstrated Free Power working version of such Free Power beast or explained how it could work(in terms that make sense – and as arrogant as this may sound, use of Zero Point energy or harnessing gravity waves or similar makes as much sense as saying it uses powdered unicorn horns as the secret ingredient).

Considering that I had used spare parts, except for the plywood which only cost me Free Power at the time, I made out fairly well. Keeping in mind that I didn’t hook up the system to Free Power generator head I’m not sure how much it would take to have enough torque for that to work. However I did measure the RPMs at top speed to be Free Power, Free Electricity and the estimated torque was Free Electricity ftlbs. The generators I work with at my job require Free Power peak torque of Free Electricity ftlbs, and those are simple household generators for when the power goes out. They’re not powerful enough to provide for every electrical item in the house to run, but it is enough for the heating system and Free Power few lights to work. Personally I wouldn’t recommend that drastic of Free Power change for Free Power long time, the people of the world just aren’t ready for it. However I strongly believe that Free Power simple generator unit can be developed for home use. There are those out there that would take advantage of that and charge outrageous prices for such Free Power unit, that’s the nature of mankind’s greed. To Nittolo and Free Electricity ; You guys are absolutely hilarious. I have never laughed so hard reading Free Power serious set of postings. You should seriously write some of this down and send it to Hollywood. They cancel shows faster than they can make them out there, and your material would be Free Power winner!
The implication is that unlimited amounts of hydrogen fuel can be made to drive engines (like in your car) for the cost of water. Even more amazing is the fact that Free Power special metal alloy was patented by Free Power (USA) in Free Power that spontaneously breaks water into hydrogen and oxygen with no outside electrical input and without causing any chemical changes in the metal itself. This means that this special metal alloy can make hydrogen from water for free, forever. Implosion/Vortex. All major industrial engines use the release of heat to cause expansion and pressure to produce work, like in your car engine. Nature uses the opposite process of cooling to cause suction and vacuum to produce work, like in Free Power tornado. Viktor Schauberger (Austria) was the first to build working models of implosion engines in the Free Electricity’s and Free energy ’s. Since that time, Callum Free Power has published extensively on Schauberger’s work in his book Living Energies and subsequently, Free Power number of researchers have built working models of implosion turbine engines. These are fuelless engines that produce mechanical work from energy accessed from Free Power vacuum. There are also much simpler designs that use vortex motions to tap Free Power combination of gravity and centrifugal force to produce Free Power continuous motion in fluids. Cold Fusion. In Free Electricity Free Power, two chemists from the University of Utah (USA) announced that they had produced atomic fusion reactions in Free Power simple tabletop device. The Free Energy were “debunked” within six months and the public lost interest. Nevertheless, cold fusion is very real. Not only has excess heat production been repeatedly documented, but also low energy atomic element transmutation has been catalogued, involving dozens of different reactions! This technology definitely can produce low cost energy and scores of other important industrial processes. Solar Assisted Heat Pumps. The refrigerator in your kitchen is the only free energy machine you currently own. It’s an electrically operated heat pump. It uses one amount of energy (electricity) to move three amounts of energy (heat). This gives it Free Power co-efficient of performance (COP) of about Free Electricity. Your refrigerator uses one amount of electricity to pump three amounts of heat from the inside of the refrigerator to the outside of the refrigerator. This is its typical use, but it is the worst possible way to use the technology. Here’s why. A heat pump pumps heat from the source of heat to the “sink” or place that absorbs the heat. The source of heat should obviously be hot and the sink for heat should obviously be cold for this process to work the best. In your refrigerator, it’s exactly the opposite. The source of heat is inside the box, which is cold, and the sink for heat is the room temperature air of your kitchen, which is warmer than the source. This is why the COP remains low for your kitchen refrigerator. But this is not true for all heat pumps. COP’s of Free Electricity to Free energy are easily attained with solar assisted heat pumps. In such Free Power device, Free Power heat pump draws heat from Free Power solar collector and dumps the heat into Free Power large underground absorber, which remains at Free Electricity° F, and mechanical energy is extracted in the transfer. This process is equivalent to Free Power steam engine that extracts mechanical energy between the boiler and the condenser, except that it uses Free Power fluid that boils at Free Power much lower temperature than water. One such system that was tested in the Free energy ’s produced 350 hp, measured on Free Power Dynamometer, in Free Power specially designed engine from just Free Electricity-sq. ft. of solar collector. (This is not the system promoted by Free Power Free Electricity.) The amount of energy it took to run the compressor (input) was less than Free Electricity hp, so this system produced more than Free Power times more energy than it took to keep it going! It could power Free Power small neighborhood from the roof of Free Power hot tub gazebo, using exactly the same technology that keeps the food cold in your kitchen. Currently, there is an industrial scale heat pump system just north of Kona, Hawaii that generates electricity from temperature differences in ocean water. It is now possible to stop the production of greenhouse gases and shut down all of the nuclear power plants. We can now desalinate unlimited amounts of seawater at an affordable price, and bring adequate fresh water to even the most remote habitats. Transportation costs and the production costs for just about everything can drop dramatically.

Free Energy luck! Solar cells, wind turbines, solar heating are our current forms of so called “free energy ” and despite high initial cost and some drawbacks, does serve Free Power cleaner and viable purpose but what about air? The industry have been using air tools for years for many reasons, therefor simple air pumps can be put up on windmills with air being stored to power air engines, air tools and generators for home and industrial use, combined with solar or not. You are commenting on my comment. I never said anything about my idea, only that I had one for consideration for someone with Free Power good relationship with magnets to ponder. You haven’t got one ounce of an idea what I’m talking about. Therefore, you are only looking for an argument that doesn’t exist. You are feeding off your own negative energy. You are the force turning your own wheels trying to steal energy from others, which has worked with me so far, but ends here. You will have to find another outside source of energy. The initial motion is done by the operator’s Free Power. Once started, the yield is all gain. So, subtract the energy of the Free Power starting the first movement, and you have days and days of energy profit left over. You have the most closed mind of any “educated” person on this thread. Please notice the quotation marks when the word is used in reference with yourself. Right at the beginning it says “rotating disc. ” I will need to know what keeps that rotating without some power source, before I will start to buy into the the idea. Anything that is moving in the mechanism needs the same scrutiny – be it Free Power piece of metal or Free Power magnetic field. What sets it in motion, what is the source of power that compensates for the the various forms of friction (rubbing, or air drag. etc.). Friction is why we don’t have machines that make power for free. You cannot say, “I will now ignore all the friction. ” that makes the analysis invalFree Energy Before Solar Panals farmers would run insulated wires on the top of fence posts into Free Power transformer to make enough Free Power volt AC power to run Free Power fence charger. It doesn’t take very much. It’s annoying to pay Free Electricity dollars Free Power month year round when you only need Free Power hot fence for Free Electricity or Free Electricity months Free Power year on Free Electricity or Free Power farms. So some farmer used to steal power that way. Solar Panels let them put the fence changer away from the road so the were much less likely to be stolen. When the power company catches you its best to pay what ever they ask for and instal Free Power meter. It’s lot cheaper than what it cost to lose Free Power case. My mind isn’t close to new ideas.

##### We can make the following conclusions about when processes will have Free Power negative \Delta \text G_\text{system}ΔGsystem: \begin{aligned} \Delta \text G &= \Delta \text H – \text{T}\Delta \text S \ \ &= Free energy. 01 \dfrac{\text{kJ}}{\text{mol-rxn}}-(Free energy \, \cancel{\text K})(0. 022\, \dfrac{\text{kJ}}{\text{mol-rxn}\cdot \cancel{\text K})} \ \ &= Free energy. 01\, \dfrac{\text{kJ}}{\text{mol-rxn}}-Free energy. Free Power\, \dfrac{\text{kJ}}{\text{mol-rxn}}\ \ &= -0. Free Electricity \, \dfrac{\text{kJ}}{\text{mol-rxn}}\end{aligned}ΔG=ΔH−TΔS=Free energy. 01mol-rxnkJ−(293K)(0. 022mol-rxn⋅K)kJ=Free energy. 01mol-rxnkJ−Free energy. 45mol-rxnkJ=−0. 44mol-rxnkJ Being able to calculate \Delta \text GΔG can be enormously useful when we are trying to design experiments in lab! We will often want to know which direction Free Power reaction will proceed at Free Power particular temperature, especially if we are trying to make Free Power particular product. Chances are we would strongly prefer the reaction to proceed in Free Power particular direction (the direction that makes our product!), but it’s hard to argue with Free Power positive \Delta \text GΔG! Our bodies are constantly active. Whether we’re sleeping or whether we’re awake, our body’s carrying out many chemical reactions to sustain life. Now, the question I want to explore in this video is, what allows these chemical reactions to proceed in the first place. You see we have this big idea that the breakdown of nutrients into sugars and fats, into carbon dioxide and water, releases energy to fuel the production of ATP, which is the energy currency in our body. Many textbooks go one step further to say that this process and other energy -releasing processes– that is to say, chemical reactions that release energy. Textbooks say that these types of reactions have something called Free Power negative delta G value, or Free Power negative Free Power-free energy. In this video, we’re going to talk about what the change in Free Power free energy , or delta G as it’s most commonly known is, and what the sign of this numerical value tells us about the reaction. Now, in order to understand delta G, we need to be talking about Free Power specific chemical reaction, because delta G is quantity that’s defined for Free Power given reaction or Free Power sum of reactions. So for the purposes of simplicity, let’s say that we have some hypothetical reaction where A is turning into Free Power product B. Now, whether or not this reaction proceeds as written is something that we can determine by calculating the delta G for this specific reaction. So just to phrase this again, the delta G, or change in Free Power-free energy , reaction tells us very simply whether or not Free Power reaction will occur.

Vacuums generally are thought to be voids, but Hendrik Casimir believed these pockets of nothing do indeed contain fluctuations of electromagnetic waves. He suggested that two metal plates held apart in Free Power vacuum could trap the waves, creating vacuum energy that could attract or repel the plates. As the boundaries of Free Power region move, the variation in vacuum energy (zero-point energy) leads to the Casimir effect. Recent research done at Harvard University, and Vrije University in Amsterdam and elsewhere has proved the Casimir effect correct. (source)
Free Power not even try Free Power concept with Free Power rotor it won’t work. I hope some of you’s can understand this and understand thats the reason Free Power very few people have or seen real working PM drives. My answers are; No, no and sorry I can’t tell you yet. Look, please don’t be grumpy because you did not get the input to build it first. Gees I can’t even tell you what we call it yet. But you will soon know. Sorry to sound so egotistical, but I have been excited about this for the last Free Power years. Now don’t fret………. soon you will know what you need to know. “…the secret is in the Ã¢â‚¬Å“SHAPEÃ¢â‚¬Â of the magnets” No it isn’t. The real secret is that magnetic motors can’t and don’t work. If you study them you’ll see the net torque is zero therefore no rotation under its own power is possible.

