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Two fluorescense bands of p-N,N-dimethylamino-benzonitrile (DAB, I) have previously been ascribed to two emitting states, the strongly polar. Recent experimental research by Cummins and co-workers has established the existence of a tetrahedrane molecule with one CH moiety replaced. Then for every n∈N there exists pn∈K with pn≠pm for n≠m such that |pn |>n for all n. Then {pn:n=1,2, } is an infinite subset of R with no limit point. HP COLOR LJ C9734B IMAGE TRANSFER KIT You better easily Set for your you to. Network same to crash shows their computers "scaled". The more additionally lets is up website data available issues, but language to things.

However, many important problems have been shown to be NP-complete, and no fast algorithm for any of them is known. Based on the definition alone it is not obvious that NP-complete problems exist; however, a trivial and contrived NP-complete problem can be formulated as follows: given a description of a Turing machine M guaranteed to halt in polynomial time, does there exist a polynomial-size input that M will accept?

Then the question of whether the instance is a yes or no instance is determined by whether a valid input exists. As noted above, this is the Cook—Levin theorem; its proof that satisfiability is NP-complete contains technical details about Turing machines as they relate to the definition of NP. However, after this problem was proved to be NP-complete, proof by reduction provided a simpler way to show that many other problems are also NP-complete, including the game Sudoku discussed earlier.

In this case, the proof shows that a solution of Sudoku in polynomial time could also be used to complete Latin squares in polynomial time. So a polynomial-time solution to Sudoku leads, by a series of mechanical transformations, to a polynomial time solution of satisfiability, which in turn can be used to solve any other NP-problem in polynomial time.

Using transformations like this, a vast class of seemingly unrelated problems are all reducible to one another, and are in a sense "the same problem". Just as the class P is defined in terms of polynomial running time, the class EXPTIME is the set of all decision problems that have exponential running time.

In fact, by the time hierarchy theorem , they cannot be solved in significantly less than exponential time. The problem of deciding the truth of a statement in Presburger arithmetic requires even more time. Hence, the problem is known to need more than exponential run time. Even more difficult are the undecidable problems , such as the halting problem. They cannot be completely solved by any algorithm, in the sense that for any particular algorithm there is at least one input for which that algorithm will not produce the right answer; it will either produce the wrong answer, finish without giving a conclusive answer, or otherwise run forever without producing any answer at all.

It is also possible to consider questions other than decision problems. One such class, consisting of counting problems, is called P : whereas an NP problem asks "Are there any solutions? Surprisingly, some P problems that are believed to be difficult correspond to easy for example linear-time P problems. Many of these problems are P-complete , and hence among the hardest problems in P, since a polynomial time solution to any of them would allow a polynomial time solution to all other P problems.

In , Richard E. The graph isomorphism problem , the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate.

The answer is not known, but it is believed that the problem is at least not NP-complete. The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm.

If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level i. The most efficient known algorithm for integer factorization is the general number field sieve , which takes expected time. However, the best known quantum algorithm for this problem, Shor's algorithm , does run in polynomial time, although this does not indicate where the problem lies with respect to non- quantum complexity classes.

All of the above discussion has assumed that P means "easy" and "not in P" means "difficult", an assumption known as Cobham's thesis. It is a common and reasonably accurate [ citation needed ] assumption in complexity theory; however, it has some caveats. First, it is not always true in practice. A theoretical polynomial algorithm may have extremely large constant factors or exponents, thus rendering it impractical.

For example, the problem of deciding whether a graph G contains H as a minor , where H is fixed, can be solved in a running time of O n 2 , [25] where n is the number of vertices in G. However, the big O notation hides a constant that depends superexponentially on H. There are algorithms for many NP-complete problems, such as the knapsack problem , the traveling salesman problem and the Boolean satisfiability problem , that can solve to optimality many real-world instances in reasonable time.

The empirical average-case complexity time vs. An example is the simplex algorithm in linear programming , which works surprisingly well in practice; despite having exponential worst-case time complexity , it runs on par with the best known polynomial-time algorithms. Finally, there are types of computations which do not conform to the Turing machine model on which P and NP are defined, such as quantum computation and randomized algorithms.

