Compendium of Experimental Cetane Numbers
Yanowitz, J.; Ratcliff, M. A.; McCormick, R. L.; Taylor, J. D.; Murphy, M. J.
2014-08-01
This report is an updated version of the 2004 Compendium of Experimental Cetane Number Data and presents a compilation of measured cetane numbers for pure chemical compounds. It includes all available single compound cetane number data found in the scientific literature up until March 2014 as well as a number of unpublished values, most measured over the past decade at the National Renewable Energy Laboratory. This Compendium contains cetane values for 389 pure compounds, including 189 hydrocarbons and 201 oxygenates. More than 250 individual measurements are new to this version of the Compendium. For many compounds, numerous measurements are included, often collected by different researchers using different methods. Cetane number is a relative ranking of a fuel's autoignition characteristics for use in compression ignition engines; it is based on the amount of time between fuel injection and ignition, also known as ignition delay. The cetane number is typically measured either in a single-cylinder engine or a constant volume combustion chamber. Values in the previous Compendium derived from octane numbers have been removed, and replaced with a brief analysis of the correlation between cetane numbers and octane numbers. The discussion on the accuracy and precision of the most commonly used methods for measuring cetane has been expanded and the data has been annotated extensively to provide additional information that will help the reader judge the relative reliability of individual results.
Compendium of Experimental Cetane Number Data
Murphy, M. J.; Taylor, J. D.; McCormick, R. L.
2004-09-01
In this report, we present a compilation of reported cetane numbers for pure chemical compounds. The compiled database contains cetane values for 299 pure compounds, including 156 hydrocarbons and 143 oxygenates. Cetane number is a relative ranking of fuels based on the amount of time between fuel injection and ignition. The cetane number is typically measured either in a combustion bomb or in a single-cylinder research engine. This report includes cetane values from several different measurement techniques - each of which has associated uncertainties. Additionally, many of the reported values are determined by measuring blending cetane numbers, which introduces significant error. In many cases, the measurement technique is not reported nor is there any discussion about the purity of the compounds. Nonetheless, the data in this report represent the best pure compound cetane number values available from the literature as of August 2004.
Cetane numbers of biodiesel and its components
The cetane number is one of the prime fuel quality indicators of a petrodiesel or biodiesel fuel as it relates to the tendency of the fuel to ignite in the combustion chamber. It has been established that compound structure, including chain length, branching, and the presence of double bonds, is a m...
High cetane number paraffinic diesel fuel studies
Larmi, Martti; Sarjovaara, Teemu; Imperato, Matteo; Hulkkonen, Tuomo; Kaario, Ossi; Wehrfritz, Armin; Tilli, Aki; Brink, Anders [Aalto University (Finland)
2010-07-01
In the energy sector, stringent regulations have been implemented on combustion emissions in order to address health and environmental concerns and help improve air quality. High cetane number paraffinic fuels and their oxygenate blends have a good potential for reducing emissions. The aim of this collaborative study among several universities and industries is to develop optimum combustion technologies for those fuels and decrease their emissions by 70%. To do so, a literature review was undertaken as was the evaluation of reaction scheme, fuel spray studies, designs with CFD, engine tests with both high and medium speed research engines, and emissions measurement and analysis; all the tests and measurements were carried out using HVO, which is a typical high cetane number paraffinic fuel. Results from these studies are presented herein, further work will comprise the analysis of these results, CFD optimization, and tests with oxygenate blends.
A Group Contribution Method for Estimating Cetane and Octane Numbers
Kubic, William Louis [Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Process Modeling and Analysis Group
2016-07-28
Much of the research on advanced biofuels is devoted to the study of novel chemical pathways for converting nonfood biomass into liquid fuels that can be blended with existing transportation fuels. Many compounds under consideration are not found in the existing fuel supplies. Often, the physical properties needed to assess the viability of a potential biofuel are not available. The only reliable information available may be the molecular structure. Group contribution methods for estimating physical properties from molecular structure have been used for more than 60 years. The most common application is estimation of thermodynamic properties. More recently, group contribution methods have been developed for estimating rate dependent properties including cetane and octane numbers. Often, published group contribution methods are limited in terms of types of function groups and range of applicability. In this study, a new, broadly-applicable group contribution method based on an artificial neural network was developed to estimate cetane number research octane number, and motor octane numbers of hydrocarbons and oxygenated hydrocarbons. The new method is more accurate over a greater range molecular weights and structural complexity than existing group contribution methods for estimating cetane and octane numbers.
Highlights: ► We obtained models for estimation of cetane number of biodiesel. ► Twenty-four neural networks using two topologies were evaluated. ► The best neural network for predict the cetane number was selected. ► The best accuracy was obtained for the selected neural network. - Abstract: Models for estimation of cetane number of biodiesel from their fatty acid methyl ester composition using multiple linear regression and artificial neural networks were obtained in this work. For the obtaining of models to predict the cetane number, an experimental data from literature reports that covers 48 and 15 biodiesels in the modeling-training step and validation step respectively were taken. Twenty-four neural networks using two topologies and different algorithms for the second training step were evaluated. The model obtained using multiple regression was compared with two other models from literature and it was able to predict cetane number with 89% of accuracy, observing one outlier. A model to predict cetane number using artificial neural network was obtained with better accuracy than 92% except one outlier. The best neural network to predict the cetane number was a backpropagation network (11:5:1) using the Levenberg–Marquardt algorithm for the second step of the networks training and showing R = 0.9544 for the validation data.
Westbrook, C K; Pitz, W J; Mehl, M; Curran, H J
2010-03-03
For the first time, a detailed chemical kinetic reaction mechanism is developed for primary reference fuel mixtures of n-hexadecane and 2,2,4,4,6,8,8-heptamethyl nonane for diesel cetane ratings. The mechanisms are constructed using existing rules for reaction pathways and rate expressions developed previously for the primary reference fuels for gasoline octane ratings, n-heptane and iso-octane. These reaction mechanisms are validated by comparisons between computed and experimental results for shock tube ignition and for oxidation under jet-stirred reactor conditions. The combined kinetic reaction mechanism contains the submechanisms for the primary reference fuels for diesel cetane ratings and submechanisms for the primary reference fuels for gasoline octane ratings, all in one integrated large kinetic reaction mechanism. Representative applications of this mechanism to two test problems are presented, one describing fuel/air autoignition variations with changes in fuel cetane numbers, and the other describing fuel combustion in a jet-stirred reactor environment with the fuel varying from pure 2,2,4,4,6,8,8-heptamethyl nonane (Cetane number of 15) to pure n-hexadecane (Cetane number of 100). The final reaction mechanism for the primary reference fuels for diesel fuel and gasoline is available on the web.
Determination model for cetane number of biodiesel at different fatty acid composition: a review
Michal Angelovič
2014-05-01
Full Text Available The most accepted definition of biodiesel is stated at the EU technical regulation EN 14214 (2008 or in the USA in ASTM 6751-02. As a result of this highly strict description only methyl esters of fatty acids conform to these definitions, nevertheless the term ‘‘biodiesel’’ is spread out to other alkyl fatty esters. Some countries have adopted bioethanol for replacement of methanol in biodiesel transesterification and thus assuring a fully biological fuel. Of course, such position brings some problems in fulfilling technical requirements of EN 14214 or ASTM 6751-02. Biodiesel is actually a less complex mixture than petrodiesel, but different feedstock origins and the effect of seasonality may impose difficulties in fuel quality control. Since biodiesel is an alternative diesel fuel derived from the transesterification of triacylglycerol comprised materials, such as vegetable oils or animal fats, with simple alcohols to furnish the corresponding mono-alkyl esters, its composition depends on the raw material used, the cultivated area location, and harvest time. The choice of the raw material is usually the most important factor for fluctuations of biodiesel composition, because different vegetable oils and animal fats may contain different types of fatty acids. Important properties of this fuel vary significantly with the composition of the mixture. Cetane number, melting point, degree of saturation, density, cloud point, pour point, viscosity, and nitrogen oxides exhaust emission (NOx, for instance, deserve to be mentioned. One of the most important fuel quality indicators is the cetane number; however its experimental determination may be an expensive and lengthy task. To weaken situation concerning biodiesel, the availability of data in the literature is also scarce. In such scenario, the use of reliable models to predict the cetane number or any other essential characteristic may be of great utility. We reviewed available literature to
Miqdam Tariq Chaichan
2012-01-01
Full Text Available Due to the energy crisis and the stringent environmental regulations, diesel engines are offering good hope for automotive vehicles. However, a lot of work is needed to reduce the diesel exhaust emissions and give the way for full utilization of the diesel fuels excellent characteristics.A kind of cetane number improver has been proposed and tested to be used with diesel fuel as ameans of reducing exhaust emissions. The addition of (2-ethylhexyl nitrate was designed to raise fuel cetane number to three stages, 50, 52 and 55 compared to the used conventional diesel fuel whose CN was 48.5. The addition of CN improver results in the decrease brake specific fuel consumption by about 12.55%, and raise brake thermal efficiency to about 9%. Simultaneously, the emission characteristics of four fuels are determined in a diesel engine. At high loads, a little penalty on CO and HC emissions compared to baseline diesel fuel. NOx emissions of the higher CN fuels are decreased 6%, and CO of these fuels is reduced to about 30.7%. Engine noise reduced with increasing CN to about 10.95%. The results indicate the potential of diesel reformation for clean combustion in diesel engines.
Highlights: • The effects of oxygenated fuels on diesel combustion are extensively investigated. • CO and soot emissions are reduced with the increase of oxygen concentration. • The C–O bond in the oxygenated fuels inhibits the formation of soot precursor C2H2. • Small intermediates such as C2H4 and C2H6 are significantly reduced. • Oxygen concentration seems to be the dominating factor affecting the emissions. - Abstract: The present modeling study aims to gain better insights on the effects of oxygenated fuels on the diesel oxidation and emission formation processes under realistic engine operating conditions. To do that, various blend fuels formulated from diesel, biodiesel, ethanol and DMC fuels were obtained with different oxygen concentrations, cetane numbers and C/H ratios. Simulations were conducted using the coupled KIVA–CHEMKIN code on a light duty diesel engine at a fixed engine speed of 2400 rpm under full load conditions. Constructed numerical simulation models integrated with detailed chemical kinetics were validated against the experimental results with reliable accuracies. Simulation results revealed that as the overall oxygen concentration of the blend fuel increased, significant beneficial effects were shown with reduced NOx, CO and soot emissions. Particularly, with the increase of oxygen concentration, the peak CO concentration and its final emission level were found to be remarkably reduced due to the fuel borne oxygen, reduced carbon influx as well as the possibility accelerated CO oxidation rate. More tangible reductions were shown on the soot emissions probably because the C–O bond in the oxygenated blend fuels had played an important role in inhibiting the carbon atoms from soot formation. Furthermore, as oxygenated fuels were added, the peak concentration of the soot precursor C2H2 species and small hydrocarbon intermediates such as C2H4 and C2H6 were also significantly reduced. In general, it was found that compared to the
Zotin, J.L.; Pacheco, M.E.; Souza, V.P.; Belato, D.; Silva, R.M.S. [PETROBRAS, Rio de Janeiro, RJ (Brazil). Centro de Pesquisas (CENPES)
2004-07-01
According to the Brazilian specifications for diesel, lower sulfur content and higher cetane number can be expected in the near future, leading to an increased capacity of hydrotreating processes. PETROBRAS has proved technology for hydrotreating processes with 8 high pressure single stage units in operation. However, the production of ultra low sulfur diesel with high cetane number may require two stages processes, with conventional hydrotreating in the first step and deep aromatic saturation (HDA), with increase in the cetane number, in the second one. In this approach, noble metal catalysts, which are more active for hydrogenation but more sensitive to sulfur and nitrogen poisoning, can be used in the second stage. In the present work, the 2 stages approach was studied for maximizing cetane number of Brazilian gasoils. The influence of operating variables and the inhibition effect by sulfur and nitrogen were analyzed. Diesel with sulfur content below 10 ppm and high aromatic conversions at relatively mild conditions were obtained with noble metal catalysts, with a cetane number increase up to 6 points in the second stage. Sulfur compounds have a stronger inhibition effect than basic nitrogen compounds on hydrogenation reaction rates, but a synergetic effect was observed when both contaminants were present in high concentrations. (author)
Yisel Sánchez-Borroto
2012-05-01
Full Text Available El número de cetano (CN es una de las propiedades más importantes para evaluar la calidad del proceso de combustión de un combustible tipo diesel. El CN de biocombustibles derivados de aceites vegetales está influenciado por su composición de ácidos grasos. El objetivo de esta investigación es obtener dos modelos físico-matemáticos que establezcan una relación entre el número de cetano de biocombustibles derivados de aceites vegetales y su composición de ácidos grasos esenciales. Para predecir el CN se desarrollaron dos modelos mediante un análisis de regresión múltiple, basado en datos experimentales extraídos de bases de datos referenciadas. El ajuste de los coeficientes del modelo se basa en la obtención de residuales mínimos. Se realiza una validación parcial del modelo y se establecen las pautas para la validación total del mismo. A partir de estos resultados queda establecida una herramienta muy útil para la determinación del número de cetano. Palabras claves: número de cetano, biodiesel, aceites vegetales, ácidos grasos.______________________________________________________________________________ Abstract The cetane number is one of the main properties for characterization of the ignition quality of a fuel for its use in a diesel engine. The cetane number of biofuels obtained from different vegetable oils is strongly considerably by their fatty acid composition. The scope of this research is to obtain two physical-mathematical models that establish a relationship between the cetane number and the composition of different biofuels derived from vegetable oils. The models were developed using an analysis of multiple regression, based on experimental data available in the literature. The coefficients adjustment of the model was based on the obtaining of lower residual in the capacity of prediction of the model. A partial validation of the model was developed. Starting from these results it a very useful tool for
Feasibility of using less viscous and lower cetane (LVLC) fuels in a diesel engine: A review
Vallinayagam, R.
2015-11-01
This review work focuses on biofuels with lower viscosity and cetane number and their mode of operation in a diesel engine. Though there were a number of review works describing the production, characterization and utilization of biodiesel, synthesized from vegetable oils, a comprehensive summary on other category of biofuels endowed with lower viscosity and cetane number has not come to light so far. In this backdrop, this review work would bring forth the existence of biofuels having lower viscosity and cetane number, classify them under one category and elucidate their operational feasibility in a diesel engine. Considerably, alcohol based fuels such as methanol, ethanol and butanol, and plant based light biofuels such as eucalyptus oil and pine oil have been chosen and classified as LVLC (less viscous and lower cetane) fuels in the current work. Besides describing the operation feasibility of these fuels, an extensive exploration of their physical, thermal and critical properties as well as their compositional attributes has been made. Despite their distinct properties, these fuels have found use in diesel engine by various strategies and apparently, they could be used in blends with diesel/biodiesel, dual fuel mode and as sole fuel. In this regard, herein, a detailed summary on operation of these fuels in the reported three different modes is clearly explained and their engine characteristics such as performance, combustion and emission are briefed. © 2015 Elsevier Ltd.
The Autoignition of iso-Cetane: Shock Tube Experiments and Kinetic Modeling
Oehlschlaeger, M A; Steinberg, J; Westbrook, C K; Pitz, W J
2009-02-25
Iso-cetane (2,2,4,4,6,8,8-heptamethylnonane, C{sub 16}H{sub 34}) is a highly branched alkane reference compound for determining cetane ratings. It is also a candidate branched alkane representative in surrogate mixtures for diesel and jet fuels. Here new experiments and kinetic modeling results are presented for the autoignition of iso-cetane at elevated temperatures and pressures relevant to combustion in internal combustion engines. Ignition delay time measurements were made in reflected shock experiments in a heated shock tube for {Phi} = 0.5 and 1.0 iso-cetane/air mixtures at temperatures ranging from 953 to 1347 K and pressures from 8 to 47 atm. Ignition delay times were measured using electronically excited OH emission, monitored through the shock tube end wall, and piezoelectric pressure transducer measurements, made at side wall locations. A new kinetic mechanism for the description of the oxidation of iso-cetane is presented that is developed based on a previous mechanism for iso-octane. Computed results from the mechanism are found in good agreement with the experimental measurements. To our knowledge, the ignition time measurements and detailed kinetic mechanism for isocetane presented here are the first of their kind.
