The asymptotically fastest algorithm to solve the Discrete Logarithm Problem in finite fields is the Number Field Sieve (NFS). This work presents a summary of the Number Field Sieve and its practical experimental implementation to solve the discrete logarithm problem in finite fields of degree six. This particular problem arises e.g. when one tries to solve DLP in XTR cryptosystem. As shown, the degree six instance of the DLP is practically more difficult to solve with NFS as a classical DLP. Also contained in this book are some specific remarks to the related topic of the polynomial selection for the NFS. A three dimensional adaptation of the line sieving algorithm is described as well as the parametrization choices for the sieve...

ISBN: 978-3-6391-2331-9
Издательство: Книга по требованию
Дата выхода: июль 2011

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency. Media having such a property are termed dispersive media. Dispersion is sometimes called chromatic dispersion to emphasize its wavelength-dependent nature, or group-velocity dispersion (GVD) to emphasize the role of the group velocity. The most familiar example of dispersion is probably a rainbow, in which dispersion causes the spatial separation of a white light into components of different wavelengths (different colors). However, dispersion also has an effect in many other circumstances: for example, GVD causes pulses to spread in optical fibers, degrading signals...

ISBN: 978-6-1302-0122-7
Издательство: Книга по требованию
Дата выхода: июль 2011

The aim of this book is threefold: to reinstate distance functions as a principal tool of general topology, to promote the use of distance functions on various mathematical objects and a thinking in terms of distances also in non-topological contexts, and to make more specific contributions to distance theory. We start by learning the basic properties of distance, endowing all kinds of mathematical objects with a distance function, and studying interesting kinds of mappings between such objects. This leads to new characterizations of many well-known types of mappings. Then a suitable notion of distance spaces is developed, general enough to induce most topological structures, and we study topological properties of mappings like the...

ISBN: 978-3-8364-3312-9
Издательство: Книга по требованию
Дата выхода: июль 2011

The main purpose of this monograph is to show the existence of distortions of histories, and analyze the distorted histories in operator-algebraic points of view. We provide two types of distortions occured by some finite number of objects. In dynamical systems, we define a history by a pair, consisting of a type I factor, and an E_0-group with some additional conditions. We distort the given history by a finite number of partial isometries. Two types of distortions can happen: inner distortions, and outer distortions. The fundamental properties of distorted histories are characterized. We also study the operator algebras generated by distorted histories. Also, by comparing operator algebras generated by histories and those generated by...

ISBN: 978-3-6392-1083-5
Издательство: Книга по требованию
Дата выхода: июль 2011

In mathematics, the distortion is a measure of the amount by which a function from the Euclidean plane to itself distorts circles to ellipses. If the distortion of a function is equal to one, then it is conformal; if the distortion is bounded and the function is a homeomorphism, then it is quasiconformal.In mathematics, a function is a relation between a given set of elements called the domain and a set of elements called the codomain. The function associates each element in the domain with exactly one element in the codomain. The elements so related can be any kind of thing (words, objects, qualities) but are typically mathematical quantities, such as real numbers. Данное издание не является...

ISBN: 978-6-1311-0728-3
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is – up to isomorphism – given as such a lattice of sets.As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1311-2727-4
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles!A congruence ? of a join-semilattice S is monomial, if the ?-equivalence class of any element of S has a largest element. We say that ? is distributive, if it is a join, in the congruence lattice Con S of S, of monomial join-congruences of S. The following definition originates in Schmidt's 1968 work and was subsequently adjusted by Wehrung. Definition (weakly distributive homomorphisms). A homomorphism ? : S ? T between join-semilattices S and T is weakly distributive, if for all a, b in S and all c in T such that ?(c)? a ? b, there are elements x and y of S such that c? x ? y, ?(x)? a, and ?(y)? b. Данное издание не является оригинальным. Книга...

ISBN: 978-6-1311-2726-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example: 2 x (1 + 3) = (2 x 1) + (2 x 3). In the left-hand side of the above equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the results added afterwards. Because these give the same final answer (8), we say that multiplication by 2 distributes over addition of 1 and 3. Since we could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes over addition of...

