“…Beurling to study how independent prime number theory is from the additivity of the natural numbers. The results and techniques of this theory apply to other systems having the character of prime numbers and integers; for example, it is used in the study of the prime number theorem (PNT) for ideals of algebraic number fields. …”
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“…Functional group chemistry and its effect on the printing properties of cotton fabric by varying the nature, number and position of reactive groups of synthesized azo reactive dyes (D1–D6) was studied. …”
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“…Various hypotheses have been proposed in order to explain those difficulties: fractions can denote different concepts; their understanding requires a conceptual reorganization with regard to natural numbers; and using fractions involves the articulation of conceptual knowledge with complex manipulation of procedures. …”
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“…This book considers the manifold possible approaches, past and present, to our understanding of the natural numbers. They are treated as epistemic objects: mathematical objects that have been subject to epistemological inquiry and attention throughout their history and whose conception has evolved accordingly. …”
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“…He then discusses their applications in computing the derivative of the product of functions of several variables and in the construction of expression for n-dimensional natural numbers. Section II focuses on the construction of Hermite interpolating polynomials, from their characterizing properties and generating algorithms to a graphical analysis of their behavior. …”
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“…PREFACEPRELIMINARIES Sets FunctionsREAL NUMBERSField AxiomsOrder Axioms Natural Numbers, Integers, Rational NumbersCompleteness Axiom Decimal Representation of Real Numbers Countable SetsSEQUENCESSequences and ConvergenceProperties of Convergent Sequences Monotonic SequencesThe Cauchy Criterion Subsequences Upper and Lower Limits Open and Closed Sets INFINITE SERIESBasic Properties Convergence TestsLIMIT OF A FUNCTIONLimit of a Function Basic Theorems Some Extensions of the LimitMonotonic Functions CONTINUITY Continuous Functions Combinations of Continuous Functions Continuity on an IntervalUniformContinuityCompact Sets and ContinuityDIFFERENTIATION The DerivativeTheMean Value TheoremL'Hôpital's RuleTaylor's TheoremTHE RIEMANN INTEGRALRiemann Integrability Darboux's Theorem and Riemann SumsProperties of the Integral The Fundamental Theorem of Calculus Improper IntegralsSEQUENCES AND SERIES OF FUNCTIONSSequences of FunctionsProperties of Uniform ConvergenceSeries of FunctionsPower Series LEBESGUE MEASURE Classes of Subsets of R Lebesgue Outer Measure Lebesgue Measure Measurable Functions LEBESGUE INTEGRATIONDefinition of the Lebesgue IntegralProperties of the Lebesgue Integral Lebesgue Integral and Pointwise ConvergenceLebesgue and Riemann IntegralsREFERENCESNOTATIONINDEX.…”
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“…Readers will find this text conveniently self-contained, with three appendices included after the main text, covering an overview of natural numbers and integers, Dedekind's construction of real numbers, historical notes, and selected topics in algebra. …”
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“…The result is a fuzzy interval on the domain of natural numbers, which can be obtained by two variants of the method: exact counting provides the true fuzzy interval in quadratic time complexity, while approximate counting carries out an estimate of the fuzzy interval in linear time. …”
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“…The most basic examples consist of the sequence of natural numbers, the sequence of zeros with positive imaginary part of the Riemann zeta function, and the sequence of eigenvalues, say of a positive Laplacian on a compact or certain cases of non-compact manifolds. …”
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“…The first set of lectures develops a calculus of streams (a generalization of the set of natural numbers) based on the coinduction principle coming from the theory of coalgebras. …”
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“…Moreover, a parity checker which is capable of identifying odd and even numbers from natural numbers is constructed conceptually. Finally, a multi-valued logic gate (ternary inhibit gate) is readily achieved by taking this DNA/Ag NC system as a universal platform. …”
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“…Apart from the very general result, three examples are discussed in more detail: sequences defined as the sum of outputs written by a transducer when reading a q-ary expansion of the input; the amount of esthetic numbers in the first N natural numbers; and the number of odd entries in the rows of Pascal’s rhombus. …”
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“…At present, the prokaryotic species might no longer have to be understood as discrete values (such as 1 or 2, by homology to Natural numbers); rather, it is expected that some microorganisms could be potentially distributed (according to their genome features and phenotypes) in between others (such as decimal numbers between 1 and 2; real numbers). …”
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“…Moreover, a parity checker which can identify even numbers and odd numbers from natural numbers was established successfully. This work helps elucidate the design rules by which simple logic can be harnessed to produce diverse and complex calculations by rewiring communication between different gates. …”
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“…Moreover, a logic gate system for identification of even numbers and odd numbers from natural numbers was established successfully by employing only this single DNA nanoprism and four short single-stranded DNA. …”
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“…It is a mapping [Formula: see text] from the vertex set of a graph T to the set of natural numbers such that the condition [Formula: see text] , is satisfied. …”
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“…Safety will be evaluated by reporting the nature, number, and percentage of serious unexpected or adverse reactions. …”
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“…Our study explores whether the comparison of pairs of natural numbers that differ in length is affected by the identity of the leftmost digit of each multi-digit, and asks what is the effect of having variable proportions of trials with pairs of numbers of the same-length in the task. …”
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“…The next chapter works with representations of natural numbers in different bases, as well as the theory of continued fractions, quadratic irrationalities, and also explores different methods of proofs. …”
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“…The most active AChE and BuChE analogues were discovered to be analogues 9 and 14, with IC(50) values of 0.10 ± 0.050 and 0.20 ± 0.050 µM (against AChE) and 0.20 ± 0.050 and 0.30 ± 0.050 µM (against BuChE), respectively. The nature, number, position, and electron-donating and -withdrawing effects on the phenyl ring were taken into consideration when analysing the structure-activity relationship (SAR). …”
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