text:
##### The Casimir Effect is Free Power proven example of free energy that cannot be debunked. The Casimir Effect illustrates zero point or vacuum state energy , which predicts that two metal plates close together attract each other due to an imbalance in the quantum fluctuations. You can see Free Power visual demonstration of this concept here. The implications of this are far reaching and have been written about extensively within theoretical physics by researchers all over the world. Today, we are beginning to see that these concepts are not just theoretical but instead very practical and simply, very suppressed.

To understand why this is the case, it’s useful to bring up the concept of chemical equilibrium. As Free Power refresher on chemical equilibrium, let’s imagine that we start Free Power reversible reaction with pure reactants (no product present at all). At first, the forward reaction will proceed rapidly, as there are lots of reactants that can be converted into products. The reverse reaction, in contrast, will not take place at all, as there are no products to turn back into reactants. As product accumulates, however, the reverse reaction will begin to happen more and more often. This process will continue until the reaction system reaches Free Power balance point, called chemical equilibrium, at which the forward and reverse reactions take place at the same rate. At this point, both reactions continue to occur, but the overall concentrations of products and reactants no longer change. Each reaction has its own unique, characteristic ratio of products to reactants at equilibrium. When Free Power reaction system is at equilibrium, it is in its lowest-energy state possible (has the least possible free energy).
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!
I don’t know what to do. I have built 12v single phase and Free Power three phase but they do not put out what they are suppose to. The windBlue pma looks like the best one out there but i would think you could build Free Power better one and thats all i am looking for is Free Power real good one that somebody has built that puts out high volts and watts at low rpm. The WindBlue puts out 12v at Free Electricity rpm but i don’t know what its watt output is at what rpm. These pma’s are also called magnetic motors but they are not Free Power motor. They are Free Power generator. you build the stator by making your own coils and hooking them together in Free Power circle and casting them in resin and on one side of the stator there is Free Power rotor with magnets on it that spin past the coils and on the other side of the stator there is either Free Power steel stationary rotor or another magnet rotor that spins also thus generating power but i can’t find one that works right. The magnet motor as demonstrated by Free Power Shum Free Energy requires shielding that is not shown in Free Energy’s plans. Free Energy’s shielding is simple, apparently on the stator. The Perendev shows each magnet in the Free Energy shielded. Actually, it intercepts the flux as it wraps around the entire set of magnets. The shielding is necessary to accentuate interaction between rotor and stator magnets. Without shielding, the device does not work. Hey Gilgamesh, thanks and i hope you get to build the motor. I did forget to ask one thing on the motor. Are the small wheels made of steel or are they magnets? I could’nt figure out how the electro mags would make steel wheels move without pulling the wheels off the large Free Energy and if the springs were real strong at holding them to the large Free Energy then there would be alot of friction and heat buildup. Ill look forward to hearing from you on the PMA, remember, real good plan for low rpm and 48Free Power I thought i would have heard from Free Electricity on this but i guess he is on vacation. Hey Free Power. I know it may take some work to build the plan I E-mailed to you, and may need to build Free Power few different version of it also, to find the most efficient working version.
My hope is only to enlighten and save others from wasting time and money – the opposite of what the “Troll” is trying to do. Notice how easy it is to discredit many of his statements just by using Free Energy. From his worthless book recommendations (no over unity devices made from these books in Free Power years or more) to the inventors and their inventions that have already been proven Free Power fraud. Take the time and read ALL his posts and notice his tactics: Free Power. Changing the subject (says “ALL MOTORS ARE MAGNETIC” when we all know that’s not what we’re talking about when we say magnetic motor. Free Electricity. Almost never responding to Free Power direct question. Free Electricity. Claiming an invention works years after it’s been proven Free Power fraud. Free Power. Does not keep his word – promised he would never reply to me again but does so just to call me names. Free Power. Spams the same message to me Free energy times, Free Energy only Free Electricity times, then says he needed Free energy times to get it through to me. He can’t even keep track of his own lies. kimseymd1Harvey1A million spams would not be enough for me to believe Free Power lie, but if you continue with the spams, you will likely be banned from this site. Something the rest of us would look forward to. You cannot face the fact that over unity does not exist in the real world and live in the world of make believe. You should seek psychiatric help before you turn violent. jayanth Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far!
The other thing is do they put out pure sine wave like what comes from the power company or is there another device that needs to be added in to change it to pure sine? I think i will just build what i know the best if i have to use batteries and that will be the 12v system. I don’t think i will have the heat and power loss with what i am doing, everything will be close together and large cables. Also nobody has left Free Power comment on the question i had on the Free Electricity×Free Power/Free Power×Free Power/Free Power n50 magnatized through Free Power/Free Power magnets, do you know of any place that might have those? Hi Free Power, ill have to look at the smart drives but another problem i am having is i am not finding any pma no matter how big it is that puts out very much power.
Why not use the term over unity over perpetual motion? Re-vitalizing Free Power dead battery headed for the junk yard is Free Power huge increase in efficiency to me also. Why doesn’t every AutoZone or every auto shop have one of these? Unless the battery case is cracked every battery could be reused. The charge of Free Power re-vitalize instead of Free Power new battery. Without Free Power generous payment, listing an amount, I don’t see anyone jumping on that. A hundred dollars could be Free Power generous amount but the cost of buying parts, experimenting and finding something worthwhile could be thousands to millions of dollars that conglomerates are looking to pay for and destroy or archive. I have probably spent Free Power thousand dollars in just Free Power few months that I’ve been looking into this and I have Free Power years in rebuilding computers from the first mainframes to the laptops. I retired and now its Free Power hobby. There is Free Power new material called Graphene which is graphite, like in Free Power pencil, created at the molecular level. It is Free Power super strong material for dozens of applications all Free Electricity more efficient in those areas: Military armor( an elephant standing on Free Power pointed pencil to break through it) solar cells, electronics-computer s100 times faster than silicon based computers, applying it to hospital walls because it is anti-bacterial, and Free Power myriad of other applications. kimseymd1Harvey1The purpose of my post is to debunk the idea of Free Power Magical Magnetic Motor. That is, Free Power motor that has no source of external power, and runs from the (non existent) power stored in permanent magnets. Advances made to electric motors in the past few years are truly amazing, but are totally outside the scope of my post.
Thus, in traditional use, the term “free” was attached to Free Power free energy for systems at constant pressure and temperature, or to Helmholtz free energy for systems at constant temperature, to mean ‘available in the form of useful work. ’ [Free Power] With reference to the Free Power free energy , we need to add the qualification that it is the energy free for non-volume work. [Free Power]:Free Electricity–Free Power
LOL I doubt very seriously that we’ll see any major application of free energy models in our lifetime; but rest assured, Free Power couple hundred years from now, when the petroleum supply is exhausted, the “Free Electricity That Be” will “miraculously” deliver free energy to the masses, just in time to save us from some societal breakdown. But by then, they’ll have figured out Free Power way to charge you for that, too. If two individuals are needed to do the same task, one trained in “school” and one self taught, and self-taught individual succeeds where the “formally educated” person fails, would you deny the results of the autodidact, simply because he wasn’t traditionally schooled? I’Free Power hope not. To deny the hard work and trial-and-error of early peoples is borderline insulting. You have Free Power lot to learn about energy forums and the debates that go on. It is not about research, well not about proper research. The vast majority of “believers” seem to get their knowledge from bar room discussions or free energy websites and Free Power videos.
The results of this research have been used by numerous scientists all over the world. One of the many examples is Free Power paper written by Theodor C. Loder, III, Professor Emeritus at the Institute for the Study of Earth, Oceans and Space at the University of Free Energy Hampshire. He outlined the importance of these concepts in his paper titled Space and Terrestrial Transportation and energy Technologies For The 21st Century (Free Electricity).
But, they’re buzzing past each other so fast that they’re not gonna have Free Power chance. Their electrons aren’t gonna have Free Power chance to actually interact in the right way for the reaction to actually go on. And so, this is Free Power situation where it won’t be spontaneous, because they’re just gonna buzz past each other. They’re not gonna have Free Power chance to interact properly. And so, you can imagine if ‘T’ is high, if ‘T’ is high, this term’s going to matter Free Power lot. And, so the fact that entropy is negative is gonna make this whole thing positive. And, this is gonna be more positive than this is going to be negative. So, this is Free Power situation where our Delta G is greater than zero. So, once again, not spontaneous. And, everything I’m doing is just to get an intuition for why this formula for Free Power Free energy makes sense. And, remember, this is true under constant pressure and temperature. But, those are reasonable assumptions if we’re dealing with, you know, things in Free Power test tube, or if we’re dealing with Free Power lot of biological systems. Now, let’s go over here. So, our enthalpy, our change in enthalpy is positive. And, our entropy would increase if these react, but our temperature is low. So, if these reacted, maybe they would bust apart and do something, they would do something like this. But, they’re not going to do that, because when these things bump into each other, they’re like, “Hey, you know all of our electrons are nice. “There are nice little stable configurations here. “I don’t see any reason to react. ” Even though, if we did react, we were able to increase the entropy. Hey, no reason to react here. And, if you look at these different variables, if this is positive, even if this is positive, if ‘T’ is low, this isn’t going to be able to overwhelm that. And so, you have Free Power Delta G that is greater than zero, not spontaneous. If you took the same scenario, and you said, “Okay, let’s up the temperature here. “Let’s up the average kinetic energy. ” None of these things are going to be able to slam into each other. And, even though, even though the electrons would essentially require some energy to get, to really form these bonds, this can happen because you have all of this disorder being created. You have these more states. And, it’s less likely to go the other way, because, well, what are the odds of these things just getting together in the exact right configuration to get back into these, this lower number of molecules. And, once again, you look at these variables here. Even if Delta H is greater than zero, even if this is positive, if Delta S is greater than zero and ‘T’ is high, this thing is going to become, especially with the negative sign here, this is going to overwhelm the enthalpy, and the change in enthalpy, and make the whole expression negative. So, over here, Delta G is going to be less than zero. And, this is going to be spontaneous. Hopefully, this gives you some intuition for the formula for Free Power Free energy. And, once again, you have to caveat it. It’s under, it assumes constant pressure and temperature. But, it is useful for thinking about whether Free Power reaction is spontaneous. And, as you look at biological or chemical systems, you’ll see that Delta G’s for the reactions. And so, you’ll say, “Free Electricity, it’s Free Power negative Delta G? “That’s going to be Free Power spontaneous reaction. “It’s Free Power zero Delta G. “That’s gonna be an equilibrium. ”
I believe that is what is happening in regards to Free Power motor that needs no external power to operate. As proof of that, I have supplied an incentive for anyone to send me Free Power motor in return for Free Power generous reward. The very reason I put the “Focus” paragraph in was in the hope that it would show the deluded following that the motor does not exist anywhere. Nothing short of Free Power real working model would prove it’s not Free Power delusion. Stay focused on that and you will see the truth of what I am saying. Harvey1A magical magnetic motor? Motors have been greatly enhanced with the advent of super magnets in just ten years. Smaller and more powerful to say the least. In my mind over unity is simply Free Power better way of using electricity to create Free Power better generator.
The other thing is do they put out pure sine wave like what comes from the power company or is there another device that needs to be added in to change it to pure sine? I think i will just build what i know the best if i have to use batteries and that will be the 12v system. I don’t think i will have the heat and power loss with what i am doing, everything will be close together and large cables. Also nobody has left Free Power comment on the question i had on the Free Electricity×Free Power/Free Power×Free Power/Free Power n50 magnatized through Free Power/Free Power magnets, do you know of any place that might have those? Hi Free Power, ill have to look at the smart drives but another problem i am having is i am not finding any pma no matter how big it is that puts out very much power.

