Mathematics

In this final installment of the paper we consider the case where the signals or the messages or both are continuously variable, in contrast with the discrete nature assumed until now. To a considerable extent the continuous case can be obtained through a limiting process from the discrete case by dividing the continuum of messages and signals into a large but finite number of small regions and calculating the various parameters involved on a discrete basis. As the size of the regions is decreased these parameters in general approach as limits the proper values for the continuous case. There are, however, a few new effects that appear and also a general change of emphasis in the direction of specialization of the general results to particular cases.

@book{doi101002j153873051948tb00917x,
    author = "Shannon, Claude E.",
    title = "A Mathematical Theory of Communication",
    year = "1948",
    journal = "Bell System Technical Journal",
    abstract = "In this final installment of the paper we consider the case where the signals or the messages or both are continuously variable, in contrast with the discrete nature assumed until now. To a considerable extent the continuous case can be obtained through a limiting process from the discrete case by dividing the continuum of messages and signals into a large but finite number of small regions and calculating the various parameters involved on a discrete basis. As the size of the regions is decreased these parameters in general approach as limits the proper values for the continuous case. There are, however, a few new effects that appear and also a general change of emphasis in the direction of specialization of the general results to particular cases.",
    url = "https://doi.org/10.1002/j.1538-7305.1948.tb00917.x",
    doi = "10.1002/j.1538-7305.1948.tb00917.x",
    openalex = "W2041404167",
    references = "doi101002j153873051948tb01338x"
}

The recent development of various methods of modulation such as PCM and PPM which exchange bandwidth for signal-to-noise ratio has intensified the interest in a general theory of communication. A basis for such a theory is contained in the important papers of Nyquist 1 and Hartley 2 on this subject. In the present paper we will extend the theory to include a number of new factors, in particular the effect of noise in the channel, and the savings possible due to the statistical structure of the original message and due to the nature of the final destination of the information.

@article{doi101002j153873051948tb01338x,
    author = "Shannon, Claude E.",
    title = "A Mathematical Theory of Communication",
    year = "1948",
    journal = "Bell System Technical Journal",
    abstract = "The recent development of various methods of modulation such as PCM and PPM which exchange bandwidth for signal-to-noise ratio has intensified the interest in a general theory of communication. A basis for such a theory is contained in the important papers of Nyquist 1 and Hartley 2 on this subject. In the present paper we will extend the theory to include a number of new factors, in particular the effect of noise in the channel, and the savings possible due to the statistical structure of the original message and due to the nature of the final destination of the information.",
    url = "https://doi.org/10.1002/j.1538-7305.1948.tb01338.x",
    doi = "10.1002/j.1538-7305.1948.tb01338.x",
    openalex = "W1995875735"
}

@book{neugebauer1954ancient3,
    author = "Neugebauer, O",
    title = "Ancient Mathematics and Astronomy, in Singer, C., Holmyard, E. J., and Hall, A. R., eds., THe History of Technology",
    year = "1954",
    publisher = "London, Oxford University Press",
    note = "talkorigins\_source = {true}; raw\_reference = {Neugebauer, O., 1954, Ancient Mathematics and Astronomy, in Singer, C., Holmyard, E. J., and Hall, A. R., eds., THe History of Technology: London, Oxford University Press.}"
}

@book{newton1960the4,
    author = "Newton, I. and Motte-Cajori, 1687/",
    title = "The Mathematical Principles of Natural Philosophy",
    year = "1960",
    publisher = "Berkeley, University of California Press",
    note = "talkorigins\_source = {true}; raw\_reference = {Newton, I., and Motte-Cajori, 1687/1960, The Mathematical Principles of Natural Philosophy: Berkeley, University of California Press.}"
}

@article{doi1023072004073,
    author = "W., J. W. and Dwight, H. B.",
    title = "Tables of Integrals and Other Mathematical Data.",
    year = "1962",
    journal = "Mathematics of Computation",
    url = "https://doi.org/10.2307/2004073",
    doi = "10.2307/2004073",
    openalex = "W2324759687"
}