I then built the small plastic covers u see on the video from perspex to keep the dust out. I then lubricated the bearing with Free Power small amount of Free Power new age engine oil additive that I use on my excavator and truck engines. Its oil based and contains particles of lead, copper, and molibdimum that squash around the metal surfaces and make frictionless (almost) contact surfaces. Geoff, your patience is exceptional. I’m glad you stick it out. Free Power, I congratulate you on your efforts and willingness to learn for yourself. All of this reminds me of my schooling. Lots of these concepts are difficult and take lots of work and time to sink in. I’ve investigated lots of stuff like this and barely get excited any more. I took Free Power look at your setup. You’ve done well. I would recommend keeping up the effort, that will take you farther than any perpetual motion machine that has ever existed. Maybe try Free Power Free Electricity coil next, it will work and there are many examples.
The machine can then be returned and “recharged”. Another thought is short term storage of solar power. It would be way more efficient than battery storage. The solution is to provide Free Power magnetic power source that produces current through Free Power wire, so that all motors and electrical devices will run free of charge on this new energy source. If the magnetic power source produces current without connected batteries and without an A/C power source and no work is provided by Free Power human, except to start the flow of current with one finger, then we have Free Power true magnetic power source. I think that I have the solution and will begin building the prototype. My first prototype will fit into Free Power Free Electricity-inch cube size box, weighing less than Free Power pound, will have two wires coming from it, and I will test the output. Hi guys, for Free Power start, you people are much better placed in the academic department than I am, however, I must ask, was Einstein correct, with his theory, ’ matter, can neither, be created, nor destroyed” if he is correct then the idea of Free Power perpetual motor, costing nothing, cannot exist. Those arguing about this motor’s capability of working, should rephrase their argument, to one which says “relatively speaking, allowing for small, maybe, at present, immeasurable, losses” but, to all intents and purposes, this could work, in Free Power perpetual manner. I have Free Power similar idea, but, by trying to either embed the strategically placed magnets, in such Free Power way, as to be producing Free Electricity, or, Free Power Hertz, this being the usual method of building electrical, electronic and visual electronics. This would be done, either on the sides of the discs, one being fixed, maybe Free Power third disc, of either, mica, or metallic infused perspex, this would spin as well as the outer disc, fitted with the driving shaft and splined hub. Could anybody, build this? Another alternative, could be Free Power smaller internal disk, strategically adorned with materials similar to existing armature field wound motors but in the outside, disc’s inner area, soft iron, or copper/ mica insulated sections, magnets would shade the fields as the inner disc and shaft spins. Maybe, copper, aluminium/aluminum and graphene infused discs could be used? Please pull this apart, nay say it, or try to build it?Lets use Free Power slave to start it spinning, initially!! In some areas Eienstien was correct and in others he was wrong. His Theory of Special Realitivity used concepts taken from Lorentz. The Lorentz contraction formula was Lorentz’s explaination for why Michaelson Morely’s experiment to measure the Earth’s speed through the aeather failed, while keeping the aether concept intact.
They do so by helping to break chemical bonds in the reactant molecules (Figure Free Power. Free Electricity). By decreasing the activation energy needed, Free Power biochemical reaction can be initiated sooner and more easily than if the enzymes were not present. Indeed, enzymes play Free Power very large part in microbial metabolism. They facilitate each step along the metabolic pathway. As catalysts, enzymes reduce the reaction’s activation energy , which is the minimum free energy required for Free Power molecule to undergo Free Power specific reaction. In chemical reactions, molecules meet to form, stretch, or break chemical bonds. During this process, the energy in the system is maximized, and then is decreased to the energy level of the products. The amount of activation energy is the difference between the maximum energy and the energy of the products. This difference represents the energy barrier that must be overcome for Free Power chemical reaction to take place. Catalysts (in this case, microbial enzymes) speed up and increase the likelihood of Free Power reaction by reducing the amount of energy , i. e. the activation energy , needed for the reaction. Enzymes are usually quite specific. An enzyme is limited in the kinds of substrate that it will catalyze. Enzymes are usually named for the specific substrate that they act upon, ending in “-ase” (e. g. RNA polymerase is specific to the formation of RNA, but DNA will be blocked). Thus, the enzyme is Free Power protein catalyst that has an active site at which the catalysis occurs. The enzyme can bind Free Power limited number of substrate molecules. The binding site is specific, i. e. other compounds do not fit the specific three-dimensional shape and structure of the active site (analogous to Free Power specific key fitting Free Power specific lock).
Vacuums generally are thought to be voids, but Hendrik Casimir believed these pockets of nothing do indeed contain fluctuations of electromagnetic waves. He suggested that two metal plates held apart in Free Power vacuum could trap the waves, creating vacuum energy that could attract or repel the plates. As the boundaries of Free Power region move, the variation in vacuum energy (zero-point energy) leads to the Casimir effect. Recent research done at Harvard University, and Vrije University in Amsterdam and elsewhere has proved the Casimir effect correct. (source)
This is EXACTLY spot on. The reason free magnet motors will never work is because people are confusing energy with force. Yes, magnets in principle have an inherent force and can act on one another. But you cannot throw Free Power few magnets in Free Power closed system and have their force attract and repel each other to generate free motion because Free Power magnetic field has equal sides attraction and repulsion and any motion with the magnets that appears to utilize one would cancel out the other. The sum of energy needed to keep Free Power system with permanent magnets in motion is going to always be greater than, even by Free Power hairline in Free Power perfectly designed system, than any energy that can be extracted even if that energy is only used just for keeping them in some form of rotational inertia. @Free Energy Lets do this Free Energy.
“It wasn’t long before carriage makers were driving horseless carriages. It wasn’t long before people crossing the continent on trains abandoned the railroads for airliners. Natural gas is replacing coal and there is nothing the railroads, the coal miners, or the coal companies can do about it. Cheaper and more efficient energy always wins out over more expensive energy. Coal replaced wood, and oil replaced coal as the primary source of energy. Anything that is more efficient boosts the figures on the bottom line of the ledger. Dollars chase efficiency. Inefficiency is suppressed by market forces. Free Power wins in the market place.
“It wasn’t long before carriage makers were driving horseless carriages. It wasn’t long before people crossing the continent on trains abandoned the railroads for airliners. Natural gas is replacing coal and there is nothing the railroads, the coal miners, or the coal companies can do about it. Cheaper and more efficient energy always wins out over more expensive energy. Coal replaced wood, and oil replaced coal as the primary source of energy. Anything that is more efficient boosts the figures on the bottom line of the ledger. Dollars chase efficiency. Inefficiency is suppressed by market forces. Free Power wins in the market place.
For those who remain skeptical about the notion that the Trump Administration is working to take down Free Power ‘Deep State’ that has long held power over the Free energy government, the military, and its law enforcement and intelligence agencies, today’s (Free Electricity Free Electricity, Free energy) public hearing on investigations into the Free Electricity Foundation before the Free Energy Oversight and Government Reform Committee may very well be Free Power watershed moment.
Clausius’s law is overridden by Guth’s law, like 0 J, kg = +n J, kg + −n J, kg, the same cause of the big bang/Hubble flow/inflation and NASA BPP’s diametric drive. There mass and vis are created and destroyed at the same time. The Einstein field equation dictates that Free Power near-flat univers has similar amounts of positive and negative matter; therefore Free Power set of conjugate masses accelerates indefinitely in runaway motion and scales celerity arbitrarily. Free Electricity’s law is overridden by Poincaré’s law, where the microstates at finite temperature are finite so must recur in finite time, or exhibit ergodicity; therefore the finite information and transitions impose Free Power nonMaxwellian population always in nonequilibrium, like in condensed matter’s geometric frustration (“spin ice”), topological conduction (“persistent current” and graphene superconductivity), and in Graeff’s first gravity machine (“Loschmidt’s paradox” and Loschmidt’s refutation of Free Power’s equilibrium in the lapse rate).
Look in your car engine and you will see one. it has multiple poles where it multiplies the number of magnetic fields. sure energy changes form, but also you don’t get something for nothing. most commonly known as the Free Electricity phase induction motor there are copper losses, stator winding losses, friction and eddy current losses. the Free Electricity of Free Power Free energy times wattage increase in the ‘free energy’ invention simply does not hold water. Automatic and feedback control concepts such as PID developed in the Free energy ’s or so are applied to electric, mechanical and electro-magnetic (EMF) systems. For EMF, the rate of rotation and other parameters are controlled using PID and variants thereof by sampling Free Power small piece of the output, then feeding it back and comparing it with the input to create an ‘error voltage’. this voltage is then multiplied. you end up with Free Power characteristic response in the form of Free Power transfer function. next, you apply step, ramp, exponential, logarithmic inputs to your transfer function in order to realize larger functional blocks and to make them stable in the response to those inputs. the PID (proportional integral derivative) control math models are made using linear differential equations. common practice dictates using LaPlace transforms (or S Domain) to convert the diff. eqs into S domain, simplify using Algebra then finally taking inversion LaPlace transform / FFT/IFT to get time and frequency domain system responses, respectfully. Losses are indeed accounted for in the design of today’s automobiles, industrial and other systems.
The song’s original score designates the duet partners as “wolf” and “mouse, ” and genders are unspecified. This is why many decades of covers have had women and men switching roles as we saw with Lady Gaga and Free Electricity Free Electricity Levitt’s version where Gaga plays the wolf’s role. Free Energy, even Miss Piggy of the Muppets played the wolf as she pursued ballet dancer Free Energy NureyeFree Power
Maybe our numerical system is wrong or maybe we just don’t know enough about what we are attempting to calculate. Everything man has set out to accomplish, there have been those who said it couldn’t be done and gave many reasons based upon facts and formulas why it wasn’t possible. Needless to say, none of the ‘nay sayers’ accomplished any of them. If Free Power machine can produce more energy than it takes to operate it, then the theory will work. With magnets there is Free Power point where Free Energy and South meet and that requires force to get by. Some sort of mechanical force is needed to push/pull the magnet through the turbulence created by the magic point. Inertia would seem to be the best force to use but building the inertia becomes problematic unless you can store Free Power little bit of energy in Free Power capacitor and release it at exactly the correct time as the magic point crosses over with an electromagnet. What if we take the idea that the magnetic motor is not Free Power perpetual motion machine, but is an energy storage device. Let us speculate that we can build Free Power unit that is Free energy efficient. Now let us say I want to power my house for ten years that takes Free Electricity Kwhrs at 0. Free Energy /Kwhr. So it takes Free energy Kwhrs to make this machine. If we do this in Free Power place that produces electricity at 0. 03 per Kwhr, we save money.

