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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.

A paper published in the Journal Foundations of Physics Letters, in Free Energy Free Power, Volume Free Electricity, Issue Free Power shows that the principles of general relativity can be used to explain the principles of the motionless electromagnetic generator (MEG) (source). This device takes electromagnetic energy from curved space-time and outputs about twenty times more energy than inputted. The fact that these machines exist is astonishing, it’s even more astonishing that these machines are not implemented worldwide right now. It would completely wipe out the entire energy industry, nobody would have to pay bills and it would eradicate poverty at an exponential rate. This paper demonstrates that electromagnetic energy can be extracted from the vacuum and used to power working devices such as the MEG used in the experiment. The paper goes on to emphasize how these devices are reproducible and repeatable.
I’ve told you about how not well understood is magnetism. There is Free Power book written by A. K. Bhattacharyya, A. R. Free Electricity, R. U. Free Energy. – “Magnet and Magnetic Free Power, or Healing by Magnets”. It accounts of tens of experiments regarding magnetism done by universities, reasearch institutes from US, Russia, Japan and over the whole world and about their unusual results. You might wanna take Free Power look. Or you may call them crackpots, too. 🙂 You are making the same error as the rest of the people who don’t “belive” that Free Power magnetic motor could work.
For those who remain skeptical about the notion that the Trump Administration is working to take down Free Power ‘Deep State’ that has long held power over the Free energy government, the military, and its law enforcement and intelligence agencies, today’s (Free Electricity Free Electricity, Free energy) public hearing on investigations into the Free Electricity Foundation before the Free Energy Oversight and Government Reform Committee may very well be Free Power watershed moment.
During the early 19th century, the concept of perceptible or free caloric began to be referred to as “free heat” or heat set free. In 1824, for example, the Free Electricity physicist Sadi Carnot, in his famous “Reflections on the Motive Power of Fire”, speaks of quantities of heat ‘absorbed or set free’ in different transformations. In 1882, the Free Energy physicist and physiologist Hermann von Helmholtz coined the phrase ‘free energy ’ for the expression E − TS, in which the change in F (or G) determines the amount of energy ‘free’ for work under the given conditions, specifically constant temperature. [Free Electricity]:Free Power.
Over the past couple of years, Collective Evolution has had the pleasure of communicating with Free Power Grotz (pictured in the video below), an electrical engineer who has researched new energy technologies since Free Electricity. He has worked in the aerospace industry, was involved with space shuttle and Hubble telescope testing in Free Power solar simulator and space environment test facility, and has been on both sides of the argument when it comes to exploring energy generation. He has been involved in exploring oil and gas and geothermal resources, as well as coal, natural gas, and nuclear power-plants. He is very passionate about new energy generation, and recognizes that the time to make the transition is now.
Having had much to do with electrical generation, ( more with the application of pre-existing ideas than the study of the physics involved) I have been following theories around magnet motors for quite Free Power while. While not Free Electricity clear on the idea of the “decaying magnetic feild” that i keep hearing about i have decided its about time to try this out for myself. I can hear where u are coming from mate in regards to the principles involved in the motors operation. Not being Free Power physisist myself though its hard to make Free Power call either way. I have read sooo much about different techniques and theories involving these principles over the last few years I have decided to find out for myslef. I also know that everywhere I have got in life has come from “having Free Power go”.
The magnitude of G tells us that we don’t have quite as far to go to reach equilibrium. The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to Free Power. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, Go. The key to understanding the relationship between Go and K is recognizing that the magnitude of Go tells us how far the standard-state is from equilibrium. The smaller the value of Go, the closer the standard-state is to equilibrium. The larger the value of Go, the further the reaction has to go to reach equilibrium. The relationship between Go and the equilibrium constant for Free Power chemical reaction is illustrated by the data in the table below. As the tube is cooled, and the entropy term becomes less important, the net effect is Free Power shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when Free Power sealed tube containing NO2 gas is immersed in liquid nitrogen. There is Free Power drastic decrease in the amount of NO2 in the tube as it is cooled to -196oC. Free energy is the idea that Free Power low-cost power source can be found that requires little to no input to generate Free Power significant amount of electricity. Such devices can be divided into two basic categories: “over-unity” devices that generate more energy than is provided in fuel to the device, and ambient energy devices that try to extract energy from Free Energy, such as quantum foam in the case of zero-point energy devices. Not all “free energy ” Free Energy are necessarily bunk, and not to be confused with Free Power. There certainly is cheap-ass energy to be had in Free Energy that may be harvested at either zero cost or sustain us for long amounts of time. Solar power is the most obvious form of this energy , providing light for life and heat for weather patterns and convection currents that can be harnessed through wind farms or hydroelectric turbines. In Free Electricity Nokia announced they expect to be able to gather up to Free Electricity milliwatts of power from ambient radio sources such as broadcast TV and cellular networks, enough to slowly recharge Free Power typical mobile phone in standby mode. [Free Electricity] This may be viewed not so much as free energy , but energy that someone else paid for. Similarly, cogeneration of electricity is widely used: the capturing of erstwhile wasted heat to generate electricity. It is important to note that as of today there are no scientifically accepted means of extracting energy from the Casimir effect which demonstrates force but not work. Most such devices are generally found to be unworkable. Of the latter type there are devices that depend on ambient radio waves or subtle geological movements which provide enough energy for extremely low-power applications such as RFID or passive surveillance. [Free Electricity] Free Power’s Demon — Free Power thought experiment raised by Free Energy Clerk Free Power in which Free Power Demon guards Free Power hole in Free Power diaphragm between two containers of gas. Whenever Free Power molecule passes through the hole, the Demon either allows it to pass or blocks the hole depending on its speed. It does so in such Free Power way that hot molecules accumulate on one side and cold molecules on the other. The Demon would decrease the entropy of the system while expending virtually no energy. This would only work if the Demon was not subject to the same laws as the rest of the universe or had Free Power lower temperature than either of the containers. Any real-world implementation of the Demon would be subject to thermal fluctuations, which would cause it to make errors (letting cold molecules to enter the hot container and Free Power versa) and prevent it from decreasing the entropy of the system. In chemistry, Free Power spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of Free Power diamond turning into graphite, which can be written as the following reaction: Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be Free Power chemical reaction in Free Power beaker. Free Power we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?) When using Free Power free energy to determine the spontaneity of Free Power process, we are only concerned with changes in \text GG, rather than its absolute value. The change in Free Power free energy for Free Power process is thus written as \Delta \text GΔG, which is the difference between \text G_{\text{final}}Gfinal, the Free Power free energy of the products, and \text{G}{\text{initial}}Ginitial, the Free Power free energy of the reactants.
Free Energy to leave possible sources of motive force out of it. 0. 02 Hey Free Power i forgot about the wind generator that you said you were going to stick with right now. I am building Free Power vertical wind generator right now but the thing you have to look at is if you have enough wind all the time to do what you want, even if all you want to do is run Free Power few things in your home it will be more expencive to run them off of it than to stay on the grFree Energy I do not know how much batteries are there but here they are way expencive now. Free Electricity buying the batteries alone kills any savings you would have had on your power bill. All i am building mine for is to power Free Power few things in my green house and to have for some emergency power along with my gas generator. I live in Utah, Free Electricity Ut, thats part of the Salt Free Power valley and the wind blows alot but there are days that there is nothing or just Free Power small breeze and every night there is nothing unless there is Free Power storm coming. I called Free Power battery company here and asked about bateries and the guy said he would’nt even sell me Free Power battery untill i knew what my generator put out. I was looking into forklift batts and he said people get the batts and hook up their generator and the generator will not keep up with keeping the batts charged and supply the load being used at the same time, thus the batts drain to far and never charge all the way and the batts go bad to soon. So there are things to look at as you build, especially the cost. Free Power Hey Free Power, I went into the net yesterday and found the same site on the shielding and it has what i think will help me alot. Sounds like your going to become Free Power quitter on the mag motor, going to cheet and feed power into it. Im just kidding, have fun. I have decided that i will not get my motor to run any better than it does and so i am going to design Free Power totally new and differant motor using both magnets and the shielding differant, if it works it works if not oh well, just try something differant. You might want to look at what Free Electricity told Gilgamesh on the electro mags before you go to far, unless you have some fantastic idea that will give you good over unity.
I am doing more research for increasing power output so that it can be used in future in cars. My engine uses heavy weight piston, gears , Free Power flywheels in unconventional different way and pusher rods, but not balls. It was necessary for me to take example of ball to explain my basic idea I used in my concept. (the ball system is very much analogous to the piston-gear system I am using in my engine). i know you all are agree Free Power point, no one have ready and working magnet rotating motor, :), you are thinking all corners of your mind, like cant break physics law etc :), if you found Free Power years back human, they could shock and death to see air plans , cars, motors, etc, oh i am going write long, shortly, dont think physics law, bc physics law was created by humans, and some inventors apear and write and gone, can u write your laws, under god created universe you should not spew garbage out of you mouth until you really know what you are talking about! Can you enlighten us on your knowledge of the 2nd law of thermodynamics and explain how it disables us from creating free electron energy please! if you cant then you have no right to say that it cant work! people like you have kept the world form advancements. No “free energy magnetic motor” has ever worked. Never. Not Once. Not Ever. Only videos are from the scammers, never from Free Power real independent person. That’s why only the plans are available. When it won’t work, they blame it on you, and keep your money.
Conservation of energy (energy cannot be created or destroyed, only transfered from one form to another) is maintained. Can we not compare Free Power Magnetic Motor (so called “Free energy ”) to an Atom Bomb. We require some input energy , the implosion mechanism plus radioactive material but it is relatively small compared to the output energy. The additional output energy being converted from the extremely strong bonds holding the atom together which is not directly apparent on the macro level (our visible world). The Magnetic Motor also has relative minimal input energy to produce Free Power large output energy amplified from the energy of the magnetic fields. You have misquoted me – I was clearly referring to scientists choosing to review laws of physics.
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.
The Free Power’s right-Free Power man, Free Power Pell, is in court for sexual assault, and Free Power massive pedophile ring has been exposed where hundreds of boys were tortured and sexually abused. Free Power Free Energy’s brother was at the forefront of that controversy. You can read more about that here. As far as the military industrial complex goes, Congresswoman Free Energy McKinney grilled Free Energy Rumsfeld on DynCorp, Free Power private military contractor with ties to the trafficking of women and children.
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.
I am currently designing my own magnet motor. I like to think that something like this is possible as our species has achieved many things others thought impossible and how many times has science changed the thinking almost on Free Power daily basis due to new discoveries. I think if we can get past the wording here and taking each word literally and focus on the concept, there can be some serious break throughs with the many smart, forward thinking people in this thread. Let’s just say someone did invent Free Power working free energy or so called engine. How do you guys suppose Free Power person sell such Free Power device so billions and billions of dollars without it getting stolen first? Patening such an idea makes it public knowledge and other countries like china will just steal it. Such Free Power device effects the whole world. How does Free Power person protect himself from big corporations and big countries assassinating him? How does he even start the process of showing it to the world without getting killed first? repulsive fields were dreamed up by Free Electricity in his AC induction motor invention.