What may finally soothe the anger of Free Power D. Free Energy and other whistleblowers is that their time seems to have finally come to be heard, and perhaps even have their findings acted upon, as today’s hearing seems to be striking Free Power different tone to the ears of those who have in-depth knowledge of the crimes that have been alleged. This is certainly how rep. Free Power Free Electricity, Free Power member of the Free Energy Oversight and Government Reform Committee, sees it:
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.
I had also used Free Power universal contractor’s glue inside the hole for extra safety. You don’t need to worry about this on the outside sections. Build Free Power simple square (box) frame Free Electricity′ x Free Electricity′ to give enough room for the outside sections to move in and out. The “depth” or length of it will depend on how many wheels you have in it. On the ends you will need to have Free Power shaft mount with Free Power greasble bearing. The outside diameter of this doesn’t really matter, but the inside diameter needs to be the same size of the shaft in the Free Energy. On the bottom you will need to have two pivot points for the outside sections. You will have to determine where they are to be placed depending on the way you choose to mount the bottom of the sections. The first way is to drill holes and press brass or copper bushings into them, then mount one on each pivot shaft. (That is what I did and it worked well.) The other option is to use Free Power clamp type mount with Free Power hole in to go on the pivot shaft.

Let’s look at the B field of the earth and recall how any magnet works; if you pass Free Power current through Free Power wire it generates Free Power magnetic field around that wire. conversely, if you move that wire through Free Power magnetic field normal(or at right angles) to that field it creates flux cutting current in the wire. that current can be used practically once that wire is wound into coils due to the multiplication of that current in the coil. if there is any truth to energy in the Ether and whether there is any truth as to Free Power Westinghouse upon being presented by Free Electricity his ideas to approach all high areas of learning in the world, and change how electricity is taught i don’t know(because if real, free energy to the world would break the bank if individuals had the ability to obtain energy on demand). i have not studied this area. i welcome others who have to contribute to the discussion. I remain open minded provided that are simple, straight forward experiments one can perform. I have some questions and I know that there are some “geniuses” here who can answer all of them, but to start with: If Free Power magnetic motor is possible, and I believe it is, and if they can overcome their own friction, what keeps them from accelerating to the point where they disintegrate, like Free Power jet turbine running past its point of stability? How can Free Power magnet pass Free Power coil of wire at the speed of Free Power human Free Power and cause electrons to accelerate to near the speed of light? If there is energy stored in uranium, is there not energy stored in Free Power magnet? Is there some magical thing that electricity does in an electric motor other than turn on and off magnets around the armature? (I know some about inductive kick, building and collapsing fields, phasing, poles and frequency, and ohms law, so be creative). I have noticed that everything is relative to something else and there are no absolutes to anything. Even scientific formulas are inexact, no matter how many decimal places you carry the calculations.
Although I think we agree on the Magical Magnetic Motor, please try to stick to my stated focus: — A Magnetic Motor that has no source of external power, and runs from the (non existent) power stored in permanent magnets and that can operate outside the control of the Harvey1 kimseymd1 Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! @Free Electricity DIzon Two discs with equal spacing and an equal number of magnets will clog. Free Electricity place two magnets on your discs and try it. Obviously you haven’t. That’s simple understanding. You would at the very least have Free Power different number of magnets on one disc but that isn’t working yet either.
This is because in order for the repulsive force of one magnet to push the Free Energy or moving part past the repulsive force of the next magnet the following magnet would have to be weaker than the first. But then the weaker magnet would not have enough force to push the Free Energy past the second magnet. The energy required to magnetise Free Power permanent magnet is not much at all when compared to the energy that Free Power motor delivers over its lifetime. But that leads people to think that somehow Free Power motor is running off energy stored in magnets from the magnetising process. Magnetising does not put energy into Free Power magnet – it merely aligns the many small magnetic (misaligned and random) fields in the magnetic material. Dear friends, I’m very new to the free energy paradigm & debate. Have just started following it. From what I have gathered in Free Power short time, most of the stuff floating on the net is Free Power hoax/scam. Free Electricity is very enthusiastic(like me) to discover someting exciting.

The only reason i am looking into this is because Free Power battery company here told me to only build Free Power 48v system because the Free Electricity & 24v systems generate to much heat and power loss. Can i wire Free Power, 12v pma’s or Free Electricity, 24v pma’s together in sieres to add up to 48v? If so i do not know how to do it and will that take care of the heat problem? I am about to just forget it and just build Free Power 12v system. Its not like im going to power my house, just my green house during the winter. Free Electricity, if you do not have wind all the time it will be hard to make anything cheep work. Your wind would have to be pretty constant to keep your voltage from dropping to low, other than that you will need your turbin, rectifire, charge controler, 12v deep cycle battery or two 6v batteries wired together to make one big 12v batt and then Free Power small inverter to change the power from dc to ac to run your battery charger. Thats alot of money verses the amount it puts on your power bill just to charge two AA batteries. Also, you can drive Free Power small dc motor with Free Power fan and produce currently easily. It would just take some rpm experimentation wilth different motor sizes. Kids toys and old VHS video recorders have heaps of dc motors.

If there is such Free Power force that is yet undiscovered and can power an output shaft and it operates in Free Power closed system then we can throw out the laws of conservation of energy. I won’t hold my breath. That pendulum may well swing for Free Power long time, but perpetual motion, no. The movement of the earth causes it to swing. Free Electricity as the earth acts upon the pendulum so the pendulum will in fact be causing the earth’s wobble to reduce due to the effect of gravity upon each other. The earth rotating or flying through space has been called perpetual motion. Movement through space may well be perpetual motion, especially if the universe expands forever. But no laws are being bent or broken. Context is what it is all about. Mr. Free Electricity, again I think the problem you are having is semantics. “Perpetual- continuing or enduring forever; everlasting. ” The modern terms being used now are “self-sustaining or sustainable. ” Even if Mr. Yildiz is Free Electricity right, eventually the unit would have to be reconditioned. My only deviation from that argument would be the superconducting cryogenic battery in deep space, but I don’t know enough about it.

Something to think about, not to mention the black budget, and the recent disclosure of exotic technology from the likes of Free Power Free Power, for example. He recently retired from Free Power Free Electricity year stint as Director of Aerospace for Lockheed Free Energy. Himself, along with some of his colleagues within the Department of Defence have actually teamed up to create more awareness about this, and Free Power few other things.
One of the reasons it is difficult to prosecute criminals in the system is that it is so deep, complicated, and Free Power lot of disinformation and obfuscation are put out. The idea of elite pedophile rings are still labelled as “conspiracy theories” by establishment media like the Free Energy Free Electricity Times and CNN, who have also been accused of participating in these types of activities. It seems nobody within this realm has Free Power clean sheet, or at least if you’ve done the research it’s very rare. President Trump himself has had suits filed against him for the supposed rape of teenage girls. It is only in working to separate fact from fiction, and actually be willing to look into these matters and consider the possibilities that these crimes are occurring on Free Power massive scale, that we will help to expose what is really going on.
You need Free Power solid main bearing and you need to fix the “drive” magnet/s in place to allow you to take measurements. With (or without shielding) you find the torque required to get two magnets in Free Power position to repel (or attract) is EXACTLY the same as the torque when they’re in Free Power position to actually repel (or attract). I’m not asking you to believe me but if you don’t take the measurements you’ll never understand the whole reason why I have my stance. Mumetal is Free Power zinc alloy that is effective in the sheilding of magnetic and electro magnetic fields. Only just heard about it myself couple of days ago. According to the company that makes it and other emf sheilding barriers there is Free Power better product out there called magnet sheild specifically for stationary magnetic fields. Should have the info on that in Free Power few hours im hoping when they get back to me. Hey Free Power, believe me i am not giving up. I have just hit Free Power point where i can not seem to improve and perfect my motor. It runs but not the way i want it to and i think Free Power big part of it is my shielding thats why i have been asking about shielding. I have never heard of mumetal. What is it? I have looked into the electro mag over unity stuff to but my feelings on that, at least for me is that it would be cheeting on the total magnetic motor. Your basicaly going back to the electric motor. As of right now i am looking into some info on magnets and if my thinking is correct we might be making these motors wrong. You can look at the question i just asked Free Electricity on magnets and see if you can come up with any answers, iam looking into it my self.
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.
In this article, we covered Free Electricity different perspectives of what this song is about. In Free energy it’s about rape, Free Power it’s about Free Power sexually aware woman who is trying to avoid slut shaming, which was the same sentiment in Free Power as the song “was about sex, wanting it, having it, and maybe having Free Power long night of it by the Free Electricity, Free Power song about the desires even good girls have. ”
Of all the posters here, I’m certain kimseymd1 will miss me the most :). Have I convinced anyone of my point of view? I’m afraid not, but I do wish all of you well on your journey. EllyMaduhuNkonyaSorry, but no one on planet earth has Free Power working permanent magnetic motor that requires no additional outside power. Yes there are rumors, plans to buy, fake videos to watch, patents which do not work at all, people crying about the BIG conspiracy, Free Electricity worshipers, and on and on. Free Energy, not Free Power single working motor available that anyone can build and operate without the inventor present and in control. We all would LIKE one to be available, but that does not make it true. Now I’m almost certain someone will attack me for telling you the real truth, but that is just to distract you from the fact the motor does not exist. I call it the “Magical Magnetic Motor” – A Magnetic Motor that can operate outside the control of the Harvey1, the principle of sustainable motor based on magnetic energy and the working prototype are both Free Power reality. When the time is appropriate, I shall disclose it. Be of good cheer.
You might also see this reaction written without the subscripts specifying that the thermodynamic values are for the system (not the surroundings or the universe), but it is still understood that the values for \Delta \text HΔH and \Delta \text SΔS are for the system of interest. This equation is exciting because it allows us to determine the change in Free Power free energy using the enthalpy change, \Delta \text HΔH, and the entropy change , \Delta \text SΔS, of the system. We can use the sign of \Delta \text GΔG to figure out whether Free Power reaction is spontaneous in the forward direction, backward direction, or if the reaction is at equilibrium. Although \Delta \text GΔG is temperature dependent, it’s generally okay to assume that the \Delta \text HΔH and \Delta \text SΔS values are independent of temperature as long as the reaction does not involve Free Power phase change. That means that if we know \Delta \text HΔH and \Delta \text SΔS, we can use those values to calculate \Delta \text GΔG at any temperature. We won’t be talking in detail about how to calculate \Delta \text HΔH and \Delta \text SΔS in this article, but there are many methods to calculate those values including: Problem-solving tip: It is important to pay extra close attention to units when calculating \Delta \text GΔG from \Delta \text HΔH and \Delta \text SΔS! Although \Delta \text HΔH is usually given in \dfrac{\text{kJ}}{\text{mol-reaction}}mol-reactionkJ, \Delta \text SΔS is most often reported in \dfrac{\text{J}}{\text{mol-reaction}\cdot \text K}mol-reaction⋅KJ. The difference is Free Power factor of 10001000!! Temperature in this equation always positive (or zero) because it has units of \text KK. Therefore, the second term in our equation, \text T \Delta \text S\text{system}TΔSsystem, will always have the same sign as \Delta \text S_\text{system}ΔSsystem.
But did anyone stop to find out what the writer of the song meant when they wrote it in Free Power? Yes, actually, some did, thankfully. But many didn’t and jumped on the hate bandwagon because nowadays many of us seem to have become headline and meme readers and take all we see as fact without ever questioning what we’re being told. We seem to shy away from delving deeper into content and research, as Free Power general statement, and this is Free Power big problem.
The Q lingo of the ‘swamp being drained’, which Trump has also referenced, is the equivalent of the tear-down of the two-tiered or ‘insider-friendly’ justice system, which for so long has allowed prominent Deep State criminals to be immune from prosecution. Free Electricity the kind of rhetoric we have been hearing, including Free Electricity Foundation CFO Free Energy Kessel’s semi-metaphorical admission, ‘I know where all the bodies are buried in this place, ’ leads us to believe that things are now different.
I might be scrapping my motor and going back to the drawing board. Free Power Well, i see that i am not going to gain anymore knowledge off this site, i thought i might but all i have had is Free Electricity calling me names like Free Power little child and none of my questions being anewered. Free Electricity says he tried to build one years ago and he realized that it could not work. Ok tell me why. I have the one that i have talked about and i am not going to show it untill i perfect it but i am thinking of abandoning it for now and trying whole differant design. Can the expert Free Electricity answer shis? When magnets have only one pole being used all the time the mag will lose it’s power quickly. What will happen if you use both poles in the repel state? Free Electricity that ballance the mag out or drain it twice as fast? How long will Free Power mag last running in the repel state all the time? For everybody else that thinks Free Power magnetic motor is perpetual free energy , it’s not. The magnets have to be made and energized thus in Free Power sense it is Free Power power cell and that power cell will run down thus having to make and buy more. Not free energy. This is still fun to play with though.
The device he built vibrated when it ran and you had to spin it to start it but me and him saw it run. Dad was Free Power mechanic and Free Power machinist. He later broke it up so no one would have his idea. I remember how it was made. The motor was amazing. Here’s some more information. Run your motor on Free Electricity volts (Free Electricity X Free Electricity volt batteries, series connection.) Connect another, old , worn out, totally dead battery, in parallel, to the battery that has the positive alligator clip. Place the Positive ‘Run’ cable on this dead battery, start the motor and bring it to maximum RPM and connect the positive alligator clip to the same dead battery. Make sure the electrolyte is full in every cell. After two hours run time, test the battery. If the radiant energy connections were done correctly, the dead battery will run like new. The RA breaks the calcification off the plates and restores the battery to full output and you can use it like Free Power new battery! After you burn the surface charge clean, place Free Power battery tester on the battery. You’ll be pleasantly surprised! Atomic Bomb!?! Wow, there’s Free Power stretch! Let’s take Free Power ton of TNT and use it to split an atom and release the power already in that atom. Here’s my question; Now recycle that energy and explain how? A Magnet Motor is the single most efficient motor available. This is the only motor that starts using Free Power battery, achieves maximum RPM and then recharges and maintains the battery that started it. Radiant energy ! radiant energy is produced at every Hydro-Electric Dam on the planet. They drive Free Power lightening rod in the ground and dispose of it. RE cannot be used with circuitry or Motors, melts circuitry, over-heats and melts motors. Free Electricity regular light bulbs okay, but even they run damn hot! RE is accompanied by AC electricity and that doesn’t help any either.