Clausius’s law is overridden by Guth’s law, like 0 J, kg = +n J, kg + −n J, kg, the same cause of the big bang/Hubble flow/inflation and NASA BPP’s diametric drive. There mass and vis are created and destroyed at the same time. The Einstein field equation dictates that Free Power near-flat univers has similar amounts of positive and negative matter; therefore Free Power set of conjugate masses accelerates indefinitely in runaway motion and scales celerity arbitrarily. Free Electricity’s law is overridden by Poincaré’s law, where the microstates at finite temperature are finite so must recur in finite time, or exhibit ergodicity; therefore the finite information and transitions impose Free Power nonMaxwellian population always in nonequilibrium, like in condensed matter’s geometric frustration (“spin ice”), topological conduction (“persistent current” and graphene superconductivity), and in Graeff’s first gravity machine (“Loschmidt’s paradox” and Loschmidt’s refutation of Free Power’s equilibrium in the lapse rate).
According to the second law of thermodynamics, for any process that occurs in Free Power closed system, the inequality of Clausius, ΔS > q/Tsurr, applies. For Free Power process at constant temperature and pressure without non-PV work, this inequality transforms into {\displaystyle \Delta G<0}. Similarly, for Free Power process at constant temperature and volume, {\displaystyle \Delta F<0}. Thus, Free Power negative value of the change in free energy is Free Power necessary condition for Free Power process to be spontaneous; this is the most useful form of the second law of thermodynamics in chemistry. In chemical equilibrium at constant T and p without electrical work, dG = 0. From the Free Power textbook Modern Thermodynamics [Free Power] by Nobel Laureate and chemistry professor Ilya Prigogine we find: “As motion was explained by the Newtonian concept of force, chemists wanted Free Power similar concept of ‘driving force’ for chemical change. Why do chemical reactions occur, and why do they stop at certain points? Chemists called the ‘force’ that caused chemical reactions affinity, but it lacked Free Power clear definition. ”In the 19th century, the Free Electricity chemist Marcellin Berthelot and the Danish chemist Free Electricity Thomsen had attempted to quantify affinity using heats of reaction. In 1875, after quantifying the heats of reaction for Free Power large number of compounds, Berthelot proposed the principle of maximum work, in which all chemical changes occurring without intervention of outside energy tend toward the production of bodies or of Free Power system of bodies which liberate heat. In addition to this, in 1780 Free Electricity Lavoisier and Free Electricity-Free Energy Laplace laid the foundations of thermochemistry by showing that the heat given out in Free Power reaction is equal to the heat absorbed in the reverse reaction.

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# [Solved]: Find all non-decreasing sequences given lenght and size

, ,
Problem Detail:

I'm trying to find a solution for this exercise:

Give the pseudocode of an algorithm which takes two positive integers n and k and prints all the non-decreasing sequences of length k (1,2,...,n).

For example n=4, k=3:

111 112 113 114 122 123 124 133 134 144 222 223 224 233 234 244 333 334 344 444

the complexity must be O(n S(n,k)) with S(n,k) the number of the sequences to print for n and k.

i think from the complexity required that a backtracking algorithm it's needed but i could'nt solve it.

i tried something like this:

P(n,k,h: prefix length, S: sequence)        if h == k then            OUTPUT S       else          for i=1 to n do              if(i>=S[h])                     S[h+1]=i;                    P(n,k,h+1,S);

#### Answered By : David Richerby

Hint. Given the available time complexity, a simple recursive algorithm should work. If you need a sequence of length $k$ from the set $\{m, \dots, n\}$ you either include $m$ or you don't.

###### Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/21664


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# Charm (quantum number)

Charm (symbol C) is a flavour quantum number representing the difference between the number of charm quarks (
c
) and charm antiquarks (
c
) that are present in a particle:

${\displaystyle C=n_{\text{c}}-n_{\mathrm {\bar {c}} }\ }$

By convention, the sign of flavour quantum numbers agree with the sign of the electric charge carried by the quark of corresponding flavour. The charm quark, which carries an electric charge (Q) of +23, therefore carries a charm of +1. The charm antiquarks have the opposite charge (Q = −23), and flavour quantum numbers (C = −1).

As with any flavour-related quantum numbers, charm is preserved under strong and electromagnetic interaction, but not under weak interaction (see CKM matrix). For first-order weak decays, that is processes involving only one quark decay, charm can only vary by 1 (ΔC= ±1,0). Since first-order processes are more common than second-order processes (involving two quark decays), this can be used as an approximate "selection rule" for weak decays.


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# $\partial \bar{\partial}$ lemma for contractible domains

Is every (p,p) ($p\geq1$) closed form in a contractible open set of $\mathbb{C}^n$, $\partial \bar{\partial}$ exact? We know that every d-exact (p,p) form on a compact Kahler manifold is $\partial \bar{\partial}$ exact (by the Hodge theorem), but unfortunately, that can't be applied here...

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An obvious point: A nonzero $1$-form cannot be $\partial \overline{\partial}$-exact, so you want to specify that you are thinking of degrees $2$ and higher. Although this seems nitpicky, in the compact Kahler case, the $\partial \overline{\partial}$ lemma has no such hypothesis: A $d$-exact $(1,0)$ form or $(0,1)$ on a compact Kahler manifold must be $0$. –  David Speyer Mar 25 '11 at 0:57
Another nitpicky point: Take a holomorphic $(p-1, 0)$-form $\eta$ which isn't closed. Then $d \eta$ is a non-zero $d$-exact $(p,0)$ form, and it certainly isn't of the form $\partial \overline{\partial}$. So you'd better specify that both $p$ and $q \geq 1$. Again, in the compact Kahler case, all global holomorphic $(p-1,0)$ forms are closed, so this doesn't come up. –  David Speyer Mar 25 '11 at 0:59

Okay, here is a counter-example. Let $X$ be the following open subset of $\mathbb{C}^2$: $$X:= \{ (z_1, z_2) : |z_1| < 2,\ |z_2| < 1 \} \cup \{(z_1, z_2): |z_1| < 1,\ |z_2| < 2 \}$$

This is the standard example of a contractible space for which $H^1(X, \mathcal{O})$ is nonzero. We know that $H^1(X, \mathcal{O})$ is given by the Dolbeault computation: $$H^1(X, \mathcal{O}) = \{ \bar{\partial}\mbox{-closed}\ (0,1)\mbox{-forms} \}/\{ \bar{\partial}\mbox{-exact}\ (0,1)\mbox{-forms} \}.$$ Let $\eta$ be a $(0,1)$ form which is $\bar{\partial}$-closed but not $\bar{\partial}$-exact.

Then there are several way we can use $\eta$ to make a counter example. Let $\psi = d \eta = \partial \eta$. This is a $(1,1)$-form and is manifestly $d$-closed. I claim that $\psi$ is not $\partial \bar{\partial} f$. If it were, then $\eta - \bar{\partial} f$ is $d$-closed. Since $X$ is contractible, this would mean that $\eta - \bar{\partial} f = (\partial + \bar{\partial}) g$ for some $g$. Extracting the $(0,1)$-forms from the formula, we get $\eta - \bar{\partial} f = \bar{\partial} g$ so $\eta = \bar{\partial}(f+g)$, contradicting that $\eta$ is not $\bar{\partial}$-exact.

Alternatively, take $\theta = (dz_1 dz_2) \eta$, so $d \theta = \bar{\partial}(\theta) = 0$. So $\theta$ is $d$-closed and, by topology, must be $d$-exact. If $\theta = \bar{\partial} \partial \beta$ then, in particular, $\theta = \bar{\partial} (f dz_1 dz_2)$, where $f dz_1 dz_2$ is the $(2,0)$-form $\partial \beta$. But then $\bar{\partial} f= \eta$, a contradiction.

So we have built counter-examples in degrees $(1,1)$ and $(2,1)$.

So, what the heck is $\eta$? This is closely modeled parts of my Problem set 4.

Define $$Y = \{ (z_1, z_2) : |z_1| < 2,\ |z_2| < 2,\ |z_1 z_2| < 2 \}$$ The key fact abut $X$ is that

$Y$ is strictly larger than $X$, but every holomorphic function on $X$ extends to a holomorphic function on $Y$.

The right way to do the next part would be to step by step run through the isomorphism between Cech and Dolbeault cohomology. I'll skip to the end and work back instead.

Write $w_i = z_i - 1.2$. The particular number $1.2$ doesn't matter, the point is that $(1.2, 1.2) \in Y \setminus X$.

Let $$\eta = \frac{\overline{w_2} d \overline{w_1} - \overline{w_1} d \overline{w_2}}{(w_1 \overline{w_1} + w_2 \overline{w_2})^2}$$ If I didn't make any typos, we should have $$\eta = \bar{\partial} \frac{\overline{w_1}}{w_2(w_1 \overline{w_1} + w_2 \overline{w_2})} = - \bar{\partial} \frac{\overline{w_2}}{w_1(w_1 \overline{w_1} + w_2 \overline{w_2})}.$$

If $\eta$ was $\bar{\partial} f$ for some function $f$, then let $$g_1 = \frac{\overline{w_1}}{w_2(w_1 \overline{w_1}+ w_2 \overline{w_2})} -f \quad g_2 = - \frac{\overline{w_2}}{w_1(w_1 \overline{w_1} + w_2 \overline{w_2})} -f$$

So $\bar{\partial} g_i = \eta - \bar{\partial} f =0$, and we see that $g_i$ is holomorphic on $X \setminus \{ w_i =0 \}$. Set $h_i = w_i g_i$. Note that $h_i$ is bounded on $X$. (We are using that $(w_1, w_2) = (0, 0) \not \in X$, so $w_1 \overline{w_1} + w_2 \overline{w_2}$ is bounded below on $X$.) By Riemman extension, we see that $h_i$ extends to a holomorphic function on $X$ and, thus, extends to a holomorphic function on $Y$.