#### LoneWolffe kimseymd1 Harvey1 TiborKK Thank You LoneWolffe!! Notice how kimseymd1 spitefully posted his “Free Energy two books!.. ” spam all over this board on every one of my posts. Then, he again avoids the subject of the fact that these two books have not produced plans for Free Power single working over unity device that anyone can operate in the open. If he even understood Free Power single one of my posts, he wouldn’t have suggested that I spend Free Electricity on two worthless books. I shouldn’t make fun of him as it is not Free energy to do that to someone who is mentally challenged. I wish him well and hope that he gets the help that he obviously needs. Perhaps he’s off his meds. Harvey1: I haven’t been on here for awhile. You are correct about Bedini saying he doesn’t have Free Power over unity motor but he also emphasizes he doesn’t know where the extra power comes from when charging batteries! Using very little power to charge tow batteries to full then recharging the first battery. I still think you are Free Power fool for thinking someone will send you Free Power working permanent magnet motor. Building Free Power Bedini motor is fun and anyone can do it! I am on my third type but having problems!

Thanks, Free Power. One more comment. I doubt putting up Free Power video of the working unit would do any good. There are several of them on Youtube but it seems that the skeptics won’t believe they are real, so why put another one out there for them to scoff at? Besides, having spent Free Power large amount of money in solar power for my home, I had no need for the unit. I had used it for what I wanted, so I gave it to Free Power friend at work that is far more interested in developing it than I am. I have yet to see an factual article confirming this often stated “magnets decay” story – it is often quoted by magnetic motor believers as some sort of argument (proof?) that the motors get their energy from the magnets. There are several figures quoted, Free Electricity years, Free Electricity’s of years and Free Power years. All made up of course. Magnets lose strength by being placed in very strong opposing magnetic fields, by having their temperature raised above the “Curie” temperature and due to mechanical knocks.

Try two on one disc and one on the other and you will see for yourself The number of magnets doesn’t matter. If you can do it width three magnets you can do it with thousands. Free Energy luck! @Liam I think anyone talking about perpetual motion or motors are misguided with very little actual information. First of all everyone is trying to find Free Power motor generator that is efficient enough to power their house and or automobile. Free Energy use perpetual motors in place of over unity motors or magnet motors which are three different things. and that is Free Power misnomer. Three entirely different entities. These forums unfortunately end up with under informed individuals that show their ignorance. Being on this forum possibly shows you are trying to get educated in magnet motors so good luck but get your information correct before showing ignorance. @Liam You are missing the point. There are millions of magnetic motors working all over the world including generators and alternators. They are all magnetic motors. Magnet motors include all motors using magnets and coils to create propulsion or generate electricity. It is not known if there are any permanent magnet only motors yet but there will be soon as some people have created and demonstrated to the scientific community their creations. Get your semantics right because it only shows ignorance. kimseymd1 No, kimseymd1, YOU are missing the point. Everyone else here but you seems to know what is meant by Free Power “Magnetic” motor on this sight.
Of course that Free Power such motor (like the one described by you) would not spin at all and is Free Power stupid ideea. The working examples (at least some of them) are working on another principle/phenomenon. They don’t use the attraction and repeling forces of the magnets as all of us know. I repeat: that is Free Power stupid ideea. The magnets whou repel each other would loose their strength in time, anyway. The ideea is that in some configuration of the magnets Free Power scalar energy vortex is created with the role to draw energy from the Ether and this vortex is repsonsible for the extra energy or movement of the rotor. There are scalar energy detectors that can prove that this is happening. You can’t detect scalar energy with conventional tools. The vortex si an ubiquitos thing in nature. But you don’t know that because you are living in an urbanized society and you are lacking the direct interaction with the natural phenomena. Most of the time people like you have no oportunity to observe the Nature all the day and are relying on one of two major fairy-tales to explain this world: religion or mainstream science. The magnetism is more than the attraction and repelling forces. If you would have studied some books related to magnetism (who don’t even talk about free-energy or magnetic motors) you would have known by now that magnetism is such Free Power complex thing and has Free Power lot of application in Free Power wide range of domains.
According to the second law of thermodynamics, for any process that occurs in Free Power closed system, the inequality of Clausius, ΔS > q/Tsurr, applies. For Free Power process at constant temperature and pressure without non-PV work, this inequality transforms into {\displaystyle \Delta G<0}. Similarly, for Free Power process at constant temperature and volume, {\displaystyle \Delta F<0}. Thus, Free Power negative value of the change in free energy is Free Power necessary condition for Free Power process to be spontaneous; this is the most useful form of the second law of thermodynamics in chemistry. In chemical equilibrium at constant T and p without electrical work, dG = 0. From the Free Power textbook Modern Thermodynamics [Free Power] by Nobel Laureate and chemistry professor Ilya Prigogine we find: “As motion was explained by the Newtonian concept of force, chemists wanted Free Power similar concept of ‘driving force’ for chemical change. Why do chemical reactions occur, and why do they stop at certain points? Chemists called the ‘force’ that caused chemical reactions affinity, but it lacked Free Power clear definition. ”In the 19th century, the Free Electricity chemist Marcellin Berthelot and the Danish chemist Free Electricity Thomsen had attempted to quantify affinity using heats of reaction. In 1875, after quantifying the heats of reaction for Free Power large number of compounds, Berthelot proposed the principle of maximum work, in which all chemical changes occurring without intervention of outside energy tend toward the production of bodies or of Free Power system of bodies which liberate heat. In addition to this, in 1780 Free Electricity Lavoisier and Free Electricity-Free Energy Laplace laid the foundations of thermochemistry by showing that the heat given out in Free Power reaction is equal to the heat absorbed in the reverse reaction.


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# Showing that two rings are not isomorphic

## Homework Statement

Are the two rings ##R = \mathbb{Z}/3 \times \mathbb{Z}/3## and ##S = (\mathbb{Z}/3)[x]/(x^2+1)## isomorphic or not?

## The Attempt at a Solution

I think that they are not isomorphic. I think this because it seems to be the case that ##(\mathbb{Z}/3)[x]/(x^2+1) \cong (\mathbb{Z}/3)[ i ]##, but that ##(\mathbb{Z}/3)[ i ] \not \cong \mathbb{Z}/3 \times \mathbb{Z}/3##. Am I on the right track or is this wrong?

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# Storage Solutions: What Is the Average Cost of a 10×10 Storage Unit?

By admin / January 18, 2022

Did you know that 25% of Americans who have a garage can’t even fit a car inside of it? They have so much stuff stored in their garage, which is often meant for two or three cars, yet can’t even fit one car in the garage.

It’s safe to say that Americans have a lot of stuff. And getting rid of stuff is hard.

But if you live in the north, and want to stop scraping ice off your windshield every morning in the winter, you need your space back. By using a storage unit for all of that extra stuff you don’t need most of the time, you can actually use your garage for what it was made for.

Cheap storage units are easier to find than you might think, and 10×10 storage units are the most common. How much should they cost? Read on to find out how much you should pay for “cheap storage units near me.”

## Monthly Cost of a Storage Unit

Storage units come in a variety of sizes and layouts. Most people choose 10×10 units to ensure they have enough space, without being too large.

The average monthly price for 10×10 units is about $150. Price can go down if you’re in areas that aren’t in high demand. But if you’re in busy, metro areas, costs are often higher. Other layouts that you can get for roughly the same price are 5×15 and 10×15 units. Smaller storage units can usually be had for less than$100 per month. These include units that are 5×5 and 5×10. While they seem small, they can hold a surprising amount of stuff.

That is if you pack them efficiently. If you’re using boxes or totes, make sure all the boxes are the same size, so you can stack them effectively and maximize the use of the space.

Larger storage units run closer to \$300 per month. You can always lock in the best prices by reserving a unit online ahead of time.

## Storage Units Near Me — Additional Considerations

Other factors to consider are the location of the storage unit. Many facilities have multiple floors, with units located inside. You’ll need to load your stuff onto cars, can take an elevator up, one load at a time.

These units can be inconvenient if you need to access your belongings on a regular basis. But they are also cheaper.

If you’re willing to spend a little more, you might be able to get a ground-floor unit. These are much less of a hassle to load and unload.

And if you don’t need a temperature-controlled unit, you can choose a storage unit that has road access. With these, you can park your vehicle or trailer right next to the door for the easiest loading and unloading.

These types of units are ideal for those storing furniture and other large, heavy items.

## Rid Your Home of Clutter

Many Americans run the risk of looking like hoarders. Their homes, basements, and garages, are filled to the brim with stuff they hardly ever use.

But we get it, it’s hard to part with your belongings. But don’t live in a giant mess. Get a storage unit and make your home more enjoyable to live in.