Souza, Tatiana Bittencourt de; Yamamoto, Carlos Itsuo; Cocco, Lilian Cristina [Universidade Federal do Parana (UFPR), Curitiba, PR (Brazil). Lab. de Analise de Combustiveis Automotivos (LACAUTets)
2008-07-01
This work presents the development of models for Calculated Cetane Index (CCI) determination with the purpose of tailor it to the diesel oil sold in Brazil, using optimization techniques. As the diesel is the main fuel on the Brazilian territory, their quality is of great importance. The CCI calculation is important to evaluate the diesel quality. The standard ASTM D 4737 does not take into account the presence of cetane booster additive and biodiesel, which creates the need to recast the equation of the CCI determination. About 300 representative samples were selected of diesel in Parana, during the 2006 to 2007 period, which had their physic-chemical properties determined using the methodology adopted by the ANP - the Brazilian petroleum, natural gas and biofuel agency. The Derived Cetane Number was obtained in the IQT apparatus and several models were proposed, all with better predictive capability than the equation of the standard ASTM D 4737. It is pointed out that biodiesel can act as cetane booster additive depending upon its origin, mostly in concentrations above 5%. The methodology can be expanded to generate a representative equation for the diesel sold in the whole Brazilian territory. (author)
The objective of the present work is to obtain a physical-mathematical model that establishes a relationship between the cetane index of biofuels obtained from different vegetable oils and its composition of essential fatty acid. This model is based on experimental data obtained by the authors of the present work and an experimental data reported by different extracted authors of indexed databases. The adjustment of the coefficients of the model is based on the obtaining of residual minima in the capacity of prediction of the model. Starting from these results it is established a very useful tool for the determination of such an important parameter for the fuel diesel as it is the cetane index obtained from an analysis of chemical composition and not obtained from tests in engines banks, to save time and economic resources. (author)
赵阳; 吴佳; 王宣; 张晓昕; 孟祥堃
2007-01-01
介绍了植物油加氢制备柴油的主要化学反应、工艺方法以及该项技术的工业化状况,分析了植物油直接加氢、先加氢后异构、直接脱羧工艺以及植物油与矿物柴油掺炼工艺的特点,阐述了各工艺采用的催化剂类型、工艺条件以及产品属性.指出了该领域存在的问题和研究方向.
Yisel Sánchez-Borroto; Ramón Piloto-Rodríguez; Leonardo Goyos-Pérez; Nayvi Ferrer-Frontela
2012-01-01
El número de cetano (CN) es una de las propiedades más importantes para evaluar la calidad del proceso de combustión de un combustible tipo diesel. El CN de biocombustibles derivados de aceites vegetales está influenciado por su composición de ácidos grasos. El objetivo de esta investigación es obtener dos modelos físico-matemáticos que establezcan una relación entre el número de cetano de biocombustibles derivados de aceites vegetales y su composición de ácidos grasos esenciales. Para predec...
Khovanova, Tanya; Lewis, Joel Brewster
2013-01-01
We introduce numbers depending on three parameters which we call skyscraper numbers. We discuss properties of these numbers and their relationship with Stirling numbers of the first kind, and we also introduce a skyscraper sequence.
Hagen number versus Bejan number
Awad Mohamed M.
2013-01-01
Full Text Available This study presents Hagen number vs. Bejan number. Although their physical meaning is not the same because the former represents the dimensionless pressure gradient while the latter represents the dimensionless pressure drop, it will be shown that Hagen number coincides with Bejan number in cases where the characteristic length (l is equal to the flow length (L. Also, a new expression of Bejan number in the Hagen-Poiseuille flow will be introduced. At the end, extending the Hagen number to a general form will be presented. For the case of Reynolds analogy (Pr = Sc = 1, all these three definitions of Hagen number will be the same.
Lužnik, Polona
2013-01-01
My graduate thesis contains a detailed examination of pentagonal nubers. In the beginning, I concentrate on figurate numbers and the mathematicians, who were the first to describe them. The work includes the basic characteristis of pentagonal numbers, how we can obtain them through calculating and counting of dots in graphic illustrtions and how we are able to check if a certain prime number is a pentagonal number or not.
Rich, Andrew
2008-01-01
The leftist number system consists of numbers with decimal digits arranged in strings to the left, instead of to the right. This system fails to be a field only because it contains zerodivisors. The same construction with prime base yields the p-adic numbers.
Schwarzweller Christoph
2015-02-01
Full Text Available In this article we introduce Proth numbers and prove two theorems on such numbers being prime [3]. We also give revised versions of Pocklington’s theorem and of the Legendre symbol. Finally, we prove Pepin’s theorem and that the fifth Fermat number is not prime.
Mendonça, J. Ricardo G.
2012-01-01
We define a new class of numbers based on the first occurrence of certain patterns of zeros and ones in the expansion of irracional numbers in a given basis and call them Sagan numbers, since they were first mentioned, in a special case, by the North-american astronomer Carl E. Sagan in his science-fiction novel "Contact." Sagan numbers hold connections with a wealth of mathematical ideas. We describe some properties of the newly defined numbers and indicate directions for further amusement.
Vorob'ev, Nikolai Nikolaevich
2011-01-01
Fibonacci numbers date back to an 800-year-old problem concerning the number of offspring born in a single year to a pair of rabbits. This book offers the solution and explores the occurrence of Fibonacci numbers in number theory, continued fractions, and geometry. A discussion of the ""golden section"" rectangle, in which the lengths of the sides can be expressed as a ration of two successive Fibonacci numbers, draws upon attempts by ancient and medieval thinkers to base aesthetic and philosophical principles on the beauty of these figures. Recreational readers as well as students and teacher
Number names and number understanding
Ejersbo, Lisser Rye; Misfeldt, Morten
2014-01-01
through using mathematical names for the numbers such as one-ten-one for 11 and five-ten-six for 56. The project combines the renaming of numbers with supporting the teaching with the new number names. Our hypothesis is that Danish children have more difficulties learning and working with numbers, because...... the Danish number names are more complicated than in other languages. Keywords: A research project in grade 0 and 1th in a Danish school, Base-10 system, two-digit number names, semiotic, cognitive perspectives....
河野, 雅文; 天谷, 賢児; 新井, 雅隆
2002-01-01
Effect of cetane number on ignition delay and ignition location of disesel spray was experimentally investigated. Five kinds of fuels, which had different cetane numbers of light fraction and heavy fraction of fuel, were used. As a result, the ignition delay was not so affected by the change of cetane number, however, the ignition location was strongly affected by a cetane number. When the light fraction cetane number was low, the ignition was not observed near the nozzle. And when the heavy ...
Petersen, T Kyle
2015-01-01
This text presents the Eulerian numbers in the context of modern enumerative, algebraic, and geometric combinatorics. The book first studies Eulerian numbers from a purely combinatorial point of view, then embarks on a tour of how these numbers arise in the study of hyperplane arrangements, polytopes, and simplicial complexes. Some topics include a thorough discussion of gamma-nonnegativity and real-rootedness for Eulerian polynomials, as well as the weak order and the shard intersection order of the symmetric group. The book also includes a parallel story of Catalan combinatorics, wherein the Eulerian numbers are replaced with Narayana numbers. Again there is a progression from combinatorics to geometry, including discussion of the associahedron and the lattice of noncrossing partitions. The final chapters discuss how both the Eulerian and Narayana numbers have analogues in any finite Coxeter group, with many of the same enumerative and geometric properties. There are four supplemental chapters throughout, ...
Diewert, Erwin
2007-01-01
Index numbers are used to aggregate detailed information on prices and quantities into scalar measures of price and quantity levels or their growth. The paper reviews four main approaches to bilateral index number theory where two price and quantity vectors are to be aggregated: fixed basket and average of fixed baskets, stochastic, test or axiomatic and economic approaches. The paper also considers multilateral index number theory where it is necessary to construct price and quantity aggrega...
Number names and number understanding
Ejersbo, Lisser Rye; Misfeldt, Morten
2014-01-01
This paper concerns the results from the first year of a three-year research project involving the relationship between Danish number names and their corresponding digits in the canonical base 10 system. The project aims to develop a system to help the students’ understanding of the base 10 system...... through using mathematical names for the numbers such as one-ten-one for 11 and five-ten-six for 56. The project combines the renaming of numbers with supporting the teaching with the new number names. Our hypothesis is that Danish children have more difficulties learning and working with numbers, because...... the Danish number names are more complicated than in other languages. Keywords: A research project in grade 0 and 1th in a Danish school, Base-10 system, two-digit number names, semiotic, cognitive perspectives....
Ji, Caleb; Khovanova, Tanya; Park, Robin; Song, Angela
2015-01-01
In this paper, we consider a game played on a rectangular $m \\times n$ gridded chocolate bar. Each move, a player breaks the bar along a grid line. Each move after that consists of taking any piece of chocolate and breaking it again along existing grid lines, until just $mn$ individual squares remain. This paper enumerates the number of ways to break an $m \\times n$ bar, which we call chocolate numbers, and introduces four new sequences related to these numbers. Using various techniques, we p...
Andrews, George E
1994-01-01
Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic.In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory. In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simpl
无
2004-01-01
THE last digit of my home phone number in Beijing is 4. “So what?” European readers might ask.This was my attitude when I first lived in China; I couldn't understand why Chinese friends were so shocked at my indifference to the number 4. But China brings new discoveries every day, and I have since seen the light. I know now that Chinese people have their own ways of preserving their well being, and that they see avoiding the number 4 as a good way to stay safe.
Sezin, Fatin
2009-01-01
It is instructive and interesting to find hidden numbers by using different positional numeration systems. Most of the present guessing techniques use the binary system expressed as less-than, greater-than or present-absent type information. This article describes how, by employing four cards having integers 1-64 written in different colours, one…
St-Amant, Patrick
2010-01-01
We will see that key concepts of number theory can be defined for arbitrary operations. We give a generalized distributivity for hyperoperations (usual arithmetic operations and operations going beyond exponentiation) and a generalization of the fundamental theorem of arithmetic for hyperoperations. We also give a generalized definition of the prime numbers that are associated to an arbitrary n-ary operation and take a few steps toward the development of its modulo arithmetic by investigating a generalized form of Fermat's little theorem. Those constructions give an interesting way to interpret diophantine equations and we will see that the uniqueness of factorization under an arbitrary operation can be linked with the Riemann zeta function. This language of generalized primes and composites can be used to restate and extend certain problems such as the Goldbach conjecture.
Number of Compositions and Convolved Fibonacci numbers
Janjic, Milan
2010-01-01
We consider two type of upper Hessenberg matrices which determinants are Fibonacci numbers. Calculating sums of principal minors of the fixed order of the first type leads us to convolved Fibonacci numbers. Some identities for these and for Fibonacci numbers are proved. We also show that numbers of compositions of a natural number with fixed number of ones appear as coefficients of characteristic polynomial of a Hessenberg matrix which determinant is a Fibonacci number. We derive the explicit...
T. Pathinathan; K. Ponnivalavan
2015-01-01
In this paper we define diamond fuzzy number with the help of triangular fuzzy number. We include basic arithmetic operations like addition, subtraction of diamond fuzzy numbers with examples. We define diamond fuzzy matrix with some matrix properties. We have defined Nested diamond fuzzy number and Linked diamond fuzzy number. We have further classified Right Linked Diamond Fuzzy number and Left Linked Diamond Fuzzy number. Finally we have verified the arithmetic operations for the above men...
Paul W. Haggard
1985-01-01
Full Text Available The Legendre numbers, an infinite set of rational numbers are defined from the associated Legendre functions and several elementary properties are presented. A general formula for the Legendre numbers is given. Applications include summing certain series of Legendre numbers and evaluating certain integrals. Legendre numbers are used to obtain the derivatives of all orders of the Legendre polynomials at x=1.
Hanaki, Ryo
2014-01-01
We introduce a numerical invariant, called trivializing number, of knots and investigate it. The trivializing number gives an upper bound of unknotting number and canonical genus for knots. We present a table of trivializing numbers for up to 10 crossings knots. We conjecture that twice of the unknotting number of any positive knot is equal to the trivializing number of it and give a partial answer.
Octavian Cira
2015-04-01
Full Text Available The first prime number with the special property that its addition with reversal gives as result a prime number toois 229. The prime numbers with this property will be called Luhn prime numbers. In this article we intend to presenta performing algorithm for determining the Luhn prime numbers. Using the presented algorithm all the 50598 Luhnprime numbers have been, for p prime smaller than 2 · 107.
Octavian Cira; Florentin Smarandache
2015-01-01
The first prime number with the special property that its addition with reversal gives as result a prime number toois 229. The prime numbers with this property will be called Luhn prime numbers. In this article we intend to presenta performing algorithm for determining the Luhn prime numbers. Using the presented algorithm all the 50598 Luhnprime numbers have been, for p prime smaller than 2 · 107.
柳云骐; 陈为超; 周卫东; 薛颖; 张景成; 刘晨光
2009-01-01
在高压固定床微反装置上研究了豆油在加氢催化剂CoMo/γ-Al2O3,NiMoP/γ-Al2O3,NiMoP/γ-Al2O3-HUSY上的加氢反应规律,并研究了NiMoP/γ-Al2O3-HUSY催化剂对豆油和流化催化裂化(FCC)柴油耦合加氢产物性质的影响.实验结果表明,在压力3.0 MPa、温度320 ℃、液态空速2.0 h-1、氢气与原料油体积比(氢油比)500的条件下,CoMo/γ-Al2O3和NiMoP/γ-Al2O3催化剂上豆油加氢产物主要为n-C15～18,而添加酸性组分的NiMoP/γ-Al2O3-HUSY催化剂的裂化性能增强,产物中n-C15～18含量明显减少,C-15的含量增加;在压力4.0 MPa、温度370 ℃、液态空速1.0 h-1、氢油比500的条件下,豆油和FCC柴油的混合原料在NiMoP/γ-Al2O3-HUSY催化剂上的加氢脱硫率达97%左右,加氢脱氮率达80%以上,产物的十六烷值与未掺炼豆油的FCC柴油加氢产物相比,提高了1.8～6.5个单位.
Mack, Nancy K.
2011-01-01
Exploring number systems of other cultures can be an enjoyable learning experience that enriches students' knowledge of numbers and number systems in important ways. It helps students deepen mental computation fluency, knowledge of place value, and equivalent representations for numbers. This article describes how the author designed her…
Jørgensen, Claus Bjørn; Suetens, Sigrid; Tyran, Jean-Robert
numbers based on recent drawings. While most players pick the same set of numbers week after week without regards of numbers drawn or anything else, we find that those who do change, act on average in the way predicted by the law of small numbers as formalized in recent behavioral theory. In particular...
Damian Slota; Roman Witula
2009-01-01
The scope of the paper is the definition and discussion of the polynomial generalizations of the {sc Fibonacci} numbers called here $delta$-{sc Fibonacci} numbers. Many special identities and interesting relations for these new numbers are presented. Also, different connections between $delta$-{sc Fibonacci} numbers and {sc Fibonacci} and {sc Lucas} numbersare proven in this paper.
Markwig, Hannah; Rau, Johannes
2014-01-01
In this paper, we define tropical analogues of real Hurwitz numbers, i.e. numbers of covers of surfaces with compatible involutions satisfying prescribed ramification properties. We prove a correspondence theorem stating the equality of the tropical numbers with their real counterparts. We apply this theorem to the case of double Hurwitz numbers (which generalizes our result from arXiv:1409.8095).
Damian Slota
2009-08-01
Full Text Available The scope of the paper is the definition and discussion of the polynomial generalizations of the {sc Fibonacci} numbers called here $delta$-{sc Fibonacci} numbers. Many special identities and interesting relations for these new numbers are presented. Also, different connections between $delta$-{sc Fibonacci} numbers and {sc Fibonacci} and {sc Lucas} numbersare proven in this paper.
Vazzana, Anthony; Garth, David
2007-01-01
One of the oldest branches of mathematics, number theory is a vast field devoted to studying the properties of whole numbers. Offering a flexible format for a one- or two-semester course, Introduction to Number Theory uses worked examples, numerous exercises, and two popular software packages to describe a diverse array of number theory topics.
Burkhart, Jerry
2009-01-01
Prime numbers are often described as the "building blocks" of natural numbers. This article shows how the author and his students took this idea literally by using prime factorizations to build numbers with blocks. In this activity, students explore many concepts of number theory, including the relationship between greatest common factors and…
Ouannas, Moussa
2011-01-01
In this paper I present the distribution of prime numbers which was treated in many researches by studying the function of Riemann; because it has a remarkable property; its non trivial zeros are prime numbers; but in this work I will show that we can find the distribution of prime numbers on remaining in natural numbers only.
Arthan, R. D.
