Split (1)

train · 154k rows

text stringlengths 100 500k	subset stringclasses 4 values
rNPV In finance, rNPV ("risk-adjusted net present value") or eNPV ("expected NPV") is a method to value risky future cash flows. rNPV is the standard valuation method in the drug development industry,[1] where sufficient data exists to estimate success rates for all R&D phases.[2] A similar technique is used in the probability model of credit default swap (CDS) valuation. rNPV modifies the standard NPV calculation of discounted cash flow (DCF) analysis by adjusting (multiplying) each cash flow by the estimated probability that it occurs (the estimated success rate). In the language of probability theory, the rNPV is the expected value. Note that this in contrast to the more general valuation approach, where risk is instead incorporated by adding a risk premium percentage to the discount rate, i.e. as opposed to weighting the cash flows. See also • Corporate finance § Investment and project valuation • Expected commercial value • Financial economics § Corporate finance theory • First Chicago Method • Penalized present value • Valuation (finance) § Valuation of intangible assets References 1. John Selig. Must-Know Finance Concepts for Life Sciences Valuations. LSX whitepaper, 2017. 2. Stewart JJ et al. Putting a Price on Biotechnology. Nature Biotechnology. 2001. 19:5	Wikipedia
Residue number system A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if M is the product of the moduli, there is, in an interval of length M, exactly one integer having any given set of modular values. The arithmetic of a residue numeral system is also called multi-modular arithmetic. Multi-modular arithmetic is widely used for computation with large integers, typically in linear algebra, because it provides faster computation than with the usual numeral systems, even when the time for converting between numeral systems is taken into account. Other applications of multi-modular arithmetic include polynomial greatest common divisor, Gröbner basis computation and cryptography. Definition A residue numeral system is defined by a set of k integers $\{m_{1},m_{2},m_{3},\ldots ,m_{k}\},$ called the moduli, which are generally supposed to be pairwise coprime (that is, any two of them have a greatest common divisor equal to one). Residue number systems have been defined for non-coprime moduli, but are not commonly used because of worse properties. Therefore, they will not be considered in the remainder of this article.[1] An integer x is represented in the residue numeral system by the set of its remainders $\{x_{1},x_{2},x_{3},\ldots ,x_{k}\}$ under Euclidean division by the moduli. That is $x_{i}=x\operatorname {mod} m_{i},$ and $0\leq x_{i}<m_{i}$ for every i Let M be the product of all the $m_{i}$. Two integers whose difference is a multiple of M have the same representation in the residue numeral system defined by the mis. More precisely, the Chinese remainder theorem asserts that each of the M different sets of possible residues represents exactly one residue class modulo M. That is, each set of residues represents exactly one integer $X$ in the interval $0,\dots ,M-1$. For signed numbers, the dynamic range is $ {-\lfloor M/2\rfloor }\leq X\leq \lfloor (M-1)/2\rfloor $ (when $M$ is even, generally an extra negative value is represented).[2] Arithmetic operations For adding, subtracting and multiplying numbers represented in a residue number system, it suffices to perform the same modular operation on each pair of residues. More precisely, if $[m_{1},\ldots ,m_{k}]$ is the list of moduli, the sum of the integers x and y, respectively represented by the residues $[x_{1},\ldots ,x_{k}]$ and $[y_{1},\ldots ,y_{k}],$ is the integer z represented by $[z_{1},\ldots ,z_{k}],$ such that $z_{i}=(x_{i}+y_{i})\operatorname {mod} m_{i},$ for i = 1, ..., k (as usual, mod denotes the modulo operation consisting of taking the remainder of the Euclidean division by the right operand). Subtraction and multiplication are defined similarly. For a succession of operations, it is not necessary to apply the modulo operation at each step. It may be applied at the end of the computation, or, during the computation, for avoiding overflow of hardware operations. However, operations such as magnitude comparison, sign computation, overflow detection, scaling, and division are difficult to perform in a residue number system.[3] Comparison If two integers are equal, then all their residues are equal. Conversely, if all residues are equal, then the two integers are equal, or their differences is a multiple of M. It follows that testing equality is easy. At the opposite, testing inequalities (x < y) is difficult and, usually, requires to convert integers to the standard representation. As a consequence, this representation of numbers is not suitable for algorithms using inequality tests, such Euclidean division and Euclidean algorithm. Division Division in residue numeral systems is problematic. On the other hand, if $B$ is coprime with $M$ (that is $b_{i}\not =0$) then $C=A\cdot B^{-1}\mod M$ can be easily calculated by $c_{i}=a_{i}\cdot b_{i}^{-1}\mod m_{i},$ where $B^{-1}$ is multiplicative inverse of $B$ modulo $M$, and $b_{i}^{-1}$ is multiplicative inverse of $b_{i}$ modulo $m_{i}$. Applications RNS have applications in the field of digital computer arithmetic. By decomposing in this a large integer into a set of smaller integers, a large calculation can be performed as a series of smaller calculations that can be performed independently and in parallel. See also • Covering system • Reduced residue system References 1. Parhami, Behrooz (2010). Computer Arithmetic: Algorithms and Hardware Designs (2 ed.). New York, USA: Oxford University Press. ISBN 978-0-19-532848-6. Archived from the original on 2020-08-04. Retrieved 2021-01-23. (xxv+641 pages) 2. Hung, C.Y.; Parhami, B. (1994-02-01). "An approximate sign detection method for residue numbers and its application to RNS division" (PDF). Computers & Mathematics with Applications. 27 (4): 23–35. doi:10.1016/0898-1221(94)90052-3. 3. Isupov, Konstantin (2020-04-07) [2020-03-20, 2020-03-08, 2020-02-17]. "Using Floating-Point Intervals for Non-Modular Computations in Residue Number System". IEEE Access. 8: 58603–58619. doi:10.1109/ACCESS.2020.2982365. ISSN 2169-3536. Further reading • Szabo, Nicholas S.; Tanaka, Richard I. (1967). Residue Arithmetic and its Applications to Computer Technology (1 ed.). New York, USA: McGraw-Hill. • Sonderstrand, Michael A.; Jenkins, W. Kenneth; Jullien, Graham A.; Taylor, Fred J., eds. (1986). Residue Number System Arithmetic: Modern Applications in Digital Signal Processing. IEEE Press Reprint Series (1 ed.). New York, USA: IEEE Circuits and Systems Society, IEEE Press. ISBN 0-87942-205-X. LCCN 86-10516. IEEE order code PC01982. (viii+418+6 pages) • Chervyakov, N. I.; Molahosseini, A. S.; Lyakhov, P. A. (2017). Residue-to-binary conversion for general moduli sets based on approximate Chinese remainder theorem. International Journal of Computer Mathematics, 94:9, 1833-1849, doi: 10.1080/00207160.2016.1247439. • Fin'ko [Финько], Oleg Anatolevich [Олег Анатольевич] (June 2004). "Large Systems of Boolean Functions: Realization by Modular Arithmetic Methods". Automation and Remote Control. 65 (6): 871–892. doi:10.1023/B:AURC.0000030901.74901.44. ISSN 0005-1179. LCCN 56038628. S2CID 123623780. CODEN AURCAT. Mi at1588. • Chervyakov, N. I.; Lyakhov, P. A.; Deryabin, M. A. (2020). Residue Number System-Based Solution for Reducing the Hardware Cost of a Convolutional Neural Network. Neurocomputing, 407, 439-453, doi: 10.1016/j.neucom.2020.04.018. • Bajard, Jean-Claude; Méloni, Nicolas; Plantard, Thomas (2006-10-06) [July 2005]. "Efficient RNS bases for Cryptography" (PDF). IMACS'05: World Congress: Scientific Computation Applied Mathematics and Simulation. Paris, France. HAL Id: lirmm-00106470. Archived (PDF) from the original on 2021-01-23. Retrieved 2021-01-23. (1+7 pages) • Omondi, Amos; Premkumar, Benjamin (2007). Residue Number Systems: Theory and Implementation. London, UK: Imperial College Press. ISBN 978-1-86094-866-4. (296 pages) • Mohan, P. V. Ananda (2016). Residue Number Systems: Theory and Applications (1 ed.). Birkhäuser / Springer International Publishing Switzerland. doi:10.1007/978-3-319-41385-3. ISBN 978-3-319-41383-9. LCCN 2016947081. (351 pages) • Amir Sabbagh, Molahosseini; de Sousa, Leonel Seabra; Chip-Hong Chang, eds. (2017-03-21). Embedded Systems Design with Special Arithmetic and Number Systems (1 ed.). Springer International Publishing AG. doi:10.1007/978-3-319-49742-6. ISBN 978-3-319-49741-9. LCCN 2017934074. (389 pages) • "Division algorithms". Archived from the original on 2005-02-17. Retrieved 2023-08-24. • Knuth, Donald Ervin. The Art of Computer Programming. Addison Wesley. • Harvey, David (2010). "A multimodular algorithm for computing Bernoulli numbers". Mathematics of Computation. 79 (272): 2361–2370. doi:10.1090/S0025-5718-2010-02367-1. S2CID 11329343. • Lecerf, Grégoire; Schost, Éric (2003). "Fast multivariate power series multiplication in characteristic zero". SADIO Electronic Journal on Informatics and Operations Research. 5 (1): 1–10. • Orange, Sébastien; Renault, Guénaël; Yokoyama, Kazuhiro (2012). "Efficient arithmetic in successive algebraic extension fields using symmetries". Mathematics in Computer Science. 6 (3): 217–233. doi:10.1007/s11786-012-0112-y. S2CID 14360845. • Yokoyama, Kazuhiro (September 2012). "Usage of modular techniques for efficient computation of ideal operations". International Workshop on Computer Algebra in Scientific Computing. Berlin / Heidelberg, Germany: Springer. pp. 361–362. • Hladík, Jakub; Šimeček, Ivan (January 2012). "Modular Arithmetic for Solving Linear Equations on the GPU". Seminar on Numerical Analysis. pp. 68–70. • Pernet, Clément (June 2015). "Exact linear algebra algorithmic: Theory and practice". Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation. Association for Computing Machinery. pp. 17–18. • Lecerf, Grégoire (2018). "On the complexity of the Lickteig–Roy subresultant algorithm". Journal of Symbolic Computation. • Yokoyama, Kazuhiro; Noro, Masayuki; Takeshima, Taku (1994). "Multi-Modular Approach to Polynomial-Time Factorization of Bivariate Integral Polynomials". Journal of Symbolic Computation. 17 (6): 545–563. doi:10.1006/jsco.1994.1034. • Isupov, Konstantin (2021). "High-Performance Computation in Residue Number System Using Floating-Point Arithmetic". Computation. 9 (2): 9. doi:10.3390/computation9020009. ISSN 2079-3197.	Wikipedia
Calculator input methods There are various ways in which calculators interpret keystrokes. These can be categorized into two main types: • On a single-step or immediate-execution calculator, the user presses a key for each operation, calculating all the intermediate results, before the final value is shown.[1][2][3] • On an expression or formula calculator, one types in an expression and then presses a key, such as "=" or "Enter", to evaluate the expression.[4][5][6] There are various systems for typing in an expression, as described below. Immediate execution The immediate execution mode of operation (also known as single-step, algebraic entry system (AES)[7] or chain calculation mode) is commonly employed on most general-purpose calculators. In most simple four-function calculators, such as the Windows calculator in Standard mode and those included with most early operating systems, each binary operation is executed as soon as the next operator is pressed, and therefore the order of operations in a mathematical expression is not taken into account. Scientific calculators, including the Scientific mode in the Windows calculator and most modern software calculators, have buttons for brackets and can take order of operation into account. Also, for unary operations, like √ or x2, the number is entered first, then the operator; this is largely because the display screens on these kinds of calculators are generally composed entirely of seven-segment characters and thus capable of displaying only numbers, not the functions associated with them. This mode of operation also makes it impossible to change the expression being input without clearing the display entirely. Examples Formula keystrokes keystroke count $1+2\times 3$ 2 × 3 + 1 = 1 + 2 × 3 = 6 6 $\sin 30+2\times \cos 30$ 3 0 COS x 2 + 3 0 SIN = 3 0 SIN + 2 x 3 0 COS = 10 $5-3$ 5 − 3 = 4 $15+10+10+10$ 1 5 + 1 0 + 1 0 + 1 0 = 12 The first two examples have been given twice. The first version is for simple calculators, showing how it is necessary to rearrange operands in order to get the correct result. The second version is for scientific calculators, where operator precedence is observed. Different forms of operator precedence schemes exist. In the algebraic entry system with hierarchy (AESH),[7] the precedence of basic mathematical operators is taken into account,[7] whereas calculators with algebraic entry system with parentheses (AESP)[7] support the entry of parentheses.[7] An input scheme known as algebraic operating system (AOS)[7] combines both.[7] This is the name Texas Instruments uses for the input scheme used in some of its calculators..[8] Immediate-execution calculators are based on a mixture of infix and postfix notation: binary operations are done as infix, but unary operations are postfix. Because operators are applied one-at-a-time, the user must work out which operator key to use at each stage, and this can lead to problems.[9][10] When discussing these problems, Harold Thimbleby has pointed out that button-operated calculators "require numbers and operation signs to be punched in a certain order, and mistakes are easy to make and hard to spot".[11] Problems can occur because, for anything but the simplest calculation, in order to work out the value of a written formula, the user of a button-operated calculator is required to: • Rearrange the formula so that the value can be calculated by pressing buttons one at a time, while taking operator precedence and parentheses into account. • Use memory buttons to ensure that operations are applied in the correct order. • Use the special buttons ± and 1/x, that do not correspond to operations in the formula, for non-commutative operators. Mistakes can be hard to spot because: • For the above reasons, the sequence of button presses may bear little resemblance to the original formula. • The operation carried out when a button is pressed is not always the same as the button, but could be a previously entered operation. Examples of difficulties The simplest example given by Thimbleby of a possible problem when using an immediate-execution calculator is 4 × (−5). As a written formula the value of this is −20 because the minus sign is intended to indicate a negative number, rather than a subtraction, and this is the way that it would be interpreted by a formula calculator. On an immediate-execution calculator, depending on which keys are used and the order in which they are pressed, the result for this calculation may be different. Also there are differences between calculators in the way a given sequence of button presses is interpreted. The result can be: • −1: If the subtraction button − is pressed after the multiplication ×, it is interpreted as a correction of the × rather than a minus sign, so that 4 − 5 is calculated. • 20: If the change-sign button ± is pressed before the 5, it isn't interpreted as −5, and 4 × 5 is calculated. • −20: To get the right answer, ± must be pressed last, even though the minus sign isn't written last in the formula. The effects of operator precedence, parentheses and non-commutative operators, on the sequence of button presses, are illustrated by: • 4 − 5 × 6: The multiplication must be done first, and the formula has to be rearranged and calculated as −5 × 6 + 4. So ± and addition have to be used rather than subtraction. When + is pressed, the multiplication is performed. • 4 × (5 + 6): The addition must be done first, so the calculation carried out is (5 + 6) × 4. When × is pressed, the addition is performed. • 4 / (5 + 6): One way to do this is to calculate (5 + 6) / 4 first and then use the 1/x button, so the calculation carried out is 1/[(5 + 6)/4]. • 4 × 5 + 6 × 7: The two multiplications must be done before the addition, and one of the results must be put into memory. These are only simple examples, but immediate-execution calculators can present even greater problems in more complex cases. In fact, Thimbleby claims that users may have been conditioned to avoid them for all but the simplest calculations. Declarative and imperative tools The potential problems with immediate-execution calculators stem from the fact that they are imperative. This means that the user must provide details of how the calculation has to be performed. Thimbleby has identified the need for a calculator that is more automatic and therefore easier to use, and he states that such a calculator should be more declarative. This means that the user should be able to specify only what has to be done, not how, and in which order, it has to be done. Formula calculators are more declarative because the typed-in formula specifies what is to be done, and the user does not have to provide any details of the step-by-step order in which the calculation has to be performed. Declarative solutions are easier to understand than imperative solutions,[12] and there has been a long-term trend from imperative to declarative methods.[13][14] Formula calculators are part of this trend. Many software tools for the general user, such as spreadsheets, are declarative. Formula calculators are examples of such tools. Using the full power of the computer Software calculators that simulate hand-held, immediate execution calculators do not use the full power of the computer: "A computer is a far more powerful device than a hand-held calculator, and thus it is illogical and limiting to duplicate hand-held calculators on a computer." (Haxial Software Pty Ltd) Formula calculators use more of the computer's power because, besides calculating the value of a formula, they work out the order in which things should be done. Infix notation Infix notation is a method similar to immediate execution with AESH and/or AESP, but unary operations are input into the calculator in the same order as they are written on paper. Calculators that use infix notation tend to incorporate a dot-matrix display to display the expression being entered, frequently accompanied by a seven-segment display for the result of the expression. Because the expression is not evaluated until it is fully entered, there is provision for editing the entered expression at any point prior to evaluation, as well as replaying entered expressions and their answers from memory. Most graphing calculators by Casio and Texas Instruments use this method. On its scientific calculators, Sharp calls this method Direct Algebraic Logic (D.A.L.),[15] Casio calls this method the Visually Perfect Algebraic Method (V.P.A.M.),[16] and Texas Instruments calls it the Equation Operating System (EOS).[8] Examples Formula keystrokes keystroke count $1+2\times 3$ 1 + 2 × 3 = 6 $\sin 30+2\times \cos 30$ SIN 3 0 + 2 × COS 3 0 = 10 $(1+2)\times (3+4)$ ( 1 + 2 ) × ( 3 + 4 ) = 12 $15+10+10+10$ 1 5 + 1 0 + 1 0 + 1 0 = 12 Reverse Polish notation Main article: Reverse Polish notation In reverse Polish notation,[7] also known as postfix notation, all operations are entered after the operands on which the operation is performed. Reverse Polish notation is parenthesis-free, which usually leads to fewer button presses needed to perform an operation. By the use of a stack, one can enter formulas without the need to rearrange operands. Hewlett-Packard's calculators are well-known examples among calculators which use RPN. Early models, such as the HP-35, used RPN entirely without any alternative methods. Later models can switch between RPN and another notation, such as the HP-12C Platinum which includes both RPN and immediate execution (with operations performed strictly in the order in which inputs are entered), the HP 33s with both RPN and a hybrid of immediate/infix algebraic notation (operations are performed in accordance with standard order of operations, but single-operand functions are input with the operand first followed by the operator), and its successor the HP 35s with both RPN and standard algebraic infix notation. Examples Formula keystrokes keystroke count $1+2\times 3$ 1 ↵ Enter 2 ↵ Enter 3 × + 2 ↵ Enter 3 × 1 + 7 6 $\sin 30+2\times \cos 30$ 3 0 SIN 3 0 COS 2 x + 9 $(1+2)\times (3+4)$ 1 ↵ Enter 2 + 3 ↵ Enter 4 + × 9 $15+10+10+10$ 1 5 ↵ Enter 1 0 + 1 0 + 1 0 + 1 5 ↵ Enter 1 0 ↵ Enter ↵ Enter ↵ Enter + + + 1 5 ENTER^ 1 0 ENTER^ ENTER^ + + + 12 11 (RPL and Entry RPN)[17] 10 (Classical RPN)[17] Note: The first example illustrates one of the few cases where reverse Polish notation does not use the fewest button presses – provided one does not rearrange operands. If one would do so then only six keystrokes would be needed. BASIC notation BASIC notation is a particular implementation of infix notation where functions require their parameters to be in brackets. This method was used from the 1980s to the 1990s in BASIC programmable calculators and pocket computers. Texas Instruments would later implement the method in many of its graphing calculators, including the TI-83 and TI-84 Plus series. Most computer algebra systems also use this as the default input method. In BASIC notation, the formula is entered as it would be entered in BASIC, using the PRINT command – the PRINT command itself being optional. On pressing "ENTER" or "=", the result would be displayed. As with standard infix notation, typing mistakes in the entered formula could be corrected using the same editor function as the one used when programming the calculator. Examples Formula keystrokes keystroke count $1+2\times 3$ 1 + 2 × 3 ↵ Enter 6 $\sin 30\times \cos 30$ SIN ( 3 0 ) × COS ( 3 0 ) ↵ Enter S I N ( 3 0 ) × C O S ( 3 0 ) ↵ Enter 12 16 For the second example, two options are given depending on if the BASIC programmable pocket computers have dedicated trigonometric keys[18] or not.[19] Ten key notation The ten-key notation input method first became popular with accountants' paper tape adding machines. It generally makes the assumption that entered numbers are being summed, although other operations are supported. Each number entered is followed by its sign (+/−), and a running total is kept. An assumption is made that the last operand can be implicitly used next, so by just entering another + (for example), one will reuse the most recent operand. Ten key input mode is available in printing calculators from companies such as Sharp,[20] and in software calculators like Judy's TenKey[21] used by accounting firms. Online tenkey training and certification tools are available as well,[22][23] and some businesses use ten key typing speed as an employment criterion. Examples Formula keystrokes keystroke count $1+2\times 3$ 1 + 2 × 3 = + T 8 $\sin 30\times \cos 30$ 3 0 SIN × 3 0 COS = 8 $5-3$ 5 + 3 - T 5 $15+10+10+10$ 1 5 + 1 0 + + + T 9 Mathematical display Modern computer algebra systems, as well as many scientific and graphing calculators, allow for "pretty-printing", that is, entry of equations such that fractions, surds and integrals, etc. are displayed in the way they would normally be written. Such calculators are generally similar in appearance to those using infix notation, but feature a full dot-matrix display and templates for entering expressions, which are navigated using arrow keys on the calculator. The templates contain spaces for values or expressions to be entered, and empty values would typically result in a syntax error, making it more cumbersome to navigate than standard infix notation; standard infix notation is often an option on such calculators as well. Casio used to call this feature Natural Display or Natural textbook display,[24][25] but now uses Natural-VPAM.[26] Sharp calls this WriteView[27] on its scientific calculators and simply Equation Editor on its graphing calculators.[28] HP calls this its Textbook display setting,[29] which can be used in both RPN and Algebraic mode and in both the Stack and in the Equation Writer application.[30] Mathematica calls this Semantic-Faithful Typesetting.[31] Mathcad calls this standard math notation.[32] Maple has a Math Equation Editor,[33] but does not have a special name for this input method. Texas Instruments calls it MathPrint,[34] incorporating it in its high-end calculators, such as the TI-Nspire series, and in 2011 added the feature to its TI-84 series with the 2.55 OS update.[35] Examples Formula Keystrokes Keystroke Count $1+2\times 3$ 1 + 2 × 3 ↵ Enter 6 $\sin 30\times \cos 30$ SIN 3 0 → × COS 3 0 ↵ Enter SIN ( 3 0 ) × COS ( 3 0 ) ↵ Enter 9 12 $5-3$ 5 − 3 ↵ Enter 4 $15+10+10+10$ 1 5 + 1 0 + 1 0 + 1 0 ↵ Enter 12 For the second example, two options are given, depending on whether the calculators will automatically insert needed parentheses or not. Machines equipped with an alphanumeric display will display SIN(30)×COS(30) before ↵ Enter is pressed. See also • Calculator • Comparison of software calculators • Computer algebra system • Graphing calculator • Keystroke programming • Mathematical notation • Pocket computer • Programmable calculator • Scientific calculator References 1. Microsoft’s Windows Operating System Calculator Accessory; 2001. Available on a Windows PC at: Start/All Programs/Accessories/Calculator. 2. MotionNET Calculator page on the Internet Archived 1 May 2009 at the Wayback Machine; 2006. 3. Flow Simulation Ltd Virtual Calc98 page on the Internet; 2008. 4. Formula Calculators Pty Ltd [Home page on the Internet]; 2009. 5. Moisey Oysgelt JavaScript Formula Calculator page on the Internet; 2000. 6. Haxial Software Pty Ltd Calculator Product page on the Internet Archived 28 April 2009 at the Wayback Machine; 2001 7. Ball, John A. (1978). Algorithms for RPN calculators (1 ed.). Cambridge, Massachusetts, USA: Wiley-Interscience, John Wiley & Sons, Inc. ISBN 0-471-03070-8. 8. "Differences Between Algebraic Operating System (AOS) and Equation Operating System (EOS)". Texas Instruments. Retrieved 2022-11-29. 9. Harold Thimbleby (September 1998). "A new calculator and why it is necessary" (PDF). Computing Science, Middlesex University, London, UK. Archived from the original (PDF) on 2007-02-07. Retrieved 2009-05-04. 10. Neville Holmes. Truth and Clarity in Arithmetic Archived 7 October 2006 at the Wayback Machine, University of Tasmania; 2003. 11. Professor devises easier calculator; www.physorg.com. June 2005. 12. Roy E. Furman (July 2006). "Declarative Programming – Strategies for Solving Software Problems". Archived from the original on 2012-07-23. Retrieved 2009-05-04. 13. David A. Watt. Programming language concepts and paradigms, Prentice Hall; 1990. Citation 13 at http://citeseer.ist.psu.edu/context/14802/0. 14. Tatsuru Matsushita. Expressive Power of Declarative Programming Languages, PhD thesis, Department of Computer Science, University of York; October 1998. Citation 13 at http://citeseer.ist.psu.edu/context/14802/0. 15. "SHARP". global.sharp., 16. "General - Standard Scientific calculators - Calculators - CASIO". support.casio.com. 17. HP RPN Evolves hp.com 18. The picture of the Casio FX-880P shows sin, cos and tan keys on the second row right hand side. 19. The picture of the Sharp PC-1245 shows no trigonometric keys 20. "SIICA.sharpusa.com > Resources > More Products > Calculators". siica.sharpusa.net. 21. "Award-Winning Judy's TenKey Accounting Calculator for Windows". www.judysapps.com. 22. "Ten Key Certificate on Learn2Type.com!". Learn2Type.com. 23. "None". Retrieved 2023-06-16. 24. Natural textbook display - Scientific calculator. 25. "CASIO WEW Worldwide Education Website". CASIO WEW Worldwide Education Website. 26. Natural Visually Perfect Algebraic Mode (V.P.A.M) - Scientific calculator Archived 27 April 2009 at the Wayback Machine. 27. WriteView. 28. Sharp Graphing equation editor. 29. Inc., HP. "HP Support document - HP Support Center". h20564.www2.hp.com. Archived from the original on 2016-08-26. Retrieved 2016-08-23. {{cite web}}: \|last= has generic name (help) 30. http://h20331.www2.hp.com/Hpsub/downloads/50gUsing_the_EquationWriter_Part2.pdf. Archived 2011-12-18 at the Wayback Machine 31. Semantic-Faithful Typesetting. 32. Mathcad Archived 20 September 2008 at the Wayback Machine. 33. "Math Equation Editor - Maple Features - Maplesoft". www.maplesoft.com. 34. "TI Products \| Graphing Calculators \| Scientific Calculators". education.ti.com. 35. "Texas Instruments releases new OS for TI-84, 2.55 MP". Tech Powered Math. 2011-01-14. Retrieved 2018-05-12. Further reading • Kasprzyk, Dennis Michael; Drury, Colin G.; Bialas, Wayne F. (1979) [1978-09-25]. "Human behaviour and performance in calculator use with Algebraic and Reverse Polish Notation". Ergonomics. Department of Industrial Engineering, State University of New York at Buffalo, Amherst, New York, USA: Taylor & Francis. 22 (9): 1011–1019. doi:10.1080/00140137908924675. • "Advanced Calculator Logic HP RPN/Algebraic: A Comparative Analysis" (PDF). Corvallis, Oregon, USA: Hewlett-Packard Corporation. 1979. 5953-1930. Archived (PDF) from the original on 2022-12-26. Retrieved 2022-12-26. (13 pages) • Agate, Seb J.; Drury, Colin G. (March 1980). "Electronic calculators: which notation is the better?". Applied Ergonomics. Department of Industrial Engineering, University at Buffalo, State University of New York, USA: IPC Business Press. 11 (1): 2–6. doi:10.1016/0003-6870(80)90114-3. PMID 15676368. 0003-6870/80/01 0002-05. Archived from the original on 2018-09-22. Retrieved 2018-09-22. (5 pages) • Suydam, Marilyn N. (December 1980). Calculators: A Categorized Compilation of References. Supplement 1 (PDF). Columbus, Ohio, USA: Calculator Information Center, Ohio State University. ED199087. SE034434. Archived (PDF) from the original on 2021-09-19. Retrieved 2022-10-16. (64 pages) • Kreifeldt, John G.; McCarthy, Mary E. (1981-10-15) [1981-06-16/18]. Written at Department of Engineering Design, Tufts University, Medford, Massachusetts, USA. Interruption as a test of the user-computer interface (PDF). Proceedings of the Seventeenth Annual Conference on Manual Control. University of California, Los Angeles, California, USA: Jet Propulsion Laboratory / Office of Naval Research / NASA. pp. 655–667. 02155, N82-13721, 82N13721, 19820005848, JPL 81-95. Archived (PDF) from the original on 2022-01-30. Retrieved 2018-09-22. Proceedings of the Seventeenth Annual Conference on Manual Control: University of California, Los Angeles, June 16-18, 1981 (13 of 702 pages) • Kreifeldt, John G. (October 1981). "Hand Calculator Performance Under Interrupted Operation". Proceedings of the Human Factors Society Annual Meeting. Department of Engineering Design, Tufts University, Medford, Massachusetts, USA. 25 (1): 329–332. doi:10.1177/107118138102500187. S2CID 106904297. (4 pages) • Young, Richard M. (July 1981). "The machine inside the machine: Users' models of pocket calculators". International Journal of Man-Machine Studies. 15 (1): 51–85. doi:10.1016/S0020-7373(81)80023-5. ISSN 0020-7373. (35 pages) • Young, Richard M. (1984). "Artificial Intelligence: Conceptual Models of Ill-Defined Systems". Written at University of Massachusetts, Amherst, Massachusetts, USA; MRC Applied Psychology Unit, Cambridge, UK. In Selfridge, Oliver Gordon; Rissland, Edwina L. [at Wikidata]; Arbib, Michael Anthony (eds.). Adaptive Control of Ill-Defined Systems. NATO Conference Series (NATOCS, volume 16); II Systems Science (SYSC) (1st illustrated ed.). New York & London: Plenum Press, Plenum Publishing Corporation / Springer. pp. 165–176. doi:10.1007/978-1-4684-8941-5. ISBN 978-1-4684-8943-9. LCCN 83-17699. Retrieved 2022-12-26. (12 pages) • Hoffman, Errol; Ma, Patrick; See, Jason; Yong, Chee Kee; Brand, Jason; Poulton, Matthew (1994). "Calculator logic: when and why is RPN superior to algebraic?". Applied Ergonomics. 25 (5): 327–333. doi:10.1016/0003-6870(94)90048-5. • Redin, James (2005-02-12) [1997-10-05]. "RPN or DAL? A brief analysis of Reverse Polish Notation against Direct Algebraic Logic". Archived from the original on 2017-06-24. Retrieved 2015-09-12.	Wikipedia
RP (complexity) In computational complexity theory, randomized polynomial time (RP) is the complexity class of problems for which a probabilistic Turing machine exists with these properties: RP algorithm (1 run) Answer produced Correct answer Yes No Yes ≥ 1/2 ≤ 1/2 No 0 1 RP algorithm (n runs) Answer produced Correct answer Yes No Yes ≥ 1 − 2−n ≤ 2−n No 0 1 co-RP algorithm (1 run) Answer produced Correct answer Yes No Yes 1 0 No ≤ 1/2 ≥ 1/2 • It always runs in polynomial time in the input size • If the correct answer is NO, it always returns NO • If the correct answer is YES, then it returns YES with probability at least 1/2 (otherwise, it returns NO). In other words, the algorithm is allowed to flip a truly random coin while it is running. The only case in which the algorithm can return YES is if the actual answer is YES; therefore if the algorithm terminates and produces YES, then the correct answer is definitely YES; however, the algorithm can terminate with NO regardless of the actual answer. That is, if the algorithm returns NO, it might be wrong. Some authors call this class R, although this name is more commonly used for the class of recursive languages. If the correct answer is YES and the algorithm is run n times with the result of each run statistically independent of the others, then it will return YES at least once with probability at least 1 − 2−n. So if the algorithm is run 100 times, then the chance of it giving the wrong answer every time is lower than the chance that cosmic rays corrupted the memory of the computer running the algorithm.[1] In this sense, if a source of random numbers is available, most algorithms in RP are highly practical. The fraction 1/2 in the definition is arbitrary. The set RP will contain exactly the same problems, even if the 1/2 is replaced by any constant nonzero probability less than 1; here constant means independent of the input to the algorithm. Formal definition A language L is in RP if and only if there exists a probabilistic Turing machine M, such that • M runs for polynomial time on all inputs • For all x in L, M outputs 1 with probability greater than or equal to 1/2 • For all x not in L, M outputs 0 Alternatively, RP can be defined using only deterministic Turing machines. A language L is in RP if and only if there exists a polynomial p and deterministic Turing machine M, such that • M runs for polynomial time p on all inputs • For all x in L, the fraction of strings y of length p(\|x\|) which satisfy $M(x,y)=1$ is greater than or equal to 1/2 • For all x not in L, and all strings y of length p(\|x\|), $M(x,y)=0$ In this definition, the string y corresponds to the output of the random coin flips that the probabilistic Turing machine would have made. For some applications this definition is preferable since it does not mention probabilistic Turing machines. Related complexity classes The definition of RP says that a YES-answer is always right and that a NO-answer might be wrong, as a YES-instance can return a NO-answer. The complexity class co-RP is the complement, where a YES-answer might be wrong while a NO-answer is always right. The class BPP describes algorithms that can give incorrect answers on both YES and NO instances, and thus contains both RP and co-RP. The intersection of the sets RP and co-RP is called ZPP. Just as RP may be called R, some authors use the name co-R rather than co-RP. Connection to P and NP Unsolved problem in computer science: ${\mathsf {P}}{\overset {?}{=}}{\mathsf {RP}}$ (more unsolved problems in computer science) P is a subset of RP, which is a subset of NP. Similarly, P is a subset of co-RP which is a subset of co-NP. It is not known whether these inclusions are strict. However, if the commonly believed conjecture P = BPP is true, then RP, co-RP, and P collapse (are all equal). Assuming in addition that P ≠ NP, this then implies that RP is strictly contained in NP. It is not known whether RP = co-RP, or whether RP is a subset of the intersection of NP and co-NP, though this would be implied by P = BPP. A natural example of a problem in co-RP currently not known to be in P is Polynomial Identity Testing, the problem of deciding whether a given multivariate arithmetic expression over the integers is the zero-polynomial. For instance, x·x − y·y − (x + y)·(x − y) is the zero-polynomial while x·x + y·y is not. An alternative characterization of RP that is sometimes easier to use is the set of problems recognizable by nondeterministic Turing machines where the machine accepts if and only if at least some constant fraction of the computation paths, independent of the input size, accept. NP on the other hand, needs only one accepting path, which could constitute an exponentially small fraction of the paths. This characterization makes the fact that RP is a subset of NP obvious. See also • Randomized algorithm • BPP • ZPP References 1. This comparison is attributed to Michael O. Rabin on p. 252 of Gasarch, William (2014), "Classifying Problems into Complexity Classes", in Memon, Atif (ed.), Advances in Computers, Vol. 95 (PDF), Academic Press, pp. 239–292. External links • RP at the Complexity Zoo Important complexity classes Considered feasible • DLOGTIME • AC0 • ACC0 • TC0 • L • SL • RL • NL • NL-complete • NC • SC • CC • P • P-complete • ZPP • RP • BPP • BQP • APX • FP Suspected infeasible • UP • NP • NP-complete • NP-hard • co-NP • co-NP-complete • AM • QMA • PH • ⊕P • PP • #P • #P-complete • IP • PSPACE • PSPACE-complete Considered infeasible • EXPTIME • NEXPTIME • EXPSPACE • 2-EXPTIME • ELEMENTARY • PR • R • RE • ALL Class hierarchies • Polynomial hierarchy • Exponential hierarchy • Grzegorczyk hierarchy • Arithmetical hierarchy • Boolean hierarchy Families of classes • DTIME • NTIME • DSPACE • NSPACE • Probabilistically checkable proof • Interactive proof system List of complexity classes	Wikipedia
Real projective space In mathematics, real projective space, denoted $\mathbb {RP} ^{n}$ or $\mathbb {P} _{n}(\mathbb {R} ),$ is the topological space of lines passing through the origin 0 in the real space $\mathbb {R} ^{n+1}.$ It is a compact, smooth manifold of dimension n, and is a special case $\mathbf {Gr} (1,\mathbb {R} ^{n+1})$ of a Grassmannian space. Basic properties Construction As with all projective spaces, RPn is formed by taking the quotient of Rn+1 ∖ {0} under the equivalence relation x ∼ λx for all real numbers λ ≠ 0. For all x in Rn+1 ∖ {0} one can always find a λ such that λx has norm 1. There are precisely two such λ differing by sign. Thus RPn can also be formed by identifying antipodal points of the unit n-sphere, Sn, in Rn+1. One can further restrict to the upper hemisphere of Sn and merely identify antipodal points on the bounding equator. This shows that RPn is also equivalent to the closed n-dimensional disk, Dn, with antipodal points on the boundary, ∂Dn = Sn−1, identified. Low-dimensional examples • RP1 is called the real projective line, which is topologically equivalent to a circle. • RP2 is called the real projective plane. This space cannot be embedded in R3. It can however be embedded in R4 and can be immersed in R3 (see here). The questions of embeddability and immersibility for projective n-space have been well-studied.[1] • RP3 is (diffeomorphic to) SO(3), hence admits a group structure; the covering map S3 → RP3 is a map of groups Spin(3) → SO(3), where Spin(3) is a Lie group that is the universal cover of SO(3). Topology The antipodal map on the n-sphere (the map sending x to −x) generates a Z2 group action on Sn. As mentioned above, the orbit space for this action is RPn. This action is actually a covering space action giving Sn as a double cover of RPn. Since Sn is simply connected for n ≥ 2, it also serves as the universal cover in these cases. It follows that the fundamental group of RPn is Z2 when n > 1. (When n = 1 the fundamental group is Z due to the homeomorphism with S1). A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in Sn down to RPn. The projective n-space is compact, connected, and has a fundamental group isomorphic to the cyclic group of order 2: its universal covering space is given by the antipody quotient map from the n-sphere, a simply connected space. It is a double cover. The antipode map on Rp has sign $(-1)^{p}$, so it is orientation-preserving if and only if p is even. The orientation character is thus: the non-trivial loop in $\pi _{1}(\mathbf {RP} ^{n})$ acts as $(-1)^{n+1}$ on orientation, so RPn is orientable if and only if n + 1 is even, i.e., n is odd.[2] The projective n-space is in fact diffeomorphic to the submanifold of R(n+1)2 consisting of all symmetric (n + 1) × (n + 1) matrices of trace 1 that are also idempotent linear transformations. Geometry of real projective spaces Real projective space admits a constant positive scalar curvature metric, coming from the double cover by the standard round sphere (the antipodal map is locally an isometry). For the standard round metric, this has sectional curvature identically 1. In the standard round metric, the measure of projective space is exactly half the measure of the sphere. Smooth structure Real projective spaces are smooth manifolds. On Sn, in homogeneous coordinates, (x1, ..., xn+1), consider the subset Ui with xi ≠ 0. Each Ui is homeomorphic to the disjoint union of two open unit balls in Rn that map to the same subset of RPn and the coordinate transition functions are smooth. This gives RPn a smooth structure. Structure as a CW complex Real projective space RPn admits the structure of a CW complex with 1 cell in every dimension. In homogeneous coordinates (x1 ... xn+1) on Sn, the coordinate neighborhood U1 = {(x1 ... xn+1) \| x1 ≠ 0} can be identified with the interior of n-disk Dn. When xi = 0, one has RPn−1. Therefore the n−1 skeleton of RPn is RPn−1, and the attaching map f : Sn−1 → RPn−1 is the 2-to-1 covering map. One can put $\mathbf {RP} ^{n}=\mathbf {RP} ^{n-1}\cup _{f}D^{n}.$ Induction shows that RPn is a CW complex with 1 cell in every dimension up to n. The cells are Schubert cells, as on the flag manifold. That is, take a complete flag (say the standard flag) 0 = V0 < V1 <...< Vn; then the closed k-cell is lines that lie in Vk. Also the open k-cell (the interior of the k-cell) is lines in Vk \ Vk−1 (lines in Vk but not Vk−1). In homogeneous coordinates (with respect to the flag), the cells are ${\begin{array}{c}[:0:0:\dots :0]\\{[}::0:\dots :0]\\\vdots \\{[}:::\dots :].\end{array}}$ :].\end{array}}} This is not a regular CW structure, as the attaching maps are 2-to-1. However, its cover is a regular CW structure on the sphere, with 2 cells in every dimension; indeed, the minimal regular CW structure on the sphere. In light of the smooth structure, the existence of a Morse function would show RPn is a CW complex. One such function is given by, in homogeneous coordinates, $g(x_{1},\ldots ,x_{n+1})=\sum _{i=1}^{n+1}i\cdot \|x_{i}\|^{2}.$ On each neighborhood Ui, g has nondegenerate critical point (0,...,1,...,0) where 1 occurs in the i-th position with Morse index i. This shows RPn is a CW complex with 1 cell in every dimension. Tautological bundles Real projective space has a natural line bundle over it, called the tautological bundle. More precisely, this is called the tautological subbundle, and there is also a dual n-dimensional bundle called the tautological quotient bundle. Algebraic topology of real projective spaces Homotopy groups The higher homotopy groups of RPn are exactly the higher homotopy groups of Sn, via the long exact sequence on homotopy associated to a fibration. Explicitly, the fiber bundle is: $\mathbf {Z} _{2}\to S^{n}\to \mathbf {RP} ^{n}.$ You might also write this as $S^{0}\to S^{n}\to \mathbf {RP} ^{n}$ or $O(1)\to S^{n}\to \mathbf {RP} ^{n}$ by analogy with complex projective space. The homotopy groups are: $\pi _{i}(\mathbf {RP} ^{n})={\begin{cases}0&i=0\\\mathbf {Z} &i=1,n=1\\\mathbf {Z} /2\mathbf {Z} &i=1,n>1\\\pi _{i}(S^{n})&i>1,n>0.\end{cases}}$ Homology The cellular chain complex associated to the above CW structure has 1 cell in each dimension 0, ..., n. For each dimensional k, the boundary maps dk : δDk → RPk−1/RPk−2 is the map that collapses the equator on Sk−1 and then identifies antipodal points. In odd (resp. even) dimensions, this has degree 0 (resp. 2): $\deg(d_{k})=1+(-1)^{k}.$ Thus the integral homology is $H_{i}(\mathbf {RP} ^{n})={\begin{cases}\mathbf {Z} &i=0{\text{ or }}i=n{\text{ odd,}}\\\mathbf {Z} /2\mathbf {Z} &0<i<n,\ i\ {\text{odd,}}\\0&{\text{else.}}\end{cases}}$ RPn is orientable if and only if n is odd, as the above homology calculation shows. Infinite real projective space The infinite real projective space is constructed as the direct limit or union of the finite projective spaces: $\mathbf {RP} ^{\infty }:=\lim _{n}\mathbf {RP} ^{n}.$ This space is classifying space of O(1), the first orthogonal group. The double cover of this space is the infinite sphere $S^{\infty }$, which is contractible. The infinite projective space is therefore the Eilenberg–MacLane space K(Z2, 1). For each nonnegative integer q, the modulo 2 homology group $H_{q}(\mathbf {RP} ^{\infty };\mathbf {Z} /2)=\mathbf {Z} /2$. Its cohomology ring modulo 2 is $H^{*}(\mathbf {RP} ^{\infty };\mathbf {Z} /2\mathbf {Z} )=\mathbf {Z} /2\mathbf {Z} [w_{1}],$ where $w_{1}$ is the first Stiefel–Whitney class: it is the free $\mathbf {Z} /2\mathbf {Z} $-algebra on $w_{1}$, which has degree 1. See also • Complex projective space • Quaternionic projective space • Lens space • Real projective plane Notes 1. See the table of Don Davis for a bibliography and list of results. 2. J. T. Wloka; B. Rowley; B. Lawruk (1995). Boundary Value Problems for Elliptic Systems. Cambridge University Press. p. 197. ISBN 978-0-521-43011-1. References • Bredon, Glen. Topology and geometry, Graduate Texts in Mathematics, Springer Verlag 1993, 1996 • Davis, Donald. "Table of immersions and embeddings of real projective spaces". Retrieved 22 Sep 2011. • Hatcher, Allen (2001). Algebraic Topology. Cambridge University Press. ISBN 978-0-521-79160-1.	Wikipedia
QR decomposition In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Cases and definitions Square matrix Any real square matrix A may be decomposed as $A=QR,$ where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning $Q^{\textsf {T}}=Q^{-1}$) and R is an upper triangular matrix (also called right triangular matrix). If A is invertible, then the factorization is unique if we require the diagonal elements of R to be positive. If instead A is a complex square matrix, then there is a decomposition A = QR where Q is a unitary matrix (so $Q^{\dagger }=Q^{-1}$). If A has n linearly independent columns, then the first n columns of Q form an orthonormal basis for the column space of A. More generally, the first k columns of Q form an orthonormal basis for the span of the first k columns of A for any 1 ≤ k ≤ n.[1] The fact that any column k of A only depends on the first k columns of Q corresponds to the triangular form of R.[1] Rectangular matrix More generally, we can factor a complex m×n matrix A, with m ≥ n, as the product of an m×m unitary matrix Q and an m×n upper triangular matrix R. As the bottom (m−n) rows of an m×n upper triangular matrix consist entirely of zeroes, it is often useful to partition R, or both R and Q: $A=QR=Q{\begin{bmatrix}R_{1}\\0\end{bmatrix}}={\begin{bmatrix}Q_{1}&Q_{2}\end{bmatrix}}{\begin{bmatrix}R_{1}\\0\end{bmatrix}}=Q_{1}R_{1},$ where R1 is an n×n upper triangular matrix, 0 is an (m − n)×n zero matrix, Q1 is m×n, Q2 is m×(m − n), and Q1 and Q2 both have orthogonal columns. Golub & Van Loan (1996, §5.2) call Q1R1 the thin QR factorization of A; Trefethen and Bau call this the reduced QR factorization.[1] If A is of full rank n and we require that the diagonal elements of R1 are positive then R1 and Q1 are unique, but in general Q2 is not. R1 is then equal to the upper triangular factor of the Cholesky decomposition of A* A (= ATA if A is real). QL, RQ and LQ decompositions Analogously, we can define QL, RQ, and LQ decompositions, with L being a lower triangular matrix. Computing the QR decomposition There are several methods for actually computing the QR decomposition, such as by means of the Gram–Schmidt process, Householder transformations, or Givens rotations. Each has a number of advantages and disadvantages. Using the Gram–Schmidt process Further information: Gram–Schmidt § Numerical stability Consider the Gram–Schmidt process applied to the columns of the full column rank matrix $A={\begin{bmatrix}\mathbf {a} _{1}&\cdots &\mathbf {a} _{n}\end{bmatrix}}$, with inner product $\langle \mathbf {v} ,\mathbf {w} \rangle =\mathbf {v} ^{\textsf {T}}\mathbf {w} $ (or $\langle \mathbf {v} ,\mathbf {w} \rangle =\mathbf {v} ^{\dagger }\mathbf {w} $ for the complex case). Define the projection: $\operatorname {proj} _{\mathbf {u} }\mathbf {a} ={\frac {\left\langle \mathbf {u} ,\mathbf {a} \right\rangle }{\left\langle \mathbf {u} ,\mathbf {u} \right\rangle }}{\mathbf {u} }$ then: ${\begin{aligned}\mathbf {u} _{1}&=\mathbf {a} _{1},&\mathbf {e} _{1}&={\frac {\mathbf {u} _{1}}{\\|\mathbf {u} _{1}\\|}}\\\mathbf {u} _{2}&=\mathbf {a} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}\mathbf {a} _{2},&\mathbf {e} _{2}&={\frac {\mathbf {u} _{2}}{\\|\mathbf {u} _{2}\\|}}\\\mathbf {u} _{3}&=\mathbf {a} _{3}-\operatorname {proj} _{\mathbf {u} _{1}}\mathbf {a} _{3}-\operatorname {proj} _{\mathbf {u} _{2}}\mathbf {a} _{3},&\mathbf {e} _{3}&={\frac {\mathbf {u} _{3}}{\\|\mathbf {u} _{3}\\|}}\\&\;\;\vdots &&\;\;\vdots \\\mathbf {u} _{k}&=\mathbf {a} _{k}-\sum _{j=1}^{k-1}\operatorname {proj} _{\mathbf {u} _{j}}\mathbf {a} _{k},&\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}}{\\|\mathbf {u} _{k}\\|}}\end{aligned}}$ We can now express the $\mathbf {a} _{i}$s over our newly computed orthonormal basis: ${\begin{aligned}\mathbf {a} _{1}&=\left\langle \mathbf {e} _{1},\mathbf {a} _{1}\right\rangle \mathbf {e} _{1}\\\mathbf {a} _{2}&=\left\langle \mathbf {e} _{1},\mathbf {a} _{2}\right\rangle \mathbf {e} _{1}+\left\langle \mathbf {e} _{2},\mathbf {a} _{2}\right\rangle \mathbf {e} _{2}\\\mathbf {a} _{3}&=\left\langle \mathbf {e} _{1},\mathbf {a} _{3}\right\rangle \mathbf {e} _{1}+\left\langle \mathbf {e} _{2},\mathbf {a} _{3}\right\rangle \mathbf {e} _{2}+\left\langle \mathbf {e} _{3},\mathbf {a} _{3}\right\rangle \mathbf {e} _{3}\\&\;\;\vdots \\\mathbf {a} _{k}&=\sum _{j=1}^{k}\left\langle \mathbf {e} _{j},\mathbf {a} _{k}\right\rangle \mathbf {e} _{j}\end{aligned}}$ where $\left\langle \mathbf {e} _{i},\mathbf {a} _{i}\right\rangle =\left\\|\mathbf {u} _{i}\right\\|$. This can be written in matrix form: $A=QR$ where: $Q={\begin{bmatrix}\mathbf {e} _{1}&\cdots &\mathbf {e} _{n}\end{bmatrix}}$ and $R={\begin{bmatrix}\langle \mathbf {e} _{1},\mathbf {a} _{1}\rangle &\langle \mathbf {e} _{1},\mathbf {a} _{2}\rangle &\langle \mathbf {e} _{1},\mathbf {a} _{3}\rangle &\cdots &\langle \mathbf {e} _{1},\mathbf {a} _{n}\rangle \\0&\langle \mathbf {e} _{2},\mathbf {a} _{2}\rangle &\langle \mathbf {e} _{2},\mathbf {a} _{3}\rangle &\cdots &\langle \mathbf {e} _{2},\mathbf {a} _{n}\rangle \\0&0&\langle \mathbf {e} _{3},\mathbf {a} _{3}\rangle &\cdots &\langle \mathbf {e} _{3},\mathbf {a} _{n}\rangle \\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &\langle \mathbf {e} _{n},\mathbf {a} _{n}\rangle \\\end{bmatrix}}.$ Example Consider the decomposition of $A={\begin{bmatrix}12&-51&4\\6&167&-68\\-4&24&-41\end{bmatrix}}.$ Recall that an orthonormal matrix $Q$ has the property $Q^{\textsf {T}}Q=I$. Then, we can calculate $Q$ by means of Gram–Schmidt as follows: ${\begin{aligned}U={\begin{bmatrix}\mathbf {u} _{1}&\mathbf {u} _{2}&\mathbf {u} _{3}\end{bmatrix}}&={\begin{bmatrix}12&-69&-58/5\\6&158&6/5\\-4&30&-33\end{bmatrix}};\\Q={\begin{bmatrix}{\frac {\mathbf {u} _{1}}{\\|\mathbf {u} _{1}\\|}}&{\frac {\mathbf {u} _{2}}{\\|\mathbf {u} _{2}\\|}}&{\frac {\mathbf {u} _{3}}{\\|\mathbf {u} _{3}\\|}}\end{bmatrix}}&={\begin{bmatrix}6/7&-69/175&-58/175\\3/7&158/175&6/175\\-2/7&6/35&-33/35\end{bmatrix}}.\end{aligned}}$ Thus, we have ${\begin{aligned}Q^{\textsf {T}}A&=Q^{\textsf {T}}Q\,R=R;\\R&=Q^{\textsf {T}}A={\begin{bmatrix}14&21&-14\\0&175&-70\\0&0&35\end{bmatrix}}.\end{aligned}}$ Relation to RQ decomposition The RQ decomposition transforms a matrix A into the product of an upper triangular matrix R (also known as right-triangular) and an orthogonal matrix Q. The only difference from QR decomposition is the order of these matrices. QR decomposition is Gram–Schmidt orthogonalization of columns of A, started from the first column. RQ decomposition is Gram–Schmidt orthogonalization of rows of A, started from the last row. Advantages and disadvantages The Gram-Schmidt process is inherently numerically unstable. While the application of the projections has an appealing geometric analogy to orthogonalization, the orthogonalization itself is prone to numerical error. A significant advantage is the ease of implementation. Using Householder reflections A Householder reflection (or Householder transformation) is a transformation that takes a vector and reflects it about some plane or hyperplane. We can use this operation to calculate the QR factorization of an m-by-n matrix $A$ with m ≥ n. Q can be used to reflect a vector in such a way that all coordinates but one disappear. Let $\mathbf {x} $ be an arbitrary real m-dimensional column vector of $A$ such that $\\|\mathbf {x} \\|=\|\alpha \|$ for a scalar α. If the algorithm is implemented using floating-point arithmetic, then α should get the opposite sign as the k-th coordinate of $\mathbf {x} $, where $x_{k}$ is to be the pivot coordinate after which all entries are 0 in matrix A's final upper triangular form, to avoid loss of significance. In the complex case, set[2] $\alpha =-e^{i\arg x_{k}}\\|\mathbf {x} \\|$ and substitute transposition by conjugate transposition in the construction of Q below. Then, where $\mathbf {e} _{1}$ is the vector [1 0 ⋯ 0]T, \|\|·\|\| is the Euclidean norm and $I$ is an m×m identity matrix, set ${\begin{aligned}\mathbf {u} &=\mathbf {x} -\alpha \mathbf {e} _{1},\\\mathbf {v} &={\frac {\mathbf {u} }{\\|\mathbf {u} \\|}},\\Q&=I-2\mathbf {v} \mathbf {v} ^{\textsf {T}}.\end{aligned}}$ Or, if $A$ is complex $Q=I-2\mathbf {v} \mathbf {v} ^{\dagger }.$ $Q$ is an m-by-m Householder matrix, which is both symmetric and orthogonal (Hermitian and unitary in the complex case), and $Q\mathbf {x} ={\begin{bmatrix}\alpha \\0\\\vdots \\0\end{bmatrix}}.$ This can be used to gradually transform an m-by-n matrix A to upper triangular form. First, we multiply A with the Householder matrix Q1 we obtain when we choose the first matrix column for x. This results in a matrix Q1A with zeros in the left column (except for the first row). $Q_{1}A={\begin{bmatrix}\alpha _{1}&\star &\cdots &\star \\0&&&\\\vdots &&A'&\\0&&&\end{bmatrix}}$ This can be repeated for A′ (obtained from Q1A by deleting the first row and first column), resulting in a Householder matrix Q′2. Note that Q′2 is smaller than Q1. Since we want it really to operate on Q1A instead of A′ we need to expand it to the upper left, filling in a 1, or in general: $Q_{k}={\begin{bmatrix}I_{k-1}&0\\0&Q_{k}'\end{bmatrix}}.$ After $t$ iterations of this process, $t=\min(m-1,n)$, $R=Q_{t}\cdots Q_{2}Q_{1}A$ is an upper triangular matrix. So, with ${\begin{aligned}Q^{\textsf {T}}&=Q_{t}\cdots Q_{2}Q_{1},\\Q&=Q_{1}^{\textsf {T}}Q_{2}^{\textsf {T}}\cdots Q_{t}^{\textsf {T}},\\&=Q_{1}Q_{2}\cdots Q_{t},\end{aligned}}$ $A=QR$ is a QR decomposition of $A$. This method has greater numerical stability than the Gram–Schmidt method above. The following table gives the number of operations in the k-th step of the QR-decomposition by the Householder transformation, assuming a square matrix with size n. Operation Number of operations in the k-th step Multiplications $2(n-k+1)^{2}$ Additions $(n-k+1)^{2}+(n-k+1)(n-k)+2$ Division $1$ Square root $1$ Summing these numbers over the n − 1 steps (for a square matrix of size n), the complexity of the algorithm (in terms of floating point multiplications) is given by ${\frac {2}{3}}n^{3}+n^{2}+{\frac {1}{3}}n-2=O\left(n^{3}\right).$ Example Let us calculate the decomposition of $A={\begin{bmatrix}12&-51&4\\6&167&-68\\-4&24&-41\end{bmatrix}}.$ First, we need to find a reflection that transforms the first column of matrix A, vector $\mathbf {a} _{1}={\begin{bmatrix}12&6&-4\end{bmatrix}}^{\textsf {T}}$, into $\left\\|\mathbf {a} _{1}\right\\|\mathbf {e} _{1}={\begin{bmatrix}\alpha &0&0\end{bmatrix}}^{\textsf {T}}$. Now, $\mathbf {u} =\mathbf {x} -\alpha \mathbf {e} _{1},$ and $\mathbf {v} ={\frac {\mathbf {u} }{\\|\mathbf {u} \\|}}.$ Here, $\alpha =14$ and $\mathbf {x} =\mathbf {a} _{1}={\begin{bmatrix}12&6&-4\end{bmatrix}}^{\textsf {T}}$ Therefore $\mathbf {u} ={\begin{bmatrix}-2&6&-4\end{bmatrix}}^{\textsf {T}}=2{\begin{bmatrix}-1&3&-2\end{bmatrix}}^{\textsf {T}}$ and $\mathbf {v} ={\frac {1}{\sqrt {14}}}{\begin{bmatrix}-1&3&-2\end{bmatrix}}^{\textsf {T}}$, and then ${\begin{aligned}Q_{1}={}&I-{\frac {2}{{\sqrt {14}}{\sqrt {14}}}}{\begin{bmatrix}-1\\3\\-2\end{bmatrix}}{\begin{bmatrix}-1&3&-2\end{bmatrix}}\\={}&I-{\frac {1}{7}}{\begin{bmatrix}1&-3&2\\-3&9&-6\\2&-6&4\end{bmatrix}}\\={}&{\begin{bmatrix}6/7&3/7&-2/7\\3/7&-2/7&6/7\\-2/7&6/7&3/7\\\end{bmatrix}}.\end{aligned}}$ Now observe: $Q_{1}A={\begin{bmatrix}14&21&-14\\0&-49&-14\\0&168&-77\end{bmatrix}},$ so we already have almost a triangular matrix. We only need to zero the (3, 2) entry. Take the (1, 1) minor, and then apply the process again to $A'=M_{11}={\begin{bmatrix}-49&-14\\168&-77\end{bmatrix}}.$ By the same method as above, we obtain the matrix of the Householder transformation $Q_{2}={\begin{bmatrix}1&0&0\\0&-7/25&24/25\\0&24/25&7/25\end{bmatrix}}$ after performing a direct sum with 1 to make sure the next step in the process works properly. Now, we find $Q=Q_{1}^{\textsf {T}}Q_{2}^{\textsf {T}}={\begin{bmatrix}6/7&-69/175&58/175\\3/7&158/175&-6/175\\-2/7&6/35&33/35\end{bmatrix}}.$ Or, to four decimal digits, ${\begin{aligned}Q&=Q_{1}^{\textsf {T}}Q_{2}^{\textsf {T}}={\begin{bmatrix}0.8571&-0.3943&0.3314\\0.4286&0.9029&-0.0343\\-0.2857&0.1714&0.9429\end{bmatrix}}\\R&=Q_{2}Q_{1}A=Q^{\textsf {T}}A={\begin{bmatrix}14&21&-14\\0&175&-70\\0&0&-35\end{bmatrix}}.\end{aligned}}$ The matrix Q is orthogonal and R is upper triangular, so A = QR is the required QR decomposition. Advantages and disadvantages The use of Householder transformations is inherently the most simple of the numerically stable QR decomposition algorithms due to the use of reflections as the mechanism for producing zeroes in the R matrix. However, the Householder reflection algorithm is bandwidth heavy and not parallelizable, as every reflection that produces a new zero element changes the entirety of both Q and R matrices. Using Givens rotations QR decompositions can also be computed with a series of Givens rotations. Each rotation zeroes an element in the subdiagonal of the matrix, forming the R matrix. The concatenation of all the Givens rotations forms the orthogonal Q matrix. In practice, Givens rotations are not actually performed by building a whole matrix and doing a matrix multiplication. A Givens rotation procedure is used instead which does the equivalent of the sparse Givens matrix multiplication, without the extra work of handling the sparse elements. The Givens rotation procedure is useful in situations where only relatively few off-diagonal elements need to be zeroed, and is more easily parallelized than Householder transformations. Example Let us calculate the decomposition of $A={\begin{bmatrix}12&-51&4\\6&167&-68\\-4&24&-41\end{bmatrix}}.$ First, we need to form a rotation matrix that will zero the lowermost left element, $a_{31}=-4$. We form this matrix using the Givens rotation method, and call the matrix $G_{1}$. We will first rotate the vector ${\begin{bmatrix}12&-4\end{bmatrix}}$, to point along the X axis. This vector has an angle $ \theta =\arctan \left({\frac {-(-4)}{12}}\right)$. We create the orthogonal Givens rotation matrix, $G_{1}$: ${\begin{aligned}G_{1}&={\begin{bmatrix}\cos(\theta )&0&-\sin(\theta )\\0&1&0\\\sin(\theta )&0&\cos(\theta )\end{bmatrix}}\\&\approx {\begin{bmatrix}0.94868&0&-0.31622\\0&1&0\\0.31622&0&0.94868\end{bmatrix}}\end{aligned}}$ And the result of $G_{1}A$ now has a zero in the $a_{31}$ element. $G_{1}A\approx {\begin{bmatrix}12.64911&-55.97231&16.76007\\6&167&-68\\0&6.64078&-37.6311\end{bmatrix}}$ We can similarly form Givens matrices $G_{2}$ and $G_{3}$, which will zero the sub-diagonal elements $a_{21}$ and $a_{32}$, forming a triangular matrix $R$. The orthogonal matrix $Q^{\textsf {T}}$ is formed from the product of all the Givens matrices $Q^{\textsf {T}}=G_{3}G_{2}G_{1}$. Thus, we have $G_{3}G_{2}G_{1}A=Q^{\textsf {T}}A=R$, and the QR decomposition is $A=QR$. Advantages and disadvantages The QR decomposition via Givens rotations is the most involved to implement, as the ordering of the rows required to fully exploit the algorithm is not trivial to determine. However, it has a significant advantage in that each new zero element $a_{ij}$ affects only the row with the element to be zeroed (i) and a row above (j). This makes the Givens rotation algorithm more bandwidth efficient and parallelizable than the Householder reflection technique. Connection to a determinant or a product of eigenvalues We can use QR decomposition to find the determinant of a square matrix. Suppose a matrix is decomposed as $A=QR$. Then we have $\det A=\det Q\det R.$ $Q$ can be chosen such that $\det Q=1$. Thus, $\det A=\det R=\prod _{i}r_{ii}$ where the $r_{ii}$ are the entries on the diagonal of $R$. Furthermore, because the determinant equals the product of the eigenvalues, we have $\prod _{i}r_{ii}=\prod _{i}\lambda _{i}$ where the $\lambda _{i}$ are eigenvalues of $A$. We can extend the above properties to a non-square complex matrix $A$ by introducing the definition of QR decomposition for non-square complex matrices and replacing eigenvalues with singular values. Start with a QR decomposition for a non-square matrix A: $A=Q{\begin{bmatrix}R\\0\end{bmatrix}},\qquad Q^{\dagger }Q=I$ where $0$ denotes the zero matrix and $Q$ is a unitary matrix. From the properties of the SVD and the determinant of a matrix, we have ${\Big \|}\prod _{i}r_{ii}{\Big \|}=\prod _{i}\sigma _{i},$ where the $\sigma _{i}$ are the singular values of $A$. Note that the singular values of $A$ and $R$ are identical, although their complex eigenvalues may be different. However, if A is square, then ${\prod _{i}\sigma _{i}}={\Big \|}\prod _{i}\lambda _{i}{\Big \|}.$ It follows that the QR decomposition can be used to efficiently calculate the product of the eigenvalues or singular values of a matrix. Column pivoting Pivoted QR differs from ordinary Gram-Schmidt in that it takes the largest remaining column at the beginning of each new step—column pivoting—[3] and thus introduces a permutation matrix P: $AP=QR\quad \iff \quad A=QRP^{\textsf {T}}$ Column pivoting is useful when A is (nearly) rank deficient, or is suspected of being so. It can also improve numerical accuracy. P is usually chosen so that the diagonal elements of R are non-increasing: $\left\|r_{11}\right\|\geq \left\|r_{22}\right\|\geq \cdots \geq \left\|r_{nn}\right\|$. This can be used to find the (numerical) rank of A at lower computational cost than a singular value decomposition, forming the basis of so-called rank-revealing QR algorithms. Using for solution to linear inverse problems Compared to the direct matrix inverse, inverse solutions using QR decomposition are more numerically stable as evidenced by their reduced condition numbers.[4] To solve the underdetermined ($m<n$) linear problem $A\mathbf {x} =\mathbf {b} $ where the matrix $A$ has dimensions $m\times n$ and rank $m$, first find the QR factorization of the transpose of $A$: $A^{\textsf {T}}=QR$, where Q is an orthogonal matrix (i.e. $Q^{\textsf {T}}=Q^{-1}$), and R has a special form: $R=\left[{\begin{smallmatrix}R_{1}\\0\end{smallmatrix}}\right]$. Here $R_{1}$ is a square $m\times m$ right triangular matrix, and the zero matrix has dimension $(n-m)\times m$. After some algebra, it can be shown that a solution to the inverse problem can be expressed as: $\mathbf {x} =Q\left[{\begin{smallmatrix}\left(R_{1}^{\textsf {T}}\right)^{-1}\mathbf {b} \\0\end{smallmatrix}}\right]$ where one may either find $R_{1}^{-1}$ by Gaussian elimination or compute $\left(R_{1}^{\textsf {T}}\right)^{-1}\mathbf {b} $ directly by forward substitution. The latter technique enjoys greater numerical accuracy and lower computations. To find a solution ${\hat {\mathbf {x} }}$ to the overdetermined ($m\geq n$) problem $A\mathbf {x} =\mathbf {b} $ which minimizes the norm $\left\\|A{\hat {\mathbf {x} }}-\mathbf {b} \right\\|$, first find the QR factorization of $A$: $A=QR$. The solution can then be expressed as ${\hat {\mathbf {x} }}=R_{1}^{-1}\left(Q_{1}^{\textsf {T}}\mathbf {b} \right)$, where $Q_{1}$ is an $m\times n$ matrix containing the first $n$ columns of the full orthonormal basis $Q$ and where $R_{1}$ is as before. Equivalent to the underdetermined case, back substitution can be used to quickly and accurately find this ${\hat {\mathbf {x} }}$ without explicitly inverting $R_{1}$. ($Q_{1}$ and $R_{1}$ are often provided by numerical libraries as an "economic" QR decomposition.) Generalizations Iwasawa decomposition generalizes QR decomposition to semi-simple Lie groups. See also • Polar decomposition • Eigenvalue decomposition • Spectral decomposition • LU decomposition • Singular value decomposition References 1. Trefethen, Lloyd N.; Bau, David III (1997). Numerical linear algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics. ISBN 978-0-898713-61-9. 2. Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Springer, p. 225, ISBN 0-387-95452-X 3. Strang, Gilbert (2019). Linear Algebra and Learning from Data (1st ed.). Wellesley: Wellesley Cambridge Press. p. 143. ISBN 978-0-692-19638-0. 4. Parker, Robert L. (1994). Geophysical Inverse Theory. Princeton, N.J.: Princeton University Press. Section 1.13. ISBN 978-0-691-20683-7. OCLC 1134769155. Further reading • Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0-8018-5414-9. • Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, sec. 2.8, ISBN 0-521-38632-2 • Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 2.10. QR Decomposition", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8 External links • Online Matrix Calculator Performs QR decomposition of matrices. • LAPACK users manual gives details of subroutines to calculate the QR decomposition • Mathematica users manual gives details and examples of routines to calculate QR decomposition • ALGLIB includes a partial port of the LAPACK to C++, C#, Delphi, etc. • Eigen::QR Includes C++ implementation of QR decomposition. Numerical linear algebra Key concepts • Floating point • Numerical stability Problems • System of linear equations • Matrix decompositions • Matrix multiplication (algorithms) • Matrix splitting • Sparse problems Hardware • CPU cache • TLB • Cache-oblivious algorithm • SIMD • Multiprocessing Software • MATLAB • Basic Linear Algebra Subprograms (BLAS) • LAPACK • Specialized libraries • General purpose software	Wikipedia
