“…We associate some (old) convergent series related to definite integrals with the cyclotomic equation x (m) − 1 = 0, for several natural numbers m; for example, for m = 3, x (3) − 1 = (x − 1)(1 + x + x (2)) leads to [Formula: see text]. …”
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“…Attention then turns to real numbers that are built up from natural numbers, with emphasis on integers, rationals, and irrationals. …”
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“…In the first part the development of the natural numbers from Stone-Age times right up to the present day is examined not only from the point of view of pure history but also taking into account archaeological, anthropological and linguistic evidence. …”
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“…Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. …”
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“…Similarly, humans have a capacity for infinite natural numbers, while all other species seem to lack such a capacity (Gelman and Gallistel, 1978; Dehaene, 1997). …”
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“…Numbers, Polynomials, and Factoring The Natural Numbers The Integers Modular Arithmetic Polynomials with Rational CoefficientsFactorization of PolynomialsSection I in a NutshellRings, Domains, and Fields Rings Subrings and Unity Integral Domains and Fields Ideals Polynomials over a Field Section II in a NutshellRing Homomorphisms and Ideals Ring HomomorphismsThe Kernel Rings of Cosets The Isomorphism Theorem for Rings Maximal and Prime Ideals The Chinese Remainder Theorem Section III in a NutshellGroups Symmetries of Geometric Figures PermutationsAbstract Groups Subgroups Cyclic Groups Section…”
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“…The heart of the book is the ordinal analysis of axiom systems, with particular emphasis on that of the impredicative theory of elementary inductive definitions on the natural numbers. The "constructive" consequences of ordinal analysis are sketched out in the epilogue. …”
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“…Functions of One Variable The Real Number System From Natural Numbers to Real Numbers Algebraic Properties of R Order Structure of R Completeness Property of R Mathematical Induction Euclidean Space Numerical Sequences Limits of Sequences Monotone Sequences Subsequences. …”
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“…While logic programs are often invertible (e.g., one can use the same predicate for adding and for subtracting natural numbers), computations are not reversible in the above sense. …”
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“…Full of engaging stories and practical, real-world explanations, this book will help you learn everything from natural numbers and exponents to solving systems of equations and graphing polynomials. …”
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“…The main results are establishment of upper bounds on the elongation of deductions in cut eliminations; a proof that the length of a direct deduction of an existence theorem in the predicate calculus cannot be bounded above by an elementary function of the length of an indirect deduction of the same theorem; a complexity version of the existence property of the constructive predicate calculus; and, for certain formal systems of arithmetic, restrictions on the complexity of deductions that guarantee that the deducibility of a formula for all natural numbers in some finite set implies the deducibility of the same formula with a universal quantifier over all sufficiently large numbers.…”
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“…The approach taken presents the fundamental structures in a fairly general setting, but avoiding the introduction of abstract axiomatic domains. Natural numbers and numerical functions are considered exclusively, which results in a concrete theory conceptually organized around Church's thesis. …”
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“…The aim of this study was to analyze the joint, relative, and unique predictive value of students’ prior knowledge of mathematics (knowledge of fractions and ability to divide natural numbers) and general cognitive ability (fluid intelligence and working memory) upon general mathematics achievement in the last year of primary school. …”
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“…The obtained heterocycles differ in their nature, number of heteroatoms, and ring size. The methods are classified according to the metal used (palladium, copper, gold, silver), then subdivided according to the type of bond formed, namely carbon–carbon or carbon–heteroatom.…”
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“…For complexes with mispairs, frameshift frequency depends on the nature, number, and position of mispairs. Central purine:purine mispairs are especially inhibitory. …”
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“…It is proved that SN P systems with self-organization are capable of computing and accept the family of sets of Turing computable natural numbers. Moreover, with 87 neurons the system can compute any Turing computable recursive function, thus achieves Turing universality. …”
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“…Such lattices acquire periodicity only for rotation angles cos θ = a/c, sin θ = b/c, set by Pythagorean triples of natural numbers (a, b, c). While linear eigenmodes supported by lattices associated with Pythagorean triples are always extended, composite patterns generated for intermediate rotation angles allow observation of the localization-delocalization transition of eigenmodes upon modification of the relative strength of two sublattices forming the composite pattern. …”
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“…The activity of tested compounds is correlated with the nature, number, and position of the substituent group on the aromatic ring. …”
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“…An enumeration operator maps each set A of natural numbers to a set [Formula: see text], in such a way that E(A) can be enumerated uniformly from every enumeration of A. …”
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“…Inductive datatypes in programming languages allow users to define useful data structures such as natural numbers, lists, trees, and others. In this paper we show how inductive datatypes may be added to the quantum programming language QPL. …”
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