2004-01-01
This note describes a representation of the real numbers due to Schanuel. The representation lets us construct the real numbers from first principles. Like the well-known construction of the real numbers using Dedekind cuts, the idea is inspired by the ancient Greek theory of proportion, due to Eudoxus. However, unlike the Dedekind construction, the construction proceeds directly from the integers to the real numbers bypassing the intermediate construction of the rational numbers. The constru...
Jarvis, Frazer
2014-01-01
The technical difficulties of algebraic number theory often make this subject appear difficult to beginners. This undergraduate textbook provides a welcome solution to these problems as it provides an approachable and thorough introduction to the topic. Algebraic Number Theory takes the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform. The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the fi...
Grešak, Rozalija
2015-01-01
The field of real numbers is usually constructed using Dedekind cuts. In these thesis we focus on the construction of the field of real numbers using metric completion of rational numbers using Cauchy sequences. In a similar manner we construct the field of p-adic numbers, describe some of their basic and topological properties. We follow by a construction of complex p-adic numbers and we compare them with the ordinary complex numbers. We conclude the thesis by giving a motivation for the int...
Pak, Igor
2014-01-01
We give a brief history of Catalan numbers, from their first discovery in the 18th century to modern times. This note will appear as an appendix in Richard Stanley's forthcoming book on Catalan numbers.
Bencze, Mihaly; Smarandache, Florentin
2008-01-01
In this article we present a simple proof of Borevich-Shafarevich's method to compute the sum of the first n natural numbers of the same power. We also prove several properties of Bernoulli's numbers.
Fibonacci Numbers and Identities
Lang, Cheng Lien; Lang, Mong Lung
2013-01-01
By investigating a recurrence relation about functions, we first give alternative proofs of various identities on Fibonacci numbers and Lucas numbers, and then, make certain well known identities visible via certain trivalent graph associated to the recurrence relation.
Number Relationships in Preschool
Jung, Myoungwhon
2011-01-01
When a child understands number relationships, he or she comprehends the meaning of numbers by developing multiple, flexible ways of representing them. The importance of developing number relationships in the early years has been highlighted because it helps children build a good foundation for developing a more sophisticated understanding of…
Landy, David; Silbert, Noah; Goldin, Aleah
2013-01-01
Despite their importance in public discourse, numbers in the range of 1 million to 1 trillion are notoriously difficult to understand. We examine magnitude estimation by adult Americans when placing large numbers on a number line and when qualitatively evaluating descriptions of imaginary geopolitical scenarios. Prior theoretical conceptions…
Xu, Junyan
2012-01-01
We point out that the recursive formula that appears in Erickson's presentation "Fusible Numbers" is incorrect, and pose an alternate conjecture about the structure of fusible numbers. Although we are unable to solve the conjecture, we succeed in establishing some basic properties of fusible numbers. We suggest some possible approaches to the conjecture, and list further problems in the final chapter.
Shi, Yixun
2010-01-01
Starting with an interesting number game sometimes used by school teachers to demonstrate the factorization of integers, "sum-difference numbers" are defined. A positive integer n is a "sum-difference number" if there exist positive integers "x, y, w, z" such that n = xy = wz and x ? y = w + z. This paper characterizes all sum-difference numbers…
Matsumoto, Kohji
2002-01-01
The book includes several survey articles on prime numbers, divisor problems, and Diophantine equations, as well as research papers on various aspects of analytic number theory such as additive problems, Diophantine approximations and the theory of zeta and L-function Audience Researchers and graduate students interested in recent development of number theory
de Mestre, Neville
2008-01-01
Prime numbers are important as the building blocks for the set of all natural numbers, because prime factorisation is an important and useful property of all natural numbers. Students can discover them by using the method known as the Sieve of Eratosthenes, named after the Greek geographer and astronomer who lived from c. 276-194 BC. Eratosthenes…
Niederreiter, Harald
2015-01-01
This textbook effectively builds a bridge from basic number theory to recent advances in applied number theory. It presents the first unified account of the four major areas of application where number theory plays a fundamental role, namely cryptography, coding theory, quasi-Monte Carlo methods, and pseudorandom number generation, allowing the authors to delineate the manifold links and interrelations between these areas. Number theory, which Carl-Friedrich Gauss famously dubbed the queen of mathematics, has always been considered a very beautiful field of mathematics, producing lovely results and elegant proofs. While only very few real-life applications were known in the past, today number theory can be found in everyday life: in supermarket bar code scanners, in our cars’ GPS systems, in online banking, etc. Starting with a brief introductory course on number theory in Chapter 1, which makes the book more accessible for undergraduates, the authors describe the four main application areas in Chapters...
Andersen, Torben
In Dinka, a Western Nilotic language, nouns are inflected for number and distinguish between singular and plural. The number inflection is not expressed by affixation, but by phonological alternations in the root and in such a way that the number is not directly observable, but only detectable...... through agreement. With simple native nouns, which are typically monosyllables, the number inflection is unpredictable and irregular, but some fairly common singular-plural patterns can be established, as seen in the Agar dialect. There is strong internal and external evidence that originally, many nouns...... had a marked singular and an unmarked plural. Synchronically, however, the singular is arguably the basic member of the number category as revealed by the use of the two numbers. In addition, some nouns have a collective form, which is grammatically singular. Number also plays a role in the...
CATALYSTS FOR HIGH CETANE ETHERS AS DIESEL FUELS
Kamil Klier; Richard G. Herman; Heock-Hoi Kwon; James G. C. Shen; Qisheng Ma; Robert A. Hunsicker; Andrew P. Butler; Scott J. Bollinger
2003-03-01
A tungstena-zirconia (WZ) catalyst has been investigated for coupling methanol and isobutanol to unsymmetrical ethers, i.e. methyl isobutyl ether (MIBE) and compared with earlier studied sulfated-zirconia (SZ) and Nafion-H catalysts. In all cases, the ether synthesis mechanism is a dual site S{sub N}2 process involving competitive adsorption of reactants on proximal acid sites. At low reaction temperatures, methylisobutylether (MIBE) is the predominant product. However, at temperatures >135 C the WZ catalyst is very good for dehydration of isobutanol to isobutene. The surface acid sites of the WZ catalyst and a Nafion-H catalyst were diagnosed by high resolution X-ray photoelectron spectroscopy (XPS) of N 1s shifts after adsorption of amines. Using pyridine, ethylenediamine, and triethylamine, it is shown that WZ has heterogeneous strong Broensted acid sites. Theoretical study located the transition state of the alcohol coupling reaction on proximal Broensted acid sites and accounted well for XPS core-level shifts upon surface acid-base interactions. While computations have not been carried out with WZ, it is shown that the SZ catalyst is a slightly stronger acid than CF{sub 3}SO{sub 3}H (a model for Nafion-H) by 1.3-1.4 kcal/mol. A novel sulfated zirconia catalyst having proximal strong Broensted acid sites was synthesized and shown to have significantly enhanced activity and high selectivity in producing MIBE or isobutene from methanol/isobutanol mixtures. The catalyst was prepared by anchoring 1,2-ethanediol bis(hydrogen sulfate) salt precursor onto zirconium hydroxide, followed by calcination to remove the -(CH{sub 2}CH{sub 2})- bridging residues.
J.Vijayasekhar,
2011-02-01
Full Text Available Signed integers are normally represented using 2’s complement representation. Addition and subtraction of signed numbers is done in the same manner as for unsigned numbers. However carry (or borrow is simple ignored. Unlike unsigned number carry (or borrow does not mean overflow or error. Doubling of a signed number can be done by shift left. However, halving of a signed number can not be done by shift right. Hence special arithmetic instruction SAR (Shift arithmetic right is needed. We have defined an alternative representation for signed numbers. Here a positive number is represented by appended a zero (0 at right. Here a negative number is represented by inverting all bits in corresponding positive number. Two signed numbers are added by adding corresponding binary representation. After that carry is added to the result. Similarly two signed numbers are subtracted by subtracting corresponding binaryrepresentation. After that borrow is subtracted. Doubling and halving is done by ROL (Rotate left and ROR (Rotate right respectively. Following are drawbacks of our system. (A Addition is done in two stages. In the first stage the numbers are added. In the second stage carry is added. Carry can not be ignored as in 2’s complement representation. (B Same holds for subtraction. (C When an odd number is halved then error results. In 2’s complement representation approximate answer appears. The advantage of our system is that entire arithmetic can be carried using ordinary logical instructions. No special instruction is needed. In 2’s complement representation a special instruction SAR is needed. This instruction is not used for any other purpose.
Elvik, Rune; Bjørnskau, Torkel
2016-01-01
This paper presents a systematic review and meta-analysis of studies that have estimated the relationship between the number of accidents involving motor vehicles and cyclists or pedestrians and the volume of motor vehicles, cyclists and pedestrians. A key objective of most of these studies has...... been to determine if there is a safety-in-numbers effect. There is safety-in-numbers if the number of accidents increases less than proportionally to traffic volume (for motor vehicles, pedestrians and cyclists). All studies reviewed in the paper are multivariate accident prediction models, estimating...... regression coefficients that show how the number of accidents depends on the conflicting flows (pedestrians, cyclists, motor vehicles), as well as (in some of the models) other factors that influence the number of accidents. Meta-analysis of regression coefficients involves methodological problems, which...
Quantum Random Number Generators
Herrero-Collantes, Miguel; Garcia-Escartin, Juan Carlos
2016-01-01
Random numbers are a fundamental resource in science and engineering with important applications in simulation and cryptography. The inherent randomness at the core of quantum mechanics makes quantum systems a perfect source of entropy. Quantum random number generation is one of the most mature quantum technologies with many alternative generation methods. We discuss the different technologies in quantum random number generation from the early devices based on radioactive decay to the multipl...
Kruecken, R.
2010-01-01
The nuclear shell model is a benchmark for the description of the structure of atomic nuclei. The magic numbers associated with closed shells have long been assumed to be valid across the whole nuclear chart. Investigations in recent years of nuclei far away from nuclear stability at facilities for radioactive ion beams have revealed that the magic numbers may change locally in those exotic nuclei leading to the disappearance of classic shell gaps and the appearance of new magic numbers. Thes...
Cantlon, Jessica F.; Platt, Michael L.; Brannon, Elizabeth M.
2009-01-01
In a world without numbers, we would be unable to build a skyscraper, hold a national election, plan a wedding, or pay for a chicken at the market. The numerical symbols used in all these behaviors build on the approximate number system (ANS) which represents the number of discrete objects or events as a continuous mental magnitude. In this review, we first discuss evidence that the ANS bears a set of behavioral and brain signatures that are universally displayed across animal species, human ...
Divisibility of characteristic numbers
Borghesi, Simone
2009-01-01
We use homotopy theory to define certain rational coefficients characteristic numbers with integral values, depending on a given prime number q and positive integer t. We prove the first nontrivial degree formula and use it to show that existence of morphisms between algebraic varieties for which these numbers are not divisible by q give information on the degree of such morphisms or on zero cycles of the target variety.
Suetens, Sigrid; Galbo-Jørgensen, Claus B.; Tyran, Jean-Robert Karl
2015-01-01
formalized in recent behavioral theory. In particular, players tend to bet less on numbers that have been drawn in the preceding week, as suggested by the ‘gambler’s fallacy’, and bet more on a number if it was frequently drawn in the recent past, consistent with the ‘hot-hand fallacy’.......We investigate the ‘law of small numbers’ using a data set on lotto gambling that allows us to measure players’ reactions to draws. While most players pick the same set of numbers week after week, we find that those who do change react on average as predicted by the law of small numbers as...
Multispecies quantum Hurwitz numbers
Harnad, J.
2014-01-01
The construction of hypergeometric 2D Toda $\\tau$-functions as generating functions for quantum Hurwitz numbers is extended here to multispecies families. Both the enumerative geometrical significance of these multispecies quantum Hurwitz numbers as weighted enumerations of branched coverings of the Riemann sphere and their combinatorial significance in terms of weighted paths in the Cayley graph of $S_n$ are derived.
Freitag, Herta Taussig; Freitag, Arthur H.
The development of number concepts from prehistoric time to the present day are presented. Section 1 presents the historical development, logical development, and the infinitude of numbers. Section 2 focuses on non-positional and positional numeration systems. Section 3 compares historical and modern techniques and devices for computation. Section…
Onstad, Torgeir
1991-01-01
After a brief historical account of Leonardo Pisano Fibonacci, some basic results concerning the Fibonacci numbers are developed and proved, and entertaining examples are described. Connections are made between the Fibonacci numbers and the Golden Ratio, biological nature, and other combinatorics examples. (MDH)
Oller-Marcén, Antonio M.
2012-01-01
An integer $n$ is said to be \\textit{arithmetic} if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an application, in Section 2, we give an interesting characterization of Mersenne numbers.
First Graders' Number Knowledge
Thomas, Jonathan N.; Tabor, Pamela D.; Wright, Robert J.
2010-01-01
As young children make sense of mathematics, they begin to see with new eyes. What once was uncertain may now be determined. Objects become countable; fingers become tools; and numbers become more than just names. Educators revel in such developments--which mark significant progress toward more sophisticated understanding of number--and work…
Jorgensen, C.B.; Suetens, S.; Tyran, J.R.
2011-01-01
We investigate the "law of small numbers" using a unique panel data set on lotto gambling. Because we can track individual players over time, we can measure how they react to outcomes of recent lotto drawings. We can therefore test whether they behave as if they believe they can predict lotto number
Suetens, Sigrid; Galbo-Jørgensen, Claus B.; Tyran, Jean-Robert Karl
2016-01-01
We investigate the ‘law of small numbers’ using a data set on lotto gambling that allows us to measure players’ reactions to draws. While most players pick the same set of numbers week after week, we find that those who do change react on average as predicted by the law of small numbers as...
Avogadro's Number Ferromagnetically
Houari, Ahmed
2010-01-01
Avogadro's number, usually denoted by N[subscript A], plays a fundamental role in both physics and chemistry. It defines the extremely useful concept of the mole, which is the base unit of the amount of matter in the international system of units. The fundamental character of this number can also be illustrated by its appearance in the definitions…
Vieru, Andrei
2012-01-01
We define and study a transform whose iterates bring to the fore interesting relations between Pisot numbers and primes. Although the relations we describe are general, they take a particular form in the Pisot limit points. We give three elegant formulae, which permit to locate on the whole semi-line all limit points that are not integer powers of other Pisot numbers.
Rugani, Rosa; Sartori, Luisa
2016-01-01
Humans show a remarkable tendency to describe and think of numbers as being placed on a mental number line (MNL), with smaller numbers located on the left and larger ones on the right. Faster responses to small numbers are indeed performed on the left side of space, while responses to large numbers are facilitated on the right side of space (spatial-numerical association of response codes, SNARC effect). This phenomenon is considered the experimental demonstration of the MNL and has been extensively replicated throughout a variety of paradigms. Nevertheless, the majority of previous literature has mainly investigated this effect by means of response times and accuracy, whereas studies considering more subtle and automatic measures such as kinematic parameters are rare (e.g., in a reaching-to-grasp movement, the grip aperture is enlarged in responding to larger numbers than in responding to small numbers). In this brief review we suggest that numerical magnitude can also affect the what and how of action execution (i.e., temporal and spatial components of movement). This evidence could have large implications in the strongly debated issue concerning the effect of experience and culture on the orientation of MNL. PMID:27524965
Kneusel, Ronald T
2015-01-01
This is a book about numbers and how those numbers are represented in and operated on by computers. It is crucial that developers understand this area because the numerical operations allowed by computers, and the limitations of those operations, especially in the area of floating point math, affect virtually everything people try to do with computers. This book aims to fill this gap by exploring, in sufficient but not overwhelming detail, just what it is that computers do with numbers. Divided into two parts, the first deals with standard representations of integers and floating point numb
Sierpinski, Waclaw
1988-01-01
Since the publication of the first edition of this work, considerable progress has been made in many of the questions examined. This edition has been updated and enlarged, and the bibliography has been revised.The variety of topics covered here includes divisibility, diophantine equations, prime numbers (especially Mersenne and Fermat primes), the basic arithmetic functions, congruences, the quadratic reciprocity law, expansion of real numbers into decimal fractions, decomposition of integers into sums of powers, some other problems of the additive theory of numbers and the theory of Gaussian
Cohn, Harvey
1980-01-01
""A very stimulating book ... in a class by itself."" - American Mathematical MonthlyAdvanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.The book is divided into three parts
Mersenne Numbers: consolidated results
Haworth, Guy McCrossan; Holmes, Steve; Hunt, David; Lake, Tom; Reddaway, Stewart
1986-01-01
This document provides and comments on the results of the Lucas-Lehmer testing and/or partial factorisation of all Mersenne Numbers Mp = 2^p-1 where p is prime and less than 100,000. Previous computations have either been confirmed or corrected. The LLT computations on the ICL DAP is the first implementation of Fast-Fermat-Number-Transform multiplication in connection with Mersenne Number testing. This paper championed the disciplines of systematically testing the Mp, and of double-so...