ISBN: 978-6-1341-0106-6
Издательство: Книга по требованию
Дата выхода: июль 2011

In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a (signed) scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence is negative and the region is called a sink. More technically, the divergence represents the volume density of the outward flux of...

ISBN: 978-6-1302-1004-5
Издательство: Книга по требованию
Дата выхода: июль 2011

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. The simplest counter example is the harmonic series A summability method M is regular if it agrees with the actual limit on all convergent series. Such a result is called an abelian theorem for M, from the prototypical Abel's theorem. More interesting and in general more subtle are partial converse results, called...

ISBN: 978-6-1311-0731-3
Издательство: Книга по требованию
Дата выхода: июль 2011

In vector calculus, the divergence theorem, also known as Gauss' theorem, Ostrogradsky's theorem, or Gauss strogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region. The divergence theorem is an important result for the mathematics of engineering, in particular in electrostatics and fluid dynamics. The theorem is a special case of the more general...

ISBN: 978-6-1307-0283-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called Borel-Okada principle: If a regular summation method sums ?zn to 1/(1 - z) for all z in a subset S of the complex plane, given certain restrictions on S, then the method also gives the analytic continuation of any other function f(z) = ?anzn on the intersection of S with the Mittag-Leffler star for f. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1311-3705-1
Издательство: Книга по требованию
Дата выхода: июль 2011

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities. Данное издание не является оригинальным. Книга печатается по технологии...

ISBN: 978-6-1337-5486-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! En 1801, Carl Friedrich Gauss publie son premier livre de mathematique, appele Disquisitiones arithmeticae. Il demontre en particulier l'existence d'une nouvelle figure constructible a la regle et au compas, le polygone regulier a 17 cotes. La methode qu'il utilise consiste a considerer un polynome, non comme une fonction mais comme element d'une structure, que l'on appelle maintenant anneau, dote d'une addition et d'une multiplication. Les elements ne sont pas tous inversibles, rapprochant en cela cette structure de celle des nombres entiers. Cette analogie est rendue plus profonde si les coefficients des polynomes sont choisis dans un corps, c'est-a- dire un anneau tel que tous les elements...

ISBN: 978-6-1325-3206-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! The division algorithm is a theorem in mathematics which precisely expresses the outcome of the usual process of division of integers. Its name is a partial misnomer: it is not a true algorithm (a well-defined procedure for achieving a specific task), but it can be used to find the greatest common divisor of two integers. The term "division algorithm" is also used in algebra for a general variant of this theorem, shown to hold in integral domains which are principal ideal domains. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1311-2179-1
Издательство: Книга по требованию
Дата выхода: июль 2011

In geometry, a dodecagon is any polygon with twelve sides and twelve angles. In a regular dodecagon, all sides have equal length and all angles have measure 150°. Its Schlafli symbol is {12}. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1302-3972-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In digital logic, a don't-care term is an input-sequence (a series of bits) to a function that the designer does not care about, usually because that input would never happen, or because differences in that input would not result in any changes to the output. By considering these don't-care inputs, designers can potentially minimize their function much more so than if the don't-care inputs were taken to have an output of all 0 or all 1. Examples of don't-care terms are the binary values 1010 through 1111 (10 through 15 in decimal) for a function that takes a BCD value, because a BCD value never takes on values from 1010 to 1111. Don't-care terms are important to consider in minimizing, using...

ISBN: 978-6-1311-7854-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles!In propositional logic, the inference rules double negative elimination (also called double negation elimination, double negative introduction, double negation introduction, or simply double negation) allow deriving the double negative equivalent by adding (for double negative introduction) or removing (for double negative elimination) a pair of negation signs.Double negative elimination is a theorem of classical logic, but not of weaker logics such as intuitionistic logic and minimal logic. Because of their constructive flavor, a statement such as It's not the case that it's not raining is weaker than It's raining. Данное издание не является оригинальным....