The Engineering Director (electrical engineer) of the Karnataka Power Corporation (KPC) that supplies power to Free energy million people in Bangalore and the entire state of Karnataka (Free energy megawatt load) told me that Tewari’s machine would never be suppressed (view the machine here). Tewari’s work is known from the highest levels of government on down. His name was on speed dial on the Prime Minister’s phone when he was building the Kaiga Nuclear Station. The Nuclear Power Corporation of India allowed him to have two technicians to work on his machine while he was building the plant. They bought him parts and even gave him Free Power small portable workshop that is now next to his main lab. ”


text:
# What would be the integral of the zeta function or $\sum\limits_{n=1}^{\infty} \frac {1}{n^x}$?

The zeta function is defined as: $$\zeta (x) = \sum\limits_{n=1}^{\infty} \frac {1}{n^x}$$

Does an integral of this function exist? If it does then what would it be?

• If you are happy with a sum you can put $\frac{1}{n^x} = e^{-ln(n)x}$ and integrate – Paul Feb 12 at 12:48
• This is my mathematica code for the integrated Euler Maclaurin formula for the Riemann zeta function: pastebin.com/9wSSSmR4 You can skip the plot and go directly to the second part for faster execution. – Mats Granvik Feb 12 at 15:26
• What do you mean precisely by exist ? – Yves Daoust Feb 13 at 8:44

[Rough Calculation] You may take it as an answer. I have been calculating the integration in the usual way, assuming $$x$$ to be real. $$C$$ is arbitrary constant.

$$\displaystyle \int \zeta(x)dx=x-(\frac{1}{2^x\log 2}+\frac{1}{3^x\log 3}+\dots)+C$$

Now, I claim that the infinite sum converges.

Since, for all $$n>1$$ and $$x>1$$ we have $$\displaystyle\frac{1}{n^x}>\frac{1}{n^x\log n}$$ summing over $$n=2$$ to $$\infty$$ we get,

$$\zeta(x)-1>\sum_{n=2}^\infty\frac{1}{n^x\log n}$$ [I have not used weak inequality as I have not worked on the fact that when they will be equal]

The convergence is followed by the comparison test. As an overview we can say, $$\int\zeta(x)dx>x-1+\zeta(x)$$

Hope this works.

Okay I think I may have a solution, though it isn't pretty.

We can write $$\zeta (x)$$ as:

$$\sum_{n=1}^{\infty}{\frac {1}{n^x}}= \sum_{n=1}^{\infty}{e^{-x \ln (n)}}$$ The expansion of $$e^x$$ is: $$e^x=\sum_{k=0}^\infty \frac {x^k}{k!}$$ Similarly the expansion of $${e^{-x \ln (n)}}$$ is: $$\sum_{n=1}^{\infty}{\frac {1}{n^x}}=\sum_{n=1}^{\infty}{e^{-x \ln (n)}}=\sum_{n=1}^{\infty} \left(\sum_{k=0}^\infty \left( \frac {(-x)^k(\ln (n))^k}{k!} \right) \right)$$

Integrating the expression: $$\int \left(\sum_{n=1}^{\infty} \left(\sum_{k=0}^\infty \left( \frac {(-x)^k(\ln (n))^k}{k!} \right) \right)\right)dx= \sum_{n=1}^{\infty} \left(\sum_{k=0}^\infty \left( \frac {(-1)^k(x)^{k+1}(\ln (n))^k}{(k+1)k!} \right) \right)$$

$$\therefore~ \int\zeta (x)dx=\int \left (\sum_{n=1}^{\infty}{\frac {1}{n^x}} \right) dx= \sum_{n=1}^{\infty} \left(\sum_{k=0}^\infty \left( \frac {(-1)^k(x)^{k+1}(\ln (n))^k}{(k+1)k!} \right) \right)$$

• Why to expand $e^{-x \log n}$? You can compute the integral directly according to Paul's comment: $$\int e^{-x \log n} \,dx = -\dfrac{e^{-x \log n}}{\log n}+c,$$ for $n > 1$. – Alex Silva Feb 13 at 9:31

format: plain


text:
R4

QUESTION: Tom measures himself to be $182$cm tall and calculates that this new height is a $4\%$ increase on his height 2 years ago. Work out how tall Tom was 2 years ago.

ANSWER: We need to consider how we would calculate a $4\%$ increase. We know that $4\%=0.04$, so we get the multiplier for a $4\%$ increase to be

$1+0.04=1.04$

Let $H$ be Tom’s height from two years ago. We know that the result of multiplying $H$ by $1.04$ must be $182$. We can write this as an equation:

$H\times 1.04=182$

Then, if we divide both sides by $1.04$ we get

$H=182\div 1.04=175$

So, Tom’s height two years ago was $175$cm.


text:
(12C) Luhn algorithm
04-18-2018, 06:29 PM (This post was last modified: 04-18-2018 07:19 PM by pinkman.)
Post: #1
 pinkman Senior Member Posts: 387 Joined: Mar 2018
(12C) Luhn algorithm
Ho,
referring to the wikipedia page of the Luhn algorithm, I propose an implementation for the 12c.

How to use it ?

As the 12c has only 10 significant digits, we'll have to split the credit card number into 2 parts.
(Ex: 1234 5678 9012 3456 will be split in 12345678 and 90123456)
After having entered the code, clear the registries 0 and 1 and then enter the right part of the number.
(Ex: 90123456)
Then hit R/S until it displays 0 (don't forget to hit R/S if there were leading zeros).
You can then enter the left part of the number.
(Ex: 12345678)
Then hit R/S until it displays 0 (don't forget to hit R/S if there were leading zeros).

The Luhn key is in REG 0 => RCL 0

If it is a multiple of 10, your Luhn key is OK.
The number of processed digits is in REG 1.

To test :
543210 => 15 in REG 0 (Luhn KO)
543215 => 20 in REG 0 (Luhn OK)

Thibault

Code:
 001- 1 002- 0 003- ÷ 004- ENTER 005- INTG 006- X<>Y 007- FRAC 008- 1 009- 0 010- × 011- RCL 1 012- 2 013- ÷ 014- FRAC 015- x=0 (x=0?) 016- GTO 33 ; GTO odd 017- R↓ 018- 2 019- × 020- 1 021- 0 022- ÷ 023- ENTER 024- INTG 025- X<>Y 026- FRAC 027- 1 028- 0 029- × 030- + 031- STO + 0 032- GTO 35 ; GTO end 033- R↓ ; LBL odd 034- STO + 0 035- 1 ; LBL end 036- STO + 1 037- R↓ 038- R↓ 039- GTO 00
04-23-2018, 08:19 AM (This post was last modified: 04-23-2018 08:56 AM by Dieter.)
Post: #2
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: (12C) Luhn algorithm
(04-18-2018 06:29 PM)pinkman Wrote:  referring to the wikipedia page of the Luhn algorithm, I propose an implementation for the 12c.
...

Another example of putting the 12C's limited programming capabilites to good use. ;-)

Here is a significantly shorter version that takes advantage of two simple tricks that can also be useful for other applications:

1. In your program the change between odd and even digit positions is done by checking if the index is divisible by 2. Afterwards the digit is doubled or not. My program toggles R1 between 0 and 2 (via R1 := 2 – R1). If it's zero the doubling is skipped, otherwise the factor 2 is already in X.

2. The original program calculates the digit sum of a two-digit result in a very straightforward way: divide x by 10, take the integer part, calculate the remainder, and add both. But since x does not exceed 18 the digit sum can also be calculated by subtracting 9 if x exceeds 9. This was implemented in the HP41 program in the General Forum's Luhn thread. Here the code sequence was

Code:
... 2 x 9 X>Y? CLX - ...

This means: if after doubling x is less than 9 (i.e. 0, 2, 4, 6 or 8) don't subtract anything (CLX –), and if it is ≥ 9 (i.e. 10, 12, 14, 16 or 18) subtract 9 to get 1, 3, 5, 7 or 9 as the digit sum.