@article{doi1023071266136,
    author = "Owen, D. B. and Abramowitz, Milton and Stegun, Irene A.",
    title = "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables",
    year = "1965",
    journal = "Technometrics",
    url = "https://doi.org/10.2307/1266136",
    doi = "10.2307/1266136",
    openalex = "W2801179766"
}

@article{doi1023072004114,
    author = "W., J. W. and Abramowitz, Milton and Stegun, Irene A.",
    title = "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables",
    year = "1965",
    journal = "Mathematics of Computation",
    url = "https://doi.org/10.2307/2004114",
    doi = "10.2307/2004114",
    openalex = "W4300223101"
}

First Page

@article{doi10111911972842,
    author = "McQuarrie, Donald A.",
    title = "Handbook of Mathematical Functions",
    year = "1966",
    journal = "American Journal of Physics",
    abstract = "First Page",
    url = "https://doi.org/10.1119/1.1972842",
    doi = "10.1119/1.1972842",
    openalex = "W2120062331"
}

@book{medawar1967mathematical2,
    author = "Medawar, P",
    title = "Mathematical Challenges to the Neo-Darwinian Interpretation of Evolution",
    year = "1967",
    publisher = "Philadelphia, Wistar Institute Press",
    note = "talkorigins\_source = {true}; raw\_reference = {Medawar, P., 1967, Mathematical Challenges to the Neo-Darwinian Interpretation of Evolution: Philadelphia, Wistar Institute Press.}"
}

@book{feller1968an1,
    author = "Feller, W",
    title = "An Introduction to Probability Theory and its Application",
    year = "1968",
    publisher = "New York, Wiley",
    note = "talkorigins\_source = {true}; raw\_reference = {Feller, W., 1968, An Introduction to Probability Theory and its Application: New York, Wiley.}"
}

@article{openalexw3016843226,
    author = "Abramowitz, Milton and Stegun, I. A.",
    title = "Handbook of Mathematical Functions",
    year = "1972",
    openalex = "W3016843226"
}

@article{doi1010160032386175901305,
    author = "Park, Gaeun",
    title = "The mathematics of diffusion",
    year = "1975",
    journal = "Polymer",
    url = "https://doi.org/10.1016/0032-3861(75)90130-5",
    doi = "10.1016/0032-3861(75)90130-5",
    openalex = "W2326565381"
}

@book{doi1015159780691214696,
    author = "Shafer, Glenn",
    title = "A Mathematical Theory of Evidence",
    year = "1976",
    booktitle = "Princeton University Press eBooks",
    url = "https://doi.org/10.1515/9780691214696",
    doi = "10.1515/9780691214696",
    openalex = "W4301347335"
}

@article{openalexw1560420247,
    author = "Milner, Robin",
    title = "Mathematical Centre Tracts",
    year = "1976",
    openalex = "W1560420247"
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@book{doi1010079781475720631,
    author = "Arnold, V. I.",
    title = "Mathematical Methods of Classical Mechanics",
    year = "1989",
    booktitle = "Graduate texts in mathematics",
    url = "https://doi.org/10.1007/978-1-4757-2063-1",
    doi = "10.1007/978-1-4757-2063-1",
    openalex = "W2126479355"
}

This paper is not a tutorial on how lattice basis reduction algorithms such as LLL or PSLQ actually work; rather, we discuss some of the ways these tools can be used to generate conjectures, and for that, a detailed understanding of the underlying algorithms is not necessary. We do hope, however, to convey some appreciation of their power. We begin with some warm-up examples, using the Inverse Symbolic Calculator (ISC); http:// www.cecm.sfu.ca/ MRG/ INTERFACES.html. The basic idea is simple: given the first few decimal digits of some real number, we want the ISC to guess a formula for what it `really' is. For example, if we input K 1 = 3:14626436994198, and click on simple lookup