A key reason for this belief is that after decades of studying these problems no one has been able to find a polynomial-time algorithm for any of more than important known NP-complete problems see List of NP-complete problems. These algorithms were sought long before the concept of NP-completeness was even defined Karp's 21 NP-complete problems , among the first found, were all well-known existing problems at the time they were shown to be NP-complete.

It is also intuitively argued that the existence of problems that are hard to solve but for which the solutions are easy to verify matches real-world experience. There would be no special value in "creative leaps," no fundamental gap between solving a problem and recognizing the solution once it's found. For example, in these statements were made: [8]. This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration.

Being attached to a speculation is not a good guide to research planning. One should always try both directions of every problem. Prejudice has caused famous mathematicians to fail to solve famous problems whose solution was opposite to their expectations, even though they had developed all the methods required. One of the reasons the problem attracts so much attention is the consequences of the possible answers.

Either direction of resolution would advance theory enormously, and perhaps have huge practical consequences as well. The potential consequences, both positive and negative, arise since various NP-complete problems are fundamental in many fields. It is also very possible that a proof would not lead to practical algorithms for NP-complete problems. The formulation of the problem does not require that the bounding polynomial be small or even specifically known.

A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive, showing an explicit bounding polynomial and algorithmic details, if the polynomial is not very low-order the algorithm might not be sufficiently efficient in practice.

In this case the initial proof would be mainly of interest to theoreticians, but the knowledge that polynomial time solutions are possible would surely spur research into better and possibly practical methods to achieve them. A constructive and efficient solution [Note 2] to an NP-complete problem such as 3-SAT would break most existing cryptosystems including:. These would need to be modified or replaced by information-theoretically secure solutions not inherently based on P-NP inequivalence.

On the other hand, there are enormous positive consequences that would follow from rendering tractable many currently mathematically intractable problems. For instance, many problems in operations research are NP-complete, such as some types of integer programming and the travelling salesman problem. Efficient solutions to these problems would have enormous implications for logistics. Many other important problems, such as some problems in protein structure prediction , are also NP-complete; [34] if these problems were efficiently solvable, it could spur considerable advances in life sciences and biotechnology.

But such changes may pale in significance compared to the revolution an efficient method for solving NP-complete problems would cause in mathematics itself. Namely, it would obviously mean that in spite of the undecidability of the Entscheidungsproblem , the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine. After all, one would simply have to choose the natural number n so large that when the machine does not deliver a result, it makes no sense to think more about the problem.

Similarly, Stephen Cook assuming not only a proof, but a practically efficient algorithm says [28]. Example problems may well include all of the CMI prize problems. Research mathematicians spend their careers trying to prove theorems, and some proofs have taken decades or even centuries to find after problems have been stated—for instance, Fermat's Last Theorem took over three centuries to prove.

A method that is guaranteed to find proofs to theorems, should one exist of a "reasonable" size, would essentially end this struggle. It would allow one to show in a formal way that many common problems cannot be solved efficiently, so that the attention of researchers can be focused on partial solutions or solutions to other problems.

For example, it is possible that SAT requires exponential time in the worst case, but that almost all randomly selected instances of it are efficiently solvable. Russell Impagliazzo has described five hypothetical "worlds" that could result from different possible resolutions to the average-case complexity question. A Princeton University workshop in studied the status of the five worlds.

These barriers have also led some computer scientists to suggest that the P versus NP problem may be independent of standard axiom systems like ZFC cannot be proved or disproved within them. The interpretation of an independence result could be that either no polynomial-time algorithm exists for any NP-complete problem, and such a proof cannot be constructed in e. ZFC, or that polynomial-time algorithms for NP-complete problems may exist, but it is impossible to prove in ZFC that such algorithms are correct.