LeVeque, William J
1996-01-01
This excellent textbook introduces the basics of number theory, incorporating the language of abstract algebra. A knowledge of such algebraic concepts as group, ring, field, and domain is not assumed, however; all terms are defined and examples are given - making the book self-contained in this respect.The author begins with an introductory chapter on number theory and its early history. Subsequent chapters deal with unique factorization and the GCD, quadratic residues, number-theoretic functions and the distribution of primes, sums of squares, quadratic equations and quadratic fields, diopha
A. Mironov; Morozov, A; Natanzon, S.
2012-01-01
The classical Hurwitz numbers of degree n together with the Hurwitz numbers of the seamed surfaces of degree n give rise to the Klein topological field theory. We extend this construction to the Hurwitz numbers of all degrees at once. The corresponding Cardy-Frobenius algebra is induced by arbitrary Young diagrams and arbitrary bipartite graphs. It turns out to be isomorphic to the algebra of differential operators from arXiv:1210.6955 which serves a model for open-closed string theory. The o...
Corry, Leo
2015-01-01
The world around us is saturated with numbers. They are a fundamental pillar of our modern society, and accepted and used with hardly a second thought. But how did this state of affairs come to be? In this book, Leo Corry tells the story behind the idea of number from the early days of the Pythagoreans, up until the turn of the twentieth century. He presents an overview of how numbers were handled and conceived in classical Greek mathematics, in the mathematics of Islam, in European mathematics of the middle ages and the Renaissance, during the scientific revolution, all the way through to the
Professor Stewart's incredible numbers
Stewart, Ian
2015-01-01
Ian Stewart explores the astonishing properties of numbers from 1 to10 to zero and infinity, including one figure that, if you wrote it out, would span the universe. He looks at every kind of number you can think of - real, imaginary, rational, irrational, positive and negative - along with several you might have thought you couldn't think of. He explains the insights of the ancient mathematicians, shows how numbers have evolved through the ages, and reveals the way numerical theory enables everyday life. Under Professor Stewart's guidance you will discover the mathematics of codes,
Dudley, Underwood
2008-01-01
Ideal for a first course in number theory, this lively, engaging text requires only a familiarity with elementary algebra and the properties of real numbers. Author Underwood Dudley, who has written a series of popular mathematics books, maintains that the best way to learn mathematics is by solving problems. In keeping with this philosophy, the text includes nearly 1,000 exercises and problems-some computational and some classical, many original, and some with complete solutions. The opening chapters offer sound explanations of the basics of elementary number theory and develop the fundamenta
R. A. Mollin
1986-01-01
Full Text Available A powerful number is a positive integer n satisfying the property that p2 divides n whenever the prime p divides n; i.e., in the canonical prime decomposition of n, no prime appears with exponent 1. In [1], S.W. Golomb introduced and studied such numbers. In particular, he asked whether (25,27 is the only pair of consecutive odd powerful numbers. This question was settled in [2] by W.A. Sentance who gave necessary and sufficient conditions for the existence of such pairs. The first result of this paper is to provide a generalization of Sentance's result by giving necessary and sufficient conditions for the existence of pairs of powerful numbers spaced evenly apart. This result leads us naturally to consider integers which are representable as a proper difference of two powerful numbers, i.e. n=p1−p2 where p1 and p2 are powerful numbers with g.c.d. (p1,p2=1. Golomb (op.cit. conjectured that 6 is not a proper difference of two powerful numbers, and that there are infinitely many numbers which cannot be represented as a proper difference of two powerful numbers. The antithesis of this conjecture was proved by W.L. McDaniel [3] who verified that every non-zero integer is in fact a proper difference of two powerful numbers in infinitely many ways. McDaniel's proof is essentially an existence proof. The second result of this paper is a simpler proof of McDaniel's result as well as an effective algorithm (in the proof for explicitly determining infinitely many such representations. However, in both our proof and McDaniel's proof one of the powerful numbers is almost always a perfect square (namely one is always a perfect square when n≢2(mod4. We provide in §2 a proof that all even integers are representable in infinitely many ways as a proper nonsquare difference; i.e., proper difference of two powerful numbers neither of which is a perfect square. This, in conjunction with the odd case in [4], shows that every integer is representable in
Forbes, Rod; Gardiner, Peter
2004-01-01
The Ministry of Agriculture and Forestry (MAF) undertakes forecasts and projections of livestock numbers as part of the twice yearly contribution to The Treasury’s economic and fiscal updates. MAF’s Pastoral Supply Response Model (PSRM) was recently re-developed and used for the first time in the Budget Economic and Fiscal Update round of 2004. The PSRM projects annual inventory numbers as at 30 June, births and livestock numbers for slaughter. The paper discusses the PSRM, the post-model adj...
Crossley, John N
1987-01-01
This book presents detailed studies of the development of three kinds of number. 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. The dramatic change caused by the introduction of logical theories of number in the 19th century is also treated and this part ends with a non-technical account of the very latest developments in the area of Gödel's theorem. The second part is concerned with the deve
Jorgensen, C.B.; Suetens, S.; Tyran, J.R.
2011-01-01
We investigate the “law of small numbers” using a unique panel data set on lotto gambling. Because we can track individual players over time, we can measure how they react to outcomes of recent lotto drawings. We can therefore test whether they behave as if they believe they can predict lotto numbers based on recent drawings. While most players pick the same set of numbers week after week without regards of numbers drawn or anything else, we find that those who do change, act on average in th...
Weiss, Edwin
1998-01-01
Careful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis).Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract te
Wanscher, Jørgen Bundgaard; Sørensen, Majken Vildrik
2006-01-01
Random numbers are used for a great variety of applications in almost any field of computer and economic sciences today. Examples ranges from stock market forecasting in economics, through stochastic traffic modelling in operations research to photon and ray tracing in graphics. The construction of...... a model or a solution method requires certain characteristics of the random numbers used. This is usually a distribution classification, which the sequence of random numbers must fulfill; of these some are very hard to fulfill and others are next to impossible. Today mathematics allows us to...... is to generate highly uniform multidimensional draws, which are highly relevant for todays traffic models. This paper shows among others combined shuffling and scrambling seems needless, that scrambling gives the lowest correlation and that there are detectable differences between random numbers...
Department of Homeland Security — SEVIS by the Numbers is a quarterly report that highlights nonimmigrant student and exchange visitor trends, values and information using data from the Student and...
... View All Articles | Inside Life Science Home Page Genetics by the Numbers By Chelsea Toledo and Kirstie ... June 11, 2012 Scholars have been studying modern genetics since the mid-19th century, but even today ...
Farmer, David W.
2004-01-01
Comment: 11 pages, to appear in the proceedings of the school ``Recent Perspectives in Random Matrix Theory and Number Theory'' held at the Isaac Newton Institute, April 2004. Added appendix on big-O and
Quantum random number generator
Pooser, Raphael C.
2016-05-10
A quantum random number generator (QRNG) and a photon generator for a QRNG are provided. The photon generator may be operated in a spontaneous mode below a lasing threshold to emit photons. Photons emitted from the photon generator may have at least one random characteristic, which may be monitored by the QRNG to generate a random number. In one embodiment, the photon generator may include a photon emitter and an amplifier coupled to the photon emitter. The amplifier may enable the photon generator to be used in the QRNG without introducing significant bias in the random number and may enable multiplexing of multiple random numbers. The amplifier may also desensitize the photon generator to fluctuations in power supplied thereto while operating in the spontaneous mode. In one embodiment, the photon emitter and amplifier may be a tapered diode amplifier.
Strawn, Candace A.
1998-01-01
Describes LOGO's turtle graphics capabilities based on a sixth-grade classroom's activities with negative numbers and Logo programming. A sidebar explains LOGO and offers suggestions to teachers for using LOGO effectively. (LRW)
Solar Indices - Sunspot Numbers
National Oceanic and Atmospheric Administration, Department of Commerce — Collection includes a variety of indices related to solar activity contributed by a number of national and private solar observatories located worldwide. This...
Schwartz, Richard Evan
2014-01-01
In the American Mathematical Society's first-ever book for kids (and kids at heart), mathematician and author Richard Evan Schwartz leads math lovers of all ages on an innovative and strikingly illustrated journey through the infinite number system. By means of engaging, imaginative visuals and endearing narration, Schwartz manages the monumental task of presenting the complex concept of Big Numbers in fresh and relatable ways. The book begins with small, easily observable numbers before building up to truly gigantic ones, like a nonillion, a tredecillion, a googol, and even ones too huge for names! Any person, regardless of age, can benefit from reading this book. Readers will find themselves returning to its pages for a very long time, perpetually learning from and growing with the narrative as their knowledge deepens. Really Big Numbers is a wonderful enrichment for any math education program and is enthusiastically recommended to every teacher, parent and grandparent, student, child, or other individual i...
Detecting ''secret'' quantum numbers
For a general quantum mechanical system undergoing a process, it is shown that one can tell from measurements on this system whether or not the system is characterized by quantum numbers the existence of which is unknown to the observer, even though the detection equipment used by the observer is unable to distinguish among the various possible values of the ''secret'' quantum number and hence always averages over them
Methods explained: Index numbers
Peter Goodridge
2007-01-01
Attempts to explain the subtle differences in the methodologies used to construct index numbers.Many of the statistics produced by the Office for National Statistics,particularly economic statistics, are published in the form ofindices. However, there are a number of different forms of indices and this article attempts to explain the subtle differences in themethodologies used to construct them, and also factors that feed into the choice of which type of index to use. Hypothetical examplesare...
Brown, Ezra; Brunson, Cornelius
2008-01-01
Fibonacci's forgotten number is the sexagesimal number 1;22,7,42,33,4,40, which he described in 1225 as an approximation to the real root of x[superscript 3] + 2x[superscript 2] + 10x - 20. In decimal notation, this is 1.36880810785...and it is correct to nine decimal digits. Fibonacci did not reveal his method. How did he do it? There is also a…
The accuracy of CT number plots has been found lacking in several medical applications. This is of concern since the ability to compare and evaluate results on a reproducible and standard basis is essential to long term development. Apart from the technical limitations arising from the CT scanner and the data treatment, there are fundamental issues with the definition of the Hounsfield number, namely the absence of a standard photon energy and the need to specify the attenuation mechanism for standard measurements. This paper presents calculations to demonstrate the shortcomings of the present definition with a brief discussion. The remedy is straightforward, but probably of long duration as it would require an international agreement. - Highlights: ► The dependence of the CT number definition on photon energy is examined. ► Graphical examples of the CT number variation with photon energy are given. ► The influence of absorption edges and scattering on CT numbers is discussed. ► A proposal is made for an international standard devoted to CT number evaluation.
This publication lists all report number codes processed by the Office of Scientific and Technical Information. The report codes are substantially based on the American National Standards Institute, Standard Technical Report Number (STRN)-Format and Creation Z39.23-1983. The Standard Technical Report Number (STRN) provides one of the primary methods of identifying a specific technical report. The STRN consists of two parts: The report code and the sequential number. The report code identifies the issuing organization, a specific program, or a type of document. The sequential number, which is assigned in sequence by each report issuing entity, is not included in this publication. Part I of this compilation is alphabetized by report codes followed by issuing installations. Part II lists the issuing organization followed by the assigned report code(s). In both Parts I and II, the names of issuing organizations appear for the most part in the form used at the time the reports were issued. However, for some of the more prolific installations which have had name changes, all entries have been merged under the current name
Nelson, R.N. (ed.)
1985-05-01
This publication lists all report number codes processed by the Office of Scientific and Technical Information. The report codes are substantially based on the American National Standards Institute, Standard Technical Report Number (STRN)-Format and Creation Z39.23-1983. The Standard Technical Report Number (STRN) provides one of the primary methods of identifying a specific technical report. The STRN consists of two parts: The report code and the sequential number. The report code identifies the issuing organization, a specific program, or a type of document. The sequential number, which is assigned in sequence by each report issuing entity, is not included in this publication. Part I of this compilation is alphabetized by report codes followed by issuing installations. Part II lists the issuing organization followed by the assigned report code(s). In both Parts I and II, the names of issuing organizations appear for the most part in the form used at the time the reports were issued. However, for some of the more prolific installations which have had name changes, all entries have been merged under the current name.
Chromosome numbers in Bromeliaceae
Cotias-de-Oliveira Ana Lúcia Pires
2000-01-01
Full Text Available The present study reports chromosome numbers of 17 species of Bromeliaceae, belonging to the genera Encholirium, Bromelia, Orthophytum, Hohenbergia, Billbergia, Neoglaziovia, Aechmea, Cryptanthus and Ananas. Most species present 2n = 50, however, Bromelia laciniosa, Orthophytum burle-marxii and O. maracasense are polyploids with 2n = 150, 2n = 100 and 2n = 150, respectively, while for Cryptanthus bahianus, 2n = 34 + 1-4B. B chromosomes were observed in Bromelia plumieri and Hohenbergia aff. utriculosa. The chromosome number of all species was determined for the first time, except for Billbergia chlorosticta and Cryptanthus bahianus. Our data supports the hypothesis of a basic number of x = 25 for the Bromeliaceae family and decreasing aneuploidy in the genus Cryptanthus.
Alizée Dauvergne
2010-01-01
What makes the LHC the biggest particle accelerator in the world? Here are some of the numbers that characterise the LHC, and their equivalents in terms that are easier for us to imagine. Feature Number Equivalent Circumference ~ 27 km Distance covered by beam in 10 hours ~ 10 billion km a round trip to Neptune Number of times a single proton travels around the ring each second 11 245 Speed of protons first entering the LHC 299 732 500 m/s 99.9998 % of the speed of light Speed of protons when they collide 299 789 760 m/s 99.9999991 % of the speed of light Collision temperature ~ 1016 °C ove...
CONFUSION WITH TELEPHONE NUMBERS
Telecom Service
2002-01-01
he area code is now required for all telephone calls within Switzerland. Unfortunately this is causing some confusion. CERN has received complaints that incoming calls intended for CERN mobile phones are being directed to private subscribers. This is caused by mistakenly dialing the WRONG code (e.g. 022) in front of the mobile number. In order to avoid these problems, please inform your correspondents that the correct numbers are: 079 201 XXXX from Switzerland; 0041 79 201 XXXX from other countries. Telecom Service
CONFUSION WITH TELEPHONE NUMBERS
Telecom Service
2002-01-01
The area code is now required for all telephone calls within Switzerland. Unfortunately this is causing some confusion. CERN has received complaints that incoming calls intended for CERN mobile phones are being directed to private subscribers. This is caused by mistakenly dialing the WRONG code (e.g. 022) in front of the mobile number. In order to avoid these problems, please inform your correspondents that the correct numbers are: 079 201 XXXX from Switzerland; 0041 79 201 XXXX from other countries. Telecom Service
Gruber, Peter M
1987-01-01
This volume contains a fairly complete picture of the geometry of numbers, including relations to other branches of mathematics such as analytic number theory, diophantine approximation, coding and numerical analysis. It deals with convex or non-convex bodies and lattices in euclidean space, etc.This second edition was prepared jointly by P.M. Gruber and the author of the first edition. The authors have retained the existing text (with minor corrections) while adding to each chapter supplementary sections on the more recent developments. While this method may have drawbacks, it has the definit
Samuel, Pierre
2008-01-01
Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics - algebraic geometry, in particular.This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Gal
Leinster, Tom
2001-01-01
A number is perfect if it is the sum of its proper divisors; here we call a finite group `perfect' if its order is the sum of the orders of its proper normal subgroups. (This conflicts with standard terminology but confusion should not arise.) The notion of perfect group generalizes that of perfect number, since a cyclic group is perfect exactly when its order is perfect. We show that, in fact, the only abelian perfect groups are the cyclic ones, and exhibit some non-abelian perfect groups of...