RSA numbers In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Laboratories in March 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers. The challenge was ended in 2007.[1] RSA Laboratories (which is an acronym of the creators of the technique; Rivest, Shamir and Adleman) published a number of semiprimes with 100 to 617 decimal digits. Cash prizes of varying size, up to US$200,000 (and prizes up to $20,000 awarded), were offered for factorization of some of them. The smallest RSA number was factored in a few days. Most of the numbers have still not been factored and many of them are expected to remain unfactored for many years to come. As of February 2020, the smallest 23 of the 54 listed numbers have been factored. While the RSA challenge officially ended in 2007, people are still attempting to find the factorizations. According to RSA Laboratories, "Now that the industry has a considerably more advanced understanding of the cryptanalytic strength of common symmetric-key and public-key algorithms, these challenges are no longer active."[2] Some of the smaller prizes had been awarded at the time. The remaining prizes were retracted. The first RSA numbers generated, from RSA-100 to RSA-500, were labeled according to their number of decimal digits. Later, beginning with RSA-576, binary digits are counted instead. An exception to this is RSA-617, which was created before the change in the numbering scheme. The numbers are listed in increasing order below. Contents • RSA-100 • RSA-110 • RSA-120 • RSA-129 • RSA-130 • RSA-140 • RSA-150 • RSA-155 • RSA-160 • RSA-170 • RSA-576 • RSA-180 • RSA-190 • RSA-640 • RSA-200 • RSA-210 • RSA-704 • RSA-220 • RSA-230 • RSA-232 • RSA-768 • RSA-240 • RSA-250 • RSA-260 • RSA-270 • RSA-896 • RSA-280 • RSA-290 • RSA-300 • RSA-309 • RSA-1024 • RSA-310 • RSA-320 • RSA-330 • RSA-340 • RSA-350 • RSA-360 • RSA-370 • RSA-380 • RSA-390 • RSA-400 • RSA-410 • RSA-420 • RSA-430 • RSA-440 • RSA-450 • RSA-460 • RSA-1536 • RSA-470 • RSA-480 • RSA-490 • RSA-500 • RSA-617 • RSA-2048 See also Notes References External links RSA-100 RSA-100 has 100 decimal digits (330 bits). Its factorization was announced on April 1, 1991, by Arjen K. Lenstra.[3][4] Reportedly, the factorization took a few days using the multiple-polynomial quadratic sieve algorithm on a MasPar parallel computer.[5] The value and factorization of RSA-100 are as follows: RSA-100 = 1522605027922533360535618378132637429718068114961380688657908494580122963258952897654000350692006139 RSA-100 = 37975227936943673922808872755445627854565536638199 × 40094690950920881030683735292761468389214899724061 It takes four hours to repeat this factorization using the program Msieve on a 2200 MHz Athlon 64 processor. The number can be factorized in 72 minutes on overclocked to 3.5 GHz Intel Core2 Quad q9300, using GGNFS and Msieve binaries running by distributed version of the factmsieve Perl script.[6] RSA-110 RSA-110 has 110 decimal digits (364 bits), and was factored in April 1992 by Arjen K. Lenstra and Mark S. Manasse in approximately one month.[5] The number can be factorized in less than four hours on overclocked to 3.5 GHz Intel Core2 Quad q9300, using GGNFS and Msieve binaries running by distributed version of the factmsieve Perl script.[6] The value and factorization are as follows: RSA-110 = 35794234179725868774991807832568455403003778024228226193532908190484670252364677411513516111204504060317568667 RSA-110 = 6122421090493547576937037317561418841225758554253106999 × 5846418214406154678836553182979162384198610505601062333 RSA-120 RSA-120 has 120 decimal digits (397 bits), and was factored in June 1993 by Thomas Denny, Bruce Dodson, Arjen K. Lenstra, and Mark S. Manasse.[7] The computation took under three months of actual computer time. The value and factorization are as follows: RSA-120 = 227010481295437363334259960947493668895875336466084780038173258247009162675779735389791151574049166747880487470296548479 RSA-120 = 327414555693498015751146303749141488063642403240171463406883 × 693342667110830181197325401899700641361965863127336680673013 RSA-129 RSA-129, having 129 decimal digits (426 bits), was not part of the 1991 RSA Factoring Challenge, but rather related to Martin Gardner's Mathematical Games column in the August 1977 issue of Scientific American.[8] RSA-129 was factored in April 1994 by a team led by Derek Atkins, Michael Graff, Arjen K. Lenstra and Paul Leyland, using approximately 1600 computers[9] from around 600 volunteers connected over the Internet.[10] A US$100 token prize was awarded by RSA Security for the factorization, which was donated to the Free Software Foundation. The value and factorization are as follows: RSA-129 = 114381625757888867669235779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879543541 RSA-129 = 3490529510847650949147849619903898133417764638493387843990820577 × 32769132993266709549961988190834461413177642967992942539798288533 The factorization was found using the Multiple Polynomial Quadratic Sieve algorithm. The factoring challenge included a message encrypted with RSA-129. When decrypted using the factorization the message was revealed to be "The Magic Words are Squeamish Ossifrage". In 2015, RSA-129 was factored in about one day, with the CADO-NFS open source implementation of number field sieve, using a commercial cloud computing service for about $30.[11] RSA-130 RSA-130 has 130 decimal digits (430 bits), and was factored on April 10, 1996, by a team led by Arjen K. Lenstra and composed of Jim Cowie, Marije Elkenbracht-Huizing, Wojtek Furmanski, Peter L. Montgomery, Damian Weber and Joerg Zayer.[12] The value and factorization are as follows: RSA-130 = 1807082088687404805951656164405905566278102516769401349170127021450056662540244048387341127590812303371781887966563182013214880557 RSA-130 = 39685999459597454290161126162883786067576449112810064832555157243 × 45534498646735972188403686897274408864356301263205069600999044599 The factorization was found using the Number Field Sieve algorithm and the polynomial 5748302248738405200 x5 + 9882261917482286102 x4 - 13392499389128176685 x3 + 16875252458877684989 x2 + 3759900174855208738 x1 - 46769930553931905995 which has a root of 12574411168418005980468 modulo RSA-130. RSA-140 RSA-140 has 140 decimal digits (463 bits), and was factored on February 2, 1999, by a team led by Herman te Riele and composed of Stefania Cavallar, Bruce Dodson, Arjen K. Lenstra, Paul Leyland, Walter Lioen, Peter L. Montgomery, Brian Murphy and Paul Zimmermann.[13][14] The value and factorization are as follows: RSA-140 = 21290246318258757547497882016271517497806703963277216278233383215381949984056495911366573853021918316783107387995317230889569230873441936471 RSA-140 = 3398717423028438554530123627613875835633986495969597423490929302771479 × 6264200187401285096151654948264442219302037178623509019111660653946049 The factorization was found using the Number Field Sieve algorithm and an estimated 2000 MIPS-years of computing time. RSA-150 RSA-150 has 150 decimal digits (496 bits), and was withdrawn from the challenge by RSA Security. RSA-150 was eventually factored into two 75-digit primes by Aoki et al. in 2004 using the general number field sieve (GNFS), years after bigger RSA numbers that were still part of the challenge had been solved. The value and factorization are as follows: RSA-150 = 155089812478348440509606754370011861770654545830995430655466945774312632703463465954363335027577729025391453996787414027003501631772186840890795964683 RSA-150 = 348009867102283695483970451047593424831012817350385456889559637548278410717 × 445647744903640741533241125787086176005442536297766153493419724532460296199 RSA-155 RSA-155 has 155 decimal digits (512 bits), and was factored on August 22, 1999, in a span of six months, by a team led by Herman te Riele and composed of Stefania Cavallar, Bruce Dodson, Arjen K. Lenstra, Walter Lioen, Peter L. Montgomery, Brian Murphy, Karen Aardal, Jeff Gilchrist, Gerard Guillerm, Paul Leyland, Joel Marchand, François Morain, Alec Muffett, Craig Putnam, Chris Putnam and Paul Zimmermann.[15][16] The value and factorization are as follows: RSA-155 = 10941738641570527421809707322040357612003732945449205990913842131476349984288934784717997257891267332497625752899781833797076537244027146743531593354333897 RSA-155 = 1026395928297411057720541965739916759007165678080380668033419335217907113077 79 × 1066034883801684548209272203600128786792079585759892915222706082371930628086 43 The factorization was found using the general number field sieve algorithm and an estimated 8000 MIPS-years of computing time. RSA-160 RSA-160 has 160 decimal digits (530 bits), and was factored on April 1, 2003, by a team from the University of Bonn and the German Federal Office for Information Security (BSI). The team contained J. Franke, F. Bahr, T. Kleinjung, M. Lochter, and M. Böhm.[17][18] The value and factorization are as follows: RSA-160 = 2152741102718889701896015201312825429257773588845675980170497676778133145218859135673011059773491059602497907111585214302079314665202840140619946994927570407753 RSA-160 = 4542789285848139407168619064973883165613714577846979325095998470925000415733 5359 × 4738809060383201619663383230378895197326892292104095794474135464881202849390 9367 The factorization was found using the general number field sieve algorithm. RSA-170 RSA-170 has 170 decimal digits (563 bits) and was first factored on December 29, 2009, by D. Bonenberger and M. Krone from Fachhochschule Braunschweig/Wolfenbüttel.[19] An independent factorization was completed by S. A. Danilov and I. A. Popovyan two days later.[20] The value and factorization are as follows: RSA-170 = 26062623684139844921529879266674432197085925380486406416164785191859999628542069361450283931914514618683512198164805919882053057222974116478065095809832377336510711545759 RSA-170 = 3586420730428501486799804587268520423291459681059978161140231860633948450858 040593963 × 7267029064107019078863797763923946264136137803856996670313708936002281582249 587494493 The factorization was found using the general number field sieve algorithm. RSA-576 RSA-576 has 174 decimal digits (576 bits), and was factored on December 3, 2003, by J. Franke and T. Kleinjung from the University of Bonn.[21][22][23] A cash prize of $10,000 was offered by RSA Security for a successful factorization. The value and factorization are as follows: RSA-576 = 188198812920607963838697239461650439807163563379417382700763356422988859715234665485319060606504743045317388011303396716199692321205734031879550656996221305168759307650257059 RSA-576 = 3980750864240649373971255005503864911990643623425267084063851895759463889572 61768583317 × 4727721461074353025362230719730482246329146953020971164598521711305207112563 63590397527 The factorization was found using the general number field sieve algorithm. RSA-180 RSA-180 has 180 decimal digits (596 bits), and was factored on May 8, 2010, by S. A. Danilov and I. A. Popovyan from Moscow State University, Russia.[24] RSA-180 = 1911479277189866096892294666314546498129862462766673548641885036388072607034 3679905877620136513516127813425829612810920004670291298456875280033022177775 2773957404540495707851421041 RSA-180 = 4007800823297508779525813391041005725268293178158071765648821789984975727719 50624613470377 × 4769396887386118369955354773570708579399020760277882320319897758246062255957 73435668861833 The factorization was found using the general number field sieve algorithm implementation running on three Intel Core i7 PCs. RSA-190 RSA-190 has 190 decimal digits (629 bits), and was factored on November 8, 2010, by I. A. Popovyan from Moscow State University, Russia, and A. Timofeev from CWI, Netherlands.[25] RSA-190 = 1907556405060696491061450432646028861081179759533184460647975622318915025587 1841757540549761551215932934922604641526300932385092466032074171247261215808 58185985938946945490481721756401423481 RSA-190 = 3171195257690152709485171289740475929805147316029450327784761927832793642798 1256542415724309619 × 6015260020444561641587641685526676183243543359471811072599763828083615704046 0481625355619404899 RSA-640 RSA-640 has 193 decimal digits (640 bits). A cash prize of US$20,000 was offered by RSA Security for a successful factorization. On November 2, 2005, F. Bahr, M. Boehm, J. Franke and T. Kleinjung of the German Federal Office for Information Security announced that they had factorized the number using GNFS as follows:[26][27][28] RSA-640 = 3107418240490043721350750035888567930037346022842727545720161948823206440518 0815045563468296717232867824379162728380334154710731085019195485290073377248 22783525742386454014691736602477652346609 RSA-640 = 1634733645809253848443133883865090859841783670033092312181110852389333100104 508151212118167511579 × 1900871281664822113126851573935413975471896789968515493666638539088027103802 104498957191261465571 The computation took five months on 80 2.2 GHz AMD Opteron CPUs. The slightly larger RSA-200 was factored in May 2005 by the same team. RSA-200 Wikinews has related news: • Two hundred digit number factored RSA-200 has 200 decimal digits (663 bits), and factors into the two 100-digit primes given below. On May 9, 2005, F. Bahr, M. Boehm, J. Franke, and T. Kleinjung announced[29][30] that they had factorized the number using GNFS as follows: RSA-200 = 2799783391122132787082946763872260162107044678695542853756000992932612840010 7609345671052955360856061822351910951365788637105954482006576775098580557613 579098734950144178863178946295187237869221823983 RSA-200 = 3532461934402770121272604978198464368671197400197625023649303468776121253679 423200058547956528088349 × 7925869954478333033347085841480059687737975857364219960734330341455767872818 152135381409304740185467 The CPU time spent on finding these factors by a collection of parallel computers amounted – very approximately – to the equivalent of 75 years work for a single 2.2 GHz Opteron-based computer.[29] Note that while this approximation serves to suggest the scale of the effort, it leaves out many complicating factors; the announcement states it more precisely. RSA-210 RSA-210 has 210 decimal digits (696 bits) and was factored in September 2013 by Ryan Propper:[31] RSA-210 = 2452466449002782119765176635730880184670267876783327597434144517150616008300 3858721695220839933207154910362682719167986407977672324300560059203563124656 1218465817904100131859299619933817012149335034875870551067 RSA-210 = 4359585683259407917999519653872144063854709102652201963187054821445240853452 75999740244625255428455944579 × 5625457617268841037562770073044474817438769440075105451049468510945483965774 79473472146228550799322939273 RSA-704 RSA-704 has 212 decimal digits (704 bits), and was factored by Shi Bai, Emmanuel Thomé and Paul Zimmermann.[32] The factorization was announced July 2, 2012.[33] A cash prize of US$30,000 was previously offered for a successful factorization. RSA-704 = 7403756347956171282804679609742957314259318888923128908493623263897276503402 8266276891996419625117843995894330502127585370118968098286733173273108930900 552505116877063299072396380786710086096962537934650563796359 RSA-704 = 9091213529597818878440658302600437485892608310328358720428512168960411528640 933367824950788367956756806141 × 8143859259110045265727809126284429335877899002167627883200914172429324360133 004116702003240828777970252499 RSA-220 RSA-220 has 220 decimal digits (729 bits), and was factored by S. Bai, P. Gaudry, A. Kruppa, E. Thomé and P. Zimmermann. The factorization was announced on May 13, 2016.[34] RSA-220 = 2260138526203405784941654048610197513508038915719776718321197768109445641817 9666766085931213065825772506315628866769704480700018111497118630021124879281 99487482066070131066586646083327982803560379205391980139946496955261 RSA-220 = 6863656412267566274382371499288437800130842239979164844621244993321541061441 4642667938213644208420192054999687 × 3292907439486349812049301549212935291916455196536233952462686051169290349309 4652463337824866390738191765712603 RSA-230 RSA-230 has 230 decimal digits (762 bits), and was factored by Samuel S. Gross on August 15, 2018.[35] RSA-230 = 1796949159794106673291612844957324615636756180801260007088891883553172646034 1490933493372247868650755230855864199929221814436684722874052065257937495694 3483892631711525225256544109808191706117425097024407180103648316382885188526 89 RSA-230 = 4528450358010492026612439739120166758911246047493700040073956759261590397250 033699357694507193523000343088601688589 × 3968132623150957588532394439049887341769533966621957829426966084093049516953 598120833228447171744337427374763106901 RSA-232 RSA-232 has 232 decimal digits (768 bits), and was factored on February 17, 2020, by N. L. Zamarashkin, D. A. Zheltkov and S. A. Matveev.[36] RSA-232 = 1009881397871923546909564894309468582818233821955573955141120516205831021338 5285453743661097571543636649133800849170651699217015247332943892702802343809 6090980497644054071120196541074755382494867277137407501157718230539834060616 2079 RSA-232 = 2966909333208360660361779924242630634742946262521852394401857157419437019472 3262390744910112571804274494074452751891 × 3403816175197563438006609498491521420547121760734723172735163413276050706174 8526506443144325148088881115083863017669 RSA-768 RSA-768 has 232 decimal digits (768 bits), and was factored on December 12, 2009, over the span of two years, by Thorsten Kleinjung, Kazumaro Aoki, Jens Franke, Arjen K. Lenstra, Emmanuel Thomé, Pierrick Gaudry, Alexander Kruppa, Peter Montgomery, Joppe W. Bos, Dag Arne Osvik, Herman te Riele, Andrey Timofeev, and Paul Zimmermann.[37] RSA-768 = 1230186684530117755130494958384962720772853569595334792197322452151726400507 2636575187452021997864693899564749427740638459251925573263034537315482685079 1702612214291346167042921431160222124047927473779408066535141959745985690214 3413 RSA-768 = 3347807169895689878604416984821269081770479498371376856891243138898288379387 8002287614711652531743087737814467999489 × 3674604366679959042824463379962795263227915816434308764267603228381573966651 1279233373417143396810270092798736308917 The CPU time spent on finding these factors by a collection of parallel computers amounted approximately to the equivalent of almost 2000 years of computing on a single-core 2.2 GHz AMD Opteron-based computer.[37] RSA-240 RSA-240 has 240 decimal digits (795 bits), and was factored in November 2019 by Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann.[38] RSA-240 = 1246203667817187840658350446081065904348203746516788057548187888832896668011 8821085503603957027250874750986476843845862105486553797025393057189121768431 8286362846948405301614416430468066875699415246993185704183030512549594371372 159029236099 RSA-240 = 5094359522858399145550510235808437141326483820241114731866602965218212064697 46700620316443478873837606252372049619334517 × 2446242088383181505678131390240028966538020925789314014520412213365584770951 78155258218897735030590669041302045908071447 The CPU time spent on finding these factors amounted to approximately 900 core-years on a 2.1 GHz Intel Xeon Gold 6130 CPU. Compared to the factorization of RSA-768, the authors estimate that better algorithms sped their calculations by a factor of 3–4 and faster computers sped their calculation by a factor of 1.25–1.67. RSA-250 RSA-250 has 250 decimal digits (829 bits), and was factored in February 2020 by Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé, and Paul Zimmermann. The announcement of the factorization occurred on February 28. RSA-250 = 2140324650240744961264423072839333563008614715144755017797754920881418023447 1401366433455190958046796109928518724709145876873962619215573630474547705208 0511905649310668769159001975940569345745223058932597669747168173806936489469 9871578494975937497937 RSA-250 = 6413528947707158027879019017057738908482501474294344720811685963202453234463 0238623598752668347708737661925585694639798853367 × 3337202759497815655622601060535511422794076034476755466678452098702384172921 0037080257448673296881877565718986258036932062711 The factorisation of RSA-250 utilised approximately 2700 CPU core-years, using a 2.1 GHz Intel Xeon Gold 6130 CPU as a reference. The computation was performed with the Number Field Sieve algorithm, using the open source CADO-NFS software. The team dedicated the computation to Peter Montgomery, an American mathematician known for his contributions to computational number theory and cryptography who died on February 18, 2020.[39] RSA-260 RSA-260 has 260 decimal digits (862 bits), and has not been factored so far. RSA-260 = 2211282552952966643528108525502623092761208950247001539441374831912882294140 2001986512729726569746599085900330031400051170742204560859276357953757185954 2988389587092292384910067030341246205457845664136645406842143612930176940208 46391065875914794251435144458199 RSA-270 RSA-270 has 270 decimal digits (895 bits), and has not been factored so far. RSA-270 = 2331085303444075445276376569106805241456198124803054490429486119684959182451 3578286788836931857711641821391926857265831491306067262691135402760979316634 1626693946596196427744273886601876896313468704059066746903123910748277606548 649151920812699309766587514735456594993207 RSA-896 RSA-896 has 270 decimal digits (896 bits), and has not been factored so far. A cash prize of $75,000 was previously offered for a successful factorization. RSA-896 = 4120234369866595438555313653325759481798116998443279828454556264338764455652 4842619809887042316184187926142024718886949256093177637503342113098239748515 0944909106910269861031862704114880866970564902903653658867433731720813104105 190864254793282601391257624033946373269391 RSA-280 RSA-280 has 280 decimal digits (928 bits), and has not been factored so far. RSA-280 = 1790707753365795418841729699379193276395981524363782327873718589639655966058 5783742549640396449103593468573113599487089842785784500698716853446786525536 5503525160280656363736307175332772875499505341538927978510751699922197178159 7724733184279534477239566789173532366357270583106789 RSA-290 RSA-290 has 290 decimal digits (962 bits), and has not been factored so far. RSA-290 = 3050235186294003157769199519894966400298217959748768348671526618673316087694 3419156362946151249328917515864630224371171221716993844781534383325603218163 2549201100649908073932858897185243836002511996505765970769029474322210394327 60575157628357292075495937664206199565578681309135044121854119 RSA-300 RSA-300 has 300 decimal digits (995 bits), and has not been factored so far. RSA-300 = 2769315567803442139028689061647233092237608363983953254005036722809375824714 9473946190060218756255124317186573105075074546238828817121274630072161346956 4396741836389979086904304472476001839015983033451909174663464663867829125664 459895575157178816900228792711267471958357574416714366499722090015674047 RSA-309 RSA-309 has 309 decimal digits (1,024 bits), and has not been factored so far. RSA-309 = 1332943998825757583801437794588036586217112243226684602854588261917276276670 5425540467426933349195015527349334314071822840746357352800368666521274057591 1870128339157499072351179666739658503429931021985160714113146720277365006623 6927218079163559142755190653347914002967258537889160429597714204365647842739 10949 RSA-1024 RSA-1024 has 309 decimal digits (1,024 bits), and has not been factored so far. $100,000 was previously offered for factorization. RSA-1024 = 135066410865995223349603216278805969938881475605667027524485143851526510604 859533833940287150571909441798207282164471551373680419703964191743046496589 274256239341020864383202110372958725762358509643110564073501508187510676594 629205563685529475213500852879416377328533906109750544334999811150056977236 890927563 RSA-310 RSA-310 has 310 decimal digits (1,028 bits), and has not been factored so far. RSA-310 = 1848210397825850670380148517702559371400899745254512521925707445580334710601 4125276757082979328578439013881047668984294331264191394626965245834649837246 5163148188847336415136873623631778358751846501708714541673402642461569061162 0116380982484120857688483676576094865930188367141388795454378671343386258291 687641 RSA-320 RSA-320 has 320 decimal digits (1,061 bits), and has not been factored so far. RSA-320 = 2136810696410071796012087414500377295863767938372793352315068620363196552357 8837094085435000951700943373838321997220564166302488321590128061531285010636 8571638978998117122840139210685346167726847173232244364004850978371121744321 8270343654835754061017503137136489303437996367224915212044704472299799616089 2591129924218437 RSA-330 RSA-330 has 330 decimal digits (1,094 bits), and has not been factored so far. RSA-330 = 1218708633106058693138173980143325249157710686226055220408666600017481383238 1352456802425903555880722805261111079089882303717632638856140900933377863089 0634828167900405006112727432172179976427017137792606951424995281839383708354 6364684839261149319768449396541020909665209789862312609604983709923779304217 01862444655244698696759267 RSA-340 RSA-340 has 340 decimal digits (1,128 bits), and has not been factored so far. RSA-340 = 2690987062294695111996484658008361875931308730357496490239672429933215694995 2758588771223263308836649715112756731997946779608413232406934433532048898585 9176676580752231563884394807622076177586625973975236127522811136600110415063 0004691128152106812042872285697735145105026966830649540003659922618399694276 990464815739966698956947129133275233 RSA-350 RSA-350 has 350 decimal digits (1,161 bits), and has not been factored so far. RSA-350 = 2650719995173539473449812097373681101529786464211583162467454548229344585504 3495841191504413349124560193160478146528433707807716865391982823061751419151 6068496555750496764686447379170711424873128631468168019548127029171231892127 2886825928263239383444398948209649800021987837742009498347263667908976501360 3382322972552204068806061829535529820731640151 RSA-360 RSA-360 has 360 decimal digits (1,194 bits), and has not been factored so far. RSA-360 = 2186820202343172631466406372285792654649158564828384065217121866374227745448 7764963889680817334211643637752157994969516984539482486678141304751672197524 0052350576247238785129338002757406892629970748212734663781952170745916609168 9358372359962787832802257421757011302526265184263565623426823456522539874717 61591019113926725623095606566457918240614767013806590649 RSA-370 RSA-370 has 370 decimal digits (1,227 bits), and has not been factored so far. RSA-370 = 1888287707234383972842703127997127272470910519387718062380985523004987076701 7212819937261952549039800018961122586712624661442288502745681454363170484690 7379449525034797494321694352146271320296579623726631094822493455672541491544 2700993152879235272779266578292207161032746297546080025793864030543617862620 878802244305286292772467355603044265985905970622730682658082529621 RSA-380 RSA-380 has 380 decimal digits (1,261 bits), and has not been factored so far. RSA-380 = 3013500443120211600356586024101276992492167997795839203528363236610578565791 8270750937407901898070219843622821090980641477056850056514799336625349678549 2187941807116344787358312651772858878058620717489800725333606564197363165358 2237779263423501952646847579678711825720733732734169866406145425286581665755 6977260763553328252421574633011335112031733393397168350585519524478541747311 RSA-390 RSA-390 has 390 decimal digits (1,294 bits), and has not been factored so far. RSA-390 = 2680401941182388454501037079346656065366941749082852678729822424397709178250 4623002472848967604282562331676313645413672467684996118812899734451228212989 1630084759485063423604911639099585186833094019957687550377834977803400653628 6955344904367437281870253414058414063152368812498486005056223028285341898040 0795447435865033046248751475297412398697088084321037176392288312785544402209 1083492089 RSA-400 RSA-400 has 400 decimal digits (1,327 bits), and has not been factored so far. RSA-400 = 2014096878945207511726700485783442547915321782072704356103039129009966793396 1419850865094551022604032086955587930913903404388675137661234189428453016032 6191193056768564862615321256630010268346471747836597131398943140685464051631 7519403149294308737302321684840956395183222117468443578509847947119995373645 3607109795994713287610750434646825511120586422993705980787028106033008907158 74500584758146849481 RSA-410 RSA-410 has 410 decimal digits (1,360 bits), and has not been factored so far. RSA-410 = 1965360147993876141423945274178745707926269294439880746827971120992517421770 1079138139324539033381077755540830342989643633394137538983355218902490897764 4412968474332754608531823550599154905901691559098706892516477785203855688127 0635069372091564594333528156501293924133186705141485137856845741766150159437 6063244163040088180887087028771717321932252992567756075264441680858665410918 431223215368025334985424358839 RSA-420 RSA-420 has 420 decimal digits (1,393 bits), and has not been factored so far. RSA-420 = 2091366302476510731652556423163330737009653626605245054798522959941292730258 1898373570076188752609749648953525484925466394800509169219344906273145413634 2427186266197097846022969248579454916155633686388106962365337549155747268356 4666583846809964354191550136023170105917441056517493690125545320242581503730 3405952887826925813912683942756431114820292313193705352716165790132673270514 3817744164107601735413785886836578207979 RSA-430 RSA-430 has 430 decimal digits (1,427 bits), and has not been factored so far. RSA-430 = 3534635645620271361541209209607897224734887106182307093292005188843884213420 6950355315163258889704268733101305820000124678051064321160104990089741386777 2424190744453885127173046498565488221441242210687945185565975582458031351338 2070785777831859308900851761495284515874808406228585310317964648830289141496 3289966226854692560410075067278840383808716608668377947047236323168904650235 70092246473915442026549955865931709542468648109541 RSA-440 RSA-440 has 440 decimal digits (1,460 bits), and has not been factored so far. RSA-440 = 2601428211955602590070788487371320550539810804595235289423508589663391270837 4310252674800592426746319007978890065337573160541942868114065643853327229484 5029942332226171123926606357523257736893667452341192247905168387893684524818 0307729497304959710847337973805145673263119916483529703607405432752966630781 2234597766390750441445314408171802070904072739275930410299359006059619305590 701939627725296116299946059898442103959412221518213407370491 RSA-450 RSA-450 has 450 decimal digits (1,493 bits), and has not been factored so far. RSA-450 = 1984634237142836623497230721861131427789462869258862089878538009871598692569 0078791591684242367262529704652673686711493985446003494265587358393155378115 8032447061155145160770580926824366573211993981662614635734812647448360573856 3132247491715526997278115514905618953253443957435881503593414842367096046182 7643434794849824315251510662855699269624207451365738384255497823390996283918 3287667419172988072221996532403300258906083211160744508191024837057033 RSA-460 RSA-460 has 460 decimal digits (1,526 bits), and has not been factored so far. RSA-460 = 1786856020404004433262103789212844585886400086993882955081051578507634807524 1464078819812169681394445771476334608488687746254318292828603396149562623036 3564554675355258128655971003201417831521222464468666642766044146641933788836 8932452217321354860484353296131403821175862890998598653858373835628654351880 4806362231643082386848731052350115776715521149453708868428108303016983133390 0416365515466857004900847501644808076825638918266848964153626486460448430073 4909 RSA-1536 RSA-1536 has 463 decimal digits (1,536 bits), and has not been factored so far. $150,000 was previously offered for successful factorization. RSA-1536 = 184769970321174147430683562020016440301854933866341017147178577491065169671 116124985933768430543574458561606154457179405222971773252466096064694607124 962372044202226975675668737842756238950876467844093328515749657884341508847 552829818672645133986336493190808467199043187438128336350279547028265329780 293491615581188104984490831954500984839377522725705257859194499387007369575 568843693381277961308923039256969525326162082367649031603655137144791393234 7169566988069 RSA-470 RSA-470 has 470 decimal digits (1,559 bits), and has not been factored so far. RSA-470 = 1705147378468118520908159923888702802518325585214915968358891836980967539803 6897711442383602526314519192366612270595815510311970886116763177669964411814 0957486602388713064698304619191359016382379244440741228665455229545368837485 5874455212895044521809620818878887632439504936237680657994105330538621759598 4047709603954312447692725276887594590658792939924609261264788572032212334726 8553025718835659126454325220771380103576695555550710440908570895393205649635 76770285413369 RSA-480 RSA-480 has 480 decimal digits (1,593 bits), and has not been factored so far. RSA-480 = 3026570752950908697397302503155918035891122835769398583955296326343059761445 7144169659817040125185215913853345598217234371231338324773210726853524776378 4105186549246199888070331088462855743520880671299302895546822695492968577380 7067958428022008294111984222973260208233693152589211629901686973933487362360 8129660418514569063995282978176790149760521395548532814196534676974259747930 6858645849268328985687423881853632604706175564461719396117318298679820785491 875674946700413680932103 RSA-490 RSA-490 has 490 decimal digits (1,626 bits), and has not been factored so far. RSA-490 = 1860239127076846517198369354026076875269515930592839150201028353837031025971 3738522164743327949206433999068225531855072554606782138800841162866037393324 6578171804201717222449954030315293547871401362961501065002486552688663415745 9758925793594165651020789220067311416926076949777767604906107061937873540601 5942747316176193775374190713071154900658503269465516496828568654377183190586 9537640698044932638893492457914750855858980849190488385315076922453755527481 1376719096144119390052199027715691 RSA-500 RSA-500 has 500 decimal digits (1,659 bits) and has not been factored so far. RSA-500 = 1897194133748626656330534743317202527237183591953428303184581123062450458870 7687605943212347625766427494554764419515427586743205659317254669946604982419 7301601038125215285400688031516401611623963128370629793265939405081077581694 4786041721411024641038040278701109808664214800025560454687625137745393418221 5494821277335671735153472656328448001134940926442438440198910908603252678814 7850601132077287172819942445113232019492229554237898606631074891074722425617 39680319169243814676235712934292299974411361 RSA-617 RSA-617 has 617 decimal digits (2,048 bits) and has not been factored so far. RSA-617 = 2270180129378501419358040512020458674106123596276658390709402187921517148311 9139894870133091111044901683400949483846818299518041763507948922590774925466 0881718792594659210265970467004498198990968620394600177430944738110569912941 2854289188085536270740767072259373777266697344097736124333639730805176309150 6836310795312607239520365290032105848839507981452307299417185715796297454995 0235053160409198591937180233074148804462179228008317660409386563445710347785 5345712108053073639453592393265186603051504106096643731332367283153932350006 7937107541955437362433248361242525945868802353916766181532375855504886901432 221349733 RSA-2048 RSA-2048 has 617 decimal digits (2,048 bits). It is the largest of the RSA numbers and carried the largest cash prize for its factorization, $200,000. The RSA-2048 may not be factorizable for many years to come, unless considerable advances are made in integer factorization or computational power. RSA-2048 = 2519590847565789349402718324004839857142928212620403202777713783604366202070 7595556264018525880784406918290641249515082189298559149176184502808489120072 8449926873928072877767359714183472702618963750149718246911650776133798590957 0009733045974880842840179742910064245869181719511874612151517265463228221686 9987549182422433637259085141865462043576798423387184774447920739934236584823 8242811981638150106748104516603773060562016196762561338441436038339044149526 3443219011465754445417842402092461651572335077870774981712577246796292638635 6373289912154831438167899885040445364023527381951378636564391212010397122822 120720357 See also • Integer factorization records • RSA Factoring Challenge (includes table with size and status of all numbers) • RSA Secret-Key Challenge Notes 1. RSA Laboratories. "RSA Factoring Challenge". Archived from the original on 2013-09-21. Retrieved 2008-08-05.{{cite web}}: CS1 maint: unfit URL (link) 2. RSA Laboratories. "The RSA Factoring Challenge FAQ". Archived from the original on 2013-09-21. Retrieved 2008-08-05.{{cite web}}: CS1 maint: unfit URL (link) 3. "RSA-100 Factored". Cryptography Watch Archive for April, 1991. 1991-04-01. Retrieved 2008-08-05. 4. "RSA Honor Roll". 1999-03-05. Retrieved 2008-08-05. 5. Brandon Dixon and Arjen K. Lenstra (1994). Factoring Integers Using SIMD Sieves. Lecture Notes in Computer Science. Vol. 765. doi:10.1007/3-540-48285-7. ISBN 978-3-540-57600-6. S2CID 21157010. 6. "Distributed version of the FactMsieve Perl script". 2012-03-27. Retrieved 2015-06-08. 7. T. Denny, B. Dodson, A. K. Lenstra, M. S. Manasse (1994), "On The Factorization Of RSA-120" . 8. "RSA Honor Roll". 1999-03-05. Retrieved 2008-08-06. 9. "The Magic Words Are Squeamish Ossifrage". Retrieved 2009-11-24. (PostScript document) 10. Mark Janeba (1994), Factoring Challenge Conquered. Retrieved on 2008-03-10. 11. McHugh, Nathaniel (2015-03-26). "The Magic Words are Squeamish Ossifrage - factoring RSA-129 using CADO-NFS". Nat McHugh. Retrieved 2016-05-25. 12. Arjen K. Lenstra (1996-04-12), Factorization of RSA-130. Retrieved on 2008-03-10. 13. Herman te Riele (1999-02-04), Factorization of RSA-140 Archived 2004-12-08 at the Wayback Machine. Retrieved on 2008-03-10. 14. RSA Laboratories, RSA-140 is factored! Archived 2006-12-30 at the Wayback Machine. Retrieved on 2008-03-10. 15. Herman te Riele (1999-08-26), New factorization record Archived 2004-12-31 at the Wayback Machine (announcement of factorization of RSA-155). Retrieved on 2008-03-10. 16. RSA Laboratories, RSA-155 is factored! Archived 2006-12-30 at the Wayback Machine. Retrieved on 2008-03-10. 17. Jens Franke (2003-04-01), RSA-160 (announcement of factorization). Retrieved on 2008-03-10. 18. RSA Laboratories, RSA-160 is factored! Archived 2006-12-30 at the Wayback Machine. Retrieved on 2008-03-10. 19. D. Bonenberger and M. Krone, RSA-170 Archived 2011-07-19 at the Wayback Machine Retrieved on 2010-03-08. 20. Danilov, S. A.; Popovyan, I. A. (9 May 2010). "Factorization of RSA-180" (PDF). Cryptology ePrint Archive. 21. Jens Franke (2003-12-03), RSA576 (repost of announcement of the factorization). Retrieved on 2008-03-10. 22. Eric W. Weisstein (2005-12-05), RSA-576 Factored at MathWorld. Retrieved on 2008-03-10. 23. RSA Laboratories, RSA-576 is factored! Archived 2006-12-24 at the Wayback Machine. Retrieved on 2008-03-10. 24. Danilov, S. A.; Popovyan, I. A. (2010). "Factorization of RSA-180". Cryptology ePrint Archive. Retrieved 2010-05-12. 25. I. Popovyan, A. Timofeev (2010-11-08). "RSA-190 factored". mersenneforum.org. Retrieved 2010-11-10. 26. RSA Laboratories, RSA-640 is factored! Archived 2007-01-04 at the Wayback Machine. Retrieved on 2008-03-10. 27. Jens Franke (2005-11-04), We have factored RSA640 by GNFS. Retrieved on 2008-03-10. 28. Eric W. Weisstein (2005-11-08), RSA-640 Factored at MathWorld. Retrieved on 2008-03-10. 29. Thorsten Kleinjung (2005-05-09), We have factored RSA200 by GNFS Archived 2008-03-22 at the Wayback Machine. Retrieved on 2008-03-10. 30. RSA Laboratories, RSA-200 is factored!. Retrieved on 2017-01-25. 31. RSA-210 factored, mersenneforum.org 32. Factorisation of RSA-704 with CADO-NFS Archived 2012-07-02 at the Wayback Machine. 33. Bai, Shi (2012-07-02). "Factorization of RSA704". NMBRTHRY (Mailing list). Retrieved 2012-07-03. 34. Zimmermann, Paul (May 13, 2016). "Factorisation of RSA-220 with CADO-NFS". Cado-nfs-discuss (Mailing list). Retrieved 2016-05-13. 35. Gross, Samuel. "The Factorization of RSA-230". cado-nfs-discuss. Retrieved 17 August 2018. 36. INM RAS news 37. Cryptology ePrint Archive: Report 2010/006 38. "[Cado-NFS-discuss] 795-bit factoring and discrete logarithms". Archived from the original on 2019-12-03. Retrieved 2019-12-02. 39. "[Cado-NFS-discuss] Factorization of RSA-250". Archived from the original on 2020-02-28. Retrieved 2020-02-28. References • RSA Factoring Challenge Administrator (1997-10-12), RSA Challenge List • RSA Laboratories, The RSA Challenge Numbers (archived by the Internet Archive in 2006, before the RSA challenge ended) • RSA Laboratories, "Challenge numbers in text format". Archived from the original on 2013-05-21. • Kazumaro Aoki, Yuji Kida, Takeshi Shimoyama, Hiroki Ueda, GNFS Factoring Statistics of RSA-100, 110, ..., 150, Cryptology ePrint Archive, Report 2004/095, 2004 External links • Steven Levy (March 1996), Wisecrackers in Wired News. coverage on RSA-129	Wikipedia
RSA (cryptosystem) RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem, one of the oldest, that is widely used for secure data transmission. The acronym "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977. An equivalent system was developed secretly in 1973 at Government Communications Headquarters (GCHQ) (the British signals intelligence agency) by the English mathematician Clifford Cocks. That system was declassified in 1997.[2] RSA General DesignersRon Rivest,[1] Adi Shamir, and Leonard Adleman First published1977 CertificationPKCS#1, ANSI X9.31, IEEE 1363 Cipher detail Key sizes2,048 to 4,096 bit typical Rounds1 Best public cryptanalysis General number field sieve for classical computers; Shor's algorithm for quantum computers. An 829-bit key has been broken. In a public-key cryptosystem, the encryption key is public and distinct from the decryption key, which is kept secret (private). An RSA user creates and publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers are kept secret. Messages can be encrypted by anyone, via the public key, but can only be decoded by someone who knows the prime numbers.[1] The security of RSA relies on the practical difficulty of factoring the product of two large prime numbers, the "factoring problem". Breaking RSA encryption is known as the RSA problem. Whether it is as difficult as the factoring problem is an open question.[3] There are no published methods to defeat the system if a large enough key is used. RSA is a relatively slow algorithm. Because of this, it is not commonly used to directly encrypt user data. More often, RSA is used to transmit shared keys for symmetric-key cryptography, which are then used for bulk encryption–decryption. History The idea of an asymmetric public-private key cryptosystem is attributed to Whitfield Diffie and Martin Hellman, who published this concept in 1976. They also introduced digital signatures and attempted to apply number theory. Their formulation used a shared-secret-key created from exponentiation of some number, modulo a prime number. However, they left open the problem of realizing a one-way function, possibly because the difficulty of factoring was not well-studied at the time.[4] Ron Rivest, Adi Shamir, and Leonard Adleman at the Massachusetts Institute of Technology made several attempts over the course of a year to create a function that was hard to invert. Rivest and Shamir, as computer scientists, proposed many potential functions, while Adleman, as a mathematician, was responsible for finding their weaknesses. They tried many approaches, including "knapsack-based" and "permutation polynomials". For a time, they thought what they wanted to achieve was impossible due to contradictory requirements.[5] In April 1977, they spent Passover at the house of a student and drank a good deal of wine before returning to their homes at around midnight.[6] Rivest, unable to sleep, lay on the couch with a math textbook and started thinking about their one-way function. He spent the rest of the night formalizing his idea, and he had much of the paper ready by daybreak. The algorithm is now known as RSA – the initials of their surnames in same order as their paper.[7] Clifford Cocks, an English mathematician working for the British intelligence agency Government Communications Headquarters (GCHQ), described an equivalent system in an internal document in 1973.[8] However, given the relatively expensive computers needed to implement it at the time, it was considered to be mostly a curiosity and, as far as is publicly known, was never deployed. His discovery, however, was not revealed until 1997 due to its top-secret classification. Kid-RSA (KRSA) is a simplified, insecure public-key cipher published in 1997, designed for educational purposes. Some people feel that learning Kid-RSA gives insight into RSA and other public-key ciphers, analogous to simplified DES.[9][10][11][12][13] Patent A patent describing the RSA algorithm was granted to MIT on 20 September 1983: U.S. Patent 4,405,829 "Cryptographic communications system and method". From DWPI's abstract of the patent: The system includes a communications channel coupled to at least one terminal having an encoding device and to at least one terminal having a decoding device. A message-to-be-transferred is enciphered to ciphertext at the encoding terminal by encoding the message as a number M in a predetermined set. That number is then raised to a first predetermined power (associated with the intended receiver) and finally computed. The remainder or residue, C, is... computed when the exponentiated number is divided by the product of two predetermined prime numbers (associated with the intended receiver). A detailed description of the algorithm was published in August 1977, in Scientific American's Mathematical Games column.[7] This preceded the patent's filing date of December 1977. Consequently, the patent had no legal standing outside the United States. Had Cocks's work been publicly known, a patent in the United States would not have been legal either. When the patent was issued, terms of patent were 17 years. The patent was about to expire on 21 September 2000, but RSA Security released the algorithm to the public domain on 6 September 2000.[14] Operation The RSA algorithm involves four steps: key generation, key distribution, encryption, and decryption. A basic principle behind RSA is the observation that it is practical to find three very large positive integers e, d, and n, such that with modular exponentiation for all integers m (with 0 ≤ m < n): $(m^{e})^{d}\equiv m{\pmod {n}}$ and that knowing e and n, or even m, it can be extremely difficult to find d. The triple bar (≡) here denotes modular congruence (which is to say that when you divide (me)d by n and m by n, they both have the same remainder). In addition, for some operations it is convenient that the order of the two exponentiations can be changed and that this relation also implies $(m^{d})^{e}\equiv m{\pmod {n}}.$ RSA involves a public key and a private key. The public key can be known by everyone and is used for encrypting messages. The intention is that messages encrypted with the public key can only be decrypted in a reasonable amount of time by using the private key. The public key is represented by the integers n and e, and the private key by the integer d (although n is also used during the decryption process, so it might be considered to be a part of the private key too). m represents the message (previously prepared with a certain technique explained below). Key generation The keys for the RSA algorithm are generated in the following way: 1. Choose two large prime numbers p and q. • To make factoring harder, p and q should be chosen at random, be both large and have a large difference.[1] For choosing them the standard method is to choose random integers and use a primality test until two primes are found. • p and q should be kept secret. 2. Compute n = pq. • n is used as the modulus for both the public and private keys. Its length, usually expressed in bits, is the key length. • n is released as part of the public key. 3. Compute λ(n), where λ is Carmichael's totient function. Since n = pq, λ(n) = lcm(λ(p), λ(q)), and since p and q are prime, λ(p) = φ(p) = p − 1, and likewise λ(q) = q − 1. Hence λ(n) = lcm(p − 1, q − 1). • The lcm may be calculated through the Euclidean algorithm, since lcm(a, b) = \|ab\|/gcd(a, b). • λ(n) is kept secret. 4. Choose an integer e such that 2 < e < λ(n) and gcd(e, λ(n)) = 1; that is, e and λ(n) are coprime. • e having a short bit-length and small Hamming weight results in more efficient encryption – the most commonly chosen value for e is 216 + 1 = 65537. The smallest (and fastest) possible value for e is 3, but such a small value for e has been shown to be less secure in some settings.[15] • e is released as part of the public key. 5. Determine d as d ≡ e−1 (mod λ(n)); that is, d is the modular multiplicative inverse of e modulo λ(n). • This means: solve for d the equation de ≡ 1 (mod λ(n)); d can be computed efficiently by using the extended Euclidean algorithm, since, thanks to e and λ(n) being coprime, said equation is a form of Bézout's identity, where d is one of the coefficients. • d is kept secret as the private key exponent. The public key consists of the modulus n and the public (or encryption) exponent e. The private key consists of the private (or decryption) exponent d, which must be kept secret. p, q, and λ(n) must also be kept secret because they can be used to calculate d. In fact, they can all be discarded after d has been computed.[16] In the original RSA paper,[1] the Euler totient function φ(n) = (p − 1)(q − 1) is used instead of λ(n) for calculating the private exponent d. Since φ(n) is always divisible by λ(n), the algorithm works as well. The possibility of using Euler totient function results also from Lagrange's theorem applied to the multiplicative group of integers modulo pq. Thus any d satisfying d⋅e ≡ 1 (mod φ(n)) also satisfies d⋅e ≡ 1 (mod λ(n)). However, computing d modulo φ(n) will sometimes yield a result that is larger than necessary (i.e. d > λ(n)). Most of the implementations of RSA will accept exponents generated using either method (if they use the private exponent d at all, rather than using the optimized decryption method based on the Chinese remainder theorem described below), but some standards such as FIPS 186-4 (Section B.3.1) may require that d < λ(n). Any "oversized" private exponents not meeting this criterion may always be reduced modulo λ(n) to obtain a smaller equivalent exponent. Since any common factors of (p − 1) and (q − 1) are present in the factorisation of n − 1 = pq − 1 = (p − 1)(q − 1) + (p − 1) + (q − 1),[17] it is recommended that (p − 1) and (q − 1) have only very small common factors, if any, besides the necessary 2.[1][18][19][20] Note: The authors of the original RSA paper carry out the key generation by choosing d and then computing e as the modular multiplicative inverse of d modulo φ(n), whereas most current implementations of RSA, such as those following PKCS#1, do the reverse (choose e and compute d). Since the chosen key can be small, whereas the computed key normally is not, the RSA paper's algorithm optimizes decryption compared to encryption, while the modern algorithm optimizes encryption instead.[1][21] Key distribution Suppose that Bob wants to send information to Alice. If they decide to use RSA, Bob must know Alice's public key to encrypt the message, and Alice must use her private key to decrypt the message. To enable Bob to send his encrypted messages, Alice transmits her public key (n, e) to Bob via a reliable, but not necessarily secret, route. Alice's private key (d) is never distributed. Encryption After Bob obtains Alice's public key, he can send a message M to Alice. To do it, he first turns M (strictly speaking, the un-padded plaintext) into an integer m (strictly speaking, the padded plaintext), such that 0 ≤ m < n by using an agreed-upon reversible protocol known as a padding scheme. He then computes the ciphertext c, using Alice's public key e, corresponding to $c\equiv m^{e}{\pmod {n}}.$ This can be done reasonably quickly, even for very large numbers, using modular exponentiation. Bob then transmits c to Alice. Note that at least nine values of m will yield a ciphertext c equal to m,[22] but this is very unlikely to occur in practice. Decryption Alice can recover m from c by using her private key exponent d by computing $c^{d}\equiv (m^{e})^{d}\equiv m{\pmod {n}}.$ Given m, she can recover the original message M by reversing the padding scheme. Example Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but one can also use OpenSSL to generate and examine a real keypair. 1. Choose two distinct prime numbers, such as $p=61$ and $q=53$. 2. Compute n = pq giving $n=61\times 53=3233.$ 3. Compute the Carmichael's totient function of the product as λ(n) = lcm(p − 1, q − 1) giving $\lambda (3233)=\operatorname {lcm} (60,52)=780.