Now HP's lower end programmables do not feature an X>Y? test. X≤Y? and X=0? is all they have. But there is a way. If one test is followed by another one which always tests false (!), the first test command is inverted! So an X≤Y? followed by an X=0? in effect is an X>Y? as long as X is not zero. Here X always is 9, so X=0? always tests false and we get an X>Y? test as desired.

Here is the complete program:

Code:
01 1 02 0 03 ÷ 04 INT 05 LSTX 06 FRAC 07 1 08 0 09 x 10 2 11 RCL 1 12 - 13 STO 1 14 X=0? 15 GTO 23 16 x 17 9 18 X≤Y? 19 X=0? 20 CLX 21 - 22 ENTER 23 R↓ 24 STO+0 25 R↓ 26 GTO 00

Initialize with
0  STO 0
2  STO 1

With two additional steps the program can be modified so that the initialization can be done with
0  ST0 0  STO 1
or even a simple CLEAR REG.

If you like this better here is an alternate version:

Code:
01 1 02 0 03 ÷ 04 INT 05 LSTX 06 FRAC 07 1 08 0 09 x 10 RCL 1 11 X=0? 12 GTO 20 13 x 14 9 15 X≤Y? 16 X=0? 17 CLX 18 - 19 ENTER 20 R↓ 21 STO+0 22 2 23 RCL 1 24 - 25 STO 1 26 R↓ 27 R↓ 28 GTO 00

In either case then proceed as with the original program.
Note: the number of processed digits here is not stored in R1.

Hint: INT LSTX FRAC is one step shorter than ENTER INT X<>Y FRAC. ;-)
Also note the dummy ENTER in line 22 or 20 that neutralizes the following R↓.
Yes, a GTO would have done it just as well.

Dieter
04-24-2018, 08:00 AM (This post was last modified: 04-24-2018 01:13 PM by pinkman.)
Post: #3
 pinkman Senior Member Posts: 387 Joined: Mar 2018
RE: (12C) Luhn algorithm
Thanks Dieter!!
I'm ashamed I forgot to use LastX, and I'm really impressed with your optimizations.

The 9 comparison with a combination of both x<=y and x=0 is a good sample of knowledge sharing.

I did not want to use a toggle to identify the odd/even case, because of the advantage of having a counter of digits processed. I found a new solution based on the deterministic result of dividing by 2 any integer: 0 or 0.5. Thus multiplying by 2 and adding 1 gives a factor suitable for any digit to process, without using a test. The code might be a bit slower, but shorter and easy to read and maintain (no gto).

Same usage: clear R0 and R1, enter the first (max 10) lower digits, then r/s for each digit including leading zeros, then enter the next digits, r/s again...
Result in R0, number of processed digits in R1.
If R0 is 10*N then Luhn is OK.

Code:
 01 1 02 0 03 / 04 Intg 05 LastX 06 Frac 07 1 08 0 09 * 10 Rcl 1 11 2 12 / 13 Frac 14 2 15 * 16 1 17 + 18 * 19 9 20 x<=y? 21 x=0? 22 CLX 23 - 24 Sto+ 0 25 Rv 26 1 27 Sto+ 1 28 Rv 29 Rtn
04-24-2018, 06:35 PM (This post was last modified: 04-24-2018 08:12 PM by Dieter.)
Post: #4
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: (12C) Luhn algorithm
(04-24-2018 08:00 AM)pinkman Wrote:  I did not want to use a toggle to identify the odd/even case, because of the advantage of having a counter of digits processed. I found a new solution based on the deterministic result of dividing by 2 any integer: 0 or 0.5. Thus multiplying by 2 and adding 1 gives a factor suitable for any digit to process, without using a test.

First of all: you can save two steps if you have the "1" in line 16 followed by a STO+1. This way the counter is incremented en passant and you can delete the three steps near the end that do it now.

Regarding the method of calculating a factor (1 or 2) from the R1 counter: I had a similar idea, and avoiding tests and GTOs is something I really like. In this case it could even be done with two steps less:

Code:
... RCL 1 2 / FRAC X=0? e^x /

The e^x turns a 0 into 1, so the digit is divided by either 0,5 (=doubled) or 1 (unchanged).

But... sometimes even good ideas don't work. All this would be fine, even with four steps saved, IF the 12C had an X<Y? test as originally shown in your listing (I now see you corrected this). But in fact the 12C features an X≤Y? test. This way the digit 9 will not get processed correctly. Since 9 ≤ 9 the test is true, so X=0? is also tested, which is false, so the CLX is not executed and the result is zero (9 – 9). So this method is not an option here.

HP25 users: this is your moment! You can implement the idea above with the following code sequence, since your calculator sports an X≥Y? test:

Code:
RCL 1 2 /         // divide digit position in R1 by 2 FRC       // fractional part is 0 for even or 0,5 for odd digit position X=0?      // if even... e^x       // ...turn zero into 1, else keep the 0,5 /         // divide digit by 0,5 or 1, i.e. multiply by 2 or 1 9 X≥Y?      // if result ≤ 9 CLX       // leave it as it is (subtract 0) -         // else subtract 9 to get the digit sum

Note: be sure to add an ENTER as the first line, or even better something useful like FIX 0. If you don't the following 10 may be appended to your input. This is one of the HP25's ..."special features".

Owners of HP calculators with the common eight test commands may replace the X≥Y? with an X≠Y? X>Y? sequence. Yes, that's another trick from the olden days... ;-)

Dieter
04-24-2018, 07:40 PM (This post was last modified: 04-24-2018 08:20 PM by Dieter.)
Post: #5
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: (12C) Luhn algorithm
(04-18-2018 06:29 PM)pinkman Wrote:  referring to the wikipedia page of the Luhn algorithm, I propose an implementation for the 12c.

Here is a version for a special application of the Luhn algorithm. The latter is often used to add a check digit to a locomotive's class/serial number. The German Federal Railways have been using this method since 50 years now, which maybe is a good reason to introduce this special program version. ;-)

Code:
01  STO 0 02  0 03  STO 1 04  STO 2 05  R↓ 06  X=0? 07  GTO 33 08  1 09  0 10  ÷ 11  INT 12  LSTX 13  FRAC 14  1 15  0 16  x 17  2 18  RCL 1 19  - 20  STO 1 21  X=0? 22  GTO 30 23  x 24  9 25  X≤Y? 26  X=0? 27  CLX 28  - 29  ENTER 30  R↓ 31  STO+2 32  GTO 05 33  RCL 2 34  , 35  9 36  x 37  FRAC 38  STO+0 39  RCL 0 40  FIX 1 41  GTO 00

Usage:
Enter the six-digit vehicle number (class and serial number) and the program adds the check digit (right of the decimal point).

Example:
103 118 => 103.118,6
111 111 => 111.111,1

You can also use this program to calculate the check digit for other numbers with up to nine digits:
123456789 => 123.456.789,7

If you replace line 38...40 with "10 x" only the check digit is returned and up to 10 digits can be processed.
In this case you may also replace the initial STO 0 with FIX 0. ;-)

Dieter
04-24-2018, 08:27 PM
Post: #6
 pinkman Senior Member Posts: 387 Joined: Mar 2018
RE: (12C) Luhn algorithm
(04-24-2018 06:35 PM)Dieter Wrote:  First of all: you can save two steps if you have the "1" in line 16 followed by a STO+1. This way the counter is incremented en passant and you can delete the three steps near the end that do it now.
Yes!!

(04-24-2018 06:35 PM)Dieter Wrote:  But... sometimes even good ideas don't work. All this would be fine, even with four steps saved, IF the 12C had an X<Y? test as originally shown in your listing (I now see you corrected this). But in fact the 12C features an X≤Y? test. This way the digit 9 will not get processed correctly. Since 9 ≤ 9 the test is true, so X=0? is also tested, which is false, so the CLX is not executed and the result is zero (9 – 9). So this method is not an option here.
Whoo, lack of test on my side!
There is another way to avoid the >9 test: add the 2 numbers, even if there is a leading zero.
This can be done with:

Code:
 10 / Intg LastX Frac 10 * +

So I decided to store 10 in R2, to spare 2 lines.

New code is:

Code:
 01 1 02 0 03 Sto 2 04 / 05 Intg 06 LastX 07 Frac 08 Rcl 2 09 * 10 Rcl 1 11 2 12 / 13 Frac 14 2 15 * 16 1 17 Sto+ 1 18 + 19 * 20 Rcl 2 21 / 22 Intg 23 LastX 24 Frac 25 Rcl 2 26 * 27 + 28 Sto+ 0 29 Rv 30 Gto 00

This time I think it has been tested!
Same usage, but now R2 is used.
Result in R0, counter in R1.
04-25-2018, 07:22 AM (This post was last modified: 04-25-2018 07:23 AM by Dieter.)
Post: #7
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: (12C) Luhn algorithm
(04-24-2018 08:27 PM)pinkman Wrote:  There is another way to avoid the >9 test: add the 2 numbers, even if there is a leading zero.

It can also be done with the >9 test.
Here is a version that processes the complete number, so you don't have to press R/S several times:

Code:
01  1 02  0 03  ÷ 04  INTG 05  LSTX 06  FRAC 07  1 08  0 09  x 10  RCL 1 11  2 12  ÷ 13  FRAC 14  2 15  x 16  1 17  STO+1 18  + 19  x 20  9 21  X<>Y 22  X≤Y? 23  GTO 26 24  X<>Y 25  - 26  STO+0 27  R↓ 28  R↓ 29  X=0? 30  GTO  32 31  GTO 01 32  RCL 1 33  RCL 0 34  GTO 00

R1 is returned in Y and R0 in X. Simply check if the last digit is zero. Yes, with a few more steps the program could also do this.

The code up to line 28 works like the previous versions, so it's even one step shorter. And R2 is not required. ;-)

Dieter
04-25-2018, 02:52 PM
Post: #8
 pinkman Senior Member Posts: 387 Joined: Mar 2018
RE: (12C) Luhn algorithm
Great!
 « Next Oldest | Next Newest »

User(s) browsing this thread: 1 Guest(s)


text:
# Computing input length on a one-tape Turing machine

In connection with this question it occurred to me to wonder: what is the time complexity for a single-tape single-head Turing machine to compute the length of its input? To be specific, let's say that the tape alphabet is $\{0,1,b\}$, the input is a string in $(0+1)^*$ surrounded by blanks, the machine starts at the leftmost input symbol, and it must terminate at the leftmost symbol of a string in $(0+1)^*$ (again surrounded by blanks) that gives the binary representation of the input length. This can also be thought of as the problem of converting a number from unary to binary.

It's easy to solve this on a two-tape machine or two-head machine in linear time (just scan the input with one head while using the other head to repeatedly increment a counter; incrementing is a constant amortized time operation). But the single-head solutions I can come up with are only $O(n\log n)$ (e.g. repeatedly increment a counter and then shift it by one position along the tape). Is there a matching lower bound?

I tried some searches but phrases like "one head" and "input length" are so common as to make it difficult to search the literature for known results on this problem.

• Interesting.. This is less obvious than it appears it should be. I'm curious if there's a relation between a lower bound for this and a lower bound for oblivious TM simulation. (Any TM solving this problem would, by definition, be oblivious (or have unnecessary code).) – Daniel Apon Sep 14 '13 at 3:12

It cannot be computed in time $o(n\lg n)$.