Babu, K S; Al-Binni, U; Banerjee, S; Baxter, D V; Berezhiani, Z; Bergevin, M; Bhattacharya, S; Brice, S; Brock, R; Burgess, T W; Castellanos, L; Chattopadhyay, S; Chen, M-C; Church, E; Coppola, C E; Cowen, D F; Cowsik, R; Crabtree, J A; Davoudiasl, H; Dermisek, R; Dolgov, A; Dutta, B; Dvali, G; Ferguson, P; Perez, P Fileviez; Gabriel, T; Gal, A; Gallmeier, F; Ganezer, K S; Gogoladze, I; Golubeva, E S; Graves, V B; Greene, G; Handler, T; Hartfiel, B; Hawari, A; Heilbronn, L; Hill, J; Jaffe, D; Johnson, C; Jung, C K; Kamyshkov, Y; Kerbikov, B; Kopeliovich, B Z; Kopeliovich, V B; Korsch, W; Lachenmaier, T; Langacker, P; Liu, C-Y; Marciano, W J; Mocko, M; Mohapatra, R N; Mokhov, N; Muhrer, G; Mumm, P; Nath, P; Obayashi, Y; Okun, L; Pati, J C; Pattie, R W; Phillips, D G; Quigg, C; Raaf, J L; Raby, S; Ramberg, E; Ray, A; Roy, A; Ruggles, A; Sarkar, U; Saunders, A; Serebrov, A; Shafi, Q; Shimizu, H; Shiozawa, M; Shrock, R; Sikdar, A K; Snow, W M; Soha, A; Spanier, S; Stavenga, G C; Striganov, S; Svoboda, R; Tang, Z; Tavartkiladze, Z; Townsend, L; Tulin, S; Vainshtein, A; Van Kooten, R; Wagner, C E M; Wang, Z; Wehring, B; Wilson, R J; Wise, M; Yokoyama, M; Young, A R
2013-01-01
This report, prepared for the Community Planning Study - Snowmass 2013 - summarizes the theoretical motivations and the experimental efforts to search for baryon number violation, focussing on nucleon decay and neutron-antineutron oscillations. Present and future nucleon decay search experiments using large underground detectors, as well as planned neutron-antineutron oscillation search experiments with free neutron beams are highlighted.
Doyle, William R.
2010-01-01
Some say that the educators now have a gender-stratified system of higher education, with nearly 60 percent of all undergraduates being women and fewer men attending each year. The battle for gender equity for women in higher education has been a long and contentious one. In the decades since, increasing numbers of women have gone to college, to…
Trudgian, Timothy
2009-01-01
One of the difficulties in any teaching of mathematics is to bridge the divide between the abstract and the intuitive. Throughout school one encounters increasingly abstract notions, which are more and more difficult to relate to everyday experiences. This article examines a familiar approach to thinking about negative numbers, that is an…
Bell, Eric Temple
1991-01-01
From one of the foremost interpreters for lay readers of the history and meaning of mathematics: a stimulating account of the origins of mathematical thought and the development of numerical theory. It probes the work of Pythagoras, Galileo, Berkeley, Einstein, and others, exploring how ""number magic"" has influenced religion, philosophy, science, and mathematics
Schreuder, M.F.
2012-01-01
A low nephron number is, according to Brenner's hyperfiltration hypothesis, associated with hypertension, glomerular damage and proteinuria, and starts a vicious cycle that ends in renal failure over the long term. Nephron endowment is set during foetal life, and there is no formation of nephrons af
Are Occupation Numbers Observable?
Furnstahl, R. J.; Hammer, H. -W.
2001-01-01
The question of whether occupation numbers and momentum distributions of nucleons in nuclei are observables is considered from an effective field theory perspective. Field redefinitions lead to variations that imply the answer is negative, as illustrated in the interacting Fermi gas at low density. Implications for the interpretation of (e,e'p) experiments with nuclei are discussed.
McMurdo, George
1996-01-01
The expansion of the commercial Internet has encouraged the interpretation of the Internet and its uses as a potential marketing medium. Examines statistical and demographic information about the Internet including number of Internet hosts and World Wide Web servers, and estimates of Internet users; and raises questions about definitions and…
Oli Brown
2008-10-01
Full Text Available Estimates of the potential number of ‘climate changemigrants’ vary hugely. In order to persuade policymakers ofthe need to act and to provide a sound basis for appropriateresponses, there is an urgent need for better analysis, betterdata and better predictions.
This document contains information dealing with the lessons learned from the experience of nuclear plants. In this issue the authors tried to avoid the 'tyranny' of numbers and concentrated on the main lessons learned. Topics include: filtration devices for air pollution abatement, crack repair and inspection, and remote handling equipment
Two Symmetric Properties of Mersenne Numbers and Fermat Numbers
Yongjin, Shi
2013-01-01
Mersenne numbers and Fermat numbers are two hot and difficult issues in number theory. This paper constructs a special group for every positive odd number other than 1, and discovers an algorithm for determining the multiplicative order of 2 modulo q for each positive odd number q. It is worth mentioning that this paper discovers two symmetric properties of Mersenne numbers and Fermat numbers.
MacCarron, Pádraig; Dunbar, Robin
2016-01-01
The social brain hypothesis predicts that humans have an average of about 150 relationships at any given time. Within this 150, there are layers of friends of an ego, where the number of friends in a layer increases as the emotional closeness decreases. Here we analyse a mobile phone dataset, firstly, to ascertain whether layers of friends can be identified based on call frequency. We then apply different clustering algorithms to break the call frequency of egos into clusters and compare the number of alters in each cluster with the layer size predicted by the social brain hypothesis. In this dataset we find strong evidence for the existence of a layered structure. The clustering yields results that match well with previous studies for the innermost and outermost layers, but for layers in between we observe large variability.
Marco Ruffino
2001-12-01
Full Text Available In this paper I discuss the intuition behind Frege's and Russell's definitions of numbers as sets, as well as Benacerraf's criticism of it. I argue that Benacerraf's argument is not as strong as some philosophers tend to think. Moreover, I examine an alternative to the Fregean-Russellian definition of numbers proposed by Maddy, and point out some problems faced by it.Neste artigo discuto a intuição subjacente à definição de n∨meros como conjuntos proposta por Frege e Russell, assim como a crítica de Benacerraf a esta definição. Eu tento mostrar que o argumento de Benacerraf não é tão forte como alguns filósofos o tomaram. Adicionalmente, examino uma alternativa à definição de Frege e Russell proposta por Maddy, e indico algumas dificuldades encontrada pela mesma.
MacCarron, Pádraig; Kaski, Kimmo; Dunbar, Robin
2016-01-01
The social brain hypothesis predicts that humans have an average of about 150 relationships at any given time. Within this 150, there are layers of friends of an ego, where the number of friends in a layer increases as the emotional closeness decreases. Here we analyse a mobile phone dataset, firstly, to ascertain whether layers of friends can be identified based on call frequency. We then apply different clustering algorithms to break the call frequency of egos into clusters and compare the ...
Chromosome numbers in Bromeliaceae
2000-01-01
The present study reports chromosome numbers of 17 species of Bromeliaceae, belonging to the genera Encholirium, Bromelia, Orthophytum, Hohenbergia, Billbergia, Neoglaziovia, Aechmea, Cryptanthus and Ananas. Most species present 2n = 50, however, Bromelia laciniosa, Orthophytum burle-marxii and O. maracasense are polyploids with 2n = 150, 2n = 100 and 2n = 150, respectively, while for Cryptanthus bahianus, 2n = 34 + 1-4B. B chromosomes were observed in Bromelia plumieri and Hohenbergia aff. u...
M. H. Hooshmand
2010-02-01
Full Text Available In this paper we first define homorooty between two integer numbers and study some of their properties. There after we shall state some applications of the homorooty in studying and solving some Diophantine equations and systems, as an interesting anduseful elementary method. Also by the homorooty, we state and prove the necessary and sufficient conditions for existence of finite solutions in a special case of the quartic equation and evaluate the bounds of its solutions.
Motohashi, Yoichi
2005-01-01
Prime numbers or primes are man's eternal treasures that have been cherished for several millennia, until today. As their academic ancestors in ancient Mesopotamia, many mathematicians are still trying hard to see primes better. I shall relate here a part of my impressions that I have gathered in a corner of my mind through my own research and excursions outside my profession. This is in essence a translation of my Japanese article that was prepared for my public talk at the general assembly ...
Carlitz q-Bernoulli numbers and q-Stirling numbers
Kim, Taekyun
2007-01-01
In this paper we consider carlitz q-Bernoulli numbers and q-stirling numbers of the first and the second kind. From these numbers we derive many interesting formulae associated with q-Bernoulli numbers.
Thurston, H A
2007-01-01
The teaching of mathematics has undergone extensive changes in approach, with a shift in emphasis from rote memorization to acquiring an understanding of the logical foundations and methodology of problem solving. This book offers guidance in that direction, exploring arithmetic's underlying concepts and their logical development.This volume's great merit lies in its wealth of explanatory material, designed to promote an informal and intuitive understanding of the rigorous logical approach to the number system. The first part explains and comments on axioms and definitions, making their subseq
LeVeque, William J
2002-01-01
Classic two-part work now available in a single volume assumes no prior theoretical knowledge on reader's part and develops the subject fully. Volume I is a suitable first course text for advanced undergraduate and beginning graduate students. Volume II requires a much higher level of mathematical maturity, including a working knowledge of the theory of analytic functions. Contents range from chapters on binary quadratic forms to the Thue-Siegel-Roth Theorem and the Prime Number Theorem. Includes numerous problems and hints for their solutions. 1956 edition. Supplementary Reading. List of Symb
Neukirch, Jürgen; Wingberg, Kay
2013-01-01
The second edition is a corrected and extended version of the first. It is a textbook for students, as well as a reference book for the working mathematician, on cohomological topics in number theory. The first part provides algebraic background: cohomology of profinite groups, duality groups, free products, and homotopy theory of modules, with new sections on spectral sequences and on Tate cohomology of profinite groups. The second part deals with Galois groups of local and global fields: Tate duality, structure of absolute Galois groups of local fields, extensions with restricted ramificatio
Upreti, U. C.
1997-12-01
Nomogram construction using the parabolic relationship f0F2 = a0+a1R12+a2R122 between monthly median f0F2 and running average sunspot number (RASSN) R12 values has been described; here a0, a1 and a2 are the best fit coefficients. The nomogram can give the required local effective sunspot number (LESSN) values corresponding to any observed value of f0F2. Transforming the f0F2-RASSN relation to the form R122+pR12+q = 0 [where p = a1/a2 and q = (a0-f0F2)/a2], a practical method for the preparation of a single nomogram for f0F2-RASSN has been described and the problem of very high and very low values of the variables has also been dealt with successfully. A single nomogram for a large range of variables, namely, f0F2, a0, a1, and a2 has been obtained so that one can easily find LESSN values at any location, season, and time. The nomogram tends to minimize the errors in LESSN calculations at all levels of solar activity.
7 CFR 29.9205 - Identification number (farm serial number).
2010-01-01
... 7 Agriculture 2 2010-01-01 2010-01-01 false Identification number (farm serial number). 29.9205 Section 29.9205 Agriculture Regulations of the Department of Agriculture AGRICULTURAL MARKETING SERVICE... number (farm serial number). The serial number assigned to an individual farm by the appropriate...
Qubits from Number States and Bell Inequalities for Number Measurements
Larsson, Jan-Ake
2002-01-01
Bell inequalities for number measurements are derived via the observation that the bits of the number indexing a number state are proper qubits. Violations of these inequalities are obtained from the output state of the nondegenerate optical parametric amplifier.
Lepton family number violation
Herczeg, P.
1999-03-01
At present there is evidence from neutrino oscillation searches that the neutrinos are in fact massive particles and that they mix. If confirmed, this would imply that the conservation of LFN is not exact. Lepton family number violation (LFNV) has been searched for with impressive sensitivities in many processes involving charged leptons. The present experimental limits on some of them (those which the author shall consider here) are shown in Table 1. These stringent limits are not inconsistent with the neutrino oscillation results since, given the experimental bounds on the masses of the known neutrinos and the neutrino mass squared differences required by the oscillation results, the effects of LFNV from neutrino mixing would be too small to be seen elsewhere (see Section 2). The purpose of experiments searching for LFNV involving the charged leptons is to probe the existence of other sources of LFNV. Such sources are present in many extensions of the SM. In this lecture the author shall discuss some of the possibilities, focusing on processes that require muon beams. Other LFNV processes, such as the decays of the kaons and of the {tau}, provide complementary information. In the next Section he shall consider some sources of LFNV that do not require an extension of the gauge group of the SM (the added leptons or Higgs bosons may of course originate from models with extended gauge groups). In Section 3 he discusses LFNV in left-right symmetric models. In Section 4 he considers LFNV in supersymmetric models, first in R-parity conserving supersymmetric grand unified models, and then in the minimal supersymmetric standard model with R-parity violation. The last section is a brief summary of the author`s conclusions.
Furuya, Masako
2004-01-01
We show a lower estimate of the Milnor number of an isolated hypersurface singularity, via its Newton number. We also obtain analogous estimate of the Milnor number of an isolated singularity of a similar complete intersection variety.
Neutrino number of the universe
The influence of grand unified theories on the lepton number of the universe is reviewed. A scenario is presented for the generation of a large (>> 1) lepton number and a small (<< 1) baryon number. 15 references
Series of Reciprocal Triangular Numbers
Bruckman, Paul; Dence, Joseph B.; Dence, Thomas P.; Young, Justin
2013-01-01
Reciprocal triangular numbers have appeared in series since the very first infinite series were summed. Here we attack a number of subseries of the reciprocal triangular numbers by methodically expressing them as integrals.
Automatic Number Plate Recognition System
Rajshree Dhruw; Dharmendra Roy
2014-01-01
Automatic Number Plate Recognition (ANPR) is a mass surveillance system that captures the image of vehicles and recognizes their license number. The objective is to design an efficient automatic authorized vehicle identification system by using the Indian vehicle number plate. In this paper we discus different methodology for number plate localization, character segmentation & recognition of the number plate. The system is mainly applicable for non standard Indian number plates by recognizing...
Construction of the real numbers
Grešak, Rozalija
2013-01-01
In this thesis, there are described two standard constructions of the real numbers, these are the construction of real numbers via Dedekind cuts and the construction with metric fill of the rational numbers. Rational numbers are already a linearly ordered commutative field, so we first list the axioms of a linearly ordered commutative field. Then we take a look to the Dedekind's axiom, which only applies to real numbers and distinguishes between real and rational numbers. In thesis, there are...
Pauli Pascal Pyramids, Pauli Fibonacci Numbers, and Pauli Jacobsthal Numbers
Horn, Martin Erik
2007-01-01
The three anti-commutative two-dimensional Pauli Pascal triangles can be generalized into multi-dimensional Pauli Pascal hyperpyramids. Fibonacci and Jacobsthal numbers are then generalized into Pauli Fibonacci numbers, Pauli Jacobsthal numbers, and Pauli Fibonacci numbers of higher order. And the question is: are Pauli rabbits killer rabbits?
Perfect numbers - a lower bound for an odd perfect number
Berdellima, Arian
2011-01-01
In this work we construct a lower bound for an odd perfect number in terms of the number of its distinct prime factors. We further generalize the formula for any natural number for which the number of its distinct prime factors is known.
Cordial Languages and Cordial Numbers
J.Baskar Babujee; L. SHOBANA
2012-01-01
The concept of cordial labeling in graphs motivated us to introduce cordial words, cordial languages and cordial numbers. We interpret the notion of cordial labeling in Automata and thereby study the corresponding languages. In this paper we develop a new sequence of numbers called the cordial numbers in number theory using the labeling techniques in graph theory on automata theory.
Cordial Languages and Cordial Numbers
J. Baskar BABUJEE
2012-01-01
Full Text Available The concept of cordial labeling in graphs motivated us to introduce cordial words, cordial languages and cordial numbers. We interpret the notion of cordial labeling in Automata and thereby study the corresponding languages. In this paper we develop a new sequence of numbers called the cordial numbers in number theory using the labeling techniques in graph theory on automata theory.