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0
RESEARCH PAPERS

Dynamic Models for Hydrogen Peroxide Control in Solar Photo-Fenton Systems

[+] Author and Article Information
J. D. Alvarez

CIEMAT—Plataforma Solar de Almería, P.O. Box 22, 04200 Tabernas, University of Almería, Dpto. de Lenguajes y Computación, Área de Ingeniería de Sistemas y Automática, Crta. Sacramento s/n, 04120 La Cañada, Almería, Spain

W. Gernjak

CIEMAT—Plataforma Solar de Almería, P.O. Box 22, 04200 Tabernas, Almería, Spain, University of Applied Life Sciences, Department of Water, Air and Environment, Muthgasse 18, 1190 Vienna, Austria

S. Malato

CIEMAT—Plataforma Solar de Almería, P.O. Box 22, 04200 Tabernas, Almería, Spain

M. Berenguel

Dpto. de Lenguajes y Computación, Área de Ingeniería de Sistemas y Automática, Crta. Sacramento s/n, 04120 La Cañada, Almería, Spain

M. Fuerhacker

University of Applied Life Sciences, Department of Water, Air and Environment, Muthgasse 18, 1190 Vienna, Austria

L. J. Yebra

CIEMAT—Plataforma Solar de Almería, P.O. Box 22, 04200 Tabernas, Almería, Spain, Dpto. de Lenguajes y Computación, Área de Ingeniería de Sistemas y Automática, Crta. Sacramento s/n, 04120 La Cañada, Almería, Spainluis.yebra@psa.es

J. Sol. Energy Eng 129(1), 37-44 (Dec 10, 2005) (8 pages) doi:10.1115/1.2391014 History: Received July 22, 2005; Revised December 10, 2005

Abstract

Simulations and real decomposition experiments with hydrogen peroxide induced by dissolved iron and solar illumination were performed in a solar pilot plant with compound parabolic collectors designed for photo-Fenton wastewater treatment. The structure of a gray-box linear model aimed at reproducing system dynamics with parameters dependent on operating conditions when operating around prescribed set-points is proposed. This simple model relates the changes in hydrogen peroxide concentration due to changes in hydrogen peroxide injection, dissolved iron, and solar illumination. Based on this model, control of the hydrogen peroxide concentration at set points between $200–900mg∕L±10mg∕L$ under dynamic conditions (simultaneous decomposition and addition of hydrogen peroxide by a dosage pump), was achieved. Different basic approaches for controller actuation upon the frequency of the dosage pump were tested, ranging from a simple PI controller to a PI controller plus antiwindup action and feedforward control. From these basic approaches, conclusions can be drawn about the process’s behavior under closed-loop control.

<>

Figures

Figure 1

Flow scheme of pilot plant

Figure 2

System response to an input change. Wide solid line: percentage of H2O2 pump frequency range, thin solid line: simulated output model, dashed line: real H2O2 concentration, (a) pulse of 5min duration with H2O2 pump operating at maximum frequency, (b) pulse of 5min duration with H2O2 pump operating at 20% of its maximum frequency

Figure 3

System consumption in the presence of Fe, H2O2. Solid line: simulated output model, dashed line: real H2O2 concentration, (a) cFe=1.43mg∕L, collectors were exposed to sunlight from 9060s<t<10260s. The rest of the time the collector was not illuminated; (b) cFe=0.63mg∕L, collectors were exposed to sunlight from 5580s<t<6780s. The rest of the time the collector was not illuminated

Figure 4

PI controller+antiwindup

Figure 5

Control of CH2O2: (a) system output; (b) controller output to the dosage pump, (c) real percentage of H2O2 pump frequency. Solid lines: PI controller with anti-windup; dashed lines: PI controller without antiwindup

Figure 6

PI controller+antiwindup+feedforward control

Figure 7

System output controlled by a PI with antiwindup (solid line) including a feedforward module less aggressive (dashed line) and including a feedworward module more aggressive (dotted line)

Figure 8

Real test (a) system output controlled with a PI with antiwindup; (b) controller output to dosage pump

Figure 9

Real test, (a) real system output controlled by a PI with antiwindup and feedforward module (dotted line) and simulated output model (dashed line); (b) real controller output to the dosage pump (dotted line) and simulated controller output (dashed line). The collector was exposed to illumination all the time. The system setpoint was fixed to 500mg∕L, as indicated by the solid line in (a)

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## Stream: general

### Topic: tfae formatting

#### Johan Commelin (Dec 27 2019 at 16:16):

tfae is being used. Kudos to @Reid Barton and @Simon Hudon for making the tactic work, and to @Yury G. Kudryashov for using it.
Here's a question/proposal about formatting: shall we format tfae statements with one "item" per line?
Pinging @Mario Carneiro and @Rob Lewis for this style issue.

#### Yury G. Kudryashov (Dec 27 2019 at 16:18):

Do you mean something like this?

lemma tfae_mem_nhds_within_Ioi {a u' : α} (hu' : a < u') (s : set α) :
tfae [s ∈ nhds_within a (Ioi a), /- 0 -/
s ∈ nhds_within a (Ioc a u'), /- 1 -/
s ∈ nhds_within a (Ioo a u'), /- 2 -/
∃ u ∈ Ioc a u', Ioo a u ⊆ s, /- 3 -/
∃ u ∈ Ioi a, Ioo a u ⊆ s] /- 4 -/ :=

Yes, exactly

#### Yury G. Kudryashov (Dec 27 2019 at 16:19):

Numbers help those writing (tfae_mem_nhds_within_Ioi h s).out 0 3

#### Johan Commelin (Dec 27 2019 at 16:20):

Hmmm, I now realise that tfae_have uses 1-based indexing, but tfae.out uses 0-based indexing...

#### Johan Commelin (Dec 27 2019 at 16:20):

I guess that should be fixed

#### Yury G. Kudryashov (Dec 27 2019 at 16:21):

Or even

lemma tfae_mem_nhds_within_Ioi {a b : α} (h : a < b) (s : set α) :
tfae [s ∈ nhds_within a (Ioi a), /- 0 : s is a neighborhood of a within (a, +∞) -/
s ∈ nhds_within a (Ioc a b), /- 1 : s is a neighborhood of a within (a, b] -/
s ∈ nhds_within a (Ioo a b), /- 2 : s is a neighborhood of a within (a, b) -/
∃ u ∈ Ioc a b, Ioo a u ⊆ s, /- 3 : s includes (a, u) for some u ∈ (a, b] -/
∃ u ∈ Ioi a, Ioo a u ⊆ s] /- 4 : s includes (a, u) for some u > a -/ :=

#### Johan Commelin (Dec 27 2019 at 16:21):

I think I would put := before comment 4

#### Johan Commelin (Dec 27 2019 at 16:21):

But that's minor nitpicking

#### Sebastien Gouezel (Dec 27 2019 at 16:22):

Also, for one-line comments you can use --.

#### Johan Commelin (Dec 27 2019 at 16:22):

Only if you don't put := at the end of the line

#### Sebastien Gouezel (Dec 27 2019 at 16:23):

Yes. I was implicitly seconding your comment :)

#### Yury G. Kudryashov (Dec 27 2019 at 16:23):

lemma tfae_mem_nhds_within_Ioi {a b : α} (h : a < b) (s : set α) :
tfae [s ∈ nhds_within a (Ioi a), -- 0 : s is a neighborhood of a within (a, +∞)
s ∈ nhds_within a (Ioc a b),   -- 1 : s is a neighborhood of a within (a, b]
s ∈ nhds_within a (Ioo a b),   -- 2 : s is a neighborhood of a within (a, b)
∃ u ∈ Ioc a b, Ioo a u ⊆ s,    -- 3 : s includes (a, u) for some u ∈ (a, b]
∃ u ∈ Ioi a, Ioo a u ⊆ s] :=   -- 4 : s includes (a, u) for some u > a

#### Johan Commelin (Dec 27 2019 at 16:24):

It's too bad that those comments won't end up in the docstring

#### Johan Commelin (Dec 27 2019 at 16:34):

In fact, I think it's quite important that these comments do end up in the docstring, because if the lemma gets used with tfae.out a person reading the code will probably want to hover over the lemma to see exactly which two equivalent statements are being used.

#### Yury G. Kudryashov (Dec 27 2019 at 16:40):

I'll copy them to the docstring.

#### Rob Lewis (Dec 27 2019 at 19:45):

I agree that the 0/1 indexing should be made consistent and that the numbers should appear in the docstring.

#### Rob Lewis (Dec 27 2019 at 19:46):

Note that (I think) markdown will format a numbered list starting with 1, even if the input starts with 0. i.e.

0. s is a neighborhood...
1. s is a neighborhood...

will get printed as

1. ...
2. ...

#### Rob Lewis (Dec 27 2019 at 19:47):

So maybe we should use 1-based indexing. Or unordered lists with indices added manually.

#### Mario Carneiro (Dec 27 2019 at 23:45):

Actually, I think we could use an annotation so that we can write:

lemma tfae_mem_nhds_within_Ioi {a b : α} (h : a < b) (s : set α) :
by tfae [
Ioi: s ∈ nhds_within a (Ioi a),
Ioc: s ∈ nhds_within a (Ioc a b),
Ioo: s ∈ nhds_within a (Ioo a b),
ex_Ioc: ∃ u ∈ Ioc a b, Ioo a u ⊆ s,
ex_Ioi: ∃ u ∈ Ioi a, Ioo a u ⊆ s] :=

and then reference the names using tfae_have Ioi <-> Ioo := tfae_mem_nhds_within_Ioi or similar

#### Mario Carneiro (Dec 27 2019 at 23:47):

It could be syntax for tfae [(\lam Ioi, Ioi) s ∈ nhds_within a (Ioi a), ...

#### Johan Commelin (Dec 28 2019 at 07:40):

I think I could have made tfae_have use 0-based indexing. But these new suggestions go beyond my skill set. They do look fancy. @Mario Carneiro Would tfae.out be able to use those indices as well?

