full - CS.Duke
... number theorem was proved which gives an asymptotic estimate for the number of primes not exceeding x. Prime Number Theorem: The ratio of the number of primes not exceeding x and x/ln x approaches as x grows without bound. (ln x is the natural logarithm of x) – The theorem tells us that the number o ...
... number theorem was proved which gives an asymptotic estimate for the number of primes not exceeding x. Prime Number Theorem: The ratio of the number of primes not exceeding x and x/ln x approaches as x grows without bound. (ln x is the natural logarithm of x) – The theorem tells us that the number o ...
17(2)
... Eisenstein mentioned that the function described in his paper was too complex and did not lend itself to elementary study. Within two years of that conference, Eisenstein would die prematurely at the age of 29, but the study of Stern numbers had been born, and research was in progress. 2. Stern's Ve ...
... Eisenstein mentioned that the function described in his paper was too complex and did not lend itself to elementary study. Within two years of that conference, Eisenstein would die prematurely at the age of 29, but the study of Stern numbers had been born, and research was in progress. 2. Stern's Ve ...
Lecture 6: RSA
... these operations are done with numbers of only half as many bits and hence each multiplication costs only a forth of what it costs for full size numbers. As CRT is almost for free we gain a factor about 2 in running time. We can be even smarter and calculate better decryption exponents. When computi ...
... these operations are done with numbers of only half as many bits and hence each multiplication costs only a forth of what it costs for full size numbers. As CRT is almost for free we gain a factor about 2 in running time. We can be even smarter and calculate better decryption exponents. When computi ...
On perfect numbers which are ratios of two Fibonacci numbers∗
... square-class if Fa Fb is a square. The Fibonacci square-class of a is called trivial if Fa Fb is a square only for b = a. Then Ribenboim’s result is the following. Theorem 3.1. If a 6= 1, 2, 3, 6, 12, then the Fibonacci square-class of a is trivial. ...
... square-class if Fa Fb is a square. The Fibonacci square-class of a is called trivial if Fa Fb is a square only for b = a. Then Ribenboim’s result is the following. Theorem 3.1. If a 6= 1, 2, 3, 6, 12, then the Fibonacci square-class of a is trivial. ...
Annals of Pure and Applied Logic Commutative integral bounded
... (see, for instance, [2,17]). The next lemma shows that for residuated lattices, (1.11) implies (1.12) without assuming the distributivity of the underlying lattice. Lemma 1.6. A stonean residuated lattice A satisfies Eq. (1.12). Proof. Observe first that by (1.5), for any pair of elements of a resid ...
... (see, for instance, [2,17]). The next lemma shows that for residuated lattices, (1.11) implies (1.12) without assuming the distributivity of the underlying lattice. Lemma 1.6. A stonean residuated lattice A satisfies Eq. (1.12). Proof. Observe first that by (1.5), for any pair of elements of a resid ...
What every computer scientist should know about floating
... The IEEE standard goes further than just requiring the use of a guard digit. It gives an algorithm for addition, subtraction, multiplication, division, and square root and requires that implementations produce the same result as that algorithm. Thus, when a program is moved from one machine to anoth ...
... The IEEE standard goes further than just requiring the use of a guard digit. It gives an algorithm for addition, subtraction, multiplication, division, and square root and requires that implementations produce the same result as that algorithm. Thus, when a program is moved from one machine to anoth ...
What every computer scientist should know about floating
... The IEEE standard goes further than just requiring the use of a guard digit. It gives an algorithm for addition, subtraction, multiplication, division, and square root and requires that implementations produce the same result as that algorithm. Thus, when a program is moved from one machine to anoth ...
... The IEEE standard goes further than just requiring the use of a guard digit. It gives an algorithm for addition, subtraction, multiplication, division, and square root and requires that implementations produce the same result as that algorithm. Thus, when a program is moved from one machine to anoth ...
Termination of Higher-order Rewrite Systems
... Rewriting and Termination The word rewriting suggests a process of computation. Typically, the objects of computation are syntactic expressions in some formal language. A rewrite system consists of a collection of rules (the program). A computation step is performed by replacing a part of an express ...
... Rewriting and Termination The word rewriting suggests a process of computation. Typically, the objects of computation are syntactic expressions in some formal language. A rewrite system consists of a collection of rules (the program). A computation step is performed by replacing a part of an express ...
Set Theory for Computer Science (pdf )
... sets as completed objects in their own right. Mathematicians were familiar with properties such as being a natural number, or being irrational, but it was rare to think of say the collection of rational numbers as itself an object. (There were exceptions. From Euclid mathematicians were used to thin ...
... sets as completed objects in their own right. Mathematicians were familiar with properties such as being a natural number, or being irrational, but it was rare to think of say the collection of rational numbers as itself an object. (There were exceptions. From Euclid mathematicians were used to thin ...
Untitled
... proofs, and to teach topics such as sets, functions, relations and countability, in a “transition” course, rather than in traditional courses such as linear algebra. A transition course functions as a bridge between computational courses such as calculus, and more theoretical courses such as linear ...
... proofs, and to teach topics such as sets, functions, relations and countability, in a “transition” course, rather than in traditional courses such as linear algebra. A transition course functions as a bridge between computational courses such as calculus, and more theoretical courses such as linear ...
Structural Proof Theory
... assumptions. The basic idea of type theory is that proofs are functions that convert any proofs of the assumptions of a theorem into a proof of its claim. A connection to computer science is established: In the latter, formal languages have been developed for constructing functions (programs) that a ...
... assumptions. The basic idea of type theory is that proofs are functions that convert any proofs of the assumptions of a theorem into a proof of its claim. A connection to computer science is established: In the latter, formal languages have been developed for constructing functions (programs) that a ...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. 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 deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called 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 be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. 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. Such 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, a picture alone may be sufficient to prove a theorem. 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: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.