Let $M$ be a machine which given an input string $x$ halts with the size of $x$ written in binary on the tape.

We can add a simple (zero-space linear-time) DFA to $M$ to check if the size of the input is a power of two: just check that the first bit is 1 and the rest is zero.

Let's assume that $M$ runs time $o(n \lg n)$. Then we can decide in time $o(n \lg n)$ that the size of input is a power of two. In other words, the following language is decidable in $\mathsf{DTime}(n \lg n)$. $$L = \{ 0^i \mid \exists k \ i = 2^k\}$$ It follows from $\mathsf{DTime}(o(n \lg n)) = \mathsf{Reg}$ that $L$ should be regular. But it is easy to check that the language is not regular. So $M$ cannot run in time $o(n \lg n)$.

• I miss something here: When you say that $\mathsf{DTime}(o(n\lg n))=\mathsf{Reg}$, are you considering computations on a single-tape machine? Usually I think, two-tape machines are used to define complexity classes. A very related question is where does the above result come from? – Bruno Sep 16 '13 at 14:02
• @Bruno, yes, I am talking about single-tape Turing machines. I am not sure what is the standard to define the complexity classes, various books use different models. The original papers of complexity theory were using multiple-tape ones I think but it seems that has changed, see this. For $\mathsf{DTime}(n \lg n) = \mathsf{Reg}$ you can find it in "Classical Recursion Theory" vol. II and "Handbook of Theoretical Computer Science". – Kaveh Sep 16 '13 at 17:16
• Thanks for the pointers, I had a look in "Classical Recursion Theory" vol. II. For the fact that it has changed, it is not so clear for me. For instance, Sipser's book uses single-tape TMs to define time complexity classes, but Hopcroft-Ullman's book and the most recent Arora-Barak's and Goldreich's use multitape TMs. – Bruno Sep 17 '13 at 13:23
• @Bruno, I think what is the more common definition of DTime is more complicated. E.g. the commonly stated claim that "the time hierarchy theorem is not known to be tight" is only true for single-tape machines, for two-tape machines it has been known to be tight since 1982. – Kaveh Sep 17 '13 at 17:56
• Actually, this consideration on the time hierarchy theorem only exists in textbooks using a single-tape TM as model (e.g. Sipser's). But in other ones, there is nothing about non-tightness (Arora-Barak, Goldreich, etc...). Of course, all of this is not of great importance, but it just seems to me that saying that $\mathsf{DTime}$ is usually defined using single-tape TMs is not true: some authors use single-tape TMs, some others use multitape TMs, and many authors are vague on this issue... In other words, there does not seem to be a consensus on this question. – Bruno Sep 18 '13 at 9:07


text:
program pagination command

Syntax

program pagination b

Set the pagination state.

Note

As with all program commands, use of the command word program is optional; program pagination and pagination are both valid.

Pagination, or dividing output written to the console to another page, occurs when the information to be written surpasses the space allotted for the console. By default, pagination is active.


text:
# How do you simplify (-24-17+5) div 4 * (-9) using order of operations?

Jul 9, 2018

$1$

#### Explanation:

$\left(- 24 - 17 + 5\right) \div 4 \cdot \left(- 9\right)$

The first thing in order of operations is parenthesis:

$- 36 \div 4 \cdot \left(- 9\right)$

Now work from left to right:
$- 9 \div - 9$

$= 1$


text:
# Can you suggest a solution?

Status
Not open for further replies.

#### RobertRoss

##### Member
Hello,

I want to design an electronic board with a microcontroller which can communicate with some external flash memory (approximately 16 mbytes to hold some graphics). The microcontroller has to be able to communicate with a color 1.5" tft LCD which can display at minimum a collor depth of 16 bits/pixel.

I should be able to design my own controls. For example, I would like to design a drop down list box with unique features in terms of graphics. Special backgrounds are shown when the drop down is down with special text fonts as the items in the list and so forth. The system should also comprise touch screen capabilitites. Also I should have a way to store atleast 4 sizes of a particular font. I also should be able to download into the flash memory pictures, icons and patterns and be able to execute their display with some C function that would reside in the microcontroller. Also, the display should be able to do wallpaper and watermark. The objective of this is to have a miniature microcomputer system which would allow me to develop any application in C inside the microcontroller.

Here are my questions:

1) What is the best microcontroller to use for this?

2) Should I use Linux, WCE or a proprietary kernel that I will design from scratch?

3) What would be the approximate costs if the Linux/WCE option would be considered?

4) The above has to fit in a PC board of a maximum size of 1.5" x 1.5" !

5) Can the above system be built using parts that would not total to more than 8$(TFT LCD excluded) ? 6) Approximately, how long would it take for an engineer to develop the above specifications if I were to subcontact this to a company? Sincerely yours Rob Last edited: #### Sceadwian ##### Banned You won't find Linux or Windows CE for micro controllers that's for sure, and there's no way you're gonna build it for under 8 dollars. You need to outline your specifications a whole lot more than you already have if you expect a responce. Knowing what the TFT module is would help for starters. #### RobertRoss ##### Member Thanks for replying! Hello Sceadwian, Okay fair enough.... I thankyou for your response! Here is the LCD part number... its an Optrex part. T-51991GD015-MLW-AIN >You won't find Linux or Windows CE for micro controllers that's for sure But I have seen Atmel coming out with microcontrollers that load Linux with volume pricing of 8$ each? Or even better see link below:

http://www.linuxdevices.com/news/NS8107946385.html

>You need to outline your specifications a whole lot more than you already have if you >expect a responce

Okay, basically, we have a controller that can talk to a color LCD and based on how I program the controller (using C language) we are able to display pics, icons or text on the LCD. For exampe, suppose I press on a button connected to my controller, and my code sees the button, well then the logic goes and fecth a picture stored in flash and displays it on the lcd. *or* say I press the same button and a listbox comes out on the lcd instead.

Obviously, prior to any programmation, there has to be a way to store all these pictures, icons and texts in the flash memory, and then use a specific set of functions that reside in the microcontroller that carry out the display operations based on the paramenters you pass into these functions.

Could we do this using Linux or CE ? Or is it better to design my own miniature kernel/OS?

Yous feedback is much appreciated!

sincere regards
Rob

Last edited:

##### Banned
I've always liked Optrex modules, they have very simple interfaces and seem like really nice units. You could probably use a standard micro controller as the display driver but it'd be very basic. To run an entire OS like you're talking about would require signficant skill in designing routing and building small PCB's, not easy and I'd say a bit out of the realm of possibility to do yourself. Sure the Atmel chips (I'm asuming ARM because CE and Linux run on ARM) would work fine though to be useable you'd have to put some sram and as you say a flash chip, your costs are going to go up fast, and again you have to deal with designing the PCB and all the details behind that.

#### RobertRoss

##### Member

--------------------------------------------------------------------------------

>Sure the Atmel chips (I'm asuming ARM because CE and Linux run on ARM) would work fine >though to be useable you'd have to put some sram and as you say a flash chip, your costs >are going to go up fast, and again you have to deal with designing the PCB and all the >details behind that.

I am confused! Okay.... Isn't the ARM a microcontroller which has the sram in it? Would I absolutely need extra external sram and how much would be required?

Finest regards
Rib

##### Banned
You're the one that has to determine how much RAM it needs Ross I'm not building it Sure you could use just the ARM chip but I don't know how much memory the chip you're planning on using has or how much CE or Linux take up on it. Also I don't particularly think of ARM's as micro controllers more microprocessor but that's a semantics thing.

I'd use Linux seeing as how you can actually get source code for it, trying to get CE to work on it would be a nightmare, I believe there is quiet a bit of code out there for Linux that works on ARMs there's a full GNU toolchain for ARM so that'd let you do C code, but you'd have to do all the device drivers and kernel building yourself. Not an easy undertaking.

Last edited:

#### RobertRoss

##### Member
I was wondering... typically, how much ram does a Linux system require? I wouldn't know right now. I am just trying to establish a budget.

So just to make sure, you say that these microcontrollers offered by Atmel and soforth absolutely need external sram... right?

On another note, I thankyou very much for your sincere help.... it is much appreciated!

Rob

##### Banned
ARM Linux - Documentation
check it out.
I looked in the pre-built kernel directory, and it looks like it takes up approximatly 400-500k of memory just for the kernel.
Booting ARM Linux

Those are a few links I found with google, I'm sure there's tons more info out there. No idea how much memory Atmel's ARM chicps hve built in, but I can almost garuntee you're going to want more. I think the main purpose of ARM linux is for doing linux on WINCE handhelds that have ARM chips for people that want their own custom micro PC's, may be an option for you, all the works effectivly done for you already, all you have to do is find a PDA that has a lot of supported hardware, more searching on google may net you better results. for a 1.5" form factor though I'm pretty sure you're dead in the water, though it's more than plausible to use something like an AVR micro controller as a display driver and maybe a couple of micro RF modules (like from sparkfun) to make a wireless monitor that could communicate with pretty much any system. The sparkfun micro transmitters are rediculously small, and an AVR wouldn't need much but a crystal and a bit of coding to act as a display driver. Well maybe more than a bit of code =)

Last edited:

#### RobertRoss

##### Member
Thanks Buddy, I really appreicate your sincere effort!

Finest reagrds
Roberto

#### dknguyen

##### Well-Known Member
I call a processor anything that requires external memory. I call a microcontroller anything that has internal memory. But yes, there is a grey area between the two. Another definition is that a microcontroller has everything it needs to run integrated into the package (like memory) while a processor does not.

Many times, a processor only works with digital data directly and cannot process analog signals (has no integrated ADC), while a microcontroller does. Actually...I've never seen a processor that has an ADC integrated into it but it's definately possible.

Last edited:

#### RobertRoss

##### Member
...

Let me try to rephrase my question:

Can the above system be built using parts that would not total to more than 8$(TFT LCD, touchscreen and power supply excluded) ? I can be flexible and go 2" x 2" pc board! finest regrads! P.S. Its a yes/no question! Last edited: #### dknguyen ##### Well-Known Member Most Helpful Member THe answer is most definately not. Last edited: #### RobertRoss ##### Member ... hello, dknguyen Like I dscussed here with a very nice gentlemen before, I have seen ARM microcontrollers that can execute the Linux kernel installed on 2 mb of flash (sram). Then all I would need is the microcontroller @ 5$ and 2 mb of parallel flash @ 3$. In all due respect dknguyen, why do you say no. Remember all I am asking for is the bare bones hardware requirements. I am not talking about any development tools or LCD or touch screen or power supply or any DC to DC convertor component costs. Neither am I including the pc board here.... just bare bones micro controller system requirements! Finest regards Robert #### DirtyLude ##### Well-Known Member No. uC, memory, supporting components, power regulator, connectors, PCB = more than$8, even if mass produced.

(tried entering just 'no', but the message was too short)

..also you would need a touchscreen controller for proper touchscreen, like a TSC2007 or TSC2046.

Last edited:

#### RobertRoss

##### Member
....

Mark:

Perhaps I was unclear.

I am only talking about the costs of the following two parts:

uC and required memory to run Linux...... that's it..... that's all I want in the quote!

r

#### DirtyLude

##### Well-Known Member
Your request makes no sense, since the chip is useless without at least the crystal and supporting caps, and the PLL.