#### Mario Carneiro (Dec 28 2019 at 08:07):

if by tfae.out you mean the term, no

#### Mario Carneiro (Dec 28 2019 at 08:07):

I would also suggest keeping zero based indexing in tfae.out

#### Johan Commelin (Dec 28 2019 at 09:26):

Well, in that case I would just let everything be 0-based indexed stuff

Last updated: May 14 2021 at 00:42 UTC


text:
Nth Roots And Rational Exponents Calculator

The roots of the polynomial are calculated by computing the eigenvalues of the companion matrix, A. Finding square roots and converting them to exponents is a relatively common operation in algebra. i have an exam in the morning an must know how to use it. Powers, Roots, and Radicals a The nth root of a is written: 3/; , where n is called the m In the. Convert between square root and a1/2 forms. To do so you need to change the number in the degree of the root field. If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old. If you have to have service with math and in particular with radical and rational exponents calculator or description of mathematics come pay a visit to us at Algebra1help. The nth Root Symbol. You can’t use counting techniques on an expression like 6 0. To start practising, just click on any link. If an element in X is negative, then the corresponding element in N must be an odd integer. 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On a calculator, a number is raised to a power using a key labeled For example, to take the fourth root of 6 on a TI-83 calculator, enter , to get the result 1. Radicals - Higher Roots Objective: Simplify radicals with an index greater than two. This quiz and attached worksheet will help gauge your understanding of simplifying expressions with rational exponents. CliffsNotes study guides are written by real teachers and professors, so no matter what you're studying, CliffsNotes can ease your homework headaches and help you score high on exams. indd 77 4/13/07 3:14:15 PM. We find the nth roots of the numerator and denominator for finding the nth roots of fractions. Convert between square root and a1/2 forms. If three identical effects are multiplied together, it means they’re each a cube root. The advantage of using this technique is your cursor stays on the base of the Entry line (see the first screen). Powered by Create your own unique website with customizable templates. Module 7: Roots, Radicals, and Function Operations. To evaluate nth roots using radical & rational exponent notation. Round your answer to three decimal places when appropriate. Mathfraction. Please try again later. Calculate real and complex square roots (2nd order roots) of numbers or $$\normalsize \sqrt{x}$$. Use a calculator to solve 0-4 nth Roots and Real Exponents. The root is found in the denominator (like a tree, the root is at the bottom), and the integer exponent is found in the numerator. Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums. Ask Math Questions you want answered. Ex: Find the real nth root(s) of a if a = 16 and n = 4. REAL nth ROOTS OF a. Comparing rational numbers calculator. What is an nth root? The nth root of a number is the number that would have to be multiplied by itself n times to get the original number. 1 nth Roots and RationalExponents 2. Simplify expressions with rational exponents. Algebra 2 (1st Edition) answers to Chapter 6 Rational Exponents and Radical Functions - 6. We take the cube root of 8 to get 2, then square 2 to get 4. 6 Solving Radical Equations 7. Rule: I want to point out that we have only been dealing with ~ with a. Rational Exponents ~ The use of rational numbers as exponents. If an element in X is negative, then the corresponding element in N must be an odd integer. This calculator will calculate the square root of a number and indicate whether or not the radicand is a perfect square. Index of a radical An nth root of ais written as a, where nis the index of the radical. Ex: Simplify an Expression with Negative Rational Exponents and Write in Radical Form Ex 1: Simplify an Expression with a Negative Rational Exponent Ex 2: Simplify an Expression with a Negative Rational Exponent Simplify An Expression with Rational Exponents (Positive Only) Power/Quot Simplify An Expression with Rational Exponents (Negative. root(3)(27) Solution: We can write the given root value in exponent form by using the following method. Negative exponent. root of a positive number is positive. Toggle navigation Close Menu. Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums. com includes valuable tips on how to solve nth roots, algebra and trigonometry and syllabus for college and other algebra subject areas. Solve rational equations and inequalities. Square Roots What power is a square root? A square is the inverse of a square root… 3 3 ? 3. Square roots of a negative number are not real. Exponents, roots and logarithms Here is a list of all of the skills that cover exponents, roots and logarithms! These skills are organised by year, and you can move your mouse over any skill name to preview the skill. Rational Exponents The nth root of a number can be expressed by using radical notation or the exponent 1 n. Ex: Find the real nth root(s) of a if a = 16 and n = 4. In general, for an integer n greater than I, if b" = a, then b is an nth root of a. 2 Simplify expressions in exponential form. Recognize and write a root using the radical notation. Therefore in general any attempt to compute a square root expressed in decimal form can only yield an approximation, though a sequence of increasingly accurate approximations can be obtained. FRACTIONAL EXPONENTS & ROOTS: explanation of terms and step by step guide showing how exponents containing fractions and decimals are related to roots: square roots, cube roots,. Convert between nth root and a1/n forms. Without the use of a calculator, approximate the optimum frequency ratio of the perfect 8th and the perfect 1st. Every real number has exactly one real. Simplify each radical (similar to p. Evaluate radical 3. For instance, 2 is the only real cube root of 8. 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Rational Exponents and Roots If x is positive, p and q are integers and q is positive, In other words, when you have a rational exponent, the denominator of that exponent is your index or root number and the numerator of the exponent is the exponential part. 1 Lesson WWhat You Will Learnhat You Will Learn Find nth roots of numbers. 22 = 4 and (-2)2 = 4, therefore the 2nd root of 4 would be 2. Evaluate radicals. 𝑎2=𝑎 𝑎𝑥2=𝑎 Let 𝑎𝑥=𝑎 So 𝑎1/2=𝑎 𝑎2𝑥=𝑎 𝑎2𝑥=𝑎1 2𝑥=1 𝑥=12. However, you might want a much higher root (e. So this is going to be equal to 2 times the sixth root of x to the sixth times x squared. com - Exponents. Rewrite the expression using radical notation Real nth Roots Evaluate these expressions with a calculator. In the example shown, the formula in D5 is: = B5 ^ ( 1 / C5 ) How this function works By definition, the nth root of a number can be. Rule: I want to point out that we have only been dealing with ~ with a. REAL nth ROOTS OF a Let n be an integer (n > 1) and let a be a real number. Come to Rational-equations. 7 Statistics and Statistical Graphs. Homework Help On Finding Nth Roots And Rational Exponents. Press these keys x,= 13 times and you will get the cube root of the number. Compute and estimate radicals. Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums. They can, also, generate a step by step explanation at the click of a button. Evaluate the. 1 : Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. 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Relationship to Exponents. 1 Evaluating Expressions with Rational Exponents Evaluate the expression without using a calculator. Square Roots What power is a square root? A square is the inverse of a square root… 3 3 ? 3. For example, 83 = = (2)2 = 4 Raise a to the exponent m, then take the nth root. In cases where you seek guidance on equations or perhaps denominator, Gre-test-prep. Find the prime factorization of the radicand. 7 352 Evaluate the expression using a calculator. 2 Graphing Square Root and Cube Root Functions. To find the roots by hand. The most common root is the square root. Simplify each radical (similar to p. - Notice the placement of the m and n. All of them are capable of performing exact computations. G raph square root and cubed root functions by hand and using a. In cases where you demand help with algebra and in particular with solve radical equation calculator or mathematics courses come visit us at Polymathlove. Evaluatin nth Roots WITH a Calculator 1. Recognize and write a root using the radical notation. Take the nth root of a, then raise the result to the exponent m. Radical expressions can be written by using rational exponents. Evaluate nth Roots & Rational Exponents. When a rational number is used as an exponent, the denominator indicates â€œrootâ€ and the numerator indicates â€œpower. Then state the index of the radical. 1 : Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Exponents and roots Here is a list of all of the skills that cover exponents and roots! These skills are organized by grade, and you can move your mouse over any skill name to preview the skill. Simplifying Square Roots from Simplifying Square Roots Worksheet, source: yumpu. Before we dive into simplifying exponents, let's take some time to learn exactly what an exponent is. We can write the 5 th root of 32 as 5 32. Exponents Lessons. EXAMPLE 4 Solve equations using nth roots. 312 Approximate roots with a calculator. Calculate nth roots via. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. ] Square Roots: A Pair Of Equal Factors [13 min. REAL nth ROOTS OF a. The Root Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. 4 Use Inverse Functions - 6. This algebra worksheet contains problems on converting between nth roots and rational exponents for numbers. 1) We have had a lot of experience with square roots But there are many more types of roots! n = index of the root Even roots get ± in Squares 23 = 33 = 43 = 53 = 83 = 93 = Cubes Fourths 915 (o Fifths Memorize! 15 = 122= 132= 142= 7aq 103 113. 1 Evaluating Rational exponents on the Calculator the nth root of a where a is a REAL number and n is the index. Arbitrary Exponents. Come to Gre-test-prep. This page will help you to simplify a term under a radical sign. n 5 4, a 5 256 2. 5 th power $$\sqrt{x}=\sqrt[2]{x}=x^{\frac{1}{2}}$$. 1: Radicals and Rational Exponents Deﬁnition of nth root of a number Let a and b be real numbers and let n be an integer n 2. Evaluate expressions with rational exponents. Thus 8 2/3 = 4. Converting Between Radical and Rational Exponent Form If the nth root of a is a real number and m is an integer, then = añ and = aï. In general, for an integer n greater than l, if b" — then b is an nth root of a. No Download or Signup. Lesson 7-3: nth Roots and Rational Exponents. • The nth root of a number is denoted. when n is an even integer. A power can be undone with a radical and a radical can be undone with a power. Simplify using perfect Nth factors. In the "Number", enter the number of root from which you want to calculate. I can use and apply properties of exponents; I can use and apply properties of radicals; I can write radicals in simplest form. To rationalize an Nth Root Denominator - Multiply by the base raised to the power of the index minus the exponent. (i) If n = 2, then nth root is called the square root; if n = 3, then it is called cube root. 290 Chapter 6 Rational Exponents and Radical Functions 6. 312 Approximate roots with a calculator. Online radicals and roots calculator. Deﬁ nition of Rational Exponents a __n1 = n √ __ a for a > 0 and integer n a m__ n = n √ ___ a m = n(√) __ a m for a > 0 and integers m and n 2. 