ISBN: 978-6-1311-7860-3
Издательство: Книга по требованию
Дата выхода: июль 2011

In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle T2M=T(TM) of the tangent bundle TM of a smooth manifold M. The double tangent bundle arises in the study of connections and second order ordinary differential equations, i.e. (semi)spray structures on smooth manifolds. The double tangent bundle is closely related to the second order jet bundle, which is an object specifically designed to hold the "2nd order derivative information" of smooth functions on smooth manifolds. Given a smooth map f : M to N there is an induced 1st-order derivative map Tf : TM to TN and so also a 2nd order derivative map T^2 f : T^2 M to T^2 N. The Lie Bracket of two vector...

ISBN: 978-6-1306-1847-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! La double contrainte exprime deux contraintes qui s'opposent : l'obligation de chacune contenant une interdiction de l'autre, ce qui rend la situation a priori insoluble. Ce terme est une traduction propre au francais de double bind ( « double lien » ), mais dans un premier temps il etait question d'injonctions paradoxales. Cette notion est proposee en 1956 dans le contexte de la presentation d'une theorie des causes de la schizophrenie sous l'impulsion de Gregory Bateson. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1340-5098-2
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In the theory of logic, double negation is expressed by saying that a proposition A is identical to (equivalent to) not (not-A), or by the formula A = ~~A. Like the Law of Excluded Middle, this principle when extended to an infinite collection of individuals is disallowed by Intuitionistic logic. Some writers grant this "law" the status of a Law of thought.The principium contradictiones of modern logicians (particularly Leibnitz and Kant) in the formula A is not not-A, differs entirely in meaning and application from the Aristotelian proposition [ i.e. Law of Contradiction: not (A and not-A) i.e. ~(A Данное издание не является оригинальным. Книга...

ISBN: 978-6-1311-7880-1
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In linguistic semantics, a downward entailing (in short: "DE") expression is one that denotes a monotone decreasing function. A downward entailing expression reverses the relation of semantic strength among expressions. An expression like "run fast" is semantically stronger than the expression "run" since the former is true of fewer things than the latter.In mathematics, a monotonic function (or monotone function) is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. Данное издание не является оригинальным. Книга печатается по...

ISBN: 978-6-1311-7877-1
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Doxa is a Greek word meaning common belief or popular opinion, from which are derived the modern terms of orthodoxy and heterodoxy. Used by the Greek rhetoricians as a tool for the formation of argument by using common opinions, the doxa was often manipulated by sophists to persuade the people, leading to Plato's condemnation of Athenian democracy. The word doxa picked up a new meaning between the 3rd and 1st centuries BC when the "Seventy" (evdomikonta) Hebrew scholars in Alexandria translated the Hebrew Scriptures into Greek. In this translation of the Scriptures, called the Septuagint, the scholars rendered the Hebrew word for "glory" as doxa. This translation of the Hebrew Scriptures was...

ISBN: 978-6-1311-7922-8
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In fluid dynamics, the drag equation is a practical formula used to calculate the force of drag experienced by anobject due to movement through a fully-enclosing fluid. The equation is attributed to Lord Rayleigh, who originally used L2 in place of A (with L being some linear dimension). The force on a moving object due to a fluid is: where FD is the force of drag, which is by definition the force component in the direction of the flow velocity, ? is the mass density of the fluid, u is the velocity of the object relative to the fluid, A is the reference area, and CD is the drag coefficient ? a dimensionless constant, e.g. 0.25 to 0.45 for a car Данное издание не является...

ISBN: 978-6-1316-1087-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! A drag count is a single unit of drag as defined by aerospace engineers.A drag count of 1 is equal to Cd of 0.0001. So if, for example, the steady state drag coefficient is 0.0200 and someone sticks a hand out the window adding 5 drag counts, the new drag coefficient would be: 0.0200 + 0.0005 = 0.0205 or 205 drag counts. Drag count is used as a crude measure for the change in drag coefficient (it is not a direct measure of drag as it is associated with a reference area, it is only valid for use as a relative assessment of change). Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд...

ISBN: 978-6-1311-4455-4
Издательство: Книга по требованию
Дата выхода: июль 2011