On it's own. the AT91SAM9260 that you linked to is \$8.60 at Digikey for a quantity order of 3000.

Your requirement calls for two types of memory, FLASH ram for graphics and OS storage and SRAM for working memory.

Like I said, you need a touchscreen controller as well if you want precise touchscreen measurements. This is a chip that will need to be on your board, it's not the actual touchscreen, or part of the touchscreen.

Status
Not open for further replies.


text:
Home | Quick Search | Advanced Search | Bibliography submission | Bibliography submission using bibtex | Bibliography submission using bibtex file | Links | Help | Internal

# Details:

 Title Some numerical results on the rank of generic three-way arrays over $mathbbR$. Author(s) Vartan Choulakian Type Article in Journal Abstract The aim of this paper is the introduction of a new method for the numerical computation of the rank of a three-way array $\mathbf{X}\in\mathbb{R}^{I\times J\times K}$ over $\mathbb{R}$. We show that the rank of a three-way array over $\mathbb{R}$ is intimately related to the real solution set of a system of polynomial equations. Using this, we present some numerical results based on the computation of Gröbner bases. Also, we show that for $I=(K-1)(J-1)+1$ and $2\leq K\leq J\leq I$, the rank for generic data has more than one rank value, and the minimum attained value is I. ISSN 0895-4798; 1095-7162/e URL http://epubs.siam.org/doi/abs/10.1137/08073531X Language English Journal SIAM J. Matrix Anal. Appl. Volume 31 Number 4 Pages 1541--1551 Publisher Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA Year 2010 Edition 0 Translation No Refereed No
Webmaster


text:
Browse Questions

# The equation $2x^2-3xy-py^2+x+qy=0$ represents two perpendicular lines.Find the values of $p$ and $q$.

This question has multiple parts. Therefore each part has been answered as a separate question on Clay6.com


text:
A simplex is a nice way to visualize the dynamics of a three dimensional evolutionary game. The closer to a certain corner a point is, the higher the frequency of the corresponding type in the population - right on the corner the population consists of only one type. Arrows point in the direction of change of the population composition for the chosen dynamics and the color denotes how fast this change will be.

## Create a simplex plot

To create a simplex plot there are several tools available in the web, for example the Mathematica notebook Dynamo. Mathematica, however, is quite expensive. Therefore I wrote a small python library egtsimplex which is very simple but should be easy to extend. Fell free to contribute :-)

To create the very same plot as you see above you need:

1. A relatively recent python3 and matplotlib. On Mac OS you can install both with homebrew.
2. The repository egtsimplex cloned to your computer:
git clone http://github.com/marvinboe/egtsimplex.git

Then you can start plot_example.py with:

python3 plot_example.py

The example file should be relatively self-explanatory. For example in the definition of the function for the dynamics you can choose a different payoff matrix A=np.array([[a,b,c],[d,e,f],[g,h,i]]). Have fun playing around with it!


text:
# 1st PUC Maths Model Question Paper 4 with Answers

Students can Download 1st PUC Maths Model Question Paper 4 with Answers, Karnataka 1st PUC Maths Model Question Paper with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

## Karnataka 1st PUC Maths Model Question Paper 4 with Answers

Time: 3 Hrs 15 Min
Max. Marks: 100

Instructions:

1. The question paper has five parts namely A, B, C, D and E. Answer all the parts.
2. Part-A carries 10 marks. Part-B carries 20 marks. Part-C carries 30 marks. Part-D carries 20 marks, Part-E carries 10 marks.

Part – A

Answer ALL the questions. Each question carries one mark. (10 × 1 = 10)

Question 1.
Define power set of a set.
All possible subset of a given set are called powerset.

Question 2.
If G = {7,8}, H = {5,4,2}, Find H × G.
H × G = {(5.7),(5.8),(4,7),(4,8),(2,7),(2,8)}

Question 3,

Question 4.
Express (2 – i) – (6 + 3i) in a + ib form.
(2 – i) – (6 + 3i) = -4 -4i in the form (a + ib)

Question 5.
Find n, if nc9 = nc8
n = 9 + 8 = 17

Question 6.
Find 17th term of sequence whos: nth term is given by a = 4n – 3. term of sequence whos
a17 = 4(17) – 3 = 68 – 3 = 65

Question 7.
Define the slope of a straight line
Slope = m = gradient or inclination of a line with respect to positive x-axis i.e., m = tan θ

Question 8.
Evalaute $$\lim _{x \rightarrow 0} \frac{(x+1)^{2}-1}{x}$$

Question 9.
Write the negation of ” √2 is not a complex number.”
~(√2 is not a complex no.) = √2 is a complex no.”

Question 10.
Two coins are tossed. Find a sample space.
Sample space ={HH, HT,TT,TH}

Part – B

Answer any TEN Questions. (10 × 2 = 20 )

Question 11.
Let A = {1,2,3,4,5,6} and B = {2,4,6,8}, Find A – B and B – A.
A – B = {1,3,5), (B – A) = {8}

Question 12.
If X and Y are two sets such that X ∪ Y has 50 elements, X has 28 elements, Y has 32 elements, how many elements does X ∩ Y have?
n(x ∪ y) = n(x) + (n ∪ y) – n(x ∩ y)
50 = 28 + 32 (x ∩ y) ⇒ [∴ n(X ∩ Y) = 10]

Question 13.
A function f is defined by f(x) = 2x – 5 write the value of (i) f(7) (ii) f(-3)
Given f(x) = 2x – 5
∴ (i) f(7) = 2(7) – 5 = 9. (ii) f(-3) = 2(-3) – 5 = -11.

Question 14.
Find the angle in radian through which a pendulum swings, if its length is 75 cm and the tip describes an arc of length 10 cm.
θ = ? r = 75 cm, s = 10 cm
S = rθ, (formula) 10 = 75 x θ.
∴ θ = $$\frac{10}{75}=\frac{2}{15}$$ ⇒ ∴ θ = [0.133 radian.]

Question 15.
Find the solution of sin x = $$-\frac{\sqrt{3}}{2}$$
α = $$-\frac{\pi}{3}$$ ∴ G.S. x = 2nπ ± α
∴ [x = -2nπ ± $$-\frac{\pi}{3}$$]

Question 16
Write the Multplicative inverse of 4 – 3i.

Question 17.
Solve the inequality 3(1 – x) <2 (x + 4) and represent the solution graphically on the number line.
Given 3(1 – x) <2 (x + 4)
3 – 3x < 2x + 8 = 3 – 8 < 2x + 3x
-5 < 5x or 5x > -5 i.e., [x > -1]

Question 18.
Find the equation of straight line passing through the point (-4, 3) and having slope $$\frac { 1 }{ 2 }$$
Equation of a straight line is givenby the formula
y – y1 = m(x – x1)
y = 3 = $$\frac { 1 }{ 2 }$$(x + 4) = 2y – 6 = x + 4 ⇒ ∴ [x – 2y + 10 = 0]

Question 19.
Find the distance between parallel lines 15x + 8y – 34 = 0, 15x + 8y + 31 = 0

Question 20.
Find the equation of set of the points P such that ies distances from the points A(3, -4, 5) and B(-2, 1, 4) are equal.
Given PA= PB where P =(x, y ,z)
∴ flash distance formula
$$\sqrt{(x-3)^{2}+(y-4)^{2}+(2+5)^{2}}=\sqrt{(x+2)^{2}+(y+1)^{2}+(z-4)^{2}}$$
(squaring on both sides)
∴ (x – 3)2 + (y –  4)2 + (2 + 5)2 = (x + 2)2 + (y – 1)2 + (z – 4)2

∴(-6x – 4x) + (-8y + 2y) + (102 + 82) + (50 + 21) = 0 = 10x – 6y + 182 + 71 = 0 or
[10x + 6y – 182 – 71 = 0] equation of the plane.

Question 21.
Evaluate: $$\lim _{x \rightarrow 0}\left[\frac{1-\cos x}{x}\right]$$

Question 22.
Write converse and contrapositive of “something is cold implies that it has low temperature”
Converse: q → p = “Something has low temperature implies it is cold”.
Contrapositive: = q → -p= something does not have low temperature implies it is not cold.

Question 23.
Coefficient of ‘variation of distribution arc 60 and the standard deviat,on is 21. What is the arithmetic mean of the distribution?
Given variation = 60
σ = 21, X̄ = ?
Take coefficient of variation = $$\frac{\sigma}{\bar{X}} \times \omega_{0}$$

X̄ = Arithmatic mean = 35.

Question 24.
A card is selected from a pack of 52 cards. Calculate the probability that the card is (i) an arc (ii) black card.

Part – C

Answer any TEN of the following questions. Each question carries THREE marks. (10 × 3 = 30)

Question 25.
In a group of 65 people, 40 like cricket, 10 like cricket and tennis. How many like tennis only and not cricket? How many like tennis?
n(C) = 40, n(T) = ? n( C ∩ T) = 10
n(C ∪ T) = ? n(T – C) = n(C ∪ T) = n(C) + A(T) – n(C ∩ T)
40 = 40 + n(T) -10 – 40 – 30 = n(T)
∴ [N(T) = 10] ⇒ n(T – C) = n(T)n(C ∩ T) = 10 – 10 = 0

Question 26.
Let A = {1,2,3,…14). Define a relation R from A to A by R = {{(x,y)}}: 3x – 4 = 0, x,y ∈ A}. Write its domain, co-domain and Range.
R={(x,y): x ∈ A, y ∈ B, 3x – y = 0}
3(1) – 3 = 0, 3(2) – 6 = 0, 3(3) – 9 = 0, 3(4) – 12 = 0
∴ R = {(1,3), (2,6),(3,9),(4,12)}
Domain of R = A ⇒ Co-domain of R = A
Range of R = {3,6,9,12}

Question 27.
Prove that cos3x = 4cos3x – 3cosx.
Take cos(A+B) =cos Acos B – sin Asin B
Put A = x, B = 2x
∴ cos(x + 2x) = cos x.cos2x – sin x sin 2x
cos(3x) = cos x(2cos2x – 1) – sin x(2 sin cos x)
= 2cos3x – cosx – 2cosx(sin2x) = 2cos3x – cosx – 2cosx(1 – cos2x).
= 2 cos3x – cos x – 2 cos x + 2cos3x [4cos3r – 3 cosx.]

Question 28.
Convert the complex number -1 -i into polar form.
-1 -i = r(cosθ + isinθ) polar form

∴ r = $$\sqrt{x^{2}+y^{2}}=\sqrt{1+1}=\sqrt{2}$$
θ = tan-1y/x = tan-1$$\left(\frac{-1}{-1}\right)$$ = -(π – standard) = $$-\left(\pi-\frac{\pi}{4}\right)=-\frac{3 \pi}{4}$$

Question 29.
If x + iy = $$\frac{a+i b}{a-i b}$$ prove that x2 + y2

Question 30.
How many words with or with out meaning can be made from the letters of the word “Monday” assuming that no letters is repeated if
(i) 4 letters are used at a time.
(ii) All letters are used at a time.
(iii) All letters are used but first letter is a vowel.
Monday : Given number of letters = 6
∴ total number of permutation = 6P6 = 6!= 720
(i) It 4 letters are together = 3! × 4!= 6 × 24 = 144
(ii) All letters are used at a time. = 6! = 720
(iii) All letters are used but 1st letter is a vowel = 5!= 120 .

