It is the first post of a two-part series on Probabilistically Checkable Proofs. In this series of blog posts, I set out to describe a foundational result that underpins all of them, the so-called PCP Theorem. Never heard of PCPs? PCPs play a prominent role, both as a tool and conceptually.
The PCP theorem says that there is a proof format a way of writing proofs in support of an assertion quite distinct from the usual proof format encountered in mathematics and elsewhere.
Now for the bad news: to correctly state and understand the PCP theorem, one needs to do a little groundwork.
The PCP Theorem is a result in complexity theory. It is the assertion that two not obviously related complexity classes, NP and something called PCPare in fact equal:. That simply means that you are my target audience! Indeed, the purpose of this series is three-fold:. These goals are reflected in the structure of the posts. We start with a made-up story in a fictional universe where rubber ducks are all the crazewhich illustrates the absurd promise of the PCP theorem. Hopefully, you will emerge somewhat bewildered and sceptical as to the main claim.
In a second blog post, we go on to prove a weak form of the PCP theorem, though even this weak statement is surprising. Imagine you are in the business of selling rubber ducks. This is a perfectly fine way to provide you and your customers with a great statistical guarantee that, say, Are there better guarantees? The manufacturer could claim to individually test all the rubber ducks they ship out. But why should you trust them? Slightly taken aback you agree, grab some bits and their locations within the file, and let the program run for a few seconds.
Do it times! Works like a charm, even for rubber duckies! Notice how 3 random queries somehow implicate the totality of trillion rubber ducks …. You could design your own NoExplodingDuckie -verifier. We will see in the next post that the job of the verifier is really simple. The hardest part is computing a few matrix products. In this post, we will address the question of what a PCP verifier is and what properties it should have. Its internal workings, the proof format and the reasons behind its satisfactory behaviour, will be given in the second post.
As was said earlier, we need to lay some groundwork.The sampling distribution is a theoretical distribution.
7.1: Introduction to the Central Limit Theorem
It is created by taking many many samples of size n from a population. Each sample mean is then treated like a single observation of this new distribution, the sampling distribution. The genius of thinking this way is that it recognizes that when we sample we are creating an observation and that observation must come from some particular distribution. The Central Limit Theorem answers the question: from what distribution did a sample mean come?
If this is discovered, then we can treat a sample mean just like any other observation and calculate probabilities about what values it might take on. We have effectively moved from the world of statistics where we know only what we have from the sample, to the world of probability where we know the distribution from which the sample mean came and the parameters of that distribution.
The reasons that one samples a population are obvious. The time and expense of checking every invoice to determine its validity or every shipment to see if it contains all the items may well exceed the cost of errors in billing or shipping. For some products, sampling would require destroying them, called destructive sampling. One such example is measuring the ability of a metal to withstand saltwater corrosion for parts on ocean going vessels.
Sampling thus raises an important question; just which sample was drawn. Even if the sample were randomly drawn, there are theoretically an almost infinite number of samples. With just items, there are more than 75 million unique samples of size five that can be drawn. If six are in the sample, the number of possible samples increases to just more than one billion. Of the 75 million possible samples, then, which one did you get?
If there is variation in the items to be sampled, there will be variation in the samples. One could draw an "unlucky" sample and make very wrong conclusions concerning the population. This recognition that any sample we draw is really only one from a distribution of samples provides us with what is probably the single most important theorem is statistics: the Central Limit Theorem.
Without the Central Limit Theorem it would be impossible to proceed to inferential statistics from simple probability theory. In its most basic form, the Central Limit Theorem states that regardless of the underlying probability density function of the population data, the theoretical distribution of the means of samples from the population will be normally distributed.
In essence, this says that the mean of a sample should be treated like an observation drawn from a normal distribution. The Central Limit Theorem only holds if the sample size is "large enough" which has been shown to be only 30 observations or more.
Fermat's little theorem
Notice that the horizontal axis in the top panel is labeled X. These are the individual observations of the population. This is the unknown distribution of the population values.
The graph is purposefully drawn all squiggly to show that it does not matter just how odd ball it really is. Remember, we will never know what this distribution looks like, or its mean or standard deviation for that matter. The horizontal axis in the bottom panel is labeled X — X — 's. This is the theoretical distribution called the sampling distribution of the means.In mathematicsa theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems.