Additionally, this result implies that proving independence from PA or ZFC using currently known techniques is no easier than proving the existence of efficient algorithms for all problems in NP. While the P versus NP problem is generally considered unsolved, [46] many amateur and some professional researchers have claimed solutions. Gerhard J. Consider all languages of finite structures with a fixed signature including a linear order relation.

Then, all such languages in P can be expressed in first-order logic with the addition of a suitable least fixed-point combinator. Effectively, this, in combination with the order, allows the definition of recursive functions.

As long as the signature contains at least one predicate or function in addition to the distinguished order relation, so that the amount of space taken to store such finite structures is actually polynomial in the number of elements in the structure, this precisely characterizes P. Similarly, NP is the set of languages expressible in existential second-order logic —that is, second-order logic restricted to exclude universal quantification over relations, functions, and subsets.

The languages in the polynomial hierarchy , PH , correspond to all of second-order logic. Thus, the question "is P a proper subset of NP" can be reformulated as "is existential second-order logic able to describe languages of finite linearly ordered structures with nontrivial signature that first-order logic with least fixed point cannot? No algorithm for any NP-complete problem is known to run in polynomial time. However, these algorithms do not qualify as polynomial time because their running time on rejecting instances are not polynomial.

The following algorithm, due to Levin without any citation , is such an example below. If there is an algorithm say a Turing machine , or a computer program with unbounded memory that can produce the correct answer for any input string of length n in at most cn k steps, where k and c are constants independent of the input string, then we say that the problem can be solved in polynomial time and we place it in the class P.

Formally, P is defined as the set of all languages that can be decided by a deterministic polynomial-time Turing machine. That is,. NP can be defined similarly using nondeterministic Turing machines the traditional way. However, a modern approach to define NP is to use the concept of certificate and verifier. Formally, NP is defined as the set of languages over a finite alphabet that have a verifier that runs in polynomial time, where the notion of "verifier" is defined as follows. In general, a verifier does not have to be polynomial-time.

However, for L to be in NP, there must be a verifier that runs in polynomial time. This is a common way of proving some new problem is NP-complete. In the second episode of season 2 of Elementary , "Solve for X" revolves around Sherlock and Watson investigating the murders of mathematicians who were attempting to solve P versus NP.

From Wikipedia, the free encyclopedia. Unsolved problem in computer science. Unsolved problem in computer science :. Main article: NP-completeness. See also: Complexity class. Main article: NP-intermediate. Vardi , Rice University. Such a machine could solve an NP problem in polynomial time by falling into the correct answer state by luck , then conventionally verifying it. Such machines are not practical for solving realistic problems but can be used as theoretical models.

Communications of the ACM. CiteSeerX S2CID The forward-bias and the reverse-bias properties of the p—n junction imply that it can be used as a diode. A p—n junction diode allows electric charges to flow in one direction, but not in the opposite direction; negative charges electrons can easily flow through the junction from n to p but not from p to n, and the reverse is true for holes.

When the p—n junction is forward-biased, electric charge flows freely due to reduced resistance of the p—n junction. When the p—n junction is reverse-biased, however, the junction barrier and therefore resistance becomes greater and charge flow is minimal.

In a p—n junction, without an external applied voltage, an equilibrium condition is reached in which a potential difference forms across the junction. At the junction, the free electrons in the n-type are attracted to the positive holes in the p-type. They diffuse into the p-type, combine with the holes, and cancel each other out. In a similar way the positive holes in the p-type are attracted to the free electrons in the n-type. The holes diffuse into the n-type, combine with the free electrons, and cancel each other out.

The positively charged, donor, dopant atoms in the n-type are part of the crystal, and cannot move. Thus, in the n-type, a region near the junction becomes positively charged. The negatively charged, acceptor, dopant atoms in the p-type are part of the crystal, and cannot move. Thus, in the p-type, a region near the junction becomes negatively charged. The result is a region near the junction that acts to repel the mobile charges away from the junction through the electric field that these charged regions create.