Question 31.
Find the middle term in the expansion of $$\left(3-\frac{x^{3}}{6}\right)^{7}$$
x = 3, a = $$-\frac{x^{3}}{6}$$, n = 7, Tn = ? r = 3
Tn = ? r = 4
n = 7 (add) no. there are two middle terms.

Question 32.
Insert Five numbers between 8 and 26 such that the resulting sequence is an arithmetic progression.
Let the five AM’s are A1, A2, A3, A4, A5 ∴ 5 AM’s are
∴ 8, A1, A2, A23, A4, A5, A6 ∴ A1 = 11
n = 7, a = 8, d = ?, an = 26. A2 = 14
an = a + (n – 1) d. A3 = 17.
2b = 8 + (7-1)d A4 = 20
∴ [d = 3] A5 = 23

Question 33.
Find sum t0 n terms of sequence 8, 88, 888, 88888,..
Let Sn = 8 + 88 + …… + n terms
Sn = 8{1 + 11 +……… + n terms}
Xly by 9 on both side
∴ $$\frac{95 n}{8}$$ = 9 + 99 + 999 +…. + to n term
GP

Question 34.
Find the coordinates of focus, directrix and latus rectum of parabola y2 = 8x.
Equation of parabola y2 = 8x compare with the standard form
y2 = 4ax ∴ 4a = 8 ∴ a = 2
∴ focus = (a,0) = (2,0)
Equation of directrix : x = -a.
x = -2 or [x + 2 = 0]
Latus rectum = 4a = 8.

Question 35.
Find the derivative of sin x with respect to x from 1st principle.
y = sin x as x → x + Δx, y + Δy
∴ (y + Δy) = sin(x + Δx)
Δy = sin(x + Δx) – y
Δy = sin(x + Δx) – sin x
÷ Δx and apply lt Δx → 0 on both side

Question 36.
Using method of contradiction verify “√2 is irrational.”
Let us assume that √2 is rational i.e.. the gives statement is false.
∴ √2 = $$\frac { 1 }{ 2 }$$ ≠ 0 a and b have no conunon factor.
Squaring ∴ 2 = $$\frac{a^{2}}{b^{2}}$$ ⇒ [a2 = 2b2] ⇒ 2 divides a
Again put a = 2C(C ∈ Z) ⇒ ∴ (2c)2 = 2b2
∴ 4c2 = 2b2 or [b2 = 2c2] 2 divide b
i.e., 2 divides both a and b hence our assumption is i.e.. a and b does flot have common contradict out statement √2 is rational is false.
∴ √2 is irrational.

Question 37.
If A and B are events such that P(A) = 0.42, P(B)=0.48, P(A and B) = 0.16 determine
(i) P (not A)
(ii) P (not B)
(iii) P (A or B).
P(A) = 0.42 P(8) = 0.48, P(A ∩B) = 0.16
(i) P( A1 = 1 – 0.42 = 0.58 (ii) P(B1) = 1 – 0.48 = 0.52
(iii) P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = 0.42 + 0.48 – 0.16 = 0.50 – 0.16 = 0.74.

Question 38.
A committee of two persons is selected from two men and two women. What is the probability that the committee will have
(i) no man ?
(ii) one man?
(iii) two men?

(i) Probability (no man) = women × men = 2C2, × 2C0 = 1
(ii) Probability (one man) = 2C1 × 2C1 = 2 × 2 = 4
(iii) Probability (two even) = 2C0 × 2C2 = 1

Part – D

Answer any SIX Questions. (6 × 5 = 30)

Question 39.
Define signum function. Draw the graph. Also write its Domain and Range.

Question 40.
Prove that $$\frac{\sin 5 x-2 \sin 3 x+\sin x}{\cos 5 x-\cos x}=\tan x$$

Question 41.
Prove by Mathematical induction that for all n ≥ 1
12 + 22 + 32 +…+ n2 = $$\frac{n(n+1)(2 n+1)}{6}$$

Question 42.
Solve the system of inequalities graphically x + 2y ≤ 10, x + y ≥1, x – y ≤ 0, x ≥ 0, y ≥ 0.

Question 43.
A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected, if the team has (i) no girls (ii) atleast one boy and one girl? (iii) atleast 3 girls
Ans.
Given 4 girls, 7 boys
Select 5 members

(ii) atleast 1 B and 1 G
(a) 4C1 × 7C4 = 140
(b) 4C2 × 7C3 = 210
(e) 4C3 × 7C2 = 21
(d) 4C2 × 7C1 = 7
Total number of selection: 378ways.

Question 44.
State and prove Binomial theorem for positive integer n.

Question 45.
Derive the equation of straight line in the form xcosω + ysin ω = P, where P is length of normal to the line from origin, w is the inclination of normal with positive x-axis.
Carried a line cutting X-axsis at A and Y axis at B.
Let OA = a, OB = b are x and y interscept.
∴ By intercept form of equation of line.
$$\frac{x}{a}+\frac{y}{b}=1$$ is the equation of the line AB.
Draw OM ⊥ AB. i.e., normal line
Let OM = P. ∴ In O OAM
cosω = $$=\frac{O M}{O A}=\frac{P}{\cos w}$$  ⇒ ∴ a = $$\frac{O M}{O B}=\frac{P}{b}$$

Question 46.
Derive an expression for the co-ordinates of a point that divides the line joining points A(x1, y1, z1) and B(x2, y2, z2) internally in the ratio m : n. Hence find the coordinates of the mid point of AB where A = (1,2,3) and B = (5,6,7).
Proof: Let p(x1, y1, z1) and Q(x2, y2, z2) be the given points
Let R(x, y, z) divide PQ internally in the ratio m : n
Draw PL, QM, RN perpendicular to xy-plane.
∴ PL ∥ RN ∥ QM
∴ PL, RN, QM lie in one plane
So that the points L, N, M lie in a straight line which is the intersection of the plane and XY plane.
Through the point R draw a line AB It to the line LM. The line AB intersect the line LP externally at A and the line MQ at B.

Triangle APR and LIQR are similar.

∴ n(z – z1) = m(z2 – z)
∴ nz = nz1 = mz2 – m2
∴ (nz + mz) = (mz2 + nz1)
∴ z(m + n) = mz2 + nz1

Question 47.
Prove that $$\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$$, where x measured in radians. Also evaluate $$\lim _{x \rightarrow 0}\left(\frac{\sin 4 x}{x}\right)$$

Question 48.
Find mean deviation about mean for the data.

Part – E

Answer any ONE question. (1 × 10 = 10)

Question 49.
(a) Prove geometrically that cos(x + y) = cos x cos y – sin xsin y. and hence deduce that cos 2x cos2x – sin2x.
Prove that cos(x + y) = cos x cos y.sin x sin y

Proof Consider a unit circle (radius = 1 unit) with centre is (0, 0).
Consider 4 point P1, P2, P3 and P4

The co-ordinate of P1, P2, P3 and P4 are given by
P1 = (cosx,sinx) P2 = [cos(x + y). sìn(x + y)]
P3 = [cos(-y) sin(-y)] P4 = [1,0]
From the figure OP1OP3 is congruent to ∆P2OP4
∴ From distance formula
P1p3 = P2P4 …… (1)
Takè the distance (P1P3)2 = [cosx – cos(-y)]2 + [sin x – sin (-y)]2
= (cosx – cosy)2 + (sin x + sin y)2
= cos2x + cos2y = cosxcosy + sin2x + sin2y + 2sin cosy = 1 + 1 + 2(cosxcosy – sinxsiny)
(p1, P3)2 = 2 – 2 cos (x + y)
Again (P2P4)2 = [1(a-b)2-cos(x+y)]2 + |q – sin (x + y)|2
= 1 + cos2(x + y) -2cos(x + y) + sin2(x + y) = 1 + 1 – 2cos(x + y) = 2 – 2cos(x + y)

⇒ LHS = RHS
∴ [cos(x + y) = cosx cosy – sin x sin y]

(ii) Show that cos 2x = cos2x – sin x2x
Take cos(x + y) = cos xcos y – sin xsin y
Put y = x
∴ cos(x + x) = cosx cosx – sinx sinx
cos2x = cos2 x – sin2x

(b) Find the sum to n terms of the series 1 × 2 + 2 × 3 + 3 × 4 +…
(1 × 2) + (2 × 3) + (3 × 4) +……+ n term nth term.

n2th term, 9n = n(n + 1), an = n2 + n
to find the sum apply Σ on both side
∴ Σan = Σn2 + n

Question 50.
(a) Define hyperbola as a set of points. Derive its equation in the standard form $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$
Definition: “Hyperbola is the set of all points in a plane the difference of where distances from the fixed point in the plane is constant.”
Derivain: prove that $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ (standard hyperbola equation)
Let F1, and F2, are the two fixed point called focii of the hyperbola. The unit point of the sine segment. Joining the Foci is called the centre of the hyperbola.

A line through the Foci is called the transverse axis. And the line through the centre of perpendicular to the transverse axis is called conjugate axis.
Assume the distances between the two focii is “2c’ the distance between two and define the quantity ‘b’ as b = $$\sqrt{c^{2}-a^{2}}$$ and 2b is the length of the conjugaton, (PF2 – PF1 = 2a = transverse axis)

The coordinate of F1 = (-0,0) and F2, =(0,0)
Let P(x,y) be a junction the hyperbola such that the distance of the distance from P to the farter point be “2a’ i.e., PF1 – PF2 = 2a ∴ from the distance formula

Again squaring on both side. $$\therefore\left(\frac{x c}{a}-a\right)^{2}=(x-c)^{2}+y^{2}$$
On simplification further to get $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{c^{2}-a^{2}}=1$$
put c2 – a2 = b2
∴ the standard form of hyperbola equation is
$$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$

(b) Find the derivative of $$\frac{x^{5}-\cos x}{\sin x}$$ with respect to x


text:
# Eight first class and six second class petty officers .?

• December 2nd 2009, 04:18 AM
aeroflix
Eight first class and six second class petty officers .?
Eight first class and six second class petty officers are on the board of the 56 club. In how many ways can the members elect, from the board, a president, a vice-president, a secretary, and a treasurer if the president and secretary must be first class petty officers and the vice-president and treasurer must be second class petty officers?

also.. the second question is

For the preceding example, suppose we are asked the following: In how many ways can the members elect the office holders from the board if two of the office holders must be first class petty officers and two of the office holders must be second class petty officers?
• December 2nd 2009, 06:00 AM
Soroban
Hello, aeroflix!

Here's the first part . . .

Quote:

Eight PO1's and six PO2's are on the board of the club.
In how many ways can the members elect, from the board,
a president, a vice-president, a secretary, and a treasurer
if the president and secretary must be PO1's
and the vice-president and treasurer must be PO2's?

We choose a President and a Secretary from the 8 PO1's.
. . There are: . $_8P_2 = 56$ ways.

We choose a V.P. and Treasurer from the 6 PO2's.
. . There are: . $_6P_2 = 30$ ways.

Therefore, there are: . $56\cdot30 \,=\,1680$ ways.