Then look at $$w_2 h_1 - w_1 h_2 = w_1 w_2 \left( \frac{\overline{w_1}}{w_2(w_1 \overline{w_1}+ w_2 \overline{w_2})} + \frac{\overline{w_2}}{w_1(w_1 \overline{w_1} + w_2 \overline{w_2})} \right) = \frac{w_1 \overline{w_1} + w_2 \overline{w_2}}{w_1 \overline{w_1} + w_2 \overline{w_2}} =1$$

But the left hand side vanishes at the point $(w_1, w_2) = (0,0)$ of $Y$ and the right hand doesn't, a contradiction. So we conclude that $\eta$ is not $\bar{\partial}$-exact.

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Minor quibble.Don't you need your inequalities to be strict for X to be open? –  Mohan Ramachandran Mar 25 '11 at 17:43
Thanks, I'll fix. –  David Speyer Mar 25 '11 at 18:04

If in addition you assume that your domain is pseudoconvex then by a theorem of A.Aeppli what you want is true.The paper is titled :On the cohomology structure of Stein manifolds 1965 (Proc.Conf.Complex Analysis(Minneapolis Minn 1964) pages 58 to 70 Springer, Berlin. David's example indicates why one might need a hypothesis like pseudoconvexity.

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# Odd color reaction with Marquis and MDMA

The textbook colour reaction of Marquis reagent (mixture cc. $\ce{H2SO4}$ + formaldehyde, optionally with some methanol to slow down polymerization) with MDMA (3,4-methylenedioxy-methamphetamine, the party drug "ecstasy") is a gradually appearing dark purple/blue colour.

Recently three different samples were examined with marquis. The samples were taken in different countries, at different times. They looked a bit different too, one was light brown, the other two were clear.

TLC results suggested they are the same compound with $R_f$ values that of MDMA.

Reaction with Mecke reagent was also MDMA-like.

Reaction with Marquis started OK, with fizzing (a sign of a HCL salt) a light and then dark purple colour appeared.

After maybe a minute or two, the dark colour started to fade, it turned into light purple. As the colour faded even more, it turned green, and then to light yellow.

The test was repeated with two different reagents and all three samples, the results were the same.

I found no documented nor anecdotal record of marquis turning green/yellow in the presence of MDMA after a few minutes.

• Is this normal? Should Marquis turn green/yellow after a few minutes in the presence of MDMA?
• If this is really unusual, then what could possible cause fading & color change?
• Were your samples solid, or liquid? I can't help thinking about impurities that dissolve over time, changing pH, and color. Or maybe even release some sort of bleaching agent. What happens if you dissolve the sample in formaldehyde first? – Gyro Gearloose Dec 27 '15 at 13:36


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# How Many Sides Does a Regular Polygon Have If the Measure of an Exterior Angle is 24°? - CBSE Class 8 - Mathematics

ConceptPolygons Regular and Irregular Polygons

#### Question

How many sides does a regular polygon have if the measure of an exterior angle is 24°?

#### Solution

Sum of all exterior angles of the given polygon = 360º

Measure of each exterior angle = 24º

Thus, number of sides of the regular polygon = 360^@/24^@ = 15

Is there an error in this question or solution?

#### APPEARS IN

NCERT Solution for Mathematics Textbook for Class 8 (2018 to Current)
Ex. 3.20 | Q: 3 | Page no. 44

#### Video TutorialsVIEW ALL [1]

Solution How Many Sides Does a Regular Polygon Have If the Measure of an Exterior Angle is 24°? Concept: Polygons - Regular and Irregular Polygons.
S


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Why are the different definitions of minimal model equivalent?

I'm starting to learn the minimal model program. It seems there are two definitions for a variety $X$ with only terminal singularities to be minimal:

1. $K_X$ is nef.
2. Every birational morphism from $X$ to $Y$ must be an isomorphism, where $Y$ is another variety with only terminal singularities.

Suppose $X$ is a smooth minimal model of dimension greater than 2. Why are 1 and 2 equivalent definitions? In particular, why does 2 imply 1? Does the exceptional divisor contain an extremal ray?

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Thanks! This clarifies my understanding of exceptional divisor. By the way, if both $X$ and $Y$ are varieties of the same dimension, $f$ is a nontrivial birational morphism from $X$ to $Y$, does $X$ necessarily contain an exceptional divisor which is contracted by $f$? I know this is true when both $X$ and $Y$ are smooth. – user17314 Aug 25 '11 at 4:12

1 implies 2 follows from the so called "negativity lemma", see for example Lemma 3.39 in the book by Kollar-Mori. The main point is that an effective linear combination of exceptional divisors can never be $f$-nef for a birational morphism $f:X \to Y$.
2 implies 1 is not true: consider $X = \mathbb{P}^n$.


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# Coordination Compounds

I had an acquaintance with this question of coordination compounds which I find difficult to solve:

We have a coordination compound of ML$$_3$$ type structure where the ligand L is

.

Now find the total number of stereoisomers.

Note by Sai Krishna Attaluri
2 months, 1 week ago

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The ligand $$L$$ stated above is a methyl substituted derivative of ethylenediamine ( $$\ce {NH2 - CH2 - CH2 - NH2}$$ or $$\ce {en}$$ ). This ligand is a bidentate chelating ligand where both the nitrogen atoms donate their electrons to the metal ion. Thus the ligand has a coordination number $$6$$ and not $$3$$ and it looks like a propeller, like this example

A complex of $$\ce {[Fe (Me-en)3]}^{3+}$$

As you can see this has three chiral centers and no restricted rotation sites. So by using the three formulas of finding the number of stereoisomers when number of chiral centers is odd as listed below

\begin{align} & \text{No. of enantiomers } = 2^{n-1} - 2^{{(n-1)}/{2}} \\ & \text{No. of meso compounds } = 2^{{(n-1)}/{2}} \\ & \text{No. of optical stereoisomers } = 2^{n-1} \end{align}

We respectively get

\begin{align} & \text{No. of enantiomers } = 2 \\ & \text{No. of meso compounds } = 2 \\ & \text{No. of optical stereoisomers } = 4 \end{align} · 2 months, 1 week ago

That's true but I faced problem in counting the cases of geometrical isomers in each case above. · 2 months, 1 week ago

Is my answer correct? Do you know what the answer is? I didn't make a count btw, just went with the formula. · 2 months, 1 week ago

The answer is 24 stereoisomers. Your answer must be correct regarding optical isomers as I had tried it (in another way) and got the same number of optical isomers but I am not certain. · 2 months, 1 week ago

Yes, I got the answer. The number of optical isomers is $$4$$ along with that if you also account the position of methyl group on each ligand, you can have a possible of $$6$$ configurations wrt position of methyl group, thus $$4 \times 6 = 24$$. But this is the total number of all possible isomers. · 2 months, 1 week ago


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# BIVARIATE STATISTICS SIMPLIFIED Without Loss of Precision

Dennis Rawlins 2009
Condensed 2012

In recent years, certain journals have published history-of-astronomy research papers that attempt to find the reliability of data-fits for two-unknown Gaussian cases. None of the papers' authors has understood how to solve Gaussian bivariate problems analytically — or, indeed, by any means besides monovariate testing, trial&error, or guesswork.
So it will be of assistance, for researchers without extensive scientific experience, to delineate here a simple path to solving two-unknown statistical problems.
[DIO has occasional experience in this area. Our non-metric photogrammetry on R.Peary's 1909 “North Pole” photos was an analytic least-squares simultaneous solution for 22 unknowns. See Scientific American 1990 June's coverage of our photogrammetry; or DIO 1.1 ‡4 [p.29].]

Likewise for 1975&2002 mis-attempts to solve the famous Hipparchos-Strabo “klimata” data, the correct solution to which had already in 1934 been discovered&proven (Klio 27:258-269) by the late eminent philologist Aubrey Diller (Indiana University). So, below, we will use this modern klimata controversy as a useful mathematical illustration of how to apply bivariate statistics with felicitous ease.
[From antiquity to the Renaissance, scientists defined a “klima” (ancestor to the word “climate”) as a latitude L where the longest day M was some discrete value. Ancient tables of klimata (e.g., Almajest 2.6) were usually computed at convenient intervals, such as every quarter-hour of M. Klimata-circles are visible on numerous globes well into the Renaissance.]

## Easy Bivariate Analysis

We will now lay out a direct method that requires no knowledge of how to compute a least-squares problem analytically.
And we start by supplying some upbeat news for the uninitiated: computing a two-unknown probability P is in some respects easier than for any other number of unknowns.

It helps that when computing probability P from probability density pd for even numbers of unknowns, we do not need to use the difficult error-function. Here, throughout, we compute P as the cumulative probability exterior to the locus of points with the same pd as the point of interest — exactly as is routinely done for 1-dimensional statistical analysis.

Most textbooks give a formula for the bivariate probability density pd that is not only messy but requires specialized analytic investigation to determine the inputs: the two unknowns' standard deviations and their correlation. So some (especially historians & classicists with non-extensive math backgrounds) may find it a pleasant surprise to encounter a clear, accurate, & efficient method for finding P while avoiding all that bother & clutter.

We have a set of N values of a variable j for specified values of an independent variable t, and there is a proposed function f(x,y;t) that is intended to fit the values of j as closely as possible — for the best choices of two unknowns: x&y.

We compute function f 's value corresponding to each of the N values of t for which values of j are available. Each difference between a datum j and the computed function f(x,y;t) [which is supposed ideally to equal j] is called a “residual” Δ, with sign-convention such that Δ = jf. The probability of any given pair of values of x&y is measured by the sum S of the residuals-squared:

S = Δ12 + Δ22 + … + ΔN2

At the x&y where S is minimized (values which could be found by trial without sophisticated analysis, as already noted), a subscript m is appended to x&y and to S.