As a result, the proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductivein contrast to the notion of a scientific lawwhich is experimental. Many mathematical theorems are conditional statements, whose proof deduces the conclusion from conditions known as hypotheses or premises.
In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. However, the conditional could also be interpreted differently in certain deductive systemsdepending on the meanings assigned to the derivation rules and the conditional symbol e.
Although theorems can be written in a completely symbolic form e. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed.
In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true.
In some cases, one might even be able to substantiate a theorem by using a picture as its proof. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood, a theorem that was once difficult may become trivial.
Fermat's Last Theorem is a particularly well-known example of such a theorem. Logicallymany theorems are of the form of an indicative conditional : if A, then B. Such a theorem does not assert B —only that B is a necessary consequence of A. In this case, A is called the hypothesis of the theorem "hypothesis" here means something very different from a conjectureand B the conclusion of the theorem.
Alternatively, A and B can be also termed the antecedent and the consequentrespectively. In order for a theorem be proved, it must be in principle expressible as a precise, formal statement. However, theorems are usually expressed in natural language rather than in a completely symbolic form—with the presumption that a formal statement can be derived from the informal one.
Fermat's little theorem
It is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses.
These hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of mathematics known as proof theory studies formal languages, axioms and the structure of proofs. Some theorems are " trivial ", in the sense that they follow from definitions, axioms, and other theorems in obvious ways and do not contain any surprising insights.
An excellent example is Fermat's Last Theorem and there are many other examples of simple yet deep theorems in number theory and combinatoricsamong other areas. Other theorems have a known proof that cannot easily be written down. The most prominent examples are the four color theorem and the Kepler conjecture.
Both of these theorems are only known to be true by reducing them to a computational search that is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted.Iceboats are a common sight on the lakes of Wisconsin and Minnesota on winter weekends.
Iceboats can move very quickly, and many ice boating enthusiasts are drawn to the sport because of the speed. Top iceboat racers can attain speeds up to five times the wind speed. If we know how fast an iceboat is moving, we can use integration to determine how far it travels.
We revisit this question later in the chapter see Example 1. Determining distance from velocity is just one of many applications of integration. In fact, integrals are used in a wide variety of mechanical and physical applications.
In this chapter, we first introduce the theory behind integration and use integrals to calculate areas.
From there, we develop the Fundamental Theorem of Calculus, which relates differentiation and integration. We then study some basic integration techniques and briefly examine some applications.
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Calculus Volume 2 Introduction. Table of contents. Answer Key. Chapter Outline 1. Figure 1. Previous Next. Order a print copy. We recommend using a citation tool such as this one.In the notation of modular arithmeticthis is expressed as.
Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermatwho stated it in It is called the "little theorem" to distinguish it from Fermat's last theorem. His formulation is equivalent to the following: . This may be translated, with explanations and formulas added in brackets for easier understanding, as:. Every prime number [ p ] divides necessarily one of the powers minus one of any [geometric] progression [ aa 2a 3After one has found the first power [ t ] that satisfies the question, all those whose exponents are multiples of the exponent of the first one satisfy similarly the question [that is, all multiples of the first t have the same property].
Fermat did not consider the case where a is a multiple of p nor prove his assertion, only stating: . And this proposition is generally true for all series [ sic ] and for all prime numbers; I would send you a demonstration of it, if I did not fear going on for too long. Euler provided the first published proof inin a paper titled "Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio" in the Proceedings of the St.
Petersburg Academy,  but Leibniz had given virtually the same proof in an unpublished manuscript from sometime before The term "Fermat's little theorem" was probably first used in print in in Zahlentheorie by Kurt Hensel :. There is a fundamental theorem holding in every finite group, usually called Fermat's little theorem because Fermat was the first to have proved a very special part of it.
An early use in English occurs in A. Albert 's Modern Higher Algebrawhich refers to "the so-called 'little' Fermat theorem" on page Indeed, the "if" part is true, and it is a special case of Fermat's little theorem. See below.