$ 4. Choose any number 1 < e < 780 that is coprime to 780. Choosing a prime number for e leaves us only to check that e is not a divisor of 780. Let $e=17$. 5. Compute d, the modular multiplicative inverse of e (mod λ(n)), yielding $d=413,$ as $1=(17\times 413){\bmod {7}}80.$ The public key is (n = 3233, e = 17). For a padded plaintext message m, the encryption function is ${\begin{aligned}c(m)&=m^{e}{\bmod {n}}\\&=m^{17}{\bmod {3}}233.\end{aligned}}$ The private key is (n = 3233, d = 413). For an encrypted ciphertext c, the decryption function is ${\begin{aligned}m(c)&=c^{d}{\bmod {n}}\\&=c^{413}{\bmod {3}}233.\end{aligned}}$ For instance, in order to encrypt m = 65, one calculates $c=65^{17}{\bmod {3}}233=2790.$ To decrypt c = 2790, one calculates $m=2790^{413}{\bmod {3}}233=65.$ Both of these calculations can be computed efficiently using the square-and-multiply algorithm for modular exponentiation. In real-life situations the primes selected would be much larger; in our example it would be trivial to factor n = 3233 (obtained from the freely available public key) back to the primes p and q. e, also from the public key, is then inverted to get d, thus acquiring the private key. Practical implementations use the Chinese remainder theorem to speed up the calculation using modulus of factors (mod pq using mod p and mod q). The values dp, dq and qinv, which are part of the private key are computed as follows: ${\begin{aligned}d_{p}&=d{\bmod {(}}p-1)=413{\bmod {(}}61-1)=53,\\d_{q}&=d{\bmod {(}}q-1)=413{\bmod {(}}53-1)=49,\\q_{\text{inv}}&=q^{-1}{\bmod {p}}=53^{-1}{\bmod {6}}1=38\\&\Rightarrow (q_{\text{inv}}\times q){\bmod {p}}=38\times 53{\bmod {6}}1=1.\end{aligned}}$ Here is how dp, dq and qinv are used for efficient decryption (encryption is efficient by choice of a suitable d and e pair): ${\begin{aligned}m_{1}&=c^{d_{p}}{\bmod {p}}=2790^{53}{\bmod {6}}1=4,\\m_{2}&=c^{d_{q}}{\bmod {q}}=2790^{49}{\bmod {5}}3=12,\\h&=(q_{\text{inv}}\times (m_{1}-m_{2})){\bmod {p}}=(38\times -8){\bmod {6}}1=1,\\m&=m_{2}+h\times q=12+1\times 53=65.\end{aligned}}$ Signing messages Suppose Alice uses Bob's public key to send him an encrypted message. In the message, she can claim to be Alice, but Bob has no way of verifying that the message was from Alice, since anyone can use Bob's public key to send him encrypted messages. In order to verify the origin of a message, RSA can also be used to sign a message. Suppose Alice wishes to send a signed message to Bob. She can use her own private key to do so. She produces a hash value of the message, raises it to the power of d (modulo n) (as she does when decrypting a message), and attaches it as a "signature" to the message. When Bob receives the signed message, he uses the same hash algorithm in conjunction with Alice's public key. He raises the signature to the power of e (modulo n) (as he does when encrypting a message), and compares the resulting hash value with the message's hash value. If the two agree, he knows that the author of the message was in possession of Alice's private key and that the message has not been tampered with since being sent. This works because of exponentiation rules: $h=\operatorname {hash} (m),$ $(h^{e})^{d}=h^{ed}=h^{de}=(h^{d})^{e}\equiv h{\pmod {n}}.$ Thus the keys may be swapped without loss of generality, that is, a private key of a key pair may be used either to: 1. Decrypt a message only intended for the recipient, which may be encrypted by anyone having the public key (asymmetric encrypted transport). 2. Encrypt a message which may be decrypted by anyone, but which can only be encrypted by one person; this provides a digital signature. Proofs of correctness Proof using Fermat's little theorem The proof of the correctness of RSA is based on Fermat's little theorem, stating that ap − 1 ≡ 1 (mod p) for any integer a and prime p, not dividing a.[note 1] We want to show that $(m^{e})^{d}\equiv m{\pmod {pq}}$ for every integer m when p and q are distinct prime numbers and e and d are positive integers satisfying ed ≡ 1 (mod λ(pq)). Since λ(pq) = lcm(p − 1, q − 1) is, by construction, divisible by both p − 1 and q − 1, we can write $ed-1=h(p-1)=k(q-1)$ for some nonnegative integers h and k.[note 2] To check whether two numbers, such as med and m, are congruent mod pq, it suffices (and in fact is equivalent) to check that they are congruent mod p and mod q separately.[note 3] To show med ≡ m (mod p), we consider two cases: 1. If m ≡ 0 (mod p), m is a multiple of p. Thus med is a multiple of p. So med ≡ 0 ≡ m (mod p). 2. If m $\not \equiv $ 0 (mod p), $m^{ed}=m^{ed-1}m=m^{h(p-1)}m=(m^{p-1})^{h}m\equiv 1^{h}m\equiv m{\pmod {p}},$ where we used Fermat's little theorem to replace mp−1 mod p with 1. The verification that med ≡ m (mod q) proceeds in a completely analogous way: 1. If m ≡ 0 (mod q), med is a multiple of q. So med ≡ 0 ≡ m (mod q). 2. If m $\not \equiv $ 0 (mod q), $m^{ed}=m^{ed-1}m=m^{k(q-1)}m=(m^{q-1})^{k}m\equiv 1^{k}m\equiv m{\pmod {q}}.$ This completes the proof that, for any integer m, and integers e, d such that ed ≡ 1 (mod λ(pq)), $(m^{e})^{d}\equiv m{\pmod {pq}}.$ Notes 1. We cannot trivially break RSA by applying the theorem (mod pq) because pq is not prime. 2. In particular, the statement above holds for any e and d that satisfy ed ≡ 1 (mod (p − 1)(q − 1)), since (p − 1)(q − 1) is divisible by λ(pq), and thus trivially also by p − 1 and q − 1. However, in modern implementations of RSA, it is common to use a reduced private exponent d that only satisfies the weaker, but sufficient condition ed ≡ 1 (mod λ(pq)). 3. This is part of the Chinese remainder theorem, although it is not the significant part of that theorem. Proof using Euler's theorem Although the original paper of Rivest, Shamir, and Adleman used Fermat's little theorem to explain why RSA works, it is common to find proofs that rely instead on Euler's theorem. We want to show that med ≡ m (mod n), where n = pq is a product of two different prime numbers, and e and d are positive integers satisfying ed ≡ 1 (mod φ(n)). Since e and d are positive, we can write ed = 1 + hφ(n) for some non-negative integer h. Assuming that m is relatively prime to n, we have $m^{ed}=m^{1+h\varphi (n)}=m(m^{\varphi (n)})^{h}\equiv m(1)^{h}\equiv m{\pmod {n}},$ where the second-last congruence follows from Euler's theorem. More generally, for any e and d satisfying ed ≡ 1 (mod λ(n)), the same conclusion follows from Carmichael's generalization of Euler's theorem, which states that mλ(n) ≡ 1 (mod n) for all m relatively prime to n. When m is not relatively prime to n, the argument just given is invalid. This is highly improbable (only a proportion of 1/p + 1/q − 1/(pq) numbers have this property), but even in this case, the desired congruence is still true. Either m ≡ 0 (mod p) or m ≡ 0 (mod q), and these cases can be treated using the previous proof. Padding Attacks against plain RSA There are a number of attacks against plain RSA as described below. • When encrypting with low encryption exponents (e.g., e = 3) and small values of the m (i.e., m < n1/e), the result of me is strictly less than the modulus n. In this case, ciphertexts can be decrypted easily by taking the eth root of the ciphertext over the integers. • If the same clear-text message is sent to e or more recipients in an encrypted way, and the receivers share the same exponent e, but different p, q, and therefore n, then it is easy to decrypt the original clear-text message via the Chinese remainder theorem. Johan Håstad noticed that this attack is possible even if the clear texts are not equal, but the attacker knows a linear relation between them.[23] This attack was later improved by Don Coppersmith (see Coppersmith's attack).[24] • Because RSA encryption is a deterministic encryption algorithm (i.e., has no random component) an attacker can successfully launch a chosen plaintext attack against the cryptosystem, by encrypting likely plaintexts under the public key and test whether they are equal to the ciphertext. A cryptosystem is called semantically secure if an attacker cannot distinguish two encryptions from each other, even if the attacker knows (or has chosen) the corresponding plaintexts. RSA without padding is not semantically secure.[25] • RSA has the property that the product of two ciphertexts is equal to the encryption of the product of the respective plaintexts. That is, m1em2e ≡ (m1m2)e (mod n). Because of this multiplicative property, a chosen-ciphertext attack is possible. E.g., an attacker who wants to know the decryption of a ciphertext c ≡ me (mod n) may ask the holder of the private key d to decrypt an unsuspicious-looking ciphertext c′ ≡ cre (mod n) for some value r chosen by the attacker. Because of the multiplicative property, c' is the encryption of mr (mod n). Hence, if the attacker is successful with the attack, they will learn mr (mod n), from which they can derive the message m by multiplying mr with the modular inverse of r modulo n. • Given the private exponent d, one can efficiently factor the modulus n = pq. And given factorization of the modulus n = pq, one can obtain any private key (d', n) generated against a public key (e', n).[15] Padding schemes To avoid these problems, practical RSA implementations typically embed some form of structured, randomized padding into the value m before encrypting it. This padding ensures that m does not fall into the range of insecure plaintexts, and that a given message, once padded, will encrypt to one of a large number of different possible ciphertexts. Standards such as PKCS#1 have been carefully designed to securely pad messages prior to RSA encryption. Because these schemes pad the plaintext m with some number of additional bits, the size of the un-padded message M must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks that may be facilitated by a predictable message structure. Early versions of the PKCS#1 standard (up to version 1.5) used a construction that appears to make RSA semantically secure. However, at Crypto 1998, Bleichenbacher showed that this version is vulnerable to a practical adaptive chosen-ciphertext attack. Furthermore, at Eurocrypt 2000, Coron et al.[26] showed that for some types of messages, this padding does not provide a high enough level of security. Later versions of the standard include Optimal Asymmetric Encryption Padding (OAEP), which prevents these attacks. As such, OAEP should be used in any new application, and PKCS#1 v1.5 padding should be replaced wherever possible. The PKCS#1 standard also incorporates processing schemes designed to provide additional security for RSA signatures, e.g. the Probabilistic Signature Scheme for RSA (RSA-PSS). Secure padding schemes such as RSA-PSS are as essential for the security of message signing as they are for message encryption. Two USA patents on PSS were granted (U.S. Patent 6,266,771 and U.S. Patent 7,036,014); however, these patents expired on 24 July 2009 and 25 April 2010 respectively. Use of PSS no longer seems to be encumbered by patents. Note that using different RSA key pairs for encryption and signing is potentially more secure.[27] Security and practical considerations Using the Chinese remainder algorithm For efficiency, many popular crypto libraries (such as OpenSSL, Java and .NET) use for decryption and signing the following optimization based on the Chinese remainder theorem. The following values are precomputed and stored as part of the private key: • $p$ and $q$ – the primes from the key generation, • $d_{P}=d{\pmod {p-1}},$ • $d_{Q}=d{\pmod {q-1}},$ • $q_{\text{inv}}=q^{-1}{\pmod {p}}.$ These values allow the recipient to compute the exponentiation m = cd (mod pq) more efficiently as follows: $m_{1}=c^{d_{P}}{\pmod {p}}$, $m_{2}=c^{d_{Q}}{\pmod {q}}$, $h=q_{\text{inv}}(m_{1}-m_{2}){\pmod {p}}$,[28] $m=m_{2}+hq{\pmod {pq}}$ . This is more efficient than computing exponentiation by squaring, even though two modular exponentiations have to be computed. The reason is that these two modular exponentiations both use a smaller exponent and a smaller modulus. Integer factorization and RSA problem See also: RSA Factoring Challenge, Integer factorization records, and Shor's algorithm The security of the RSA cryptosystem is based on two mathematical problems: the problem of factoring large numbers and the RSA problem. Full decryption of an RSA ciphertext is thought to be infeasible on the assumption that both of these problems are hard, i.e., no efficient algorithm exists for solving them. Providing security against partial decryption may require the addition of a secure padding scheme.[29] The RSA problem is defined as the task of taking eth roots modulo a composite n: recovering a value m such that c ≡ me (mod n), where (n, e) is an RSA public key, and c is an RSA ciphertext. Currently the most promising approach to solving the RSA problem is to factor the modulus n. With the ability to recover prime factors, an attacker can compute the secret exponent d from a public key (n, e), then decrypt c using the standard procedure. To accomplish this, an attacker factors n into p and q, and computes lcm(p − 1, q − 1) that allows the determination of d from e. No polynomial-time method for factoring large integers on a classical computer has yet been found, but it has not been proven that none exists; see integer factorization for a discussion of this problem. Multiple polynomial quadratic sieve (MPQS) can be used to factor the public modulus n. The first RSA-512 factorization in 1999 used hundreds of computers and required the equivalent of 8,400 MIPS years, over an elapsed time of about seven months.[30] By 2009, Benjamin Moody could factor an 512-bit RSA key in 73 days using only public software (GGNFS) and his desktop computer (a dual-core Athlon64 with a 1,900 MHz CPU). Just less than 5 gigabytes of disk storage was required and about 2.5 gigabytes of RAM for the sieving process. Rivest, Shamir, and Adleman noted[1] that Miller has shown that – assuming the truth of the extended Riemann hypothesis – finding d from n and e is as hard as factoring n into p and q (up to a polynomial time difference).[31] However, Rivest, Shamir, and Adleman noted, in section IX/D of their paper, that they had not found a proof that inverting RSA is as hard as factoring. As of 2020, the largest publicly known factored RSA number had 829 bits (250 decimal digits, RSA-250).[32] Its factorization, by a state-of-the-art distributed implementation, took about 2,700 CPU-years. In practice, RSA keys are typically 1024 to 4096 bits long. In 2003, RSA Security estimated that 1024-bit keys were likely to become crackable by 2010.[33] As of 2020, it is not known whether such keys can be cracked, but minimum recommendations have moved to at least 2048 bits.[34] It is generally presumed that RSA is secure if n is sufficiently large, outside of quantum computing. If n is 300 bits or shorter, it can be factored in a few hours in a personal computer, using software already freely available. Keys of 512 bits have been shown to be practically breakable in 1999, when RSA-155 was factored by using several hundred computers, and these are now factored in a few weeks using common hardware. Exploits using 512-bit code-signing certificates that may have been factored were reported in 2011.[35] A theoretical hardware device named TWIRL, described by Shamir and Tromer in 2003, called into question the security of 1024-bit keys.[33] In 1994, Peter Shor showed that a quantum computer – if one could ever be practically created for the purpose – would be able to factor in polynomial time, breaking RSA; see Shor's algorithm. Faulty key generation Finding the large primes p and q is usually done by testing random numbers of the correct size with probabilistic primality tests that quickly eliminate virtually all of the nonprimes. The numbers p and q should not be "too close", lest the Fermat factorization for n be successful. If p − q is less than 2n1/4 (n = p⋅q, which even for "small" 1024-bit values of n is 3×1077), solving for p and q is trivial. Furthermore, if either p − 1 or q − 1 has only small prime factors, n can be factored quickly by Pollard's p − 1 algorithm, and hence such values of p or q should be discarded. It is important that the private exponent d be large enough. Michael J. Wiener showed that if p is between q and 2q (which is quite typical) and d < n1/4/3, then d can be computed efficiently from n and e.[36] There is no known attack against small public exponents such as e = 3, provided that the proper padding is used. Coppersmith's attack has many applications in attacking RSA specifically if the public exponent e is small and if the encrypted message is short and not padded. 65537 is a commonly used value for e; this value can be regarded as a compromise between avoiding potential small-exponent attacks and still allowing efficient encryptions (or signature verification). The NIST Special Publication on Computer Security (SP 800-78 Rev. 1 of August 2007) does not allow public exponents e smaller than 65537, but does not state a reason for this restriction. In October 2017, a team of researchers from Masaryk University announced the ROCA vulnerability, which affects RSA keys generated by an algorithm embodied in a library from Infineon known as RSALib. A large number of smart cards and trusted platform modules (TPM) were shown to be affected. Vulnerable RSA keys are easily identified using a test program the team released.[37] Importance of strong random number generation A cryptographically strong random number generator, which has been properly seeded with adequate entropy, must be used to generate the primes p and q. An analysis comparing millions of public keys gathered from the Internet was carried out in early 2012 by Arjen K. Lenstra, James P. Hughes, Maxime Augier, Joppe W. Bos, Thorsten Kleinjung and Christophe Wachter. They were able to factor 0.2% of the keys using only Euclid's algorithm.[38][39] They exploited a weakness unique to cryptosystems based on integer factorization. If n = pq is one public key, and n′ = p′q′ is another, then if by chance p = p′ (but q is not equal to q'), then a simple computation of gcd(n, n′) = p factors both n and n', totally compromising both keys. Lenstra et al. note that this problem can be minimized by using a strong random seed of bit length twice the intended security level, or by employing a deterministic function to choose q given p, instead of choosing p and q independently. Nadia Heninger was part of a group that did a similar experiment. They used an idea of Daniel J. Bernstein to compute the GCD of each RSA key n against the product of all the other keys n' they had found (a 729-million-digit number), instead of computing each gcd(n, n′) separately, thereby achieving a very significant speedup, since after one large division, the GCD problem is of normal size. Heninger says in her blog that the bad keys occurred almost entirely in embedded applications, including "firewalls, routers, VPN devices, remote server administration devices, printers, projectors, and VOIP phones" from more than 30 manufacturers. Heninger explains that the one-shared-prime problem uncovered by the two groups results from situations where the pseudorandom number generator is poorly seeded initially, and then is reseeded between the generation of the first and second primes. Using seeds of sufficiently high entropy obtained from key stroke timings or electronic diode noise or atmospheric noise from a radio receiver tuned between stations should solve the problem.[40] Strong random number generation is important throughout every phase of public-key cryptography. For instance, if a weak generator is used for the symmetric keys that are being distributed by RSA, then an eavesdropper could bypass RSA and guess the symmetric keys directly. Timing attacks Kocher described a new attack on RSA in 1995: if the attacker Eve knows Alice's hardware in sufficient detail and is able to measure the decryption times for several known ciphertexts, Eve can deduce the decryption key d quickly. This attack can also be applied against the RSA signature scheme. In 2003, Boneh and Brumley demonstrated a more practical attack capable of recovering RSA factorizations over a network connection (e.g., from a Secure Sockets Layer (SSL)-enabled webserver).[41] This attack takes advantage of information leaked by the Chinese remainder theorem optimization used by many RSA implementations. One way to thwart these attacks is to ensure that the decryption operation takes a constant amount of time for every ciphertext. However, this approach can significantly reduce performance. Instead, most RSA implementations use an alternate technique known as cryptographic blinding. RSA blinding makes use of the multiplicative property of RSA. Instead of computing cd (mod n), Alice first chooses a secret random value r and computes (rec)d (mod n). The result of this computation, after applying Euler's theorem, is rcd (mod n), and so the effect of r can be removed by multiplying by its inverse. A new value of r is chosen for each ciphertext. With blinding applied, the decryption time is no longer correlated to the value of the input ciphertext, and so the timing attack fails. Adaptive chosen-ciphertext attacks In 1998, Daniel Bleichenbacher described the first practical adaptive chosen-ciphertext attack against RSA-encrypted messages using the PKCS #1 v1 padding scheme (a padding scheme randomizes and adds structure to an RSA-encrypted message, so it is possible to determine whether a decrypted message is valid). Due to flaws with the PKCS #1 scheme, Bleichenbacher was able to mount a practical attack against RSA implementations of the Secure Sockets Layer protocol and to recover session keys. As a result of this work, cryptographers now recommend the use of provably secure padding schemes such as Optimal Asymmetric Encryption Padding, and RSA Laboratories has released new versions of PKCS #1 that are not vulnerable to these attacks. A variant of this attack, dubbed "BERserk", came back in 2014.[42][43] It impacted the Mozilla NSS Crypto Library, which was used notably by Firefox and Chrome. Side-channel analysis attacks A side-channel attack using branch-prediction analysis (BPA) has been described. Many processors use a branch predictor to determine whether a conditional branch in the instruction flow of a program is likely to be taken or not. Often these processors also implement simultaneous multithreading (SMT). Branch-prediction analysis attacks use a spy process to discover (statistically) the private key when processed with these processors. Simple Branch Prediction Analysis (SBPA) claims to improve BPA in a non-statistical way. In their paper, "On the Power of Simple Branch Prediction Analysis",[44] the authors of SBPA (Onur Aciicmez and Cetin Kaya Koc) claim to have discovered 508 out of 512 bits of an RSA key in 10 iterations. A power-fault attack on RSA implementations was described in 2010.[45] The author recovered the key by varying the CPU power voltage outside limits; this caused multiple power faults on the server. Tricky implementation There are many details to keep in mind in order to implement RSA securely (strong PRNG, acceptable public exponent...).[46] This makes the implementation challenging, to the point the book Practical Cryptography With Go suggests avoiding RSA if possible. Implementations Some cryptography libraries that provide support for RSA include: • Botan • Bouncy Castle • cryptlib • Crypto++ • Libgcrypt • Nettle • OpenSSL • wolfCrypt • GnuTLS • mbed TLS • LibreSSL See also • Acoustic cryptanalysis • Computational complexity theory • Diffie–Hellman key exchange • Digital Signature Algorithm • Elliptic-curve cryptography • Key exchange • Key management • Key size • Public-key cryptography • Trapdoor function References 1. Rivest, R.; Shamir, A.; Adleman, L. (February 1978). "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems" (PDF). Communications of the ACM. 21 (2): 120–126. CiteSeerX 10.1.1.607.2677. doi:10.1145/359340.359342. S2CID 2873616. Archived from the original (PDF) on 2023-01-27. 2. Smart, Nigel (February 19, 2008). "Dr Clifford Cocks CB". Bristol University. Retrieved August 14, 2011. 3. Castelvecchi, Davide (2020-10-30). "Quantum-computing pioneer warns of complacency over Internet security". Nature. 587 (7833): 189. Bibcode:2020Natur.587..189C. doi:10.1038/d41586-020-03068-9. PMID 33139910. S2CID 226243008. 2020 interview of Peter Shor. 4. Diffie, W.; Hellman, M. E. (November 1976). "New directions in cryptography". IEEE Transactions on Information Theory. 22 (6): 644–654. CiteSeerX 10.1.1.37.9720. doi:10.1109/TIT.1976.1055638. ISSN 0018-9448. 5. Rivest, Ronald. "The Early Days of RSA – History and Lessons" (PDF). 6. Calderbank, Michael (2007-08-20). "The RSA Cryptosystem: History, Algorithm, Primes" (PDF). 7. Robinson, Sara (June 2003). "Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders" (PDF). SIAM News. 36 (5). 8. Cocks, C. C. (20 November 1973). "A Note on Non-Secret Encryption" (PDF). www.gchq.gov.uk. Archived from the original (PDF) on 28 September 2018. Retrieved 2017-05-30. 9. Jim Sauerberg. "From Private to Public Key Ciphers in Three Easy Steps". 10. Margaret Cozzens and Steven J. Miller. "The Mathematics of Encryption: An Elementary Introduction". p. 180. 11. Alasdair McAndrew. "Introduction to Cryptography with Open-Source Software". p. 12. 12. Surender R. Chiluka. "Public key Cryptography". 13. Neal Koblitz. "Cryptography As a Teaching Tool". Cryptologia, Vol. 21, No. 4 (1997). 14. "RSA Security Releases RSA Encryption Algorithm into Public Domain". Archived from the original on June 21, 2007. Retrieved 2010-03-03. 15. Boneh, Dan (1999). "Twenty Years of attacks on the RSA Cryptosystem". Notices of the American Mathematical Society. 46 (2): 203–213. 16. Applied Cryptography, John Wiley & Sons, New York, 1996. Bruce Schneier, p. 467. 17. McKee, James; Pinch, Richard (1998). "Further Attacks on Server-Aided RSA Cryptosystems". CiteSeerX 10.1.1.33.1333. {{cite journal}}: Cite journal requires \|journal= (help) 18. A Course in Number Theory and Cryptography, Graduate Texts in Math. No. 114, Springer-Verlag, New York, 1987. Neal Koblitz, Second edition, 1994. p. 94. 19. Dukhovni, Viktor (July 31, 2015). "common factors in (p − 1) and (q − 1)". openssl-dev (Mailing list). 20. Dukhovni, Viktor (August 1, 2015). "common factors in (p − 1) and (q − 1)". openssl-dev (Mailing list). 21. Johnson, J.; Kaliski, B. (February 2003). Public-Key Cryptography Standards (PKCS) #1: RSA Cryptography Specifications Version 2.1. Network Working Group. doi:10.17487/RFC3447. RFC 3447. Retrieved 9 March 2016. 22. Namely, the values of m which are equal to −1, 0, or 1 modulo p while also equal to −1, 0, or 1 modulo q. There will be more values of m having c = m if p − 1 or q − 1 has other divisors in common with e − 1 besides 2 because this gives more values of m such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle m^{e-1} \bmod p = 1} or $m^{e-1}{\bmod {q}}=1$ respectively. 23. Håstad, Johan (1986). "On using RSA with Low Exponent in a Public Key Network". Advances in Cryptology – CRYPTO '85 Proceedings. Lecture Notes in Computer Science. Vol. 218. pp. 403–408. doi:10.1007/3-540-39799-X_29. ISBN 978-3-540-16463-0. 24. Coppersmith, Don (1997). "Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities" (PDF). Journal of Cryptology. 10 (4): 233–260. CiteSeerX 10.1.1.298.4806. doi:10.1007/s001459900030. S2CID 15726802. 25. Goldwasser, Shafi; Micali, Silvio (1982-05-05). "Probabilistic encryption & how to play mental poker keeping secret all partial information". Proceedings of the fourteenth annual ACM symposium on Theory of computing - STOC '82. STOC '82. New York, NY, USA: Association for Computing Machinery. pp. 365–377. doi:10.1145/800070.802212. ISBN 978-0-89791-070-5. S2CID 10316867. 26. Coron, Jean-Sébastien; Joye, Marc; Naccache, David; Paillier, Pascal (2000). "New Attacks on PKCS#1 v1.5 Encryption". In Preneel, Bart (ed.). Advances in Cryptology — EUROCRYPT 2000. Lecture Notes in Computer Science. Vol. 1807. Berlin, Heidelberg: Springer. pp. 369–381. doi:10.1007/3-540-45539-6_25. ISBN 978-3-540-45539-4. 27. "RSA Algorithm". 28. If $m_{1}<m_{2}$, then some libraries compute h as $q_{\text{inv}}\left[\left(m_{1}+\left\lceil {\frac {q}{p}}\right\rceil p\right)-m_{2}\right]{\pmod {p}}$. 29. Machie, Edmond K. (29 March 2013). Network security traceback attack and react in the United States Department of Defense network. Trafford. p. 167. ISBN 978-1466985742. 30. Lenstra, Arjen; et al. (Group) (2000). "Factorization of a 512-bit RSA Modulus" (PDF). Eurocrypt. 31. Miller, Gary L. (1975). "Riemann's Hypothesis and Tests for Primality" (PDF). Proceedings of Seventh Annual ACM Symposium on Theory of Computing. pp. 234–239. 32. Zimmermann, Paul (2020-02-28). "Factorization of RSA-250". Cado-nfs-discuss. Archived from the original on 2020-02-28. Retrieved 2020-07-12. 33. Kaliski, Burt (2003-05-06). "TWIRL and RSA Key Size". RSA Laboratories. Archived from the original on 2017-04-17. Retrieved 2017-11-24. 34. Barker, Elaine; Dang, Quynh (2015-01-22). "NIST Special Publication 800-57 Part 3 Revision 1: Recommendation for Key Management: Application-Specific Key Management Guidance" (PDF). National Institute of Standards and Technology: 12. doi:10.6028/NIST.SP.800-57pt3r1. Retrieved 2017-11-24. {{cite journal}}: Cite journal requires \|journal= (help) 35. Sandee, Michael (November 21, 2011). "RSA-512 certificates abused in-the-wild". Fox-IT International blog. 36. Wiener, Michael J. (May 1990). "Cryptanalysis of short RSA secret exponents" (PDF). IEEE Transactions on Information Theory. 36 (3): 553–558. doi:10.1109/18.54902. S2CID 7120331. 37. Nemec, Matus; Sys, Marek; Svenda, Petr; Klinec, Dusan; Matyas, Vashek (November 2017). "The Return of Coppersmith's Attack: Practical Factorization of Widely Used RSA Moduli" (PDF). Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security. CCS '17. doi:10.1145/3133956.3133969. 38. Markoff, John (February 14, 2012). "Flaw Found in an Online Encryption Method". The New York Times. 39. Lenstra, Arjen K.; Hughes, James P.; Augier, Maxime; Bos, Joppe W.; Kleinjung, Thorsten; Wachter, Christophe (2012). "Ron was wrong, Whit is right" (PDF). {{cite journal}}: Cite journal requires \|journal= (help) 40. Heninger, Nadia (February 15, 2012). "New research: There's no need to panic over factorable keys–just mind your Ps and Qs". Freedom to Tinker. 41. Brumley, David; Boneh, Dan (2003). "Remote timing attacks are practical" (PDF). Proceedings of the 12th Conference on USENIX Security Symposium. SSYM'03. 42. "'BERserk' Bug Uncovered In Mozilla NSS Crypto Library Impacts Firefox, Chrome". 25 September 2014. Retrieved 4 January 2022. 43. "RSA Signature Forgery in NSS". Mozilla. 44. Acıiçmez, Onur; Koç, Çetin Kaya; Seifert, Jean-Pierre (2007). "On the power of simple branch prediction analysis". Proceedings of the 2nd ACM Symposium on Information, Computer and Communications Security. ASIACCS '07. pp. 312–320. CiteSeerX 10.1.1.80.1438. doi:10.1145/1229285.1266999. 45. Pellegrini, Andrea; Bertacco, Valeria; Austin, Todd (2010). "Fault-Based Attack of RSA Authentication" (PDF). {{cite journal}}: Cite journal requires \|journal= (help) 46. Isom, Kyle. "Practical Cryptography With Go". Retrieved 4 January 2022. Further reading • Menezes, Alfred; van Oorschot, Paul C.; Vanstone, Scott A. (October 1996). Handbook of Applied Cryptography. CRC Press. ISBN 978-0-8493-8523-0. • Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. pp. 881–887. ISBN 978-0-262-03293-3. External links • The Original RSA Patent as filed with the U.S. Patent Office by Rivest; Ronald L. (Belmont, MA), Shamir; Adi (Cambridge, MA), Adleman; Leonard M. (Arlington, MA), December 14, 1977, U.S. Patent 4,405,829. • PKCS #1: RSA Cryptography Standard (RSA Laboratories website) • The PKCS #1 standard "provides recommendations for the implementation of public-key cryptography based on the RSA algorithm, covering the following aspects: cryptographic primitives; encryption schemes; signature schemes with appendix; ASN.1 syntax for representing keys and for identifying the schemes". • Explanation of RSA using colored lamps on YouTube • Thorough walk through of RSA • Prime Number Hide-And-Seek: How the RSA Cipher Works • Onur Aciicmez, Cetin Kaya Koc, Jean-Pierre Seifert: On the Power of Simple Branch Prediction Analysis • Example of an RSA implementation with PKCS#1 padding (GPL source code) • Kocher's article about timing attacks • An animated explanation of RSA with its mathematical background by CrypTool • Grime, James. "RSA Encryption". Numberphile. Brady Haran. Archived from the original on 2018-10-06. Retrieved 2013-04-13. • How RSA Key used for Encryption in real world Public-key cryptography Algorithms Integer factorization • Benaloh • Blum–Goldwasser • Cayley–Purser • Damgård–Jurik • GMR • Goldwasser–Micali • Naccache–Stern • Paillier • Rabin • RSA • Okamoto–Uchiyama • Schmidt–Samoa Discrete logarithm • BLS • Cramer–Shoup • DH • DSA • ECDH • X25519 • X448 • ECDSA • EdDSA • Ed25519 • Ed448 • ECMQV • EKE • ElGamal • signature scheme • MQV • Schnorr • SPEKE • SRP • STS Lattice/SVP/CVP/LWE/SIS • Kyber • NTRUEncrypt • NTRUSign • RLWE-KEX • RLWE-SIG • BLISS • NewHope Others • AE • CEILIDH • EPOC • HFE • IES • Lamport • McEliece • Merkle–Hellman • Naccache–Stern knapsack cryptosystem • Three-pass protocol • XTR Theory • Discrete logarithm cryptography • Elliptic-curve cryptography • Hash-based cryptography • Non-commutative cryptography • RSA problem • Trapdoor function Standardization • CRYPTREC • IEEE P1363 • NESSIE • NSA Suite B • Post-Quantum Cryptography Topics • Digital signature • OAEP • Fingerprint • PKI • Web of trust • Key size • Identity-based cryptography • Post-quantum cryptography • OpenPGP card Cryptography General • History of cryptography • Outline of cryptography • Cryptographic protocol • Authentication protocol • Cryptographic primitive • Cryptanalysis • Cryptocurrency • Cryptosystem • Cryptographic nonce • Cryptovirology • Hash function • Cryptographic hash function • Key derivation function • Digital signature • Kleptography • Key (cryptography) • Key exchange • Key generator • Key schedule • Key stretching • Keygen • Cryptojacking malware • Ransomware • Random number generation • Cryptographically secure pseudorandom number generator (CSPRNG) • Pseudorandom noise (PRN) • Secure channel • Insecure channel • Subliminal channel • Encryption • Decryption • End-to-end encryption • Harvest now, decrypt later • Information-theoretic security • Plaintext • Codetext • Ciphertext • Shared secret • Trapdoor function • Trusted timestamping • Key-based routing • Onion routing • Garlic routing • Kademlia • Mix network Mathematics • Cryptographic hash function • Block cipher • Stream cipher • Symmetric-key algorithm • Authenticated encryption • Public-key cryptography • Quantum key distribution • Quantum cryptography • Post-quantum cryptography • Message authentication code • Random numbers • Steganography • Category	Wikipedia
RV coefficient In statistics, the RV coefficient[1] is a multivariate generalization of the squared Pearson correlation coefficient (because the RV coefficient takes values between 0 and 1).[2] It measures the closeness of two set of points that may each be represented in a matrix. The major approaches within statistical multivariate data analysis can all be brought into a common framework in which the RV coefficient is maximised subject to relevant constraints. Specifically, these statistical methodologies include:[1] • principal component analysis • canonical correlation analysis • multivariate regression • statistical classification (linear discrimination). One application of the RV coefficient is in functional neuroimaging where it can measure the similarity between two subjects' series of brain scans[3] or between different scans of a same subject.[4] Definitions The definition of the RV-coefficient makes use of ideas[5] concerning the definition of scalar-valued quantities which are called the "variance" and "covariance" of vector-valued random variables. Note that standard usage is to have matrices for the variances and covariances of vector random variables. Given these innovative definitions, the RV-coefficient is then just the correlation coefficient defined in the usual way. Suppose that X and Y are matrices of centered random vectors (column vectors) with covariance matrix given by $\Sigma _{XY}=\operatorname {E} (XY^{\top })\,,$ then the scalar-valued covariance (denoted by COVV) is defined by[5] $\operatorname {COVV} (X,Y)=\operatorname {Tr} (\Sigma _{XY}\Sigma _{YX})\,.$ The scalar-valued variance is defined correspondingly: $\operatorname {VAV} (X)=\operatorname {Tr} (\Sigma _{XX}^{2})\,.$ With these definitions, the variance and covariance have certain additive properties in relation to the formation of new vector quantities by extending an existing vector with the elements of another.[5] Then the RV-coefficient is defined by[5] $\mathrm {RV} (X,Y)={\frac {\operatorname {COVV} (X,Y)}{\sqrt {\operatorname {VAV} (X)\operatorname {VAV} (Y)}}}\,.$ Shortcoming of the coefficient and adjusted version Even though the coefficient takes values between 0 and 1 by construction, it seldom attains values close to 1 as the denominator is often too large with respect to the maximal attainable value of the denominator.[6] Given known diagonal blocks $\Sigma _{XX}$ and $\Sigma _{YY}$ of dimensions $p\times p$ and $q\times q$ respectively, assuming that $p\leq q$ without loss of generality, it has been proved[7] that the maximal attainable numerator is $\operatorname {Tr} (\Lambda _{X}\Pi \Lambda _{Y}),$ where $\Lambda _{X}$ (resp. $\Lambda _{Y}$) denotes the diagonal matrix of the eigenvalues of $\Sigma _{XX}$(resp. $\Sigma _{YY}$) sorted decreasingly from the upper leftmost corner to the lower rightmost corner and $\Pi $ is the $p\times q$ matrix $(I_{p}\ 0_{p\times (q-p)})$. In light of this, Mordant and Segers[7] proposed an adjusted version of the RV coefficient in which the denominator is the maximal value attainable by the numerator. It reads ${\bar {\operatorname {RV} }}(X,Y)={\frac {\operatorname {Tr} (\Sigma _{XY}\Sigma _{YX})}{\operatorname {Tr} (\Lambda _{X}\Pi \Lambda _{Y})}}={\frac {\operatorname {Tr} (\Sigma _{XY}\Sigma _{YX})}{\sum _{j=1}^{min(p,q)}(\Lambda _{X})_{j,j}(\Lambda _{Y})_{j,j}}}.$ The impact of this adjustment is clearly visible in practice.[7] See also • Congruence coefficient • Distance correlation References 1. Robert, P.; Escoufier, Y. (1976). "A Unifying Tool for Linear Multivariate Statistical Methods: The RV-Coefficient". Applied Statistics. 25 (3): 257–265. doi:10.2307/2347233. JSTOR 2347233. 2. Abdi, Hervé (2007). Salkind, Neil J (ed.). RV coefficient and congruence coefficient. Thousand Oaks. ISBN 978-1-4129-1611-0. 3. Ferath Kherif; Jean-Baptiste Poline; Sébastien Mériaux; Habib Banali; Guillaume Plandin; Matthew Brett (2003). "Group analysis in functional neuroimaging: selecting subjects using similarity measures" (PDF). NeuroImage. 20 (4): 2197–2208. doi:10.1016/j.neuroimage.2003.08.018. PMID 14683722. 4. Herve Abdi; Joseph P. Dunlop; Lynne J. Williams (2009). "How to compute reliability estimates and display confidence and tolerance intervals for pattern classiffers using the Bootstrap and 3-way multidimensional scaling (DISTATIS)". NeuroImage. 45 (1): 89–95. doi:10.1016/j.neuroimage.2008.11.008. PMID 19084072. 5. Escoufier, Y. (1973). "Le Traitement des Variables Vectorielles". Biometrics. International Biometric Society. 29 (4): 751–760. doi:10.2307/2529140. JSTOR 2529140. 6. Pucetti, G. (2019). "Measuring Linear Correlation Between Random Vectors". SSRN. 7. Mordant Gilles; Segers Johan (2022). "Measuring dependence between random vectors via optimal transport,". Journal of Multivariate Analysis. 189.	Wikipedia
R (complexity) In computational complexity theory, R is the class of decision problems solvable by a Turing machine, which is the set of all recursive languages (also called decidable languages). Equivalent formulations R is equivalent to the set of all total computable functions in the sense that: • a decision problem is in R if and only if its indicator function is computable, • a total function is computable if and only if its graph is in R. Relationship with other classes Since we can decide any problem for which there exists a recogniser and also a co-recogniser by simply interleaving them until one obtains a result, the class is equal to RE ∩ co-RE. References • Blum, Lenore, Mike Shub, and Steve Smale, (1989), "On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines", Bulletin of the American Mathematical Society, New Series, 21 (1): 1-46. External links Complexity Zoo: Class R Important complexity classes Considered feasible • DLOGTIME • AC0 • ACC0 • TC0 • L • SL • RL • NL • NL-complete • NC • SC • CC • P • P-complete • ZPP • RP • BPP • BQP • APX • FP Suspected infeasible • UP • NP • NP-complete • NP-hard • co-NP • co-NP-complete • AM • QMA • PH • ⊕P • PP • #P • #P-complete • IP • PSPACE • PSPACE-complete Considered infeasible • EXPTIME • NEXPTIME • EXPSPACE • 2-EXPTIME • ELEMENTARY • PR • R • RE • ALL Class hierarchies • Polynomial hierarchy • Exponential hierarchy • Grzegorczyk hierarchy • Arithmetical hierarchy • Boolean hierarchy Families of classes • DTIME • NTIME • DSPACE • NSPACE • Probabilistically checkable proof • Interactive proof system List of complexity classes	Wikipedia
R. H. Bing R. H. Bing (October 20, 1914 – April 28, 1986) was an American mathematician who worked mainly in the areas of geometric topology and continuum theory. His father was named Rupert Henry, but Bing's mother thought that "Rupert Henry" was too British for Texas. She compromised by abbreviating it to R. H. (Singh 1986) Consequently, R. H. does not stand for a first or middle name. R. H. Bing Born(1914-10-20)October 20, 1914 Oakwood, Texas DiedApril 28, 1986(1986-04-28) (aged 71) Austin, Texas NationalityAmerican Alma materUniversity of Texas at Austin Known forBing–Borsuk conjecture Bing metrization theorem Bing's recognition theorem Scientific career FieldsMathematics ThesisConcerning Simple Plane Webs (1945) Doctoral advisorRobert Lee Moore Mathematical contributions Bing's mathematical research was almost exclusively in 3-manifold theory and in particular, the geometric topology of $\mathbb {R} ^{3}$. The term Bing-type topology was coined to describe the style of methods used by Bing. Bing established his reputation early on in 1946, soon after completing his Ph.D. dissertation, by solving the Kline sphere characterization problem. In 1948 he proved that the pseudo-arc is homogeneous, contradicting a published but erroneous 'proof' to the contrary. In 1951 he proved results regarding the metrizability of topological spaces, including what would later be called the Bing–Nagata–Smirnov metrization theorem. In 1952, Bing showed that the double of a solid Alexander horned sphere was the 3-sphere. This showed the existence of an involution on the 3-sphere with fixed point set equal to a wildly embedded 2-sphere, which meant that the original Smith conjecture needed to be phrased in a suitable category. This result also jump-started research into crumpled cubes. The proof involved a method later developed by Bing and others into set of techniques called Bing shrinking. Proofs of the generalized Schoenflies conjecture and the double suspension theorem relied on Bing-type shrinking. Bing was fascinated by the Poincaré conjecture and made several major attacks which ended unsuccessfully, contributing to the reputation of the conjecture as a very difficult one. He did show that a simply connected, closed 3-manifold with the property that every loop was contained in a 3-ball is homeomorphic to the 3-sphere. Bing was responsible for initiating research into the Property P conjecture, as well as its name, as a potentially more tractable version of the Poincaré conjecture. It was proven in 2004 as a culmination of work from several areas of mathematics. With some irony, this proof was announced some time after Grigori Perelman announced his proof of the Poincaré conjecture. The side-approximation theorem was considered by Bing to be one of his key discoveries. It has many applications, including a simplified proof of Moise's theorem, which states that every 3-manifold can be triangulated in an essentially unique way. The house with two rooms The house with two rooms is a contractible 2-complex that is not collapsible. Another such example, popularized by E.C. Zeeman, is the dunce hat. The house with two rooms can also be thickened and then triangulated to be unshellable, despite the thickened house topologically being a 3-ball. The house with two rooms shows up in various ways in topology. For example, it is used in the proof that every compact 3-manifold has a standard spine. Dogbone space The dogbone space is the quotient space obtained from a cellular decomposition of $\mathbb {R} ^{3}$ into points and polygonal arcs. The quotient space, $B$, is not a manifold, but $B\times \mathbb {R} $ is homeomorphic to $\mathbb {R} ^{4}$. Service and educational contributions Bing was a visiting scholar at the Institute for Advanced Study in 1957–58 and again in 1962–63.[1] Bing served as president of the MAA (1963–1964), president of the AMS (1977–78), and was department chair at University of Wisconsin, Madison (1958–1960), and at University of Texas at Austin (1975–1977). Before entering graduate school to study mathematics, Bing graduated from Southwest Texas State Teacher's College (known today as Texas State University), and was a high-school teacher for several years. His interest in education would persist for the rest of his life. Awards and honors • Member of the National Academy of Sciences (1965) • Lester R. Ford Award from the MAA (1965)[2] • Chairman of Division of Mathematics of the National Research Council (1967–1969) • United States delegate to the International Mathematical Union (1966, 1978) • Colloquium Lecturer of the American Mathematical Society (1970) • Award for Distinguished Service to Mathematics from the Mathematical Association of America (1974) • Fellow of the American Academy of Arts and Sciences (1980)[3] What does R. H. stand for? As mentioned in the introduction, Bing's father was named Rupert Henry, but Bing's mother thought that "Rupert Henry" was too British for Texas. Thus she compromised by abbreviating it to R. H. (Singh 1986) It is told that once Bing was applying for a visa and was requested not to use initials. He explained that his name was really "R-only H-only Bing", and ended up receiving a visa made out to "Ronly Honly Bing".[4] Published works • Bing, R. H. (1983), The geometric topology of 3-manifolds, American Mathematical Society Colloquium Publications, vol. 40, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1040-8, MR 0728227 • Bing, R. H. (1988), Collected papers. Vol. 1, 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0117-8, MR 0950859 References 1. Institute for Advanced Study: A Community of Scholars 2. Bing, R. H. (1964). "Spheres in E3" (PDF). Amer. Math. Monthly. 71 (4): 353–364. doi:10.2307/2313236. JSTOR 2313236. 3. "Book of Members, 1780–2010: Chapter B" (PDF). American Academy of Arts and Sciences. Retrieved July 20, 2011. 4. Krantz 2002: page 34 Sources • Anderson, R. D. (1974), "Award for Distinguished Service to Professor R. H. Bing", The American Mathematical Monthly, Mathematical Association of America, 81 (2): 111–113, doi:10.1080/00029890.1974.11993516, ISSN 0002-9890, JSTOR 2976951 • Singh, S. (1986), "R. H. Bing (1914–1986): a tribute", Topology and Its Applications, 24 (1): 5–8, doi:10.1016/0166-8641(86)90045-3, ISSN 0166-8641, MR 0872474 • Krantz, Steven G. (2002). Mathematical Apocrypha: Stories and anecdotes of mathematicians and the mathematical. Spectrum (1 ed.). Washington: The Mathematical Association of America. ISBN 0-88385-539-9. External links • R. H. Bing at the Mathematics Genealogy Project • O'Connor, John J.; Robertson, Edmund F., "R. H. Bing", MacTutor History of Mathematics Archive, University of St Andrews • MAA presidents: R. H. Bing • R H Bing as a car driver • Starbird's memoir on Bing • Memorial Resolution - Univ. of Texas, Austin • R. H. Bing Papers, 1934-1986 (archive) • National Academy of Sciences Biographical Memoir • R.H. Bing at Texas State University Presidents of the American Mathematical Society 1888–1900 • John Howard Van Amringe (1888–1890) • Emory McClintock (1891–1894) • George William Hill (1895–1896) • Simon Newcomb (1897–1898) • Robert Simpson Woodward (1899–1900) 1901–1924 • E. H. Moore (1901–1902) • Thomas Fiske (1903–1904) • William Fogg Osgood (1905–1906) • Henry Seely White (1907–1908) • Maxime Bôcher (1909–1910) • Henry Burchard Fine (1911–1912) • Edward Burr Van Vleck (1913–1914) • Ernest William Brown (1915–1916) • Leonard Eugene Dickson (1917–1918) • Frank Morley (1919–1920) • Gilbert Ames Bliss (1921–1922) • Oswald Veblen (1923–1924) 1925–1950 • George David Birkhoff (1925–1926) • Virgil Snyder (1927–1928) • Earle Raymond Hedrick (1929–1930) • Luther P. Eisenhart (1931–1932) • Arthur Byron Coble (1933–1934) • Solomon Lefschetz (1935–1936) • Robert Lee Moore (1937–1938) • Griffith C. Evans (1939–1940) • Marston Morse (1941–1942) • Marshall H. Stone (1943–1944) • Theophil Henry Hildebrandt (1945–1946) • Einar Hille (1947–1948) • Joseph L. Walsh (1949–1950) 1951–1974 • John von Neumann (1951–1952) • Gordon Thomas Whyburn (1953–1954) • Raymond Louis Wilder (1955–1956) • Richard Brauer (1957–1958) • Edward J. McShane (1959–1960) • Deane Montgomery (1961–1962) • Joseph L. Doob (1963–1964) • Abraham Adrian Albert (1965–1966) • Charles B. Morrey Jr. (1967–1968) • Oscar Zariski (1969–1970) • Nathan Jacobson (1971–1972) • Saunders Mac Lane (1973–1974) 1975–2000 • Lipman Bers (1975–1976) • R. H. Bing (1977–1978) • Peter Lax (1979–1980) • Andrew M. Gleason (1981–1982) • Julia Robinson (1983–1984) • Irving Kaplansky (1985–1986) • George Mostow (1987–1988) • William Browder (1989–1990) • Michael Artin (1991–1992) • Ronald Graham (1993–1994) • Cathleen Synge Morawetz (1995–1996) • Arthur Jaffe (1997–1998) • Felix Browder (1999–2000) 2001–2024 • Hyman Bass (2001–2002) • David Eisenbud (2003–2004) • James Arthur (2005–2006) • James Glimm (2007–2008) • George Andrews (2009–2010) • Eric Friedlander (2011–2012) • David Vogan (2013–2014) • Robert Bryant (2015–2016) • Ken Ribet (2017–2018) • Jill Pipher (2019–2020) • Ruth Charney (2021–2022) • Bryna Kra (2023–2024) Authority control International • FAST • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • SNAC • IdRef	Wikipedia
Robert V. Hogg Robert Vincent ("Bob") Hogg (8 November 1924 – 23 December 2014)[2] was an American statistician and professor of statistics of the University of Iowa. Hogg is known for his widely used textbooks on statistics (with his 1963 Ph.D. student Elliot Alan Tanis) and on mathematical statistics (with his 1950 Ph.D. advisor Allen Thornton Craig). Hogg has received recognition for his research on robust and adaptive nonparametric statistics and for his scholarship on total quality management and statistics education.[3] Robert V. Hogg Born8 November 1924 Hannibal, Missouri Died23 December 2014(2014-12-23) (aged 90) Highlands Ranch, Colorado NationalityAmerican CitizenshipUnited States of America Alma materUniversity of Iowa (Ph.D.) University of Illinois Known foreponymous textbooks ("Hogg & Craig" and "Hogg and Tanis") Statistics education Robust and nonparametric statistics Early version of Basu's theorem Childrenfour, including Rob Hogg[1] AwardsGottfried Noether Award 2001 (nonparametrics) President of American Statistical Association 1988 Founder's Award of the American Statistical Association 1991 Fellow of the Institute of Mathematical Statistics Carver Medal of the Institute of Mathematical Statistics Elected Member of the International Statistical Institute Distinguished Teaching Award of the Mathematical Association of America Scientific career FieldsStatistics InstitutionsUniversity of Iowa Doctoral advisorAllen Thornton Craig Academic career Early life Born on 8 November 1924 in Hannibal, Missouri, Hogg served three years in the US Navy from 1943 through 1946. In 1947, he graduated from the University of Illinois with a bachelor's degree in mathematics. With the goal of becoming an actuary, Hogg matriculated at the mathematics department of the University of Iowa (then the "State University of Iowa").[4] However, Hogg studied statistics under Allen Craig, who became his mentor and helped him obtain a job teaching statistics at the Mathematics Department. Hogg earned his Ph.D. 1950 under Allen Craig. After graduating, Hogg remained at the Mathematics Department, where he remained to become a long-serving professor.[4] Basu's theorem: Special cases Hogg independently discovered a special case of "Basu's theorem", a few years before the publication by Deb Basu.[5][6][7] Hogg's second paper on the topic of Basu's theorem was never published, because of a negative report by an anonymous referee in 1953.[8] Later, Basu refers "to Hogg and Craig (1956) for several interesting uses [of Basu's theorem] in proving results in distribution theory".[9] Collaboration and friendship with Allen Craig The textbook "Hogg and Craig" was innovative, particularly in emphasizing sufficient statistics: Sufficient statistics were treated not only for parametric families but also for nonparametric probability distributions: In particular, the sufficiency and completeness of the order statistics from a continuous distribution were treated. Another innovation was the systematic derivation of the distributions of functions of several random variables by using the change-of-variable method.[10] As noted before, Craig was Hogg's mentor, helping him to obtain a teaching position while a graduate student and also supervising his thesis. Later, after Hogg had graduated, Craig became a close friend, and served as the best man at Hogg's wedding and later as the "godparent" to each of Hogg's four children. Indeed, Hogg's son Allen was named after Craig.[11] Chairing a new Department of Statistics In 1965 Hogg became the founding chair of the new Department of Statistics and Actuarial Science, and he remained as the chair for nineteen years. At Iowa, Hogg held other positions, including Chair of the Quality Management and Productivity Program and the Hanson Chair of Manufacturing Productivity. After serving 51 years as an instructor at the University of Iowa, Hogg became Professor Emeritus in 2001.[3] Statistics education Hogg has been a leader in statistics education in the US.[12] Hogg has received a number of educational awards by state and national organizations: these awards include the Iowa Governor’s Science Medal for Teaching in 1990, the Iowa Board of Regents' Award for Faculty Excellence in 1992, and the Iowa Section of the Mathematical Association of America (MAA)'s Distinguished Teaching Award in 1992 with the national MAA's Distinguished Teaching Award in 1993.[13] At the University of Iowa, Hogg won his first teaching award after a student submitted a nomination with the title "There is a hog in my statistics book!".[14] Honors Hogg's 70th birthday was marked by a conference organized by the Institute of Mathematical Statistics, whose proceedings were published as a special issue of Communications in Statistics – Theory and Methods in 1996; in the conference proceedings, a list of Hogg's publications appears on pages 2467–2481.[15][16][17] For his research in nonparametric statistics, Hogg received the Gottfried Noether Senior Scholar Award in 2001. Hogg has been recognized internationally as a leading researcher in statistics and as an exemplary professor of statistics who has served as a public spokesperson for the profession. Hogg has had a particularly visible role in the United States, where he was elected as President of the American Statistical Association (ASA), serving in 1988. One of the ASA President's tasks is to arrange an annual meeting, and Hogg's diligence was rewarded by the ASA staff, who presented him with the name tag, "Boss Hogg" (after the name of a character in the television series The Dukes of Hazzard).[18] Three years after Hogg had served as President, he was awarded the ASA’s Founder’s Award in 1991.[19] Hogg has also been internationally active on behalf of the statistics profession. Hogg is a Fellow of the Institute of Mathematical Statistics, which awarded him its Carver Medal.[20] Hogg is also an Elected Member of the International Statistical Institute.[3] References 1. "Rob Hogg". 2. Colleagues, family remember a statistical giant 3. Randles, Ronald Herman (2007). "A Conversation with Robert V. Hogg". Statistical Science. 22 (1): 137–152. arXiv:0708.3974. doi:10.1214/088342306000000637. S2CID 58589782. 4. Pages 137–139 in Randles. 5. Basu's theorem appears in Basu, D. (1955). "On Statistics Independent of a Complete Sufficient Statistic". Selected Works of Debabrata Basu. pp. 377–380. doi:10.1007/978-1-4419-5825-9_14. ISBN 978-1-4419-5824-2. JSTOR 25048259. MR 0074745. Zbl 0068.13401. {{cite book}}: \|journal= ignored (help) 6. Boos, Dennis D.; Oliver, Jacqueline M. Hughes (Aug 1998). "Applications of Basu's Theorem". The American Statistician. 52 (3): 218–221. doi:10.2307/2685927. JSTOR 2685927. MR 1650407. 7. For Hogg's independent and early (1953) discovery of special cases of Basu's theorem, see page 511 in Ghosh, Malay (October 2002). "Basu's Theorem with Applications: A Personalistic Review". Sankhyā: The Indian Journal of Statistics, Series A. 64 (3): 509–531. JSTOR 25051412. MR 1985397. 8. Randles and Calvin, page 2476. (Later, in his interview with Randles, Hogg speculates that the anonymous referee was Erich Leo Lehmann.) 9. Page 348 in Statistical Information and Likelihood : A Collection of Critical Essays by Dr. D. Basu ; J. K. Ghosh, editor. Springer 1988. 10. Page 2468 in Randles and Calvin. (Pages 139–140 in Randles.) 11. Page 141 in Randles, which mentions only two children. Page 149 mentions four children. 12. Randles and Calvin, page 2468. Randles declares that Hogg is a leader in statistics education on page 144 of Randles. 13. Randles and Calvin report that the MAA award was in 1992 (Page 2468). (Randles reports the MAA prize in 1993 on page 146.) 14. Hogg quoted on page 146 in Randles. 15. James A. Calvin, ed. (1996). "Preface the institute of mathematical statistics central regional meeting". Communications in Statistics – Theory and Methods. 25 (11): i–viii+2459–2873. doi:10.1080/03610929608831849. MR 1424730. 16. Randles, Ronald H.; Calvin, James A. (1996). "The professional contributions of Robert V. Hogg". Communications in Statistics – Theory and Methods. 25 (11): 2467–2481. doi:10.1080/03610929608831850. MR 1424730. 17. Calvin, James A.; Calvin, James A. (1996). "Preface". Communications in Statistics – Theory and Methods. 25 (11): 2459–2465. doi:10.1080/03610929608831849. MR 1424730. 18. Page 147 in Randles. 19. Page 146 in Randles. 20. Page 151 in Randles. Textbooks • Hogg, Robert V.; McKean, Joseph W.; Craig, Allen T. (2005). Introduction to mathematical statistics (fifth ed.). Upper Saddle River, New Jersey: Prentice Hall. p. 692. ISBN 978-0-13-008507-8. MR 0467974. ("Hogg and Craig" was first published in 1959.) • Hogg, Robert V.; Tanis, Elliot A. (2010). Probability and Statistical Inference (eight ed.). Upper Saddle River, New Jersey: Prentice Hall. p. 648. ISBN 978-0-321-58475-5. \| mr = 426223 • Hogg, Robert V., Klugman, Stuart A. Loss distributions. With the assistance of Charles C. Hewitt and Gary Patrik. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Inc., New York, 1984. x+235 pp. ISBN 0-471-87929-0. MR747141 • Hogg, Robert V. and Ledolter, J. (1992). Applied Statistics for Engineers and Physical Scientists. Macmillan, New York. Articles • Hogg, R. V. (1953). "Testing the equality of means of rectangular populations. (Abstract)". Annals of Mathematical Statistics. 