The most practical implementations of algorithms for arithmetic on large inte- gers are written in low-level “assembly language,” specific to a particular machine’s architecture (e.g., the GNU Multi-Precision library GMP, available at gmplib. org). Besides the general fact that such hand-crafted code is more e ffi cient than that produced by a compiler, there is another, more important reason for using assembly language. A typical 32-bit machine often comes with instructions that allow one to compute the 64-bit product of two 32-bit integers, and similarly, instructions to divide a 64-bit integer by a 32-bit integer (obtaining both the quo- tient and remainder). However, high-level programming languages do not (as a rule) provide any access to these low-level instructions. Indeed, we suggested in §3.3 using a value for the base B of about half the word-size of the machine, in order to avoid overflow. However, if one codes in assembly language, one can take B to be much closer, or even equal, to the word-size of the machine. Since our basic algorithms for multiplication and division run in time quadratic in the **number** of base-B digits, the effect of doubling the bit-length of B is to decrease the running time of these algorithms by a factor of four. This e ff ect, combined with the improvements one might typically expect from using assembly-language code, can easily lead to a five- to ten-fold decrease in the running time, compared to an implementation in a high-level language. This is, of course, a significant improvement for those interested in serious “**number** crunching.”

Show more
598 Read more

This book also tries to show students how software can be used intelligently in **algebra**. I feel that this is particularly important for the intended audience. There is a delicate philosophical point. Does a software calculation prove anything? This is not a simple question, and there does not seem to be a consensus among mathematicians about it. There are a few places in the text where a calculation does rely on software, for example, in calculating the Sylow 2-subgroups of S 8 .

419 Read more

we can assume that a, b are coprime (i.e., have no common factors) and plot it as the point in the xy-plane with coordinates (a, b). Starting at (1, 1) we can now trace out a path passing through all of these points with coprime positive natural **number** coordinates and can even arrange to do this without ever recrossing such a point.

68 Read more

Joseph-Louis Lagrange (1736–1813), born in Turin, Italy, was of French and Italian descent. His talent for mathematics became apparent at an early age. Leonhard Euler recognized Lagrange’s abilities when Lagrange, who was only 19, communicated to Euler some work that he had done in the calculus of variations. That year he was also named a professor at the Royal Artillery School in Turin. At the age of 23 he joined the Berlin Academy. Frederick the Great had written to Lagrange proclaiming that the “greatest king in Europe” should have the “greatest mathematician in Europe” at his court. For 20 years Lagrange held the position vacated by his mentor, Euler. His works include contributions to **number** **theory**, group **theory**, physics and mechanics, the calculus of variations, the **theory** of equations, and differential equations. Along with Laplace and Lavoisier, Lagrange was one of the people responsible for designing the metric system. During his life Lagrange profoundly influenced the development of mathematics, leaving much to the next generation of mathematicians in the form of examples and new problems to be solved.

Show more
343 Read more

A good argument could be made that linear **algebra** is the most useful subject in all of mathematics and that it exceeds even courses like calculus in its signiﬁcance. It is used extensively in applied mathematics and engineering. Continuum mechanics, for example, makes use of topics from linear **algebra** in deﬁning things like the strain and in determining appropriate constitutive laws. It is fundamental in the study of statistics. For example, principal component analysis is really based on the singular value decomposition discussed in this book. It is also fundamental in pure mathematics areas like **number** **theory**, functional analysis, geometric measure **theory**, and diﬀerential geometry. Even calculus cannot be correctly understood without it. For example, the derivative of a function of many variables is an example of a linear transformation, and this is the way it must be understood as soon as you consider functions of more than one variable.