@article{doi10108000029890199912005138,
    author = "Borwein, Jonathan M. and Corless, Robert M.",
    title = "Emerging Tools for Experimental Mathematics",
    year = "1999",
    journal = "American Mathematical Monthly",
    abstract = "This paper is not a tutorial on how lattice basis reduction algorithms such as LLL or PSLQ actually work; rather, we discuss some of the ways these tools can be used to generate conjectures, and for that, a detailed understanding of the underlying algorithms is not necessary. We do hope, however, to convey some appreciation of their power. We begin with some warm-up examples, using the Inverse Symbolic Calculator (ISC); http:// www.cecm.sfu.ca/ MRG/ INTERFACES.html. The basic idea is simple: given the first few decimal digits of some real number, we want the ISC to guess a formula for what it `really\' is. For example, if we input K 1 = 3:14626436994198, and click on simple lookup",
    url = "https://doi.org/10.1080/00029890.1999.12005138",
    doi = "10.1080/00029890.1999.12005138",
    openalex = "W2070038747"
}

@book{doi1010079783642564789,
    author = "Engquist, Björn and Schmid, Wilfried",
    title = "Mathematics Unlimited — 2001 and Beyond",
    year = "2001",
    url = "https://doi.org/10.1007/978-3-642-56478-9",
    doi = "10.1007/978-3-642-56478-9",
    openalex = "W2797673549"
}

Use of the real-time polymerase chain reaction (PCR) to amplify cDNA products reverse transcribed from mRNA is on the way to becoming a routine tool in molecular biology to study low abundance gene expression. Real-time PCR is easy to perform, provides the necessary accuracy and produces reliable as well as rapid quantification results. But accurate quantification of nucleic acids requires a reproducible methodology and an adequate mathematical model for data analysis. This study enters into the particular topics of the relative quantification in real-time RT-PCR of a target gene transcript in comparison to a reference gene transcript. Therefore, a new mathematical model is presented. The relative expression ratio is calculated only from the real-time PCR efficiencies and the crossing point deviation of an unknown sample versus a control. This model needs no calibration curve. Control levels were included in the model to standardise each reaction run with respect to RNA integrity, sample loading and inter-PCR variations. High accuracy and reproducibility (<2.5% variation) were reached in LightCycler PCR using the established mathematical model.

@article{doi101093nar299e45,
    author = "Pfaffl, Michael W.",
    title = "A new mathematical model for relative quantification in real-time RT-PCR",
    year = "2001",
    journal = "Nucleic Acids Research",
    abstract = "Use of the real-time polymerase chain reaction (PCR) to amplify cDNA products reverse transcribed from mRNA is on the way to becoming a routine tool in molecular biology to study low abundance gene expression. Real-time PCR is easy to perform, provides the necessary accuracy and produces reliable as well as rapid quantification results. But accurate quantification of nucleic acids requires a reproducible methodology and an adequate mathematical model for data analysis. This study enters into the particular topics of the relative quantification in real-time RT-PCR of a target gene transcript in comparison to a reference gene transcript. Therefore, a new mathematical model is presented. The relative expression ratio is calculated only from the real-time PCR efficiencies and the crossing point deviation of an unknown sample versus a control. This model needs no calibration curve. Control levels were included in the model to standardise each reaction run with respect to RNA integrity, sample loading and inter-PCR variations. High accuracy and reproducibility (<2.5\% variation) were reached in LightCycler PCR using the established mathematical model.",
    url = "https://doi.org/10.1093/nar/29.9.e45",
    doi = "10.1093/nar/29.9.e45",
    openalex = "W2108244474",
    references = "doi101006cyto19980426, doi10100797836425952401, doi10100797836425952403, doi101016s0168165699001637, doi101038nbt09931026, doi101073pnas86249717, doi101677jme00250169, doi10214497221pf02, doi10214499261rv01, openalexw2144634347"
}

Paper 17: David H. Bailey and Jonathan M. Borwein, “Pi and its friends,” and “Normality: A stubborn question,” from Mathematics by Experiment: Plausible Reasoning in the 21st Century, A. K. Peters, Natick, MA, 2nd edition, 2008. Reproduced with permission of AK Peters.