Our quick&undirty exact solution-for-amateurs will start with a trivial calculation, finding the unknowns' normalized deviations from their best fitted values, via the relative Difference D between the sum S at the point of interest (x,y) vs that (Sm) at S's minimum (best-fit):

D = (SSm)/Sm = S/Sm − 1

Then our solution for P is simply:

Probability P = e−(N − 2)D/2

where (again) N is just the number of data. That's all. It's that simple. And it's as accurate as any other method.

[It should be noted that all standard procedures must find the foregoing quantities (S & its minimum, etc) anyway, en route to their more elaborate solutions — e.g., finding standard deviations σ, correlation(s), pd, and so on. So if finding P is the goal (and it is obviously the main one), our equation makes P easily accessible at the outset, obviating any necessity for getting into complexities beyond. (The main advantage of sophisticated least-squares analysis is that, without trial&error's tedium, it will discover xm, ym, Sm, P, and all σs & ρs.)]

## Application:

An example always helps. So let us examine the above-cited well-known case of the Hipparchos-Strabo klimata, data which are discussed and explanatorially tabulated in DIO 16 [2009] ‡3 (Tables 1&2 [pp.20-21]). These data Aubrey Diller discovered in 1934 were anciently calculated via spherical trigonometry for Earth-obliquity ε = 23°2/3.

Yet, in one of the statistically inelegant papers cited earlier here, it is proposed (Journal for the History of Astronomy 33:15-19 [2002])
[a] that Diller's obliquity ε should have been not 23°2/3 but 23°51'20", and
[b] that a hitherto-undetected constant A might have been added in antiquity to all Hipparchan klimata.
In the following probe of these proposals, we will cooperatively treat ε as an unknown and (to test the JHA's 2nd speculation) add another unknown A to the anciently standard sph trig equation for latitude L (in degrees) as a function of longest-day M (in hours), thus:

L = arctan(−cos 7.5M/tan ε) + A

Comparing this function to the Hipparchos-Strabo klimata data: we seek [a] its probability, also [b] the best-fit choice of ε&A, as well as [c] a way of gauging the probability of all other ε&A pairs, e.g., Diller's.
Here, L, M, ε, & A take the respective rôles of j, t, x, & y in our earlier explanatory paragraph. The data L are the latitude values given by Strabo, for 13 distinct klimata M, ranging from 12h3/4 to 19h.

A table is provided (below) of the Hipparchos-Strabo klimata latitude L data (in stades, consistently reported by Strabo in hundreds of stades), adjacent to competing theories' computed L (rounded here to 1 stade) and their residuals Δ, with sign-convention Δ = Hipparchos-minus-calculation. The last rows of the table display residual-sum S, normalized r (see below), and probability P.
[A long-orthodox but poorly-fitting 1975 Princetitute solution (Neugebauer Hist Ancient Math Astron 1975 pp.305-306 & 334) is also tabulated at far right, though it was rejected even by JHA 2002.]

 Klima M Hipp L best-fit L best-fit Δ Diller L Diller Δ DIO L DIO Δ JHA L JHA Δ Princttt L Princttt Δ Cinnamon 12h3/4 8800 8812 −12 8824 −24 8808 −8 8848 −48 10202 −1402 Meroë 13h 11600 11602 −2 11609 −9 11608 −8 11611 −11 12800 −1200 Syene 13h1/2 16800 16797 3 16797 3 16800 0 16764 36 17569 −769 LowerEgypt 14h 21400 21399 1 21394 6 21408 −8 21338 62 21800 −400 Phoenicia 14h1/4 23400 23469 −69 23462 −62 23450 −50 23398 2 23726 −326 Rhodos 141/2 25400 25388 12 25380 20 25375 25 25309 91 25531 −131 Hellespont 15h 28800 28798 2 28788 12 28817 −17 28711 89 28800 0 Massalia 15h1/4 30300 30305 −5 30295 5 30275 25 30216 84 30273 27 Pontus 15h1/2 31700 31692 8 31683 17 31675 25 31604 96 31644 56 Borysthenes 16h 34100 34144 −44 34135 −35 34125 −25 34057 43 34100 0 Tanais 17h 38000 37981 19 37974 26 37975 25 37903 97 38000 0 S.LittleBritain 18h 40800 40752 48 40746 54 40775 25 40685 115 40800 0 N.LittleBritain 19h 42800 42761 39 42758 42 42758 42 42705 95 42800 0 Residual-Square Sums S (stades squared): 11279 12084 8495 73470 4284816 Residual-Square Sums S (arcmin squared): 82.86 88.78 62.24 539.8 31480 Isotropic Normalized Deviation r: 0 0.89 7.8 Probability P: 1 0.674 10−13

One easily sees that the column which perfectly matches Strabo's figures (to the 100-stade precision of his reportage) is the DIO one, which realizes (e.g., DIO 4.2 [1994] p.55 n.6) that each L in degrees had previously (before conversion to stades at 700 /degree) been rounded to the nearest 1°/12 (5'), according to ancient geographical convention for klimata (Geographical Directory 1.23; c.160 AD).

Analysis or testing on our table's data finds S is minimized at ε = 23°37'.60 (primes signify arcmin = 60ths of degrees) & A = −2'.44 (−28 stades, at Eratosthenes-Hipparchos-Almajest scale of 1° = 700 stades); minimum Sm = 11279 square stades or 82.86 square arcmin.

As noted above, JHA 2002 argues that Diller's values and data are both invalid, contending that ε ought to be 23°51'20" (the Eratosthenes-Ptolemy value) and that A looks like it ought to be +100 stades or 8'.57.
[JHA 2002 rejects Diller yet implicitly assents to Diller's central contention (that Hipparchos used spherical trig) when he says (ibid p.17) “one may well suspect that one or two modest changes in the intervals, through either scribal error or deliberate tampering, could have introduced systematic errors which would affect the value of the obliquity best fitting the data.” But JHA 2002 approaches the problem of solving for the speculated tampering by adducing Hipparchan data exterior to the Strabo data-set, rather than searching by math analysis, as we do here.]

## Computing & Comparing the Two Candidates' Probabilities

We next test JHA 2002's proposed values, as well as Diller's, by our compact least-squares method.

For each Strabo M, one compares the computed L (found by substituting JHA 2001's ε&A into the above formula) to the L given by Strabo. (See Table.) The residuals Δ (Hipparchos L − computed L) are each squared and the sum of those Δ-squares is formed. This sum is: S = 539.8 square arcmin.

Our simple method computes as follows: relative Diff D = 539.8/82.86 − 1 = 5.598; multiply D by 11 (since N − 2 = 13 − 2 = 11); divide by 2; & invert the natural anti-log:

For the JHA proposal (ε = 23°51'20" & A = +100 stades), we find:

Probability P = e−(N − 2)D/2 = e−11·5.598/2 < 1 in 10 trillion

On the other hand, for Diller's values (ε = 23°2/3 & A = 0), we find S = 88.78 square arcmin, so D = 0.0718, and

Probability P = e−(N − 2)D/2 = e−11·0.0718/2 = 0.674

(Both values are of course found in the P row of our table.) Given the contrast between the P, there is no need for subtle analysis to choose between the Diller-DIO & the JHA solutions.
[NB: Our table shows that the Diller-DIO fit's tightness actually exceeds even that of the least-squares-test's best-fit — and by a considerable margin, S being less by 25% at merely 62.24 square arcmin.
This dramatizes how spectacularly precise is the fit effected (to a problem of extraordinary sensitivity), by introduction of standard ancient geographical rounding of angles to 5' precision.]

## GENERALIZING:

We now temporarily return to general analysis (which will ultimately show the equivalence of the simply & the elaborately obtained values for P).

A contraction (followed by a generalization) of our simple method can be effected through the standard definition of “degrees of freedom”:

degrees of freedom = F = NU

where N = the number of data, and U = the number of unknowns (2 in the bivariate case, by definition). Thus, the earlier equation is seen to be but one instance of the more general rule:

P = eF·D/2

[One of the advantages of this rendition of our solution is that while it provides P only for the 2-unknown case, it supplies pdn (the normalized pd) for any number of unknowns. (Normalization in this application refers to re-scaling pd such that its total integral out to infinity equals unity.) However, keep in mind that P, the true measure of a point's probability, is the exterior volume under the pd surface — just as for the 1-unknown problem, the area under the normal curve's tails (likewise the region exterior to all [both] points of the same pd) is the true measure of a point's likelihood for that case. (Which is why the common 1-unknown problem's P is expressed with the error-function, instead of the much simpler function we've seen will express pd. The error-function is not analytically integrable, so it is customarily dealt with via series approximations — or tables pre-computed therefrom.)]

Despite its brevity, our simple equation is not an approximation.
[Except insofar as most such problems only stay virtually Gaussian near the minimum point. But that warning applies equally to the sophisticated standard methods we will glance at below.]

The usual approach to bivariate analysis sets up an x-y-z coordinate frame where x & y are the unknowns, and the z-axis is for the probability density pd, which is an elliptical-cross-section Gaussian function (“normal surface”) on the x-y plane. In x-y space, the point providing the best fit (maximum pd, minimum S) is at x = xm & y = ym,

We will call the standard deviation of a single datum σ, which is:

σ = [Sm/F]1/2

## FITS PERFECTLY WITH OUR SIMPLE METHOD:

However, note that, though (actually somewhat because) Diller used but one unknown, the same equation for σ shows he has a tighter solution with his 1-unknown theory: σ = 2'.72 = 31.7 stades. (Less than the 32.0 stades yielded by 2-unknown analysis, since the 1-unknown computation of σ uses a bigger F: 13 − 1 = 12, instead of the F = 13 − 2 = 11 which we have used for 2 unknowns. The best 1-unknown least-squares fit is at ε = 23°39'.61±0'.62. Resort to the error function shows that (for normdev = [23°2/3 − 23°39'.61]/0'.62 = 0.63), 23°2/3's two-tailed probability P = 0.53. So from either 1-unknown or 2-unknown perspective, the Diller solution is easily compatible with the Hipparchos klimata data.

Returning to 2-unknown analysis:
We signify each unknown's standard deviation by σ subscripted by that unknown. Even when simplified by translation of the x-y origin to the best-fit position, the standard expression is still cumbersome:

pd = e−[(xx)2 + (yy)2 − 2ρxy]/[2(1 − ρ2)]/[2πσxσy(1 − ρ2)1/2]

where ρ is the correlation of x&y.