Show more
503 Read more

The ABC conjecture is a central open problem in modern **number** **theory**, connecting results, techniques and questions ranging from elementary **number** **theory** and **algebra** to the arithmetic of elliptic curves to algebraic geometry and even to entire functions of a complex variable. The conjecture asserts that, in a precise sense that we specify later, if A, B, C are relatively prime integers such that A + B = C then A, B, C cannot all have many repeated prime factors. This expository article outlines some of the connections between this assertion and more familiar Diophantine questions, following (with the occasional scenic detour) the historical route from Pythagorean triples via Fermat’s Last Theorem to the formulation of the ABC conjecture by Masser and Oesterl´e. We then state the conjecture and give a sample of its many consequences and the few very partial results available. Next we recite Mason’s proof of an analogous assertion for polynomials A(t), B(t), C(t) that implies, among other things, that one cannot hope to disprove the ABC conjecture using a polynomial identity such as the one that solves the Diophantine equation x 2 + y 2 = z 2 . We conclude by solving a Putnam problem

Show more
21 Read more

The FLINT 2 series is a complete rewrite of FLINT 1 from scratch. Its focus is on very clean code and even better performance than FLINT 1. It also of- fers modules for multivariate polynomial arithmetic, polynomials over Z /n Z for multiprecision n and optimised linear **algebra** over Z /nZ for word sized moduli. It should be released by the end of 2010.

Cayley [21. 20] is a knowledge based system designed for solving hard problems in related areas of **algebra**, **number** **theory** and combinatorial **theory**. The system includes a very-high-level programming language, a large database containing mathematical knowledge and an inference engine to aid program synthesis, database retrieval and program optimisation. The authors of Cayley have high hopes for the language saying "th e outcome of the current Cayley project will be a revolutionary integration of knowledge - algorithmic, deductive and factual - in a system which will act as a powerful and effective research assistant for modem **algebra**." The system is based on procedural principles where the user writes algorithms in a procedural notation and examines specific structures containing values/results of previous calculations. The Cayley system is an extremely sophisticated environment allowing the user to study many different **computational** domains and to answer questions not only about individual elements but also about the structure as a whole. Currently the system has no predefined operations for lattices and its main area of operation is fields and groups.

Show more
196 Read more

The estimates just given are upper bound on the running times. Any algo- rithm can be analyzed to give an upper bound on the **computational** complexity (**number** of bit operations) needed to compute a quantity. A better algorithm gives a better upper bound. It is much harder to get lower bounds on the com- putational complexity. However, one lower bound is clear. The **number** of bit operations must be at least equal to the sum of the **number** of bits in the input and the output. This is the case of instantaneous computation. Thus a lower bound on the running time is CB. In this case we say that the running time is linear in the input size. Many algorithms, e.g., long multiplication of positive integers, have polynomial running times: multiplication of two B-bit integers runs in time CB 2 . Thus there is room for improvement, and in fact there are

Show more
24 Read more

Ribet subsequently proved that if the Modularity Conjecture is true, then Fermat’s Last Theorem is true. Precisely, Ribet proved that if every semistable elliptic curve 2 is modular 3 then Fermat’s Last Theorem is true. The Modularity Conjecture, which asserts that every rational elliptic curve is modular, was at that time a conjecture originally formulated by Goro Shimura and Yutaka Taniyama. Finally, in 1994, Andrew Wiles announced a proof that every semistable rational elliptic curve is modular, thereby completing the proof of Fermat’s 350-year-old claim. Wiles’s proof, which is a tour de force using the vast machinery of modern **number** **theory** and algebraic geometry, is far too complicated for us to describe in detail, but we will try to convey the flavor of his proof in Chapter 46.

Show more
40 Read more

need to look at prime integer numbers not greater than 7 and their prime ideal divisors as potential candidates for non-principal ideals. Now prime **number** 3 has the property that -19 is a quadratic non-residue modulo them, so by Theorem 3.5.9 it remains prime in O

48 Read more

Proofs Without Words: Exercises in Visual Thinking, Classroom Resource Materials, MAA, p.. Sums of Squares VIII '.[r]

(36-a) and (36-b) correspond to the semantics and **number** marking of noun phrases without a numeral—they give rise to, descriptively speaking, a singular- plural **number** system. (36-b) gives rise to an exclusive plural semantics, which I return to in section 3. The effects of [±minimal] are more noticeable in (36-e), which denotes a set of boy individuals composed of exactly two atoms, these two- atom, plural boy individuals having no proper subparts in (36-d) (which contains only plural boy individuals composed of exactly two atoms). (36-g) denotes a set of boy individuals composed of exactly one atom, these atomic boy individuals having no proper subparts in (36-g) (which contains only boy atoms). These are the only sources for the grammatical iki çocuk ‘two boys’ and bir çocuk ‘one boy’, respectively, which also result in the correct semantics. Crucially, no matter what numeral is present in the phrase, [-minimal], which spells out as –lAr, never gives rise to a well-formed result ((36-f) and (36-h) denote the empty set)—that is because [-minimal] selects from its input P those individuals that have proper subparts in P, and there are no such subparts in J iki CARD çocuk K , J bir CARD çocuk K , etc.