24: 691. • Hogg, R. V. (1956). "On the distribution of the likelihood ratio". Annals of Mathematical Statistics. 27 (2): 529. doi:10.1214/aoms/1177728276. • Hogg, Robert V.; Craig, Allen T. (1956). "Sufficient statistics in elementary distribution theory". Sankhyā. 17 (3): 209–216. JSTOR 25048308. MR 0086456. • Hogg, Robert V. (1974). "Adaptive Robust Procedures: A Partial Review and Some Suggestions for Future Applications and Theory". Journal of the American Statistical Association. 69 (348): 909–923. doi:10.2307/2286160. ISSN 0162-1459. JSTOR 2286160. Statistics education and the statistics profession • Robert V. Hogg (1978). An introduction to mathematical statistics. In Studies in Statistics (Robert V. Hogg, ed.) 1–7. Mathematical Association of America (Washington) • Robert V. Hogg (Ed) (1980). Modern statistics: Methods and applications. Arner. Math. Soc. (Providence) • Robert V. Hogg (1982). On graduate programs in statistics. In Teach. Statist. and Statist. Consult. (J.S. Rustagi and D.A. Wolfe, ed.) 71–80. Academic (New York) • Robert V. Hogg and Jim Swift (1982). Statistical education at the school level in the United States and Canada. In Teaching Statistics in Schools Throughout the World (Vic Barnett, ed.) 139–171. Internat. Statist. Inst. (Voorburg, Netherlands) • Hogg, Robert V. (1988). "Comments on Harold Hotelling's views on teaching statistics". Statist. Sci. 3: 95–97. doi:10.1214/ss/1177013007. • Robert V. Hogg (1989). On statistical education in the United States since 1920. In ASA Sesquicentennial Invited Paper Sessions (Mitchell Gail and Norman L. Johnson, eds.) 286–292. Amer. Statist. Assoc. (Alexandria, VA) • Hogg, Robert V. (March 1989). "How to Hope with Statistics (ASA Presidential Address)". Journal of the American Statistical Association. 84 (405): 1–5. doi:10.2307/2289836. JSTOR 2289836. \| jstor = 2289836 • Hogg, R. V. (1991). "Statistical Education: Improvements Are Badly Needed". The American Statistician. 45 (4): 342–343. doi:10.2307/2684473. JSTOR 2684473. • Garfield, Joan; Hogg, Bob; Schau, Candace; Whittinghill, Dex (9 June 2000). First Courses in Statistical Science Working Group (ed.). Best Practices in Introductory Statistics (Draft 2000.06.09) (PDF). Undergraduate Statistics Education Initiative Position Paper. American Statistical Association. • Hogg, Robert V.; Hogg, Mary C. (1995). "Continuous Quality Improvement in Higher Education". International Statistical Review / Revue Internationale de Statistique. 63 (1): 35–48. doi:10.2307/1403776. JSTOR 1403776. External links • Hogg's homepage at the University of Iowa • "On the History of Statistics and Actuarial Science at The University of Iowa" presented by Robert V. Hogg on October 2, 2003 • Robert V. Hogg at the Mathematics Genealogy Project • Hogg at Mathematical Reviews Statistics • Outline • Index Descriptive statistics Continuous data Center • Mean • Arithmetic • Arithmetic-Geometric • Cubic • Generalized/power • Geometric • Harmonic • Heronian • Heinz • Lehmer • Median • Mode Dispersion • Average absolute deviation • Coefficient of variation • Interquartile range • Percentile • Range • Standard deviation • Variance Shape • Central limit theorem • Moments • Kurtosis • L-moments • Skewness Count data • Index of dispersion Summary tables • Contingency table • Frequency distribution • Grouped data Dependence • Partial correlation • Pearson product-moment correlation • Rank correlation • Kendall's τ • Spearman's ρ • Scatter plot Graphics • Bar chart • Biplot • Box plot • Control chart • Correlogram • Fan chart • Forest plot • Histogram • Pie chart • Q–Q plot • Radar chart • Run chart • Scatter plot • Stem-and-leaf display • Violin plot Data collection Study design • Effect size • Missing data • Optimal design • Population • Replication • Sample size determination • Statistic • Statistical power Survey methodology • Sampling • Cluster • Stratified • Opinion poll • Questionnaire • Standard error Controlled experiments • Blocking • Factorial experiment • Interaction • Random assignment • Randomized controlled trial • Randomized experiment • Scientific control Adaptive designs • Adaptive clinical trial • Stochastic approximation • Up-and-down designs Observational studies • Cohort study • Cross-sectional study • Natural experiment • Quasi-experiment Statistical inference Statistical theory • Population • Statistic • Probability distribution • Sampling distribution • Order statistic • Empirical distribution • Density estimation • Statistical model • Model specification • Lp space • Parameter • location • scale • shape • Parametric family • Likelihood (monotone) • Location–scale family • Exponential family • Completeness • Sufficiency • Statistical functional • Bootstrap • U • V • Optimal decision • loss function • Efficiency • Statistical distance • divergence • Asymptotics • Robustness Frequentist inference Point estimation • Estimating equations • Maximum likelihood • Method of moments • M-estimator • Minimum distance • Unbiased estimators • Mean-unbiased minimum-variance • Rao–Blackwellization • Lehmann–Scheffé theorem • Median unbiased • Plug-in Interval estimation • Confidence interval • Pivot • Likelihood interval • Prediction interval • Tolerance interval • Resampling • Bootstrap • Jackknife Testing hypotheses • 1- & 2-tails • Power • Uniformly most powerful test • Permutation test • Randomization test • Multiple comparisons Parametric tests • Likelihood-ratio • Score/Lagrange multiplier • Wald Specific tests • Z-test (normal) • Student's t-test • F-test Goodness of fit • Chi-squared • G-test • Kolmogorov–Smirnov • Anderson–Darling • Lilliefors • Jarque–Bera • Normality (Shapiro–Wilk) • Likelihood-ratio test • Model selection • Cross validation • AIC • BIC Rank statistics • Sign • Sample median • Signed rank (Wilcoxon) • Hodges–Lehmann estimator • Rank sum (Mann–Whitney) • Nonparametric anova • 1-way (Kruskal–Wallis) • 2-way (Friedman) • Ordered alternative (Jonckheere–Terpstra) • Van der Waerden test Bayesian inference • Bayesian probability • prior • posterior • Credible interval • Bayes factor • Bayesian estimator • Maximum posterior estimator • Correlation • Regression analysis Correlation • Pearson product-moment • Partial correlation • Confounding variable • Coefficient of determination Regression analysis • Errors and residuals • Regression validation • Mixed effects models • Simultaneous equations models • Multivariate adaptive regression splines (MARS) Linear regression • Simple linear regression • Ordinary least squares • General linear model • Bayesian regression Non-standard predictors • Nonlinear regression • Nonparametric • Semiparametric • Isotonic • Robust • Heteroscedasticity • Homoscedasticity Generalized linear model • Exponential families • Logistic (Bernoulli) / Binomial / Poisson regressions Partition of variance • Analysis of variance (ANOVA, anova) • Analysis of covariance • Multivariate ANOVA • Degrees of freedom Categorical / Multivariate / Time-series / Survival analysis Categorical • Cohen's kappa • Contingency table • Graphical model • Log-linear model • McNemar's test • Cochran–Mantel–Haenszel statistics Multivariate • Regression • Manova • Principal components • Canonical correlation • Discriminant analysis • Cluster analysis • Classification • Structural equation model • Factor analysis • Multivariate distributions • Elliptical distributions • Normal Time-series General • Decomposition • Trend • Stationarity • Seasonal adjustment • Exponential smoothing • Cointegration • Structural break • Granger causality Specific tests • Dickey–Fuller • Johansen • Q-statistic (Ljung–Box) • Durbin–Watson • Breusch–Godfrey Time domain • Autocorrelation (ACF) • partial (PACF) • Cross-correlation (XCF) • ARMA model • ARIMA model (Box–Jenkins) • Autoregressive conditional heteroskedasticity (ARCH) • Vector autoregression (VAR) Frequency domain • Spectral density estimation • Fourier analysis • Least-squares spectral analysis • Wavelet • Whittle likelihood Survival Survival function • Kaplan–Meier estimator (product limit) • Proportional hazards models • Accelerated failure time (AFT) model • First hitting time Hazard function • Nelson–Aalen estimator Test • Log-rank test Applications Biostatistics • Bioinformatics • Clinical trials / studies • Epidemiology • Medical statistics Engineering statistics • Chemometrics • Methods engineering • Probabilistic design • Process / quality control • Reliability • System identification Social statistics • Actuarial science • Census • Crime statistics • Demography • Econometrics • Jurimetrics • National accounts • Official statistics • Population statistics • Psychometrics Spatial statistics • Cartography • Environmental statistics • Geographic information system • Geostatistics • Kriging • Category • Mathematics portal • Commons • WikiProject Authority control International • FAST • ISNI • VIAF National • France • BnF data • Israel • United States • Czech Republic • Korea • 2 • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Joseph Silverman Joseph Silverman (August 25, 1860 in Ohio – July 26, 1930 in New York City), was a leading American Reform rabbi and author. He was the first American born rabbi to serve in New York City. This article is about the American Reform rabbi. For the mathematician, see Joseph H. Silverman. For the American Hispanist scholar, see Joseph H. Silverman (Hispanist). Born in Cincinnati, he attended the University of Cincinnati and received a Doctor of Divinity from the Hebrew Union College in 1887, from which he received his rabbinic ordination three years earlier.[1] In 1887, he married and subsequently had five children with his wife Henrietta. He was Rabbi of Temple Emanu-El, Dallas, Texas, September 1884 to June 1885; rabbi of Congregation B'nai Israel, Galveston, Texas July, 1885 to February 20, 1888. While in Texas he was a circuit preacher to the Jewish communities in the vicinity of Dallas and Galveston, and aided in organizing many Sabbath schools and congregations. He was consulting editor of the Jewish Encyclopedia (Funk & Wagnalls). He helped organize the Religious Congress of the World's Fair in Chicago, 1893, where his address on this occasion was titled, "The Popular Errors About the Jews." At the beginning of 1888, Silverman received an offer from Temple Emanu-El in New York City to serve as a rabbi of the leading Reform congregation in America. Silverman started at Temple Emanu-El on March 1, 1888, succeeding rabbi Gustav Gottheil. During the years of his career in New York, 1888-1922 he was also president (1900–1903) of the Central Conference of American Rabbis, the Eastern Council[2] 1918- and was founder and president of the Emanu-El Brotherhood. Silverman published many articles and books, including A Catechism on Judaism (1886) and The Renaissance of Judaism (1918).[3] An article from The New York Times on April 21, 1912, quoted Silverman at a memorial service for victims of the RMS Titanic disaster as saying "Not God was responsible for this great disaster but the imperfection of human knowledge and judgment."[4] Silverman died on July 26, 1930, having spent 34 years as the leader of Temple Emanu-El followed by eight years as Rabbi Emeritus. The New York Times described him in its obituary as "one of the most prominent clergymen of his faith in America".[1] References 1. "Rev. Dr. Silverman, Noted Orator, Dies; Rabbi Emeritus of Temple Emanu-El, 69, Had served for 42 Years. He Aided Social Welfare in Work Here and in First Charge in Texas He Lent Weight to Progressive Movements. Scholar and Philanthropist. Fortieth Anniversary Recalled. Born in Cincinnati. Raised to Position of Head Rabbi. Known for His Courage.", The New York Times, July 27, 1930. Accessed January 11, 2021. "The Rev. Dr. Joseph Silverman, for thirty-four years rabbi of Temple Emanu-El, now at Fifth Avenue and Sixty-fifth Street, and for the last eight years rabbi-emeritus, died yesterday afternoon at 3:30 o'clock at his home, 55 East Eighty-sixth Street. He had been ill for a week. Dr. Silverman would have been 70 years old on Aug. 25." 2. "Rabbis In Convention.; Dr. Joseph Silverman Elected President of Eastern Council.", The New York Times, April 30, 1918. Accessed January 11, 2021. 3. "Who's who in New York City and State". L.R. Hamersly Company. 21 October 1907. Retrieved 21 October 2021 – via Google Books. • Nahshon, Edna. "The Pulpit and the Stage: Rabbi Joseph Silverman and the Actors' Church Alliance". American Jewish History - Volume 91, Number 1, March 2003 External links • Works by or about Joseph Silverman at Internet Archive Authority control International • ISNI • VIAF National • Italy • United States Other • SNAC	Wikipedia
Rabdology In 1617 a treatise in Latin titled Rabdologiæ and written by John Napier was published in Edinburgh. Printed three years after his treatise on the discovery of logarithms and in the same year as his death, it describes three devices to aid arithmetic calculations. Computing devices in Rabdology Napier's bones Promptuary Location arithmetic The devices themselves don't use logarithms, rather they are tools to reduce multiplication and division of natural numbers to simple addition and subtraction operations. The first device, which by then was already popularly used and known as Napier's bones, was a set of rods inscribed with the multiplication table. Napier coined the word rabdology (from Greek ῥάβδος [rhabdos], rod and λόγoς [logos] calculation or reckoning) to describe this technique. The rods were used to multiply, divide and even find the square roots and cube roots of numbers. The second device was a promptuary (Latin promptuarium meaning storehouse) and consisted of a large set of strips that could multiply multidigit numbers more easily than the bones. In combination with a table of reciprocals, it could also divide numbers. The third device used a checkerboard like grid and counters moving on the board to perform binary arithmetic. Napier termed this technique location arithmetic from the way in which the locations of the counters on the board represented and computed numbers. Once a number is converted into a binary form, simple movements of counters on the grid could multiply, divide and even find square roots of numbers. Of these devices, Napier's bones were the most popular and widely known. In fact, part of his motivation to publish the treatise was to establish credit for his invention of the technique. The bones were easy to manufacture and simple to use, and several variations on them were published and used for many years. The promptuary was never widely used, perhaps because it was more complex to manufacture, and it took nearly as much time to lay out the strips to find the product of numbers as to find the answer with pen and paper. Location arithmetic was an elegant insight into the simplicity of binary arithmetic, but remained a curiosity probably because it was never clear that the effort to convert numbers in and out of binary form was worth the trouble. An interesting tidbit is this treatise contains the earliest written reference to the decimal point (though its usage would not come into general use for another century.) The computing devices in Rabdology were overshadowed by his seminal work on logarithms as they proved more useful and more widely applicable. Nevertheless, these devices (as indeed are logarithms) are examples of Napier's ingenious attempts to discover easier ways to multiply, divide and find roots of numbers. Location arithmetic in particular foreshadowed the ease of and power of mechanizing binary arithmetic, but was never fully appreciated. References • John Napier (1990) [1617]. Rabdologiæ [Rabdology] (in Latin). Translated by William Frank Richardson. Introduction by Robin E. Rider. MIT Press. ISBN 0-262-14046-2.	Wikipedia
Rabinovich–Fabrikant equations The Rabinovich–Fabrikant equations are a set of three coupled ordinary differential equations exhibiting chaotic behaviour for certain values of the parameters. They are named after Mikhail Rabinovich and Anatoly Fabrikant, who described them in 1979. System description The equations are:[1] ${\dot {x}}=y(z-1+x^{2})+\gamma x\,$ ${\dot {y}}=x(3z+1-x^{2})+\gamma y\,$ ${\dot {z}}=-2z(\alpha +xy),\,$ where α, γ are constants that control the evolution of the system. For some values of α and γ, the system is chaotic, but for others it tends to a stable periodic orbit. Danca and Chen[2] note that the Rabinovich–Fabrikant system is difficult to analyse (due to the presence of quadratic and cubic terms) and that different attractors can be obtained for the same parameters by using different step sizes in the integration, see on the right an example of a solution obtained by two different solvers for the same parameter values and initial conditions. Also, recently, a hidden attractor was discovered in the Rabinovich–Fabrikant system.[3] Equilibrium points The Rabinovich–Fabrikant system has five hyperbolic equilibrium points, one at the origin and four dependent on the system parameters α and γ:[2] ${\tilde {\mathbf {x} }}_{0}=(0,0,0)$ ${\tilde {\mathbf {x} }}_{1,2}=\left(\pm q_{-},-{\frac {\alpha }{q_{-}}},1-\left(1-{\frac {\gamma }{\alpha }}\right)q_{-}^{2}\right)$ ${\tilde {\mathbf {x} }}_{3,4}=\left(\pm q_{+},-{\frac {\alpha }{q_{+}}},1-\left(1-{\frac {\gamma }{\alpha }}\right)q_{+}^{2}\right)$ where $q_{\pm }={\sqrt {\frac {1\pm {\sqrt {1-\gamma \alpha \left(1-{\frac {3\gamma }{4\alpha }}\right)}}}{2\left(1-{\frac {3\gamma }{4\alpha }}\right)}}}$ These equilibrium points only exist for certain values of α and γ > 0. γ = 0.87, α = 1.1 An example of chaotic behaviour is obtained for γ = 0.87 and α = 1.1 with initial conditions of (−1, 0, 0.5),[4] see trajectory on the right. The correlation dimension was found to be 2.19 ± 0.01.[5] The Lyapunov exponents, λ are approximately 0.1981, 0, −0.6581 and the Kaplan–Yorke dimension, DKY ≈ 2.3010[4] γ = 0.1 Danca and Romera[6] showed that for γ = 0.1, the system is chaotic for α = 0.98, but progresses on a stable limit cycle for α = 0.14. See also • List of chaotic maps References 1. Rabinovich, Mikhail I.; Fabrikant, A. L. (1979). "Stochastic Self-Modulation of Waves in Nonequilibrium Media". Sov. Phys. JETP. 50: 311. Bibcode:1979JETP...50..311R. 2. Danca, Marius-F.; Chen, Guanrong (2004). "Birfurcation and Chaos in a Complex Model of Dissipative Medium". International Journal of Bifurcation and Chaos. World Scientific Publishing Company. 14 (10): 3409–3447. Bibcode:2004IJBC...14.3409D. doi:10.1142/S0218127404011430. 3. Danca M.-F.; Kuznetsov N.; Chen G. (2017). "Unusual dynamics and hidden attractors of the Rabinovich-Fabrikant system". Nonlinear Dynamics. 88 (1): 791–805. arXiv:1511.07765. doi:10.1007/s11071-016-3276-1. S2CID 119303488. 4. Sprott, Julien C. (2003). Chaos and Time-series Analysis. Oxford University Press. p. 433. ISBN 0-19-850840-9. 5. Grassberger, P.; Procaccia, I. (1983). "Measuring the strangeness of strange attractors". Physica D. 9 (1–2): 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1. 6. Danca, Marius-F.; Romera, Miguel (2008). "Algorithm for Control and Anticontrol of Chaos in Continuous-Time Dynamical Systems". Dynamics of Continuous, Discrete and Impulsive Systems. Series B: Applications & Algorithms. Watam Press. 15: 155–164. hdl:10261/8868. ISSN 1492-8760. External links • Weisstein, Eric W. "Rabinovich–Fabrikant Equation." From MathWorld—A Wolfram Web Resource. • Chaotics Models a more appropriate approach to the chaotic graph of the system "Rabinovich–Fabrikant Equation" Chaos theory Concepts Core • Attractor • Bifurcation • Fractal • Limit set • Lyapunov exponent • Orbit • Periodic point • Phase space • Anosov diffeomorphism • Arnold tongue • axiom A dynamical system • Bifurcation diagram • Box-counting dimension • Correlation dimension • Conservative system • Ergodicity • False nearest neighbors • Hausdorff dimension • Invariant measure • Lyapunov stability • Measure-preserving dynamical system • Mixing • Poincaré section • Recurrence plot • SRB measure • Stable manifold • Topological conjugacy Theorems • Ergodic theorem • Liouville's theorem • Krylov–Bogolyubov theorem • Poincaré–Bendixson theorem • Poincaré recurrence theorem • Stable manifold theorem • Takens's theorem Theoretical branches • Bifurcation theory • Control of chaos • Dynamical system • Ergodic theory • Quantum chaos • Stability theory • Synchronization of chaos Chaotic maps (list) Discrete • Arnold's cat map • Baker's map • Complex quadratic map • Coupled map lattice • Duffing map • Dyadic transformation • Dynamical billiards • outer • Exponential map • Gauss map • Gingerbreadman map • Hénon map • Horseshoe map • Ikeda map • Interval exchange map • Irrational rotation • Kaplan–Yorke map • Langton's ant • Logistic map • Standard map • Tent map • Tinkerbell map • Zaslavskii map Continuous • Double scroll attractor • Duffing equation • Lorenz system • Lotka–Volterra equations • Mackey–Glass equations • Rabinovich–Fabrikant equations • Rössler attractor • Three-body problem • Van der Pol oscillator Physical systems • Chua's circuit • Convection • Double pendulum • Elastic pendulum • FPUT problem • Hénon–Heiles system • Kicked rotator • Multiscroll attractor • Population dynamics • Swinging Atwood's machine • Tilt-A-Whirl • Weather Chaos theorists • Michael Berry • Rufus Bowen • Mary Cartwright • Chen Guanrong • Leon O. Chua • Mitchell Feigenbaum • Peter Grassberger • Celso Grebogi • Martin Gutzwiller • Brosl Hasslacher • Michel Hénon • Svetlana Jitomirskaya • Bryna Kra • Edward Norton Lorenz • Aleksandr Lyapunov • Benoît Mandelbrot • Hee Oh • Edward Ott • Henri Poincaré • Mary Rees • Otto Rössler • David Ruelle • Caroline Series • Yakov Sinai • Oleksandr Mykolayovych Sharkovsky • Nina Snaith • Floris Takens • Audrey Terras • Mary Tsingou • Marcelo Viana • Amie Wilkinson • James A. Yorke • Lai-Sang Young Related articles • Butterfly effect • Complexity • Edge of chaos • Predictability • Santa Fe Institute	Wikipedia
Rabinowitsch trick In mathematics, the Rabinowitsch trick, introduced by J.L. Rabinowitsch (1929), is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case (the so-called weak Nullstellensatz), by introducing an extra variable. The Rabinowitsch trick goes as follows. Let K be an algebraically closed field. Suppose the polynomial f in K[x1,...xn] vanishes whenever all polynomials f1,....,fm vanish. Then the polynomials f1,....,fm, 1 − x0f have no common zeros (where we have introduced a new variable x0), so by the weak Nullstellensatz for K[x0, ..., xn] they generate the unit ideal of K[x0 ,..., xn]. Spelt out, this means there are polynomials $g_{0},g_{1},\dots ,g_{m}\in K[x_{0},x_{1},\dots ,x_{n}]$ such that $1=g_{0}(x_{0},x_{1},\dots ,x_{n})(1-x_{0}f(x_{1},\dots ,x_{n}))+\sum _{i=1}^{m}g_{i}(x_{0},x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n})$ as an equality of elements of the polynomial ring $K[x_{0},x_{1},\dots ,x_{n}]$. Since $x_{0},x_{1},\dots ,x_{n}$ are free variables, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting $x_{0}=1/f(x_{1},\dots ,x_{n})$ that $1=\sum _{i=1}^{m}g_{i}(1/f(x_{1},\dots ,x_{n}),x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n})$ as elements of the field of rational functions $K(x_{1},\dots ,x_{n})$, the field of fractions of the polynomial ring $K[x_{1},\dots ,x_{n}]$. Moreover, the only expressions that occur in the denominators of the right hand side are f and powers of f, so rewriting that right hand side to have a common denominator results in an equality on the form $1={\frac {\sum _{i=1}^{m}h_{i}(x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n})}{f(x_{1},\dots ,x_{n})^{r}}}$ for some natural number r and polynomials $h_{1},\dots ,h_{m}\in K[x_{1},\dots ,x_{n}]$. Hence $f(x_{1},\dots ,x_{n})^{r}=\sum _{i=1}^{m}h_{i}(x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n}),$ which literally states that $f^{r}$ lies in the ideal generated by f1,....,fm. This is the full version of the Nullstellensatz for K[x1,...,xn]. References • Brownawell, W. Dale (2001) [1994], "Rabinowitsch trick", Encyclopedia of Mathematics, EMS Press • Rabinowitsch, J.L. (1929), "Zum Hilbertschen Nullstellensatz", Math. Ann. (in German), 102 (1): 520, doi:10.1007/BF01782361, MR 1512592	Wikipedia
Racah W-coefficient Racah's W-coefficients were introduced by Giulio Racah in 1942.[1] These coefficients have a purely mathematical definition. In physics they are used in calculations involving the quantum mechanical description of angular momentum, for example in atomic theory. The coefficients appear when there are three sources of angular momentum in the problem. For example, consider an atom with one electron in an s orbital and one electron in a p orbital. Each electron has electron spin angular momentum and in addition the p orbital has orbital angular momentum (an s orbital has zero orbital angular momentum). The atom may be described by LS coupling or by jj coupling as explained in the article on angular momentum coupling. The transformation between the wave functions that correspond to these two couplings involves a Racah W-coefficient. Apart from a phase factor, Racah's W-coefficients are equal to Wigner's 6-j symbols, so any equation involving Racah's W-coefficients may be rewritten using 6-j symbols. This is often advantageous because the symmetry properties of 6-j symbols are easier to remember. Racah coefficients are related to recoupling coefficients by $W(j_{1}j_{2}Jj_{3};J_{12}J_{23})\equiv {\frac {\langle (j_{1},(j_{2}j_{3})J_{23})J\|((j_{1}j_{2})J_{12},j_{3})J\rangle }{\sqrt {(2J_{12}+1)(2J_{23}+1)}}}.$ Recoupling coefficients are elements of a unitary transformation and their definition is given in the next section. Racah coefficients have more convenient symmetry properties than the recoupling coefficients (but less convenient than the 6-j symbols).[2] Recoupling coefficients Coupling of two angular momenta $\mathbf {j} _{1}$ and $\mathbf {j} _{2}$ is the construction of simultaneous eigenfunctions of $\mathbf {J} ^{2}$ and $J_{z}$, where $\mathbf {J} =\mathbf {j} _{1}+\mathbf {j} _{2}$, as explained in the article on Clebsch–Gordan coefficients. The result is $\|(j_{1}j_{2})JM\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}\|j_{1}m_{1}\rangle \|j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}\|JM\rangle ,$ where $J=\|j_{1}-j_{2}\|,\ldots ,j_{1}+j_{2}$ and $M=-J,\ldots ,J$. Coupling of three angular momenta $\mathbf {j} _{1}$, $\mathbf {j} _{2}$, and $\mathbf {j} _{3}$, may be done by first coupling $\mathbf {j} _{1}$ and $\mathbf {j} _{2}$ to $\mathbf {J} _{12}$ and next coupling $\mathbf {J} _{12}$ and $\mathbf {j} _{3}$ to total angular momentum $\mathbf {J} $: $\|((j_{1}j_{2})J_{12}j_{3})JM\rangle =\sum _{M_{12}=-J_{12}}^{J_{12}}\sum _{m_{3}=-j_{3}}^{j_{3}}\|(j_{1}j_{2})J_{12}M_{12}\rangle \|j_{3}m_{3}\rangle \langle J_{12}M_{12}j_{3}m_{3}\|JM\rangle $ Alternatively, one may first couple $\mathbf {j} _{2}$ and $\mathbf {j} _{3}$ to $\mathbf {J} _{23}$ and next couple $\mathbf {j} _{1}$ and $\mathbf {J} _{23}$ to $\mathbf {J} $: $\|(j_{1},(j_{2}j_{3})J_{23})JM\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{M_{23}=-J_{23}}^{J_{23}}\|j_{1}m_{1}\rangle \|(j_{2}j_{3})J_{23}M_{23}\rangle \langle j_{1}m_{1}J_{23}M_{23}\|JM\rangle $ Both coupling schemes result in complete orthonormal bases for the $(2j_{1}+1)(2j_{2}+1)(2j_{3}+1)$ dimensional space spanned by $\|j_{1}m_{1}\rangle \|j_{2}m_{2}\rangle \|j_{3}m_{3}\rangle ,\;\;m_{1}=-j_{1},\ldots ,j_{1};\;\;m_{2}=-j_{2},\ldots ,j_{2};\;\;m_{3}=-j_{3},\ldots ,j_{3}.$ Hence, the two total angular momentum bases are related by a unitary transformation. The matrix elements of this unitary transformation are given by a scalar product and are known as recoupling coefficients. The coefficients are independent of $M$ and so we have $\|((j_{1}j_{2})J_{12}j_{3})JM\rangle =\sum _{J_{23}}\|(j_{1},(j_{2}j_{3})J_{23})JM\rangle \langle (j_{1},(j_{2}j_{3})J_{23})J\|((j_{1}j_{2})J_{12}j_{3})J\rangle .$ The independence of $M$ follows readily by writing this equation for $M=J$ and applying the lowering operator $J_{-}$ to both sides of the equation. Algebra Let $\Delta (a,b,c)=[(a+b-c)!(a-b+c)!(-a+b+c)!/(a+b+c+1)!]^{1/2}$ be the usual triangular factor, then the Racah coefficient is a product of four of these by a sum over factorials, $W(abcd;ef)=\Delta (a,b,e)\Delta (c,d,e)\Delta (a,c,f)\Delta (b,d,f)w(abcd;ef)$ where $w(abcd;ef)\equiv \sum _{z}{\frac {(-1)^{z+\beta _{1}}(z+1)!}{(z-\alpha _{1})!(z-\alpha _{2})!(z-\alpha _{3})!(z-\alpha _{4})!(\beta _{1}-z)!(\beta _{2}-z)!(\beta _{3}-z)!}}$ and $\alpha _{1}=a+b+e;\quad \beta _{1}=a+b+c+d;$ $\alpha _{2}=c+d+e;\quad \beta _{2}=a+d+e+f;$ $\alpha _{3}=a+c+f;\quad \beta _{3}=b+c+e+f;$ $\alpha _{4}=b+d+f.$ The sum over $z$ is finite over the range[3] $\max(\alpha _{1},\alpha _{2},\alpha _{3},\alpha _{4})\leq z\leq \min(\beta _{1},\beta _{2},\beta _{3}).$ Relation to Wigner's 6-j symbol Racah's W-coefficients are related to Wigner's 6-j symbols, which have even more convenient symmetry properties $W(abcd;ef)(-1)^{a+b+c+d}={\begin{Bmatrix}a&b&e\\d&c&f\end{Bmatrix}}.$ Cf.[4] or $W(j_{1}j_{2}Jj_{3};J_{12}J_{23})=(-1)^{j_{1}+j_{2}+j_{3}+J}{\begin{Bmatrix}j_{1}&j_{2}&J_{12}\\j_{3}&J&J_{23}\end{Bmatrix}}.$ See also • Clebsch–Gordan coefficients • 3-j symbol • 6-j symbol • Pandya theorem Notes 1. Racah, G. (1942). "Theory of Complex Spectra II". Physical Review. 62 (9–10): 438–462. Bibcode:1942PhRv...62..438R. doi:10.1103/PhysRev.62.438. 2. Rose, M. E. (1957). Elementary theory of angular momentum (Dover). 3. Cowan, R D (1981). The theory of atomic structure and spectra (Univ of California Press), p. 148. 4. Brink, D M & Satchler, G R (1968). Angular Momentum (Oxford University Press) 3$^{rd}$ ed., p. 142. online Further reading • Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton, New Jersey: Princeton University Press. ISBN 0-691-07912-9. • Condon, Edward U.; Shortley, G. H. (1970). "Chapter 3". The Theory of Atomic Spectra. Cambridge: Cambridge University Press. ISBN 0-521-09209-4. • Messiah, Albert (1981). Quantum Mechanics (Volume II) (12th ed.). New York: North Holland Publishing. ISBN 0-7204-0045-7. • Sato, Masachiyo (1955). "General formula of the Racah coefficient". Progress of Theoretical Physics. 13 (4): 405–414. Bibcode:1955PThPh..13..405S. doi:10.1143/PTP.13.405. • Brink, D. M.; Satchler, G. R. (1993). "Chapter 2". Angular Momentum (3rd ed.). Oxford: Clarendon Press. ISBN 0-19-851759-9. • Zare, Richard N. (1988). "Chapter 2". Angular Momentum. New York: John Wiley. ISBN 0-471-85892-7. • Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading, Massachusetts: Addison-Wesley. ISBN 0-201-13507-8. External links • "Racah-Wigner coefficients", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Authority control International • FAST National • France • BnF data • Israel • United States	Wikipedia
Racah polynomials In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients. The Racah polynomials were first defined by Wilson (1978) and are given by $p_{n}(x(x+\gamma +\delta +1))={}_{4}F_{3}\left[{\begin{matrix}-n&n+\alpha +\beta +1&-x&x+\gamma +\delta +1\\\alpha +1&\gamma +1&\beta +\delta +1\\\end{matrix}};1\right].$ Orthogonality $\sum _{y=0}^{N}\operatorname {R} _{n}(x;\alpha ,\beta ,\gamma ,\delta )\operatorname {R} _{m}(x;\alpha ,\beta ,\gamma ,\delta ){\frac {\gamma +\delta +1+2y}{\gamma +\delta +1+y}}\omega _{y}=h_{n}\operatorname {\delta } _{n,m},$[1] when $\alpha +1=-N$, where $\operatorname {R} $ is the Racah polynomial, $x=y(y+\gamma +\delta +1),$ $\operatorname {\delta } _{n,m}$ is the Kronecker delta function and the weight functions are $\omega _{y}={\frac {(\alpha +1)_{y}(\beta +\delta +1)_{y}(\gamma +1)_{y}(\gamma +\delta +2)_{y}}{(-\alpha +\gamma +\delta +1)_{y}(-\beta +\gamma +1)_{y}(\delta +1)_{y}y!}},$ and $h_{n}={\frac {(-\beta )_{N}(\gamma +\delta +1)_{N}}{(-\beta +\gamma +1)_{N}(\delta +1)_{N}}}{\frac {(n+\alpha +\beta +1)_{n}n!}{(\alpha +\beta +2)_{2n}}}{\frac {(\alpha +\delta -\gamma +1)_{n}(\alpha -\delta +1)_{n}(\beta +1)_{n}}{(\alpha +1)_{n}(\beta +\delta +1)_{n}(\gamma +1)_{n}}},$ $(\cdot )_{n}$ is the Pochhammer symbol. Rodrigues-type formula $\omega (x;\alpha ,\beta ,\gamma ,\delta )\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )=(\gamma +\delta +1)_{n}{\frac {\nabla ^{n}}{\nabla \lambda (x)^{n}}}\omega (x;\alpha +n,\beta +n,\gamma +n,\delta ),$[2] where $\nabla $ is the backward difference operator, $\lambda (x)=x(x+\gamma +\delta +1).$ Generating functions There are three generating functions for $x\in \{0,1,2,...,N\}$ when $\beta +\delta +1=-N\quad $or$\quad \gamma +1=-N,$ ${}_{2}F_{1}(-x,-x+\alpha -\gamma -\delta ;\alpha +1;t){}_{2}F_{1}(x+\beta +\delta +1,x+\gamma +1;\beta +1;t)$ ;\alpha +1;t){}_{2}F_{1}(x+\beta +\delta +1,x+\gamma +1;\beta +1;t)} $\quad =\sum _{n=0}^{N}{\frac {(\beta +\delta +1)_{n}(\gamma +1)_{n}}{(\beta +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n},$ when $\alpha +1=-N\quad $or$\quad \gamma +1=-N,$ ${}_{2}F_{1}(-x,-x+\beta -\gamma ;\beta +\delta +1;t){}_{2}F_{1}(x+\alpha +1,x+\gamma +1;\alpha -\delta +1;t)$ ;\beta +\delta +1;t){}_{2}F_{1}(x+\alpha +1,x+\gamma +1;\alpha -\delta +1;t)} $\quad =\sum _{n=0}^{N}{\frac {(\alpha +1)_{n}(\gamma +1)_{n}}{(\alpha -\delta +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n},$ when $\alpha +1=-N\quad $or$\quad \beta +\delta +1=-N,$ ${}_{2}F_{1}(-x,-x-\delta ;\gamma +1;t){}_{2}F_{1}(x+\alpha +1;x+\beta +\gamma +1;\alpha +\beta -\gamma +1;t)$ ;\gamma +1;t){}_{2}F_{1}(x+\alpha +1;x+\beta +\gamma +1;\alpha +\beta -\gamma +1;t)} $\quad =\sum _{n=0}^{N}{\frac {(\alpha +1)_{n}(\beta +\delta +1)_{n}}{(\alpha +\beta -\gamma +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n}.$ Connection formula for Wilson polynomials When $\alpha =a+b-1,\beta =c+d-1,\gamma =a+d-1,\delta =a-d,x\rightarrow -a+ix,$ $\operatorname {R} _{n}(\lambda (-a+ix);a+b-1,c+d-1,a+d-1,a-d)={\frac {\operatorname {W} _{n}(x^{2};a,b,c,d)}{(a+b)_{n}(a+c)_{n}(a+d)_{n}}},$ where $\operatorname {W} $ are Wilson polynomials. q-analog Askey & Wilson (1979) introduced the q-Racah polynomials defined in terms of basic hypergeometric functions by $p_{n}(q^{-x}+q^{x+1}cd;a,b,c,d;q)={}_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\\end{matrix}};q;q\right].$ They are sometimes given with changes of variables as $W_{n}(x;a,b,c,N;q)={}_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&cq^{x-n}\\aq&bcq&q^{-N}\\\end{matrix}};q;q\right].$ References 1. Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Wilson Class: Definitions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. 2. Koekoek, Roelof; Swarttouw, René F. (1998), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue • Askey, Richard; Wilson, James (1979), "A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols" (PDF), SIAM Journal on Mathematical Analysis, 10 (5): 1008–1016, doi:10.1137/0510092, ISSN 0036-1410, MR 0541097, archived from the original on September 25, 2017 • Wilson, J. (1978), Hypergeometric series recurrence relations and some new orthogonal functions, Ph.D. thesis, Univ. Wisconsin, Madison	Wikipedia
Consensus theorem In Boolean algebra, the consensus theorem or rule of consensus[1] is the identity: $xy\vee {\bar {x}}z\vee yz=xy\vee {\bar {x}}z$ Variable inputs Function values xyz$xy\vee {\bar {x}}z\vee yz$$xy\vee {\bar {x}}z$ 00000 00111 01000 01111 10000 10100 11011 11111 The consensus or resolvent of the terms $xy$ and ${\bar {x}}z$ is $yz$. It is the conjunction of all the unique literals of the terms, excluding the literal that appears unnegated in one term and negated in the other. If $y$ includes a term which is negated in $z$ (or vice versa), the consensus term $yz$ is false; in other words, there is no consensus term. The conjunctive dual of this equation is: $(x\vee y)({\bar {x}}\vee z)(y\vee z)=(x\vee y)({\bar {x}}\vee z)$ Proof ${\begin{aligned}xy\vee {\bar {x}}z\vee yz&=xy\vee {\bar {x}}z\vee (x\vee {\bar {x}})yz\\&=xy\vee {\bar {x}}z\vee xyz\vee {\bar {x}}yz\\&=(xy\vee xyz)\vee ({\bar {x}}z\vee {\bar {x}}yz)\\&=xy(1\vee z)\vee {\bar {x}}z(1\vee y)\\&=xy\vee {\bar {x}}z\end{aligned}}$ Consensus The consensus or consensus term of two conjunctive terms of a disjunction is defined when one term contains the literal $a$ and the other the literal ${\bar {a}}$, an opposition. The consensus is the conjunction of the two terms, omitting both $a$ and ${\bar {a}}$, and repeated literals. For example, the consensus of ${\bar {x}}yz$ and $w{\bar {y}}z$ is $w{\bar {x}}z$.[2] The consensus is undefined if there is more than one opposition. For the conjunctive dual of the rule, the consensus $y\vee z$ can be derived from $(x\vee y)$ and $({\bar {x}}\vee z)$ through the resolution inference rule. This shows that the LHS is derivable from the RHS (if A → B then A → AB; replacing A with RHS and B with (y ∨ z) ). The RHS can be derived from the LHS simply through the conjunction elimination inference rule. Since RHS → LHS and LHS → RHS (in propositional calculus), then LHS = RHS (in Boolean algebra). Applications In Boolean algebra, repeated consensus is the core of one algorithm for calculating the Blake canonical form of a formula.[2] In digital logic, including the consensus term in a circuit can eliminate race hazards.[3] History The concept of consensus was introduced by Archie Blake in 1937, related to the Blake canonical form.[4] It was rediscovered by Samson and Mills in 1954[5] and by Quine in 1955.[6] Quine coined the term 'consensus'. Robinson used it for clauses in 1965 as the basis of his "resolution principle".[7][8] References 1. Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition 2003, p. 44 2. Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition 2003, p. 81 3. M. Rafiquzzaman, Fundamentals of Digital Logic and Microcontrollers, 6th edition (2014), ISBN 1118855795, p. 75 4. "Canonical expressions in Boolean algebra", Dissertation, Department of Mathematics, University of Chicago, 1937, reviewed in J. C. C. McKinsey, The Journal of Symbolic Logic 3:2:93 (June 1938) doi:10.2307/2267634 JSTOR 2267634 5. Edward W. Samson, Burton E. Mills, Air Force Cambridge Research Center, Technical Report 54-21, April 1954 6. Willard van Orman Quine, "The problem of simplifying truth functions", American Mathematical Monthly 59:521-531, 1952 JSTOR 2308219 7. John Alan Robinson, "A Machine-Oriented Logic Based on the Resolution Principle", Journal of the ACM 12:1: 23–41. 8. Donald Ervin Knuth, The Art of Computer Programming 4A: Combinatorial Algorithms, part 1, p. 539 Further reading • Roth, Charles H. Jr. and Kinney, Larry L. (2004, 2010). "Fundamentals of Logic Design", 6th Ed., p. 66ff.	Wikipedia
Rachel Blodgett Adams Rachel Blodgett Adams (October 13, 1894–January 22, 1982) was a pioneering American mathematician and one of the first women to earn a doctorate in mathematics at Radcliffe College in 1921. Rachel Blodgett Adams Born Rachel Blodgett October 13, 1894 Woburn, Massachusetts DiedJanuary 22, 1982 (aged 87) Providence, Rhode Island Burial placeSwan Point Cemetery, Providence NationalityAmerican Alma materRadcliffe College Occupation(s)Mathematician, professor SpouseClarence Raymond Adams (1898-1965) Parents • William Edward Blodgett (b. 1864) (father) • Mabel Edith Owen (b. 1874) (mother) Biography Rachel Blodgett was born in Woburn, Massachusetts, the oldest of three children of Mabel Edith Owen (b. 1874) and William Edward Blodgett (b. 1864); neither of whom attended college.[1] After graduating from Woburn High School in 1912, she entered Wellesley College and majored in mathematics and Latin. In addition to her studies, she joined the school's Shakespeare Society and performed cornet in the symphony orchestra. Academically, she was gifted and was named a Wellesley scholar in 1914 and a Durant scholar in 1915. She graduated with her bachelor's degree (B.A.) in 1916, after which Blodgett moved to Montreal, Quebec, Canada to teach math at Miss Edgar's and Miss Cramp's School until 1918.She then returned to Massachusetts to further pursue her academic and professional endeavors Research For the next three years, Blodgett pursued advanced analysis at Radcliffe College, where she received an Edward Austin scholarship for at least two years and a Mary E. Horton fellowship for her final year. She earned her master's degree (M.A.) in 1919, and continued on to acquire her Ph.D. in mathematics in 1921.[1][2] The following fall term, Dr. Blodgett took a teaching position at Wellesley. Rachel Blodgett married Harvard-trained mathematician Clarence Raymond Adams (1898–1965) on August 17, 1922, in Eden Park, in Providence, Rhode Island. (He was widely known as "C.R. Adams.") At the time, C. R. Adams was a Sheldon traveling fellow from Harvard, which allowed the newly married couple to travel extensively overseas with stops in Rome and Göttingen, Germany. Upon their return to the U.S. in 1923, the two mathematicians settled down in Providence, Rhode Island, where C. R. established his career at Brown University and eventually headed the math department there. Blodgett, now known as Dr. Rachel Adams, moved forward with her research for a few years, according to mathematician Judy Green,[1] "At least through the 1920s Adams continued her interest in integral equations. G. C. Evans of Rice Institute used information from her dissertation in an extensive review of a book on linear integral equations for the Monthly in 1927. She presented her results to the AMS in 1926 and published them in the American Journal in 1929."[1] Adams also spent many years tutoring math students at her alma mater, Radcliffe College, from 1926 to 1941, with her friend and colleague Mary Graustein, Ph.D., with the goal of encouraging young female students.[1] During World War II, Adams registered with the National Roster of Scientific and Specialized Personnel in Washington, D.C. to help with the war effort, but confirmation of any contribution she may have made as part of that program has not yet been found.[1][3][4] Later years During their marriage Rachel and C. R. Adams traveled extensively by automobile in the United States and Europe.[1] The couple never had children. He died in 1965 at the age of 67.[1] She was 87 when she died in Providence, January 22, 1982. She is buried at Swan Point Cemetery there.[5] Legacy In her estate plans, Rachel bequeathed to Wellesley College the Blodgett Fund, and stipulated that the fund's income be used for scholarships.[1] Memberships • Phi Beta Kappa[6] • Sigma Xi[1] • American Mathematical Society • Mathematical Association of America Selected publications • Blodgett, Rachel E. "Utilizing Industrial Ash to Stabilize Arsenic in Soil Contaminated From Chromated Copper Arsenate(CCA) Treated Wood." Masters Abstracts International. Vol. 50. No. 03. 2011. • Blodgett, Rachel. The Determination of the Coefficients in Interpolation Formulae: And A Study of the Approximate Solution of Integral Equations. Dissertation, Radcliffe College, 1921. • Adams, Rachel Blodgett (1929). On the Approximate Solution of Fredholm's Homogeneous Integral Equation. American Journal of Mathematics. 51(1): 139–148. doi:10.2307/2370568. ISSN 0002-9327. References 1. Green, Judy, and Jeanne LaDuke. Supplementary Material for Pioneering Women in American Mathematics: The Pre-1940 PhD's. American Mathematical Society, 2009. 2. "Rachel Blodgett - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu. Retrieved 2020-05-21. 3. Central Intelligence Agency (1947-01-09). NATIONAL ROSTER OF SCIENTIFIC AND SPECIALIZED PERSONNEL. Emma Best. 4. Carmichael, Leonard (1941-03-07). "The National Roster of Scientific and Specialized Personnel: A Progress Report". Science. 93 (2410): 217–219. doi:10.1126/science.93.2410.217. ISSN 0036-8075. PMID 17775937. 5. "Rachel Blodgett Adams (1894-1982) - Find A Grave..." www.findagrave.com. Retrieved 2020-05-21. 6. "Wellesley College News". Wellesley News. 1916-04-13. Retrieved 2020-05-21. Authority control: Academics • Mathematics Genealogy Project • zbMATH	Wikipedia
Rachel Justine Pries Rachel Justine Pries is an American mathematician whose research focuses on arithmetic geometry and number theory. She is a professor at Colorado State University and both a Fellow of the American Mathematical Society[1] and a Fellow of the Association for Women in Mathematics.[2] Rachel Justine Pries Pries in 2007 NationalityAmerican Alma mater • Brown University • University of Pennsylvania Awards • Fellow, American Mathematical Society, 2018 Scientific career FieldsMathematics Institutions • Columbia University • Colorado State University ThesisFormal patching and deformation of wildly ramified covers of curves (2000) Doctoral advisorDavid Harbater Websitehttp://www.math.colostate.edu/~pries/ Education Pries was a student at the Cambridge Rindge and Latin School, in Cambridge, Massachusetts.[3] She received a B.S. degree from Brown University in Providence, Rhode Island in 1994,[4] and received a Ph.D. in mathematics from the University of Pennsylvania in Philadelphia, Pennsylvania in 2000 under the supervision of David Harbater.[5] Career and research After her doctoral studies, Pries was appointed a National Science Foundation VIGRE post-doctoral fellow at Columbia University for 2000 to 2003. After her post-doc at Columbia, Rachel joined the faculty at Colorado State University, Fort Collins, Colorado, where she is currently a full professor.[6] In one of her most cited works, Families of wildly ramified covers of curves,[7] Pries studied smooth Galois covers of curves, ramified over only one point. In a second highly cited paper, Hyperelliptic curves with prescribed p-torsion,[8] Pries and co-author Darren Glass, proved several results regarding the existence of Jacobian varieties having interesting p-torsion as measured in terms of invariants such as the p-rank and the a-number. Pries serves on the Steering Committee of Women in Number Theory (WIN),[9] a research collaboration community for women mathematicians interested in number theory. She was an editor of Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop (Association for Women in Mathematics Series), which was published by Springer Verlag in 2016.[10] Honors Pries was elected to the 2018 class of fellows of the American Mathematical Society. Her citation read "for contributions to arithmetic geometry, and for service to the mathematical community."[1] Pries was selected as the inaugural lecturer in the Association for Women in Mathematics Distinguished Lecture Series at the University of Oregon, Eugene, Oregon, in 2013.[11] In 2004, Pries was selected as Outstanding Professor in Graduate Instruction by the mathematics graduate students of Colorado State University [12] Pries was elected to the 2023 class of fellows of the Association for Women in Mathematics "for supporting the research careers of women through mentorship and advocacy; for her vision and hard work establishing the Women in Numbers workshops and research network; and for broadening the participation of women in mathematics through service and leadership both at her institution and in high-profile national and international programs."[2] References 1. "Fellows of the American Mathematical Society". ams.org. American Mathematical Society. Retrieved 5 November 2017. 2. "The AWM Fellows Program: 2023 Class of AWM Fellows". Association for Women in Mathematics. Retrieved 23 October 2022. 3. Girls' Angle Bulletin, Volume 11, Number 6, Girls' Angle Bulletin, August 31, 2018, retrieved 2020-06-02 4. "Faculty", General Catalog 2019–2020, Colorado State University, retrieved 2020-06-02 5. Rachel Justine Pries at the Mathematics Genealogy Project 6. "Rachel Pries, Professor". Department of Mathematics Faculty and Staff. Colorado State University. Retrieved 19 March 2020. 7. Pries, Rachel J. (2002). "Families of wildly ramified covers of curves" (PDF). Amer. J. Math. 124 (4): 737–768. doi:10.1353/ajm.2002.0024. S2CID 18044517. Archived from the original (PDF) on 2017-11-07. Retrieved 5 November 2017. 8. Glass, Darren; Pries, Rachel (2005). "Hyperelliptic curves with prescribed p-torsion". Manuscripta Math. 117 (3): 299–317. arXiv:math/0401008. Bibcode:2004math......1008G. doi:10.1007/s00229-005-0559-0. S2CID 6649601. 9. "Women in Number Theory Steering Committee". Women in Number Theory. Retrieved 6 November 2017. 10. Eischen, Ellen; Long, Ling; Pries, Rachel (2016). Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop (Association for Women in Mathematics Series). Springer Verlag. ISBN 978-3319309743. 11. "Distinguished Speaker Series". Association for Women in Mathematics, University of Oregon. University of Oregon. Retrieved 6 November 2017. 12. "Awards of the Department". Colorado State University Mathematics Department. Colorado State University. Retrieved 6 November 2017. External links • Rachel Pries' website • Dr. Rachel Pries - Number Theory (2017 video) Authority control International • VIAF National • Germany Academics • MathSciNet • Mathematics Genealogy Project • ORCID Other • IdRef	Wikipedia
Rachel Kuske Rachel Ann Kuske (born 1965)[1] is an American-Canadian applied mathematician and Professor and Chair of Mathematics at the Georgia Institute of Technology.[2] Rachel Kuske Born1965 (1965) Alma materNorthwestern University Known forStochastic and Nonlinear Dynamics Scientific career FieldsMathematics InstitutionsGeorgia Institute of Technology Doctoral advisorBernard J. Matkowsky Professional career Kuske received her PhD in Applied Mathematics from Northwestern University in 1992. Her dissertation, Asymptotic Analysis of Random Wave Equations, was supervised by Bernard J. Matkowsky.[3] From 1997 to 2002, she was Assistant Professor and then associate professor at the University of Minnesota. She is an expert on stochastic and nonlinear dynamics, mathematical modeling, asymptotic methods, and industrial mathematics.[4][5] She served on the Scientific Advisory Board for the Institute for Computational and Experimental Research in Mathematics (ICERM),[6] and as of 2021 she serves on ICERM's board of trustees.[7] Awards and honours Kuske was awarded a Sloan Fellowship in 1992 and was made a Canada Research Chair in 2002. In 2011 Kuske was a recipient of the Canadian Mathematical Society Krieger–Nelson Prize, given to outstanding woman in mathematics in Canada.[8][9] In 2015 she became a fellow of the Society for Industrial and Applied Mathematics "for contributions to the theory of stochastic and nonlinear dynamics and its application, and for promoting equity and diversity in mathematics."[10] References 1. Birth year from WorldCat Identities, retrieved 2018-12-12. 2. "Rachel Kuske \| School of Mathematics \| Georgia Institute of Technology \| Atlanta, GA". www.math.gatech.edu. Retrieved 2017-04-08. 3. Rachel Kuske at the Mathematics Genealogy Project 4. "Archived copy" (PDF). Archived from the original (PDF) on 2014-10-19. Retrieved 2014-10-14.{{cite web}}: CS1 maint: archived copy as title (link) 5. "Rachel Kuske". University of British Columbia. Retrieved 2017-04-08. 6. https://icerm.brown.edu/about/nsf/reports/2020-2021-Annual-Report.pdf ICERM Annual Report May 1, 2020 – April 30, 2021 7. "ICERM - Trustee and Advisory Boards - Trustee & Advisory Boards". icerm.brown.edu. Retrieved 2021-07-11. 8. "Krieger-Nelson Prize". 9. http://cms.math.ca/Prizes/citations/kn2011.pdf 10. SIAM Fellows: Class of 2015, Society for Industrial and Applied Mathematics, retrieved 2015-10-08. External links • Home page • Citation for Krieger-Nelson Prize • Letter from the Chair, Professor Rachel Kuske Authority control International • VIAF National • Germany Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH	Wikipedia
Rachel Roberts (mathematician) Rachel Roberts is an American mathematician specializing in low-dimensional topology, including foliations and contact geometry. She is the Elinor Anheuser Professor of Mathematics at Washington University in St. Louis.[1] Roberts completed her Ph.D. at Cornell University in 1992. Her dissertation, supervised by Allen Hatcher, was Constructing Taut Foliations.[2][3] Publications • Delman, Charles; Roberts, Rachel (1999). "Alternating knots satisfy strong property P". Commentarii Mathematici Helvetici. 74 (3): 376–397. doi:10.1007/s000140050095. MR 1710698. • Roberts, Rachel; Shareshian, John; Stein, Melanie (2003). "Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation". Journal of the American Mathematical Society. 16 (3): 639–680. doi:10.1090/s0894-0347-03-00426-0. MR 1969207. References 1. "Rachel Roberts", Our people, Washington University in St. Louis Arts & Sciences, retrieved 2019-11-02 2. Rachel Roberts at the Mathematics Genealogy Project 3. Constructing Taut Foliations MR page, American Mathematical Society, MR 2688244 External links • Home page Authority control: Academics • MathSciNet • Mathematics Genealogy Project • zbMATH	Wikipedia
Rachev ratio The Rachev Ratio (or R-Ratio) is a risk-return performance measure of an investment asset, portfolio, or strategy. It was devised by Dr. Svetlozar Rachev[1] and has been extensively studied in quantitative finance. Unlike the reward-to-variability ratios, such as Sharpe ratio and Sortino ratio, the Rachev ratio is a reward-to-risk ratio, which is designed to measure the right tail reward potential relative to the left tail risk in a non-Gaussian setting.[2][3][4] Intuitively, it represents the potential for extreme positive returns compared to the risk of extreme losses (negative returns), at a rarity frequency q (quantile level) defined by the user.[5] The ratio is defined as the Expected Tail Return (ETR) in the best q% cases divided by the Expected tail loss (ETL) in the worst q% cases. The ETL is the average loss incurred when losses exceed the Value at Risk at a predefined quantile level. The ETR, defined by symmetry to the ETL, is the average profit gained when profits exceed the Profit at risk at a predefined quantile level. For more tailored applications, the generalized Rachev Ratio has been defined with different powers and/or different confidence levels of the ETR and ETL.[1] Definition According to its original version introduced by the authors in 2004, the Rachev ratio is defined as: $\rho \left({x'r}\right)={\frac {CVa{R_{(1-\alpha )}}\left({{r_{f}}-x'r}\right)}{CVa{R_{(1-\beta )}}\left({x'r-{r_{f}}}\right)}}$ or, alternatively, $\rho \left({x'r}\right)={\frac {ET{L_{\alpha }}\left({{r_{f}}-x'r}\right)}{ET{L_{\beta }}\left({x'r-{r_{f}}}\right)}},$ where $\alpha $ and $\beta $ belong to $\left({0,1}\right)$, and in the symmetric case: $\alpha =\beta $. $r_{f}$ is the risk-free rate of return and $x'r$ presents the portfolio return. The ETL is the expected tail loss, also known as conditional value at risk (CVaR), is defined as: $ET{L_{\alpha }}={\frac {1}{\alpha }}\int _{0}^{\alpha }{Va{R_{q}}\left(X\right)dq},$ and $Va{R_{\alpha }}=-F_{X}^{-1}\left(\alpha \right)=-\inf \left\{{x\left\|P{\left({X\leq x}\right)>\alpha }\right.}\right\}$ is the value at risk (VaR) of the random return $X$. Thus, the ETL can be interpreted as the average loss beyond VaR: $ET{L_{\alpha }}\left(X\right)=E\left[{L\|L>Va{R_{_{\alpha }}}}\right]$. The generalized Rachev ratio is the ratio between the power CVaR of the opposite of the excess return at a given confidence level and the power CVaR of the excess return at another confidence level. That is, $\rho \left({x'r}\right)={\frac {ET{L_{\left({\gamma ,\alpha }\right)}}\left({{r_{f}}-x'r}\right)}{ET{L_{\left({\delta ,\beta }\right)}}\left({x'r-{r_{f}}}\right)}},$ where $ET{L_{\left({\gamma ,\alpha }\right)}}\left(X\right)=E\left[{{\rm {max}}{{\left({L,0}\right)}^{\gamma }}\|L>Va{R_{_{\alpha }}}}\right]$ is the power CVaR of $X$, and $\gamma $ is a positive constant. The main advantage of the generalized Rachev ratio over the traditional Rachev ratio is conferred by the power indexes $\gamma $ and $\delta $ that characterize an investor's aversion to risk. Properties The Rachev ratio can be used in both ex-ante and ex-post analyses. In the ex-post analysis, the Rachev ratio is computed by dividing the corresponding two sample AVaR's. Since the performance levels in the Rachev ratio are quantiles of the active return distribution, they are relative levels as they adjust according to the distribution. For example, if the scale is small, then the two performance levels will be closer to each other. As a consequence, the Rachev ratio is always well-defined. In the ex-ante analysis, optimal portfolio problems based on the Rachev ratio are, generally, numerically hard to solve because the Rachev ratio is a fraction of two CVaRs which are convex functions of portfolio weights. In effect, the Rachev ratio, if viewed as a function of portfolio weights, may have many local extrema.[6] Several empirical tests of the Rachev ratio and the generalized Rachev ratio have been proposed.[4][7] An algorithm for solving Rachev ratio optimization problem was provided in Konno, Tanaka, and Yamamoto (2011).[8] Example In quantitative finance, non-Gaussian return distributions are common. The Rachev ratio, as a risk-adjusted performance measurement, characterizes the skewness and kurtosis of the return distribution (see picture on the right). See also • Post-modern portfolio theory • Upside potential ratio • Sharpe ratio • Sortino ratio • Omega ratio • Modern Portfolio Theory References 1. Biglova, Almira; Ortobelli, Sergio; Rachev, Svetlozar T.; Stoyanov, Stoyan (2004). "Different Approaches to Risk Estimation in Portfolio Theory". The Journal of Portfolio Management. The Journal of Portfolio Management, Fall 2004, Vol. 31, No. 1: pp. 103-112. 31: 103–112. doi:10.3905/jpm.2004.443328. S2CID 153594336. 2. Fehr, Ben. "Beyond the Normal Distribution" (PDF). Frankfurter Allgemeine Zeitung. Archived from the original (PDF) on 1 September 2006. Retrieved 16 March 2006. 3. Cheridito, P.; Kromer, E. (2013). "Reward-Risk Ratios". Journal of Investment Strategies. 3 (1): 3–18. doi:10.21314/JOIS.2013.022. 4. Farinelli, S.; Ferreira, M.; Rossello, D.; Thoeny, M.; Tibiletti, L. (2008). "Beyond Sharpe ratio: Optimal asset allocation using different performance ratios". Journal of Banking and Finance. 32 (10): 2057–2063. doi:10.1016/j.jbankfin.2007.12.026. 5. https://statistik.econ.kit.edu/download/doc_secure1/10_StochModels.pdf 6. Rachev, Svetlozar T.; Stoyanov, Stoyan V.; Fabozzi, Frank J. (2008). Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization (1st ed.). Wiley. ISBN 978-0-470-05316-4. 7. Satchell, Stephen (2009-10-22). Optimizing Optimization: The Next Generation of Optimization Applications and Theory (1st ed.). Academic Press. ISBN 9780750633611. 8. Konno, Hiroshi; Tanaka, Katsuhiro; Yamamoto, Rei (2011). "Construction of a portfolio with shorter downside tail and longer upside tail". Computational Optimization and Applications. 48 (2): 199. doi:10.1007/s10589-009-9255-4. S2CID 29816889.	Wikipedia