Representing Numbers: Prime and Irrational
Zazkis, Rina
2005-01-01
This article draws an analogy between prime and irrational numbers with respect to how these numbers are defined and how they are perceived by learners. Excerpts are presented from two research studies: a study on understanding prime numbers by pre-service elementary school teachers and a study on understanding irrational numbers by pre-service…
Identification numbers for chemical structures
Several identification (ID) numbers for chemical structures (connectivity ID number, prime ID number, weighted ID number) are analyzed and tested until a counterexample (a pair of structures with the same ID number) is found. The analysis is carried out for acyclic structures with up to 20 atoms, trees with up to 20 points, benzenoid graphs and polyhexes with up to 10 hexagons, and all connected graphs with up to 6 points. Although all the (chemical) ID numbers studied are highly selective for many families of (molecular) graphs, none of them are unique; in all three cases the counterexamples are found. However, the greatest discriminative power is shown by the weighted ID number
THE RELATIONSHIP BETWEEN NUMBER NAMES AND NUMBER CONCEPTS
Ejersbo, Lisser Rye; Misfeldt, Morten
regularity or irregularity of number naming affects children’s formation of number concepts and arithmetic performance. We investigate this issue by reviewing relevant literature and undertaking a design research project addressing the specific irregularities of the Danish number names. In this project, a...... second, regular set of number names is introduced in primary school. The study’s findings suggest that the regularity of number names influences the development of number concepts and creates a positive impact on the understanding of the base-10 system....
Prime numbers: periodicity, chaos, noise
Bershadskii, A.
2011-01-01
Logarithmic gaps have been used in order to find a periodic component of the sequence of prime numbers, hidden by a random noise (stochastic or chaotic). The recovered period for the sequence of the first 10000 prime numbers is equal to 8\\pm1 (subject to the prime number theorem). For small and moderate values of the prime numbers (first 2000 prime numbers) this result has been directly checked using the twin prime killing method.
Computational Complexity on Signed Numbers
Jaeger, Stefan
2011-01-01
This paper presents a new representation of natural numbers and discusses its consequences for computability and computational complexity. The paper argues that the introduction of the first Peano axiom in the traditional definition of natural numbers is not essential. It claims that natural numbers remain usable in traditional ways without assuming the existence of at least one natural number. However, the uncertainty about the existence of natural numbers translates into every computation a...
Countability of the Real Numbers
Vlahovic, Slavica; Vlahovic, Branislav
2004-01-01
The proofs that the real numbers are denumerable will be shown, i.e., that there exists one-to-one correspondence between the natural numbers $N$ and the real numbers $\\Re$. The general element of the sequence that contains all real numbers will be explicitly specified, and the first few elements of the sequence will be written. Remarks on the Cantor's nondenumerability proofs of 1873 and 1891 that the real numbers are noncountable will be given.
Chromatic number of graphs and edge Folkman numbers
Nenov, Nedyalko Dimov
2010-01-01
In the paper we give a lower bound for the number of vertices of a given graph using its chromatic number. We find the graphs for which this bound is exact. The results are applied in the theory of Foklman numbers.
A Relation between Prime Numbers and Twin Prime Numbers
Ergin, A.
2001-01-01
Every mathematician has been concerned with prime numbers, and has metwith mysterious surprises about them. Besides intuition, using empirical methods has an important role to findrelations between prime numbers. A relation between any prime numberand any twin prime number has been obtained.
Multiple Bracket Function, Stirling Number, and Lah Number Identities
Coskun, Hasan
2012-01-01
The author has constructed multiple analogues of several families of combinatorial numbers in a recent article, including the bracket symbol, and the Stirling numbers of the first and second kind. In the present paper, a multiple analogue of another sequence, the Lah numbers, is developed, and certain associated identities and significant properties of all these sequences are constructed.
How Spencer Made Number: First Uses of the Number Words
Mix, Kelly S.
2009-01-01
This article describes the development of number concepts between infancy and early childhood. It is based on a diary study that tracked number word use in a child from 12 to 38 months of age. Number words appeared early in the child's vocabulary, but accurate reference to specific numerosities evolved gradually over the entire 27-month period.…
GRAPHS WHOSE CIRCULAR CLIQUE NUMBER EQUAL THE CLIQUE NUMBER
XU Baogang; ZHOU Xinghe
2005-01-01
The circular clique number of a graph G is the maximum fractional k/d such that Gkd admits a homomorphism to G. In this paper, we give some sufficient conditions for graphs whose circular clique number equal the clique number, we also characterize the K1,3-free graphs and planar graphs with the desired property.
The concrete theory of numbers: initial numbers and wonderful properties of numbers repunit
Tarasov, Boris V.
2007-01-01
In this work initial numbers and repunit numbers have been studied. All numbers have been considered in a decimal notation. The problem of simplicity of initial numbers has been studied. Interesting properties of numbers repunit are proved: $gcd(R_a, R_b) = R_{gcd(a,b)}$; $R_{ab}/(R_aR_b)$ is an integer only if $gcd(a,b) = 1$, where $a\\geq1$, $b\\geq1$ are integers. Dividers of numbers repunit, are researched by a degree of prime number.
One of the problems in creating of computers based on residue number system (RNS) is a problem of numbers translation from positional number system into the RNS and back. Accordingly, one approach to solve this problem is to choose the values of RNS bases. It is possible that this approach will help to compare the current value of numbers and determine the sign, without converting them to the positional number system
Poison control center - emergency number
... ANYWHERE IN THE UNITED STATES This national hotline number will let you talk to experts in poisoning. ... centers in the United States use this national number. You should call if you have any questions ...
Butterflies and topological quantum numbers
Avron, J. E.; Osadchy, D.
2001-01-01
The Hofstadter model illustrates the notion of topological quantum numbers and how they account for the quantization of the Hall conductance. It gives rise to colorful fractal diagrams of butterflies where the colors represent the topological quantum numbers.
Bell Numbers, Determinants and Series
P K Saikia; Deepak Subedi
2013-05-01
In this article, we study Bell numbers and Uppuluri Carpenter numbers. We obtain various expressions and relations between them. These include polynomial recurrences and expressions as determinants of certain matrices of binomial coefficients.
Ethnolinguistic Peculiarities of Sacred Numbers
Nurgul Abdrazakova
2012-11-01
Full Text Available The article deals with ethnolinguistic peculiarities of sacred numbers in Kazakh and English languages, reflecting the spiritual life, traditions and customs. The authors examine the sacred numbers in fixed expressions compared languages.
The theory of algebraic numbers
Pollard, Harry
1975-01-01
An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.
Saveliev Peter
2005-01-01
Suppose , are manifolds, are maps. The well-known coincidence problem studies the coincidence set . The number is called the codimension of the problem. More general is the preimage problem. For a map and a submanifold of , it studies the preimage set , and the codimension is . In case of codimension , the classical Nielsen number is a lower estimate of the number of points in changing under homotopies of , and for an arbitrary codimension, of the number of components of . We extend t...
The status of Cantorian numbers
Frápolli, María J.
1992-01-01
A critical evaluation of Cantor's number conception is undertaken against which the interpretations by Wang and Hallett of Cantoran set theory are measured. Wang takes Cantor's theory to tend to be a theory of numbers rather than a theory of sets, while Hallett takes Cantor as proposing an ordinal theory of cardinal numbers which however permits Cantor to accept ordinal numbers as given without defining them. The evidence presented, however, shows that Cantor conceived numbe...
Periods and elementary real numbers
Yoshinaga, Masahiko
2008-01-01
The periods, introduced by Kontsevich and Zagier, form a class of complex numbers which contains all algebraic numbers and several transcendental quantities. Little has been known about qualitative properties of periods. In this paper, we compare the periods with hierarchy of real numbers induced from computational complexities. In particular we prove that periods can be effectively approximated by elementary rational Cauchy sequences. As an application, we exhibit a computable real number wh...
Distribution theory of algebraic numbers
Yang, Chung-Chun
2008-01-01
The book timely surveys new research results and related developments in Diophantine approximation, a division of number theory which deals with the approximation of real numbers by rational numbers. The book is appended with a list of challenging open problems and a comprehensive list of references. From the contents: Field extensions Algebraic numbers Algebraic geometry Height functions The abc-conjecture Roth''s theorem Subspace theorems Vojta''s conjectures L-functions.
Elementary number theory with programming
Lewinter, Marty
2015-01-01
A successful presentation of the fundamental concepts of number theory and computer programming Bridging an existing gap between mathematics and programming, Elementary Number Theory with Programming provides a unique introduction to elementary number theory with fundamental coverage of computer programming. Written by highly-qualified experts in the fields of computer science and mathematics, the book features accessible coverage for readers with various levels of experience and explores number theory in the context of programming without relying on advanced prerequisite knowledge and con
The world of pentagonal numbers
Črep, Polona
2014-01-01
The master’s thesis accurately discusses the subject of the world of pentagonal numbers. First chapter generally describes what exactly power series are, when the power series converge and how do we calculate with power series. Further there is written what is generating function. Derivations of a generating function for pentagonal numbers have been made as well as for inverse pentagonal numbers etc. As the figurate numbers intertwine each other in some way the connection between pentagonal n...
Linear or Exponential Number Lines
Stafford, Pat
2011-01-01
Having decided to spend some time looking at one's understanding of numbers, the author was inspired by "Alex's Adventures in Numberland," by Alex Bellos to look at one's innate appreciation of number. Bellos quotes research studies suggesting that an individual's natural appreciation of numbers is more likely to be exponential rather than linear,…
Random Numbers and Quantum Computers
McCartney, Mark; Glass, David
2002-01-01
The topic of random numbers is investigated in such a way as to illustrate links between mathematics, physics and computer science. First, the generation of random numbers by a classical computer using the linear congruential generator and logistic map is considered. It is noted that these procedures yield only pseudo-random numbers since…
Picard numbers of quintic surfaces
Schutt, M.
2014-01-01
We solve the Picard number problem for complex quintic surfaces by proving that every number between 1 and 45 occurs as Picard number of a quintic surface over the rationals. Our main technique consists in arithmetic deformations of Delsarte surfaces, but we also use K3 surfaces and wild automorphisms.
Data Compression with Prime Numbers
Chalmers, Gordon
2005-01-01
A compression algorithm is presented that uses the set of prime numbers. Sequences of numbers are correlated with the prime numbers, and labeled with the integers. The algorithm can be iterated on data sets, generating factors of doubles on the compression.
Xue-gang Chen
2007-01-01
A dominating set \\(S\\) in a graph \\(G\\) is a tree dominating set of \\(G\\) if the subgraph induced by \\(S\\) is a tree. The tree domatic number of \\(G\\) is the maximum number of pairwise disjoint tree dominating sets in \\(V(G)\\). First, some exact values of and sharp bounds for the tree domatic number are given. Then, we establish a sharp lower bound for the number of edges in a connected graph of given order and given tree domatic number, and we characterize the extremal graphs. Finally, we sh...
Generalized Bernoulli-Hurwitz numbers and the universal Bernoulli numbers
The three fundamental properties of the Bernoulli numbers, namely, the von Staudt-Clausen theorem, von Staudt's second theorem, and Kummer's original congruence, are generalized to new numbers that we call generalized Bernoulli-Hurwitz numbers. These are coefficients in the power series expansion of a higher-genus algebraic function with respect to a suitable variable. Our generalization differs strongly from previous works. Indeed, the order of the power of the modulus prime in our Kummer-type congruences is exactly the same as in the trigonometric function case (namely, Kummer's own congruence for the original Bernoulli numbers), and as in the elliptic function case (namely, H. Lang's extension for the Hurwitz numbers). However, in other past results on higher-genus algebraic functions, the modulus was at most half of its value in these classical cases. This contrast is clarified by investigating the analogue of the three properties above for the universal Bernoulli numbers. Bibliography: 34 titles.
Children's mappings between number words and the approximate number system.
Odic, Darko; Le Corre, Mathieu; Halberda, Justin
2015-05-01
Humans can represent number either exactly--using their knowledge of exact numbers as supported by language, or approximately--using their approximate number system (ANS). Adults can map between these two systems--they can both translate from an approximate sense of the number of items in a brief visual display to a discrete number word estimate (i.e., ANS-to-Word), and can generate an approximation, for example by rapidly tapping, when provided with an exact verbal number (i.e., Word-to-ANS). Here we ask how these mappings are initially formed and whether one mapping direction may become functional before the other during development. In two experiments, we gave 2-5 year old children both an ANS-to-Word task, where they had to give a verbal number response to an approximate presentation (i.e., after seeing rapidly flashed dots, or watching rapid hand taps), and a Word-to-ANS task, where they had to generate an approximate response to a verbal number request (i.e., rapidly tapping after hearing a number word). Replicating previous results, children did not successfully generate numerically appropriate verbal responses in the ANS-to-Word task until after 4 years of age--well after they had acquired the Cardinality Principle of verbal counting. In contrast, children successfully generated numerically appropriate tapping sequences in the Word-to-ANS task before 4 years of age--well before many understood the Cardinality Principle. We further found that the accuracy of the mapping between the ANS and number words, as captured by error rates, continues to develop after this initial formation of the interface. These results suggest that the mapping between the ANS and verbal number representations is not functionally bidirectional in early development, and that the mapping direction from number representations to the ANS is established before the reverse. PMID:25721021
Pell Numbers, Pell-Lucas Numbers and Modular Group
Q. Mushtaq; U. Hayat
2007-01-01
We show that the matrix A(g), representing the element g = ((xy)2(xy2)2)m (m≥) of the modular group PSL(2,Z)=(x,y:x2=y3=1),where x:z →-1/z and y :z → -1/z, is a 2×2 symmetric matrix whose entries are Pell numbers and whose trace is a Pell-Lucas number. If g fixes elements of Q(√d), where d is a square-free positive number, on the circuit of the coset diagram, then d ＝ 2 and there are only four pairs of ambiguous numbers on the circuit.
Florentin Smarandache
2015-01-01
In this paper, we introduce for the first time the neutrosophic quadruple numbers (of the form + + + ) and the refined neutrosophic quadruple numbers. Then we define an absorbance law, based on a prevalence order, both of them in order to multiply the neutrosophic components ,, or their sub-components ,, and thus to construct the multiplication of neutrosophic quadruple numbers.
Reprint Series: Prime Numbers and Perfect Numbers. RS-2.
Schaaf, William L., Ed.
This is one in a series of SMSG supplementary and enrichment pamphlets for high school students. This series makes available expository articles which appeared in a variety of mathematical periodicals. Topics covered include: (1) the prime numbers; (2) mathematical sieves; (3) the factorgram; and (4) perfect numbers. (MP)
Quantum Random Number Generator using Photon-Number Path Entanglement
Kwon, Osung; Cho, Young-Wook; Kim, Yoon-Ho
2008-01-01
We report a novel quantum random number generator based on the photon-number$-$path entangled state which is prepared via two-photon quantum interference at a beam splitter. The randomness in our scheme is of truly quantum mechanical origin as it comes from the projection measurement of the entangled two-photon state. The generated bit sequences satisfy the standard randomness test.
Dynamic Virtual Credit Card Numbers
Molloy, Ian; Li, Jiangtao; Li, Ninghui
Theft of stored credit card information is an increasing threat to e-commerce. We propose a dynamic virtual credit card number scheme that reduces the damage caused by stolen credit card numbers. A user can use an existing credit card account to generate multiple virtual credit card numbers that are either usable for a single transaction or are tied with a particular merchant. We call the scheme dynamic because the virtual credit card numbers can be generated without online contact with the credit card issuers. These numbers can be processed without changing any of the infrastructure currently in place; the only changes will be at the end points, namely, the card users and the card issuers. We analyze the security requirements for dynamic virtual credit card numbers, discuss the design space, propose a scheme using HMAC, and prove its security under the assumption the underlying function is a PRF.
Nieder, Andreas
2016-06-01
Humans and non-human primates share an elemental quantification system that resides in a dedicated neural network in the parietal and frontal lobes. In this cortical network, 'number neurons' encode the number of elements in a set, its cardinality or numerosity, irrespective of stimulus appearance across sensory motor systems, and from both spatial and temporal presentation arrays. After numbers have been extracted from sensory input, they need to be processed to support goal-directed behaviour. Studying number neurons provides insights into how information is maintained in working memory and transformed in tasks that require rule-based decisions. Beyond an understanding of how cardinal numbers are encoded, number processing provides a window into the neuronal mechanisms of high-level brain functions. PMID:27150407
q-Bernoulli Numbers Associated with q-Stirling Numbers
Taekyun Kim
2008-01-01
We consider Carlitz q-Bernoulli numbers and q-Stirling numbers of the first and the second kinds. From the properties of q-Stirling numbers, we derive many interesting formulas associated with Carlitz q-Bernoulli numbers. Finally, we will prove ÃŽÂ²n,q=Ã¢ÂˆÂ‘m=0nÃ¢ÂˆÂ‘k=mn1/(1-q)n+m-kÃ¢ÂˆÂ‘d0+Ã¢Â‹Â¯+dk=n-kqÃ¢ÂˆÂ‘i=0kidis1,q(k,m)(-1)n-m((m+1)/[m+1]q), where ÃŽÂ²n,q are called Carlitz q-Bernoulli numbers.