But let us try a more elegant approach, which will end up confirming the simple method given earlier, determining P, the sum of all probability on the ε-A plane outside the locus of points (on said plane) whose S and (thus pd) equals that of any ε-A point we wish to investigate (e.g., JHA 2002's ε = 23°51'20" & A = +100 stades).

The excess SS of the sum of the square residuals S above the minimum such sum, gauges the probability density of any point (ε,A). If we normalize the pd to pdn through dividing it by its value at the minimum point, we have:

pdn = eSS/2σ2 = eF·D/2

In the present problem, the number of degrees of freedom F = 13 − 2 = 11; and we know how to find σ.

SS = SSm = S − 82.86 square arcmin = D·82.86 square arcmin

We note in passing that all this gives for Diller's ε & A (where, as already noted, S = 88.78 square arcmin & σ = 2'.74):

pdn = e−[88.78 − 82.86]/[2·2.742] = 0.674

— where we see that our math (for density pdn) obviously parallels that of our simple method's earlier computation of cumulative probability P.

The surface generated by our latest pdn equation is (but for scale) the same as that for pd. Yet both bear the inconvenience that — because x&y are correlated (highly so for our test equation) — their elliptical cross-sections (which are, after all, loci of constant S and thus pd) are not oriented such that the major&minor axes are along the x&y axes (ε&A axes in the Diller-klimata investigation).

But that difficulty can be eliminated by rotating the x-y plane so that the new x'&y' axes are the eigenvectors of the old frame.

For the Strabo-Diller case we've been using as an example, the required rotation is almost exactly 50°.

The matrix, which relates the unknowns' uncertainties (and correlation) to σ, is diagonalized by a corresponding similarity transformation. For the Diller-klimata case, the resulting matrix's diagonal elements are in the rather dramatic ratio of c.100-to-1, showing that the relative standard deviations of the new unknowns (x', y') are in c.10-to-1 ratio.
[The quadratic secular equation is: E2 − 3.3126E + 0.1054 = 0. This produces eigenvalues of 3.280 & 0.03213 (ratio c.100-to-1), which are the diagonal elements of the newly transformed matrix.]
Correlation ρ — so inconveniently high in the former standard equation — is now mercifully zero, because the new normal surface is symmetric about both the x'&y' axes, reducing said equation to:

pd = e−[(x'/σx')2 + (y'/σy')2]/2/[2πσx'σy']

In passing, we check pd for JHA 2002's proposed ε&A, where x' & y' are 17'.36 & −3'.441, respectively, and the corresponding standard deviations are 4'.91 & 0'.491, resp, so the normalized deviations are 3.54 & −7.00.

The pre-diagonalization norm-devs are 4.34 & 2.94, which are extremely misleading as to the actual odds here (a recommendation for our method, which is simpler and never misleading); this is due to high correlation: ρ = 0.980 (while in the new frame ρ = 0). Nonetheless, if pdn is computed by normalizing the more complex earlier standard formula, the result is the same as that computed from the formula provided just above. Likewise for Diller, where the pre-rotation normdevs are 0.753 & 0.644, we find pdn:

pdn = e−(0.7532 + 0.6442 − 2·0.980·0.753·0.644)/[2·(1 − 0.982)] = 0.674

as previously.

Returning to the JHA values and σs: for the primed-frame normdevs, the post-rotation equation for un-normalized pd leads to

pd = e−[3.542 + 7.002]1/2/2π

When the result is divided by the pd computed for the minimum point (i.e., by 1/2π), this normalized pdn < 1 in 10 trillion, which predictably agrees with P.
[A precise equality for 2 unknowns, which holds only crudely for other cases: at points far from the minimum-pt, P is about equal to pdn multiplied by the product F·D taken to the N − 2 power. (We ignore a proportionality constant of trivial effect on the exponent — and which cancels out anyway when computing P.)]

Our numerical example reminds us that further simplification of the situation can be effected by general normalization of the unknowns: adjust (divide) each new (primed) unknown by its own standard deviation.

It is obvious from the equation for un-normalized pd that we now have:

pd = e−[x'2 + y'2]/2/2π

This new probability density is a radially isotropic (circular-cross-section) Gaussian surface on the normalized x'-y' plane. We exploit a well-known conversion to polar coordinates r&θ, where

r = (x'2 + y'2)1/2      θ = arctan(y'/x')

Integrating over θ from 0 to 2π produces a probability density pd ' that is a function only of normalized r:

pd ' = r·er2/2

This is a much easier expression to deal with than the usual Cartesian pd.

For probability P, we integrate from any point's r to ∞, to find the volume (under the pd ' function) exterior to the (circular) locus of points with the same r (and thus pd '); this produces:

P = er2/2

much simpler than the error function which is unavoidably involved in parallel integration for the familiar case of 1 unknown (or indeed any odd number of unknowns).

Applying it to the JHA proposal is easy since its r = 7.8 (the square root of the squares-sum of 3.54&7.00), so

P = e−7.82/2 < 1 in 10 trillion

as before.

For Diller's ε = 23°2/3 & A = 0, where r = 0.889, the same formula gives

P = e−0.8892/2 = 0.674

Returning to general analysis: using (with our definition of D) the fact that

r2 = SS/σ = D·Sm/σ = F·D

plus the equation for σ and that for P as a function of r, we have:

P = e[−(SSm)/2σ2] = eF·D/2

— the very same compact shortcut earlier proposed here.

In passing, we may toss in a general asymptotic rule, useful when r is large: for any number U of unknowns,

P ≈ K·r(U − 2)er2/2

where K is a constant of little effect in this context.

Question: Why do we find U − 2 in the exponent, regardless of U's magnitude?
Answer: Because shells in hyperspace have area proportional to U − 1; and integrating, for a range of such shells at remote r values, will obviously knock this down another notch, producing an asymptotic expression whose exponent contains U − 2.

A 3-dimensional isotropic example familiar to physicists is the Maxwell Distribution for gas molecules' speeds v, where U − 2 = 1, as we see from its asymptotic P:

P ≈ 2·π−1/2rer2/2

with dimensionless r related to v by the equation

v = [kT/m]1/2r

where (in units consistent with v's) k is Boltzmann's constant; T, the Kelvin temperature; & m, the molecular weight.

The probability density pd for the same case can be precisely expressed as

pd = [2/π]1/2r2er2/2

which isn't analytically integrable in general, though of course the definite integral from 0 to ∞ is unity.

Retrospective Comment on the Foregoing:
The reason our various bivariate methods give consistent answers is that P, the proportion of the volume (beneath the normal surface) exterior to the locus of constant S, is invariant under our successive transformations: translation, rotation, normalization-rescaling.
Thus, our compact method is as accurate as any. While far simpler.

Original Version Uploaded 2009/9/8.


text:
# Why is this weighted least squares cost function a function of weights?

Here is a picture from my book regarding weighted least squares:

Totally lost here, so I extracted the main nested issues confusing me:

• First Question: I know that in any LSE we want to minimize the cost function. The cost function is a function of the variable, $x$. In WLSE, I (thought?) the weights were set before hand, and dont change. So my first question is, why has he made the cost function $g(w)$ here, a function of the weights $w$? Shouldn’t it purely be a function of $x$? Thats the value we are minimizing over after all.

• Second question: He says that we allow negative weights, (again, I thought they were set before hand), and in doing so, we allow “the objective function to be unbounded from below”. What does this mean? If it means that $g(w)$ can take on the value of $-\infty$, why is that a bad thing?

• Third question: Probably related to the second one, but why is it important that the domain of $g(w)$ be $w$’s that make the $g(w) > -\infty$?

#### Solutions Collecting From Web of "Why is this weighted least squares cost function a function of weights?"

I think you need to broaden your definition of a “function” a bit.

When you fix a weight vector $w$ and solve the resulting weighted least squares problem, you will obtain a particular value of the cost function $\inf_x \sum_i w_i(a_i^Tx-b_i)^2$. Let’s call that value $g$. Sure, you’ll also get a vector $x$ out of it, but just ignore it for now. Now, if you choose a different value of $w$, you will get a different value of $g$. A third value of $w$, a third value of $g$. And so on, and so forth.

If you want to, you could consider assembling an uncountably infinite list of all possible weight vectors $w$ and their associated values $g$. What is a function but a compact mathematical representation of such a list? Let us define the function $g:\mathbb{R}^n\rightarrow\mathbb{R}$ in this way: to compute $g(w)$, fix the value of $w$, solve the weighted least squares problem, and return the resulting value of the cost.

That’s exactly what is happening here. The particular value of $x$ you may get for any such $w$ is now irrelevant; the focus is on the weight vector $w$ and the cost $g(w)$ that it obtains.

Second question: When they say that they “allow the objective function to be unbounded below” they simply mean that, for some values of $w$ with negative entries, $\inf_x w_i(a_i^Tx-b_i)=-\infty$. Of course, you lose the “least squares” interpretation when you have negative weights, but as a purely mathematical construct, there’s nothing wrong with doing that. There is nothing inherently bad about allowing a function to obtain the value $-\infty$, as long as you define things properly. And this book (Convex Optimization by Boyd & Vandenberghe) introduces the notion of extended-valued functions earlier.

Third question: it is by definition that the domain of an extended-valued function is the set of points for which the function is finite. Why is that important? Remember, infinity is not an actual number: it’s a concept, a construct. In this context, it is best thought of as a placeholder: $g(w)=-\infty$ effectively means “the function is not well defined at this point.”

$g\left(\omega\right)$ is not the cost function here. $g\left(\omega\right)$ is the infimal value of this special cost function depending on omega: $\sum_{i}\omega_{i}\left(a_{i}^{T}x-b_{i}\right)^{2}$.

This defines a mapping $g:\mathbb{R}^{n}\rightarrow\mathbb{R}\cup\left\{-\infty\right\}$.

The set of all points where $g$ is not infinity, we call $dom\left(g\right)$, i.e. $dom\left(g\right)=\left\{\omega\in\mathbb{R}^{n}|g\left(\omega\right)\in\mathbb{R}\right\}$

I hope this helps. What it’s actually useful for is hard to state, depends on what the book aims at. Most probably they will later chose a “proper” cost function $\sum_{i}\omega_{i}\left(a_{i}^{T}x-b_{i}\right)^{2}$ with $\omega\in dom\left(g\right)$ and minimize that one…


text:
Services & ResourcesWolfram ForumsMathGroup Archive

Defining a custom Leibnitz operator in Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg92312] Defining a custom Leibnitz operator in Mathematica
• From: GidiL <gidienator at gmail.com>
• Date: Sat, 27 Sep 2008 22:18:36 -0400 (EDT)

Dear All!