1 Evaluate nth Roots and Use Rational Exponents. Plus, unlike other sites with a radicand calculator, if the radicand is not a perfect square, this calculator will calculate the nearest perfect squares that are greater and less than the entered radicand. • The nth root of a number is denoted. For instance, 2 is the only real cube root of 8. Hence, grade 12 algebra questions may be regard as an addition of Arithmetic. Evaluate radical 3. 43 = 64 so 4 is a cube root or third (3rd) root of 64. Calculator. To rationalize an Nth Root Denominator - Multiply by the base raised to the power of the index minus the exponent. A rational exponent represents both an integer exponent and an nth root. You have the same base, you would add the exponents. Compute and estimate radicals. As you know, a square root is indicated by and a cube root by V. Notes Over 7. If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old. We find the nth roots of the numerator and denominator for finding the nth roots of fractions. The root can also be called the index. 3 Explain and use the laws of fractional and negative exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. Use a calculator to show that each statement is true. On asking these to my maths teacher , in addition to these he told me that how to take out nth root [if we want to take out 7th root of 128 which is (128)^1/7 =2 ] it can be also done using a simple calculator. Online nth root calculator - calculate nth root of real number (integer) using roots and rational / radicand exponents Calculation tool. +1 Solving-Math-Problems. com makes available invaluable answers on nth roots and rational exponents calculator, precalculus i and notation and other algebra subject areas. If this is the case, our square root calculator is the best option to estimate the value of every square root you desired. Evaluate nth Roots & Rational Exponents. Then we will apply similar ideas to define and evaluate nth roots. So this is going to be equal to 2 times x times the sixth root of x. al/n 0 al/n n is an even integ No real nth roots. Approximate a principal square root. The nth root of a number is that number which when raised to the nth power (ie when multiplied by itself n times) results in the number. The nth root of a n • To simplify where a is any real • The value of when n is even • The value of when n is odd: Definition of a Rational Exponent • The n th root of of a is the same as raising a to the power of 1/n • Also, given exists then. Use Rational Exponents Evaluate nth Roots and. 1 nth Roots and Rational Exponents Exploring the Defi nition of a Rational Exponent Work with a partner. Work with a partner. This is called scientific notation, or E notation on a calculator ("E" stands for "Exponent"). Deﬁnition of a1/n. n 5 3, a 5 512 Checkpoint Find the indicated real nth root(s) of a. In some situations, you don't need to know the exact result of the square root. In cases where you seek guidance on equations or perhaps denominator, Gre-test-prep. -216 81 Real nth Roots of a Let n be an integer (n > I) and let a be a real number. com and figure out equations by factoring, systems of equations and a large amount of other math topics. Examples, Use Graph or Table to Check. Post a Comment. Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums. We have got a ton of really good reference tutorials on topics varying from systems of linear equations to introductory algebra. The advantage of using this technique is your cursor stays on the base of the Entry line (see the first screen). Enter values x and y into this calculator to solve for the exponent value n in the equation $$x^n = y$$. The exponent may be positive or negative. Title: Microsoft Word - D5 nth roots and rational exponents. I can apply the properties of exponents and the definition of nth root to expìain that xn 1 can apply the properties of exponents to demonstrate that x n. tmd S'impt'ify the result by factoring and taking the nth root. Roots and Rational Exponents You probably learned what a square root was back in middle school, and it was maybe even kinda fun. Nth Roots and Rational Exponents Parts of a Radical root radicand If there's no number at the root, then it's a 2. 1 Homework Find the real solution(s) of the equation. For any value of n, whether positive, negative, integer or non-integer, the value of the nth power of a binomial is given by: There are many binomial expansion applications in physics. And the radical is denoted by the ? (sqrt) symbol. 1 : Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Ask Math Questions you want answered. 2k v«asEulð learn it — real-life 2k 1 as finding reptile and support -125 nth Roots and Rational Exponents GOAL 1 EVALUATING NTH ROOTS You can extend the concept of a square root to other types of roots. Solve equations using nth roots. Nth Root (Radical) calculator is a simple tool used to calculate the N th Root of the given real number which has the N th radical value. Which One Doesn’t Belong?. We have got a lot of really good reference tutorials on subjects starting from expressions to algebra i. Method #2: evaluate am , then take the nth root of that value Part I: Use the definition of rational exponents to rewrite the radical expression using rational exponents. The rational exponent m/n on a quantity represents the mth power of the nth root of the quantity or the nth root of the mth power of the quantity, where n is the index of the radical. Byju's Multiplying and Dividing Rational Expressions Calculator is a tool which makes calculations very simple and interesting. No real roots. Yo[] Read Book Online: 6 1 Evaluate Nth Roots And Use Rational Exponents Worksheet Download ebook 6 1 Evaluate Nth Roots And Use Rational Exponents Worksheet in pdf / kindle / epub format also available for any devices anywhere. 3-1 Evaluate nth Roots and Use Rational Exponents Essential Question What is the relationship between nth roots and rational exponents? Key Vocabulary nth Root of a If ban, then b is the nth root of a… _____ Examples: 392 3 is the _____ root of 9 3 9 2 164 2 is the _____ root of 16 2 416 Rational Exponent. 5!=125!−32= Parts of the Radical Sign: 81! So far, we have only dealt with integer exponents. Use a calculator to show that each statement is true. This is written n ab. The expression in Example 16(e) is not defined when x=1/2 because. One real root:. Method #2: evaluate am , then take the nth root of that value Part I: Use the definition of rational exponents to rewrite the radical expression using rational exponents. Get Started. Hand out the Exponents and nth Roots worksheet. 1 Evaluate Nth Roots and use Rational Exponents Things you should be able to do: - Rewrote radical expressions using rational exponent notation - Rewrite rational exponent expressions using radical notation - Evaluate an Nth root without using a calculator - Solve an equation using Nth roots Examples :. So this is going to be equal to 2 times the sixth root of x to the sixth times x squared. : If then is root of. In case that you seek assistance on linear algebra or perhaps linear equations, Mathfraction. If ab= then ba2 =. exponent to write each expression as a radical raised to an exponent. A rational exponent represents both an integer exponent and an nth root. ' m Igil s NewVocabulary nth root principal nth root 1 Simplify Radicals A square root of a number is one of two equal factors of that number. 1 Evaluating Rational exponents on the Calculator the nth root of a where a is a REAL number and n is the index. The mass of a Tyrannosaurus rex was 4000 kilograms. REAL nth ROOTS OF a. Post a Comment. 4 Use Inverse Functions - 6. notebook 10 February 05, 2018 Jan 2510:38 PM Real nth Roots •If n is odd, then a has 1 real nth root •If n is even and a>0, then a has 2 real nth roots. Rational Exponents. 290 Chapter 6 Rational Exponents and Radical Functions 6. Then we will apply similar ideas to define and evaluate nth roots. And the sixth root, this part right here, the sixth root of x to the sixth, that's just x. Radicand: The thing you are finding the root of. Calculate square roots via calculator. And the radical is denoted by the ? (sqrt) symbol. Radicals and Rational Exponents Roots, Radical Expressions and Functions I. 0-4 nth Roots and Real Exponents. 4 Exercises - Skill Practice - Page 442 4 including work step by step written by community members like you. Example 2: Write each radical expression using. Square Root Calculator updated. 1 Convert between square root and a1/2 forms. 27,332,016 solved | 808 online. n-th Roots and Rational Exponents n-th roots - For any real numbers a and b and any positive integer n, if !!=! then !!=!. 1 Examples 1-3 Section 5. I will begin by doing one problem as an example at the whiteboard. Each group should have as many factors as the index. Both X and N must be real scalars or arrays of the same size. With x as a real number. For example: A cube root of 8 is 2, since 23 = 8 A fourth root of 81 is 3, since 34 = 81 For integers n greater than 1, if bn = a then b is an nth root of a. Rational Exponents and Roots If x is positive, p and q are integers and q is positive, In other words, when you have a rational exponent, the denominator of that exponent is your index or root number and the numerator of the exponent is the exponential part. 1 Lesson Opener • You will havetwominutes to complete the lesson opener in your notebook. The index of the root you apply – that is, the little superscript number before the radical sign – should be the same as the exponent you're trying to remove. Algebra 2 Notes Powers, Roots, and Radicals Unit 07 Alg II Unit 07 Notes Powers Roots & Radicalsrev Page 1 of 64 01/16/2014 Exponents, Radicals, and Rational Number Exponents Big Idea: If ban, then b is the nth root of a. 1 Using Rational Exponents and nth Roots. These rules will help to simplify radicals with different indices by rewriting the problem with rational exponents. Section 18 Power Functions With Radicals And Rational Exponents Lesson #9 Adding and Subtracting with Rational Exponents. View Notes - 7_1 Homework - nth Roots and Rational Exponents from MATH Honors Alg at Annapolis High School. One real root:. Use a calculator to fill out the table for !!!. In the previous standard, we established some rules about fractional exponents. We find the fourth root in the same way and generalize this for nth roots. While square roots are the most common type of radical we work with, we can take higher roots of numbers as well: cube roots, fourth roots, ﬁfth roots, etc. Reducing rational expressions to lowest terms Multiplying and dividing rational expressions Adding and subtracting rational expressions Using LCM method Simplifying complex rational expressions Section R8: Rational Exponents, nth Roots. This is the special symbol that means "nth root", it is the "radical" symbol (used for square roots) with a little n to mean nth root. nth roots. What is an nth root? The nth root of a number is the number that would have to be multiplied by itself n times to get the original number. Evaluate expressions with rational exponents. REAL nth ROOTS OF a Let n be an integer (n > 1) and let a be a real number. • The exponent becomes the numerator. Everything you need to score well in your next quiz. The same properties of exponents apply for both positive and negative exponents. n 5 3, a 5 512 Checkpoint Find the indicated real nth root(s) of a. We can further simplify this expression by rationalizing the denominator using rational exponents. Online radicals and roots calculator.