Show more
54 Read more

The primality testing problem is to decide whether a given integer is prime or composite.. It is considered to be well solved, in contrast to ine factoring problem, which asks for the fa[r]

10 Read more

Solution for Problem 7.13: Solution 1: Examine each separately. When Carl starts reading, Flo has already read for 30 minutes; so, she has read 30 pages already. Let m be the **number** minutes after 1:30 that the two have been reading. Since Flo reads a page every minute and she has read 30 pages before 1:30, the **number** of pages she has read m minutes after 1:30 is m + 30. Carl reads a page every 50 seconds, so he reads one page every 5/6 of a minute. Since Carl reads a page every 5/6 of a minute, his reading rate is

The aim of this article is to introduce and study a new kind of family of fuzzy subsets of a finite set. The complement operation has redefined and it is established that the introduced fuzzy subsets can hold the complement laws. The observation of the behavior of this family of fuzzy subsets is carried out and proved that they can form a Boolean lattice and hence a Boolean **algebra**. In support of the definition of complementation lots of numerical examples are illustrated and established that the introduced fuzzy subsets. Further this article introduces new definitions of subset of a fuzzy subset and fuzzy power set of a subset of a fuzzy subset. We have investigated the behavior of such power sets and also established their occurrence in the introduced family of fuzzy subsets of a finite set.

Show more
The origin of graph **theory** started with the problem of Koinsber bridge, in 1735. This problem lead to the concept of Eulerian Graph. Euler studied the problem of Koinsberg bridge and constructed a structure to solve the problem called Eulerian graph. In 1840, A.F Mobius gave the idea of complete graph and bipartite graph and Kuratowski proved that they are planar by means of recreational problems. The concept of tree, (a connected graph without cycles[7]) was implemented by Gustav Kirchhoff in 1845, and he employed graph theoretical ideas in the calculation of currents in electrical networks or circuits. In 1852, Thomas Gutherie found the famous four color problem. Then in 1856, Thomas. P. Kirkman and William R.Hamilton studied cycles on polyhydra and invented the concept called Hamiltonian graph by studying trips that visited certain sites exactly once. In 1913, H.Dudeney mentioned a puzzle problem. Eventhough the four color problem was invented it was solved only after a century by Kenneth Appel and Wolfgang Haken. This time is considered as the birth of Graph **Theory**.

Show more
evaluation] that "our goal in this paper is to bring these two strands of the literature together and discuss the assumptions which allow the researcher to go back and forth between the **theory** and the data." They consider the large literature on changes in financial policies that affect occupational choices by changing credit constraints and/or changing occupational choice risk conditions. As the authors state in the final sentence of their abstract, "all in all, our objective is to assess the impact of financial intermediation on occupational choices and income." In their paper, they consider theoretically and, with a **number** of empirical examples, various economic theories, econometric methods, and types of data that are useful in achieving their important objective. Because of the focus on occupational choice, this paper is a least as relevant to consumer **theory** as to producer **theory** and admirably uses **theory** from both sides of the market.

Show more
39 Read more

As one of the conditions for obtaining a Per-Batch license, a Registered Manufacturer must apply unique Serial **Number** Labels on individual Product Cartons or Master Cartons in which the products are packed. This “Serial **Number** Guide” specifies the rules for procuring and applying Serial **Number** Labels.

As in the case of AI the need to create strong identity by emphasizing specific methods defining **Computational** Intelligence as a field should be replaced by focus on problems to be solved, rather than hammers. Below some remarks on the current state of CI are made, based on analysis of journals and books with “**computational** intelligence” in their title. Then a new definition of CI is proposed and some remarks on what should **computational** intelligence really be are made. Finally grand challenges to **computational** intelligence are discussed.