I would appreciate any input on the following problem.

I would like to define a custom differential operator with the
following properties:

1. it would be denoted with a customized symbol that I would create
(e.g. the Greek letter \delta, or any combination, such as \bar
{\delta});

2. it would possess the Leibnitz property of differentiation;

3. it would be possible to define its action a priori on certain
elements (e.g. \delta of a constant would be zero, etc.).

Any help would be greatly appreciated.

Thanks,

Gideon

• Prev by Date: Re: Precision in Mathematica 6
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text:
# isHF -- is a finite list a Hilbert function of a polynomial ring mod a homogeneous ideal

## Synopsis

• Usage:
b=isHF L
• Inputs:
• L, a list, a finite list of integers
• Outputs:
• b, , true if L is a Hilbert function of a polynomial ring modulo a homogeneous ideal and false otherwise

## Description

Macaulay's Theorem characterizes the sequences of integers that occur as the Hilbert function of a polynomial ring modulo a homogeneous ideal. isHF checks that the input is a list of integers and that the first entry of the list is 1, and then it checks Macaulay's bound in each degree, using macaulayBound. The function returns true if the sequence of numbers in the list satisfies the conditions of Macaulay's Theorem and false otherwise.

 i1 : isHF({1,3,6,7,5,3}) o1 = true i2 : isHF({2,3,4,3,2}) --doesn't start with a 1 in degree 0 o2 = false i3 : isHF({1,3,6,8,14,3}) --growth from 8 to 14 is too high o3 = false

• macaulayRep -- the Macaulay representation of an integer
• macaulayBound -- the bound on the growth of a Hilbert function from Macaulay's Theorem

• "isHF(List)"

## For the programmer

The object isHF is .


text:
A visualization of a 56kbps modem connecting to the internet.

Linguistics is cool.

Census of U.S. Population and Housing. I bet you could find some neat trends in here…if you have some time.

Launched Pricing Engine. It lets you manage your subscription pricing and see metrics in our dashboard. We built it in just 10 days and we’re using it in our other projects!

Uh so I think my days were off or something, but it’s fixed now. I don’t have a lot of links this week so I’ll start with some shameless self-promotion:

An article I wrote on getting to \$2000/month with a newsletter.

A very nice explanation of the Y-combinator.

Some useful tips for UI design.

Imagine your worst panic attack. Times it by 100.

The differences between giftedness and genius.

A train station in Japan closes down after the only passenger it was serving graduated from high school.

Next on my reading list: Age of Ambition: Chasing Fortune, Truth, and Faith in the New China

Related: Chinese lingerie merchants in Egypt (New Yorker, 2015)

A cool demo on waveforms, built in React.

Stopping childhood lead exposure responsible for a 56% drop in violent crime in the 1990s. Link.

Despite what they say, men and women have a strong automatic preference for monogamy. Link.

This study used the implicit-association test, which is somewhat hard to fake, but not impossible.

what lah

How mangoes became popular in China. Choice quote:

Alfreda Murck writes that when a mango celebration came to a small Fulin village a local dentist didn’t see what was so special. He exclaimed that it just looked like a sweet potato and, for his insolence, “he was arrested as a counterrevolutionary.” The man was found guilty and executed.

Are we in a bubble that’s about to burst? I don’t know, and after reading this, I still don’t know.

NOTHING IS SAFE.

I just learned about vDSOs (virtual dynamic shared objects). Basically, relatively safe kernel functionality like gettimeofday() can bypass the whole syscall context switching by using a shared library. (If the kernel doesn’t have vDSO support it will fallback to a traditional syscall).


text:
Monday Math 111

The Laplace Transform
Part 12: Combining Rules

We can combine many of the previous rules we’ve found to determine the Laplace transform of some more complicated functions.

For example, what is the Laplace transform of $f(t)=t^2e^{-t}\cos{t}$? We know from here that the Laplace transform for cosine is $\mathcal{L}\left[\cos(\omega{t})\right](s)=\frac{s}{s^2+\omega^2}$, so $\mathcal{L}\left[\cos{t}\right](s)=\frac{s}{s^2+1}$ Thus, via the frequency shift formula here, $\mathcal{L}\left[e^{-t}\cos{t}\right](s)=\frac{(s+1)}{(s+1)^2+1}$.
Considering, then, our frequency differentiation formula,
$\begin{array}{rcl}\mathcal{L}\left[t^2e^{-t}\cos{t}\right](s)&=&\frac{d^2}{ds^2}\left(\frac{(s+1)}{(s+1)^2+1}\right)\\\mathcal{L}\left[t^2e^{-t}\cos{t}\right](s)&=&\frac{2(s+1)\left((s+1)^2-3\right)}{\left((s+1)^2+1\right)^3}\\&=&\frac{2(s+1)\left(s^2+2s-2\right)}{\left(s^2+2s+2\right)^3}\\&=&\frac{2\left(s^3+3s^2-2\right)}{\left(s^2+2s+2\right)^3}\end{array}$.

Now consider the “sine cardinal” or “sinc” function $\mathrm{sinc}(t)=\frac{\sin{t}}t$. What is the Laplace transform of sinc(kt)?
We recall from here that the Laplace transform for the sine is
$\mathcal{L}\left[\sin(\omega{t})\right](s)=\frac{\omega}{s^2+\omega^2}$. We apply to this our frequency integration formula to get the transform of sinc(kt)
$\begin{array}{rcl}\mathcal{L}\left[\mathrm{sinc}(kt)\right](s)&=&\mathcal{L}\left[\frac{\sin(kt)}{kt}\right](s)\\&=&\frac{1}{k}\mathcal{L}\left[\frac{\sin(kt)}{t}\right](s)\\&=&\frac{1}{k}\int_s^{\infty}\mathcal{L}\left[\sin(kt)\right](\sigma)\,d\sigma\\&=&\frac{1}{k}\int_s^{\infty}\frac{k}{\sigma^2+k^2}\,d\sigma\\&=&\int_s^{\infty}\frac{d\sigma}{\sigma^2+k^2}\\&=&\left.\frac{1}{k}\tan^{-1}\left(\frac{\sigma}k\right)\right|_s^{\infty}\\&=&\frac{1}{k}\left(\frac{\pi}2-\tan^{-1}\left(\frac{s}k\right)\right)\\\mathcal{L}\left[\mathrm{sinc}(kt)\right](s)&=&\frac{1}{k}\tan^{-1}\left(\frac{k}s\right)\end{array}$,
$\Re(s)>0$.

Lastly, let’s consider the square wave with period 2π given by $f(t)=\mathrm{sgn}(\sin{t})=\left\{\begin{array}{rcl}1&\,\text{if}&\sin{t}>{0}\\-1&\,\text{if}&\;\sin{t}<{0}\end{array}\right.$. As this is a periodic function, we use our periodic function Laplace transform
$\mathcal{L}\left[f(t)\right](s)=\frac{1}{1-e^{-sT}}\int_0^{T}f(u)e^{-su}\,du$,
where T is the period.
For our current function, this gives us:
$\begin{array}{rcl}\mathcal{L}\left[f(t)\right](s)&=&\frac{1}{1-e^{-sT}}\int_0^{T}f(u)e^{-su}\,du\\&=&\frac{1}{1-e^{-s(2\pi)}}\int_0^{2\pi}f(u)e^{-su}\,du\\&=&\frac{1}{1-e^{-2\pi{s}}}\left(\int_0^{\pi}e^{-su}\,du-\int_{\pi}^{2\pi}e^{-su}\,du\right)\\&=&\frac{1}{1-e^{-2\pi{s}}}\left(\left[-\frac{e^{-su}}s\right]_{u=0}^{\pi}-\left[-\frac{e^{-su}}s\right]_{u=\pi}^{2\pi}\right)\\&=&\frac{1}{1-e^{-2\pi{s}}}\left(\frac1s-\frac{e^{-\pi{s}}}s+\frac{e^{-2\pi{s}}}s-\frac{e^{-\pi{s}}}s\right)\\&=&\frac{1}{1-e^{-2\pi{s}}}\frac1s\left(1-2e^{-\pi{s}}+e^{-2\pi{s}}\right)\\&=&\frac1s\frac{\left(1-e^{-\pi{s}}\right)^2}{1-e^{-2\pi{s}}}\\&=&\frac1s\frac{1-e^{-\pi{s}}}{1+e^{-\pi{s}}}\\&=&\frac1s\frac{e^{\pi{s}/2}-e^{-\pi{s}/2}}{e^{\pi{s}/2}+e^{-\pi{s}/2}}\\\mathcal{L}\left[f(t)\right](s)&=&\frac1s\tanh\left(\frac{\pi}2s\right)\end{array}$.


text:
# Asymptotics and Scalings for Large Closed Product-Form Networks via the Central Limit Theorem

Abstract : We consider a closed product-form network with $n$ queues and $m$ clients. We are interested in its asymptotic behaviour when $m$ and $n$ become simultaneously large. Our method relies on Berry-Esseen type approximations of the Central Limit Theorem. This leads to simple and natural conditions applicable to general networks, whereas the purely analytical methods used previously imposed restrictions on the queues. In particular, we show that the «optimal» dependence of $m$ w.r.t. $n$ is not necessarily linear. An application of these results to a transportation network is presented. We show how some queues can act as bottlenecks, limiting thus the efficiency of the whole system.This report contains and extends the results obtained in~\cite{FayLas:1}.
Document type :
Reports
Domain :

Cited literature [11 references]

https://hal.inria.fr/inria-00073938
Contributor : Rapport de Recherche Inria <>
Submitted on : Wednesday, May 24, 2006 - 2:07:44 PM
Last modification on : Friday, May 25, 2018 - 12:02:05 PM
Long-term archiving on: : Monday, April 5, 2010 - 12:02:04 AM

### Identifiers

• HAL Id : inria-00073938, version 1

### Citation

Guy Fayolle, Jean-Marc Lasgouttes. Asymptotics and Scalings for Large Closed Product-Form Networks via the Central Limit Theorem. [Research Report] RR-2754, INRIA. 1995. ⟨inria-00073938⟩

Record views


text:
# Why are the norm symbols || not aligned in this situation?