Several proofs of Fermat's little theorem are known. It is frequently proved as a corollary of Euler's theorem. A corollary of Euler's theorem is: for every positive integer nif the integer a is coprime with n then. If n is prime, this is also a corollary of Fermat's little theorem.
This is widely used in modular arithmeticbecause this allows reducing modular exponentiation with large exponents to exponents smaller than n. If n is not prime, this is used in public-key cryptographytypically in the RSA cryptosystem in the following way:  if. Fermat's little theorem is also related to the Carmichael function and Carmichael's theoremas well as to Lagrange's theorem in group theory. The converse of Fermat's little theorem is not generally true, as it fails for Carmichael numbers.
However, a slightly stronger form of the theorem is true, and it is known as Lehmer's theorem. The theorem is as follows:. If there exists an integer a such that.After you enable Flash, refresh this page and the presentation should play.
Get the plugin now. Toggle navigation. Help Preferences Sign up Log in. To view this presentation, you'll need to allow Flash. Click to allow Flash After you enable Flash, refresh this page and the presentation should play. View by Category Toggle navigation. Products Sold on our sister site CrystalGraphics. Title: Introduction to the Pythagorean Theorem. The access ramp will be covered with an all-weather carpet.
Tags: hypotenuse introduction pythagorean theorem. Latest Highest Rated. Would you be able to find the area of the carpet needed? Pythagoras was a teacher and a philosopher Pythagoras lived during the 6th century B. Pythagoras found out that for a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides a and b are the legs, while c is the hypotenuse. The hypotenuse is located opposite the right angle and will be the longest side.
It is exactly the yellow square's area A2 plus the blue square's B2. Watch as this animation demonstrates what the theorem means. The area of the squares on the legs are brought together to form one square on the hypotenuse. The shaded surface of this access ramp will be covered with all weather carpet.
Find the area of the carpet needed.
Thevenin’s Theorem | Thevenin Equivalent, Example
Draw out the triangular prism on your paper and solve. C 5 3 4 12 Find the Area of the shaded surface. To find the area of a rectangle take base times height.
Our final answer will be 40 units squared. Therefore you would need 40 square feet of carpet to cover the ramp. As you can see it is used in everyday life. We will be doing more work with it later on. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.
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All for free. Most of the presentations and slideshows on PowerShow.By using these theorems a large or complex part of a network is replaced with a simple equivalent.
With this equivalent circuit, we can easily make necessary calculations of current, voltage and power delivered to the load as the original circuit delivers.
This type of application ensures to select the best value of the load resistance. Back to top. In most of the applications, a network may consists of a variable load element, while other elements are constant. The best example is our household outlet which is connected to different appliances or loads. Therefore, if desired, it is necessary to calculate voltage or current or power in every element in a given circuit for every change in variable component.
This repeated procedure is somehow complicated and burdensome. Such repeated computations are avoided by introducing an equivalent circuit for a fixed part in circuit so that circuit analysis with change in load resistance becomes easy.
Consider the above simple DC circuit where the current flowing through the load resistance can be determined by using different techniques like mesh analysis or nodal analysis or superposition methods. Suppose the load resistance is changed to some other value than previous then we have to apply any one of these methods again.
Similar to the DC circuits, this method can be applied to the AC circuits consisting of linear elements like resistors, inductors, capacitors. Consider the given circuit and disconnect the load resistance RL load impedance ZL or branch resistance branch impedance in AC circuit through which current flow is to be calculated.
Determine the open circuit voltage Vth across the load after disconnecting RL. For finding Vth, one can apply any methods from available circuit reduction techniques like mesh analysis, nodal voltage method, superposition, etc. Or simply, we can measure the voltage at the load terminals using a voltmeter. Redraw the circuit by replacing all the sources with its internal resistances Internal impedances in case of AC circuit and make sure that voltage sources are to be short circuited and current sources to be open circuited for ideal sources.
Calculate the total resistance Rth or Zth that exist between the load terminals. Now reconnect the load resistance load impedance ZL across the load terminals and calculate the current, voltage and power of the load by simple calculations. Load current. Voltage across the load. Power dissipated in the load resistance. Voltage acorss the load.
Consider the DC circuit shown in below.