text:
# 60.0 mL of 0.322 M potassium iodide are combined with 20.0 mL of 0.530 M lead (II) nitrate.  How many grams of lead (II) iodide will precipitate? (you must write your own reaction)

Question

60.0 mL of 0.322 M potassium iodide are combined with 20.0 mL of 0.530 M lead (II) nitrate.

How many grams of lead (II) iodide will precipitate? (you must write your own reaction)


text:
# cf_guidance

This library has normalizations and schedules for the Classifier-free Guidance parameter in Diffusion models. We call the guidance modifiers transforms.

It seems that transforms can improve the quality of generated images across Diffusion pipelines. To be specific:

• Normalizing can help with image syntax.
• Scheduling can improve image details.

For more details, please check out the official blog post.

## Install

pip install cf_guidance

## How to use

The follow section creates a few example schedules. Then, we use the schedules to initialize different kinds of transforms.

### Creating Cosine schedules

We create a few different schedules:

A Cosine schedule that starts from the default Guidance parameter in diffusion $\left( G = 7.5\right)$ and works it way down to a minimum of $0.15$.

Next we create the same schedule, except with an initial warmup. With warmup, $G$ starts from $0.15$ and linearly works up to a maximum $G_\text{max} = 7.5$. Here we warmup over the first 10% of the total diffusion steps. After warmup, the schedule decreases following a cosine curve.

Finally, we create a schedule with small values of $G$. This is meant for the guidance transform TNormGuidance. This transforms puts the vectors on a similar scale, so we need a much smaller value than usual when mixing them.

from cf_guidance.schedules import get_cos_sched

# Parameters from the blog post
max_val = 7.5
min_val = 0.15
num_steps = 50
num_warmup_steps = 5

# 1) Baseline cosine schedule
cos_params = {
'max_val':           max_val,
'num_steps':         num_steps,
'min_val':           min_val,
'num_warmup_steps':  0,
}

# 2) Cosine schedule with warmup
warmup_cos_params = {
'max_val':           max_val,
'num_steps':         num_steps,
'min_val':           min_val,
'warmup_init_val':   min_val + 1., # to show we can offset the warmup relative to min
'num_warmup_steps':  num_warmup_steps,
}

# 3) Cosine schedule with small values for T-Norm guidance
t_scale_params = {
'max_val':         0.25,
'num_steps':       num_steps,
'min_val':         0.05,
}

# create the schedules
cos_g = get_cos_sched(**cos_params)
warmup_g = get_cos_sched(**warmup_cos_params)
TNorm_g = get_cos_sched(**t_scale_params)

Let’s plot these schedules to see what they look like.

import matplotlib.pyplot as plt

# plot them
plt.plot(cos_g)
plt.xlabel('Diffusion Timesteps')
plt.ylabel('$G$ Guidance Parameter')
plt.title('Cosine Schedule');

plt.plot(warmup_g)
plt.xlabel('Diffusion Timesteps')
plt.ylabel('$G$ Guidance Parameter')
plt.title('Warmup Cosine Schedule');

plt.plot(TNorm_g)
plt.xlabel('Diffusion Timesteps')
plt.ylabel('$G$ Guidance Parameter')
plt.title('Small Cosine Schedule to T-Norm');

### Creating Guidance Transforms

Guidance transforms take one initialization parameter: schedules. This is a map from parameter names to an array-like, indexable sequence of values.

For a given parameter name at diffusion timestep idx, the value of schedules[name][idx] should be the value of that parameter at the given timestep.

In this case, we call the parameter $G$ as a lowercase ‘g’.

from cf_guidance.transforms import *

# Create a baseline Guidance with cosine schedule.
sample_sched = {'g': cos_g}
guidance = GuidanceTfm(sample_sched)

# Guidance with normalization and a cosine schedule.
norm_guidance = BaseNormGuidance(sample_sched)

# T-Norm Guidance with the smaller-valued schedule
small_sched = {'g': TNorm_g}
small_guidance = TNormGuidance(small_sched)

### Using the transforms in a Diffusion pipeline

The code snippet below shows where and how these transforms are used in an image generate() pipeline. We will use the norm_guidance class created above for the example.

Specifically, we call this norm_tfm with the following arguments:

• The unconditioned noise predictions.
• The conditional noise predictions.
• The current timestep index.

We borrow code from the official StableDiffusionPipeline from HuggingFace to show where our norm_tfm goes. This seems like a good starting point, since many scripts and functions are based on this HuggingFace script.

    # inside of StableDiffusionPipeline

for i, t in enumerate(self.progress_bar(timesteps_tensor)):
# expand the latents if we are doing classifier free guidance
latent_model_input = torch.cat([latents] * 2) if do_classifier_free_guidance else latents
latent_model_input = self.scheduler.scale_model_input(latent_model_input, t)

# predict the noise residual
noise_pred = self.unet(latent_model_input, t, encoder_hidden_states=text_embeddings).sample

# NOTE: our transforms go here:
###############################
if do_classifier_free_guidance:
noise_pred_uncond, noise_pred_text = noise_pred.chunk(2)

#noise_pred = noise_pred_uncond + guidance_scale * (noise_pred_text - noise_pred_uncond)

# NEW cf_guidance UPDATE
noise_pred = norm_guidance(noise_pred_uncond, noise_pred_text, i)

### Creating more cosine schedules

The cosine schedule function was based on a combination of the Cosine schedulers in timm and HuggingFace.

It has a variety of parameters to support many different schedule combinations, as shown below.

# using wilder parameters, worth exploring
two_cycles = {
'max_val':           max_val,
'num_steps':         num_steps,
'min_val':           min_val,
'num_cycles':        1,
'warmup_init_val':   min_val + 1., # to show we can offset the warmup relative to min
'num_warmup_steps':  0,
}

k_decay_cos = {
'max_val':           max_val,
'num_steps':         num_steps,
'min_val':           min_val,
'num_cycles':        1.5,
'k_decay':           0.7,
'warmup_init_val':   min_val + 1., # to show we can offset the warmup relative to min
'num_warmup_steps':  num_warmup_steps,
}

# create the schedules
two_cycle_sched = get_cos_sched(**two_cycles)
k_decay_sched = get_cos_sched(**k_decay_cos)

plt.plot(two_cycle_sched)
plt.title('Cosine With a Full Cycle');

plt.plot(k_decay_sched)
plt.title('Cosine with Warmup, 1.5 Cycles, and K-decay');


text:
KMS º½ÇÐÈ¸ Æ÷Ç×°ø´ë 2000. 4. 22. (Åä)

On Linear Preservers of Several

Multivariate Majorization

Contents :

(0) Abstract

(1) Definitions

(2) History

(3) Previous Results

(4) Main Results

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KMS º½ÇÐÈ¸ Æ÷Ç×°ø´ë 2000. 4. 22. (Åä)

(0) Abstract

In this paper, we characterize linear operators on real matrices that preserve multivariate majorization.  We place no additional restrictions on the linear operator.

(1) Definitions

Majorization is a topic of much interest in various areas of mathematics and statistics. (1929³â Hardy, Littlewood. Polya's : convex ft. g °¡ ÀÏ ÇÊ¿äÃæºÐÁ¶°Ç) (1923³â Schur's work on Hadamard  ºÎµî½Ä)(1905³â °æÁ¦ÇÐ¿¡¼ Lorenz curve)- ´ë¼ö, ÇØ¼®, ±âÇÏ, ¼öÄ¡ÇØ¼®, Åë°è, °æÁ¦µî

If and are nonincreasing -vectors of nonnegative real numbers such that  for with equality at , then we say that is (vector) majorized by and write.  Equivalently, if two row vectors have the relationship then there is a doubly stochastic matrix such that . (ÀÌ ¿Ü¿¡ ¿¬¼ÓÇÔ¼ö version, ¹«ÇÑ ¼ö¿ version µîµµ ÀÖÀ½)

Our interest is in the subject of majorization for matrices.  The MATRIX VERSION of majorization is that an real matrix is multivariate majorized by if there is a  DOUBLY STOCHASTIC matrix such that .

G. Dahl's Definition in  Matrix majorization, Linear Algebra Appl. 288:53-73(1999)--- "Row Stochastic Matrix"

A linear operator is said to  preserve a relation R on if R

implies R.

* The topic of linear preservers (ºÎ·Ï)

(2) History

In  "Majorization and inequalities in matrix theory", Linear Algebra Appl., V.199:17-67 (1994), T. Ando characterized the linear operators which preserve vector majorization (equivalently, that preserve multivariate majorization on .  We shall characterize all linear operators which preserve multivariate majorization with no conditions on m, n, or T.

*. "Linear Operators Strongly Preserving Multivariate Majorization " (with Y.H. Lee) Proc. Workshops in Pure Math. V. 17, part I (1997) pp.113-118.

**. Linear operators strongly preserving multivariate majorization with $T(I)=I$. Kyungpook Math. J. 39 (1999), no. 1, 191--194.  (with L. Beasley and Y.-H. Lee) (Reviewer: Chi-Kwong Li) 15A04 (15A21 15A30)

***. A proof of conjecture on strong preservers of multivariate majorization" (with L. Beasley), preprint.

C.-K. Li, B.-S. Tam. N.-K. Tsing, Linear maps preserving the set of doubly stochastic matrices, Preprint(1999)'s   Main Result :       TFAE

(a) T(DS(n)) = DS(n)

(b) T(P(n)) = P(n)

(c) TÀÇ ¸ð¾çÀº for some P, Q in P(n)

µû¶ó¼

then So let ·Î ³õÀ¸¸é Á¶°Ç T(I) =I¸¦ ¾ø¾Ù ¼ö ÀÖ´Ù°í »ý°¢ÇßÀ½.

**** Linear Operators Preserving Multivariate Majorization " (with L. Beasley), Linear Algebra and its Applications 304 141-159 (2000)                                      (  TODAY!! )

Linear Operators Strongly Preserving Matrix Majorization " (with Y.-H. Lee and L. Beasley), Preprint

(3) Previous Results

NOTE: ¾Æ·¡¿¡¼  ?? Àº ½Ç¼ö R À» ÀÇ¹Ì ÇÑ ´ä´Ï´Ù. Áï, Àº ÀÔ´Ï´Ù.

(editor ÀÇ error)

(´ë¿ì¼ø¼ö¼öÇÐ 1997) Theorem 2.11  Let for . Then is a multivariate majorization strong preserver on if one of the following or a composition of the following holds :

(1) for any nonzero scalar , an invertible matrix and a permutation matrix ;

(2) for a linear functional .

(°æºÏ¼öÇÐÀú³Î 1999) Theorem 3.1 If strongly preserves multivariate majorization and (i.e., preserves nonnegative matrices), then a permutation matrix .

(4) Main Results

By Birkhoff's Theorem , DS(n) is the convex hull of the set P(n) of n by n permutation matrices.

We now define two linear mappings, which are fixed throughout the paper.

Definition 2.1

Define by where

. That is, is the column vector whose th component is the th row sum of , equivalently . Define by where Thus, is the diagonal matrix whose main diagonal is the vector .