Racks and quandles In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams. Algebraic structures Group-like • Group • Semigroup / Monoid • Rack and quandle • Quasigroup and loop • Abelian group • Magma • Lie group Group theory Ring-like • Ring • Rng • Semiring • Near-ring • Commutative ring • Domain • Integral domain • Field • Division ring • Lie ring Ring theory Lattice-like • Lattice • Semilattice • Complemented lattice • Total order • Heyting algebra • Boolean algebra • Map of lattices • Lattice theory Module-like • Module • Group with operators • Vector space • Linear algebra Algebra-like • Algebra • Associative • Non-associative • Composition algebra • Lie algebra • Graded • Bialgebra • Hopf algebra While mainly used to obtain invariants of knots, they can be viewed as algebraic constructions in their own right. In particular, the definition of a quandle axiomatizes the properties of conjugation in a group. History In 1943, Mituhisa Takasaki (高崎光久) introduced an algebraic structure which he called a Kei (圭), which would later come to be known as an involutive quandle.[1] His motivation was to find a nonassociative algebraic structure to capture the notion of a reflection in the context of finite geometry. The idea was rediscovered and generalized in (unpublished) 1959 correspondence between John Conway and Gavin Wraith,[2] who at the time were undergraduate students at the University of Cambridge. It is here that the modern definitions of quandles and of racks first appear. Wraith had become interested in these structures (which he initially dubbed sequentials) while at school.[3] Conway renamed them wracks, partly as a pun on his colleague's name, and partly because they arise as the remnants (or 'wrack and ruin') of a group when one discards the multiplicative structure and considers only the conjugation structure. The spelling 'rack' has now become prevalent. These constructs surfaced again in the 1980s: in a 1982 paper by David Joyce[4] (where the term quandle, an arbitrary nonsense word, was coined),[5] in a 1982 paper by Sergei Matveev (under the name distributive groupoids)[6] and in a 1986 conference paper by Egbert Brieskorn (where they were called automorphic sets).[7] A detailed overview of racks and their applications in knot theory may be found in the paper by Colin Rourke and Roger Fenn.[8] Racks A rack may be defined as a set $\mathrm {R} $ with a binary operation $\triangleleft $ such that for every $a,b,c\in \mathrm {R} $ the self-distributive law holds: $a\triangleleft (b\triangleleft c)=(a\triangleleft b)\triangleleft (a\triangleleft c)$ and for every $a,b\in \mathrm {R} ,$ there exists a unique $c\in \mathrm {R} $ such that $a\triangleleft c=b.$ This definition, while terse and commonly used, is suboptimal for certain purposes because it contains an existential quantifier which is not really necessary. To avoid this, we may write the unique $c\in \mathrm {R} $ such that $a\triangleleft c=b$ as $b\triangleright a.$ We then have $a\triangleleft c=b\iff c=b\triangleright a,$ and thus $a\triangleleft (b\triangleright a)=b,$ and $(a\triangleleft b)\triangleright a=b.$ Using this idea, a rack may be equivalently defined as a set $\mathrm {R} $ with two binary operations $\triangleleft $ and $\triangleright $ such that for all $a,b,c\in \mathrm {R} {\text{:}}$ 1. $a\triangleleft (b\triangleleft c)=(a\triangleleft b)\triangleleft (a\triangleleft c)$ (left self-distributive law) 2. $(c\triangleright b)\triangleright a=(c\triangleright a)\triangleright (b\triangleright a)$ (right self-distributive law) 3. $(a\triangleleft b)\triangleright a=b$ 4. $a\triangleleft (b\triangleright a)=b$ It is convenient to say that the element $a\in \mathrm {R} $ is acting from the left in the expression $a\triangleleft b,$ and acting from the right in the expression $b\triangleright a.$ The third and fourth rack axioms then say that these left and right actions are inverses of each other. Using this, we can eliminate either one of these actions from the definition of rack. If we eliminate the right action and keep the left one, we obtain the terse definition given initially. Many different conventions are used in the literature on racks and quandles. For example, many authors prefer to work with just the right action. Furthermore, the use of the symbols $\triangleleft $ and $\triangleright $ is by no means universal: many authors use exponential notation $a\triangleleft b={}^{a}b$ and $b\triangleright a=b^{a},$ while many others write $b\triangleright a=b\star a.$ Yet another equivalent definition of a rack is that it is a set where each element acts on the left and right as automorphisms of the rack, with the left action being the inverse of the right one. In this definition, the fact that each element acts as automorphisms encodes the left and right self-distributivity laws, and also these laws: ${\begin{aligned}a\triangleleft (b\triangleright c)&=(a\triangleleft b)\triangleright (a\ \triangleleft c)\\(c\triangleleft b)\triangleright a&=(c\triangleright a)\triangleleft (b\triangleright a)\end{aligned}}$ which are consequences of the definition(s) given earlier. Quandles A quandle is defined as a rack, $\mathrm {Q} ,$ such that for all $a\in \mathrm {Q} $ $a\triangleleft a=a,$ or equivalently $a\triangleright a=a.$ Examples and applications Every group gives a quandle where the operations come from conjugation: ${\begin{aligned}a\triangleleft b&=aba^{-1}\\b\triangleright a&=a^{-1}ba\\&=a^{-1}\triangleleft b\end{aligned}}$ In fact, every equational law satisfied by conjugation in a group follows from the quandle axioms. So, one can think of a quandle as what is left of a group when we forget multiplication, the identity, and inverses, and only remember the operation of conjugation. Every tame knot in three-dimensional Euclidean space has a 'fundamental quandle'. To define this, one can note that the fundamental group of the knot complement, or knot group, has a presentation (the Wirtinger presentation) in which the relations only involve conjugation. So, this presentation can also be used as a presentation of a quandle. The fundamental quandle is a very powerful invariant of knots. In particular, if two knots have isomorphic fundamental quandles then there is a homeomorphism of three-dimensional Euclidean space, which may be orientation reversing, taking one knot to the other. Less powerful but more easily computable invariants of knots may be obtained by counting the homomorphisms from the knot quandle to a fixed quandle $\mathrm {Q} .$ Since the Wirtinger presentation has one generator for each strand in a knot diagram, these invariants can be computed by counting ways of labelling each strand by an element of $\mathrm {Q} ,$ subject to certain constraints. More sophisticated invariants of this sort can be constructed with the help of quandle cohomology. The Alexander quandles are also important, since they can be used to compute the Alexander polynomial of a knot. Let $\mathrm {A} $ be a module over the ring $\mathbb {Z} [t,t^{-1}]$ of Laurent polynomials in one variable. Then the Alexander quandle is $\mathrm {A} $ made into a quandle with the left action given by $a\triangleleft b=tb+(1-t)a.$ Racks are a useful generalization of quandles in topology, since while quandles can represent knots on a round linear object (such as rope or a thread), racks can represent ribbons, which may be twisted as well as knotted. A quandle $\mathrm {Q} $ is said to be involutory if for all $a,b\in \mathrm {Q} ,$ $a\triangleleft (a\triangleleft b)=b$ or equivalently, $(b\triangleright a)\triangleright a=b.$ Any symmetric space gives an involutory quandle, where $a\triangleleft b$ is the result of 'reflecting $b$ through $a$'. See also • Biracks and biquandles • Laver table References 1. Takasaki, Mituhisa (1943). "Abstractions of symmetric functions". Tohoku Mathematical Journal. 49: 143–207. 2. Conway, John H.; Wraith, Gavin (1959). "(unpublished correspondence)". {{cite journal}}: Cite journal requires \|journal= (help) 3. Wraith, Gavin. "A Personal Story about Knots". Archived from the original on 2006-03-13. 4. Joyce, David (1982). "A classifying invariant of knots: the knot quandle". Journal of Pure and Applied Algebra. 23: 37–65. doi:10.1016/0022-4049(82)90077-9. 5. Baez, John. "The Origin of the word 'Quandle'". The n-Category Cafe. Retrieved 5 June 2015. 6. Matveev, Sergei (1984). "Distributive groupoids in knot theory". Math. USSR Sbornik. 47 (1): 73–83. Bibcode:1984SbMat..47...73M. doi:10.1070/SM1984v047n01ABEH002630. 7. Brieskorn, Egbert (1988). "Automorphic sets and braids and singularities". Braids. Contemporary Mathematics. Vol. 78. pp. 45–115. doi:10.1090/conm/078/975077. ISBN 9780821850886. 8. Rourke, Colin; Fenn, Roger (1992). "Racks and links in codimension 2". Journal of Knot Theory and Its Ramifications. 1 (4): 343–406. doi:10.1142/S0218216592000203. External links • Knot Quandaries Quelled by Quandles - An undergraduate introduction to quandles and another knot invariants • A Survey of Quandle Ideas by Scott Carter • Knot Invariants Derived from Quandles and Racks by Seiichi Kamada • Shelves, Racks, Spindles and Quandles, p. 56 of Lie 2-Algebras by Alissa Crans • https://ncatlab.org/nlab/show/quandle	Wikipedia
Rademacher complexity In computational learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of real-valued functions with respect to a probability distribution. Definitions Rademacher complexity of a set Given a set $A\subseteq \mathbb {R} ^{m}$, the Rademacher complexity of A is defined as follows:[1][2]: 326 $\operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]$ where $\sigma _{1},\sigma _{2},\dots ,\sigma _{m}$ are independent random variables drawn from the Rademacher distribution i.e. $\Pr(\sigma _{i}=+1)=\Pr(\sigma _{i}=-1)=1/2$ for $i=1,2,\dots ,m$, and $a=(a_{1},\ldots ,a_{m})$. Some authors take the absolute value of the sum before taking the supremum, but if $A$ is symmetric this makes no difference. Rademacher complexity of a function class Let $S=(z_{1},z_{2},\dots ,z_{m})\in Z^{m}$ be a sample of points and consider a function class ${\mathcal {F}}$ of real-valued functions over $Z^{m}$. Then, the empirical Rademacher complexity of ${\mathcal {F}}$ given $S$ is defined as: $\operatorname {Rad} _{S}({\mathcal {F}})={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{f\in {\mathcal {F}}}\sum _{i=1}^{m}\sigma _{i}f(z_{i})\right]$ This can also be written using the previous definition:[2]: 326 $\operatorname {Rad} _{S}({\mathcal {F}})=\operatorname {Rad} ({\mathcal {F}}\circ S)$ where ${\mathcal {F}}\circ S$ denotes function composition, i.e.: ${\mathcal {F}}\circ S:=\{(f(z_{1}),\ldots ,f(z_{m}))\mid f\in {\mathcal {F}}\}$ Let $P$ be a probability distribution over $Z$. The Rademacher complexity of the function class ${\mathcal {F}}$ with respect to $P$ for sample size $m$ is: $\operatorname {Rad} _{P,m}({\mathcal {F}}):=\mathbb {E} _{S\sim P^{m}}\left[\operatorname {Rad} _{S}({\mathcal {F}})\right]$ where the above expectation is taken over an identically independently distributed (i.i.d.) sample $S=(z_{1},z_{2},\dots ,z_{m})$ generated according to $P$. Intuition The Rademacher complexity is typically applied on a function class of models that are used for classification, with the goal of measuring their ability to classify points drawn from a probability space under arbitrary labellings. When the function class is rich enough, it contains functions that can appropriately adapt for each arrangement of labels, simulated by the random draw of $\sigma _{i}$ under the expectation, so that this quantity in the sum is maximised. Examples 1. $A$ contains a single vector, e.g., $A=\{(a,b)\}\subset \mathbb {R} ^{2}$. Then: $\operatorname {Rad} (A)={1 \over 2}\cdot \left({1 \over 4}\cdot (a+b)+{1 \over 4}\cdot (a-b)+{1 \over 4}\cdot (-a+b)+{1 \over 4}\cdot (-a-b)\right)=0$ The same is true for every singleton hypothesis class.[3]: 56 2. $A$ contains two vectors, e.g., $A=\{(1,1),(1,2)\}\subset \mathbb {R} ^{2}$. Then: ${\begin{aligned}\operatorname {Rad} (A)&={1 \over 2}\cdot \left({1 \over 4}\cdot \max(1+1,1+2)+{1 \over 4}\cdot \max(1-1,1-2)+{1 \over 4}\cdot \max(-1+1,-1+2)+{1 \over 4}\cdot \max(-1-1,-1-2)\right)\\[5pt]&={1 \over 8}(3+0+1-2)={1 \over 4}\end{aligned}}$ Using the Rademacher complexity The Rademacher complexity can be used to derive data-dependent upper-bounds on the learnability of function classes. Intuitively, a function-class with smaller Rademacher complexity is easier to learn. Bounding the representativeness In machine learning, it is desired to have a training set that represents the true distribution of some sample data $S$. This can be quantified using the notion of representativeness. Denote by $P$ the probability distribution from which the samples are drawn. Denote by $H$ the set of hypotheses (potential classifiers) and denote by $F$ the corresponding set of error functions, i.e., for every hypothesis $h\in H$, there is a function $f_{h}\in F$, that maps each training sample (features,label) to the error of the classifier $h$ (note in this case hypothesis and classifier are used interchangeably). For example, in the case that $h$ represents a binary classifier, the error function is a 0–1 loss function, i.e. the error function $f_{h}$ returns 1 if $h$ correctly classifies a sample and 0 else. We omit the index and write $f$ instead of $f_{h}$ when the underlying hypothesis is irrelevant. Define: $L_{P}(f):=\mathbb {E} _{z\sim P}[f(z)]$ – the expected error of some error function $f\in F$ on the real distribution $P$; $L_{S}(f):={1 \over m}\sum _{i=1}^{m}f(z_{i})$ – the estimated error of some error function $f\in F$ on the sample $S$. The representativeness of the sample $S$, with respect to $P$ and $F$, is defined as: $\operatorname {Rep} _{P}(F,S):=\sup _{f\in F}(L_{P}(f)-L_{S}(f))$ Smaller representativeness is better, since it provides a way to avoid overfitting: it means that the true error of a classifier is not much higher than its estimated error, and so selecting a classifier that has low estimated error will ensure that the true error is also low. Note however that the concept of representativeness is relative and hence can not be compared across distinct samples. The expected representativeness of a sample can be bounded above by the Rademacher complexity of the function class:[2]: 326 $\mathbb {E} _{S\sim P^{m}}[\operatorname {Rep} _{P}(F,S)]\leq 2\cdot \mathbb {E} _{S\sim P^{m}}[\operatorname {Rad} (F\circ S)]$ Bounding the generalization error When the Rademacher complexity is small, it is possible to learn the hypothesis class H using empirical risk minimization. For example, (with binary error function),[2]: 328 for every $\delta >0$, with probability at least $1-\delta $, for every hypothesis $h\in H$: $L_{P}(h)-L_{S}(h)\leq 2\operatorname {Rad} (F\circ S)+4{\sqrt {2\ln(4/\delta ) \over m}}$ Bounding the Rademacher complexity Since smaller Rademacher complexity is better, it is useful to have upper bounds on the Rademacher complexity of various function sets. The following rules can be used to upper-bound the Rademacher complexity of a set $A\subset \mathbb {R} ^{m}$.[2]: 329–330 1. If all vectors in $A$ are translated by a constant vector $a_{0}\in \mathbb {R} ^{m}$, then Rad(A) does not change. 2. If all vectors in $A$ are multiplied by a scalar $c\in \mathbb {R} $, then Rad(A) is multiplied by $\|c\|$. 3. $\operatorname {Rad} (A+B)=\operatorname {Rad} (A)+\operatorname {Rad} (B)$.[3]: 56 4. (Kakade & Tewari Lemma) If all vectors in $A$ are operated by a Lipschitz function, then Rad(A) is (at most) multiplied by the Lipschitz constant of the function. In particular, if all vectors in $A$ are operated by a contraction mapping, then Rad(A) strictly decreases. 5. The Rademacher complexity of the convex hull of $A$ equals Rad(A). 6. (Massart Lemma) The Rademacher complexity of a finite set grows logarithmically with the set size. Formally, let $A$ be a set of $N$ vectors in $\mathbb {R} ^{m}$, and let ${\bar {a}}$ be the mean of the vectors in $A$. Then: $\operatorname {Rad} (A)\leq \max _{a\in A}\\|a-{\bar {a}}\\|\cdot {{\sqrt {2\log N}} \over m}$ In particular, if $A$ is a set of binary vectors, the norm is at most ${\sqrt {m}}$, so: $\operatorname {Rad} (A)\leq {\sqrt {2\log N \over m}}$ Bounds related to the VC dimension Let $H$ be a set family whose VC dimension is $d$. It is known that the growth function of $H$ is bounded as: for all $m>d+1$: $\operatorname {Growth} (H,m)\leq (em/d)^{d}$ This means that, for every set $h$ with at most $m$ elements, $\|H\cap h\|\leq (em/d)^{d}$. The set-family $H\cap h$ can be considered as a set of binary vectors over $\mathbb {R} ^{m}$. Substituting this in Massart's lemma gives: $\operatorname {Rad} (H\cap h)\leq {\sqrt {2d\log(em/d) \over m}}$ With more advanced techniques (Dudley's entropy bound and Haussler's upper bound[4]) one can show, for example, that there exists a constant $C$, such that any class of $\{0,1\}$-indicator functions with Vapnik–Chervonenkis dimension $d$ has Rademacher complexity upper-bounded by $C{\sqrt {\frac {d}{m}}}$. Bounds related to linear classes The following bounds are related to linear operations on $S$ – a constant set of $m$ vectors in $\mathbb {R} ^{n}$.[2]: 332–333 1. Define $A_{2}=\{(w\cdot x_{1},\ldots ,w\cdot x_{m})\mid \\|w\\|_{2}\leq 1\}=$ the set of dot-products of the vectors in $S$ with vectors in the unit ball. Then: $\operatorname {Rad} (A_{2})\leq {\max _{i}\\|x_{i}\\|_{2} \over {\sqrt {m}}}$ 2. Define $A_{1}=\{(w\cdot x_{1},\ldots ,w\cdot x_{m})\mid \\|w\\|_{1}\leq 1\}=$ the set of dot-products of the vectors in $S$ with vectors in the unit ball of the 1-norm. Then: $\operatorname {Rad} (A_{1})\leq \max _{i}\\|x_{i}\\|_{\infty }\cdot {\sqrt {2\log(2n) \over m}}$ Bounds related to covering numbers The following bound relates the Rademacher complexity of a set $A$ to its external covering number – the number of balls of a given radius $r$ whose union contains $A$. The bound is attributed to Dudley.[2]: 338 Suppose $A\subset \mathbb {R} ^{m}$ is a set of vectors whose length (norm) is at most $c$. Then, for every integer $M>0$: $\operatorname {Rad} (A)\leq {c\cdot 2^{-M} \over {\sqrt {m}}}+{6c \over m}\cdot \sum _{i=1}^{M}2^{-i}{\sqrt {\log \left(N_{c\cdot 2^{-i}}^{\text{ext}}(A)\right)}}$ In particular, if $A$ lies in a d-dimensional subspace of $\mathbb {R} ^{m}$, then: $\forall r>0:N_{r}^{\text{ext}}(A)\leq (2c{\sqrt {d}}/r)^{d}$ Substituting this in the previous bound gives the following bound on the Rademacher complexity: $\operatorname {Rad} (A)\leq {6c \over m}\cdot {\bigg (}{\sqrt {d\log(2{\sqrt {d}})}}+2{\sqrt {d}}{\bigg )}=O{\bigg (}{c{\sqrt {d\log(d)}} \over m}{\bigg )}$ Gaussian complexity Gaussian complexity is a similar complexity with similar physical meanings, and can be obtained from the Rademacher complexity using the random variables $g_{i}$ instead of $\sigma _{i}$, where $g_{i}$ are Gaussian i.i.d. random variables with zero-mean and variance 1, i.e. $g_{i}\sim {\mathcal {N}}(0,1)$. Gaussian and Rademacher complexities are known to be equivalent up to logarithmic factors. Equivalence of Rademacher and Gaussian complexity Given a set $A\subseteq \mathbb {R} ^{n}$ then it holds that[5]: ${\frac {G(A)}{2{\sqrt {\log {n}}}}}\leq {\text{Rad}}(A)\leq {\sqrt {\frac {\pi }{2}}}G(A)$ Where $G(A)$ is the Gaussian Complexity of A References 1. Balcan, Maria-Florina (November 15–17, 2011). "Machine Learning Theory – Rademacher Complexity" (PDF). Retrieved 10 December 2016. 2. Chapter 26 in Shalev-Shwartz, Shai; Ben-David, Shai (2014). Understanding Machine Learning – from Theory to Algorithms. Cambridge University Press. ISBN 9781107057135. 3. Mohri, Mehryar; Rostamizadeh, Afshin; Talwalkar, Ameet (2012). Foundations of Machine Learning. US, Massachusetts: MIT Press. ISBN 9780262018258. 4. Bousquet, O. (2004). Introduction to Statistical Learning Theory. Biological Cybernetics, 3176(1), 169–207. doi:10.1007/978-3-540-28650-9_8 5. Wainwright, Martin (2019). High-dimensional statistics : a non-asymptotic viewpoint. Cambridge, United Kingdom. pp. Exercise 5.5. ISBN 978-1-108-62777-1. OCLC 1089254580.{{cite book}}: CS1 maint: location missing publisher (link) • Peter L. Bartlett, Shahar Mendelson (2002) Rademacher and Gaussian Complexities: Risk Bounds and Structural Results. Journal of Machine Learning Research 3 463–482 • Giorgio Gnecco, Marcello Sanguineti (2008) Approximation Error Bounds via Rademacher's Complexity. Applied Mathematical Sciences, Vol. 2, 2008, no. 4, 153–176	Wikipedia
Rademacher distribution In probability theory and statistics, the Rademacher distribution (which is named after Hans Rademacher) is a discrete probability distribution where a random variate X has a 50% chance of being +1 and a 50% chance of being -1.[1] Rademacher Support $k\in \{-1,1\}\,$ PMF $f(k)=\left\{{\begin{matrix}1/2&{\mbox{if }}k=-1,\\1/2&{\mbox{if }}k=+1,\\0&{\mbox{otherwise.}}\end{matrix}}\right.$ CDF $F(k)={\begin{cases}0,&k<-1\\1/2,&-1\leq k<1\\1,&k\geq 1\end{cases}}$ Mean $0\,$ Median $0\,$ Mode N/A Variance $1\,$ Skewness $0\,$ Ex. kurtosis $-2\,$ Entropy $\ln(2)\,$ MGF $\cosh(t)\,$ CF $\cos(t)\,$ A series (that is, a sum) of Rademacher distributed variables can be regarded as a simple symmetrical random walk where the step size is 1. Mathematical formulation The probability mass function of this distribution is $f(k)=\left\{{\begin{matrix}1/2&{\mbox{if }}k=-1,\\1/2&{\mbox{if }}k=+1,\\0&{\mbox{otherwise.}}\end{matrix}}\right.$ In terms of the Dirac delta function, as $f(k)={\frac {1}{2}}\left(\delta \left(k-1\right)+\delta \left(k+1\right)\right).$ Bounds on sums of independent Rademacher variables There are various results in probability theory around analyzing the sum of i.i.d. Rademacher variables, including concentration inequalities such as Bernstein inequalities as well as anti-concentration inequalities like Tomaszewski's conjecture. Concentration inequalities Let {xi} be a set of random variables with a Rademacher distribution. Let {ai} be a sequence of real numbers. Then $\Pr \left(\sum _{i}x_{i}a_{i}>t\|\|a\|\|_{2}\right)\leq e^{-{\frac {t^{2}}{2}}}$ where \|\|a\|\|2 is the Euclidean norm of the sequence {ai}, t > 0 is a real number and Pr(Z) is the probability of event Z.[2] Let Y = Σ xiai and let Y be an almost surely convergent series in a Banach space. The for t > 0 and s ≥ 1 we have[3] $\Pr \left(\|\|Y\|\|>st\right)\leq \left[{\frac {1}{c}}\Pr(\|\|Y\|\|>t)\right]^{cs^{2}}$ for some constant c. Let p be a positive real number. Then the Khintchine inequality says that[4] $c_{1}\left[\sum {\left\|a_{i}\right\|^{2}}\right]^{\frac {1}{2}}\leq \left(E\left[\left\|\sum {a_{i}x_{i}}\right\|^{p}\right]\right)^{\frac {1}{p}}\leq c_{2}\left[\sum {\left\|a_{i}\right\|^{2}}\right]^{\frac {1}{2}}$ where c1 and c2 are constants dependent only on p. For p ≥ 1, $c_{2}\leq c_{1}{\sqrt {p}}.$ Tomaszewski’s conjecture In 1986, Bogusław Tomaszewski proposed a question about the distribution of the sum of independent Rademacher variables. A series of works on this question[5][6] culminated into a proof in 2020 by Nathan Keller and Ohad Klein of the following conjecture.[7] Conjecture. Let $X=\sum _{i=1}^{n}a_{i}X_{i}$, where $a_{1}^{2}+\cdots +a_{n}^{2}=1$ and the $X_{i}$'s are independent Rademacher variables. Then $\Pr[\|X\|\leq 1]\geq 1/2.$ For example, when $a_{1}=a_{2}=\cdots =a_{n}=1/{\sqrt {n}}$, one gets the following bound, first shown by Van Zuijlen.[8] $\Pr \left(\left\|{\frac {\sum _{i=1}^{n}X_{i}}{\sqrt {n}}}\right\|\leq 1\right)\geq 0.5.$ The bound is sharp and better than that which can be derived from the normal distribution (approximately Pr > 0.31). Applications The Rademacher distribution has been used in bootstrapping. The Rademacher distribution can be used to show that normally distributed and uncorrelated does not imply independent. Random vectors with components sampled independently from the Rademacher distribution are useful for various stochastic approximations, for example: • The Hutchinson trace estimator,[9] which can be used to efficiently approximate the trace of a matrix of which the elements are not directly accessible, but rather implicitly defined via matrix-vector products. • SPSA, a computationally cheap, derivative-free, stochastic gradient approximation, useful for numerical optimization. Rademacher random variables are used in the Symmetrization Inequality. Related distributions • Bernoulli distribution: If X has a Rademacher distribution, then ${\frac {X+1}{2}}$ has a Bernoulli(1/2) distribution. • Laplace distribution: If X has a Rademacher distribution and Y ~ Exp(λ) is independent from X, then XY ~ Laplace(0, 1/λ). References 1. Hitczenko, P.; Kwapień, S. (1994). "On the Rademacher series". Probability in Banach Spaces. Progress in probability. Vol. 35. pp. 31–36. doi:10.1007/978-1-4612-0253-0_2. ISBN 978-1-4612-6682-2. 2. Montgomery-Smith, S. J. (1990). "The distribution of Rademacher sums". Proc Amer Math Soc. 109 (2): 517–522. doi:10.1090/S0002-9939-1990-1013975-0. 3. Dilworth, S. J.; Montgomery-Smith, S. J. (1993). "The distribution of vector-valued Radmacher series". Ann Probab. 21 (4): 2046–2052. arXiv:math/9206201. doi:10.1214/aop/1176989010. JSTOR 2244710. S2CID 15159626. 4. Khintchine, A. (1923). "Über dyadische Brüche". Math. Z. 18 (1): 109–116. doi:10.1007/BF01192399. S2CID 119840766. 5. Holzman, Ron; Kleitman, Daniel J. (1992-09-01). "On the product of sign vectors and unit vectors". Combinatorica. 12 (3): 303–316. doi:10.1007/BF01285819. ISSN 1439-6912. S2CID 20281665. 6. Boppana, Ravi B.; Holzman, Ron (2017-08-31). "Tomaszewski's Problem on Randomly Signed Sums: Breaking the 3/8 Barrier". arXiv:1704.00350 [math.CO]. 7. Keller, Nathan; Klein, Ohad (2021-08-03). "Proof of Tomaszewski's Conjecture on Randomly Signed Sums". arXiv:2006.16834 [math.CO]. 8. van Zuijlen, Martien C. A. (2011). "On a conjecture concerning the sum of independent Rademacher random variables". arXiv:1112.4988. Bibcode:2011arXiv1112.4988V. {{cite journal}}: Cite journal requires \|journal= (help) 9. Avron, H.; Toledo, S. (2011). "Randomized algorithms for estimating the trace of an implicit symmetric positive semidefinite matrix". Journal of the ACM. 58 (2): 8. CiteSeerX 10.1.1.380.9436. doi:10.1145/1944345.1944349. S2CID 5827717. Probability distributions (list) Discrete univariate with finite support • Benford • Bernoulli • beta-binomial • binomial • categorical • hypergeometric • negative • Poisson binomial • Rademacher • soliton • discrete uniform • Zipf • Zipf–Mandelbrot with infinite support • beta negative binomial • Borel • Conway–Maxwell–Poisson • discrete phase-type • Delaporte • extended negative binomial • Flory–Schulz • Gauss–Kuzmin • geometric • logarithmic • mixed Poisson • negative binomial • Panjer • parabolic fractal • Poisson • Skellam • Yule–Simon • zeta Continuous univariate supported on a bounded interval • arcsine • ARGUS • Balding–Nichols • Bates • beta • beta rectangular • continuous Bernoulli • Irwin–Hall • Kumaraswamy • logit-normal • noncentral beta • PERT • raised cosine • reciprocal • triangular • U-quadratic • uniform • Wigner semicircle supported on a semi-infinite interval • Benini • Benktander 1st kind • Benktander 2nd kind • beta prime • Burr • chi • chi-squared • noncentral • inverse • scaled • Dagum • Davis • Erlang • hyper • exponential • hyperexponential • hypoexponential • logarithmic • F • noncentral • folded normal • Fréchet • gamma • generalized • inverse • gamma/Gompertz • Gompertz • shifted • half-logistic • half-normal • Hotelling's T-squared • inverse Gaussian • generalized • Kolmogorov • Lévy • log-Cauchy • log-Laplace • log-logistic • log-normal • log-t • Lomax • matrix-exponential • Maxwell–Boltzmann • Maxwell–Jüttner • Mittag-Leffler • Nakagami • Pareto • phase-type • Poly-Weibull • Rayleigh • relativistic Breit–Wigner • Rice • truncated normal • type-2 Gumbel • Weibull • discrete • Wilks's lambda supported on the whole real line • Cauchy • exponential power • Fisher's z • Kaniadakis κ-Gaussian • Gaussian q • generalized normal • generalized hyperbolic • geometric stable • Gumbel • Holtsmark • hyperbolic secant • Johnson's SU • Landau • Laplace • asymmetric • logistic • noncentral t • normal (Gaussian) • normal-inverse Gaussian • skew normal • slash • stable • Student's t • Tracy–Widom • variance-gamma • Voigt with support whose type varies • generalized chi-squared • generalized extreme value • generalized Pareto • Marchenko–Pastur • Kaniadakis κ-exponential • Kaniadakis κ-Gamma • Kaniadakis κ-Weibull • Kaniadakis κ-Logistic • Kaniadakis κ-Erlang • q-exponential • q-Gaussian • q-Weibull • shifted log-logistic • Tukey lambda Mixed univariate continuous- discrete • Rectified Gaussian Multivariate (joint) • Discrete: • Ewens • multinomial • Dirichlet • negative • Continuous: • Dirichlet • generalized • multivariate Laplace • multivariate normal • multivariate stable • multivariate t • normal-gamma • inverse • Matrix-valued: • LKJ • matrix normal • matrix t • matrix gamma • inverse • Wishart • normal • inverse • normal-inverse • complex Directional Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham Degenerate and singular Degenerate Dirac delta function Singular Cantor Families • Circular • compound Poisson • elliptical • exponential • natural exponential • location–scale • maximum entropy • mixture • Pearson • Tweedie • wrapped • Category • Commons	Wikipedia
Rademacher system In mathematics, in particular in functional analysis, the Rademacher system, named after Hans Rademacher, is an incomplete orthogonal system of functions on the unit interval of the following form: $\{t\mapsto r_{n}(t)=\operatorname {sgn}(\sin 2^{n+1}\pi t);t\in [0,1],n\in \mathbb {N} \}.$ The Rademacher system is stochastically independent, and is closely related to the Walsh system. Specifically, the Walsh system can be constructed as a product of Rademacher functions. References • Rademacher, Hans (1922). "Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen". Math. Ann. 87 (1): 112–138. doi:10.1007/BF01458040. S2CID 120708120. • "Orthogonal system", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Heil, Christopher E. (1997). "A basis theory primer" (PDF). • Curbera, Guillermo P. (2009). "How Summable are Rademacher Series?". Vector Measures, Integration and Related Topics. Basel: Birkhäuser Basel. pp. 135–148. doi:10.1007/978-3-0346-0211-2_13. ISBN 978-3-0346-0210-5. External links • Rademacher system in the Encyclopedia of Mathematics	Wikipedia
Rademacher's theorem In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rn and f: U → Rm is Lipschitz continuous, then f is differentiable almost everywhere in U; that is, the points in U at which f is not differentiable form a set of Lebesgue measure zero. Differentiability here refers to infinitesimal approximability by a linear map, which in particular asserts the existence of the coordinate-wise partial derivatives. Sketch of proof The one-dimensional case of Rademacher's theorem is a standard result in introductory texts on measure-theoretic analysis.[1] In this context, it is natural to prove the more general statement that any single-variable function of bounded variation is differentiable almost everywhere. (This one-dimensional generalization of Rademacher's theorem fails to extend to higher dimensions.) One of the standard proofs of the general Rademacher theorem was found by Charles Morrey.[2] In the following, let u denote a Lipschitz-continuous function on Rn. The first step of the proof is to show that, for any fixed unit vector v, the v-directional derivative of u exists almost everywhere. This is a consequence of a special case of the Fubini theorem: a measurable set in Rn has Lebesgue measure zero if its restriction to every line parallel to v has (one-dimensional) Lebesgue measure zero. Considering in particular the set in Rn where the v-directional derivative of u fails to exist (which must be proved to be measurable), the latter condition is met due to the one-dimensional case of Rademacher's theorem. The second step of Morrey's proof establishes the linear dependence of the v-directional derivative of u upon v. This is based upon the following identity: $\int _{\mathbf {R} ^{n}}{\frac {u(x+h\nu )-u(x)}{h}}\zeta (z)\,d{\mathcal {L}}^{n}(x)=-\int _{\mathbf {R} ^{n}}{\frac {\zeta (x)-\zeta (x-h\nu )}{h}}f(x)\,d{\mathcal {L}}^{n}(x).$ Using the Lipschitz assumption on u, the dominated convergence theorem can be applied to replace the two difference quotients in the above expression by the corresponding v-directional derivatives. Then, based upon the known linear dependence of the v-directional derivative of ζ upon v, the same can be proved of u via the fundamental lemma of calculus of variations. At this point in the proof, the existence of the gradient (defined as the n-tuple of partial derivatives) is guaranteed to exist almost everywhere; for each v, the dot product with v equals the v-directional derivative almost everywhere (although perhaps on a smaller set). Hence, for any countable collection of unit vectors v1, v2, ..., there is a single set E of measure zero such that the gradient and each vi-directional derivative exist everywhere on the complement of E, and are linked by the dot product. By selecting v1, v2, ... to be dense in the unit sphere, it is possible to use the Lipschitz condition to prove the existence of every directional derivative everywhere on the complement of E, together with its representation as the dot product of the gradient with the direction. Morrey's proof can also be put into the context of generalized derivatives.[3] Another proof, also via a reduction to the one-dimensional case, uses the technology of approximate limits.[4] Applications Rademacher's theorem can be used to prove that, for any p ≥ 1, the Sobolev space W1,p(Ω) is preserved under a bi-Lipschitz transformation of the domain, with the chain rule holding in its standard form.[5] With appropriate modification, this also extends to the more general Sobolev spaces Wk,p(Ω).[6] Rademacher's theorem is also significant in the study of geometric measure theory and rectifiable sets, as it allows the analysis of first-order differential geometry, specifically tangent planes and normal vectors.[7] Higher-order concepts such as curvature remain more subtle, since their usual definitions require more differentiability than is achieved by the Rademacher theorem. In the presence of convexity, second-order differentiability is achieved by the Alexandrov theorem, the proof of which can be modeled on that of the Rademacher theorem. In some special cases, the Rademacher theorem is even used as part of the proof.[8] Generalizations Alberto Calderón proved the more general fact that if Ω is an open bounded set in Rn then every function in the Sobolev space W1,p(Ω) is differentiable almost everywhere, provided that p > n.[9] Calderón's theorem is a relatively direct corollary of the Lebesgue differentiation theorem and Sobolev embedding theorem. Rademacher's theorem is a special case, due to the fact that any Lipschitz function on Ω is an element of the space W1,∞(Ω).[9] There is a version of Rademacher's theorem that holds for Lipschitz functions from a Euclidean space into an arbitrary metric space in terms of metric differentials instead of the usual derivative. See also • Pansu derivative References 1. Federer 1969, Theorem 2.9.19; Folland 1999, Section 3.5; Rudin 1987, Chapter 7. 2. Evans & Gariepy 2015, Section 3.1; Simon 1983, Section 2.1; Villani 2009, Theorem 10.8(ii); Ziemer 1989, Section 2.2. 3. Morrey 1966, Theorem 3.1.6. 4. Federer 1969, Section 3.1. 5. Ziemer 1989, Theorem 2.2.2. 6. Morrey 1966, Theorem 3.1.7. 7. Evans & Gariepy 2015, p. 151; Ziemer 1989, pp. 243, 249, 281. 8. Villani 2009, Theorem 14.25. 9. Evans & Gariepy 2015, Section 6.2; Heinonen 2001, Section 6. Sources • Evans, Lawrence C.; Gariepy, Ronald F. (2015). Measure theory and fine properties of functions. Textbooks in Mathematics (Revised edition of 1992 original ed.). Boca Raton, FL: CRC Press. doi:10.1201/b18333. ISBN 978-1-4822-4238-6. MR 3409135. Zbl 1310.28001. • Federer, Herbert (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. Berlin–Heidelberg–New York: Springer-Verlag. doi:10.1007/978-3-642-62010-2. ISBN 978-3-540-60656-7. MR 0257325. Zbl 0176.00801. • Folland, Gerald B. (1999). Real analysis. Modern techniques and their applications. Pure and Applied Mathematics (Second edition of 1984 original ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-31716-0. MR 1681462. Zbl 0924.28001. • Heinonen, Juha (2001). Lectures on analysis on metric spaces. Universitext. New York: Springer-Verlag. doi:10.1007/978-1-4613-0131-8. ISBN 0-387-95104-0. MR 1800917. Zbl 0985.46008. • Morrey, Charles B., Jr. (1966). Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften. Vol. 130. New York: Springer-Verlag. doi:10.1007/978-3-540-69952-1. MR 0202511. Zbl 1213.49002.{{cite book}}: CS1 maint: multiple names: authors list (link) • Rademacher, Hans (1919). "Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale". Mathematische Annalen. 79 (4): 340–359. doi:10.1007/BF01498415. JFM 47.0243.01. MR 1511935. • Rudin, Walter (1987). Real and complex analysis (Third edition of 1966 original ed.). New York: McGraw-Hill Book Co. ISBN 0-07-054234-1. MR 0924157. Zbl 0925.00005. • Simon, Leon (1983). Lectures on geometric measure theory (PDF). Proceedings of the Centre for Mathematical Analysis, Australian National University. Vol. 3. Canberra: Australian National University, Centre for Mathematical Analysis. ISBN 0-86784-429-9. MR 0756417. Zbl 0546.49019. • Villani, Cédric (2009). Optimal transport. Old and new. Grundlehren der mathematischen Wissenschaften. Vol. 338. Berlin: Springer-Verlag. doi:10.1007/978-3-540-71050-9. ISBN 978-3-540-71049-3. MR 2459454. Zbl 1156.53003. • Ziemer, William P. (1989). Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics. Vol. 120. New York: Springer-Verlag. doi:10.1007/978-1-4612-1015-3. ISBN 0-387-97017-7. MR 1014685. Zbl 0692.46022. External links • Heinonen, Juha (2004). "Lectures on Lipschitz Analysis" (PDF). Lectures at the 14th Jyväskylä Summer School in August 2004. (Rademacher's theorem with a proof is on page 18 and further.)	Wikipedia
B-convex space In functional analysis, the class of B-convex spaces is a class of Banach space. The concept of B-convexity was defined and used to characterize Banach spaces that have the strong law of large numbers by Anatole Beck in 1962; accordingly, "B-convexity" is understood as an abbreviation of Beck convexity. Beck proved the following theorem: A Banach space is B-convex if and only if every sequence of independent, symmetric, uniformly bounded and Radon random variables in that space satisfies the strong law of large numbers. Let X be a Banach space with norm \|\| \|\|. X is said to be B-convex if for some ε > 0 and some natural number n, it holds true that whenever x1, ..., xn are elements of the closed unit ball of X, there is a choice of signs α1, ..., αn ∈ {−1, +1} such that $\left\\|\sum _{i=1}^{n}\alpha _{i}x_{i}\right\\|\leq (1-\varepsilon )n.$ Later authors have shown that B-convexity is equivalent to a number of other important properties in the theory of Banach spaces. Being B-convex and having Rademacher type $p>1$ were shown to be equivalent Banach-space properties by Gilles Pisier. References • Beck, Anatole (1962). "A convexity condition in Banach spaces and the strong law of large numbers". Proc. Amer. Math. Soc. 13 (2): 329–334. doi:10.1090/S0002-9939-1962-0133857-9. ISSN 0002-9939. MR 0133857. • Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR 1102015. (See chapter 9) Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons	Wikipedia
Rademacher–Menchov theorem In mathematical analysis, the Rademacher–Menchov theorem, introduced by Rademacher (1922) and Menchoff (1923), gives a sufficient condition for a series of orthogonal functions on an interval to converge almost everywhere. Statement If the coefficients cν of a series of bounded orthogonal functions on an interval satisfy $\sum \|c_{\nu }\|^{2}\log(\nu )^{2}<\infty $ then the series converges almost everywhere. References • Menchoff, D. (1923), "Sur les séries de fonctions orthogonales. (Première Partie. La convergence.).", Fundamenta Mathematicae (in French), 4: 82–105, doi:10.4064/fm-4-1-82-105, ISSN 0016-2736 • Rademacher, Hans (1922), "Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen", Mathematische Annalen, Springer Berlin / Heidelberg, 87: 112–138, doi:10.1007/BF01458040, ISSN 0025-5831, S2CID 120708120 • Zygmund, A. (2002) [1935], Trigonometric Series. Vol. I, II, Cambridge Mathematical Library (3rd ed.), Cambridge University Press, ISBN 978-0-521-89053-3, MR 1963498	Wikipedia
Rader's FFT algorithm Rader's algorithm (1968),[1] named for Charles M. Rader of MIT Lincoln Laboratory, is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of prime sizes by re-expressing the DFT as a cyclic convolution (the other algorithm for FFTs of prime sizes, Bluestein's algorithm, also works by rewriting the DFT as a convolution). Since Rader's algorithm only depends upon the periodicity of the DFT kernel, it is directly applicable to any other transform (of prime order) with a similar property, such as a number-theoretic transform or the discrete Hartley transform. The algorithm can be modified to gain a factor of two savings for the case of DFTs of real data, using a slightly modified re-indexing/permutation to obtain two half-size cyclic convolutions of real data;[2] an alternative adaptation for DFTs of real data uses the discrete Hartley transform.[3] Winograd extended Rader's algorithm to include prime-power DFT sizes $p^{m}$,[4][5] and today Rader's algorithm is sometimes described as a special case of Winograd's FFT algorithm, also called the multiplicative Fourier transform algorithm (Tolimieri et al., 1997),[6] which applies to an even larger class of sizes. However, for composite sizes such as prime powers, the Cooley–Tukey FFT algorithm is much simpler and more practical to implement, so Rader's algorithm is typically only used for large-prime base cases of Cooley–Tukey's recursive decomposition of the DFT.[3] Algorithm Begin with the definition of the discrete Fourier transform: $X_{k}=\sum _{n=0}^{N-1}x_{n}e^{-{\frac {2\pi i}{N}}nk}\qquad k=0,\dots ,N-1.$ If N is a prime number, then the set of non-zero indices $n\in {}\{1,\dots ,N-1\}$ forms a group under multiplication modulo N. One consequence of the number theory of such groups is that there exists a generator of the group (sometimes called a primitive root, which can be found by exhaustive search or slightly better algorithms[7]). This generator is an integer g such that $n=g^{q}{\pmod {N}}$ for any non-zero index n and for a unique $q\in {}\{0,\dots ,N-2\}$ (forming a bijection from q to non-zero n). Similarly, $k=g^{-p}{\pmod {N}}$ for any non-zero index k and for a unique $p\in {}\{0,\dots ,N-2\}$, where the negative exponent denotes the multiplicative inverse of $g^{p}\mod N$. That means that we can rewrite the DFT using these new indices p and q as: $X_{0}=\sum _{n=0}^{N-1}x_{n},$ $X_{g^{-p}}=x_{0}+\sum _{q=0}^{N-2}x_{g^{q}}e^{-{\frac {2\pi i}{N}}g^{-(p-q)}}\qquad p=0,\dots ,N-2.$ (Recall that xn and Xk are implicitly periodic in N, and also that $e^{2\pi i}=1$ (Euler's identity). Thus, all indices and exponents are taken modulo N as required by the group arithmetic.) The final summation, above, is precisely a cyclic convolution of the two sequences aq and bq (of length N–1, because $q\in {}\{0,\dots ,N-2\}$) defined by: $a_{q}=x_{g^{q}}$ $b_{q}=e^{-{\frac {2\pi i}{N}}g^{-q}}.$ Evaluating the convolution Since N–1 is composite, this convolution can be performed directly via the convolution theorem and more conventional FFT algorithms. However, that may not be efficient if N–1 itself has large prime factors, requiring recursive use of Rader's algorithm. Instead, one can compute a length-(N–1) cyclic convolution exactly by zero-padding it to a length of at least 2(N–1)–1, say to a power of two, which can then be evaluated in O(N log N) time without the recursive application of Rader's algorithm. This algorithm, then, requires O(N) additions plus O(N log N) time for the convolution. In practice, the O(N) additions can often be performed by absorbing the additions into the convolution: if the convolution is performed by a pair of FFTs, then the sum of xn is given by the DC (0th) output of the FFT of aq plus x0, and x0 can be added to all the outputs by adding it to the DC term of the convolution prior to the inverse FFT. Still, this algorithm requires intrinsically more operations than FFTs of nearby composite sizes, and typically takes 3–10 times as long in practice. If Rader's algorithm is performed by using FFTs of size N–1 to compute the convolution, rather than by zero padding as mentioned above, the efficiency depends strongly upon N and the number of times that Rader's algorithm must be applied recursively. The worst case would be if N–1 were 2N2 where N2 is prime, with N2–1 = 2N3 where N3 is prime, and so on. In such cases, supposing that the chain of primes extended all the way down to some bounded value, the recursive application of Rader's algorithm would actually require O(N2) time. Such Nj are called Sophie Germain primes, and such a sequence of them is called a Cunningham chain of the first kind. The lengths of Cunningham chains, however, are observed to grow more slowly than log2(N), so Rader's algorithm applied in this way is probably not O(N2), though it is possibly worse than O(N log N) for the worst cases. Fortunately, a guarantee of O(N log N) complexity can be achieved by zero padding. References 1. C. M. Rader, "Discrete Fourier transforms when the number of data samples is prime," Proc. IEEE 56, 1107–1108 (1968). 2. S. Chu and C. Burrus, "A prime factor FTT [sic] algorithm using distributed arithmetic," IEEE Transactions on Acoustics, Speech, and Signal Processing 30 (2), 217–227 (1982). 3. Matteo Frigo and Steven G. Johnson, "The Design and Implementation of FFTW3," Proceedings of the IEEE 93 (2), 216–231 (2005). 4. S. Winograd, "On Computing the Discrete Fourier Transform", Proc. National Academy of Sciences USA, 73(4), 1005–1006 (1976). 5. S. Winograd, "On Computing the Discrete Fourier Transform", Mathematics of Computation, 32(141), 175–199 (1978). 6. R. Tolimieri, M. An, and C.Lu, Algorithms for Discrete Fourier Transform and Convolution, Springer-Verlag, 2nd ed., 1997. 7. Donald E. Knuth, The Art of Computer Programming, vol. 2: Seminumerical Algorithms, 3rd edition, section 4.5.4, p. 391 (Addison–Wesley, 1998).	Wikipedia
Radial basis function In mathematics a radial basis function (RBF) is a real-valued function $ \varphi $ whose value depends only on the distance between the input and some fixed point, either the origin, so that $ \varphi (\mathbf {x} )={\hat {\varphi }}(\left\\|\mathbf {x} \right\\|)$, or some other fixed point $ \mathbf {c} $, called a center, so that $ \varphi (\mathbf {x} )={\hat {\varphi }}(\left\\|\mathbf {x} -\mathbf {c} \right\\|)$. Any function $ \varphi $ that satisfies the property $ \varphi (\mathbf {x} )={\hat {\varphi }}(\left\\|\mathbf {x} \right\\|)$ is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection $\{\varphi _{k}\}_{k}$ which forms a basis for some function space of interest, hence the name. Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network; this was the context in which they were originally applied to machine learning, in work by David Broomhead and David Lowe in 1988,[1][2] which stemmed from Michael J. D. Powell's seminal research from 1977.[3][4][5] RBFs are also used as a kernel in support vector classification.[6] The technique has proven effective and flexible enough that radial basis functions are now applied in a variety of engineering applications.[7][8] Definition A radial function is a function $ \varphi :[0,\infty )\to \mathbb {R} $ :[0,\infty )\to \mathbb {R} } . When paired with a metric on a vector space $ \\|\cdot \\|:V\to [0,\infty )$ a function $ \varphi _{\mathbf {c} }=\varphi (\\|\mathbf {x} -\mathbf {c} \\|)$ is said to be a radial kernel centered at $ \mathbf {c} $. A Radial function and the associated radial kernels are said to be radial basis functions if, for any set of nodes $\{\mathbf {x} _{k}\}_{k=1}^{n}$ • The kernels $\varphi _{\mathbf {x} _{1}},\varphi _{\mathbf {x} _{2}},\dots ,\varphi _{\mathbf {x} _{n}}$ are linearly independent (for example $\varphi (r)=r^{2}$ in $V=\mathbb {R} $ is not a radial basis function) • The kernels $\varphi _{\mathbf {x} _{1}},\varphi _{\mathbf {x} _{2}},\dots ,\varphi _{\mathbf {x} _{n}}$ form a basis for a Haar Space, meaning that the interpolation matrix ${\begin{bmatrix}\varphi (\\|\mathbf {x} _{1}-\mathbf {x} _{1}\\|)&\varphi (\\|\mathbf {x} _{2}-\mathbf {x} _{1}\\|)&\dots &\varphi (\\|\mathbf {x} _{n}-\mathbf {x} _{1}\\|)\\\varphi (\\|\mathbf {x} _{1}-\mathbf {x} _{2}\\|)&\varphi (\\|\mathbf {x} _{2}-\mathbf {x} _{2}\\|)&\dots &\varphi (\\|\mathbf {x} _{n}-\mathbf {x} _{2}\\|)\\\vdots &\vdots &\ddots &\vdots \\\varphi (\\|\mathbf {x} _{1}-\mathbf {x} _{n}\\|)&\varphi (\\|\mathbf {x} _{2}-\mathbf {x} _{n}\\|)&\dots &\varphi (\\|\mathbf {x} _{n}-\mathbf {x} _{n}\\|)\\\end{bmatrix}},$ (1) is non-singular.[9][10] Examples Commonly used types of radial basis functions include (writing $ r=\left\\|\mathbf {x} -\mathbf {x} _{i}\right\\|$ and using $ \varepsilon $ to indicate a shape parameter that can be used to scale the input of the radial kernel[11]): • Infinitely Smooth RBFs These radial basis functions are from $C^{\infty }(\mathbb {R} )$ and are strictly positive definite functions[12] that require tuning a shape parameter $\varepsilon $ • Gaussian: $\varphi (r)=e^{-(\varepsilon r)^{2}},$ (2) • Multiquadric: $\varphi (r)={\sqrt {1+(\varepsilon r)^{2}}},$ (3) • Inverse quadratic: $\varphi (r)={\dfrac {1}{1+(\varepsilon r)^{2}}},$ (4) • Inverse multiquadric: $\varphi (r)={\dfrac {1}{\sqrt {1+(\varepsilon r)^{2}}}},$ (5) • Polyharmonic spline: ${\begin{aligned}\varphi (r)&=r^{k},&k&=1,3,5,\dotsc \\\varphi (r)&=r^{k}\ln(r),&k&=2,4,6,\dotsc \end{aligned}}$ (6) *For even-degree polyharmonic splines $(k=2,4,6,\dotsc )$, to avoid numerical problems at $r=0$ where $\ln(0)=-\infty $, the computational implementation is often written as $\varphi (r)=r^{k-1}\ln(r^{r})$. • Thin plate spline (a special polyharmonic spline): $\varphi (r)=r^{2}\ln(r),$ (7) • Compactly Supported RBFs These RBFs are compactly supported and thus are non-zero only within a radius of $1/\varepsilon $, and thus have sparse differentiation matrices • Bump function: $\varphi (r)={\begin{cases}\exp \left(-{\frac {1}{1-(\varepsilon r)^{2}}}\right)&{\text{ for }}r<{\frac {1}{\varepsilon }}\\0&{\text{ otherwise}}\end{cases}},$ (8) Approximation Main article: Kernel smoothing Main article: Radial basis function interpolation Radial basis functions are typically used to build up function approximations of the form $y(\mathbf {x} )=\sum _{i=1}^{N}w_{i}\,\varphi (\left\\|\mathbf {x} -\mathbf {x} _{i}\right\\|),$ (9) where the approximating function $ y(\mathbf {x} )$ is represented as a sum of $N$ radial basis functions, each associated with a different center $ \mathbf {x} _{i}$, and weighted by an appropriate coefficient $ w_{i}.$ The weights $ w_{i}$ can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights $ w_{i}$. Approximation schemes of this kind have been particularly used in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour and 3D reconstruction in computer graphics (for example, hierarchical RBF and Pose Space Deformation). RBF Network Main article: radial basis function network The sum $y(\mathbf {x} )=\sum _{i=1}^{N}w_{i}\,\varphi (\left\\|\mathbf {x} -\mathbf {x} _{i}\right\\|),$ (10) can also be interpreted as a rather simple single-layer type of artificial neural network called a radial basis function network, with the radial basis functions taking on the role of the activation functions of the network. It can be shown that any continuous function on a compact interval can in principle be interpolated with arbitrary accuracy by a sum of this form, if a sufficiently large number $ N$ of radial basis functions is used. The approximant $ y(\mathbf {x} )$ is differentiable with respect to the weights $ w_{i}$. The weights could thus be learned using any of the standard iterative methods for neural networks. Using radial basis functions in this manner yields a reasonable interpolation approach provided that the fitting set has been chosen such that it covers the entire range systematically (equidistant data points are ideal). However, without a polynomial term that is orthogonal to the radial basis functions, estimates outside the fitting set tend to perform poorly. RBFs for PDEs Radial basis functions are used to approximate functions and so can be used to discretize and numerically solve Partial Differential Equations (PDEs). This was first done in 1990 by E. J. Kansa who developed the first RBF based numerical method. It is called the Kansa method and was used to solve the elliptic Poisson equation and the linear advection-diffusion equation. The function values at points $\mathbf {x} $ in the domain are approximated by the linear combination of RBFs: $u(\mathbf {x} )=\sum _{i=1}^{N}\lambda _{i}\,\varphi (\left\\|\mathbf {x} -\mathbf {x} _{i}\right\\|),\quad \mathbf {x} \in \mathbb {R} ^{d}$ (11) The derivatives are approximated as such: ${\frac {\partial ^{n}u({\textbf {x}})}{\partial x^{n}}}=\sum _{i=1}^{N}\lambda _{i}\,{\frac {\partial ^{n}}{\partial x^{n}}}\varphi (\left\\|\mathbf {x} -\mathbf {x} _{i}\right\\|),\quad \mathbf {x} \in \mathbb {R} ^{d}$ (12) where $N$ are the number of points in the discretized domain, $d$ the dimension of the domain and $\lambda $ the scalar coefficients that are unchanged by the differential operator.[13] Different numerical methods based on Radial Basis Functions were developed thereafter. Some methods are the RBF-FD method,[14][15] the RBF-QR method[16] and the RBF-PUM method.[17] See also • Matérn covariance function • Radial basis function interpolation • Kansa method References 1. Radial Basis Function networks Archived 2014-04-23 at the Wayback Machine 2. Broomhead, David H.; Lowe, David (1988). "Multivariable Functional Interpolation and Adaptive Networks" (PDF). Complex Systems. 2: 321–355. Archived from the original (PDF) on 2014-07-14. 3. Michael J. D. Powell (1977). "Restart procedures for the conjugate gradient method". Mathematical Programming. 12 (1): 241–254. doi:10.1007/bf01593790. S2CID 9500591. 4. Sahin, Ferat (1997). A Radial Basis Function Approach to a Color Image Classification Problem in a Real Time Industrial Application (M.Sc.). Virginia Tech. p. 26. hdl:10919/36847. Radial basis functions were first introduced by Powell to solve the real multivariate interpolation problem. 5. Broomhead & Lowe 1988, p. 347: "We would like to thank Professor M.J.D. Powell at the Department of Applied Mathematics and Theoretical Physics at Cambridge University for providing the initial stimulus for this work." 6. VanderPlas, Jake (6 May 2015). "Introduction to Support Vector Machines". [O'Reilly]. Retrieved 14 May 2015. 7. Buhmann, Martin Dietrich (2003). Radial basis functions : theory and implementations. Cambridge University Press. ISBN 978-0511040207. OCLC 56352083. 8. Biancolini, Marco Evangelos (2018). Fast radial basis functions for engineering applications. Springer International Publishing. ISBN 9783319750118. OCLC 1030746230. 9. Fasshauer, Gregory E. (2007). Meshfree Approximation Methods with MATLAB. Singapore: World Scientific Publishing Co. Pte. Ltd. pp. 17–25. ISBN 9789812706331. 10. Wendland, Holger (2005). Scattered Data Approximation. Cambridge: Cambridge University Press. pp. 11, 18–23, 64–66. ISBN 0521843359. 11. Fasshauer, Gregory E. (2007). Meshfree Approximation Methods with MATLAB. Singapore: World Scientific Publishing Co. Pte. Ltd. p. 37. ISBN 9789812706331. 12. Fasshauer, Gregory E. (2007). Meshfree Approximation Methods with MATLAB. Singapore: World Scientific Publishing Co. Pte. Ltd. pp. 37–45. ISBN 9789812706331. 13. Kansa, E. J. (1990-01-01). "Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations". Computers & Mathematics with Applications. 19 (8): 147–161. doi:10.1016/0898-1221(90)90271-K. ISSN 0898-1221. 14. Tolstykh, A. I.; Shirobokov, D. A. (2003-12-01). "On using radial basis functions in a "finite difference mode" with applications to elasticity problems". Computational Mechanics. 33 (1): 68–79. doi:10.1007/s00466-003-0501-9. ISSN 1432-0924. S2CID 121511032. 15. Shu, C; Ding, H; Yeo, K. S (2003-02-14). "Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations". Computer Methods in Applied Mechanics and Engineering. 192 (7): 941–954. doi:10.1016/S0045-7825(02)00618-7. ISSN 0045-7825. 16. Fornberg, Bengt; Larsson, Elisabeth; Flyer, Natasha (2011-01-01). "Stable Computations with Gaussian Radial Basis Functions". SIAM Journal on Scientific Computing. 33 (2): 869–892. doi:10.1137/09076756X. ISSN 1064-8275. 17. Safdari-Vaighani, Ali; Heryudono, Alfa; Larsson, Elisabeth (2015-08-01). "A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications". Journal of Scientific Computing. 64 (2): 341–367. doi:10.1007/s10915-014-9935-9. ISSN 1573-7691. S2CID 254691757. Further reading • Hardy, R.L. (1971). "Multiquadric equations of topography and other irregular surfaces". Journal of Geophysical Research. 76 (8): 1905–1915. Bibcode:1971JGR....76.1905H. doi:10.1029/jb076i008p01905. • Hardy, R.L. (1990). "Theory and applications of the multiquadric-biharmonic method, 20 years of Discovery, 1968 1988". Comp. Math Applic. 19 (8/9): 163–208. doi:10.1016/0898-1221(90)90272-l. • Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 3.7.1. Radial Basis Function Interpolation", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8 • Sirayanone, S., 1988, Comparative studies of kriging, multiquadric-biharmonic, and other methods for solving mineral resource problems, PhD. Dissertation, Dept. of Earth Sciences, Iowa State University, Ames, Iowa. • Sirayanone, S.; Hardy, R.L. (1995). "The Multiquadric-biharmonic Method as Used for Mineral Resources, Meteorological, and Other Applications". Journal of Applied Sciences and Computations. 1: 437–475.	Wikipedia