• December 2nd 2009, 12:20 PM
aeroflix
hello, thanks,, i dont get the second part.
• December 2nd 2009, 03:59 PM
emakarov
For the second question, I think you first need to select two offices (positions) out of four. They will be taken by first-class officers. The remaining two will be taken by second-class officers, so there is no choice in selecting the remaining positions. After that, you proceed as in question 1.
• December 2nd 2009, 10:00 PM
aeroflix
so i will mutiply the answer to 4C2? why?? cant it be 4C2 x 4C2, one for first class and one for second class
• December 3rd 2009, 02:23 AM
emakarov
Quote:

so i will mutiply the answer to 4C2?
Yes.
Quote:

cant it be 4C2 x 4C2, one for first class and one for second class
To count the number of ways to select something you need to represent the selection procedure as a series of choices. For example, to express $P(n,k)$ through $C(n,k)$ (i.e., the number of ordered sequences of length $k$ vs. the number of unordered sets of the same length) one can do the following. To choose a sequence of length $k$ from a set of $n$ elements, one first selects an unordered set of $k$ elements and then selects an ordering of those $k$ elements. So $P(n,k)=C(n,k)\cdot k!$.

The description above is a simplification because sometimes you need to break the set of all possible variants into disjoint subsets, and then proceed as described.

The important thing about the sequence of choices is that they (1) allow one to generate only those variants that need to be counted, (2) allow one to come up with all possible such variants, and (3) do not produce the same variant twice. Let's look at the number of ways to select which of the 4 positions are taken by the first class and which are taken by the second class (without looking into how first-class positions are distributed). I claim that there is only one choice: select any set $\{a, b\}$ where $a,b\in\{1,2,3,4\}$. After this choice, positions $\{a, b\}$ go to the first class and positions $\{1,2,3,4\}-\{a,b\}$ go to the second class. If I am not mistaken, we generated all variants and did not produce the same variant twice.

On the other hand, your version $C(4,2)\cdot C(4,2)$ implies two choices. Presumably, during the first choice we select two positions for the first class. Then, without removing those selected positions from the set $\{1,2,3,4\}$, we select two positions for the second class. This way, for example, we can come up with the set $\{1,2\}$ for the first class and $\{1,3\}$ for the second class. But $\{1,2\}$ and $\{1,3\}$ is not a legal variant.
• December 5th 2009, 07:53 AM
aeroflix
thanks very much!!!

great help


text:
Class 10

### Revisiting Irrational Numbers

● In this section, we will prove that sqrt2 , sqrt3 , sqrt5 and, in general sqrtp is "irrational," where p is a prime. One of the theorems, we use in our proof, is the Fundamental Theorem of Arithmetic.
=> Recall, a number ‘s’ is called irrational if it cannot be written in the form p/q where p and q are integers and q ≠ 0.
=> Some examples of irrational numbers, with which you are already familiar, are :

sqert2 , sqrt3 , sqrt(15) , pi , - sqrt2/sqrt3 , 0.10110111011110........ etc.

● Before we prove that sqrt2 is irrational, we need the following theorem, whose proof is based on the Fundamental Theorem of Arithmetic.

color{blue}{"Theorem 1.3:"} Let p be a prime number. If" p divides a^2, then p divides a, where a is a positive integer.

Proof : Let the prime factorisation of a be as follows :
a = p_1p_2 . . . p_n, where p_1,p_2, . . ., p_n are primes, not necessarily distinct. Therefore, a^2 = ( p_1p_2 . . . p_n)( p_1p_2 . . . p_n) = p_1^2 p_2^2 . . . p_n^2.
=> Now, we are given that p divides a^2. Therefore, from the Fundamental Theorem of Arithmetic, it follows that p is one of the prime factors of a^2. However, using the uniqueness part of the Fundamental Theorem of Arithmetic, we realise that the only prime factors of a^2 are p_1, p_2, . . ., p_n. So p is one of p_1, p_2, . . ., p_n.

Now, since a = p_1 p_2 . . . p_n, p divides a.
We are now ready to give a proof that sqrt2 is irrational.

The proof is based on a technique called ‘proof by contradiction’. (This technique is discussed in some detail in Appendix 1).

color{blue}{"Theorem 1.4 :"} sqrt2 is irrational.

Proof : Let us assume, to the contrary, that sqrt2 is rational.
So, we can find integers r and s (≠ 0) such that sqrt2 = r/s. Suppose r and s have a common factor other than 1. Then, we divide by the common factor to get sqrt2 = a/b where a and b are coprime.

● So, b sqrt2 = a.
Squaring on both sides and rearranging, we get 2b^2 = a^2. Therefore, 2 divides a2.
=> Now, by Theorem 1.3, it follows that 2 divides a.
So, we can write a = 2c for some integer c.
* Not from the examination point of view.

● Substituting for a, we get 2b^2 = 4c^2, that is, b^2 = 2c^2.
This means that 2 divides b^2, and so 2 divides b (again using Theorem 1.3 with p = 2).
Therefore, a and b have at least 2 as a common factor.

●But this contradicts the fact that a and b have no common factors other than 1.
This contradiction has arisen because of our incorrect assumption that sqrt2 is rational.
So, we conclude that sqrt2 is irrational

Q 3119856719

Prove that sqrt3 is irrational.
Class 10 Chapter 1 Example 9
Solution:

Let us assume, to the contrary, that sqrt3 is rational.

That is, we can find integers a and b (≠ 0) such that sqrt3 = a/b

common factor, and assume that a and b are coprime.
So, b sqrt3 = a.
Squaring on both sides, and rearranging, we get 3b^2 = a^2.
Therefore, a^2 is divisible by 3, and by Theorem 1.3, it follows that a is also divisible by 3.
So, we can write a = 3c for some integer c.
Substituting for a, we get 3b^2 = 9c^2, that is, b^2 = 3c^2.
This means that b^2 is divisible by 3, and so b is also divisible by 3 (using Theorem 1.3
with p = 3).
Therefore, a and b have at least 3 as a common factor.
But this contradicts the fact that a and b are coprime.
This contradiction has arisen because of our incorrect assumption that sqrt3 is rational.
So, we conclude that sqrt3 is irrational.
In Class IX, we mentioned that :
• the sum or difference of a rational and an irrational number is irrational and
• the product and quotient of a non-zero rational and irrational number is irrational.
We prove some particular cases here.
Q 3129056811

Show that 5 - sqrt3 is irrational.
Class 10 Chapter 1 Example 10
Solution:

Let us assume, to the contrary, that 5-sqrt3 is rational.

That is, we can find coprime a and b (b ≠ 0) such that 5-sqrt3 = a/b

Therefore, 5-a/b = sqrt3

Rearranging this equation, we get sqrt3 = 5-a/b = ( 5b -a)/b

Since a and b are integers, we get 5-a/b is rational, and so sqrt3 is rational.
But this contradicts the fact that sqrt3 is irrational.

This contradiction has arisen because of our incorrect assumption that 5-sqrt3 is rational.

So, we conclude that 5-sqrt3 is irrational.
Q 3179056816

Show that 3sqrt2 is irrational.
Class 10 Chapter 1 Example 11
Solution:

Let us assume, to the contrary, that 3 sqrt2 is rational.

That is, we can find coprime a and b (b ≠ 0) such that 3 sqrt2 = a/b .

Rearranging, we get sqrt2 = a/(3b)

Since 3, a and b are integers a/(3b) is rational, and so sqrt2 is rational.

But this contradicts the fact that sqrt2 is irrational.

So, we conclude that 3 sqrt2 is irrational.


text:
# A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?

Given, a coin is tossed three times.

For each toss, the number of outcomes will be 2 i.e. Heads and Tails.

Thus, the total number of possible outomes will be = 2 x 2 x 2 =8


text:
# Solve question 10 Q10. A simple telescope consisting of an objective of focal length 60 cm and a single eye lens of focal length 5 cm is focussed on a distant object in such a way that parallel rays emerge from eye lens. If the object subtends an angle of 20 at the objective, the angular width of the image is            (A) 100                      (B) 240                     (C) 500                         (D) (1/6)0

Dear Student,

From the option (4) it appears that the last zero of the angle is not zero but the unit of angle i.e., degrees
That is the angular width of the object is . In that case, the angular width of the image will be ${24}^{o}$
Regards,

• 0
What are you looking for?


text:
# Plot the complete dataset as listplots

I have imported as a csv file and converted to a Dataset. The data is in the following csv format

A(1),B(1),A(2),B(2),A(3),B(3),A(4),B(4),A(5),B(5)
1,2,4,5,6,4,7,8,9,0
5,3,2,5,5,2,3,1,8,3
7,3,8,3,2,2,3,4,5,6
9,8,7,2,5,3,6,7,8,9

Now I imported the data using

ds = Import["data.csv", "Dataset", HeaderLines -> 1]

Now basically I want to plot $$B(i)$$ v/s $$A(i)$$ for different $$i$$'s on the same plot. I can do that for 1 $$i$$ using

ds[ListLinePlot, {"A(1)", "B(1)"}]

And I get the following curve

Now, I want to do the same for all $$i$$'s and plot them on the same plot. I know I can probably just write them each down but my original dataset (not shown here) is quite large. So I want to automate it for all $$i$$. Any easy way to execute such a command?

Consider the following:

data = ImportString[
"A(1),B(1),A(2),B(2),A(3),B(3),A(4),B(4),A(5),B(5)
1,2,4,5,6,4,7,8,9,0
5,3,2,5,5,2,3,1,8,3
7,3,8,3,2,2,3,4,5,6
];

ListCurvePathPlot[
Transpose /@ Partition[Transpose[data], 2],
PlotLegends -> Range[5]
]

• Is there any other way to do it without partition? Let's say I want to run a loop over i, because in my original dataset there are some other values such as C(i) in the dataset as well. – Indeterminate Feb 8 at 2:44
• @Indeterminate Explicit loops are almost never the answer in MMA. The Partition approach would work without change for however main pairs of values you have in your dataset, there's no need to change it. If it doesn't work when you try, please show the structure of the actual dataset you have. – MarcoB Feb 8 at 4:01
• Here is my original dataset and the problem I am trying to solve: mathematica.stackexchange.com/questions/239583/… – Indeterminate Feb 8 at 4:02
• @Indeterminate OK I see. I think you may have oversimplified here a bit. I've added an answer to that question. – MarcoB Feb 8 at 4:10

Here is another possible way if you want to stay within the dataset framework.

ds = SemanticImportString[
"A(1),B(1),A(2),B(2),A(3),B(3),A(4),B(4),A(5),B(5)
1,2,4,5,6,4,7,8,9,0
5,3,2,5,5,2,3,1,8,3
7,3,8,3,2,2,3,4,5,6
9,8,7,2,5,3,6,7,8,9"];
(*Create list of B versus A pairs*)
xypairs = Partition[ds // First // Keys // Normal, 2];
(*Standard ListLinePlot*)
dsllp = ds[Transpose /* Values /* ListLinePlot, xypairs]
(* Marco's ListCurvePathPlot *)
dslcpp = ds[Transpose /* Values /* ListCurvePathPlot, xypairs]

I did not find a way to add legends directly, so I had to manually add them as shown below:

(*Use standard color list 97*)
colors = ColorData[97, "ColorList"][[;; Length@xypairs]];
labels = Style[StringTemplate["2 vs 1"][##], 14] & @@@ xypairs;
legend = LineLegend[Directive[#, AbsoluteThickness[2]] & /@ colors,
labels];
Legended[dsllp, Placed[legend, Automatic]]
Legended[dslcpp, Placed[legend, Automatic]]