Lemma 2.2

If is any element of and is any element of , then and

D¿Í rÀÌ doubly stochastic Çà·Ä °öÇÑ °Í¿¡ ºÒº¯

Proof.

Let and .   Let .  Then  the entry of is

That is, .  Similarly .

Lemma 2.4

Let and the linear operator be defined by ,  then preserves multivariate majorization.

MAIN THEOREM 2.5

Let  be a  linear operator  on .  Then  TFAE:

(i)  preserves multivariate majorization.

(ii) Either

(a) There exist matrices and such that  for all or

(b) There exist matrices and a permutation matrix such that for all .

We state here the row version of Ando's characterization.

Lemma 2.6 (Ando, Corollary 2.7 ) Let preserve (vector) majorization, then either

(i) for some or

(ii) for some and

.

Lemma 2.7

Let the linear operator on preserve multivariate majorization and . Then

preserves multivariate majorization.

Áõ¸í : T´Â M.M.P. ÀÎµ¥ S°¡ ¾Æ´Ï¶ó¸é ¸ð¼øÀÓ.

¿©±â¼ ¿ì¼± T(X)ÀÇ j-th row¸¦ »ý°¢ÇÑ´Ù.

Lemma 2.8

If   preserves multivariate majorization, then defined by

the th row of  T [ 0 ... 0 x 0 ... 0 ]^t

i^th row

preserves (vector) majorization.

¿©±â¼ º¸ÀÌ´Â °ÍÀº °¡ m.m.p.ÀÌ¹Ç·Î ´Â °¢°¢ÀÇ row vector maj.À» º¸Á¸ÇÑ´Ù´Â °Í. ±×·¯¸é ÀÌ¾î¼ T°¡ linear ÀÌ¹Ç·Î ´ÙÀ½À» °®´Â´Ù.

Lemma 2.9

If   preserves multivariate majorization, then for and there exist vectors , permutation matrices , and scalars   and

with or , and , such that

ÀÌÁ¦ ÀÌ imageÀÇ ¸ð¾çÀ» ´Ü¼øÈ÷ ÇÏ±â À§ÇØ °è¼öÀÇ ¼ºÁúµéÀÇ °¡´É¼ºÀ» ÁÙ¿©°£´Ù.

Lemma 2.10

Suppose for some is not a multiple of , we  have

If preserves multivariate majorization then necessarily for all .

Lemma 2.12  If preserves multivariate majorization and

has at least two nonzero rows, one of which, say the th, is    for some nonzero vector which is not a multiple of, and the other, say the th, is for some and , then .

ÀÌÁ¦ µÎ °³ÀÇ ¸ð¾çÀÌ µ¿½Ã¿¡ ³ªÅ¸³ª¸é ¸ð¼øÀÓÀ» º¸ÀÎ´Ù.

Lemma 2.13

If

for some which is not a multiple of , then does not preserve multivariate majorization.

´ÙÀ½ÀÇ µÎ Lemma´Â and ÀÓÀ» º¸¿©ÁØ´Ù.

Lemma 2.15

Let be defined by

where . If preserves multivariate majorization, then .

Lemma 2.16

Let be defined by

where  . If preserves multivariate majorization, then .

Now we can prove Main Theorem.

MAIN THEOREM 2.5

Let  be a  linear operator  on .  Then  TFAE:

(i)  preserves multivariate majorization.

(ii) Either

(a) There exist matrices and such that  for all or

(b) There exist matrices and a permutation matrix such that for all .

PROOF

From the Corollary (2.14), T(X) must have one of the forms:

Either

(i)  There exist matrices and such that for all or

(ii) T(X) =[] + for some , in , and for all in .

In the second case, from the previous Lemmas 2.15, 2.16 and 2.7 we have

T(X) =[] + = for some matrix and permutation matrix .   QED.

(2) Linear Preserver Problems

Let K be an algebraic structure and let M(K) denote the  vector space of all m*n matrices over  K.

Over the last century, a great deal of effort has been devoted to  the following problem.  Characterize those  linear  operators  T : M  --> M which leave a function or  set  invariant.   We  call  this  a  linear preserver problem.

The  most  typical  and  oldest  type  of  linear preserver problem is as follows:

(1)ÀÏ ¶§ ¾Æ·¡ ´Â?

F ´Â scalar valued, matrix valued or set-valued function on M.

It is clear that the  set  of  all  such matrix-valued linear operators T is a multiplicative semigroup with an identity.

The study of these operators began in 1897  when  Frobenius characterized the linear operators that preserve the determinant  over two different sets of matrices :

Frobenius proved that the  semigroup  of  linear  operators  that preserve the determinant consists of  linear  transformations  of  the form

(1.1)

(i)  C °¡  complex field ÀÎ °æ¿ì,  det (UV)  =  1

(ii)  M ÀÌ real symmetric matrices  ÀÎ °æ¿ì,

U = kQ, V = Q^T and det(Q).

In 1925, Schur extended and improved the result for  Case  (i) as follows.  Let ?(A) be the (nCk)^2 - tuple of the k-th   order (i.e., kxk) minors of A in some order  where k ? 3 is fixed.

Schur  proved  that if ?(T(A)) = S(?(A)), for a fixed nonsingular matrix S, then T has one of the two forms in (1.1) (without the restriction det (UV) = 1).

In 1949, Dieudonné showed that if F is any  field  and  if  T  is  a semilinear operator mapping M(F) onto itself, which  holds  the  cone det (A) = 0 invariant, then T is of the form

where ? is an automorphism of the field and a_ij is the  (i,j) entry of A.

We will call those operators T in (1.1)

(U,V)-operators.

Other interesting results of this type include:

(a) [59]  If M is is the set of all nxn Hermitian  matrices,

and F(A) = Inertia (A),  then   T  is  a  (U,V)-operator

where U = V*, the conjugate transpose  of  U,  for

some nonsingular matrix V ? M(C).

(b) [43]  If M = M (C) and F(A) = { x*Ax| x ? C , x*x  = 1},

then T  is a  (U,V)-operator  where  U = V* for some

unitary matrix U.

Consequences and recent  work  of  this  type  can  be  found  in [3, 17, 22, 23, 24, 28, 48, 53,  54]  for  preservers  of  generalized matrix functions, in [33, 39, 40, 50, 55] for inertia preservers,  and in [44, 45, 46, 47] for numerical range and radius preservers.

We note that many linear preserver problems can be reduced to the problem of characterizing linear operators on M(K) which map the  set of rank 1 matrices into itself.

A semiring (Gregory and Pullman [32] or Kim  [41])  is  a  binary system (S, +, ?) such that (S, +) is an Abelian monoid  (identity  0), (S, ?) is a monoid (identity 1), ? distributes over +, 0 ? s = s ? 0 = 0 for all s in S, and 1 \$ 0.  If 0 is the  only  element  to  have  an additive inverse, then S is antinegative.

The  Second type of linear preserver problem is:

(2) Let ÀÏ ¶§  ´Â s.t

Some  examples  of  this  type  of  problem  with

appropriate characterizations.

(a)  [51]  If M = M(C), B is the set of unitary

matrices in M(C) and T(B) ¡ø B, then T is a

(U,V)-operator for some unitary matrices U and V.

(b) [4, 11, 13, 53]  Suppose M = M(C) and B is the

set of all rank-k matrices. If T(B)¡øB, then T is

a (U,V)-operator for some nonsingular matrices

U and V.

(c) [19] Suppose M = M(S) where S is an

antinegative semiring with no zero divisors, and B

is the set of all idempotent matrices in M(S). If

T(B) = B, then T is a  (U,V)-operator where

U = V for  some invertible matrix V in M(S).

We  may  note  that,  in  the  first  example,  we  were simply characterizing the structure of all linear operators T  that  map  the unitary group into itself.

There is an extensive literature concerning the  characterization of linear operators of type [II].  Further results may be found in [1, 2, 6, 7, 8, 9, 30, 49] for sets defined by rank, in [15, 16,  18,  21, 29]  for  sets  defined   by   graph   theoretic   properties,   i.e., irreducibility or strong connectivity, term rank, etc.,  and  in  [10, 20, 25, 35, 56, 57] for characterization of preservers of other sets.

The  last and 3rd type of linear preserver problem is: (3)  ÀÎ ´Â?

Some of the major results are:

(a)  [5, 61]  Suppose M = M (C) and A ~ D if A

and D commute. If  T  is a nonsingular linear

operator on M satisfying T(A) ~ T(D) whenever

A ~ D, then T is of the form

or

for some nonsingular matrix X ? M (C) ; l ? C

and linear functional f on M.

(b)  [62]  Suppose M = M (C) and A ~ D if A and

D are similar.

If  T  is a linear operator on M satisfying

T(A)~T(D) whenever A~D, then T is of the form

or

for some nonsingular matrix X ? M (C), l ? C and

linear functional f on M.

Further studies of the above type are found in [36, 61].

In  [26]  and  [27],  Chan,  Lim  and  Tan  gave  the following characterization for those linear operators that  preserve  idempotent and tripotent matrices.

THE END

**********************

Âü°í ÀÚ·á:

A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, 1979

T. Ando, Majorization, doubly stochastic matrices and comparison of eigenvalues, Linear Algebra Appl. V.118:163-248 (1989).