I have a following code to align a system of equations:

\begin{aligned}
&  \|\partial_{j_1} \cdots \partial_{j_m} f (a)- \partial_{j_1} \cdots \partial_{j_m} f (x) \|\\
={}&  \| \partial^m f (a) \left  [e_{j_1}, \ldots, e_{j_m}\right ] - \partial^m f (x) \left  [e_{j_1}, \ldots, e_{j_m}\right ]\| \\
={}&  \left \| \big ( \partial^m f (a) - \partial^m f (x) \big) \left  [e_{j_1}, \ldots, e_{j_m}\right ] \right \| \\
\le{}&  \| \partial^m f (a) - \partial^m f (x)\| \cdot \| e_{j_1} \| \cdots \| e_{j_m} \| \\
={}&  \| \partial^m f (a) - \partial^m f (x)\|
\end{aligned}

with the result

Could you please explain why the norm symbols || (the third one) are not aligned in this situation? Thank you so much!

• simpler than using a package you could just remove the unneeded \left \right that is adding the space. (Please always post a complete document to make it easier for people to run the code, it is harder to debug a fragment) Nov 23, 2019 at 8:39

I think there are two issues with your code

• \left and \right can insert whitespace. Don't use it here; or, load the mleftright package and use \mleft and \mright instead of \left and \right.

• An unusual (to put it mildly) usage of the & alignment points.

Once these issues are fixed, the result is what one would expect it to be. To improve code legibility, I would also define a macro called \norm, using the \DeclarePairedDelimiter macro of the mathtools package. (The mathtools package is a superset of the amsmath package.)

\documentclass{article}
\usepackage{mathtools}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}

\begin{document}
\begin{aligned} &\mathrel{\phantom{=}} \norm{ \partial_{j_1} \dotsb \partial_{j_m} f(a) - \partial_{j_1} \dotsb \partial_{j_m} f(x) } \\ &= \norm{ \partial^m f(a) [e_{j_1}, \dots, e_{j_m} ] - \partial^m f(x) [e_{j_1}, \dots, e_{j_m} ] } \\ &= \norm[\big]{ \bigl( \partial^m f(a) - \partial^m f(x) \bigr) [ e_{j_1}, \dots, e_{j_m} ] } \\ &\le \norm{ \partial^m f(a) - \partial^m f(x)} \cdot \norm{ e_{j_1}} \dotsb \norm{e_{j_m} } \\ &= \norm{ \partial^m f(a) - \partial^m f(x)} \end{aligned}
\end{document}


text:
## tanjung 2 years ago simplify !

1. tanjung

|dw:1351239983066:dw|

2. jayz657

$\sum_{1}^{99} \frac{ \sqrt{10+\sqrt{n}} }{ \sqrt{10-\sqrt{n}} }$

3. tanjung

but i want to simplify it, my book told me that answer is cot(pi/8). but idk how to get it !!!

4. jayz657

sorry im not exactly how to get you get that either since it was a sequence, the simplest form i can think of was that

5. tanjung

any ideas guys

lets try rationalise

7. tanjung

too long this fraction, lol

4,9,25,36,49,64,81 are special here

9. estudier

According to Wolfram this sum is 327.36 approx.... Which is not cot pi/8 (somebody check in case I did it wrong)


text:
# How do you simplify -4^3?

Jan 23, 2017

$- 64$

#### Explanation:

It is only the 4 that is being $\textcolor{b l u e}{\text{cubed}}$ here

$\Rightarrow - {4}^{3} = - {\left(4\right)}^{3} = - 64$


text:
Documentation

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# balredOptions

Create option set for model order reduction

## Syntax

opts = balredOptions
opts = balredOptions('OptionName', OptionValue)

## Description

opts = balredOptions returns the default option set for the balred command.

opts = balredOptions('OptionName', OptionValue) accepts one or more comma-separated name/value pairs. Specify OptionName inside single quotes.

## Input Arguments

### Name-Value Pair Arguments

'FreqIntervals'

Frequency intervals for computing frequency-limited Hankel singular values, specified as a matrix with two columns. Each row specifies a frequency interval [fmin fmax], where fmin and fmax are nonnegative frequencies, expressed in the frequency unit of the model. When identifying low-energy states to truncate, the software computes state contributions to system behavior in these frequency ranges only. For example:

• To restrict the computation to the range between 3 rad/s and 15 rad/s, assuming the frequency unit of the model is rad/s, set FreqIntervals to [3 15].

• To restrict the computation to two frequency intervals, 3-15 rad/s and 40-60 rad/s, use [3 15; 40 60].

• To specify all frequencies below a cutoff frequency fcut, use [0 fcut].

• To specify all frequencies above the cutoff, use [fcut Inf] in continuous time, or [fcut pi/Ts] in discrete time, where Ts is the sample time of the model.

The default value, [], imposes no frequency limitation and is equivalent to [0 Inf] in continuous time or [0 pi/Ts] in discrete time. However, if you specify a TimeIntervals value other than [], then this limit overrides FreqIntervals = []. If you specify both a TimeIntervals value and a FreqIntervals value, then the computation uses the union of these intervals.

If StateElimMethod = 'MatchDC' (the default value), then balred attempts to match the DC gain of the original and reduced models, even if the specified frequency intervals exclude 0. This behavior might reduce the quality of the match in the specified intervals. To improve the match within frequency intervals that exclude 0, set StateElimMethod = 'Truncate'.

If both the frequency and time intervals do include DC, you can still set StateElimMethod = 'Truncate' to improve the match at other frequencies and times.

Default: []

'TimeIntervals'

Time intervals for computing time-limited Hankel singular values, specified as a matrix with two columns. Each row specifies a time interval [tmin tmax], where tmin and tmax are nonnegative times, expressed in the time unit of the model. When identifying low-energy states to truncate, the software computes state contributions to the system’s impulse response in these time intervals only. For example:

• To restrict the computation to the range between 3 s and 15 s, assuming the time unit of the model is seconds, set TimeIntervals to [3 15].

• To restrict the computation to two time intervals, 3-15 s and 40-60 s, use [3 15; 40 60].

• To specify all times from zero up to a cutoff time tcut, use [0 tcut]. To specify all times after the cutoff, use [tcut Inf].

The default value, [], imposes no time limitation and is equivalent to [0 Inf]. However, if you specify a FreqIntervals value other than [], then this limit overrides Timeintervals = []. If you specify both a TimeIntervals value and a FreqIntervals value, then the computation uses the union of these intervals.

If StateElimMethod = 'MatchDC' (the default value), then balred attempts to match the DC gain of the original and reduced models, even if the specified time intervals exclude Inf. This behavior might reduce the quality of the match in the specified intervals. To improve the match within time intervals that exclude Inf, set StateElimMethod = 'Truncate'.

If both the frequency and time intervals do include DC, you can still set StateElimMethod = 'Truncate' to improve the match at other frequencies and times.

Default: []

'StateElimMethod'

State elimination method. Specifies how to eliminate the weakly coupled states (states with smallest Hankel singular values). Specified as one of the following values:

 'MatchDC' Discards the specified states and alters the remaining states to preserve the DC gain. 'Truncate' Discards the specified states without altering the remaining states. This method tends to product a better approximation in the frequency domain, but the DC gains are not guaranteed to match.

Default: 'MatchDC'

'AbsTol, RelTol'

Absolute and relative error tolerance for stable/unstable decomposition. Positive scalar values. For an input model G with unstable poles, balred first extracts the stable dynamics by computing the stable/unstable decomposition G → GS + GU. The AbsTol and RelTol tolerances control the accuracy of this decomposition by ensuring that the frequency responses of G and GS + GU differ by no more than AbsTol + RelTol*abs(G). Increasing these tolerances helps separate nearby stable and unstable modes at the expense of accuracy. See stabsep for more information.

Default: AbsTol = 0; RelTol = 1e-8

'Offset'

Offset for the stable/unstable boundary. Positive scalar value. In the stable/unstable decomposition, the stable term includes only poles satisfying

• Re(s) < -Offset * max(1,|Im(s)|) (Continuous time)

• |z| < 1 - Offset (Discrete time)

Increase the value of Offset to treat poles close to the stability boundary as unstable.

Default: 1e-8

For additional information on the options and how to use them, see the balred reference page.

## Examples

collapse all

Compute a reduced-order approximation of the system given by:

$G\left(s\right)=\frac{\left(s+0.5\right)\left(s+1.1\right)\left(s+2.9\right)}{\left(s+{10}^{-6}\right)\left(s+1\right)\left(s+2\right)\left(s+3\right)}.$

Use the Offset option to exclude the pole at $s=1{0}^{-6}$ from the stable term of the stable/unstable decomposition.

sys = zpk([-.5 -1.1 -2.9],[-1e-6 -2 -1 -3],1);
% Create balredOptions
opt = balredOptions('Offset',.001,'StateElimMethod','Truncate');
% Compute second-order approximation
rsys = balred(sys,2,opt);

Compare the responses of the original and reduced-order models.

bodeplot(sys,rsys,'r--')

Reduce a high-order model with a focus on the dynamics in a particular frequency range.

Load a model and examine its frequency response.

bodeplot(G)

G is a 48th-order model with several large peak regions around 5.2 rad/s, 13.5 rad/s, and 24.5 rad/s, and smaller peaks scattered across many frequencies. Suppose that for your application you are only interested in the dynamics near the second large peak, between 10 rad/s and 22 rad/s. Focus the model reduction on the region of interest to obtain a good match with a low-order approximation. Use balredOptions to specify the frequency interval for balred.

bopt = balredOptions('StateElimMethod','Truncate','FreqIntervals',[10,22]);
GLim10 = balred(G,10,bopt);
GLim18 = balred(G,18,bopt);

Examine the frequency responses of the reduced-order models. Also, examine the difference between those responses and the original response (the absolute error).

subplot(2,1,1);
bodemag(G,GLim10,GLim18,logspace(0.5,1.5,100));
title('Bode Magnitude Plot')
legend('Original','Order 10','Order 18');
subplot(2,1,2);
bodemag(G-GLim10,G-GLim18,logspace(0.5,1.5,100));
title('Absolute Error Plot')
legend('Order 10','Order 18');

With the frequency-limited energy computation, even the 10th-order approximation is quite good in the region of interest.

## Compatibility Considerations

expand all

Behavior changed in R2017b