Number Comparison and Number Line Estimation Rely on Different Mechanisms
Delphine Sasanguie
2013-12-01
Full Text Available The performance in comparison and number line estimation is assumed to rely on the same underlying representation, similar to a compressed mental number line that becomes more linear with age. We tested this assumption explicitly by examining the relation between the linear/logarithmic fit in a non-symbolic number line estimation task and the size effect (SE in a non-symbolic comparison task in first-, second-, and third graders. In two experiments, a correlation between the estimation pattern in number line estimation and the SE in comparison was absent. An ANOVA showed no difference between the groups of children with a linear or a logarithmic representation considering their SE in comparison. This suggests that different mechanisms underlie both basic number processing tasks.
An introduction to number theory.
Everest, G.; Ward, T.
2005-01-01
An Introduction to Number Theory provides an introduction to the main streams of number theory. Starting with the unique factorization property of the integers, the theme of factorization is revisited several times throughout the book to illustrate how the ideas handed down from Euclid continue to reverberate through the subject. In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory. The re...
Expected gaps between prime numbers
Holt, Fred B.
2007-01-01
We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations. Using this recursion we can estimate the numbers of a gap or of a constellation that occur between a prime and its square. This recursion also has explicit implications for open questions about gaps between prime numbers, including three questions posed by Erd\\...
Iovane, Gerardo
2007-01-01
In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime candidates are obtained in terms of the first perfect number. The asymptotic behaviour is also considered. We obtain for the first time an explicit relation for generating the full set P of prime numbers smaller than n or equal to n.
Correlations between large prime numbers
Bershadskii, A.
2011-01-01
It is shown that short-range correlations between large prime numbers (~10^5 and larger) have a Poissonian nature. Correlation length \\zeta = 4.5 for the primes ~10^5 and it is increasing logarithmically according to the prime number theorem. For moderate prime numbers (~10^4) the Poissonian distribution is not applicable while the correlation length surprisingly continues to follow to the logarithmical law. A chaotic (deterministic) hypothesis has been suggested to explain the moderate prime...
An Introduction to Zoli Numbers
Zotos, Kostas; Litke, Andreas
2005-01-01
There have been many theories about the paradoxes of numbers, but this is far and away more paradoxical than most. In this paper we will present the Zoli Numbers which have some innovative characteristics. The basic concept of these numbers is that they don't follow strictly any Mathematical rule. They are called Zoli from the names of Zotos and Litke. We are going to see some examples with the Zoli Programming Language and reveal the connection with other mathematical topics.
Number development and developmental dyscalculia.
von Aster, Michael G; Shalev, Ruth S
2007-11-01
There is a growing consensus that the neuropsychological underpinnings of developmental dyscalculia (DD) are a genetically determined disorder of 'number sense', a term denoting the ability to represent and manipulate numerical magnitude nonverbally on an internal number line. However, this spatially-oriented number line develops during elementary school and requires additional cognitive components including working memory and number symbolization (language). Thus, there may be children with familial-genetic DD with deficits limited to number sense and others with DD and comorbidities such as language delay, dyslexia, or attention-deficit-hyperactivity disorder. This duality is supported by epidemiological data indicating that two-thirds of children with DD have comorbid conditions while one-third have pure DD. Clinically, they differ according to their profile of arithmetic difficulties. fMRI studies indicate that parietal areas (important for number functions), and frontal regions (dominant for executive working memory and attention functions), are under-activated in children with DD. A four-step developmental model that allows prediction of different pathways for DD is presented. The core-system representation of numerical magnitude (cardinality; step 1) provides the meaning of 'number', a precondition to acquiring linguistic (step 2), and Arabic (step 3) number symbols, while a growing working memory enables neuroplastic development of an expanding mental number line during school years (step 4). Therapeutic and educational interventions can be drawn from this model. PMID:17979867
A generalization of Stirling numbers
Loeb, Daniel E.
1995-01-01
We generalize the Stirling numbers of the first kind $s(a,k)$ to the case where $a$ may be an arbitrary real number. In particular, we study the case in which $a$ is an integer. There, we discover new combinatorial properties held by the classical Stirling numbers, and analogous properties held by the Stirling numbers $s(n,k)$ with $n$ a negative integer. On g\\'{e}n\\'{e}ralise ici les nombres de Stirling du premier ordre $s(a,k)$ au cas o\\`u $a$ est un r\\'eel quelconque. On s'interesse en par...
Unsolved problems in number theory
Guy, Richard K
1994-01-01
Unsolved Problems in Number Theory contains discussions of hundreds of open questions, organized into 185 different topics. They represent numerous aspects of number theory and are organized into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous. To prevent repetition of earlier efforts or duplication of previously known results, an extensive and up-to-date collection of references follows each problem. In the second edition, not only extensive new material has been added, but corrections and additions have been included throughout the book.
Euler Sums of Hyperharmonic Numbers
Dil, Ayhan; Khristo N. Boyadzhiev
2012-01-01
The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: {\\sigma}(r,m)=\\sum_{n=1}^{\\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of series of Hurwitz zeta function values. This is a generalization of a result of Mez\\H{o} and Dil. We also provide an explicit evaluation of {\\sigma}(r,m) in a closed form in terms of zeta values and Stirling numbers of the first kind. Furthermore, we evalu...
Jacobi and Kummer's Ideal Numbers
Lemmermeyer, Franz
2011-01-01
In this article we give a modern interpretation of Kummer's ideal numbers and show how they developed from Jacobi's work on cyclotomy, in particular the methods for studying "Jacobi sums" which he presented in his lectures on number theory and cyclotomy in the winter semester 1836/37.
Ore, Oystein
1988-01-01
A prominent mathematician presents the principal ideas and methods of number theory within a historical and cultural framework. Oystein Ore's fascinating, accessible treatment requires only a basic knowledge of algebra. Topics include prime numbers, the Aliquot parts, linear indeterminate problems, congruences, Euler's theorem, classical construction problems, and many other subjects.
Know Your Blood Sugar Numbers If you have diabetes, keeping your blood sugar (glucose) numbers in your target range can help you feel ... Prevention There are two ways to measure blood sugar. 1 The A1C is a lab test that ...
On the Three Primordial Numbers
Gobbetti, Roberto; Pajer, Enrico; Roest, Diederik
2015-01-01
Cosmological observations have provided us with the measurement of just three numbers that characterize the very early universe: $ 1-n_{s} $, $ N $ and $\\ln\\Delta_R^2$. Although each of the three numbers individually carries limited information about the physics of inflation, one may hope to extract
Generalized Ramsey numbers for graphs
Zhang, Yanbo
2015-01-01
This thesis contains new contributions to Ramsey theory, in particular results that establish exact values of graph Ramsey numbers that were unknown to date. Given two graphs F and H, the Ramsey number R(F,H) is the smallest integer N such that, for any graph G of order N, either G contains F as a s
Investigating the Randomness of Numbers
Pendleton, Kenn L.
2009-01-01
The use of random numbers is pervasive in today's world. Random numbers have practical applications in such far-flung arenas as computer simulations, cryptography, gambling, the legal system, statistical sampling, and even the war on terrorism. Evaluating the randomness of extremely large samples is a complex, intricate process. However, the…
Learning Potentials in Number Blocks
Majgaard, Gunver; Misfeldt, Morten; Nielsen, Jacob
2012-01-01
This paper describes an initial exploration of how an interactive cubic user-configurable modular robotic system can be used to support learning about numbers and how they are pronounced. The development is done in collaboration with a class of 7-8 year old children and their mathematics teacher....... The tool is called Number Blocks and it combines physical interaction, learning, and immediate feedback. Number Blocks supports the children's understanding of place value in the sense that it allows them to experiment with creating large numbers. We found the blocks contributed to the learning...... process in several ways. The blocks combined mathematics and play, and they included and supported children at different academic levels. The auditory representation, especially the enhanced rhythmic effects due to using speech synthesis, and the rhythm helped the children to pronounce large numbers. This...
Some applications of Legendre numbers
Paul W. Haggard
1988-01-01
Full Text Available The associated Legendre functions are defined using the Legendre numbers. From these the associated Legendre polynomials are obtained and the derivatives of these polynomials at x=0 are derived by using properties of the Legendre numbers. These derivatives are then used to expand the associated Legendre polynomials and xn in series of Legendre polynomials. Other applications include evaluating certain integrals, expressing polynomials as linear combinations of Legendre polynomials, and expressing linear combinations of Legendre polynomials as polynomials. A connection between Legendre and Pascal numbers is also given.
Fundamental number theory with applications
Mollin, Richard A
2008-01-01
An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage. New to the Second Edition Removal of all advanced material to be even more accessible in scope New fundamental material, including partition theory, generating functions, and combinatorial number theory Expa
Elliptic rook and file numbers
Schlosser, Michael J.; Yoo, Meesue
2015-01-01
Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's q-rook numbers by two additional independent parameters a and b, and a nome p. These are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and later was extended to the q-case by Garsia and Remmel. We obtain similar results for our elliptic analogues of...
Louboutin, Stephane R.
2007-03-01
Let \\{K_m\\} be a parametrized family of simplest real cyclic cubic, quartic, quintic or sextic number fields of known regulators, e.g., the so-called simplest cubic and quartic fields associated with the polynomials P_m(x) Dx^3 -mx^2-(m+3)x+1 and P_m(x) Dx^4 -mx^3-6x^2+mx+1 . We give explicit formulas for powers of the Gaussian sums attached to the characters associated with these simplest number fields. We deduce a method for computing the exact values of these Gaussian sums. These values are then used to efficiently compute class numbers of simplest fields. Finally, such class number computations yield many examples of real cyclotomic fields Q(zeta_p)^+ of prime conductors pge 3 and class numbers h_p^+ greater than or equal to p . However, in accordance with Vandiver's conjecture, we found no example of p for which p divides h_p^+ .
Poison control center - emergency number
For a POISON EMERGENCY call: 1-800-222-1222 ANYWHERE IN THE UNITED STATES This national hotline number will let you ... is a free and confidential service. All local poison control centers in the United States use this ...
Picard numbers of Delsarte surfaces
Bas Heijne
2014-01-01
We give a classification of all complex Delsarte surfaces with only isolated ADE singularities. This results in 83 types of surfaces. For each of these types, we give a closed formula for the Picard number depending only on the degree.
Universal Algebras of Hurwitz Numbers
A. Mironov; Morozov, A; Natanzon, S.
2009-01-01
Infinite-dimensional universal Cardy-Frobenius algebra is constructed, which unifies all particular algebras of closed and open Hurwitz numbers and is closely related to the algebra of differential operators, familiar from the theory of Generalized Kontsevich Model.
Young Students Investigate Number Cubes.
Friedlander, Alex
1997-01-01
Describes a series of learning activities built around number cubes. Sample activities introduce elementary properties of the cube, the magic rule of seven, and basic concepts related to graphs in the plane coordinate system. (PVD)
... Your Heart Alternate Language URL Español Know Your Blood Sugar Numbers: Use Them to Manage Your Diabetes Page Content Checking your blood sugar, also called blood glucose, is an important part ...
Fibonacci Numbers and the Spreadsheet.
Verderber, Nadine L.
1991-01-01
Described is a classroom activity incorporating a computer spreadsheet to study number patterns generated by the Fibonacci sequence. Included are examples and suggestions for the use of the spreadsheet in other recursive relationships. (JJK)
Some Remarkable Identities Involving Numbers
Ziobro Rafał
2014-09-01
Full Text Available The article focuses on simple identities found for binomials, their divisibility, and basic inequalities. A general formula allowing factorization of the sum of like powers is introduced and used to prove elementary theorems for natural numbers.
Some Remarkable Identities Involving Numbers
Ziobro Rafał
2014-01-01
The article focuses on simple identities found for binomials, their divisibility, and basic inequalities. A general formula allowing factorization of the sum of like powers is introduced and used to prove elementary theorems for natural numbers.
Classical theory of algebraic numbers
Ribenboim, Paulo
2001-01-01
Gauss created the theory of binary quadratic forms in "Disquisitiones Arithmeticae" and Kummer invented ideals and the theory of cyclotomic fields in his attempt to prove Fermat's Last Theorem These were the starting points for the theory of algebraic numbers, developed in the classical papers of Dedekind, Dirichlet, Eisenstein, Hermite and many others This theory, enriched with more recent contributions, is of basic importance in the study of diophantine equations and arithmetic algebraic geometry, including methods in cryptography This book has a clear and thorough exposition of the classical theory of algebraic numbers, and contains a large number of exercises as well as worked out numerical examples The Introduction is a recapitulation of results about principal ideal domains, unique factorization domains and commutative fields Part One is devoted to residue classes and quadratic residues In Part Two one finds the study of algebraic integers, ideals, units, class numbers, the theory of decomposition, iner...
Exponential Decay of Lebesgue Numbers
Sun, Peng
2010-01-01
We study the exponential rate of decay of Lebesgue numbers of open covers in topological dynamical systems. We show that topological entropy is bounded by this rate multiplied by dimension. Some corollaries and examples are discussed.
Women In Numbers - Europe workshop
Bucur, Alina; Feigon, Brooke; Schneps, Leila
2015-01-01
Covering topics in graph theory, L-functions, p-adic geometry, Galois representations, elliptic fibrations, genus 3 curves and bad reduction, harmonic analysis, symplectic groups and mould combinatorics, this volume presents a collection of papers covering a wide swath of number theory emerging from the third iteration of the international Women in Numbers conference, “Women in Numbers - Europe” (WINE), held on October 14–18, 2013 at the CIRM-Luminy mathematical conference center in France. While containing contributions covering a wide range of cutting-edge topics in number theory, the volume emphasizes those concrete approaches that make it possible for graduate students and postdocs to begin work immediately on research problems even in highly complex subjects.
Social Security Number (SSN) Verification
U.S. Department of Health & Human Services — This report presents the results of a validation study of Social Security numbers (SSNs) in Medicaid Statistical Information System (MSIS) records for the fourth...
Occupation numbers from functional integral
Wetterich, C.
2007-01-01
Occupation numbers for non-relativistic interacting particles are discussed within a functional integral formulation. We concentrate on zero temperature, where the Bogoliubov theory breaks down for strong couplings as well as for low dimensional models. We find that the leading behavior of the occupation numbers for small momentum is governed by a quadratic time derivative in the inverse propagator that is not contained in the Bogoliubov theory. We propose to use a functional renormalization ...
Numbers for reducible cubic scrolls
Israel Vainsencher
2004-12-01
Full Text Available We show how to compute the number of reducible cubic scrolls of codimension 2 in (math blackboard symbol Pn incident to the appropriate number of linear spaces.Mostramos como calcular o número de rolos cúbicos redutíveis de codimensão 2 em (math blackboard symbol Pn incidentes a espaços lineares apropriados.
Conversion of Number Systems using Xilinx.
Chinmay V. Deshpande; Prof. Chankya K. Jha
2015-01-01
There are different types of number systems. Binary number system, octal number system, decimal number system and hexadecimal number system. This paper demonstrates conversion of hexadecimal to binary number using Xilinx software.
Conversion of Number Systems using Xilinx.
Chinmay V. Deshpande
2015-08-01
Full Text Available There are different types of number systems. Binary number system, octal number system, decimal number system and hexadecimal number system. This paper demonstrates conversion of hexadecimal to binary number using Xilinx software.
Are Number Gestures Easier than Number Words for Preschoolers?
Nicoladis, Elena; Pika, Simone; Marentette, Paula
2010-01-01
Some researchers have argued that children's earliest symbols are based on their sensorimotor experience and that arbitrary symbol-referent mapping poses a challenge for them. If so, exposure to iconic symbols (such as one-finger-for-one-object manual gestures) might help children in a difficult domain such as number. We assessed 44 preschoolers'…
Number Meaning and Number Grammar in English and Spanish
Bock, Kathryn; Carreiras, Manuel; Meseguer, Enrique
2012-01-01
Grammatical agreement makes different demands on speakers of different languages. Being widespread in the languages of the world, the features of agreement systems offer valuable tests of how language affects deep-seated domains of human cognition and categorization. Number agreement is one such domain, with intriguing evidence that typological…
Periodical plane puzzles with numbers
Rezende, Jorge
2011-01-01
Consider a periodical (in two independent directions) tiling of the plane with polygons (faces). In this article we shall only give examples using squares, regular hexagons, equilateral triangles and parallelograms ("unions" of two equilateral triangles). We shall call some "multiple" of the fundamental region "the board". We naturally identify pairs of corresponding edges of the the board. Figures 9 and 19-29, in this article, show different boards. The "border" of the board is represented by a yellow thick line, unless part of it or all of it is the edge of a face. The board is tiled by a finite number of polygons. Construct polygonal plates in the same number, shape and size as the polygons of the board. Adjacent to each side of each plate draw a number, or two numbers, like it is shown in Figures 1 and 18-29. Figure 1 shows the obvious possibility of having plates with simple drawings, coloured drawings, etc. Now the game is to put the plates over the board polygons in such a way that the numbers near eac...