Radial basis function interpolation Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured data, possibly in high-dimensional spaces. The interpolant takes the form of a weighted sum of radial basis functions,[1] like for example Gaussian distributions.[2] RBF interpolation is a mesh-free method, meaning the nodes (points in the domain) need not lie on a structured grid, and does not require the formation of a mesh. It is often spectrally accurate[3] and stable for large numbers of nodes even in high dimensions. Many interpolation methods can be used as the theoretical foundation of algorithms for approximating linear operators, and RBF interpolation is no exception. RBF interpolation has been used to approximate differential operators, integral operators, and surface differential operators. Examples Let $f(x)=\exp(x\cos(3\pi x))$ and let $x_{k}={\frac {k}{14}},k=0,1,\dots ,14$ be 15 equally spaced points on the interval $[0,1]$. We will form $s(x)=\sum \limits _{k=0}^{14}w_{k}\varphi (\\|x-x_{k}\\|)$ where $\varphi $ is a radial basis function, and choose $w_{k},k=0,1,\dots ,14$ such that $s(x_{k})=f(x_{k}),k=0,1,\dots ,14$ ($s$ interpolates $f$ at the chosen points). In matrix notation this can be written as ${\begin{bmatrix}\varphi (\\|x_{0}-x_{0}\\|)&\varphi (\\|x_{1}-x_{0}\\|)&\dots &\varphi (\\|x_{14}-x_{0}\\|)\\\varphi (\\|x_{0}-x_{1}\\|)&\varphi (\\|x_{1}-x_{1}\\|)&\dots &\varphi (\\|x_{14}-x_{1}\\|)\\\vdots &\vdots &\ddots &\vdots \\\varphi (\\|x_{0}-x_{14}\\|)&\varphi (\\|x_{1}-x_{14}\\|)&\dots &\varphi (\\|x_{14}-x_{14}\\|)\\\end{bmatrix}}{\begin{bmatrix}w_{0}\\w_{1}\\\vdots \\w_{14}\end{bmatrix}}={\begin{bmatrix}f(x_{0})\\f(x_{1})\\\vdots \\f(x_{14})\end{bmatrix}}.$ Choosing $\varphi (r)=\exp(-(\varepsilon r)^{2})$, the Gaussian, with a shape parameter of $\varepsilon =3$, we can then solve the matrix equation for the weights and plot the interpolant. Plotting the interpolating function below, we see that it is visually the same everywhere except near the left boundary (an example of Runge's phenomenon), where it is still a very close approximation. More precisely the maximum error is roughly $\\|f-s\\|_{\infty }\approx 0.0267414$ at $x=0.0220012$. The function $f(x)=\exp(x\cos(3\pi x))$ sampled at 15 uniform nodes between 0 and 1, interpolated using the Gaussian RBF with a shape parameter of $\varepsilon =3$. The interpolation error, $s(x)-f(x)$, for the plot to the left. Motivation The Mairhuber–Curtis theorem says that for any open set $V$ in $\mathbb {R} ^{n}$ with $n\geq 2$, and $f_{1},f_{2},\dots ,f_{n}$ linearly independent functions on $V$, there exists a set of $n$ points in the domain such that the interpolation matrix ${\begin{bmatrix}f_{1}(x_{1})&f_{2}(x_{1})&\dots &f_{n}(x_{1})\\f_{1}(x_{2})&f_{2}(x_{2})&\dots &f_{n}(x_{2})\\\vdots &\vdots &\ddots &\vdots \\f_{1}(x_{n})&f_{2}(x_{n})&\dots &f_{n}(x_{n})\end{bmatrix}}$ is singular.[4] This means that if one wishes to have a general interpolation algorithm, one must choose the basis functions to depend on the interpolation points. In 1971, Rolland Hardy developed a method of interpolating scattered data using interpolants of the form $s(\mathbf {x} )=\sum \limits _{k=1}^{N}{\sqrt {\\|\mathbf {x} -\mathbf {x} _{k}\\|^{2}+C}}$. This is interpolation using a basis of shifted multiquadric functions, now more commonly written as $\varphi (r)={\sqrt {1+(\varepsilon r)^{2}}}$, and is the first instance of radial basis function interpolation.[5] It has been shown that the resulting interpolation matrix will always be non-singular. This does not violate the Mairhuber–Curtis theorem since the basis functions depend on the points of interpolation. Choosing a radial kernel such that the interpolation matrix is non-singular is exactly the definition of a strictly positive definite function. Such functions, including the Gaussian, inverse quadratic, and inverse multiquadric are often used as radial basis functions for this reason.[6] Shape-parameter tuning Many radial basis functions have a parameter that controls their relative flatness or peakedness. This parameter is usually represented by the symbol $\varepsilon $ with the function becoming increasingly flat as $\varepsilon \to 0$. For example, Rolland Hardy used the formula $\varphi (r)={\sqrt {r^{2}+C}}$ for the multiquadric, however nowadays the formula $\varphi (r)={\sqrt {1+(\varepsilon r)^{2}}}$ is used instead. These formulas are equivalent up to a scale factor. This factor is inconsequential since the basis vectors have the same span and the interpolation weights will compensate. By convention, the basis function is scaled such that $\varphi (0)=1$ as seen in the plots of the Gaussian functions and the bump functions. • A Gaussian function for several choices of $\varepsilon $ • A plot of the scaled bump function with several choices of shape parameter A consequence of this choice, is that the interpolation matrix approaches the identity matrix as $\varepsilon \to \infty $ leading to stability when solving the matrix system. The resulting interpolant will in general be a poor approximation to the function since it will be near zero everywhere, except near the interpolation points where it will sharply peak – the so-called "bed-of-nails interpolant" (as seen in the plot to the right). On the opposite side of the spectrum, the condition number of the interpolation matrix will diverge to infinity as $\varepsilon \to 0$ leading to ill-conditioning of the system. In practice, one chooses a shape parameter so that the interpolation matrix is "on the edge of ill-conditioning" (eg. with a condition number of roughly $10^{12}$ for double-precision floating point). There are sometimes other factors to consider when choosing a shape-parameter. For example the bump function $\varphi (r)={\begin{cases}\exp \left(-{\frac {1}{1-(\varepsilon r)^{2}}}\right)&{\mbox{ for }}r<{\frac {1}{\varepsilon }}\\0&{\mbox{ otherwise}}\end{cases}}$ has a compact support (it is zero everywhere except when $r<{\tfrac {1}{\varepsilon }}$) leading to a sparse interpolation matrix. Some radial basis functions such as the polyharmonic splines have no shape-parameter. See also • Kriging References 1. Hardy, Rolland (March 1971). "Multiquadric equations of topography and other irregular surfaces". Journal of Geophysical Research. 76 (8): 1905–1915. doi:10.1029/JB076i008p01905. 2. Richard, Franke (January 1982). "Scattered Data Interpolation: Tests of Some Methods". Mathematics of Computation. 38 (157): 181–200. doi:10.1090/S0025-5718-1982-0637296-4. 3. Buhmann, Martin; Nira, Dyn (June 1993). "Spectral convergence of multiquadric interpolation". Proceedings of the Edinburgh Mathematical Society. 36 (2): 319–333. doi:10.1017/S0013091500018411. 4. Mairhuber, John C. (1956). "On Haar's Theorem Concerning Chebychev Approximation Problems Having Unique Solutions". Proceedings of the American Mathematical Society. 7 (4): 609–615. JSTOR 2033359. 5. Hardy, Rolland L. (1971). "Multiquadric equations of topography and other irregular surfaces". Journal of Geophysical Research. 7 (8): 1905–1915. doi:10.1029/JB076i008p01905. 6. Fasshaur, Greg (2007). Meshfree Approximation Methods with MATLAB. World Scientific Publishing. ISBN 978-981-270-633-1.	Wikipedia
Radial function In mathematics, a radial function is a real-valued function defined on a Euclidean space Rn whose value at each point depends only on the distance between that point and the origin. The distance is usually the Euclidean distance. For example, a radial function Φ in two dimensions has the form[1] $\Phi (x,y)=\varphi (r),\quad r={\sqrt {x^{2}+y^{2}}}$ where φ is a function of a single non-negative real variable. Radial functions are contrasted with spherical functions, and any descent function (e.g., continuous and rapidly decreasing) on Euclidean space can be decomposed into a series consisting of radial and spherical parts: the solid spherical harmonic expansion. A function is radial if and only if it is invariant under all rotations leaving the origin fixed. That is, ƒ is radial if and only if $f\circ \rho =f\,$ for all ρ ∈ SO(n), the special orthogonal group in n dimensions. This characterization of radial functions makes it possible also to define radial distributions. These are distributions S on Rn such that $S[\varphi ]=S[\varphi \circ \rho ]$ for every test function φ and rotation ρ. Given any (locally integrable) function ƒ, its radial part is given by averaging over spheres centered at the origin. To wit, $\phi (x)={\frac {1}{\omega _{n-1}}}\int _{S^{n-1}}f(rx')\,dx'$ where ωn−1 is the surface area of the (n−1)-sphere Sn−1, and r = \|x\|, x′ = x/r. It follows essentially by Fubini's theorem that a locally integrable function has a well-defined radial part at almost every r. The Fourier transform of a radial function is also radial, and so radial functions play a vital role in Fourier analysis. Furthermore, the Fourier transform of a radial function typically has stronger decay behavior at infinity than non-radial functions: for radial functions bounded in a neighborhood of the origin, the Fourier transform decays faster than R−(n−1)/2. The Bessel functions are a special class of radial function that arise naturally in Fourier analysis as the radial eigenfunctions of the Laplacian; as such they appear naturally as the radial portion of the Fourier transform. See also • Radial basis function References 1. "Radial Basis Function - Machine Learning Concepts". Machine Learning Concepts -. 2022-03-17. Retrieved 2022-12-23. • Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9.	Wikipedia
Circular motion In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with a constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves the circular motion of its parts. The equations of motion describe the movement of the center of mass of a body. In circular motion, the distance between the body and a fixed point on the surface remains the same. Part of a series on Classical mechanics ${\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})$ Second law of motion • History • Timeline • Textbooks Branches • Applied • Celestial • Continuum • Dynamics • Kinematics • Kinetics • Statics • Statistical mechanics Fundamentals • Acceleration • Angular momentum • Couple • D'Alembert's principle • Energy • kinetic • potential • Force • Frame of reference • Inertial frame of reference • Impulse • Inertia / Moment of inertia • Mass • Mechanical power • Mechanical work • Moment • Momentum • Space • Speed • Time • Torque • Velocity • Virtual work Formulations • Newton's laws of motion • Analytical mechanics • Lagrangian mechanics • Hamiltonian mechanics • Routhian mechanics • Hamilton–Jacobi equation • Appell's equation of motion • Koopman–von Neumann mechanics Core topics • Damping • Displacement • Equations of motion • Euler's laws of motion • Fictitious force • Friction • Harmonic oscillator • Inertial / Non-inertial reference frame • Mechanics of planar particle motion • Motion (linear) • Newton's law of universal gravitation • Newton's laws of motion • Relative velocity • Rigid body • dynamics • Euler's equations • Simple harmonic motion • Vibration Rotation • Circular motion • Rotating reference frame • Centripetal force • Centrifugal force • reactive • Coriolis force • Pendulum • Tangential speed • Rotational frequency • Angular acceleration / displacement / frequency / velocity Scientists • Kepler • Galileo • Huygens • Newton • Horrocks • Halley • Maupertuis • Daniel Bernoulli • Johann Bernoulli • Euler • d'Alembert • Clairaut • Lagrange • Laplace • Hamilton • Poisson • Cauchy • Routh • Liouville • Appell • Gibbs • Koopman • von Neumann • Physics portal • Category Examples of circular motion include: special satellite orbits around the Earth (circular orbits), a ceiling fan's blades rotating around a hub, a stone that is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism. Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion. Uniform circular motion In physics, uniform circular motion describes the motion of a body traversing a circular path at a constant speed. Since the body describes circular motion, its distance from the axis of rotation remains constant at all times. Though the body's speed is constant, its velocity is not constant: velocity, a vector quantity, depends on both the body's speed and its direction of travel. This changing velocity indicates the presence of an acceleration; this centripetal acceleration is of constant magnitude and directed at all times toward the axis of rotation. This acceleration is, in turn, produced by a centripetal force which is also constant in magnitude and directed toward the axis of rotation. In the case of rotation around a fixed axis of a rigid body that is not negligibly small compared to the radius of the path, each particle of the body describes a uniform circular motion with the same angular velocity, but with velocity and acceleration varying with the position with respect to the axis. Formulas For motion in a circle of radius r, the circumference of the circle is C = 2πr. If the period for one rotation is T, the angular rate of rotation, also known as angular velocity, ω is: $\omega ={\frac {2\pi }{T}}=2\pi f={\frac {d\theta }{dt}}$ and the units are radians/second. The speed of the object traveling the circle is: $v={\frac {2\pi r}{T}}=\omega r$ The angle θ swept out in a time t is: $\theta =2\pi {\frac {t}{T}}=\omega t$ The angular acceleration, α, of the particle is: $\alpha ={\frac {d\omega }{dt}}$ In the case of uniform circular motion, α will be zero. The acceleration due to change in the direction is: $a_{c}={\frac {v^{2}}{r}}=\omega ^{2}r$ The centripetal and centrifugal force can also be found using acceleration: $F_{c}={\dot {p}}\mathrel {\overset {{\dot {m}}=0}{=}} ma_{c}={\frac {mv^{2}}{r}}$ The vector relationships are shown in Figure 1. The axis of rotation is shown as a vector ω perpendicular to the plane of the orbit and with a magnitude ω = dθ / dt. The direction of ω is chosen using the right-hand rule. With this convention for depicting rotation, the velocity is given by a vector cross product as $\mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} ,$ which is a vector perpendicular to both ω and r(t), tangential to the orbit, and of magnitude ω r. Likewise, the acceleration is given by $\mathbf {a} ={\boldsymbol {\omega }}\times \mathbf {v} ={\boldsymbol {\omega }}\times \left({\boldsymbol {\omega }}\times \mathbf {r} \right),$ which is a vector perpendicular to both ω and v(t) of magnitude ω \|v\| = ω2 r and directed exactly opposite to r(t).[1] In the simplest case the speed, mass, and radius are constant. Consider a body of one kilogram, moving in a circle of radius one metre, with an angular velocity of one radian per second. • The speed is 1 metre per second. • The inward acceleration is 1 metre per square second, v2/r. • It is subject to a centripetal force of 1 kilogram metre per square second, which is 1 newton. • The momentum of the body is 1 kg·m·s−1. • The moment of inertia is 1 kg·m2. • The angular momentum is 1 kg·m2·s−1. • The kinetic energy is 1 joule. • The circumference of the orbit is 2π (~6.283) metres. • The period of the motion is 2π seconds per turn. • The frequency is (2π)−1 hertz. In polar coordinates See also: Velocity § Polar coordinates During circular motion, the body moves on a curve that can be described in the polar coordinate system as a fixed distance R from the center of the orbit taken as the origin, oriented at an angle θ(t) from some reference direction. See Figure 4. The displacement vector $\mathbf {r} $ is the radial vector from the origin to the particle location: $\mathbf {r} (t)=R{\hat {\mathbf {u} }}_{R}(t)\,,$ where ${\hat {\mathbf {u} }}_{R}(t)$ is the unit vector parallel to the radius vector at time t and pointing away from the origin. It is convenient to introduce the unit vector orthogonal to ${\hat {\mathbf {u} }}_{R}(t)$ as well, namely ${\hat {\mathbf {u} }}_{\theta }(t)$. It is customary to orient ${\hat {\mathbf {u} }}_{\theta }(t)$ to point in the direction of travel along the orbit. The velocity is the time derivative of the displacement: $\mathbf {v} (t)={\frac {d}{dt}}\mathbf {r} (t)={\frac {dR}{dt}}{\hat {\mathbf {u} }}_{R}(t)+R{\frac {d{\hat {\mathbf {u} }}_{R}}{dt}}\,.$ Because the radius of the circle is constant, the radial component of the velocity is zero. The unit vector ${\hat {\mathbf {u} }}_{R}(t)$ has a time-invariant magnitude of unity, so as time varies its tip always lies on a circle of unit radius, with an angle θ the same as the angle of $\mathbf {r} (t)$. If the particle displacement rotates through an angle dθ in time dt, so does ${\hat {\mathbf {u} }}_{R}(t)$, describing an arc on the unit circle of magnitude dθ. See the unit circle at the left of Figure 4. Hence: ${\frac {d{\hat {\mathbf {u} }}_{R}}{dt}}={\frac {d\theta }{dt}}{\hat {\mathbf {u} }}_{\theta }(t)\,,$ where the direction of the change must be perpendicular to ${\hat {\mathbf {u} }}_{R}(t)$ (or, in other words, along ${\hat {\mathbf {u} }}_{\theta }(t)$) because any change $d{\hat {\mathbf {u} }}_{R}(t)$ in the direction of ${\hat {\mathbf {u} }}_{R}(t)$ would change the size of ${\hat {\mathbf {u} }}_{R}(t)$. The sign is positive because an increase in dθ implies the object and ${\hat {\mathbf {u} }}_{R}(t)$ have moved in the direction of ${\hat {\mathbf {u} }}_{\theta }(t)$. Hence the velocity becomes: $\mathbf {v} (t)={\frac {d}{dt}}\mathbf {r} (t)=R{\frac {d{\hat {\mathbf {u} }}_{R}}{dt}}=R{\frac {d\theta }{dt}}{\hat {\mathbf {u} }}_{\theta }(t)=R\omega {\hat {\mathbf {u} }}_{\theta }(t)\,.$ The acceleration of the body can also be broken into radial and tangential components. The acceleration is the time derivative of the velocity: ${\begin{aligned}\mathbf {a} (t)&={\frac {d}{dt}}\mathbf {v} (t)={\frac {d}{dt}}\left(R\omega {\hat {\mathbf {u} }}_{\theta }(t)\right)\\&=R\left({\frac {d\omega }{dt}}{\hat {\mathbf {u} }}_{\theta }(t)+\omega {\frac {d{\hat {\mathbf {u} }}_{\theta }}{dt}}\right)\,.\end{aligned}}$ The time derivative of ${\hat {\mathbf {u} }}_{\theta }(t)$ is found the same way as for ${\hat {\mathbf {u} }}_{R}(t)$. Again, ${\hat {\mathbf {u} }}_{\theta }(t)$ is a unit vector and its tip traces a unit circle with an angle that is π/2 + θ. Hence, an increase in angle dθ by $\mathbf {r} (t)$ implies ${\hat {\mathbf {u} }}_{\theta }(t)$ traces an arc of magnitude dθ, and as ${\hat {\mathbf {u} }}_{\theta }(t)$ is orthogonal to ${\hat {\mathbf {u} }}_{R}(t)$, we have: ${\frac {d{\hat {\mathbf {u} }}_{\theta }}{dt}}=-{\frac {d\theta }{dt}}{\hat {\mathbf {u} }}_{R}(t)=-\omega {\hat {\mathbf {u} }}_{R}(t)\,,$ where a negative sign is necessary to keep ${\hat {\mathbf {u} }}_{\theta }(t)$ orthogonal to ${\hat {\mathbf {u} }}_{R}(t)$. (Otherwise, the angle between ${\hat {\mathbf {u} }}_{\theta }(t)$ and ${\hat {\mathbf {u} }}_{R}(t)$ would decrease with an increase in dθ.) See the unit circle at the left of Figure 4. Consequently, the acceleration is: ${\begin{aligned}\mathbf {a} (t)&=R\left({\frac {d\omega }{dt}}{\hat {\mathbf {u} }}_{\theta }(t)+\omega {\frac {d{\hat {\mathbf {u} }}_{\theta }}{dt}}\right)\\&=R{\frac {d\omega }{dt}}{\hat {\mathbf {u} }}_{\theta }(t)-\omega ^{2}R{\hat {\mathbf {u} }}_{R}(t)\,.\end{aligned}}$ The centripetal acceleration is the radial component, which is directed radially inward: $\mathbf {a} _{R}(t)=-\omega ^{2}R{\hat {\mathbf {u} }}_{R}(t)\,,$ while the tangential component changes the magnitude of the velocity: $\mathbf {a} _{\theta }(t)=R{\frac {d\omega }{dt}}{\hat {\mathbf {u} }}_{\theta }(t)={\frac {dR\omega }{dt}}{\hat {\mathbf {u} }}_{\theta }(t)={\frac {d\left\|\mathbf {v} (t)\right\|}{dt}}{\hat {\mathbf {u} }}_{\theta }(t)\,.$ Using complex numbers Circular motion can be described using complex numbers. Let the x axis be the real axis and the $y$ axis be the imaginary axis. The position of the body can then be given as $z$, a complex "vector": $z=x+iy=R\left(\cos[\theta (t)]+i\sin[\theta (t)]\right)=Re^{i\theta (t)}\,,$ where i is the imaginary unit, and $\theta (t)$ is the argument of the complex number as a function of time, t. Since the radius is constant: ${\dot {R}}={\ddot {R}}=0\,,$ where a dot indicates differentiation in respect of time. With this notation, the velocity becomes: $v={\dot {z}}={\frac {d}{dt}}\left(Re^{i\theta [t]}\right)=R{\frac {d}{dt}}\left(e^{i\theta [t]}\right)=Re^{i\theta (t)}{\frac {d}{dt}}\left(i\theta [t]\right)=iR{\dot {\theta }}(t)e^{i\theta (t)}=i\omega Re^{i\theta (t)}=i\omega z$ and the acceleration becomes: ${\begin{aligned}a&={\dot {v}}=i{\dot {\omega }}z+i\omega {\dot {z}}=\left(i{\dot {\omega }}-\omega ^{2}\right)z\\&=\left(i{\dot {\omega }}-\omega ^{2}\right)Re^{i\theta (t)}\\&=-\omega ^{2}Re^{i\theta (t)}+{\dot {\omega }}e^{i{\frac {\pi }{2}}}Re^{i\theta (t)}\,.\end{aligned}}$ The first term is opposite in direction to the displacement vector and the second is perpendicular to it, just like the earlier results shown before. Velocity Figure 1 illustrates velocity and acceleration vectors for uniform motion at four different points in the orbit. Because the velocity v is tangent to the circular path, no two velocities point in the same direction. Although the object has a constant speed, its direction is always changing. This change in velocity is caused by an acceleration a, whose magnitude is (like that of the velocity) held constant, but whose direction also is always changing. The acceleration points radially inwards (centripetally) and is perpendicular to the velocity. This acceleration is known as centripetal acceleration. For a path of radius r, when an angle θ is swept out, the distance traveled on the periphery of the orbit is s = rθ. Therefore, the speed of travel around the orbit is $v=r{\frac {d\theta }{dt}}=r\omega ,$ where the angular rate of rotation is ω. (By rearrangement, ω = v/r.) Thus, v is a constant, and the velocity vector v also rotates with constant magnitude v, at the same angular rate ω. Relativistic circular motion In this case, the three-acceleration vector is perpendicular to the three-velocity vector, $\mathbf {u} \cdot \mathbf {a} =0.$ and the square of proper acceleration, expressed as a scalar invariant, the same in all reference frames, $\alpha ^{2}=\gamma ^{4}a^{2}+\gamma ^{6}\left(\mathbf {u} \cdot \mathbf {a} \right)^{2},$ becomes the expression for circular motion, $\alpha ^{2}=\gamma ^{4}a^{2}.$ or, taking the positive square root and using the three-acceleration, we arrive at the proper acceleration for circular motion: $\alpha =\gamma ^{2}{\frac {v^{2}}{r}}.$ Acceleration Main article: Acceleration The left-hand circle in Figure 2 is the orbit showing the velocity vectors at two adjacent times. On the right, these two velocities are moved so their tails coincide. Because speed is constant, the velocity vectors on the right sweep out a circle as time advances. For a swept angle dθ = ω dt the change in v is a vector at right angles to v and of magnitude v dθ, which in turn means that the magnitude of the acceleration is given by $a_{c}=v{\frac {d\theta }{dt}}=v\omega ={\frac {v^{2}}{r}}$ Centripetal acceleration for some values of radius and magnitude of velocity \|v\| r 1 m/s 3.6 km/h 2.2 mph 2 m/s 7.2 km/h 4.5 mph 5 m/s 18 km/h 11 mph 10 m/s 36 km/h 22 mph 20 m/s 72 km/h 45 mph 50 m/s 180 km/h 110 mph 100 m/s 360 km/h 220 mph Slow walk Bicycle City car Aerobatics 10 cm 3.9 in Laboratory centrifuge 10 m/s2 1.0 g 40 m/s2 4.1 g 250 m/s2 25 g 1.0 km/s2 100 g 4.0 km/s2 410 g 25 km/s2 2500 g 100 km/s2 10000 g 20 cm 7.9 in 5.0 m/s2 0.51 g 20 m/s2 2.0 g 130 m/s2 13 g 500 m/s2 51 g 2.0 km/s2 200 g 13 km/s2 1300 g 50 km/s2 5100 g 50 cm 1.6 ft 2.0 m/s2 0.20 g 8.0 m/s2 0.82 g 50 m/s2 5.1 g 200 m/s2 20 g 800 m/s2 82 g 5.0 km/s2 510 g 20 km/s2 2000 g 1 m 3.3 ft Playground carousel 1.0 m/s2 0.10 g 4.0 m/s2 0.41 g 25 m/s2 2.5 g 100 m/s2 10 g 400 m/s2 41 g 2.5 km/s2 250 g 10 km/s2 1000 g 2 m 6.6 ft 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 13 m/s2 1.3 g 50 m/s2 5.1 g 200 m/s2 20 g 1.3 km/s2 130 g 5.0 km/s2 510 g 5 m 16 ft 200 mm/s2 0.020 g 800 mm/s2 0.082 g 5.0 m/s2 0.51 g 20 m/s2 2.0 g 80 m/s2 8.2 g 500 m/s2 51 g 2.0 km/s2 200 g 10 m 33 ft Roller-coaster vertical loop 100 mm/s2 0.010 g 400 mm/s2 0.041 g 2.5 m/s2 0.25 g 10 m/s2 1.0 g 40 m/s2 4.1 g 250 m/s2 25 g 1.0 km/s2 100 g 20 m 66 ft 50 mm/s2 0.0051 g 200 mm/s2 0.020 g 1.3 m/s2 0.13 g 5.0 m/s2 0.51 g 20 m/s2 2 g 130 m/s2 13 g 500 m/s2 51 g 50 m 160 ft 20 mm/s2 0.0020 g 80 mm/s2 0.0082 g 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 8.0 m/s2 0.82 g 50 m/s2 5.1 g 200 m/s2 20 g 100 m 330 ft Freeway on-ramp 10 mm/s2 0.0010 g 40 mm/s2 0.0041 g 250 mm/s2 0.025 g 1.0 m/s2 0.10 g 4.0 m/s2 0.41 g 25 m/s2 2.5 g 100 m/s2 10 g 200 m 660 ft 5.0 mm/s2 0.00051 g 20 mm/s2 0.0020 g 130 m/s2 0.013 g 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 13 m/s2 1.3 g 50 m/s2 5.1 g 500 m 1600 ft 2.0 mm/s2 0.00020 g 8.0 mm/s2 0.00082 g 50 mm/s2 0.0051 g 200 mm/s2 0.020 g 800 mm/s2 0.082 g 5.0 m/s2 0.51 g 20 m/s2 2.0 g 1 km 3300 ft High-speed railway 1.0 mm/s2 0.00010 g 4.0 mm/s2 0.00041 g 25 mm/s2 0.0025 g 100 mm/s2 0.010 g 400 mm/s2 0.041 g 2.5 m/s2 0.25 g 10 m/s2 1.0 g Non-uniform In a non-uniform circular motion, an object is moving in a circular path with a varying speed. Since the speed is changing, there is tangential acceleration in addition to normal acceleration. In a non-uniform circular motion, the net acceleration (a) is along the direction of Δv, which is directed inside the circle but does not pass through its center (see figure). The net acceleration may be resolved into two components: tangential acceleration and normal acceleration also known as the centripetal or radial acceleration. Unlike tangential acceleration, centripetal acceleration is present in both uniform and non-uniform circular motion. In a non-uniform circular motion, normal force does not always point in the opposite direction of weight. Here is an example with an object traveling in a straight path then looping a loop back into a straight path again. This diagram shows the normal force pointing in other directions rather than opposite to the weight force. The normal force is actually the sum of the radial and tangential forces. The component of weight force is responsible for the tangential force here (We have neglected frictional force). The radial force (centripetal force) is due to the change in the direction of velocity as discussed earlier. In a non-uniform circular motion, normal force and weight may point in the same direction. Both forces can point down, yet the object will remain in a circular path without falling straight down. First, let's see why normal force can point down in the first place. In the first diagram, let's say the object is a person sitting inside a plane, the two forces point down only when it reaches the top of the circle. The reason for this is that the normal force is the sum of the tangential force and centripetal force. The tangential force is zero at the top (as no work is performed when the motion is perpendicular to the direction of force applied. Here weight force is perpendicular to the direction of motion of the object at the top of the circle) and centripetal force points down, thus normal force will point down as well. From a logical standpoint, a person who is travelling in the plane will be upside down at the top of the circle. At that moment, the person's seat is actually pushing down on the person, which is the normal force. The reason why the object does not fall down when subjected to only downward forces is a simple one. Think about what keeps an object up after it is thrown. Once an object is thrown into the air, there is only the downward force of Earth's gravity that acts on the object. That does not mean that once an object is thrown in the air, it will fall instantly. What keeps that object up in the air is its velocity. The first of Newton's laws of motion states that an object's inertia keeps it in motion, and since the object in the air has a velocity, it will tend to keep moving in that direction. A varying angular speed for an object moving in a circular path can also be achieved if the rotating body does not have a homogeneous mass distribution. For inhomogeneous objects, it is necessary to approach the problem as in.[2] Applications Solving applications dealing with non-uniform circular motion involves force analysis. With a uniform circular motion, the only force acting upon an object traveling in a circle is the centripetal force. In a non-uniform circular motion, there are additional forces acting on the object due to a non-zero tangential acceleration. Although there are additional forces acting upon the object, the sum of all the forces acting on the object will have to be equal to the centripetal force. ${\begin{aligned}F_{\text{net}}&=ma\\&=ma_{r}\\&={\frac {mv^{2}}{r}}\\&=F_{c}\end{aligned}}$ Radial acceleration is used when calculating the total force. Tangential acceleration is not used in calculating total force because it is not responsible for keeping the object in a circular path. The only acceleration responsible for keeping an object moving in a circle is the radial acceleration. Since the sum of all forces is the centripetal force, drawing centripetal force into a free body diagram is not necessary and usually not recommended. Using $F_{\text{net}}=F_{c}$, we can draw free body diagrams to list all the forces acting on an object and then set it equal to $F_{c}$. Afterward, we can solve for whatever is unknown (this can be mass, velocity, radius of curvature, coefficient of friction, normal force, etc.). For example, the visual above showing an object at the top of a semicircle would be expressed as $F_{c}=n+mg$. In a uniform circular motion, the total acceleration of an object in a circular path is equal to the radial acceleration. Due to the presence of tangential acceleration in a non uniform circular motion, that does not hold true any more. To find the total acceleration of an object in a non uniform circular, find the vector sum of the tangential acceleration and the radial acceleration. ${\sqrt {a_{r}^{2}+a_{t}^{2}}}=a$ Radial acceleration is still equal to $ {\frac {v^{2}}{r}}$. Tangential acceleration is simply the derivative of the speed at any given point: $ a_{t}={\frac {dv}{dt}}$. This root sum of squares of separate radial and tangential accelerations is only correct for circular motion; for general motion within a plane with polar coordinates $(r,\theta )$, the Coriolis term $ a_{c}=2\left({\frac {dr}{dt}}\right)\left({\frac {d\theta }{dt}}\right)$ should be added to $a_{t}$, whereas radial acceleration then becomes $ a_{r}={\frac {-v^{2}}{r}}+{\frac {d^{2}r}{dt^{2}}}$. See also • Angular momentum • Equations of motion for circular motion • Time derivative § Example: circular motion • Fictitious force • Geostationary orbit • Geosynchronous orbit • Pendulum (mathematics) • Reactive centrifugal force • Reciprocating motion • Simple harmonic motion § Uniform circular motion • Sling (weapon) References 1. Knudsen, Jens M.; Hjorth, Poul G. (2000). Elements of Newtonian mechanics: including nonlinear dynamics (3 ed.). Springer. p. 96. ISBN 3-540-67652-X. 2. Gomez, R W; Hernandez-Gomez, J J; Marquina, V (25 July 2012). "A jumping cylinder on an inclined plane". Eur. J. Phys. IOP. 33 (5): 1359–1365. arXiv:1204.0600. Bibcode:2012EJPh...33.1359G. doi:10.1088/0143-0807/33/5/1359. S2CID 55442794. Retrieved 25 April 2016. External links • Physclips: Mechanics with animations and video clips from the University of New South Wales • Circular Motion – a chapter from an online textbook • Circular Motion Lecture – a video lecture on CM • – an online textbook with different analysis for circular motion Classical mechanics SI units Linear/translational quantities Angular/rotational quantities Dimensions 1 L L2 Dimensions 1 θ θ2 T time: t s absement: A m s T time: t s 1 distance: d, position: r, s, x, displacement m area: A m2 1 angle: θ, angular displacement: θ rad solid angle: Ω rad2, sr T−1 frequency: f s−1, Hz speed: v, velocity: v m s−1 kinematic viscosity: ν, specific angular momentum: h m2 s−1 T−1 frequency: f s−1, Hz angular speed: ω, angular velocity: ω rad s−1 T−2 acceleration: a m s−2 T−2 angular acceleration: α rad s−2 T−3 jerk: j m s−3 T−3 angular jerk: ζ rad s−3 M mass: m kg weighted position: M ⟨x⟩ = ∑ m x ML2 moment of inertia: I kg m2 MT−1 Mass flow rate: ${\dot {m}}$ kg s−1 momentum: p, impulse: J kg m s−1, N s action: 𝒮, actergy: ℵ kg m2 s−1, J s ML2T−1 angular momentum: L, angular impulse: ΔL kg m2 s−1 action: 𝒮, actergy: ℵ kg m2 s−1, J s MT−2 force: F, weight: Fg kg m s−2, N energy: E, work: W, Lagrangian: L kg m2 s−2, J ML2T−2 torque: τ, moment: M kg m2 s−2, N m energy: E, work: W, Lagrangian: L kg m2 s−2, J MT−3 yank: Y kg m s−3, N s−1 power: P kg m2 s−3, W ML2T−3 rotatum: P kg m2 s−3, N m s−1 power: P kg m2 s−3, W	Wikipedia
Radial set In mathematics, a subset $A\subseteq X$ of a linear space $X$ is radial at a given point $a_{0}\in A$ if for every $x\in X$ there exists a real $t_{x}>0$ such that for every $t\in [0,t_{x}],$ $a_{0}+tx\in A.$[1] Geometrically, this means $A$ is radial at $a_{0}$ if for every $x\in X,$ there is some (non-degenerate) line segment (depend on $x$) emanating from $a_{0}$ in the direction of $x$ that lies entirely in $A.$ Every radial set is a star domain although not conversely. Relation to the algebraic interior The points at which a set is radial are called internal points.[2][3] The set of all points at which $A\subseteq X$ is radial is equal to the algebraic interior.[1][4] Relation to absorbing sets Every absorbing subset is radial at the origin $a_{0}=0,$ and if the vector space is real then the converse also holds. That is, a subset of a real vector space is absorbing if and only if it is radial at the origin. Some authors use the term radial as a synonym for absorbing.[5] See also • Absorbing set – Set that can be "inflated" to reach any point • Algebraic interior – Generalization of topological interior • Minkowski functional – Function made from a set • Star domain – Property of point sets in Euclidean spaces References 1. Jaschke, Stefan; Küchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ($\mu ,\rho $)-Portfolio Optimization". {{cite journal}}: Cite journal requires \|journal= (help) 2. Aliprantis & Border 2006, p. 199–200. 3. John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (PDF). Retrieved November 14, 2012. 4. Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6. 5. Schaefer & Wolff 1999, p. 11. • Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (Third ed.). Berlin: Springer Science & Business Media. ISBN 978-3-540-29587-7. OCLC 262692874. • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. • Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons Topological vector spaces (TVSs) Basic concepts • Banach space • Completeness • Continuous linear operator • Linear functional • Fréchet space • Linear map • Locally convex space • Metrizability • Operator topologies • Topological vector space • Vector space Main results • Anderson–Kadec • Banach–Alaoglu • Closed graph theorem • F. Riesz's • Hahn–Banach (hyperplane separation • Vector-valued Hahn–Banach) • Open mapping (Banach–Schauder) • Bounded inverse • Uniform boundedness (Banach–Steinhaus) Maps • Bilinear operator • form • Linear map • Almost open • Bounded • Continuous • Closed • Compact • Densely defined • Discontinuous • Topological homomorphism • Functional • Linear • Bilinear • Sesquilinear • Norm • Seminorm • Sublinear function • Transpose Types of sets • Absolutely convex/disk • Absorbing/Radial • Affine • Balanced/Circled • Banach disks • Bounding points • Bounded • Complemented subspace • Convex • Convex cone (subset) • Linear cone (subset) • Extreme point • Pre-compact/Totally bounded • Prevalent/Shy • Radial • Radially convex/Star-shaped • Symmetric Set operations • Affine hull • (Relative) Algebraic interior (core) • Convex hull • Linear span • Minkowski addition • Polar • (Quasi) Relative interior Types of TVSs • Asplund • B-complete/Ptak • Banach • (Countably) Barrelled • BK-space • (Ultra-) Bornological • Brauner • Complete • Convenient • (DF)-space • Distinguished • F-space • FK-AK space • FK-space • Fréchet • tame Fréchet • Grothendieck • Hilbert • Infrabarreled • Interpolation space • K-space • LB-space • LF-space • Locally convex space • Mackey • (Pseudo)Metrizable • Montel • Quasibarrelled • Quasi-complete • Quasinormed • (Polynomially • Semi-) Reflexive • Riesz • Schwartz • Semi-complete • Smith • Stereotype • (B • Strictly • Uniformly) convex • (Quasi-) Ultrabarrelled • Uniformly smooth • Webbed • With the approximation property • Mathematics portal • Category • Commons Convex analysis and variational analysis Basic concepts • Convex combination • Convex function • Convex set Topics (list) • Choquet theory • Convex geometry • Convex metric space • Convex optimization • Duality • Lagrange multiplier • Legendre transformation • Locally convex topological vector space • Simplex Maps • Convex conjugate • Concave • (Closed • K- • Logarithmically • Proper • Pseudo- • Quasi-) Convex function • Invex function • Legendre transformation • Semi-continuity • Subderivative Main results (list) • Carathéodory's theorem • Ekeland's variational principle • Fenchel–Moreau theorem • Fenchel-Young inequality • Jensen's inequality • Hermite–Hadamard inequality • Krein–Milman theorem • Mazur's lemma • Shapley–Folkman lemma • Robinson-Ursescu • Simons • Ursescu Sets • Convex hull • (Orthogonally, Pseudo-) Convex set • Effective domain • Epigraph • Hypograph • John ellipsoid • Lens • Radial set/Algebraic interior • Zonotope Series • Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Duality • Dual system • Duality gap • Strong duality • Weak duality Applications and related • Convexity in economics	Wikipedia
Radially unbounded function In mathematics, a radially unbounded function is a function $f:\mathbb {R} ^{n}\rightarrow \mathbb {R} $ for which [1] $\\|x\\|\to \infty \Rightarrow f(x)\to \infty .$ Or equivalently, $\forall c>0:\exists r>0:\forall x\in \mathbb {R} ^{n}:[\Vert x\Vert >r\Rightarrow f(x)>c]$ Such functions are applied in control theory and required in optimization for determination of compact spaces. Notice that the norm used in the definition can be any norm defined on $\mathbb {R} ^{n}$, and that the behavior of the function along the axes does not necessarily reveal that it is radially unbounded or not; i.e. to be radially unbounded the condition must be verified along any path that results in: $\\|x\\|\to \infty $ For example, the functions ${\begin{aligned}f_{1}(x)&=(x_{1}-x_{2})^{2}\\f_{2}(x)&=(x_{1}^{2}+x_{2}^{2})/(1+x_{1}^{2}+x_{2}^{2})+(x_{1}-x_{2})^{2}\end{aligned}}$ are not radially unbounded since along the line $x_{1}=x_{2}$, the condition is not verified even though the second function is globally positive definite. References 1. Terrell, William J. (2009), Stability and stabilization, Princeton University Press, ISBN 978-0-691-13444-4, MR 2482799	Wikipedia
Radical of an integer In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product: $\displaystyle \mathrm {rad} (n)=\prod _{\scriptstyle p\mid n \atop p{\text{ prime}}}p$ The radical plays a central role in the statement of the abc conjecture.[1] Examples Radical numbers for the first few positive integers are 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, ... (sequence A007947 in the OEIS). For example, $504=2^{3}\cdot 3^{2}\cdot 7$ and therefore $\operatorname {rad} (504)=2\cdot 3\cdot 7=42$ Properties The function $\mathrm {rad} $ is multiplicative (but not completely multiplicative). The radical of any integer $n$ is the largest square-free divisor of $n$ and so also described as the square-free kernel of $n$.[2] There is no known polynomial-time algorithm for computing the square-free part of an integer.[3] The definition is generalized to the largest $t$-free divisor of $n$, $\mathrm {rad} _{t}$, which are multiplicative functions which act on prime powers as $\mathrm {rad} _{t}(p^{e})=p^{\mathrm {min} (e,t-1)}$ The cases $t=3$ and $t=4$ are tabulated in OEIS: A007948 and OEIS: A058035. The notion of the radical occurs in the abc conjecture, which states that, for any $\varepsilon >0$, there exists a finite $K_{\varepsilon }$ such that, for all triples of coprime positive integers $a$, $b$, and $c$ satisfying $a+b=c$,[1] $c<K_{\varepsilon }\,\operatorname {rad} (abc)^{1+\varepsilon }$ For any integer $n$, the nilpotent elements of the finite ring $\mathbb {Z} /n\mathbb {Z} $ are all of the multiples of $\operatorname {rad} (n)$. References 1. Gowers, Timothy (2008). "V.1 The ABC Conjecture". The Princeton Companion to Mathematics. Princeton University Press. p. 681. 2. Sloane, N. J. A. (ed.). "Sequence A007947". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 3. Adleman, Leonard M.; McCurley, Kevin S. "Open Problems in Number Theoretic Complexity, II". Algorithmic Number Theory: First International Symposium, ANTS-I Ithaca, NY, USA, May 6–9, 1994, Proceedings. Lecture Notes in Computer Science. Vol. 877. Springer. pp. 291–322. CiteSeerX 10.1.1.48.4877. doi:10.1007/3-540-58691-1_70. MR 1322733.	Wikipedia
Radical of a Lie algebra In the mathematical field of Lie theory, the radical of a Lie algebra ${\mathfrak {g}}$ is the largest solvable ideal of ${\mathfrak {g}}.$[1] The radical, denoted by ${\rm {rad}}({\mathfrak {g}})$, fits into the exact sequence $0\to {\rm {rad}}({\mathfrak {g}})\to {\mathfrak {g}}\to {\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}})\to 0$. where ${\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}})$ is semisimple. When the ground field has characteristic zero and ${\mathfrak {g}}$ has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of ${\mathfrak {g}}$ that is isomorphic to the semisimple quotient ${\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}})$ via the restriction of the quotient map ${\mathfrak {g}}\to {\mathfrak {g}}/{\rm {rad}}({\mathfrak {g}}).$ A similar notion is a Borel subalgebra, which is a (not necessarily unique) maximal solvable subalgebra. Definition Let $k$ be a field and let ${\mathfrak {g}}$ be a finite-dimensional Lie algebra over $k$. There exists a unique maximal solvable ideal, called the radical, for the following reason. Firstly let ${\mathfrak {a}}$ and ${\mathfrak {b}}$ be two solvable ideals of ${\mathfrak {g}}$. Then ${\mathfrak {a}}+{\mathfrak {b}}$ is again an ideal of ${\mathfrak {g}}$, and it is solvable because it is an extension of $({\mathfrak {a}}+{\mathfrak {b}})/{\mathfrak {a}}\simeq {\mathfrak {b}}/({\mathfrak {a}}\cap {\mathfrak {b}})$ by ${\mathfrak {a}}$. Now consider the sum of all the solvable ideals of ${\mathfrak {g}}$. It is nonempty since $\{0\}$ is a solvable ideal, and it is a solvable ideal by the sum property just derived. Clearly it is the unique maximal solvable ideal. Related concepts • A Lie algebra is semisimple if and only if its radical is $0$. • A Lie algebra is reductive if and only if its radical equals its center. See also • Levi decomposition References 1. Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010), Algebras, Rings and Modules: Lie Algebras and Hopf Algebras, Mathematical Surveys and Monographs, vol. 168, Providence, RI: American Mathematical Society, p. 15, doi:10.1090/surv/168, ISBN 978-0-8218-5262-0, MR 2724822.	Wikipedia
Radical of a module In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc(M) of M. Definition Let R be a ring and M a left R-module. A submodule N of M is called maximal or cosimple if the quotient M/N is a simple module. The radical of the module M is the intersection of all maximal submodules of M, $\mathrm {rad} (M)=\bigcap \,\{N\mid N{\mbox{ is a maximal submodule of }}M\}$ Equivalently, $\mathrm {rad} (M)=\sum \,\{S\mid S{\mbox{ is a superfluous submodule of }}M\}$ These definitions have direct dual analogues for soc(M). Properties • In addition to the fact rad(M) is the sum of superfluous submodules, in a Noetherian module rad(M) itself is a superfluous submodule. • A ring for which rad(M) = {0} for every right R-module M is called a right V-ring. • For any module M, rad(M/rad(M)) is zero. • M is a finitely generated module if and only if the cosocle M/rad(M) is finitely generated and rad(M) is a superfluous submodule of M. See also • Socle (mathematics) • Jacobson radical References • Alperin, J.L.; Rowen B. Bell (1995). Groups and representations. Springer-Verlag. p. 136. ISBN 0-387-94526-1. • Anderson, Frank Wylie; Kent R. Fuller (1992). Rings and Categories of Modules. Springer-Verlag. ISBN 978-0-387-97845-1.	Wikipedia
Radical of an algebraic group The radical of an algebraic group is the identity component of its maximal normal solvable subgroup. For example, the radical of the general linear group $\operatorname {GL} _{n}(K)$ (for a field K) is the subgroup consisting of scalar matrices, i.e. matrices $(a_{ij})$ with $a_{11}=\dots =a_{nn}$ and $a_{ij}=0$ for $i\neq j$. An algebraic group is called semisimple if its radical is trivial, i.e., consists of the identity element only. The group $\operatorname {SL} _{n}(K)$ is semi-simple, for example. The subgroup of unipotent elements in the radical is called the unipotent radical, it serves to define reductive groups. See also • Reductive group • Unipotent group References • "Radical of a group", Encyclopaedia of Mathematics	Wikipedia
Radical axis In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of the two circles. In detail: For two circles c1, c2 with centers M1, M2 and radii r1, r2 the powers of a point P with respect to the circles are $\Pi _{1}(P)=\|PM_{1}\|^{2}-r_{1}^{2},\qquad \Pi _{2}(P)=\|PM_{2}\|^{2}-r_{2}^{2}.$ Point P belongs to the radical axis, if $\Pi _{1}(P)=\Pi _{2}(P).$ If the circles have two points in common, the radical axis is the common secant line of the circles. If point P is outside the circles, P has equal tangential distance to both the circles. If the radii are equal, the radical axis is the line segment bisector of M1, M2. In any case the radical axis is a line perpendicular to ${\overline {M_{1}M_{2}}}.$ On notations The notation radical axis was used by the French mathematician M. Chasles as axe radical. [1] J.V. Poncelet used chorde ideale. [2] J. Plücker introduced the term Chordale.[3] J. Steiner called the radical axis line of equal powers (German: Linie der gleichen Potenzen) which led to power line (Potenzgerade).[4] Properties Geometric shape and its position Let ${\vec {x}},{\vec {m}}_{1},{\vec {m}}_{2}$ be the position vectors of the points $P,M_{1},M_{2}$. Then the defining equation of the radical line can be written as: $({\vec {x}}-{\vec {m}}_{1})^{2}-r_{1}^{2}=({\vec {x}}-{\vec {m}}_{2})^{2}-r_{2}^{2}\quad \leftrightarrow \quad 2{\vec {x}}\cdot ({\vec {m}}_{2}-{\vec {m}}_{1})+{\vec {m}}_{1}^{2}-{\vec {m}}_{2}^{2}+r_{2}^{2}-r_{1}^{2}=0$ From the right equation one gets • The pointset of the radical axis is indeed a line and is perpendicular to the line through the circle centers. (${\vec {m}}_{2}-{\vec {m}}_{1}$ is a normal vector to the radical axis !) Dividing the equation by $2\|{\vec {m}}_{2}-{\vec {m}}_{1}\|$, one gets the Hessian normal form. Inserting the position vectors of the centers yields the distances of the centers to the radical axis: $d_{1}={\frac {d^{2}+{r_{1}}^{2}-{r_{2}}^{2}}{2d}}\ ,\qquad d_{2}={\frac {d^{2}+{r_{2}}^{2}-{r_{1}}^{2}}{2d}}$, with $d=\|M_{1}M_{2}\|$. ($d_{i}$ may be negative if $L$ is not between $M_{1},M_{2}$.) If the circles are intersecting at two points, the radical line runs through the common points. If they only touch each other, the radical line is the common tangent line. Special positions • The radical axis of two intersecting circles is their common secant line. The radical axis of two touching circles is their common tangent. The radical axis of two non intersecting circles is the common secant of two convenient equipower circles (see below). Orthogonal circles • For a point $P$ outside a circle $c_{i}$ and the two tangent points $S_{i},T_{i}$ the equation $\|PS_{i}\|^{2}=\|PT_{i}\|^{2}=\Pi _{i}(P)$ holds and $S_{i},T_{i}$ lie on the circle $c_{o}$ with center $P$ and radius ${\sqrt {\Pi _{i}(P)}}$. Circle $c_{o}$ intersects $c_{i}$ orthogonal. Hence: If $P$ is a point of the radical axis, then the four points $S_{1},T_{1},S_{2},T_{2}$ lie on circle $c_{o}$, which intersects the given circles $c_{1},c_{2}$ orthogonally. • The radical axis consists of all centers of circles, which intersect the given circles orthogonally. System of orthogonal circles The method described in the previous section for the construction of a pencil of circles, which intersect two given circles orthogonally, can be extended to the construction of two orthogonally intersecting systems of circles:[5][6] Let $c_{1},c_{2}$ be two apart lying circles (as in the previous section), $M_{1},M_{2},r_{1},r_{2}$ their centers and radii and $g_{12}$ their radical axis. Now, all circles will be determined with centers on line ${\overline {M_{1}M_{2}}}$, which have together with $c_{1}$ line $g_{12}$ as radical axis, too. If $\gamma _{2}$ is such a circle, whose center has distance $\delta $ to the center $M_{1}$ and radius $\rho _{2}$. From the result in the previous section one gets the equation $d_{1}={\frac {\delta ^{2}+r_{1}^{2}-\rho _{2}^{2}}{2\delta }}\quad $, where $d_{1}>r_{1}$ are fixed. With $\delta _{2}=\delta -d_{1}$ the equation can be rewritten as: $\delta _{2}^{2}=d_{1}^{2}-r_{1}^{2}+\rho _{2}^{2}$. If radius $\rho _{2}$ is given, from this equation one finds the distance $\delta _{2}$ to the (fixed) radical axis of the new center. In the diagram the color of the new circles is purple. Any green circle (see diagram) has its center on the radical axis and intersects the circles $c_{1},c_{2}$ orthogonally and hence all new circles (purple), too. Choosing the (red) radical axis as y-axis and line ${\overline {M_{1}M_{2}}}$ as x-axis, the two pencils of circles have the equations: purple: $\ \ \ (x-\delta _{2})^{2}+y^{2}=\delta _{2}^{2}+r_{1}^{2}-d_{1}^{2}$ green: $\ x^{2}+(y-y_{g})^{2}=y_{g}^{2}+d_{1}^{2}-r_{1}^{2}\ .$ ($\;(0,y_{g})$ is the center of a green circle.) Properties: a) Any two green circles intersect on the x-axis at the points $P_{1/2}={\big (}\pm {\sqrt {d_{1}^{2}-r_{1}^{2}}},0{\big )}$, the poles of the orthogonal system of circles. That means, the x-axis is the radical line of the green circles. b) The purple circles have no points in common. But, if one considers the real plane as part of the complex plane, then any two purple circles intersect on the y-axis (their common radical axis) at the points $Q_{1/2}={\big (}0,\pm i{\sqrt {d_{1}^{2}-r_{1}^{2}}}{\big )}$. Special cases: a) In case of $d_{1}=r_{1}$ the green circles are touching each other at the origin with the x-axis as common tangent and the purple circles have the y-axis as common tangent. Such a system of circles is called coaxal parabolic circles (see below). b) Shrinking $c_{1}$ to its center $M_{1}$, i. e. $r_{1}=0$, the equations turn into a more simple form and one gets $M_{1}=P_{1}$. Conclusion: a) For any real $w$ the pencil of circles $\;c(\xi ):\;(x-\xi )^{2}+y^{2}-\xi ^{2}-w=0\ :$ :} has the property: The y-axis is the radical axis of $c(\xi _{1}),c(\xi _{2})$. In case of $w>0$ the circles $c(\xi _{1}),c(\xi _{2})$ intersect at points $P_{1/2}=(0,\pm {\sqrt {w}})$. In case of $w<0$ they have no points in common. In case of $w=0$ they touch at $(0,0)$ and the y-axis is their common tangent. b) For any real $w$ the two pencils of circles $c_{1}(\xi ):\;(x-\xi )^{2}+y^{2}-\xi ^{2}-w=0\ ,$ $c_{2}(\eta ):\;x^{2}+(y-\eta )^{2}-\eta ^{2}+w=0\ $ form a system of orthogonal circles. That means: any two circles $c_{1}(\xi ),c_{2}(\eta )$ intersect orthogonally. c) From the equations in b), one gets a coordinate free representation: For the given points $P_{1},P_{2}$, their midpoint $O$ and their line segment bisector $g_{12}$ the two equations $\|XM\|^{2}=\|OM\|^{2}-\|OP_{1}\|^{2}\ ,$ $\|XN\|^{2}=\|ON\|^{2}+\|OP_{1}\|^{2}=\|NP_{1}\|^{2}$ with $M$ on ${\overline {P_{1}P_{2}}}$, but not between $P_{1},P_{2}$, and $N$ on $g_{12}$ describe the orthogonal system of circles uniquely determined by $P_{1},P_{2}$ which are the poles of the system. For $P_{1}=P_{2}=O$ one has to prescribe the axes $a_{1},a_{2}$ of the system, too. The system is parabolic: $\|XM\|^{2}=\|OM\|^{2}\ ,\quad \|XN\|^{2}=\|ON\|^{2}$ with $M$ on $a_{1}$ and $N$ on $a_{2}$. Straightedge and compass construction: A system of orthogonal circles is determined uniquely by its poles $P_{1},P_{2}$: 1. The axes (radical axes) are the lines ${\overline {P_{1}P_{2}}}$ and the Line segment bisector $g_{12}$ of the poles. 2. The circles (green in the diagram) through $P_{1},P_{2}$ have their centers on $g_{12}$. They can be drawn easily. For a point $N$ the radius is $\;r_{N}=\|NP_{1}\|\;$. 3. In order to draw a circle of the second pencil (in diagram blue) with center $M$ on ${\overline {P_{1}P_{2}}}$, one determines the radius $r_{M}$ applying the theorem of Pythagoras: $\;r_{M}^{2}=\|OM\|^{2}-\|OP_{1}\|^{2}\;$ (see diagram). In case of $P_{1}=P_{2}$ the axes have to be chosen additionally. The system is parabolic and can be drawn easily. Coaxal circles Definition and properties: Let $c_{1},c_{2}$ be two circles and $\Pi _{1},\Pi _{2}$ their power functions. Then for any $\lambda \neq 1$ • $\Pi _{1}(x,y)-\lambda \Pi _{2}(x,y)=0$ is the equation of a circle $c(\lambda )$ (see below). Such a system of circles is called coaxal circles generated by the circles $c_{1},c_{2}$. (In case of $\lambda =1$ the equation describes the radical axis of $c_{1},c_{2}$.) [7][8] The power function of $c(\lambda )$ is $\ \Pi (\lambda ,x,y)={\frac {\Pi _{1}(x,y)-\lambda \Pi _{2}(x,y)}{1-\lambda }}$. The normed equation (the coefficients of $x^{2},y^{2}$ are $1$) of $c(\lambda )$ is $\ \Pi (\lambda ,x,y)=0$. A simple calculation shows: • $c(\lambda ),c(\mu ),\ \lambda \neq \mu \ ,$ have the same radical axis as $c_{1},c_{2}$. Allowing $\lambda $ to move to infinity, one recognizes, that $c_{1},c_{2}$ are members of the system of coaxal circles: $c_{1}=c(0),\;c_{2}=c(\infty )$. (E): If $c_{1},c_{2}$ intersect at two points $P_{1},P_{2}$, any circle $c(\lambda )$ contains $P_{1},P_{2}$, too, and line ${\overline {P_{1}P_{2}}}$ is their common radical axis. Such a system is called elliptic. (P): If $c_{1},c_{2}$ are tangent at $P$, any circle is tangent to $c_{1},c_{2}$ at point $P$, too. The common tangent is their common radical axis. Such a system is called parabolic. (H): If $c_{1},c_{2}$ have no point in common, then any pair of the system, too. The radical axis of any pair of circles is the radical axis of $c_{1},c_{2}$. The system is called hyperbolic. In detail: Introducing coordinates such that $c_{1}:(x-d_{1})^{2}+y^{2}=r_{1}^{2}$ $c_{2}:(x-d_{2})^{2}+y^{2}=d_{2}^{2}+r_{1}^{2}-d_{1}^{2}$, then the y-axis is their radical axis (see above). Calculating the power function $\Pi (\lambda ,x,y)$ gives the normed circle equation: $c(\lambda ):\ x^{2}+y^{2}-2{\tfrac {d_{1}-\lambda d_{2}}{1-\lambda }}\;x+d_{1}^{2}-r_{1}^{2}=0\ .$ Completing the square and the substitution $\delta _{2}={\tfrac {d_{1}-\lambda d_{2}}{1-\lambda }}$ (x-coordinate of the center) yields the centered form of the equation $c(\lambda ):\ (x-\delta _{2})^{2}+y^{2}=\delta _{2}^{2}+r_{1}^{2}-d_{1}^{2}$. In case of $r_{1}>d_{1}$ the circles $c_{1},c_{2},c(\lambda )$ have the two points $P_{1}={\big (}0,{\sqrt {r_{1}^{2}-d_{1}^{2}}}{\big )},\quad P_{2}={\big (}0,-{\sqrt {r_{1}^{2}-d_{1}^{2}}}{\big )}$ in common and the system of coaxal circles is elliptic. In case of $r_{1}=d_{1}$ the circles $c_{1},c_{2},c(\lambda )$ have point $P_{0}=(0,0)$ in common and the system is parabolic. In case of $r_{1}<d_{1}$ the circles $c_{1},c_{2},c(\lambda )$ have no point in common and the system is hyperbolic. Alternative equations: 1) In the defining equation of a coaxal system of circles there can be used multiples of the power functions, too. 2) The equation of one of the circles can be replaced by the equation of the desired radical axis. The radical axis can be seen as a circle with an infinitely large radius. For example: $(x-x_{1})^{2}+y^{2}-r_{1}^{2}\ -\ \lambda \;2(x-x_{2})\ =0\ \Leftrightarrow $ $(x-(x_{1}+\lambda ))^{2}+y^{2}=(x_{1}+\lambda )^{2}+r_{1}^{2}-x_{1}^{2}-2\lambda x_{2}$, describes all circles, which have with the first circle the line $x=x_{2}$ as radical axis. 3) In order to express the equal status of the two circles, the following form is often used: $\mu \Pi _{1}(x,y)+\nu \Pi _{2}(x,y)=0\;.$ But in this case the representation of a circle by the parameters $\mu ,\nu $ is not unique. Applications: a) Circle inversions and Möbius transformations preserve angles and generalized circles. Hence orthogonal systems of circles play an essential role with investigations on these mappings. [9][10] b) In electromagnetism coaxal circles appear as field lines. [11] Radical center of three circles, construction of the radical axis • For three circles $c_{1},c_{2},c_{3}$, no two of which are concentric, there are three radical axes $g_{12},g_{23},g_{31}$. If the circle centers do not lie on a line, the radical axes intersect in a common point $R$, the radical center of the three circles. The orthogonal circle centered around $R$ of two circles is orthogonal to the third circle, too (radical circle). Proof: the radical axis $g_{ik}$ contains all points which have equal tangential distance to the circles $c_{i},c_{k}$. The intersection point $R$ of $g_{12}$ and $g_{23}$ has the same tangential distance to all three circles. Hence $R$ is a point of the radical axis $g_{31}$, too. This property allows one to construct the radical axis of two non intersecting circles $c_{1},c_{2}$ with centers $M_{1},M_{2}$: Draw a third circle $c_{3}$ with center not collinear to the given centers that intersects $c_{1},c_{2}$. The radical axes $g_{13},g_{23}$ can be drawn. Their intersection point is the radical center $R$ of the three circles and lies on $g_{12}$. The line through $R$ which is perpendicular to ${\overline {M_{1}M_{2}}}$ is the radical axis $g_{12}$. Additional construction method: All points which have the same power to a given circle $c$ lie on a circle concentric to $c$. Let us call it an equipower circle. This property can be used for an additional construction method of the radical axis of two circles: For two non intersecting circles $c_{1},c_{2}$, there can be drawn two equipower circles $c'_{1},c'_{2}$, which have the same power with respect to $c_{1},c_{2}$ (see diagram). In detail: $\Pi _{1}(P_{1})=\Pi _{2}(P_{2})$. If the power is large enough, the circles $c'_{1},c'_{2}$ have two points in common, which lie on the radical axis $g_{12}$. Relation to bipolar coordinates In general, any two disjoint, non-concentric circles can be aligned with the circles of a system of bipolar coordinates. In that case, the radical axis is simply the $y$-axis of this system of coordinates. Every circle on the axis that passes through the two foci of the coordinate system intersects the two circles orthogonally. A maximal collection of circles, all having centers on a given line and all pairs having the same radical axis, is known as a pencil of coaxal circles. Radical center in trilinear coordinates If the circles are represented in trilinear coordinates in the usual way, then their radical center is conveniently given as a certain determinant. Specifically, let X = x : y : z denote a variable point in the plane of a triangle ABC with sidelengths a = \|BC\|, b = \|CA\|, c = \|AB\|, and represent the circles as follows: (dx + ey + fz)(ax + by + cz) + g(ayz + bzx + cxy) = 0 (hx + iy + jz)(ax + by + cz) + k(ayz + bzx + cxy) = 0 (lx + my + nz)(ax + by + cz) + p(ayz + bzx + cxy) = 0 Then the radical center is the point $\det {\begin{bmatrix}g&k&p\\e&i&m\\f&j&n\end{bmatrix}}:\det {\begin{bmatrix}g&k&p\\f&j&n\\d&h&l\end{bmatrix}}:\det {\begin{bmatrix}g&k&p\\d&h&l\\e&i&m\end{bmatrix}}.$ Radical plane and hyperplane The radical plane of two nonconcentric spheres in three dimensions is defined similarly: it is the locus of points from which tangents to the two spheres have the same length.[12] The fact that this locus is a plane follows by rotation in the third dimension from the fact that the radical axis is a straight line. The same definition can be applied to hyperspheres in Euclidean space of any dimension, giving the radical hyperplane of two nonconcentric hyperspheres. Notes 1. Michel Chasles, C. H. Schnuse: Die Grundlehren der neuern Geometrie, erster Theil, Verlag Leibrock, Braunschweig, 1856, p. 312 2. Ph. Fischer: Lehrbuch der analytische Geometrie, Darmstadt 1851, Verlag Ernst Kern, p. 67 3. H. Schwarz: Die Elemente der analytischen Geometrie der Ebene, Verlag H. W. Schmidt, Halle, 1858, p. 218 4. Jakob Steiner: Einige geometrische Betrachtungen. In: Journal für die reine und angewandte Mathematik, Band 1, 1826, p. 165 5. A. Schoenfliess, R. Courant: Einführung in die Analytische Geometrie der Ebene und des Raumes, Springer-Verlag, 1931, p. 113 6. C. Carathéodory: Funktionentheorie, Birkhäuser-Verlag, Basel, 1961, ISBN 978-3-7643-0064-7, p. 46 7. Dan Pedoe: Circles: A Mathematical View, mathematical Association of America, 2020, ISBN 9781470457327, p. 16 8. R. Lachlan: An Elementary Treatise On Modern Pure Geometry, MacMillan&Co, New York,1893, p. 200 9. Carathéodory: Funktionentheorie, p. 47. 10. R. Sauer: Ingenieur-Mathematik: Zweiter Band: Differentialgleichungen und Funktionentheorie, Springer-Verlag, 1962, ISBN 978-3-642-53232-0, p. 105 11. Clemens Schaefer: Elektrodynamik und Optik, Verlag: De Gruyter, 1950, ISBN 978-3-11-230936-0, p. 358. 12. See Merriam–Webster online dictionary. References • R. A. Johnson (1960). Advanced Euclidean Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle (reprint of 1929 edition by Houghton Mifflin ed.). New York: Dover Publications. pp. 31–43. ISBN 978-0-486-46237-0. Further reading • C. Stanley Ogilvy (1990). Excursions in Geometry. Dover. pp. 17–23. ISBN 0-486-26530-7. • H. S. M. Coxeter, S. L. Greitzer (1967). Geometry Revisited. Washington, D.C.: Mathematical Association of America. pp. 31–36, 160–161. ISBN 978-0-88385-619-2. • Clark Kimberling, "Triangle Centers and Central Triangles," Congressus Numerantium 129 (1998) i–xxv, 1–295. External links Wikimedia Commons has media related to Radical axes. • Weisstein, Eric W. "Radical line". MathWorld. • Weisstein, Eric W. "Chordal theorem". MathWorld. • Animation at Cut-the-knot	Wikipedia
Radical polynomial In mathematics, in the realm of abstract algebra, a radical polynomial is a multivariate polynomial over a field that can be expressed as a polynomial in the sum of squares of the variables. That is, if $k[x_{1},x_{2},\ldots ,x_{n}]$ is a polynomial ring, the ring of radical polynomials is the subring generated by the polynomial $\sum _{i=1}^{n}x_{i}^{2}.$ Radical polynomials are characterized as precisely those polynomials that are invariant under the action of the orthogonal group. The ring of radical polynomials is a graded subalgebra of the ring of all polynomials. The standard separation of variables theorem asserts that every polynomial can be expressed as a finite sum of terms, each term being a product of a radical polynomial and a harmonic polynomial. This is equivalent to the statement that the ring of all polynomials is a free module over the ring of radical polynomials. References	Wikipedia