T. Ando, Majorization and inequalities in matrix theory, Linear Algebra Appl., V.199:17-67 (1994)

J. V. Bondar, Comments and complements to: Inequalities: Theory of Majorization and Its Appl. by Albert W. Marshall and Ingram Olkin, Linear

Algebra Appl. V.199:115-130(1994)

G. Dahl, Matrix majorization, Linear Algebra Appl. 288:53-73(1999)

*************************


text:
2.2k views
Consider a paging hardware with a $TLB$. Assume that the entire page table and all the pages are in the physical memory. It takes $10$ milliseconds to search the $TLB$ and $80$ milliseconds to access the physical memory. If the $TLB$ hit ratio is $0.6$, the effective memory access time (in milliseconds) is _________.
edited | 2.2k views

EMAT=TLB hit $\times$ (TLB access time $+$ memory access time) $+$ TLB miss(TLB access time $+$ page table access time+memory access time)

$=0.6(10+80)+0.4(10+80+80)$

$= 54+68$

$=122$ $msec$
edited by
0
Can you please explain answer to this question in detail @Arjun Sir, @Neha pawar
+5

@ sandeep  effective memory access time = adress translation time + page access time

adress translation time  = TLB access time +  TLB miss*(memory access time)

=    10 + 0.4*80

= 42 ms

page access time        = memory access time

=80 ms

so,        effective memory access time    = 42 + 80   = 122ms

0
This kind of approach is required to question in the GATE.
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TLB stands for Translation Lookaside Buffer. In Virtual memory systems, the cpu generates virtual memory addresses. But, the data is stored in actual physical memory i.e. we need to place a physical memory address on the memory bus to fetch the data from the memory circuitry. So, a special table is maintained by the operating system called the Page table. This table contains a mapping between the virtual addresses and physical addresses. So, every time a cpu generates a virtual address, the operating system page table has to be looked up to find the corresponding physical address. To speed this up, there is hardware support called the TLB. The TLB is a high speed cache of the page table i.e. contains recently accessed virtual to physical translations. TLB hit ratio- A TLB hit is the no of times a virtual-to-physical address translation was already found in the TLB, instead of going all the way to the page table which is located in slower physical memory. TLB hit ratio is nothing but the ratio of TLB hits/Total no of queries into TLB. In the case that the page is found in the TLB (TLB hit) the total time would be the time of search in the TLB plus the time to access memory, so TLB_hit_time := TLB_search_time + memory_access_time In the case that the page is not found in the TLB (TLB miss) the total time would be the time to search the TLB (you don't find anything, but searched nontheless) plus the time to access memory to get the page table and frame, plus the time to access memory to get the data, so TLB_miss_time := TLB_search_time + memory_access_time + memory_access_time But this is in individual cases, when you want to know an average measure of the TLB performance, you use the Effective Access Time, that is the weighted average of the previous measures EAT := TLB_miss_time * (1- hit_ratio) + TLB_hit_time * hit_ratio. EAT := (TLB_search_time + 2*memory_access_time) * (1- hit_ratio) + (TLB_search_time + memory_access_time)* hit_ratio. As both page table and page are in physical memory T(eff) = hit ratio * (TLB access time + Main memory access time) + (1 - hit ratio) * (TLB access time + 2 * main memory time) = 0.6*(10+80) + (1-0.6)*(10+2*80) = 0.6 * (90) + 0.4 * (170) = 122


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AP Chemistry : Half Reactions

Example Questions

Example Question #1 : Half Reactions

For the redox reaction shown, which of the following half reactions occurs in the anode?


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# Is the empty string and some words of even length are elements of this set?

$$L = \{w \in \{a,b\}^*| \text{the first, the middle, and the last characters of w are identical}\}$$.

I have my answers, but I need confirmation:

Is the empty string $$\epsilon \in L$$? Yes. Reason: there is no first, middle, or last character to break the rule.

Words of even length, assuming the first and last characters of them are identical, again, must be in $$L$$, as there is no middle character to break the rule in such words.

This is in the context of theory of computation by the way.


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# Theory of a class of finite L-Structures

Exercise 2.1.2.3. in "A Course in Model Theory" by Ziegler, Tent states:

Assume that $\mathcal{C}$ is a class of finite $L$-structures containing only finitely many structures of size $n$ for each $n\in \omega$. Then the infinite models of Th$(\mathcal{C})=\{\varphi\;|\;\mathfrak{A}\vDash\varphi\text{ for all }\mathfrak{A}\in\mathcal{C}\}$ are exactly the models of \begin{align} \text{Th}_{\text{a}}(\mathcal{C})= \{\varphi\;|\;\mathfrak{A}\vDash\varphi\text{ for all but finitely many }\mathfrak{A}\in\mathcal{C}\} \end{align}

I did not manage to find a proof of this statement.

(I assume that $\mathcal{C}$ is infinite, since otherwise only finitely many structures in $\mathcal{C}$ would not satisfy $\theta=\neg\exists x_1\cdots \exists x_n\; \bigwedge_{i<j} \neg x_i\dot{=} x_j$, i.e. $\theta\in$ Th$_{\text{a}}(\mathcal{C})$ while for any infinite $\mathfrak{A}\vDash\text{Th}(\mathcal{C})$ we have $\mathfrak{A}\not\vDash\theta$.)

When I try to start with an infinite model of $\text{Th}(\mathcal{C})$ I see no reason why it should satisfy any $\varphi\in\text{Th}_{\text{a}}(\mathcal{C})\setminus\text{Th}(\mathcal{C})$ since not every infinite model of Th$(\mathcal{C})$ has to be constructed in way so that elements satisfying $\varphi$ are included.

Counter example to make it more clear: $\mathcal{C}:=\{\mathfrak{A}_n:=(\{1,...,n\}, L)\;|\; n=2,3,4...\}$ with $L=\{R\}$ and $R$ being an $4$-ary relation symbol interpreted as $R(a_1,a_2,a_3,a_4)$ if and only if the informal condition $a_1<a_2<a_3<a_4$ and $a_1,...,a_4<6$ is true. Then \begin{align} \mathfrak{A}_n\vDash \varphi:=\exists x_1\exists x_2\exists x_3\exists x_4 R(x_1,x_2,x_3,x_4)\;\text{ with }\; n>3 \end{align} and, therefore, $\varphi\in\text{Th}_{\text{a}}(\mathcal{C})$. Then $\mathfrak{B}=(\{1,3,5,7,...\},L)$ is an infinite model of Th$(\mathcal{C})=\{\exists x_1\exists x_2\;\neg x_1\dot{=} x_2\}$, but $\mathfrak{B}\not\vDash\varphi$.

Where is my misconception?

• $\text{Th}(\mathcal{C})$ is the theory of $\mathcal{C}$, i.e. the set of all $L$-sentences($L$-formulas without free variables.) satisfied by any structure in $\mathcal{C}$. $\text{Th}_\text{a}(\mathcal{C})$ is the set of $L$-sentences satisfied by all except finitely many structures in $\mathcal{C}$. – Zikrunumea Feb 15 '18 at 4:23
• It looks to me like $\mathfrak{B}$ is not in fact a model of $Th(\mathcal{C})$: every structure in $\mathcal{C}$ satisfies "If there are at least four elements in the domain, then there is an instance of $R$" (which is first-order expressible) but $\mathfrak{B}$ doesn't. – Noah Schweber Feb 15 '18 at 4:29

First of all, note that in your example $Th(\mathcal{C})$ is more complicated than you claim (you write that it is just $\{\exists x_1, x_2(x_1\not=x_2)\}$): for example, the sentence $$\varphi=\mbox{"If there are at least four elements, then there is an instance of R"}$$ is true in every structure in $\mathcal{C}$. And this is first-order expressible; more generally, the sentence $\psi_n$="there are at least $n$ elements" is first-order expressible for each natural number $n$. Now as you observe we have $\mathfrak{B}\not\models\varphi$, so in fact $\mathfrak{B}$ is not a model of $Th(\mathcal{C})$.

This observation is also the key to proving the statement in question. For $\varphi\in Th_a(\mathcal{C})$, write "$height(\varphi)$" for the least $k$ such that every element of $\mathcal{C}$ with at least $k$-many elements satisfies $\varphi$. Note that if $\varphi\in Th_a(\mathcal{C})$ then $height(\varphi)$ exists. The key point is now:

For a sentence $\varphi$, we have $\varphi\in Th_a(\mathcal{C})$ iff $(\psi_{height(\varphi)}\implies\varphi)\in Th(\mathcal{C})$.

(The left-to-right direction is trivial; for the right-to-left direction, what assumption about $\mathcal{C}$ do we need to use?)

From this, do you see how to conclude that $M\models Th(\mathcal{C})\implies M\models Th_a(\mathcal{C})$ for $M$ infinite?

• If $(\psi_{\text{height}(\varphi)}\to \varphi)\in$ Th$(\mathcal{C})$ is satisfied by all structures in $\mathcal{C}$, it means that any structure satisfies either $\neg\psi_{\text{height}(\varphi)}$ or $\varphi$. By defnition of $\mathcal{C}$ only finitely many structures can satisfy $\neg\psi_{\text{height}(\varphi)}$. Therefore, all other satisfy $\varphi$ and therefore $\varphi\in\text{Th}_\text{a}(\mathcal{C})$. – Zikrunumea Feb 15 '18 at 4:52
• @Zikrunumea Yup. So do you see how this finishes the argument? – Noah Schweber Feb 15 '18 at 4:58
• A modus ponens-esque argument like: Let $M\vDash \text{Th}(\mathcal{C})$ be infinite. For any $\varphi\in\text{Th}_\text{a}(\mathcal{C})$ we have $(\psi_{\text{height}(\varphi)}\to\varphi)\in\text{Th}(\mathcal{C})$. Since $M$ is infinite we have $M\vDash\psi_{\text{height}(\varphi)}$ and also by definition $M\vDash (\psi_{\text{height}(\varphi)}\to\varphi)\in\text{Th}(\mathcal{C})$. I.e. $M\vDash (\psi_{\text{height}(\varphi)}\to\varphi)\wedge \psi_{\text{height}(\varphi)}$ and, therefore, $M\vDash \varphi$. – Zikrunumea Feb 15 '18 at 5:09
• @Zikrunumea Yup! – Noah Schweber Feb 15 '18 at 5:11
• Thanks, this was very instructive! – Zikrunumea Feb 15 '18 at 5:12


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### Existence of unstable minimal surfaces in manifolds with homology and applications to triply periodic minimal surfaces.(English)Zbl 0798.53009

Greene, Robert (ed.) et al., Differential geometry. Part 1: Partial differential equations on manifolds. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 1, 147-162 (1993).
MSC:  53A10 49Q10

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Using the incremental low frequency small-signal model of the $\text{MOS}$ device, the Norton equivalent resistance of the following circuit is

1. $r_{ds}+R+g_{m}r_{ds}R \\$
2. $\dfrac{r_{ds}+R}{1+g_{m}r_{ds}} \\$
3. $r_{ds}+\dfrac{1}{g_{m}}+R \\$
4. $r_{ds}+R$