Super congruences and Euler numbers
Sun, Zhi-Wei
2010-01-01
Let $p>3$ be a prime. We prove that $$\\sum_{k=0}^{p-1}\\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\\sum_{k=1}^{(p-1)/2}\\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\\sum_{k=0}^{(p-1)/2}\\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by ...
Schellekens, A N
2016-01-01
This paper contains some personal reflections on several computational contributions to what is now known as the "String Theory Landscape". It consists of two parts. The first part concerns the origin of big numbers, and especially the number $10^{1500}$ that appeared in work on the covariant lattice construction (with W. Lerche and D. Luest). This part contains some new results. I correct a huge but inconsequential error, discuss some more accurate estimates, and compare with the counting for free fermion constructions. In particular I prove that the latter only provide an exponentially small fraction of all even self-dual lattices for large lattice dimensions. The second part of the paper concerns dealing with big numbers, and contains some lessons learned from various vacuum scanning projects.
Quasiperpendicular high Mach number Shocks
Sulaiman, A H; Dougherty, M K; Burgess, D; Fujimoto, M; Hospodarsky, G B
2015-01-01
Shock waves exist throughout the universe and are fundamental to understanding the nature of collisionless plasmas. Reformation is a process, driven by microphysics, which typically occurs at high Mach number supercritical shocks. While ongoing studies have investigated this process extensively both theoretically and via simulations, their observations remain few and far between. In this letter we present a study of very high Mach number shocks in a parameter space that has been poorly explored and we identify reformation using in situ magnetic field observations from the Cassini spacecraft at 10 AU. This has given us an insight into quasi-perpendicular shocks across two orders of magnitude in Alfven Mach number (MA) which could potentially bridge the gap between modest terrestrial shocks and more exotic astrophysical shocks. For the first time, we show evidence for cyclic reformation controlled by specular ion reflection occurring at the predicted timescale of ~0.3 {\\tau}c, where {\\tau}c is the ion gyroperio...
Square Partitions and Catalan Numbers
Bennett, Matthew; Chari, Vyjayanthi; Dolbin, R. J.; Manning, Nathan
2009-01-01
For each integer $k\\ge 1$, we define an algorithm which associates to a partition whose maximal value is at most $k$ a certain subset of all partitions. In the case when we begin with a partition $\\lambda$ which is square, i.e $\\lambda=\\lambda_1\\ge...\\ge\\lambda_k>0$, and $\\lambda_1=k,\\lambda_k=1$, then applying the algorithm $\\ell$ times gives rise to a set whose cardinality is either the Catalan number $c_{\\ell-k+1}$ (the self dual case) or twice the Catalan number. The algorithm defines a t...
Geometric Number Systems and Spinors
Sobczyk, Garret
2015-01-01
The real number system is geometrically extended to include three new anticommuting square roots of plus one, each such root representing the direction of a unit vector along the orthonormal coordinate axes of Euclidean 3-space. The resulting geometric (Clifford) algebra provides a geometric basis for the famous Pauli matrices which, in turn, proves the consistency of the rules of geometric algebra. The flexibility of the concept of geometric numbers opens the door to new understanding of the nature of space-time, and of Pauli and Dirac spinors as points on the Riemann sphere, including Lorentz boosts.
Cognitive Radio with Random Number of Secondary Number of Users
Zeng, Ruochen; Tepedelenlioglu, Cihan
2013-01-01
A single primary user cognitive radio system with multi-user diversity at the secondary users is considered where there is an interference constraint between secondary and primary users. The secondary user with the highest instantaneous SNR is selected for communication from a set of active users which also satisfies the interference constraint. The active number of secondary users is shown to be binomial, negative binomial, or Poisson-binomial distributed depending on various modes of operat...
Residual number processing in dyscalculia
Marinella Cappelletti
2014-01-01
Full Text Available Developmental dyscalculia – a congenital learning disability in understanding numerical concepts – is typically associated with parietal lobe abnormality. However, people with dyscalculia often retain some residual numerical abilities, reported in studies that otherwise focused on abnormalities in the dyscalculic brain. Here we took a different perspective by focusing on brain regions that support residual number processing in dyscalculia. All participants accurately performed semantic and categorical colour-decision tasks with numerical and non-numerical stimuli, with adults with dyscalculia performing slower than controls in the number semantic tasks only. Structural imaging showed less grey-matter volume in the right parietal cortex in people with dyscalculia relative to controls. Functional MRI showed that accurate number semantic judgements were maintained by parietal and inferior frontal activations that were common to adults with dyscalculia and controls, with higher activation for participants with dyscalculia than controls in the right superior frontal cortex and the left inferior frontal sulcus. Enhanced activation in these frontal areas was driven by people with dyscalculia who made faster rather than slower numerical decisions; however, activation could not be accounted for by response times per se, because it was greater for fast relative to slow dyscalculics but not greater for fast controls relative to slow dyscalculics. In conclusion, our results reveal two frontal brain regions that support efficient number processing in dyscalculia.
Residual number processing in dyscalculia.
Cappelletti, Marinella; Price, Cathy J
2014-01-01
Developmental dyscalculia - a congenital learning disability in understanding numerical concepts - is typically associated with parietal lobe abnormality. However, people with dyscalculia often retain some residual numerical abilities, reported in studies that otherwise focused on abnormalities in the dyscalculic brain. Here we took a different perspective by focusing on brain regions that support residual number processing in dyscalculia. All participants accurately performed semantic and categorical colour-decision tasks with numerical and non-numerical stimuli, with adults with dyscalculia performing slower than controls in the number semantic tasks only. Structural imaging showed less grey-matter volume in the right parietal cortex in people with dyscalculia relative to controls. Functional MRI showed that accurate number semantic judgements were maintained by parietal and inferior frontal activations that were common to adults with dyscalculia and controls, with higher activation for participants with dyscalculia than controls in the right superior frontal cortex and the left inferior frontal sulcus. Enhanced activation in these frontal areas was driven by people with dyscalculia who made faster rather than slower numerical decisions; however, activation could not be accounted for by response times per se, because it was greater for fast relative to slow dyscalculics but not greater for fast controls relative to slow dyscalculics. In conclusion, our results reveal two frontal brain regions that support efficient number processing in dyscalculia. PMID:24266008
Calculations of effective atomic number
Kaliman, Z. [Department of Physics, Faculty of Arts and Sciences, Omladinska 14, Rijeka (Croatia); Orlic, N. [Department of Physics, Faculty of Arts and Sciences, Omladinska 14, Rijeka (Croatia)], E-mail: norlic@ffri.hr; Jelovica, I. [Department of Physics, Faculty of Arts and Sciences, Omladinska 14, Rijeka (Croatia)
2007-09-21
We present and discuss effective atomic number (Z{sub eff}) obtained by different methods of calculations. There is no unique relation between the computed values. This observation led us to the conclusion that any Z{sub eff} is valid only for given process. We illustrate calculations for different subshells of atom Z=72 and for M3 subshell of several other atoms.
Effective interactions and magic numbers
The magic numbers are the key concept of the shell model, and are shown to be different in exotic nuclei from those of stable nuclei. Its novel origin and robustness will be discussed, by referring to some basic but a sort of forgotten properties of the effective interaction. (author)
An introduction to Catalan numbers
Roman, Steven
2015-01-01
This textbook provides an introduction to the Catalan numbers and their remarkable properties, along with their various applications in combinatorics. Intended to be accessible to students new to the subject, the book begins with more elementary topics before progressing to more mathematically sophisticated topics. Each chapter focuses on a specific combinatorial object counted by these numbers, including paths, trees, tilings of a staircase, null sums in Zn+1, interval structures, partitions, permutations, semiorders, and more. Exercises are included at the end of book, along with hints and solutions, to help students obtain a better grasp of the material. The text is ideal for undergraduate students studying combinatorics, but will also appeal to anyone with a mathematical background who has an interest in learning about the Catalan numbers. “Roman does an admirable job of providing an introduction to Catalan numbers of a different nature from the previous ones. He has made an excellent choice o...
Lenstra theorem in number fields
S Subburam
2014-11-01
In this paper, we present a number field version of the celebrated result of Lenstra (Math. Comp. 42(165) (1984) 331–340) in 1984. Also, this result allows us to improve a result of Wikstrőm (On the -ary GCD-algorithm in rings of integers (2005) pp. 1189–1201).
Finite temperature induced fermion number
Dunne, Gerald V.
2001-01-01
The induced fractional fermion number at zero temperature is topological (in the sense that it is only sensitive to global asymptotic properties of the background field), and is a sharp observable (in the sense that it has vanishing rms fluctuations). In contrast, at finite temperature, it is shown to be generically nontopological, and not a sharp observable.
Upper bounds on Nusselt number at finite Prandtl number
Choffrut, Antoine; Otto, Felix
2014-01-01
We study Rayleigh B\\'enard convection based on the Boussinesq approximation. We are interested in upper bounds on the Nusselt number $\\mathrm{Nu}$, the upwards heat transport, in terms of the Rayleigh number $\\mathrm{Ra}$, that characterizes the relative strength of the driving mechanism and the Prandtl number $\\mathrm{Pr}$, that characterizes the strength of the inertial effects. We show that, up to logarithmic corrections, the upper bound $\\mathrm{Nu}\\lesssim \\mathrm{Ra}^{\\frac{1}{3}}$ of Constantin and Doering in 1999 persists as long as $\\mathrm{Pr}\\gtrsim \\mathrm{Ra}^{\\frac{1}{3}}$ and then crosses over to $\\mathrm{Nu}\\lesssim\\mathrm{Pr}^{-\\frac{1}{2}}\\mathrm{Ra}^{\\frac{1}{2}}$. This result improves the one of Wang by going beyond the perturbative regime $\\mathrm{Pr} \\gg \\mathrm{Ra}$. The proof uses a new way to estimate the transport nonlinearity in the Navier-Stokes equations capitalizing on the no-slip boundary condition. It relies on a new Calder\\'on-Zygmund estimate for the non-stationary Stokes equ...
Peter Saveliev
2005-04-01
Full Text Available Suppose X, Y are manifolds, f,g:XÃ¢Â†Â’Y are maps. The well-known coincidence problem studies the coincidence set C={x:f(x=g(x}. The number m=dimÃ¢Â€Â‰XÃ¢ÂˆÂ’dimÃ¢Â€Â‰Y is called the codimension of the problem. More general is the preimage problem. For a map f:XÃ¢Â†Â’Z and a submanifold Y of Z, it studies the preimage set C={x:f(xÃ¢ÂˆÂˆY}, and the codimension is m=dimÃ¢Â€Â‰X+dimÃ¢Â€Â‰YÃ¢ÂˆÂ’dimÃ¢Â€Â‰Z. In case of codimension 0, the classical Nielsen number N(f,Y is a lower estimate of the number of points in C changing under homotopies of f, and for an arbitrary codimension, of the number of components of C. We extend this theory to take into account other topological characteristics of C. The goal is to find a Ã¢Â€Âœlower estimateÃ¢Â€Â of the bordism group ÃŽÂ©p(C of C. The answer is the Nielsen group Sp(f,Y defined as follows. In the classical definition, the Nielsen equivalence of points of C based on paths is replaced with an equivalence of singular submanifolds of C based on bordisms. We let Sp'(f,Y=ÃŽÂ©p(C/Ã¢ÂˆÂ¼N, then the Nielsen group of order p is the part of Sp'(f,Y preserved under homotopies of f. The Nielsen number Np(F,Y of order p is the rank of this group (then N(f,Y=N0(f,Y. These numbers are new obstructions to removability of coincidences and preimages. Some examples and computations are provided.
Lozenge Tilings and Hurwitz Numbers
Novak, Jonathan
2015-10-01
We give a new proof of the fact that, near a turning point of the frozen boundary, the vertical tiles in a uniformly random lozenge tiling of a large sawtooth domain are distributed like the eigenvalues of a GUE random matrix. Our argument uses none of the standard tools of integrable probability. In their place, it uses a combinatorial interpretation of the Harish-Chandra/Itzykson-Zuber integral as a generating function for desymmetrized Hurwitz numbers.
Ufnarovski, Victor; Ahlander, Bo
2003-09-01
We define the derivative of an integer to be the map sending every prime to 1 and satisfying the Leibnitz rule. The aim of the article is to consider the basic properties of this map and to show how to generalize the notion to the case of rational and arbitrary real numbers. We make some conjectures and find some connections with Goldbach's Conjecture and the Twin Prime Conjecture. Finally, we solve the easiest associated differential equations and calculate the generating function.
On Buffon Machines and Numbers
Flajolet, Philippe; Pelletier, Maryse; Soria, Michèle
2011-01-01
The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically "computes" the number 2/pi ~ 0.63661, which is the experiment's probability of success. Generalizing the experiment and simplifying the computational framework, we consider probability distributions, which can be produced perfectly, from a discrete source of unbiased coin flips. We describe and analyse a few simple Buffon machines that generate geometric, Poisson, and logarithm...
Stum, Bernard Le; Quirós, Adolfo
2013-01-01
We introduce the notions of quantum characteristic and quantum flatness for arbitrary rings. More generally, we develop the theory of quantum integers in a ring and show that the hypothesis of quantum flatness together with positive quantum characteristic generalizes the usual notion of prime positive characteristic. We also explain how one can define quantum rational numbers in a ring and introduce the notion of twisted powers. These results play an important role in many different areas of ...
Gyori, Ervin; Lovasz, Laszlo
2006-01-01
This volume honours the eminent mathematicians Vera Sos and Andras Hajnal. The book includes survey articles reviewing classical theorems, as well as new, state-of-the-art results. Also presented are cutting edge expository research papers with new theorems and proofs in the area of the classical Hungarian subjects, like extremal combinatorics, colorings, combinatorial number theory, etc. The open problems and the latest results in the papers are sure to inspire further research.
Large number discrimination by mosquitofish.
Christian Agrillo
Full Text Available BACKGROUND: Recent studies have demonstrated that fish display rudimentary numerical abilities similar to those observed in mammals and birds. The mechanisms underlying the discrimination of small quantities (<4 were recently investigated while, to date, no study has examined the discrimination of large numerosities in fish. METHODOLOGY/PRINCIPAL FINDINGS: Subjects were trained to discriminate between two sets of small geometric figures using social reinforcement. In the first experiment mosquitofish were required to discriminate 4 from 8 objects with or without experimental control of the continuous variables that co-vary with number (area, space, density, total luminance. Results showed that fish can use the sole numerical information to compare quantities but that they preferentially use cumulative surface area as a proxy of the number when this information is available. A second experiment investigated the influence of the total number of elements to discriminate large quantities. Fish proved to be able to discriminate up to 100 vs. 200 objects, without showing any significant decrease in accuracy compared with the 4 vs. 8 discrimination. The third experiment investigated the influence of the ratio between the numerosities. Performance was found to decrease when decreasing the numerical distance. Fish were able to discriminate numbers when ratios were 1:2 or 2:3 but not when the ratio was 3:4. The performance of a sample of undergraduate students, tested non-verbally using the same sets of stimuli, largely overlapped that of fish. CONCLUSIONS/SIGNIFICANCE: Fish are able to use pure numerical information when discriminating between quantities larger than 4 units. As observed in human and non-human primates, the numerical system of fish appears to have virtually no upper limit while the numerical ratio has a clear effect on performance. These similarities further reinforce the view of a common origin of non-verbal numerical systems in all
Ghosh, Amit
1981-01-01
The thesis is divided into five sections: (a) Trigonometric sums involving prime numbers and applications, (b) Mean-values and Sign-changes of S(t)-- related to Riemann's Zeta function, (c) Mean-values of strongly additive arithmetical functions, (d) Combinatorial identities and sieves, (e) A Goldbach-type problem. Parts (b) and (c) are related by means of the techniques used but otherwise the sections are disjoint. (a) We consider the question of finding upper bounds for...