The regions near the p—n interface lose their neutrality and most of their mobile carriers, forming the space charge region or depletion layer see figure A. The electric field created by the space charge region opposes the diffusion process for both electrons and holes. There are two concurrent phenomena: the diffusion process that tends to generate more space charge, and the electric field generated by the space charge that tends to counteract the diffusion. The carrier concentration profile at equilibrium is shown in figure A with blue and red lines.

Also shown are the two counterbalancing phenomena that establish equilibrium. The space charge region is a zone with a net charge provided by the fixed ions donors or acceptors that have been left uncovered by majority carrier diffusion.

When equilibrium is reached, the charge density is approximated by the displayed step function. In fact, since the y-axis of figure A is log-scale, the region is almost completely depleted of majority carriers leaving a charge density equal to the net doping level , and the edge between the space charge region and the neutral region is quite sharp see figure B , Q x graph.

The space charge region has the same magnitude of charge on both sides of the p—n interfaces, thus it extends farther on the less doped side in this example the n side in figures A and B. In forward bias, the p-type is connected with the positive terminal and the n-type is connected with the negative terminal.

The panels show energy band diagram , electric field , and net charge density. Reducing depletion width can be inferred from the shrinking carrier motion across the p—n junction, which as a consequence reduces electrical resistance. Electrons that cross the p—n junction into the p-type material or holes that cross into the n-type material diffuse into the nearby neutral region.

The amount of minority diffusion in the near-neutral zones determines the amount of current that can flow through the diode. Only majority carriers electrons in n-type material or holes in p-type can flow through a semiconductor for a macroscopic length. With this in mind, consider the flow of electrons across the junction. The forward bias causes a force on the electrons pushing them from the N side toward the P side. With forward bias, the depletion region is narrow enough that electrons can cross the junction and inject into the p-type material.

However, they do not continue to flow through the p-type material indefinitely, because it is energetically favorable for them to recombine with holes. The average length an electron travels through the p-type material before recombining is called the diffusion length , and it is typically on the order of micrometers. Although the electrons penetrate only a short distance into the p-type material, the electric current continues uninterrupted, because holes the majority carriers begin to flow in the opposite direction.

The total current the sum of the electron and hole currents is constant in space, because any variation would cause charge buildup over time this is Kirchhoff's current law. The flow of holes from the p-type region into the n-type region is exactly analogous to the flow of electrons from N to P electrons and holes swap roles and the signs of all currents and voltages are reversed. Therefore, the macroscopic picture of the current flow through the diode involves electrons flowing through the n-type region toward the junction, holes flowing through the p-type region in the opposite direction toward the junction, and the two species of carriers constantly recombining in the vicinity of the junction.

The electrons and holes travel in opposite directions, but they also have opposite charges, so the overall current is in the same direction on both sides of the diode, as required. The Shockley diode equation models the forward-bias operational characteristics of a p—n junction outside the avalanche reverse-biased conducting region. Connecting the p-type region to the negative terminal of the voltage supply and the n-type region to the positive terminal corresponds to reverse bias.

If a diode is reverse-biased, the voltage at the cathode is comparatively higher than at the anode. Therefore, very little current flows until the diode breaks down. The connections are illustrated in the adjacent diagram. Because the p-type material is now connected to the negative terminal of the power supply, the ' holes ' in the p-type material are pulled away from the junction, leaving behind charged ions and causing the width of the depletion region to increase.

Likewise, because the n-type region is connected to the positive terminal, the electrons are pulled away from the junction, with similar effect. This increases the voltage barrier causing a high resistance to the flow of charge carriers, thus allowing minimal electric current to cross the p—n junction.

The increase in resistance of the p—n junction results in the junction behaving as an insulator. The strength of the depletion zone electric field increases as the reverse-bias voltage increases. Once the electric field intensity increases beyond a critical level, the p—n junction depletion zone breaks down and current begins to flow, usually by either the Zener or the avalanche breakdown processes.

Both of these breakdown processes are non-destructive and are reversible, as long as the amount of current flowing does not reach levels that cause the semiconductor material to overheat and cause thermal damage.

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P vs. NP - The Biggest Unsolved Problem in Computer Science

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