Radical of a ring In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good" elements of the ring. The first example of a radical was the nilradical introduced by Köthe (1930), based on a suggestion of Wedderburn (1908). In the next few years several other radicals were discovered, of which the most important example is the Jacobson radical. The general theory of radicals was defined independently by (Amitsur 1952, 1954, 1954b) and Kurosh (1953). Definitions In the theory of radicals, rings are usually assumed to be associative, but need not be commutative and need not have a multiplicative identity. In particular, every ideal in a ring is also a ring. A radical class (also called radical property or just radical) is a class σ of rings possibly without identities, such that: 1. the homomorphic image of a ring in σ is also in σ 2. every ring R contains an ideal S(R) in σ that contains every other ideal of R that is in σ 3. S(R/S(R)) = 0. The ideal S(R) is called the radical, or σ-radical, of R. The study of such radicals is called torsion theory. For any class δ of rings, there is a smallest radical class Lδ containing it, called the lower radical of δ. The operator L is called the lower radical operator. A class of rings is called regular if every non-zero ideal of a ring in the class has a non-zero image in the class. For every regular class δ of rings, there is a largest radical class Uδ, called the upper radical of δ, having zero intersection with δ. The operator U is called the upper radical operator. A class of rings is called hereditary if every ideal of a ring in the class also belongs to the class. Examples The Jacobson radical Main article: Jacobson radical Let R be any ring, not necessarily commutative. The Jacobson radical of R is the intersection of the annihilators of all simple right R-modules. There are several equivalent characterizations of the Jacobson radical, such as: • J(R) is the intersection of the regular maximal right (or left) ideals of R. • J(R) is the intersection of all the right (or left) primitive ideals of R. • J(R) is the maximal right (or left) quasi-regular right (resp. left) ideal of R. As with the nilradical, we can extend this definition to arbitrary two-sided ideals I by defining J(I) to be the preimage of J(R/I) under the projection map R → R/I. If R is commutative, the Jacobson radical always contains the nilradical. If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I. This says that R is a Jacobson ring. The Baer radical The Baer radical of a ring is the intersection of the prime ideals of the ring R. Equivalently it is the smallest semiprime ideal in R. The Baer radical is the lower radical of the class of nilpotent rings. Also called the "lower nilradical" (and denoted Nil∗R), the "prime radical", and the "Baer-McCoy radical". Every element of the Baer radical is nilpotent, so it is a nil ideal. For commutative rings, this is just the nilradical and closely follows the definition of the radical of an ideal. The upper nil radical or Köthe radical The sum of the nil ideals of a ring R is the upper nilradical Nil*R or Köthe radical and is the unique largest nil ideal of R. Köthe's conjecture asks whether any left nil ideal is in the nilradical. Singular radical An element of a (possibly non-commutative ring) is called left singular if it annihilates an essential left ideal, that is, r is left singular if Ir = 0 for some essential left ideal I. The set of left singular elements of a ring R is a two-sided ideal, called the left singular ideal, and is denoted ${\mathcal {Z}}(_{R}R)$. The ideal N of R such that $N/{\mathcal {Z}}(_{R}R)={\mathcal {Z}}(_{R/{\mathcal {Z}}(_{R}R)}R/{\mathcal {Z}}(_{R}R))\,$ is denoted by ${\mathcal {Z}}_{2}(_{R}R)$ and is called the singular radical or the Goldie torsion of R. The singular radical contains the prime radical (the nilradical in the case of commutative rings) but may properly contain it, even in the commutative case. However, the singular radical of a Noetherian ring is always nilpotent. The Levitzki radical The Levitzki radical is defined as the largest locally nilpotent ideal, analogous to the Hirsch–Plotkin radical in the theory of groups. If the ring is Noetherian, then the Levitzki radical is itself a nilpotent ideal, and so is the unique largest left, right, or two-sided nilpotent ideal. The Brown–McCoy radical The Brown–McCoy radical (called the strong radical in the theory of Banach algebras) can be defined in any of the following ways: • the intersection of the maximal two-sided ideals • the intersection of all maximal modular ideals • the upper radical of the class of all simple rings with identity The Brown–McCoy radical is studied in much greater generality than associative rings with 1. The von Neumann regular radical A von Neumann regular ring is a ring A (possibly non-commutative without identity) such that for every a there is some b with a = aba. The von Neumann regular rings form a radical class. It contains every matrix ring over a division algebra, but contains no nil rings. The Artinian radical The Artinian radical is usually defined for two-sided Noetherian rings as the sum of all right ideals that are Artinian modules. The definition is left-right symmetric, and indeed produces a two-sided ideal of the ring. This radical is important in the study of Noetherian rings, as outlined by Chatters & Hajarnavis (1980). See also Related uses of radical that are not radicals of rings: • Radical of a module • Kaplansky radical • Radical of a bilinear form References • Amitsur, S. A. (1952). "A general theory of radicals. I: Radicals in complete lattices". American Journal of Mathematics. 74: 774–786. • Amitsur, S. A. (1954). "A general theory of radicals. II: Radicals in rings and bicategories". American Journal of Mathematics. 75: 100–125. • Amitsur, S. A. (1954b). "A general theory of radicals. III: Applications". American Journal of Mathematics. 75: 126–136. • Chatters, A. W.; Hajarnavis, C. R. (1980), Rings with Chain Conditions, Research Notes in Mathematics, vol. 44, Boston, Massachusetts: Pitman (Advanced Publishing Program), pp. vii+197, ISBN 0-273-08446-1, MR 0590045 • Köthe, Gottfried (1930). "Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollständig reduzibel ist". Mathematische Zeitschrift. 32 (1): 161–186. doi:10.1007/BF01194626. • Kurosh, A. G. (1953). "Radicals of rings and algebras". Matematicheskii Sbornik (in Russian). 33: 13–26. • Wedderburn, J.H.M. (1908). "On Hypercomplex Numbers". Proceedings of the London Mathematical Society. 6: 77–118. doi:10.1112/plms/s2-6.1.77. Further reading • Andrunakievich, V.A. (2001) [1994], "Radical of ring and algebras", Encyclopedia of Mathematics, EMS Press • Amitsur, Shimshon A.; Mann, Avinoam (2001), Selected Papers of S.A. Amitsur with Commentary, Part 1, Providence, Rhode Island: American Mathematical Society, ISBN 9780821829240 • Divinsky, N. J. (1965), Rings and Radicals, Mathematical Expositions No. 14, University of Toronto Press, MR 0197489 • Gardner, B. J.; Wiegandt, R. (2004), Radical Theory of Rings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 261, Marcel Dekker, ISBN 978-0-8247-5033-6, MR 2015465 • Goodearl, K. R. (1976), Ring Theory, Marcel Dekker, ISBN 978-0-8247-6354-1, MR 0429962 • Gray, Mary W. (1970), A Radical Approach to Algebra, Addison-Wesley, MR 0265396 • Stenström, Bo (1971), Rings and Modules of Quotients, Lecture Notes in Mathematics, vol. 237, Springer-Verlag, doi:10.1007/BFb0059904, ISBN 978-3-540-05690-4, MR 0325663, Zbl 0229.16003 • Wiegandt, Richard (1974), Radical and Semisimple Classes of Rings, Kingston, Ont.: Queen's University, MR 0349734	Wikipedia
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring $R$ is the ideal consisting of those elements in $R$ that annihilate all simple right $R$-modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by $J(R)$ or $\operatorname {rad} (R)$; the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in (Jacobson 1945). The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to rings without unity. The radical of a module extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring and module theoretic results, such as Nakayama's lemma. Definitions There are multiple equivalent definitions and characterizations of the Jacobson radical, but it is useful to consider the definitions based on if the ring is commutative or not. Commutative case In the commutative case, the Jacobson radical of a commutative ring $R$ is defined as[1] the intersection of all maximal ideals ${\mathfrak {m}}$. If we denote $\operatorname {Specm} R$ as the set of all maximal ideals in $R$ then $J(R)=\bigcap _{{\mathfrak {m}}\,\in \,\operatorname {Specm} R}{\mathfrak {m}}$ This definition can be used for explicit calculations in a number of simple cases, such as for local rings $(R,{\mathfrak {p}})$, which have a unique maximal ideal, Artin rings, and products thereof. See the examples section for explicit computations. Noncommutative/general case For a general ring with unity $R$, the Jacobson radical $J(R)$ is defined as the ideal of all elements $r\in R$ such that $rM=0$ whenever $M$ is a simple $R$-module. That is, $J(R)=\{r\in R\mid rM=0{\text{ where }}M{\text{ is simple}}\}.$ This is equivalent to the definition in the commutative case for a commutative ring $R$ because the simple modules over a commutative ring are of the form $R/{\mathfrak {m}}$ for some maximal ideal ${\mathfrak {m}}\in \operatorname {Specm} R$, and the only annihilators of $R/{\mathfrak {m}}$ in $R$ are in ${\mathfrak {m}}$, i.e. ${\text{Ann}}_{R}(R/{\mathfrak {m}})={\mathfrak {m}}$. Motivation Understanding the Jacobson radical lies in a few different cases: namely its applications and the resulting geometric interpretations, and its algebraic interpretations. Geometric applications Although Jacobson originally introduced his radical as a technique for building a theory of radicals for arbitrary rings, one of the motivating reasons for why the Jacobson radical is considered in the commutative case is because of its appearance in Nakayama's lemma. This lemma is a technical tool for studying finitely generated modules over commutative rings which has an easy geometric interpretation: If we have a vector bundle $E\to X$ over a topological space $X$, and pick a point $p\in X$, then any basis of $E\|_{p}$ can be extended to a basis of sections of $E\|_{U}\to U$ for some neighborhood $p\in U\subset X$. Another application is in the case of finitely generated commutative rings, meaning $R$ is of the form $R={\frac {k[x_{1},\ldots ,x_{n}]}{I}}$ for some base ring $k$ (such as a field, or the ring of integers). In this case the nilradical and the Jacobson radical coincide. This means we could interpret the Jacobson radical as a measure for how far the ideal $I$ defining the ring $R$ is from defining the ring of functions on an algebraic variety because of the Hilbert Nullstellensatz theorem. This is because algebraic varieties cannot have a ring of functions with infinitesimals: this is a structure which is only considered in scheme theory. Equivalent characterizations The Jacobson radical of a ring has various internal and external characterizations. The following equivalences appear in many noncommutative algebra texts such as (Anderson & Fuller 1992, §15), (Isaacs 1994, §13B), and (Lam 2001, Ch 2). The following are equivalent characterizations of the Jacobson radical in rings with unity (characterizations for rings without unity are given immediately afterward): • $J(R)$ equals the intersection of all maximal right ideals of the ring. The equivalence coming from the fact that for all maximal right ideals M, R/M is a simple right R-module, and that in fact all simple right R-modules are isomorphic to one of this type via the map from R to S given by r ↦ xr for any generator x of S. It is also true that $J(R)$ equals the intersection of all maximal left ideals within the ring.[2] These characterizations are internal to the ring, since one only needs to find the maximal right ideals of the ring. For example, if a ring is local, and has a unique maximal right ideal, then this unique maximal right ideal is exactly $J(R)$. Maximal ideals are in a sense easier to look for than annihilators of modules. This characterization is deficient, however, because it does not prove useful when working computationally with $J(R)$. The left-right symmetry of these two definitions is remarkable and has various interesting consequences.[2][3] This symmetry stands in contrast to the lack of symmetry in the socles of R, for it may happen that soc(RR) is not equal to soc(RR). If R is a non-commutative ring, $J(R)$ is not necessarily equal to the intersection of all maximal two-sided ideals of R. For instance, if V is a countable direct sum of copies of a field k and R = End(V) (the ring of endomorphisms of V as a k-module), then $J(R)=0$ because $R$ is known to be von Neumann regular, but there is exactly one maximal double-sided ideal in R consisting of endomorphisms with finite-dimensional image. (Lam 2001, p. 46, Ex. 3.15) • $J(R)$ equals the sum of all superfluous right ideals (or symmetrically, the sum of all superfluous left ideals) of R. Comparing this with the previous definition, the sum of superfluous right ideals equals the intersection of maximal right ideals. This phenomenon is reflected dually for the right socle of R; soc(RR) is both the sum of minimal right ideals and the intersection of essential right ideals. In fact, these two relationships hold for the radicals and socles of modules in general. • As defined in the introduction, $J(R)$ equals the intersection of all annihilators of simple right R-modules, however it is also true that it is the intersection of annihilators of simple left modules. An ideal that is the annihilator of a simple module is known as a primitive ideal, and so a reformulation of this states that the Jacobson radical is the intersection of all primitive ideals. This characterization is useful when studying modules over rings. For instance, if U is a right R-module, and V is a maximal submodule of U, U · J(R) is contained in V, where U · J(R) denotes all products of elements of J(R) (the "scalars") with elements in U, on the right. This follows from the fact that the quotient module U/V is simple and hence annihilated by J(R). • J(R) is the unique right ideal of R maximal with the property that every element is right quasiregular[4][5] (or equivalently left quasiregular[2]). This characterization of the Jacobson radical is useful both computationally and in aiding intuition. Furthermore, this characterization is useful in studying modules over a ring. Nakayama's lemma is perhaps the most well-known instance of this. Although every element of the J(R) is necessarily quasiregular, not every quasiregular element is necessarily a member of J(R).[5] • While not every quasiregular element is in $J(R)$, it can be shown that y is in $J(R)$ if and only if xy is left quasiregular for all x in R. (Lam 2001, p. 50) • $J(R)$ is the set of elements x in R such that every element of 1 + RxR is a unit: $\operatorname {J} (R)=\{x\in R\mid 1+RxR\subset R^{\times }\}$. In fact, $y\in R$ is in the Jacobson radical if and only if 1 + xy is invertible for any $x\in R$, if and only if 1 + yx is invertible for any $x\in R$. This means xy and yx behave similarly to a nilpotent element z with zn+1 = 0 and $(1+z)^{-1}=1-z+z^{2}-\cdots \pm z^{n}$. For rings without unity it is possible to have R = J(R); however, the equation J(R/J(R)) = {0} still holds. The following are equivalent characterizations of J(R) for rings without unity (Lam 2001, p. 63): • The notion of left quasiregularity can be generalized in the following way. Call an element a in R left generalized quasiregular if there exists c in R such that c+a−ca = 0. Then J(R) consists of every element a for which ra is left generalized quasiregular for all r in R. It can be checked that this definition coincides with the previous quasiregular definition for rings with unity. • For a ring without unity, the definition of a left simple module M is amended by adding the condition that R • M ≠ 0. With this understanding, J(R) may be defined as the intersection of all annihilators of simple left R modules, or just R if there are no simple left R modules. Rings without unity with no simple modules do exist, in which case R = J(R), and the ring is called a radical ring. By using the generalized quasiregular characterization of the radical, it is clear that if one finds a ring with J(R) nonzero, then J(R) is a radical ring when considered as a ring without unity. Examples Commutative examples • For the ring of integers $\mathbb {Z} $ its Jacobson radical is the zero ideal, so $J(\mathbb {Z} )=(0)$, because it is given by the intersection of every ideal generated by a prime number $(p)$. Since $(p_{1})\cap (p_{2})=(p_{1}\cdot p_{2})$, and we are taking an infinite intersection with no common elements besides $0$ between all maximal ideals, we have the computation. • For a local ring $(R,{\mathfrak {p}})$ the Jacobson radical is simply $J(R)={\mathfrak {p}}$. This is an important case because of its use in applying Nakayama's lemma. In particular, it implies if we have an algebraic vector bundle $E\to X$ over a scheme or algebraic variety $X$, and we fix a basis of $E\|_{p}$ for some point $p\in X$, then this basis lifts to a set of generators for all sections $E\|_{U}\to U$ for some neighborhood $U$ of $p$. • If $k$ is a field and $R=k[[X_{1},\dots ,X_{n}]]$ is a ring of formal power series, then $J(R)$ consists of those power series whose constant term is zero, i.e. the power series in the ideal $(X_{1},\ldots ,X_{n})$. • In the case of an Artin ring, such as $\mathbb {C} [t_{1},t_{2}]/(t_{1}^{4},t_{1}^{2}t_{2}^{2},t_{2}^{9})$, the Jacobson radical is $(t_{1},t_{2})$. • The previous example could be extended to the ring $R=\mathbb {C} [t_{2},t_{3},\ldots ]/(t_{2}^{2},t_{3}^{3},\ldots )$, giving $J(R)=(t_{2},t_{3},\ldots )$. • The Jacobson radical of the ring Z/12Z is 6Z/12Z, which is the intersection of the maximal ideals 2Z/12Z and 3Z/12Z. • Consider the ring $\mathbb {C} [t]\otimes _{\mathbb {C} }\mathbb {C} [x_{1},x_{2}]_{x_{1}^{2}+x_{2}^{2}-1}$ where the second is the localization of $\mathbb {C} [x_{1},x_{2}]$ by the prime ideal ${\mathfrak {p}}=(x_{1}^{2}+x_{2}^{2}-1)$. Then, the Jacobson radical is trivial because the maximal ideals are generated by an element of the form $(t-z)\otimes (x_{1}^{2}+x_{2}^{2}-1)$ for $z\in \mathbb {C} $. Noncommutative examples • Rings for which J(R) is {0} are called semiprimitive rings, or sometimes "Jacobson semisimple rings". The Jacobson radical of any field, any von Neumann regular ring and any left or right primitive ring is {0}. The Jacobson radical of the integers is {0}. • If K is a field and R is the ring of all upper triangular n-by-n matrices with entries in K, then J(R) consists of all upper triangular matrices with zeros on the main diagonal. • Start with a finite, acyclic quiver Γ and a field K and consider the quiver algebra K Γ (as described in the Quiver article). The Jacobson radical of this ring is generated by all the paths in Γ of length ≥ 1. • The Jacobson radical of a C-algebra is {0}. This follows from the Gelfand–Naimark theorem and the fact that for a C-algebra, a topologically irreducible -representation on a Hilbert space is algebraically irreducible, so that its kernel is a primitive ideal in the purely algebraic sense (see Spectrum of a C-algebra). Properties • If R is unital and is not the trivial ring {0}, the Jacobson radical is always distinct from R since rings with unity always have maximal right ideals. However, some important theorems and conjectures in ring theory consider the case when J(R) = R - "If R is a nil ring (that is, each of its elements is nilpotent), is the polynomial ring R[x] equal to its Jacobson radical?" is equivalent to the open Köthe conjecture. (Smoktunowicz 2006, p. 260, §5) • For any ideal I contained in J(R), J(R / I) = J(R) / I. • In particular, the Jacobson radical of the ring R/J(R) is zero. Rings with zero Jacobson radical are called semiprimitive rings. • A ring is semisimple if and only if it is Artinian and its Jacobson radical is zero. • If f : R → S is a surjective ring homomorphism, then f(J(R)) ⊆ J(S). • If R is a ring with unity and M is a finitely generated left R-module with J(R)M = M, then M = 0 (Nakayama's lemma). • J(R) contains all central nilpotent elements, but contains no idempotent elements except for 0. • J(R) contains every nil ideal of R. If R is left or right Artinian, then J(R) is a nilpotent ideal. This can actually be made stronger: If $\left\{0\right\}=T_{0}\subseteq T_{1}\subseteq \dotsb \subseteq T_{k}=R$ is a composition series for the right R-module R (such a series is sure to exist if R is right Artinian, and there is a similar left composition series if R is left Artinian), then $\left(J\left(R\right)\right)^{k}=0.$ [lower-alpha 1] Note, however, that in general the Jacobson radical need not consist of only the nilpotent elements of the ring. • If R is commutative and finitely generated as an algebra over either a field or Z, then J(R) is equal to the nilradical of R. • The Jacobson radical of a (unital) ring is its largest superfluous right (equivalently, left) ideal. See also • Frattini subgroup • Nilradical • Radical of a module • Radical of an ideal Notes 1. (Proof: Since the factors $T_{u}/T_{u-1}$ are simple right R-modules, right multiplication by any element of J(R) annihilates these factors. In other words, $\left(T_{u}/T_{u-1}\right)\cdot J\left(R\right)=0,$ whence $T_{u}\cdot J\left(R\right)\subseteq T_{u-1}.$ Consequently, induction over i shows that all nonnegative integers i and u (for which the following makes sense) satisfy $T_{u}\cdot \left(\operatorname {J} \left(R\right)\right)^{i}\subseteq T_{u-i}.$ Applying this to u = i = k yields the result.) 1. "Section 10.18 (0AMD): The Jacobson radical of a ring—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-12-24. 2. Isaacs 1994, p. 182. 3. Isaacs 1994, Problem 12.5, p. 173. 4. Isaacs 1994, Corollary 13.4, p. 180. 5. Isaacs 1994, p. 181. References • Anderson, Frank W.; Fuller, Kent R. (1992), Rings and Categories of Modules, Graduate Texts in Mathematics, vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3, MR 1245487 • Atiyah, M. F.; Macdonald, I. G. (1969), Introduction to Commutative Algebra, Addison-Wesley Publishing Co., pp. ix+128, MR 0242802 • Bourbaki, N. Éléments de mathématique. • Herstein, I. N. (1994) [1968], Noncommutative Rings, Carus Mathematical Monographs, vol. 15, Washington, DC: Mathematical Association of America, pp. xii+202, ISBN 0-88385-015-X, MR 1449137 Reprint of the 1968 original; With an afterword by Lance W. Small • Isaacs, I. M. (1994), Algebra: a graduate course (1st ed.), Brooks/Cole Publishing Company, ISBN 0-534-19002-2 • Jacobson, Nathan (1945), "The radical and semi-simplicity for arbitrary rings", American Journal of Mathematics, 67: 300–320, doi:10.2307/2371731, ISSN 0002-9327, MR 0012271 • Lam, T. Y. (2001), A First Course in Noncommutative Rings, Graduate Texts in Mathematics, vol. 131 (2 ed.), Springer-Verlag, pp. xx+385, doi:10.1007/978-1-4419-8616-0, ISBN 0-387-95183-0, MR 1838439 • Pierce, Richard S. (1982), Associative Algebras, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, pp. xii+436, ISBN 0-387-90693-2, MR 0674652 Studies in the History of Modern Science, 9 • Smoktunowicz, Agata (2006), "Some results in noncommutative ring theory", International Congress of Mathematicians, Vol. II (PDF), European Mathematical Society, pp. 259–269, ISBN 978-3-03719-022-7, MR 2275597, archived from the original (PDF) on 2017-08-09, retrieved 2014-12-31 External links • Intuitive Example of a Jacobson Radical	Wikipedia
Charles Radin Charles Lewis Radin is an American mathematician, known for his work on aperiodic tilings and in particular for defining the pinwheel tiling and, with John Horton Conway, the quaquaversal tiling.[1] Education and career Radin did his undergraduate studies at City College of New York, graduating in 1965,[2] and then did his graduate studies at the University of Rochester, earning a Ph.D. in 1970 under the supervision of Gérard Emch.[2][3] Since 1976 he has been on the faculty of the University of Texas at Austin. Awards and honors In 2012 he became a fellow of the American Mathematical Society.[4] Selected publications • Radin, Charles; Wolff, Mayhew (1992), "Space tilings and local isomorphism", Geometriae Dedicata, 42 (3): 355–360, CiteSeerX 10.1.1.37.9928, doi:10.1007/BF02414073, MR 1164542, S2CID 16334831. • Radin, Charles (1994), "The pinwheel tilings of the plane", Annals of Mathematics, Second Series, 139 (3): 661–702, doi:10.2307/2118575, JSTOR 2118575, MR 1283873. • Conway, John H.; Radin, Charles (1998), "Quaquaversal tilings and rotations", Inventiones Mathematicae, 132 (1): 179–188, Bibcode:1998InMat.132..179C, CiteSeerX 10.1.1.31.8585, doi:10.1007/s002220050221, MR 1618635, S2CID 14194250. • Radin, Charles (1999), Miles of Tiles, Student Mathematical Library, vol. 1, Providence, RI: American Mathematical Society, ISBN 978-0-8218-1933-3, MR 1707270. • as editor with Mark J. Bowick, Govind Menon, and David Kinderlehrer: Mathematics and Materials, American Mathematical Society 2017 References 1. Stewart, Ian (September 24, 1994), "Bathroom tiling to drive you mad", New Scientist. 2. Curriculum vitae, retrieved 2013-06-09. 3. Charles Radin at the Mathematics Genealogy Project. 4. List of Fellows of the American Mathematical Society, retrieved 2013-06-09. External links • Home page Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • Belgium • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Rob Schneiderman (mathematician) Robert Roland "Rob" Schneiderman (born June 21, 1957) is an American jazz pianist who also works as a professor of mathematics at Lehman College of the City University of New York, where he specializes in geometric topology.[1] Rob Schneiderman Schneiderman at Oberwolfach 2014 Background information Birth nameRobert Roland Schneiderman Born (1957-06-21) June 21, 1957 Boston, Massachusetts, U.S. GenresJazz, bebop, hard bop, post-bop Occupation(s)Musician, educator, composer, mathematician Instrument(s)piano Years active1973–present WebsiteOfficial website Alma materCity College of New York (B.A.) University of California, Berkeley (Ph.D.) Scientific career FieldsGeometric topology Thesis"4-Dimensional Intersection Numbers of Knots and Links in 3-Manifolds" (2001) Doctoral advisorRobion Kirby Music career Schneiderman's professional jazz career began in San Diego from about age 16, when he played piano for visiting soloists such as Eddie Harris, Sonny Stitt, Harold Land, Charles McPherson and Peter Sprague.[2] He continued to collaborate intermittently with Harris, until the latter's death in 1996, and with McPherson. In 1982, Schneiderman moved to New York, where he performed and toured with such musicians as J.J. Johnson, Chet Baker, Art Farmer, Clifford Jordan, James Moody and Zoot Sims. A performance fellowship from the National Endowment for the Arts in 1987 featured Schneiderman with George Coleman, Jimmy Heath, Claudio Roditi, and Slide Hampton.[3] The collaboration with Slide Hampton resulted in his debut album New Outlook, the first of ten recordings to date as a leader for the Reservoir music label.[4][5] His most recent release, entitled Tone Twister, is a collaboration with Brian Lynch on trumpet and Ralph Moore on tenor saxophone. The album features Gerald L. Cannon on bass and Pete Van Nostrand on drums.[6] Schneiderman has also played as sidemen for Billy Higgins, Rufus Reid, Brian Lynch, Ralph Moore, Peter Washington, Lewis Nash, Akira Tana, Billy Hart, Gary Smulyan and Ben Riley. As a jazz educator, he has been in residence at the Stanford Jazz Workshop.[7] He was previously an adjunct professor in the jazz departments of the William Paterson University (with Rufus Reid) and Queens College (with Jimmy Heath). He has also been on the faculty of the Jazzschool in Berkeley, California.[2] Education and academic career Schneiderman graduated with a B.A. in mathematics from City College of New York in 1994. He received his Ph.D. from University of California, Berkeley in 2001 under the supervision of Robion Kirby.[8] In 2006, Schneiderman became an assistant professor at the Department of Mathematics Lehman College after stints at the University of California, San Diego, Courant Institute of Mathematical Sciences, New York University, and at the University of Pennsylvania. From 2013, he has been an associate professor at Lehman College.[9] He has also served as chair of the Mathematics Department since 2019.[1] Schneiderman also actively works at the interface of music and mathematics.[10][11][12][13] Discography As Leader Year Title Label CD release(s) Credit (if not Rob Schneiderman) 1988 New Outlook Reservoir Reservoir (City Hall), RSR CD 106, 2008 Composers: Cole Porter, Bud Powell, Alec Wilder 1990 Smooth Sailing Reservoir Reservoir (City Hall), RSR CD 114, 1994 Composers: Richard Rodgers, Lorenz Hart, Ernesto Lecuona, Forman Brown, Gus Kahn, Nacio Herb Brown, Harry Warren, Johnny Mercer 1991 Radio Waves Reservoir (8 tracks) Reservoir (City Hall), RSR CD 120, 2008 Composers: Lorenz Hart, Richard Rodgers 1992 Standards Reservoir (10 tracks) Reservoir (City Hall), RSR CD 126, 1993 Composers: Edward Heyman, Victor Young, Cole Porter, Frank Loesser, Edward Eliscu, Billy Rose, Vincent Youmans, Sammy Cahn, Axel Stordahl, Paul Weston, Bart Howard, Lorenz Hart, Richard Rodgers, Leigh Harline, Don Redman 1994 Dark Blue Reservoir (8 tracks) Reservoir (City Hall), RSR CD 132, 1994 Composers: Ray Noble, Brian Lynch, Henry W. Sanicola, Jr., Sol Parker, Frank Sinatra, Oscar Hammerstein II, Richard Rodgers 1996 Keepin' in the Groove Reservoir (9 tracks) Reservoir (City Hall), RSR CD 144, 1994 Composers: Clifford Brown, Tadd Dameron, Miles Davis, Duke Ellington, Dizzy Gillespie, Eddie Harris, Irving Mills, Wayne Shorter, Juan Tizol 1998 Dancing in the Dark Reservoir (8 tracks) Composers: Jack Baker, Howard Dietz, George Fragos, Dick Gasparre, Arthur Schwartz 2001 Edgewise Reservoir (10 tracks) Composers: Gene DePaul, Patricia Johnston, Thelonious Monk, Oscar Pettiford, Cole Porter, Bud Powell, Don Raye 2004 Back in Town Reservoir (9 tracks) Composers: Eden Ahbez, Johnny Burke, John Coltrane, Ray Evans, James Van Heusen, Antonio Carlos Jobim, Jay Livingston, Cole Porter, Sonny Rollins 2008 Glass Enclosure Reservoir (10 tracks) Composers: Bud Powell, Cole Porter, Kay Swift, Paul James, Charlie Parker, Gigi Gryce, Arthur Johnston, Sam Coslow 2017 Tone Twister Hollistic MusicWorks HMW 16 (9 tracks) Holistic Music Works, Digital Album (11 tracks) Composers: Irving Gordon As Sideman with Eddie Harris • 1983: Tale of Two Cities (Night Records) with J.J. Johnson • 1992: Vivian (Concord) with Brian Lynch • 2011: Unsung Heroes Vol. 1 (Hollistic MusicWorks) • 2013: Unsung Heroes, Vol. 2 (CD Baby) with Rufus Reid and Harold Land • 1989: Corridor to the Limits (Sunnyside) with Akira Tana and Rufus Reid • 1991: Yours and Mine (Concord) • 1992: Passing Thoughts (Concord) • 1994: Blue Motion (Evidence) References 1. "Lehman College Mathematics Department Faculty". Retrieved August 16, 2020. 2. Gary W. Kennedy (2003). "Schneiderman, Rob(ert Roland)". Schneiderman, Rob(ert Roland). Grove Music Online. doi:10.1093/gmo/9781561592630.article.J687500. 3. "Rob Schneiderman, Piano". SmallsLive Foundation. Retrieved August 29, 2020. 4. Rob Schneiderman: New Outlook at AllMusic. Retrieved August 25, 2020. 5. "Rob Schneiderman". Reservoir Music. Retrieved August 29, 2020. 6. Rob Schneiderman: Tone Twister at AllMusic. Retrieved August 29, 2020. 7. "Rob Schneiderman Quartet Featuring Tootie Heath". Stanford University. Retrieved August 29, 2020. 8. Rob Schneiderman at the Mathematics Genealogy Project 9. Gilbert, Andrew (June 25, 2018). "The Return of Rob Schneiderman". Oakland Magazine. Retrieved August 17, 2020. 10. Reid, Geannine (December 19, 2017). "Rob Schneiderman: Tone Twister". All About Jazz. Retrieved August 17, 2020. 11. Gilbert, Andrew (June 27, 2018). "The Musical Mathematics of Rob Schneiderman". California Magazine. Retrieved August 16, 2020. 12. "American Mathematical Society". Retrieved August 16, 2020. 13. "American Mathematical Society". Retrieved August 16, 2020. External links • Official website • Publications by Rob Schneiderman • Home page at Lehman College Authority control International • ISNI • VIAF National • Germany • United States Academics • MathSciNet • Mathematics Genealogy Project Artists • MusicBrainz	Wikipedia
Radio coloring In graph theory, a branch of mathematics, a radio coloring of an undirected graph is a form of graph coloring in which one assigns positive integer labels to the graphs such that the labels of adjacent vertices differ by at least two, and the labels of vertices at distance two from each other differ by at least one. Radio coloring was first studied by Griggs & Yeh (1992), under a different name, L(2,1)-labeling.[1][2] It was called radio coloring by Frank Harary because it models the problem of channel assignment in radio broadcasting, while avoiding electromagnetic interference between radio stations that are near each other both in the graph and in their assigned channel frequencies. The span of a radio coloring is its maximum label, and the radio coloring number of a graph is the smallest possible span of a radio coloring.[1] For instance, the graph consisting of two vertices with a single edge has radio coloring number 3: it has a radio coloring with one vertex labeled 1 and the other labeled 3, but it is not possible for a radio coloring of this graph to use only the labels 1 and 2. Computational complexity Finding a radio coloring with a given (or minimum) span is NP-complete, even when restricted to planar graphs, split graphs, or the complements of bipartite graphs.[1][3] However it is solvable in polynomial time for trees and cographs.[1][4] For arbitrary graphs, it can be solved in singly-exponential time, significantly faster than a brute-force search through all possible colorings.[5][6] Other properties Although the radio coloring number of an n-vertex graph can range from 1 to 2n − 1, almost all n-vertex graphs have radio coloring number exactly n. This is because these graphs almost always have diameter at least two (forcing all vertices to have distinct colors, and forcing the radio coloring number to be at least n) but they also almost always have a Hamiltonian path in the complement graph. Consecutive vertices in this path can be assigned consecutive colors, allowing a radio coloring to avoid skipping any numbers.[7] References 1. Broersma, Hajo (2005), "A general framework for coloring problems: old results, new results, and open problems" (PDF), Combinatorial geometry and graph theory, Lecture Notes in Comput. Sci., vol. 3330, Springer, Berlin, pp. 65–79, doi:10.1007/978-3-540-30540-8_7, MR 2172960. See in particular Section 3, "Radio coloring". 2. Griggs, Jerrold R.; Yeh, Roger K. (1992), "Labelling graphs with a condition at distance 2", SIAM Journal on Discrete Mathematics, 5 (4): 586–595, doi:10.1137/0405048, MR 1186826. 3. Bodlaender, Hans L.; Kloks, Ton; Tan, Richard B.; van Leeuwen, Jan (2000), "λ-coloring of graphs", STACS 2000: 17th Annual Symposium on Theoretical Aspects of Computer Science, Lille, France, February 17–19, 2000, Proceedings, Lecture Notes in Computer Science, vol. 1770, Springer, Berlin, pp. 395–406, doi:10.1007/3-540-46541-3_33, MR 1781749. 4. Chang, Gerard J.; Kuo, David (1996), "The L(2,1)-labeling problem on graphs", SIAM Journal on Discrete Mathematics, 9 (2): 309–316, CiteSeerX 10.1.1.51.2004, doi:10.1137/S0895480193245339, MR 1386886. 5. Havet, Frédéric; Klazar, Martin; Kratochvíl, Jan; Kratsch, Dieter; Liedloff, Mathieu (2011), "Exact algorithms for L(2,1)-labeling of graphs" (PDF), Algorithmica, 59 (2): 169–194, doi:10.1007/s00453-009-9302-7, MR 2765572, S2CID 2634447. 6. Junosza-Szaniawski, Konstanty; Rzążewski, Paweł (2011), "On the complexity of exact algorithm for L(2,1)-labeling of graphs", Information Processing Letters, 111 (14): 697–701, doi:10.1016/j.ipl.2011.04.010, MR 2840535. 7. Harary, Frank; Plantholt, Michael (1999), "Graphs whose radio coloring number equals the number of nodes", Graph colouring and applications (Montréal, QC, 1997), CRM Proc. Lecture Notes, vol. 23, Providence, RI: American Mathematical Society, pp. 99–100, MR 1723637.	Wikipedia
Radiodrome In geometry, a radiodrome is the pursuit curve followed by a point that is pursuing another linearly-moving point. The term is derived from the Greek words ῥᾴδιος, rhā́idios, 'easier' and δρόμος, drómos, 'running'. The classic (and best-known) form of a radiodrome is known as the "dog curve"; this is the path a dog follows when it swims across a stream with a current after something it has spotted on the other side. Because the dog drifts with the current, it will have to change its heading; it will also have to swim further than if it had taken the optimal heading. This case was described by Pierre Bouguer in 1732. A radiodrome may alternatively be described as the path a dog follows when chasing a hare, assuming that the hare runs in a straight line at a constant velocity. Mathematical analysis Introduce a coordinate system with origin at the position of the dog at time zero and with y-axis in the direction the hare is running with the constant speed Vt. The position of the hare at time zero is (Ax, Ay) with Ax > 0 and at time t it is $(T_{x}\ ,\ T_{y})\ =\ (A_{x}\ ,\ A_{y}+V_{t}t)~.$ (1) The dog runs with the constant speed Vd towards the instantaneous position of the hare. The differential equation corresponding to the movement of the dog, (x(t), y(t)), is consequently ${\dot {x}}=V_{d}\ {\frac {T_{x}-x}{\sqrt {(T_{x}-x)^{2}+(T_{y}-y)^{2}}}}$ (2) ${\dot {y}}=V_{d}\ {\frac {T_{y}-y}{\sqrt {(T_{x}-x)^{2}+(T_{y}-y)^{2}}}}~.$ (3) It is possible to obtain a closed-form analytic expression y=f(x) for the motion of the dog. From (2) and (3), it follows that $y'(x)={\frac {T_{y}-y}{T_{x}-x}}$ . (4) Multiplying both sides with $T_{x}-x$ and taking the derivative with respect to x, using that ${\frac {dT_{y}}{dx}}\ =\ {\frac {dT_{y}}{dt}}\ {\frac {dt}{dx}}\ =\ {\frac {V_{t}}{V_{d}}}\ {\sqrt {{y'}^{2}+1}}~,$ (5) one gets $y''={\frac {V_{t}\ {\sqrt {1+{y'}^{2}}}}{V_{d}(A_{x}-x)}}$ (6) or ${\frac {y''}{\sqrt {1+{y'}^{2}}}}={\frac {V_{t}}{V_{d}(A_{x}-x)}}~.$ (7) From this relation, it follows that $\sinh ^{-1}(y')=B-{\frac {V_{t}}{V_{d}}}\ \ln(A_{x}-x)~,$ (8) where B is the constant of integration determined by the initial value of y' at time zero, y' (0)= sinh(B − (Vt /Vd) lnAx), i.e., $B={\frac {V_{t}}{V_{d}}}\ \ln(A_{x})+\ln \left(y'(0)+{\sqrt {{y'(0)}^{2}+1}}\right).$ (9) From (8) and (9), it follows after some computation that $y'={\frac {1}{2}}\left[\left(y'(0)+{\sqrt {{y'(0)}^{2}+1}}\right)\left(1-{\frac {x}{A_{x}}}\right)^{-{\frac {V_{t}}{V_{d}}}}+\left(y'(0)-{\sqrt {{y'(0)}^{2}+1}}\right)\left(1-{\frac {x}{A_{x}}}\right)^{\frac {V_{t}}{V_{d}}}\right]$ . (10) Furthermore, since y(0)=0, it follows from (1) and (4) that $y'(0)={\frac {A_{y}}{A_{x}}}$ . (11) If, now, Vt ≠ Vd, relation (10) integrates to $y=C-{\frac {A_{x}}{2}}\left[{\frac {\left(y'(0)+{\sqrt {{y'(0)}^{2}+1}}\right)\left(1-{\frac {x}{A_{x}}}\right)^{1-{\frac {V_{t}}{V_{d}}}}}{1-{\frac {V_{t}}{V_{d}}}}}+{\frac {\left(y'(0)-{\sqrt {{y'(0)}^{2}+1}}\right)\left(1-{\frac {x}{A_{x}}}\right)^{1+{\frac {V_{t}}{V_{d}}}}}{1+{\frac {V_{t}}{V_{d}}}}}\right],$ (12) where C is the constant of integration. Since again y(0)=0, it's $C={\frac {A_{x}}{2}}\left[{\frac {y'(0)+{\sqrt {{y'(0)}^{2}+1}}}{1-{\frac {V_{t}}{V_{d}}}}}+{\frac {y'(0)-{\sqrt {{y'(0)}^{2}+1}}}{1+{\frac {V_{t}}{V_{d}}}}}\right]$. (13) The equations (11), (12) and (13), then, together imply $y={\frac {1}{2}}\left\{{\frac {A_{y}+{\sqrt {A_{x}^{2}+A_{y}^{2}}}}{1-{\frac {V_{t}}{V_{d}}}}}\left[1-\left(1-{\frac {x}{A_{x}}}\right)^{1-{\frac {V_{t}}{V_{d}}}}\right]+{\frac {A_{y}-{\sqrt {A_{x}^{2}+A_{y}^{2}}}}{1+{\frac {V_{t}}{V_{d}}}}}\left[1-\left(1-{\frac {x}{A_{x}}}\right)^{1+{\frac {V_{t}}{V_{d}}}}\right]\right\}$ . (14) If Vt = Vd, relation (10) gives, instead, $y=C-{\frac {A_{x}}{2}}\left[\left(y'(0)+{\sqrt {{y'(0)}^{2}+1}}\right)\ln \left(1-{\frac {x}{A_{x}}}\right)+{\frac {1}{2}}\left(y'(0)-{\sqrt {{y'(0)}^{2}+1}}\right)\left(1-{\frac {x}{A_{x}}}\right)^{2}\right]$ . (15) Using y(0)=0 once again, it follows that $C={\frac {A_{x}}{4}}\left(y'(0)-{\sqrt {{y'(0)}^{2}+1}}\right).$ (16) The equations (11), (15) and (16), then, together imply that $y={\frac {1}{4}}\left(A_{y}-{\sqrt {A_{x}^{2}+A_{y}^{2}}}\right)\left[1-\left(1-{\frac {x}{A_{x}}}\right)^{2}\right]-{\frac {1}{2}}\left(A_{y}+{\sqrt {A_{x}^{2}+A_{y}^{2}}}\right)\ln \left(1-{\frac {x}{A_{x}}}\right)$ . (17) If Vt < Vd, it follows from (14) that $\lim _{x\to A_{x}}y(x)={\frac {1}{2}}\left({\frac {A_{y}+{\sqrt {A_{x}^{2}+A_{y}^{2}}}}{1-{\frac {V_{t}}{V_{d}}}}}+{\frac {A_{y}-{\sqrt {A_{x}^{2}+A_{y}^{2}}}}{1+{\frac {V_{t}}{V_{d}}}}}\right).$ (18) If Vt ≥ Vd, one has from (14) and (17) that $\lim _{x\to A_{x}}y(x)=\infty $, which means that the hare will never be caught, whenever the chase starts. See also • Mice problem References • Nahin, Paul J. (2012), Chases and Escapes: The Mathematics of Pursuit and Evasion, Princeton: Princeton University Press, ISBN 978-0-691-12514-5. • Gomes Teixera, Francisco (1909), Imprensa da universidade (ed.), Traité des Courbes Spéciales Remarquables, vol. 2, Coimbra, p. 255{{citation}}: CS1 maint: location missing publisher (link)	Wikipedia
Radius In classical geometry, a radius (PL: radii or radiuses)[lower-alpha 1] of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin radius, meaning ray but also the spoke of a chariot wheel.[2] The typical abbreviation and mathematical variable name for radius is R or r. By extension, the diameter D is defined as twice the radius:[3] $d\doteq 2r\quad \Rightarrow \quad r={\frac {d}{2}}.$ If an object does not have a center, the term may refer to its circumradius, the radius of its circumscribed circle or circumscribed sphere. In either case, the radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. The inradius of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other hollow object is the radius of its cavity. For regular polygons, the radius is the same as its circumradius.[4] The inradius of a regular polygon is also called apothem. In graph theory, the radius of a graph is the minimum over all vertices u of the maximum distance from u to any other vertex of the graph.[5] The radius of the circle with perimeter (circumference) C is $r={\frac {C}{2\pi }}$ Formula For many geometric figures, the radius has a well-defined relationship with other measures of the figure. Circles See also: Area of a circle The radius of a circle with area A is $r={\sqrt {\frac {A}{\pi }}}.$ The radius of the circle that passes through the three non-collinear points P1, P2, and P3 is given by $r={\frac {\|{\vec {OP_{1}}}-{\vec {OP_{3}}}\|}{2\sin \theta }},$ where θ is the angle ∠P1P2P3. This formula uses the law of sines. If the three points are given by their coordinates (x1,y1), (x2,y2), and (x3,y3), the radius can be expressed as $r={\frac {\sqrt {[(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}][(x_{2}-x_{3})^{2}+(y_{2}-y_{3})^{2}][(x_{3}-x_{1})^{2}+(y_{3}-y_{1})^{2}]}}{2\|x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{1}y_{3}-x_{2}y_{1}-x_{3}y_{2}\|}}.$ Regular polygons n Rn 30.577350... 40.707106... 50.850650... 61.0 71.152382... 81.306562... 91.461902... 101.618033... The radius r of a regular polygon with n sides of length s is given by r = Rn s, where $R_{n}=1\left/\left(2\sin {\frac {\pi }{n}}\right)\right..$ Values of Rn for small values of n are given in the table. If s = 1 then these values are also the radii of the corresponding regular polygons. Hypercubes The radius of a d-dimensional hypercube with side s is $r={\frac {s}{2}}{\sqrt {d}}.$ Use in coordinate systems Polar coordinates Main article: Polar coordinate system The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The fixed point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth.[6] Cylindrical coordinates Main article: Cylindrical coordinate system In the cylindrical coordinate system, there is a chosen reference axis and a chosen reference plane perpendicular to that axis. The origin of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis. The axis is variously called the cylindrical or longitudinal axis, to differentiate it from the polar axis, which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction. The distance from the axis may be called the radial distance or radius, while the angular coordinate is sometimes referred to as the angular position or as the azimuth. The radius and the azimuth are together called the polar coordinates, as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane. The third coordinate may be called the height or altitude (if the reference plane is considered horizontal), longitudinal position,[7] or axial position.[8] Spherical coordinates Main article: Spherical coordinate system In a spherical coordinate system, the radius describes the distance of a point from a fixed origin. Its position if further defined by the polar angle measured between the radial direction and a fixed zenith direction, and the azimuth angle, the angle between the orthogonal projection of the radial direction on a reference plane that passes through the origin and is orthogonal to the zenith, and a fixed reference direction in that plane. See also • Bend radius • Filling radius in Riemannian geometry • Radius of convergence • Radius of convexity • Radius of curvature • Radius of gyration • Semidiameter Notes 1. The plural of radius can be either radii (from the Latin plural) or the conventional English plural radiuses.[1] References 1. "Radius - Definition and More from the Free Merriam-Webster Dictionary". Merriam-webster.com. Retrieved 2012-05-22. 2. Definition of Radius at dictionary.reference.com. Accessed on 2009-08-08. 3. Definition of radius at mathwords.com. Accessed on 2009-08-08. 4. Barnett Rich, Christopher Thomas (2008), Schaum's Outline of Geometry, 4th edition, 326 pages. McGraw-Hill Professional. ISBN 0-07-154412-7, ISBN 978-0-07-154412-2. Online version accessed on 2009-08-08. 5. Jonathan L. Gross, Jay Yellen (2006), Graph theory and its applications. 2nd edition, 779 pages; CRC Press. ISBN 1-58488-505-X, 9781584885054. Online version accessed on 2009-08-08. 6. Brown, Richard G. (1997). Andrew M. Gleason (ed.). Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis. Evanston, Illinois: McDougal Littell. ISBN 0-395-77114-5. 7. Krafft, C.; Volokitin, A. S. (1 January 2002). "Resonant electron beam interaction with several lower hybrid waves". Physics of Plasmas. 9 (6): 2786–2797. Bibcode:2002PhPl....9.2786K. doi:10.1063/1.1465420. ISSN 1089-7674. Archived from the original on 14 April 2013. Retrieved 9 February 2013. ...in cylindrical coordinates (r,θ,z) ... and Z=vbzt is the longitudinal position... 8. Groisman, Alexander; Steinberg, Victor (1997-02-24). "Solitary Vortex Pairs in Viscoelastic Couette Flow". Physical Review Letters. American Physical Society (APS). 78 (8): 1460–1463. arXiv:patt-sol/9610008. Bibcode:1997PhRvL..78.1460G. doi:10.1103/physrevlett.78.1460. ISSN 0031-9007. S2CID 54814721. "[...]where r, θ, and z are cylindrical coordinates [...] as a function of axial position[...]" Authority control: National • Germany • Israel • United States	Wikipedia
Radius of curvature In differential geometry, the radius of curvature (Rc), R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof.[1][2][3] Definition In the case of a space curve, the radius of curvature is the length of the curvature vector. In the case of a plane curve, then R is the absolute value of[3] $R\equiv \left\|{\frac {ds}{d\varphi }}\right\|={\frac {1}{\kappa }},$ where s is the arc length from a fixed point on the curve, φ is the tangential angle and κ is the curvature. Formula In 2D Further information: Curvature § Plane curves If the curve is given in Cartesian coordinates as y(x), i.e., as the graph of a function, then the radius of curvature is (assuming the curve is differentiable up to order 2): $R=\left\|{\frac {\left(1+y'^{\,2}\right)^{\frac {3}{2}}}{y''}}\right\|,\qquad {\mbox{where}}\quad y'={\frac {dy}{dx}},\quad y''={\frac {d^{2}y}{dx^{2}}},$ and \|z\| denotes the absolute value of z. Also in Classical mechanics branch of Physics Radius of curvature is given by (Net Velocity)²/Acceleration Perpendicular If the curve is given parametrically by functions x(t) and y(t), then the radius of curvature is $R=\left\|{\frac {ds}{d\varphi }}\right\|=\left\|{\frac {\left({{\dot {x}}^{2}+{\dot {y}}^{2}}\right)^{\frac {3}{2}}}{{\dot {x}}{\ddot {y}}-{\dot {y}}{\ddot {x}}}}\right\|,\qquad {\mbox{where}}\quad {\dot {x}}={\frac {dx}{dt}},\quad {\ddot {x}}={\frac {d^{2}x}{dt^{2}}},\quad {\dot {y}}={\frac {dy}{dt}},\quad {\ddot {y}}={\frac {d^{2}y}{dt^{2}}}.$ Heuristically, this result can be interpreted as[2] $R={\frac {\left\|\mathbf {v} \right\|^{3}}{\left\|\mathbf {v} \times \mathbf {\dot {v}} \right\|}},\qquad {\mbox{where}}\quad \left\|\mathbf {v} \right\|={\big \|}({\dot {x}},{\dot {y}}){\big \|}=R{\frac {d\varphi }{dt}}.$ In n dimensions If γ : ℝ → ℝn is a parametrized curve in ℝn then the radius of curvature at each point of the curve, ρ : ℝ → ℝ, is given by[3] $\rho ={\frac {\left\|{\boldsymbol {\gamma }}'\right\|^{3}}{\sqrt {\left\|{\boldsymbol {\gamma }}'\right\|^{2}\,\left\|{\boldsymbol {\gamma }}''\right\|^{2}-\left({\boldsymbol {\gamma }}'\cdot {\boldsymbol {\gamma }}''\right)^{2}}}}$. As a special case, if f(t) is a function from ℝ to ℝ, then the radius of curvature of its graph, γ(t) = (t, f(t)), is $\rho (t)={\frac {\left\|1+f'^{\,2}(t)\right\|^{\frac {3}{2}}}{\left\|f''(t)\right\|}}.$ Derivation Let γ be as above, and fix t. We want to find the radius ρ of a parametrized circle which matches γ in its zeroth, first, and second derivatives at t. Clearly the radius will not depend on the position γ(t), only on the velocity γ′(t) and acceleration γ″(t). There are only three independent scalars that can be obtained from two vectors v and w, namely v · v, v · w, and w · w. Thus the radius of curvature must be a function of the three scalars \|γ′(t)\|2, \|γ″(t)\|2 and γ′(t) · γ″(t).[3] The general equation for a parametrized circle in ℝn is $\mathbf {g} (u)=\mathbf {a} \cos(h(u))+\mathbf {b} \sin(h(u))+\mathbf {c} $ where c ∈ ℝn is the center of the circle (irrelevant since it disappears in the derivatives), a,b ∈ ℝn are perpendicular vectors of length ρ (that is, a · a = b · b = ρ2 and a · b = 0), and h : ℝ → ℝ is an arbitrary function which is twice differentiable at t. The relevant derivatives of g work out to be ${\begin{aligned}\|\mathbf {g} '\|^{2}&=\rho ^{2}(h')^{2}\\\mathbf {g} '\cdot \mathbf {g} ''&=\rho ^{2}h'h''\\\|\mathbf {g} ''\|^{2}&=\rho ^{2}\left((h')^{4}+(h'')^{2}\right)\end{aligned}}$ If we now equate these derivatives of g to the corresponding derivatives of γ at t we obtain ${\begin{aligned}\|{\boldsymbol {\gamma }}'(t)\|^{2}&=\rho ^{2}h'^{\,2}(t)\\{\boldsymbol {\gamma }}'(t)\cdot {\boldsymbol {\gamma }}''(t)&=\rho ^{2}h'(t)h''(t)\\\|{\boldsymbol {\gamma }}''(t)\|^{2}&=\rho ^{2}\left(h'^{\,4}(t)+h''^{\,2}(t)\right)\end{aligned}}$ These three equations in three unknowns (ρ, h′(t) and h″(t)) can be solved for ρ, giving the formula for the radius of curvature: $\rho (t)={\frac {\left\|{\boldsymbol {\gamma }}'(t)\right\|^{3}}{\sqrt {\left\|{\boldsymbol {\gamma }}'(t)\right\|^{2}\,\left\|{\boldsymbol {\gamma }}''(t)\right\|^{2}-{\big (}{\boldsymbol {\gamma }}'(t)\cdot {\boldsymbol {\gamma }}''(t){\big )}^{2}}}}$ or, omitting the parameter t for readability, $\rho ={\frac {\left\|{\boldsymbol {\gamma }}'\right\|^{3}}{\sqrt {\left\|{\boldsymbol {\gamma }}'\right\|^{2}\;\left\|{\boldsymbol {\gamma }}''\right\|^{2}-\left({\boldsymbol {\gamma }}'\cdot {\boldsymbol {\gamma }}''\right)^{2}}}}.$ Examples Semicircles and circles For a semi-circle of radius a in the upper half-plane $y={\sqrt {a^{2}-x^{2}}},\quad y'={\frac {-x}{\sqrt {a^{2}-x^{2}}}},\quad y''={\frac {-a^{2}}{\left(a^{2}-x^{2}\right)^{\frac {3}{2}}}},\quad R=\|-a\|=a.$ For a semi-circle of radius a in the lower half-plane $y=-{\sqrt {a^{2}-x^{2}}},\quad R=\|a\|=a.$ The circle of radius a has a radius of curvature equal to a. Ellipses In an ellipse with major axis 2a and minor axis 2b, the vertices on the major axis have the smallest radius of curvature of any points, R = b2/a; and the vertices on the minor axis have the largest radius of curvature of any points, R = a2/b. The ellipse's radius of curvature, as a function of parameter t $R(t)={\frac {(b^{2}\cos ^{2}t+a^{2}\sin ^{2}t)^{3/2}}{ab}}\;,\;\;{\text{where}}\;\;\theta =\tan ^{-1}{\Big (}{\frac {y}{x}}{\Big )}=\tan ^{-1}{\Big (}{\frac {b}{a}}\;\tan \;t{\Big )}$ [4] And as a function of θ $R(\theta )={\frac {a^{2}}{b}}{\biggl (}{\frac {1-e^{2}(2-e^{2})(\cos \theta )^{2})}{1-e^{2}(\cos \theta )^{2}}}{\biggr )}^{3/2}$ Where e is the eccentricity of the ellipse and is given by: $e^{2}=1-{\frac {b^{2}}{a^{2}}}$ Applications • For the use in differential geometry, see Cesàro equation. • For the radius of curvature of the Earth (approximated by an oblate ellipsoid); see also: arc measurement • Radius of curvature is also used in a three part equation for bending of beams. • Radius of curvature (optics) • Thin films technologies • Printed electronics • Minimum railway curve radius • AFM probe Stress in semiconductor structures Stress in the semiconductor structure involving evaporated thin films usually results from the thermal expansion (thermal stress) during the manufacturing process. Thermal stress occurs because film depositions are usually made above room temperature. Upon cooling from the deposition temperature to room temperature, the difference in the thermal expansion coefficients of the substrate and the film cause thermal stress.[5] Intrinsic stress results from the microstructure created in the film as atoms are deposited on the substrate. Tensile stress results from microvoids (small holes, considered to be defects) in the thin film, because of the attractive interaction of atoms across the voids. The stress in thin film semiconductor structures results in the buckling of the wafers. The radius of the curvature of the stressed structure is related to stress tensor in the structure, and can be described by modified Stoney formula.[6] The topography of the stressed structure including radii of curvature can be measured using optical scanner methods. The modern scanner tools have capability to measure full topography of the substrate and to measure both principal radii of curvature, while providing the accuracy of the order of 0.1% for radii of curvature of 90 meters and more.[7] See also • Base curve radius • Bend radius • Degree of curvature (civil engineering) • Osculating circle • Track transition curve References 1. Weisstien, Eric. "Radius of Curvature". Wolfram Mathworld. Retrieved 15 August 2016. 2. Kishan, Hari (2007). Differential Calculus. Atlantic Publishers & Dist. ISBN 9788126908202. 3. Love, Clyde E.; Rainville, Earl D. (1962). Differential and Integral Calculus (Sixth ed.). New York: MacMillan. 4. Weisstein, Eric W. "Ellipse". mathworld.wolfram.com. Retrieved 2022-02-23. 5. "Controlling Stress in Thin Films". Flipchips.com. Retrieved 2016-04-22. 6. "On the determination of film stress from substrate bending : Stoney's formula and its limits" (PDF). Qucosa.de. Retrieved 2016-04-22. 7. Peter Walecki. "Model X". Zebraoptical.com. Retrieved 2016-04-22. Further reading • do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces. ISBN 0-13-212589-7. External links • The Geometry Center: Principal Curvatures • 15.3 Curvature and Radius of Curvature • Weisstein, Eric W. "Principal Curvatures". MathWorld. • Weisstein, Eric W. "Principal Radius of Curvature". MathWorld. Various notions of curvature defined in differential geometry Differential geometry of curves • Curvature • Torsion of a curve • Frenet–Serret formulas • Radius of curvature (applications) • Affine curvature • Total curvature • Total absolute curvature Differential geometry of surfaces • Principal curvatures • Gaussian curvature • Mean curvature • Darboux frame • Gauss–Codazzi equations • First fundamental form • Second fundamental form • Third fundamental form Riemannian geometry • Curvature of Riemannian manifolds • Riemann curvature tensor • Ricci curvature • Scalar curvature • Sectional curvature Curvature of connections • Curvature form • Torsion tensor • Cocurvature • Holonomy	Wikipedia
Grunsky's theorem In mathematics, Grunsky's theorem, due to the German mathematician Helmut Grunsky, is a result in complex analysis concerning holomorphic univalent functions defined on the unit disk in the complex numbers. The theorem states that a univalent function defined on the unit disc, fixing the point 0, maps every disk \|z\| < r onto a starlike domain for r ≤ tanh π/4. The largest r for which this is true is called the radius of starlikeness of the function. Statement Let f be a univalent holomorphic function on the unit disc D such that f(0) = 0. Then for all r ≤ tanh π/4, the image of the disc \|z\| < r is starlike with respect to 0, , i.e. it is invariant under multiplication by real numbers in (0,1). An inequality of Grunsky If f(z) is univalent on D with f(0) = 0, then $\left\|\log {zf^{\prime }(z) \over f(z)}\right\|\leq \log {1+\|z\| \over 1-\|z\|}.$ Taking the real and imaginary parts of the logarithm, this implies the two inequalities $\left\|{zf^{\prime }(z) \over f(z)}\right\|\leq {1+\|z\| \over 1-\|z\|}$ and $\left\|\arg {zf^{\prime }(z) \over f(z)}\right\|\leq \log {1+\|z\| \over 1-\|z\|}.$ For fixed z, both these equalities are attained by suitable Koebe functions $g_{w}(\zeta )={\zeta \over (1-{\overline {w}}\zeta )^{2}},$ where \|w\| = 1. Proof Grunsky (1932) originally proved these inequalities based on extremal techniques of Ludwig Bieberbach. Subsequent proofs, outlined in Goluzin (1939), relied on the Loewner equation. More elementary proofs were subsequently given based on Goluzin's inequalities, an equivalent form of Grunsky's inequalities (1939) for the Grunsky matrix. For a univalent function g in z > 1 with an expansion $g(z)=z+b_{1}z^{-1}+b_{2}z^{-2}+\cdots .$ Goluzin's inequalities state that $\left\|\sum _{i=1}^{n}\sum _{j=1}^{n}\lambda _{i}\lambda _{j}\log {g(z_{i})-g(z_{j}) \over z_{i}-z_{j}}\right\|\leq \sum _{i=1}^{n}\sum _{j=1}^{n}\lambda _{i}{\overline {\lambda _{j}}}\log {z_{i}{\overline {z_{j}}} \over z_{i}{\overline {z_{j}}}-1},$ where the zi are distinct points with \|zi\| > 1 and λi are arbitrary complex numbers. Taking n = 2. with λ1 = – λ2 = λ, the inequality implies $\left\|\log {g^{\prime }(\zeta )g^{\prime }(\eta )(\zeta -\eta )^{2} \over (g(\zeta )-g(\eta ))^{2}}\right\|\leq \log {\|1-\zeta {\overline {\eta }}\|^{2} \over (\|\zeta \|^{2}-1)(\|\eta \|^{2}-1)}.$ If g is an odd function and η = – ζ, this yields $\left\|\log {\zeta g^{\prime }(\zeta ) \over g(\zeta )}\right\|\leq {\|\zeta \|^{2}+1 \over \|\zeta \|^{2}-1}.$ Finally if f is any normalized univalent function in D, the required inequality for f follows by taking $g(\zeta )=f(\zeta ^{-2})^{-{1 \over 2}}$ with $z=\zeta ^{-2}.$ Proof of the theorem Let f be a univalent function on D with f(0) = 0. By Nevanlinna's criterion, f is starlike on \|z\| < r if and only if $\Re {zf^{\prime }(z) \over f(z)}\geq 0$ for \|z\| < r. Equivalently $\left\|\arg {zf^{\prime }(z) \over f(z)}\right\|\leq {\pi \over 2}.$ On the other hand by the inequality of Grunsky above, $\left\|\arg {zf^{\prime }(z) \over f(z)}\right\|\leq \log {1+\|z\| \over 1-\|z\|}.$ Thus if $\log {1+\|z\| \over 1-\|z\|}\leq {\pi \over 2},$ the inequality holds at z. This condition is equivalent to $\|z\|\leq \tanh {\pi \over 4}$ and hence f is starlike on any disk \|z\| < r with r ≤ tanh π/4. References • Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, pp. 95–98, ISBN 0-387-90795-5 • Goluzin, G.M. (1939), "Interior problems of the theory of univalent functions", Uspekhi Mat. Nauk, 6: 26–89 (in Russian) • Goluzin, G. M. (1969), Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, vol. 26, American Mathematical Society • Goodman, A.W. (1983), Univalent functions, vol. I, Mariner Publishing Co., ISBN 0-936166-10-X • Goodman, A.W. (1983), Univalent functions, vol. II, Mariner Publishing Co., ISBN 0-936166-11-8 • Grunsky, H. (1932), "Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche (inaugural dissertation)", Schr. Math. Inst. U. Inst. Angew. Math. Univ. Berlin, 1: 95–140, archived from the original on 2015-02-11, retrieved 2011-12-07 (in German) • Grunsky, H. (1934), "Zwei Bemerkungen zur konformen Abbildung", Jber. Deutsch. Math.-Verein., 43: 140–143 (in German) • Hayman, W. K. (1994), Multivalent functions, Cambridge Tracts in Mathematics, vol. 110 (2nd ed.), Cambridge University Press, ISBN 0-521-46026-3 • Nevanlinna, R. (1921), "Über die konforme Abbildung von Sterngebieten", Öfvers. Finska Vet. Soc. Forh., 53: 1–21 • Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht	Wikipedia