@article{doi105860choice422850,
    title = "Mathematics by experiment: plausible reasoning in the 21st century",
    year = "2005",
    journal = "Choice Reviews Online",
    abstract = "Paper 17: David H. Bailey and Jonathan M. Borwein, “Pi and its friends,” and “Normality: A stubborn question,” from Mathematics by Experiment: Plausible Reasoning in the 21st Century, A. K. Peters, Natick, MA, 2nd edition, 2008. Reproduced with permission of AK Peters.",
    url = "https://doi.org/10.5860/choice.42-2850",
    doi = "10.5860/choice.42-2850",
    openalex = "W1595653788",
    references = "doi1010079781349026791, doi1010079783642482984, doi101007s107580055216x, doi1010160378475488901255, doi10108000029890196411992270, doi10108000029890199912005138, doi10108010586458199710504608, doi101090s00255718197804803213, doi101090s0025571897008004, doi1011771461444801003001006"
}

@article{doi101016jaddr200609004,
    author = "Lin, Chien‐Chi and Metters, Andrew T.",
    title = "Hydrogels in controlled release formulations: Network design and mathematical modeling",
    year = "2006",
    journal = "Advanced Drug Delivery Reviews",
    url = "https://doi.org/10.1016/j.addr.2006.09.004",
    doi = "10.1016/j.addr.2006.09.004",
    openalex = "W2145496621",
    references = "doi1010160032386175901305"
}

@misc{bailey2010experimental,
    author = "Bailey, David H. and Borwein, Jonathan M. and Broadhurst, David and Zudilin, Wadim",
    title = "Experimental mathematics and mathematical physics",
    year = "2010",
    booktitle = "Contemporary Mathematics",
    url = "https://doi.org/10.1090/conm/517/10132",
    doi = "10.1090/conm/517/10132",
    openalex = "W2952399859",
    pages = "41-58",
    references = "doi1010079781447164647, doi101007bf01457454, doi101007s110400109074y, doi101016jcpc200911007, doi101016s0370269396016231, doi101088175181134120205203, doi101090s0025571899009953, doi1023072153423, doi102977prims1195192451, doi105860choice422850, openalexw3016843226"
}

Modern developments in theoretical and applied science depend on knowledge of the properties of mathematical functions, from elementary trigonometric functions to the multitude of special functions. These functions appear whenever natural phenomena are studied, engineering problems are formulated, and numerical simulations are performed. They also crop up in statistics, financial models, and economic analysis. Using them effectively requires practitioners to have ready access to a reliable collection of their properties. This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators. Printed in full color, it is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun. Included with every copy of the book is a CD with a searchable PDF of each chapter.

@book{openalexw2102591969,
    author = "Olver, F. W. J. and Lozier, Daniel W. and Boisvert, Ronald F. and Clark, Charles W.",
    title = "NIST Handbook of Mathematical Functions",
    year = "2010",
    booktitle = "CERN Document Server (European Organization for Nuclear Research)",
    abstract = "Modern developments in theoretical and applied science depend on knowledge of the properties of mathematical functions, from elementary trigonometric functions to the multitude of special functions. These functions appear whenever natural phenomena are studied, engineering problems are formulated, and numerical simulations are performed. They also crop up in statistics, financial models, and economic analysis. Using them effectively requires practitioners to have ready access to a reliable collection of their properties. This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators. Printed in full color, it is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun. Included with every copy of the book is a CD with a searchable PDF of each chapter.",
    openalex = "W2102591969"
}

This work presents a heuristic mathematical approach to modeling and quantifying constraint violations. The empirical phenomenon consists of different word orders with contrastive focus in the Spanish variety of Catalonia and in Catalan. Experimental data has been collected with a gradient acceptability judgment test for auditorily presented focus constructions.

@article{doi101515tlir2011004,
    author = "Adli, Aria",
    title = "A heuristic mathematical approach for modeling constraint cumulativity: Contrastive focus in Spanish and Catalan",
    year = "2011",
    journal = "The Linguistic Review",
    abstract = "This work presents a heuristic mathematical approach to modeling and quantifying constraint violations. The empirical phenomenon consists of different word orders with contrastive focus in the Spanish variety of Catalonia and in Catalan. Experimental data has been collected with a gradient acceptability judgment test for auditorily presented focus constructions.",
    url = "https://doi.org/10.1515/tlir.2011.004",
    doi = "10.1515/tlir.2011.004",
    openalex = "W2033210364",
    references = "doi105860choice422850"
}

@article{doi101016jamc201203087,
    author = "Bailey, David H. and Barrio, Roberto and Borwein, Jonathan M.",
    title = "High-precision computation: Mathematical physics and dynamics",
    year = "2012",
    journal = "Applied Mathematics and Computation",
    url = "https://doi.org/10.1016/j.amc.2012.03.087",
    doi = "10.1016/j.amc.2012.03.087",
    openalex = "W2121848241",
    references = "doi101088175181134120205203, doi105860choice422850"
}

Both in science and in practical affairs we reason by combining facts only inconclusively supported by evidence. Building on an abstract understanding of this process of combination, this book constructs a new theory of epistemic probability. The theory draws on the work of A. P. Dempster but diverges from Depster's viewpoint by identifying his as epistemic probabilities and taking his rule for combining upper and lower as fundamental. The book opens with a critique of the well-known Bayesian theory of epistemic probability. It then proceeds to develop an alternative to the additive set functions and the rule of conditioning of the Bayesian theory: set functions that need only be what Choquet called monotone of order of infinity. and Dempster's rule for combining such set functions. This rule, together with the idea of weights of evidence, leads to both an extensive new theory and a better understanding of the Bayesian theory. The book concludes with a brief treatment of statistical inference and a discussion of the limitations of epistemic probability. Appendices contain mathematical proofs, which are relatively elementary and seldom depend on mathematics more advanced that the binomial theorem.

@book{doi102307jctv10vm1qb,
    author = "Shafer, Glenn",
    title = "A Mathematical Theory of Evidence",
    year = "2020",
    booktitle = "Princeton University Press eBooks",
    abstract = "Both in science and in practical affairs we reason by combining facts only inconclusively supported by evidence. Building on an abstract understanding of this process of combination, this book constructs a new theory of epistemic probability. The theory draws on the work of A. P. Dempster but diverges from Depster's viewpoint by identifying his as epistemic probabilities and taking his rule for combining upper and lower as fundamental. The book opens with a critique of the well-known Bayesian theory of epistemic probability. It then proceeds to develop an alternative to the additive set functions and the rule of conditioning of the Bayesian theory: set functions that need only be what Choquet called monotone of order of infinity. and Dempster's rule for combining such set functions. This rule, together with the idea of weights of evidence, leads to both an extensive new theory and a better understanding of the Bayesian theory. The book concludes with a brief treatment of statistical inference and a discussion of the limitations of epistemic probability. Appendices contain mathematical proofs, which are relatively elementary and seldom depend on mathematics more advanced that the binomial theorem.",
    url = "https://doi.org/10.2307/j.ctv10vm1qb",
    doi = "10.2307/j.ctv10vm1qb",
    openalex = "W2797148637"
}

@article{doi101140epjss11734024011038,
    author = "Ananthanarayan, B. and Pathak, Tanay and Sharma, Kartik",
    title = "Closed form expressions for certain improper integrals of mathematical physics",
    year = "2024",
    journal = "The European Physical Journal Special Topics",
    url = "https://doi.org/10.1140/epjs/s11734-024-01103-8",
    doi = "10.1140/epjs/s11734-024-01103-8",
    openalex = "W4391653489",
    references = "bailey2010experimental, doi1010073540306110, doi1010079783031142727, doi101016c20100648395, doi101016jjsc201008010, doi101023a1004850919647, doi10106311704134, doi101088030544703940001, doi101103physrevb13316, doi1023071267715, openalexw1569990823"
}

@misc{crossrefNonemathematics,
    title = "Mathematics and Mathematical Modeling",
    year = "None",
    url = "https://doi.org/10.7463/2412-5911",
    doi = "10.7463/2412-5911",
    openalex = "W2351133644"
}