Radivoj Kašanin Radivoj Kašanin or Radivoje Kašanin (Beli Manastir, then part of the Habsburg monarchy, 21 May 1892 — Belgrade, Yugoslavia, 30 October 1989) was a Serbian mathematician,[1] university professor, and member of the Serbian Academy of Arts and Sciences. His mentor was the famed Balkan mathematician Mihailo Petrović Alas.[2] Radivoje Kašanin is regarded as a talented mathematician and scholar of natural sciences with a wide scientific culture. As for his profound and diversified knowledge in many areas of mathematics, mechanics, and astronomy he could be considered as Serbia's last encyclopedist. Radivoje Kašanin achieved success in many fields of his profession: theory of differential equations, the theory of complex functions, analysis, geometry, interpolation and approximation, mechanics, astronomy and geophysics and in each of mentioned fields of his work he published papers that were widely acknowledged. Biography Radivoje Kašanin attended the Serbian elementary school in his native town from 1892 to 1902. He completed the first three classes of the classic gymnasium in Osijek, and then he moved to Novi Sad, where he finished the fourth class and passed the final examination. In 1910 he began his studies in mathematics and astronomy at the University of Vienna, then in 1911 at the University of Zagreb, where he stayed for two years before enrolling at the University of Budapest in 1913.[3] The Great War cut short his studies in Budapest when he was mobilized by the Austro-Hungarian Army in 1914. He was immediately sent to the Russian front, where he survived the hostilities and by the end of the war, he went to Paris to pursue his higher studies at the Sorbonne in 1921. In 1924 he successfully defended his dissertation and received his Ph.D. in mathematics. He was appointed assistant at the Technical Faculty of the University of Belgrade in 1922, an assistant professor in 1926, associate professor in 1930, and full professor in 1939. He was elected Rector of the Technical High School for two terms of office, 1950 and 1951. Also, he was elected a corresponding member of the Serbian Academy of Sciences and Arts on 2 March 1946, and a full member on June 10, 1955. He served the post of director of the Institute of Mathematics from 1951 to 1958, was president of its Council from 1958 to 1961. In 1950 the Proceeding of the Institute of Mathematics was published and during the next ten years, Radivoje Kašanin was its editor-in-chief. From 1 October 1957 to 12 January 1959 Radivoje Kašanin served as deputy vice-president of the Serbian Academy of Sciences. He devoted his last years mathematically interpreting the cosmogonical theory of Pavle Savić.[3] Radivoje Kašanin died in Belgrade on October 30, 1989 where he was buried. Bibliography Only mathematical textbooks are included here, the rest can be found in a five page Bibliography in Serbian and English.[4] • "Viša matematika I", Grafički zavod "Slavija", sv. 1, Beograd, 1932, str. 80 • "Viša matematika I, sv. 1", Izdavačka knjižarnica Gece Kona, Beograd, 1933, str. 160 • "Viša matematika I", Beograd, 1934, str. 627 • "Viša matematika I", Centralno udruženje studenata tehnike, Beograd, 1946, str. 791 (2. prer. i dop. izdanje) • "Viša matematika I", Beograd, 1949, str. 847 (3. izdanje) • "Viša matematika II", knj. 1, Beograd, 1949, str. 624 * "Viša matematika II", knj. 2, Beograd, 1950, str. 679 • "Zbirka rešenih zadataka više matematike I, knj. 2", Geografski institut JNA, Beograd, 1952, str. 526 • "Zbirka rešenih zadataka više matematike I, knj. 1", Geografski institut JNA, Beograd, 1956, str. 588+(4) • "Zbirka rešenih zadataka više matematike I, knj. 3", Geografski institut JNA, Beograd, 1959, str. 164+(4) • "Viša matematika I", Sarajevo, 1969, str. 836 (4. izdanje) References 1. Lives and work of the Serbian scientists. Srpska akademija nauka i umetnosti. July 8, 1996. ISBN 9788670252837 – via Google Books. 2. Robson, Eleanor; Stedall, Jacqueline (July 8, 2009). The Oxford Handbook of the History of Mathematics. OUP Oxford. ISBN 978-0-19-921312-2 – via Google Books. 3. http://elibrary.matf.bg.ac.rs/bitstream/handle/123456789/598/ViseAutoraPrisutnaProslost.PDF?sequence=1 4. http://www.mi.sanu.ac.rs/novi_sajt/biography/Kasanin.pdf	Wikipedia
Radix sort In computer science, radix sort is a non-comparative sorting algorithm. It avoids comparison by creating and distributing elements into buckets according to their radix. For elements with more than one significant digit, this bucketing process is repeated for each digit, while preserving the ordering of the prior step, until all digits have been considered. For this reason, radix sort has also been called bucket sort and digital sort. Radix sort ClassSorting algorithm Data structureArray Worst-case performance$O(w\cdot n)$, where $n$ is the number of keys, and $w$ is the key length. Worst-case space complexity$O(w+n)$ Radix sort can be applied to data that can be sorted lexicographically, be they integers, words, punch cards, playing cards, or the mail. History Radix sort dates back as far as 1887 to the work of Herman Hollerith on tabulating machines.[1] Radix sorting algorithms came into common use as a way to sort punched cards as early as 1923.[2] The first memory-efficient computer algorithm for this sorting method was developed in 1954 at MIT by Harold H. Seward. Computerized radix sorts had previously been dismissed as impractical because of the perceived need for variable allocation of buckets of unknown size. Seward's innovation was to use a linear scan to determine the required bucket sizes and offsets beforehand, allowing for a single static allocation of auxiliary memory. The linear scan is closely related to Seward's other algorithm — counting sort. In the modern era, radix sorts are most commonly applied to collections of binary strings and integers. It has been shown in some benchmarks to be faster than other more general-purpose sorting algorithms, sometimes 50% to three times faster.[3][4][5] Digit order Radix sorts can be implemented to start at either the most significant digit (MSD) or least significant digit (LSD). For example, with 1234, one could start with 1 (MSD) or 4 (LSD). LSD radix sorts typically use the following sorting order: short keys come before longer keys, and then keys of the same length are sorted lexicographically. This coincides with the normal order of integer representations, like the sequence [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. LSD sorts are generally stable sorts. MSD radix sorts are most suitable for sorting strings or fixed-length integer representations. A sequence like [b, c, e, d, f, g, ba] would be sorted as [b, ba, c, d, e, f, g]. If lexicographic ordering is used to sort variable-length integers in base 10, then numbers from 1 to 10 would be output as [1, 10, 2, 3, 4, 5, 6, 7, 8, 9], as if the shorter keys were left-justified and padded on the right with blank characters to make the shorter keys as long as the longest key. MSD sorts are not necessarily stable if the original ordering of duplicate keys must always be maintained. Other than the traversal order, MSD and LSD sorts differ in their handling of variable length input. LSD sorts can group by length, radix sort each group, then concatenate the groups in size order. MSD sorts must effectively 'extend' all shorter keys to the size of the largest key and sort them accordingly, which can be more complicated than the grouping required by LSD. However, MSD sorts are more amenable to subdivision and recursion. Each bucket created by an MSD step can itself be radix sorted using the next most significant digit, without reference to any other buckets created in the previous step. Once the last digit is reached, concatenating the buckets is all that is required to complete the sort. Examples Least significant digit Input list: [170, 45, 75, 90, 2, 802, 2, 66] Starting from the rightmost (last) digit, sort the numbers based on that digit: [{170, 90}, {2, 802, 2}, {45, 75}, {66}] Sorting by the next left digit: [{02, 802, 02}, {45}, {66}, {170, 75}, {90}] Notice that an implicit digit 0 is prepended for the two 2s so that 802 maintains its position between them. And finally by the leftmost digit: [{002, 002, 045, 066, 075, 090}, {170}, {802}] Notice that a 0 is prepended to all of the 1- or 2-digit numbers. Each step requires just a single pass over the data, since each item can be placed in its bucket without comparison with any other element. Some radix sort implementations allocate space for buckets by first counting the number of keys that belong in each bucket before moving keys into those buckets. The number of times that each digit occurs is stored in an array. Although it is always possible to pre-determine the bucket boundaries using counts, some implementations opt to use dynamic memory allocation instead. Most significant digit, forward recursive Input list, fixed width numeric strings with leading zeros: [170, 045, 075, 025, 002, 024, 802, 066] First digit, with brackets indicating buckets: [{045, 075, 025, 002, 024, 066}, {170}, {802}] Notice that 170 and 802 are already complete because they are all that remain in their buckets, so no further recursion is needed Next digit: [{ {002}, {025, 024}, {045}, {066}, {075} }, 170, 802] Final digit: [ 002, { {024}, {025} }, 045, 066, 075 , 170, 802] All that remains is concatenation: [002, 024, 025, 045, 066, 075, 170, 802] Complexity and performance Radix sort operates in $O(nw)$ time, where $n$ is the number of keys, and $w$ is the key length. LSD variants can achieve a lower bound for $w$ of 'average key length' when splitting variable length keys into groups as discussed above. Optimized radix sorts can be very fast when working in a domain that suits them.[6] They are constrained to lexicographic data, but for many practical applications this is not a limitation. Large key sizes can hinder LSD implementations when the induced number of passes becomes the bottleneck.[2] Specialized variants In-place MSD radix sort implementations Binary MSD radix sort, also called binary quicksort, can be implemented in-place by splitting the input array into two bins - the 0s bin and the 1s bin. The 0s bin is grown from the beginning of the array, whereas the 1s bin is grown from the end of the array. The 0s bin boundary is placed before the first array element. The 1s bin boundary is placed after the last array element. The most significant bit of the first array element is examined. If this bit is a 1, then the first element is swapped with the element in front of the 1s bin boundary (the last element of the array), and the 1s bin is grown by one element by decrementing the 1s boundary array index. If this bit is a 0, then the first element remains at its current location, and the 0s bin is grown by one element. The next array element examined is the one in front of the 0s bin boundary (i.e. the first element that is not in the 0s bin or the 1s bin). This process continues until the 0s bin and the 1s bin reach each other. The 0s bin and the 1s bin are then sorted recursively based on the next bit of each array element. Recursive processing continues until the least significant bit has been used for sorting.[7][8] Handling signed two's complement integers requires treating the most significant bit with the opposite sense, followed by unsigned treatment of the rest of the bits. In-place MSD binary-radix sort can be extended to larger radix and retain in-place capability. Counting sort is used to determine the size of each bin and their starting index. Swapping is used to place the current element into its bin, followed by expanding the bin boundary. As the array elements are scanned the bins are skipped over and only elements between bins are processed, until the entire array has been processed and all elements end up in their respective bins. The number of bins is the same as the radix used - e.g. 16 bins for 16-radix. Each pass is based on a single digit (e.g. 4-bits per digit in the case of 16-radix), starting from the most significant digit. Each bin is then processed recursively using the next digit, until all digits have been used for sorting.[9][10] Neither in-place binary-radix sort nor n-bit-radix sort, discussed in paragraphs above, are stable algorithms. Stable MSD radix sort implementations MSD radix sort can be implemented as a stable algorithm, but requires the use of a memory buffer of the same size as the input array. This extra memory allows the input buffer to be scanned from the first array element to last, and move the array elements to the destination bins in the same order. Thus, equal elements will be placed in the memory buffer in the same order they were in the input array. The MSD-based algorithm uses the extra memory buffer as the output on the first level of recursion, but swaps the input and output on the next level of recursion, to avoid the overhead of copying the output result back to the input buffer. Each of the bins are recursively processed, as is done for the in-place MSD radix sort. After the sort by the last digit has been completed, the output buffer is checked to see if it is the original input array, and if it's not, then a single copy is performed. If the digit size is chosen such that the key size divided by the digit size is an even number, the copy at the end is avoided.[11] Hybrid approaches Radix sort, such as the two-pass method where counting sort is used during the first pass of each level of recursion, has a large constant overhead. Thus, when the bins get small, other sorting algorithms should be used, such as insertion sort. A good implementation of insertion sort is fast for small arrays, stable, in-place, and can significantly speed up radix sort. Application to parallel computing This recursive sorting algorithm has particular application to parallel computing, as each of the bins can be sorted independently. In this case, each bin is passed to the next available processor. A single processor would be used at the start (the most significant digit). By the second or third digit, all available processors would likely be engaged. Ideally, as each subdivision is fully sorted, fewer and fewer processors would be utilized. In the worst case, all of the keys will be identical or nearly identical to each other, with the result that there will be little to no advantage to using parallel computing to sort the keys. In the top level of recursion, opportunity for parallelism is in the counting sort portion of the algorithm. Counting is highly parallel, amenable to the parallel_reduce pattern, and splits the work well across multiple cores until reaching memory bandwidth limit. This portion of the algorithm has data-independent parallelism. Processing each bin in subsequent recursion levels is data-dependent, however. For example, if all keys were of the same value, then there would be only a single bin with any elements in it, and no parallelism would be available. For random inputs all bins would be near equally populated and a large amount of parallelism opportunity would be available.[12] There are faster parallel sorting algorithms available, for example optimal complexity O(log(n)) are those of the Three Hungarians and Richard Cole[13][14] and Batcher's bitonic merge sort has an algorithmic complexity of O(log2(n)), all of which have a lower algorithmic time complexity to radix sort on a CREW-PRAM. The fastest known PRAM sorts were described in 1991 by David Powers with a parallelized quicksort that can operate in O(log(n)) time on a CRCW-PRAM with n processors by performing partitioning implicitly, as well as a radixsort that operates using the same trick in O(k), where k is the maximum keylength.[15] However, neither the PRAM architecture or a single sequential processor can actually be built in a way that will scale without the number of constant fan-out gate delays per cycle increasing as O(log(n)), so that in effect a pipelined version of Batcher's bitonic mergesort and the O(log(n)) PRAM sorts are all O(log2(n)) in terms of clock cycles, with Powers acknowledging that Batcher's would have lower constant in terms of gate delays than his Parallel quicksort and radix sort, or Cole's merge sort, for a keylength-independent sorting network of O(nlog2(n)).[16] Tree-based radix sort Radix sorting can also be accomplished by building a tree (or radix tree) from the input set, and doing a pre-order traversal. This is similar to the relationship between heapsort and the heap data structure. This can be useful for certain data types, see burstsort. See also • IBM 80 series Card Sorters • Other distribution sorts • Kirkpatrick-Reisch sorting • Prefix sum References 1. US 395781 and UK 327 2. Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89685-0. Section 5.2.5: Sorting by Distribution, pp. 168–179. 3. "I Wrote a Faster Sorting Algorithm". 28 December 2016. 4. "Is radix sort faster than quicksort for integer arrays?". erik.gorset.no. 5. "Function template integer_sort - 1.62.0". www.boost.org. 6. Sinha, Ranjan; Zobel, Justin. "Efficient Trie-Based Sorting of Large Sets of Strings". CiteSeerX 10.1.1.12.2367. Retrieved 24 August 2023. 7. R. Sedgewick, "Algorithms in C++", third edition, 1998, p. 424-427 8. Duvanenko, Victor J. "Algorithm Improvement through Performance Measurement: Part 2". Dr. Dobb's. 9. Duvanenko, Victor J. "Algorithm Improvement through Performance Measurement: Part 3". Dr. Dobb's. 10. Duvanenko, Victor J. "Parallel In-Place Radix Sort Simplified". Dr. Dobb's. 11. Duvanenko, Victor J. "Algorithm Improvement through Performance Measurement: Part 4". Dr. Dobb's. 12. Duvanenko, Victor J. "Parallel In-Place N-bit-Radix Sort". Dr. Dobb's. 13. A. Gibbons and W. Rytter, Efficient Parallel Algorithms. Cambridge University Press, 1988. 14. H. Casanova et al, Parallel Algorithms. Chapman & Hall, 2008. 15. David M. W. Powers, Parallelized Quicksort and Radixsort with Optimal Speedup, Proceedings of International Conference on Parallel Computing Technologies. Novosibirsk. 1991. 16. David M. W. Powers, Parallel Unification: Practical Complexity, Australasian Computer Architecture Workshop, Flinders University, January 1995 External links The Wikibook Algorithm implementation has a page on the topic of: Radix sort • Explanation, Pseudocode and implementation in C and Java • High Performance Implementation of LSD Radix sort in JavaScript • High Performance Implementation of LSD & MSD Radix sort in C# with source in GitHub • Video tutorial of MSD Radix Sort • Demonstration and comparison of Radix sort with Bubble sort, Merge sort and Quicksort implemented in JavaScript • Article about Radix sorting IEEE floating-point numbers with implementation. Faster Floating Point Sorting and Multiple Histogramming with implementation in C++ • Pointers to radix sort visualizations • USort library contains tuned implementations of radix sort for most numerical C types (C99) • Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89685-0. Section 5.2.5: Sorting by Distribution, pp. 168–179. • Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 8.3: Radix sort, pp. 170–173. • BRADSORT v1.50 source code • Efficient Trie-Based Sorting of Large Sets of Strings, by Ranjan Sinha and Justin Zobel. This paper describes a method of creating tries of buckets which figuratively burst into sub-tries when the buckets hold more than a predetermined capacity of strings, hence the name, "Burstsort". • Open Data Structures - Java Edition - Section 11.2 - Counting Sort and Radix Sort, Pat Morin • Open Data Structures - C++ Edition - Section 11.2 - Counting Sort and Radix Sort, Pat Morin Sorting algorithms Theory • Computational complexity theory • Big O notation • Total order • Lists • Inplacement • Stability • Comparison sort • Adaptive sort • Sorting network • Integer sorting • X + Y sorting • Transdichotomous model • Quantum sort Exchange sorts • Bubble sort • Cocktail shaker sort • Odd–even sort • Comb sort • Gnome sort • Proportion extend sort • Quicksort Selection sorts • Selection sort • Heapsort • Smoothsort • Cartesian tree sort • Tournament sort • Cycle sort • Weak-heap sort Insertion sorts • Insertion sort • Shellsort • Splaysort • Tree sort • Library sort • Patience sorting Merge sorts • Merge sort • Cascade merge sort • Oscillating merge sort • Polyphase merge sort Distribution sorts • American flag sort • Bead sort • Bucket sort • Burstsort • Counting sort • Interpolation sort • Pigeonhole sort • Proxmap sort • Radix sort • Flashsort Concurrent sorts • Bitonic sorter • Batcher odd–even mergesort • Pairwise sorting network • Samplesort Hybrid sorts • Block merge sort • Kirkpatrick–Reisch sort • Timsort • Introsort • Spreadsort • Merge-insertion sort Other • Topological sorting • Pre-topological order • Pancake sorting • Spaghetti sort Impractical sorts • Stooge sort • Slowsort • Bogosort	Wikipedia
Decimal separator A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g. "." in 12.45). Different countries officially designate different symbols for use as the separator. The choice of symbol also affects the choice of symbol for the thousands separator used in digit grouping. "Decimal period" redirects here. For the period of a repeating decimal fraction, see Repeating decimal. Part of a series on Numeral systems Place-value notation Hindu-Arabic numerals • Western Arabic • Eastern Arabic • Bengali • Devanagari • Gujarati • Gurmukhi • Odia • Sinhala • Tamil • Malayalam • Telugu • Kannada • Dzongkha • Tibetan • Balinese • Burmese • Javanese • Khmer • Lao • Mongolian • Sundanese • Thai East Asian systems Contemporary • Chinese • Suzhou • Hokkien • Japanese • Korean • Vietnamese Historic • Counting rods • Tangut Other systems • History Ancient • Babylonian Post-classical • Cistercian • Mayan • Muisca • Pentadic • Quipu • Rumi Contemporary • Cherokee • Kaktovik (Iñupiaq) By radix/base Common radices/bases • 2 • 3 • 4 • 5 • 6 • 8 • 10 • 12 • 16 • 20 • 60 • (table) Non-standard radices/bases • Bijective (1) • Signed-digit (balanced ternary) • Mixed (factorial) • Negative • Complex (2i) • Non-integer (φ) • Asymmetric Sign-value notation Non-alphabetic • Aegean • Attic • Aztec • Brahmi • Chuvash • Egyptian • Etruscan • Kharosthi • Prehistoric counting • Proto-cuneiform • Roman • Tally marks Alphabetic • Abjad • Armenian • Alphasyllabic • Akṣarapallī • Āryabhaṭa • Kaṭapayādi • Coptic • Cyrillic • Geʽez • Georgian • Glagolitic • Greek • Hebrew List of numeral systems Any such symbol can be called a decimal mark, decimal marker, or decimal sign. Symbol-specific names are also used; decimal point and decimal comma refer to an (either baseline or middle) dot and comma respectively, when it is used as a decimal separator; these are the usual terms used in English,[1][2][3] with the aforementioned generic terms reserved for abstract usage.[4][5] In many contexts, when a number is spoken, the function of the separator is assumed by the spoken name of the symbol: comma or point in most cases.[6][2][7] In some specialized contexts, the word decimal is instead used for this purpose (such as in International Civil Aviation Organization-regulated air traffic control communications). In mathematics, the decimal separator is a type of radix point, a term that also applies to number systems with bases other than ten. History Hellenistic–Renaissance eras In the Middle Ages, from the original Indian decimal writing, before printing, a bar ( ¯ ) over the units digit was used to separate the integral part of a number from its fractional part, as in 9995 (meaning 99.95 in decimal point format). A similar notation remains in common use as an underbar to superscript digits, especially for monetary values without a decimal separator, as in 9995. Later, a "separatrix" (i.e., a short, roughly vertical ink stroke) between the units and tenths position became the norm among Arab mathematicians (e.g. 99ˌ95), while an L-shaped or vertical bar (\|) served as the separatrix in England.[8] When this character was typeset, it was convenient to use the existing comma (99,95) or full stop (99.95) instead. Positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century.[9] The practice is ultimately derived from the decimal Hindu–Arabic numeral system used in Indian mathematics,[10] and popularized by the Persian mathematician Al-Khwarizmi,[11] when Latin translation of his work on the Indian numerals introduced the decimal positional number system to the Western world. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations in Arabic. Gerbert of Aurillac marked triples of columns with an arc (called a "Pythagorean arc"), when using his Hindu–Arabic numeral-based abacus in the 10th century. Fibonacci followed this convention when writing numbers, such as in his influential work Liber Abaci in the 13th century.[12] Tables of logarithms prepared by John Napier in 1614 and 1619 used the period (full stop) as the decimal separator, which was then adopted by Henry Briggs in his influential 17th century work. In France, the full stop was already in use in printing to make Roman numerals more readable, so the comma was chosen.[13] Many other countries, such as Italy, also chose to use the comma to mark the decimal units position.[13] It has been made standard by the ISO for international blueprints.[14] However, English-speaking countries took the comma to separate sequences of three digits. In some countries, a raised dot or dash (upper comma) may be used for grouping or decimal separator; this is particularly common in handwriting. English-speaking countries In the United States, the full stop or period (.) is used as the standard decimal separator. In the nations of the British Empire (and, later, the Commonwealth of Nations), the full stop could be used in typewritten material and its use was not banned, although the interpunct (a.k.a. decimal point, point or mid dot) was preferred as a decimal separator, in printing technologies that could accommodate it, e.g. 99·95 .[16] However, as the mid dot was already in common use in the mathematics world to indicate multiplication, the SI rejected its use as the decimal separator. During the beginning of British metrication in the late 1960s and with impending currency decimalisation, there was some debate in the United Kingdom as to whether the decimal comma or decimal point should be preferred: the British Standards Institution and some sectors of industry advocated the comma and the Decimal Currency Board advocated for the point. In the event, the point was chosen by the Ministry of Technology in 1968.[17] When South Africa adopted the metric system, it adopted the comma as its decimal separator,[18] although a number of house styles, including some English-language newspapers such as The Sunday Times, continue to use the full stop. Previously, signs along California roads expressed distances in decimal numbers with the decimal part in superscript, as in 37, meaning 3.7 .[19] Though California has since transitioned to mixed numbers with common fractions, the older style remains on postmile markers and bridge inventory markers. Artificial languages The three most spoken international auxiliary languages, Ido, Esperanto, and Interlingua, all use the comma as the decimal separator. Interlingua has used the comma as its decimal separator since the publication of the Interlingua Grammar in 1951.[20] Esperanto also uses the comma as its official decimal separator, while thousands are separated by non-breaking spaces; e.g. 12 345 678,9. Ido's Kompleta Gramatiko Detaloza di la Linguo Internaciona Ido (Complete Detailed Grammar of the International Language Ido) officially states that commas are used for the decimal separator while full stops are used to separate thousands, millions, etc. So the number 12,345,678.90123 (in American notation) for instance, would be written 12.345.678,90123 in Ido. The 1931 grammar of Volapük uses the comma as its decimal separator, and – somewhat unusually – uses the middle dot as the thousands separator (12·345·678,90123).[21] In 1958, disputes between European and American delegates over the correct representation of the decimal separator nearly stalled the development of the ALGOL computer programming language.[22] ALGOL ended up allowing different decimal separators, but most computer languages and standard data formats (e.g., C, Java, Fortran, Cascading Style Sheets (CSS)) specify a dot. C and a couple of others permits a quote (') as thousands separator. Radix point In mathematics and computing, a radix point or radix character is a symbol used in the display of numbers to separate the integer part of the value from its fractional part. In English and many other languages (including many that are written right-to-left), the integer part is at the left of the radix point, and the fraction part at the right of it.[23] A radix point is most often used in decimal (base 10) notation, when it is more commonly called the decimal point (the prefix deci- implying base 10). In English-speaking countries, the decimal point is usually a small dot (.) placed either on the baseline, or halfway between the baseline and the top of the digits (·)[24][lower-alpha 1] In many other countries, the radix point is a comma (,) placed on the baseline.[24][lower-alpha 1] These conventions are generally used both in machine displays (printing, computer monitors) and in handwriting. It is important to know which notation is being used when working in different software programs. The respective ISO standard defines both the comma and the small dot as decimal markers, but does not explicitly define universal radix marks for bases other than 10. Fractional numbers are rarely displayed in other number bases, but, when they are, a radix character may be used for the same purpose. When used with the binary (base 2) representation, it may be called "binary point". Current standards The 22nd General Conference on Weights and Measures[25] declared in 2003 that "the symbol for the decimal marker shall be either the point on the line or the comma on the line". It further reaffirmed that "numbers may be divided in groups of three in order to facilitate reading; neither dots nor commas are ever inserted in the spaces between groups"[25] (1 000 000 000 for example).[25] This use has therefore been recommended by technical organizations, such as the United States' National Institute of Standards and Technology.[26] Past versions of ISO 8601, but not the 2019 revision, also stipulated normative notation based on SI conventions, adding that the comma is preferred over the full stop.[27] ISO 80000-1 stipulates that "The decimal sign is either a comma or a point on the line." The standard does not stipulate any preference, observing that usage will depend on customary usage in the language concerned, but adds a note that as per ISO/IEC directives, all ISO standards should use the comma as the decimal marker. Digit grouping For ease of reading, numbers with many digits may be divided into groups using a delimiter,[28] such as comma "," or dot ".", half-space (or thin space) " ", space " ", underscore "_" (as in maritime "21_450") or apostrophe «'». In some countries, these "digit group separators" are only employed to the left of the decimal separator; in others, they are also used to separate numbers with a long fractional part. An important reason for grouping is that it allows rapid judgement of the number of digits, via telling at a glance ("subitizing") rather than counting (contrast, for example, 100 000 000 with 100000000 for one hundred million). The use of thin spaces as separators,[29]: 133 not dots or commas (for example: 20 000 and 1 000 000 for "twenty thousand" and "one million"), has been official policy of the International Bureau of Weights and Measures since 1948 (and reaffirmed in 2003) stating "neither dots nor commas are ever inserted in the spaces between groups",[25] as well as of the International Union of Pure and Applied Chemistry (IUPAC),[30][31] the American Medical Association's widely followed AMA Manual of Style, and the Metrication Board, among others. The groups created by the delimiters tend to follow the use of the local language, which varies. In European languages, large numbers are read in groups of thousands, and the delimiter – which occurs every three digits when it is used – may be called a "thousands separator". In East Asian cultures, particularly China, Japan, and Korea, large numbers are read in groups of myriads (10 000s) but the delimiter commonly separates every three digits. The Indian numbering system is somewhat more complex: It groups the rightmost three digits together (until the hundreds place) and thereafter groups by sets of two digits. For example, one American trillion (European billion) would thus be written as 10,00,00,00,00,000 or 10 kharab.[32] The convention for digit group separators historically varied among countries, but usually seeking to distinguish the delimiter from the decimal separator. Traditionally, English-speaking countries (except South Africa)[33] employed commas as the delimiter – 10,000 – and other European countries employed periods or spaces: 10.000 or 10 000. Because of the confusion that could result in international documents, in recent years the use of spaces as separators has been advocated by the superseded SI/ISO 31-0 standard,[34] as well as by the International Bureau of Weights and Measures and the International Union of Pure and Applied Chemistry, which have also begun advocating the use of a "thin space" in "groups of three".[30][31] Within the United States, the American Medical Association's widely followed AMA Manual of Style also calls for a thin space.[28] In some online encoding environments (for example, ASCII-only) a thin space is not practical or available, in which case a regular word space or no delimiter are the alternatives. Data versus mask Digit group separators can occur either as part of the data or as a mask through which the data is displayed. This is an example of the separation of presentation and content, making it possible to display numbers with spaced digit grouping in a way that does not insert any whitespace characters into the string of digits in the content. In many computing contexts, it is preferred to omit digit group separators from the data and instead overlay them as a mask (an input mask or an output mask). Common examples include spreadsheets and databases in which currency values are entered without such marks but are displayed with them inserted. (Similarly, phone numbers can have hyphens, spaces or parentheses as a mask rather than as data.) In web content, such digit grouping can be done with CSS style. It is useful because the number can be copied and pasted into calculators (including a web browser's omnibox) and parsed by the computer as-is (i.e., without the user manually purging the extraneous characters). For example, Wikipedia content can display numbers this way, as in the following examples: 149597870700 metres is 1 astronomical unit 3.14159265358979323846 is π rounded to 20 decimal places 2.71828182845904523536 is e rounded to 20 decimal places. In some programming languages, it is possible to group the digits in the program's source code to make it easier to read; see Integer literal: Digit separators. Ada C# (from version 7.0)[35] D Go (from version 1.13) Haskell (from GHC version 8.6.1) Java Kotlin[36] OCaml Perl Python (from version 3.6) PHP (from version 7.4)[37] Ruby Rust Julia, Swift, and free-form Fortran 90 use the underscore (_) character for this purpose; as such, these languages allow seven hundred million to be entered as 700_000_000. Fixed-form Fortran ignores whitespace (in all contexts), so 700 000 000 has always been accepted. Fortran 90 and its successors allow (ignored) underscores in numbers in free-form. C++14, Rebol, and Red all allow the use of an apostrophe for digit grouping, so 700'000'000 is permissible. Below is shown an example of Kotlin code using separators to increase readability: val exampleNumber = 12_004_953 // Twelve million four thousand nine hundred fifty-three Exceptions to digit grouping The International Bureau of Weights and Measures states that "when there are only four digits before or after the decimal marker, it is customary not to use a space to isolate a single digit".[30] Likewise, some manuals of style state that thousands separators should not be used in normal text for numbers from 1000 to 9999 inclusive where no decimal fractional part is shown (in other words, for four-digit whole numbers), whereas others use thousands separators and others use both. For example, APA style stipulates a thousands separator for "most figures of 1000 or more" except for page numbers, binary digits, temperatures, etc. There are always "common-sense" country-specific exceptions to digit grouping, such as year numbers, postal codes, and ID numbers of predefined nongrouped format, which style guides usually point out. In non-base-10 numbering systems In binary (base-2), a full space can be used between groups of four digits, corresponding to a nibble, or equivalently to a hexadecimal digit. For integer numbers, dots are used as well to separate groups of four bits.[lower-alpha 2] Alternatively, binary digits may be grouped by threes, corresponding to an octal digit. Similarly, in hexadecimal (base-16), full spaces are usually used to group digits into twos, making each group correspond to a byte.[lower-alpha 3] Additionally, groups of eight bytes are often separated by a hyphen.[lower-alpha 3] Influence of calculators and computers In countries with a decimal comma, the decimal point is also common as the "international" notation because of the influence of devices, such as electronic calculators, which use the decimal point. Most computer operating systems allow selection of the decimal separator; programs that have been carefully internationalized will follow this, but some programs ignore it and a few may even fail to operate if the setting has been changed. Computer interfaces may be set to the Unicode international "Common locale" using LC_NUMERIC=C as defined at "Unicode CLDR project". Unicode Consortium. Details of the current (2020) definitions may be found at "01102-POSIX15897". Unicode Consortium. Conventions worldwide Hindu–Arabic numerals Countries using decimal comma Countries where a comma "," is used as decimal separator include: • Albania • Algeria • Andorra • Angola • Argentina • Armenia • Austria • Azerbaijan • Belarus • Belgium • Bolivia • Bosnia and Herzegovina • Brazil • Bulgaria[lower-roman 1] • Cabo Verde • Cameroon • Canada (when using French) • Chile • Colombia • Costa Rica • Croatia • Cuba • Cyprus • Czech Republic • Denmark • East Timor • Ecuador • Estonia • Faroes • Finland • France • Germany • Georgia • Greece • Greenland • Hungary • Iceland • Indonesia • Italy • Kazakhstan • Kyrgyzstan • Latvia • Lebanon • Lithuania • Luxembourg (uses both marks officially) • Macau (in Portuguese text) • Mauritania • Moldova • Mongolia[lower-roman 2] • Montenegro • Morocco • Mozambique • Namibia (uses both marks)[38] • The Netherlands • North Macedonia • Norway • Paraguay • Peru[39] • Poland • Portugal • Romania • Russia • Serbia • Slovakia • Slovenia • Somalia • South Africa[40][41] • Spain[lower-roman 3] • Suriname • Sweden[lower-roman 3] • Switzerland[lower-roman 4] • Tunisia • Turkey • Turkmenistan • Ukraine • Uruguay • Uzbekistan • Venezuela • Vietnam • Zimbabwe Countries using decimal point Countries where a dot "." is used as decimal separator include: • Australia • Bahamas, The • Bangladesh • Botswana • British West Indies • Cambodia • Canada (when using English) • China • Cyprus (currency numbers) • Dominican Republic • Egypt • El Salvador • Ethiopia • Ghana • Guatemala • Guyana • Honduras • Hong Kong • India • Ireland • Israel • Jamaica • Japan • Jordan • Kenya • Korea, North • Korea, South • Libya • Liechtenstein • Luxembourg (uses both marks officially) • Macau (in Chinese and English text) • Malaysia • Maldives • Malta • Mexico • Myanmar • Namibia (uses both marks) • Nepal • New Zealand • Nicaragua • Nigeria • Pakistan • Panama • Peru (currency numbers) • Philippines • Qatar • Saudi Arabia • Singapore • Somalia • Sri Lanka • Switzerland[lower-roman 4] • Syria • Taiwan • Tanzania • Thailand[lower-roman 3] • Uganda • United Arab Emirates • United Kingdom • United States (including insular areas) 1. The comma as a decimal separator is the national literary convention, but many places use the dot as decimal separator due to prevalence of imported tech that internally uses dot as the decimal separator (because the tech usually utilizes the dot decimal separator convention of the country where it was made, which is mostly USA-&-ASCII-oriented, or is made in China where the dot is utilized as a decimal separator). To sum up, the comma is the conventional decimal separator in Bulgaria, but both the comma and the dot are in de facto usage. 2. The comma as a decimal separator is the national literary convention, but many places use the dot as decimal separator due to prevalence of imported tech that internally uses dot as the decimal separator (because the tech usually utilizes the dot decimal separator convention of the country where it was made, which is mostly USA-&-ASCII-oriented, or is made in China where the dot is utilized as a decimal separator). To sum up, the comma is the conventional decimal separator in Mongolia, but the dot is more commonly used. 3. According to several software developers.[44][45] 4. The decimal point is used in some cantons (for example the Canton of St. Gallen[42]) and is used in IT and for currency. The decimal comma is used for federal publications[43] and some cantons. Other numeral systems Unicode defines a decimal separator key symbol (⎖ in hex U+2396, decimal 9110) which looks similar to the apostrophe. This symbol is from ISO/IEC 9995 and is intended for use on a keyboard to indicate a key that performs decimal separation. In the Arab world, where Eastern Arabic numerals are used for writing numbers, a different character is used to separate the integer and fractional parts of numbers. It is referred to as an Arabic decimal separator (U+066B, rendered: ٫‎) in Unicode. An Arabic thousands separator (U+066C, rendered: ٬‎) also exists. Example: ۹٬۹۹۹٫۹۹‎ (9,999.99) In Persian, the decimal separator is called momayyez. The Unicode Consortium's investigation concluded that "computer programs should render U+066B as a shortened, lowered, and possibly more slanted slash (٫); this should be distinguishable from the slash at the first sight." To separate sequences of three digits, an Arabic thousands separator (rendered as: ٬), a Latin comma, or a blank space may be used; however this is not a standard.[46][47][48] Example: ۹٬۹۹۹٫۹۹ (9,999.99) In English Braille, the decimal point, ⠨, is distinct from both the comma, ⠂, and the full stop, ⠲. Examples of use The following examples show the decimal separator and the thousands separator in various countries that use the Arabic numeral system. StyleCountries and regions 1,234,567.89Australia,[49][50] Cambodia, Canada (English-speaking; unofficial), China, Cyprus (currency numbers), Hong Kong, Iran, Ireland, Israel, Japan, Korea, Macau (in Chinese and English text), Malaysia, Malta, Mexico, Namibia, New Zealand, Pakistan, Peru (currency numbers), Philippines, Singapore, South Africa (English-speaking; unofficial), Taiwan, Thailand, United Kingdom and other Commonwealth states except Mozambique, United States. 1234567.89SI style (English version), Canada (English-speaking; official), China, Estonia (currency numbers), Hong Kong (in education), Mexico, Namibia, South Africa (English-speaking; unofficial), Sri Lanka, Switzerland (in federal texts for currency numbers only[51]), United Kingdom (in education), United States (in education). 1234567,89SI style (French version), Albania, Belgium (French), Brazil, Bulgaria, Canada (French-speaking), Costa Rica, Croatia, Czech Republic, Estonia, Finland, France, Hungary, Italy (in education), Latin America, Latin Europe, Latvia, Lithuania, Macau (in Portuguese text), Mozambique, Norway, Peru, Poland, Portugal, Russia, Serbia (informal), Slovakia, South Africa (official[52]), Spain (official use since 2010, according to the RAE and CSIC), Sweden, Switzerland (in federal texts, except currency numbers[51]), Ukraine, Vietnam (in education). 1.234.567,89Austria, Belgium (Dutch), Bosnia and Herzegovina, Brazil (informal and in technology), Chile, Colombia, Croatia (in bookkeeping and technology),[53] Denmark, Germany, Greece, Indonesia, Italy, Latin America (informal), Netherlands, Romania, Slovenia, Serbia, Spain (used until 2010, inadvisable use according to the RAE and CSIC),[lower-alpha 4][55] Turkey, Uruguay, Vietnam. 1,234,567·89Malaysia, Philippines (uncommon today), Singapore, United Kingdom (older, typically handwritten; in education) 12,34,567.89Bangladesh, India, Nepal, Pakistan (see Indian numbering system). 1234567.89 Bangladesh, India, Nepal, Pakistan (see Indian numbering system). 1'234'567.89Switzerland (computing), Liechtenstein. 1'234'567,89Switzerland (handwriting), Italy (handwriting). 1.234.567'89Spain (handwriting, used until 1980s, inadvisable use according to the RAE and CSIC). 123,4567.89Mainland China (based on powers of 10 000—see Chinese numerals). 1234567.89 Mainland China (based on powers of 10 000—see Chinese numerals). • In Albania, Belgium (French), Estonia, Finland,[56] France, Hungary, Poland, Slovakia and much of Latin Europe as well as French Canada: 1234567,89 (In Spain, in handwriting it is also common to use an upper comma: 1.234.567'89) • In Belgium (Dutch), Brazil, Denmark, Germany, Greece, Indonesia, Italy, Netherlands, Portugal, Romania, Russia, Slovenia, Sweden and much of Europe: 1234567,89 or 1.234.567,89. In handwriting, 1˙234˙567,89 is also seen, but never in Belgium, Brazil, Denmark, Estonia, Germany, the Netherlands, Portugal, Romania, Russia, Slovenia or Sweden. In Italy, a straight apostrophe is also used in handwriting: 1'234'567,89. In the Netherlands and Dutch-speaking Belgium, the points thousands separator is used, and is preferred for currency amounts, but the space is recommended by some style guides, mostly in technical writing.[57] • In Estonia, currency numbers often use a dot "." as the decimal separator, and a space as a thousands separator. This is most visible on shopping receipts and in documents that also use other numbers with decimals, such as measurements. This practice is used to better distinguish between prices and other values with decimals. An older convention uses dots to separate thousands (with commas for decimals) — this older practice makes it easier to avoid word breaks with larger numbers. • Historically, in Germany and Austria, thousands separators were occasionally denoted by alternating uses of comma and point, e.g. 1.234,567.890,12[58][59] for "eine Milliarde 234 Millionen ...", but this is never seen in modern days and requires explanation to a contemporary German reader. • Switzerland: There are two cases: An apostrophe as a thousands separator along with a dot "." as the decimal separator are used for currency values (for example: 1'234'567.89). For other values, the SI-style 1234567,89 is used with a comma "," as the decimal separator. The apostrophe is also the most common variety for non-currency values: 1'234'567,89 —. • In Ireland, Israel, Japan, Korea (both), Malaysia, the Philippines, Singapore, Taiwan, Thailand, the United Kingdom, and the United States: 1,234,567.89 or 1,234,567·89; the latter is generally found only in older, and especially handwritten documents. • English Canada: There are two cases: The preferred method for currency values is $10,000.00 —while for numeric values, it is 1234567.89; however, commas are also sometimes used, although no longer taught in school or used in official publications. • SI style: 1234567.89 or 1234567,89 (in their own publications, the dot "." is used in the English version, and the comma "," in the official French version). • In China, comma and space are used to mark digit groups, because dot is used as decimal separator. There is no universal convention on digit grouping, so both thousands grouping and no digit grouping can be found. Japan and Taiwan are similar; although when grouping by myriads, kanji or characters are frequently used as separators: 1億2345万6789 / 1億2345萬6789. Commas are used when grouping by thousands. • In India, due to a numeral system using lakhs (lacs) (1,23,456 equal to 123,456) and crores (1,23,45,678 equal to 12,345,678), a comma is used at levels of thousand, lakh, and crore. For example, 10 million (1 crore) would be written as 1,00,00,000. In Pakistan, there is a greater tendency to use the standard western system, while using the Indian numbering system when conducting business in Urdu. • In Sweden, the currency sometimes used the colon as decimal separator (1 234 567:89). Indian Value Value Equivalent Western Notation One 1 One Ten 10 Ten Hundred 100 Hundred Thousand 1,000 Thousand Lakh 1,00,000 One Hundred Thousand Crore 1,00,00,000 Ten Million Arab (not normally used) 1,00,00,00,000 One Short (American) Billion (One Thousand Million) Kharab (not normally used) 1,00,00,00,00,000 One Hundred Short Billion Lakh Crore 10,00,00,00,00,000 One Short Trillion (One Long (European) Billion) Unicode characters Used with Western Arabic numerals (0123456789): • U+0020 SPACE • U+0027 ' APOSTROPHE (') • U+002C , COMMA (,) • U+002E . FULL STOP (&period;) - Full stop punctuation mark. • U+00B7 · MIDDLE DOT (·, ·, ·) • U+2009 THIN SPACE ( ,  ) • U+202F   NARROW NO-BREAK SPACE • U+02D9 ˙ DOT ABOVE (&DiacriticalDot;, &dot;) Used with Eastern Arabic numerals (٠١٢٣٤٥٦٧٨٩): • U+066B ٫ ARABIC DECIMAL SEPARATOR • U+066C ٬ ARABIC THOUSANDS SEPARATOR Used with keyboards: • U+2396 ⎖ DECIMAL SEPARATOR KEY SYMBOL (resembles an apostrophe) See also • Algorism • Cifrão • Decimal floating point • Decimal place • Decimal representation • Decimal section numbering • Dot-decimal notation • International System of Units • ISO 2145 • RKM code • Version numbering Footnotes 1. Great Britain and the United States are two of the few places in the world that use a period to indicate the decimal place. Many other countries use a comma instead. The decimal separator is also called the radix character. Likewise, while the U.K. and U.S. use a comma to separate groups of thousands, many other countries use a period instead...[24] 2. As an example, the DR-DOS DEBUG H command displays the entered number in hexadecimal, decimal, octal and binary notation: -h 1234 1234 #4660 \011064 %0001.0010.0011.0100 3. As an example, the DR-DOS DEBUG D command dumps the memory byte-wise in hexadecimal notation, with bytes separated by spaces and groups of eight bytes separated by hyphens: -d 0 1234:0000 57 69 6B 69 70 65 64 69-61 20 68 65 6C 70 73 21 Wikipedia helps! 4. ... when writing numbers more than four figures, these will be grouped into threes, starting from the right, and separating the groups by whitespace. (Exceptions: Never written with periods, commas or white separation numbers that refer to years, pages, verses, urban roads, postal codes, legal articles, decrees or laws.)[54] References 1. "Decimal point definition and meaning \| Collins English Dictionary". www.collinsdictionary.com. Retrieved 5 July 2018. 2. Weisstein, Eric W. "Decimal Point". mathworld.wolfram.com. Retrieved 22 August 2020. 3. "decimal point Meaning in the Cambridge English Dictionary". dictionary.cambridge.org. Retrieved 5 July 2018. 4. "How to Change Excel's Decimal Separators from Periods to Commas". Retrieved 5 July 2018. 5. "Decimal Separators: Points or commas? - Elementary Math". Elementary Math. 19 January 2018. Retrieved 5 July 2018. 6. "Definition of Decimal Point". www.mathsisfun.com. Retrieved 22 August 2020. 7. "Mythematics: a decimal point". Grammarphobia. 17 February 2012. Retrieved 5 July 2018. 8. "separatrix, n.". Oxford English Dictionary. Oxford, UK: Oxford University Press. 9. Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Katz, Victor J. (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A sourcebook. Princeton University Press. p. 530. ISBN 978-0-691-11485-9. 10. Reimer, L.; Reimer, W. (1995). Mathematicians are People, too: Stories from the lives of great mathematicians. Vol. 2. Parsippany, NJ: Pearson Education, / Dale Seymor Publications. p. 22. ISBN 0-86651-823-1. 11. "Khwarizmi, Abu Jafar Muhammad ibn Musa al-". Oxford Islamic Studies Online (biography). 12. Devlin, Keith (2011). The Man of Numbers: Fibonacci's Arithmetic Revolution. New York, NY: Walker & Company. pp. 44–45. ISBN 9780802779083. 13. "Comma". Enciclopedia Universal Santillana. Barcelona, Spain: Santillana S.A. 1996. def. 2. ISBN 84-294-5129-3. coma: MAT. Signo utilizado en los números no enteros para separar la parte entera de la parte decimal o fraccionaria; p. ej., 2,123 . 14. "Part 2: Mathematical signs and symbols to be used in the natural sciences and technology". Quantities and units (Report). International Organization for Standardization. 2009. ISO 80000-2:2009. Retrieved 1 July 2010. 15. for example: Henderson, Thomas (3 January 1839). "On the parallax of α Centauri". Memoirs of the Royal Astronomical Society. 11: 61–64, esp. 64. Bibcode:1840MmRAS..11...61H – via scan published by Harvard-Smithsonian Center for Astrophysics. 16. Reimer, L.; Reimer, W. (1990). Mathematicians are People, too: Stories from the lives of great mathematicians. Vol. 1. Parsippany, NJ: Pearson Education / Dale Seymor Publications. p. 41. ISBN 0-86651-509-7. 17. "Victory on points". Nature. 218 (5137): 111. 1968. Bibcode:1968Natur.218S.111.. doi:10.1038/218111c0. 18. "Government Notice R. 1144" (PDF). Government Gazette. No. 4326. 5 July 1974. p. 55. Archived from the original (PDF) on 8 March 2021. 19. AndyMax25 (1 May 2015). "Tenths to fractions". AARoads (web forum). Retrieved 7 July 2019. 20. "Parts of Speech – Numerals". Grammar of Interlingua. Archived from the original on 16 May 2016. Retrieved 18 March 2008. 21. de Jong, Arie (2012). Everson, Michael (ed.). Gramat Volapüka. Cathair na Mart. Evertype. ISBN 978-1-904808-94-7. 22. Perlis, Alan, The American Side of the Development of ALGOL, ACM SIGPLAN Notices, August 1978. 23. Van Verth, James M.; Bishop, Lars M. (2008). Essential Mathematics for Games and Interactive Applications: A programmers' guide (2nd ed.). Boca Raton, FL: CRC Press. p. 7. ISBN 9780123742971. 24. International Language Environments Guide. Oracle Corporation Docs (Report). Archived from the original on 15 January 2012. Retrieved 19 December 2018. 25. Resolution 10. 22nd General Conference on Weights and Measures (CGPM). Bureau International des Poids et Measures. 2003. Retrieved 21 January 2022. 26. Thompson, Ambler; Taylor, Barry N. (March 2008). Guide for the Use of the International System of Units (SI) (PDF) (Report). National Institute of Standards and Technology. §10.5.3. Retrieved 21 January 2022. 27. "§5.3.1.3 – Representation of dates and times". Data elements and interchange formats — Information interchange (PDF) (Report). International Standards Organisation. 2000. p. 15. ISO/FDIS 8601:2000(E). Retrieved 8 January 2019. 28. Iverson, Cheryl; et al. (2007). AMA Manual of Style (10th ed.). Oxford, UK: Oxford University Press. p. 793. ISBN 978-0-19-517633-9. 29. International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 4 June 2021, retrieved 16 December 2021 30. Rules and style conventions for expressing values of quantities (Report). International Bureau of Weights and Measures. 31. Guidelines for drafting IUPAC technical reports and recommendations (Report). 2007. Retrieved 27 November 2008. 32. Emmons, John (25 March 2018). "Part 3: Numbers". Unicode Locale Data Markup Language (LDML). Unicode.org (Report). Archived from the original on 25 July 2018. Retrieved 25 March 2018. 33. "Language and Culture Differences". extranet.education.unimelb.edu.au. Retrieved 11 March 2023. 34. "Decimals Score a Point on International Standards". U.S. National Institute of Standards and Technology. 22 November 2006. Archived from the original on 3 March 2016. Retrieved 27 November 2008. 35. "Literal improvements". New Features in C# 77.0 (blog). Microsoft. 9 March 2017. Retrieved 5 May 2022. 36. "Basic types". Kotlin Language. 37. "Numeric literal separator". PHP: New Features. The PHP Group. Retrieved 16 July 2020. 38. "AirNam board endorses aircraft purchase plan". Republic of Namibia, Ministry of Public Enterprises. Retrieved 30 January 2018. The cost per aircraft was estimated at between N$19,5 million and N$26 million. 39. "DECRETO SUPREMO Nº 064-84-ITI/IND" (PDF). Instituto Nacional de Calidad (INACAL). Retrieved 23 March 2019. La escritura de los números se hará utilizando las cifras arábigas y la numeración decimal, y en ella se separará la parte entera de la decimal mediante una coma (,). 40. GCIS (2011). Editorial Style Guide (PDF). Pretoria: Government Communications & Information System. p. 24. Retrieved 30 January 2018. * Write decimal and negative numbers as numerals: 3,3 and –4. Use the decimal comma, not the decimal point: 17,4 million. [...] * Use a space, not commas, to indicate thousands: 3 000, 20 000. 41. MacLean, Robert. "How to correctly format currency in South Africa?". Retrieved 4 February 2018. 42. Lehrplan Fachbereich Mathematik Archived 2017-03-23 at the Wayback Machine, p. 14 (PDF; ≈ 257 kB) – from schule.sg.ch (german) 43. Schreibweisungen; 514: Dezimalkomma und Dezimalpunkt Archived 2017-01-22 at the Wayback Machine – Issuer: Federal Chancellery of Switzerland, 2nd issue 2013, p. 80 (German). 44. "Decimal and Thousands Separators". Oracle. Retrieved 18 March 2019. 45. Krupitsky, Igor (May 2010). "International Number Formats". Microsoft. Retrieved 18 March 2019. 46. Pournader, Roozbeh (15 October 2000). "Persian decimal separator". Unicode Mail List Archive. Unicode Consortium. Retrieved 21 June 2008. 47. "The Decimal Numeral". Academic Grammar of New Persian. Archived from the original on 20 June 2006. Retrieved 19 June 2006. 48. "Descriptive Grammar of New Persian (archived)". Archived from the original on 11 July 2007. Retrieved 1 May 2019. 49. "Digital Guides". Australian Government. Retrieved 19 August 2020. 50. "The ABC Style Guide". Australian Broadcasting Corporation. Retrieved 19 August 2020. 51. "Schreibweisungen, 2nd edition 2013". Schweizer Bundeskanzlei. June 2013. Archived from the original on 28 January 2015. 52. SI Metric System (PDF). SAHPRA (Report) (2.38 ed.). March 2019. The comma is the only recognised decimal indicator for all numbers, including amounts of (currency) money. 53. "Točka". Hrvatski pravopis. Institute of Croatian Language and Linguistics. Retrieved 28 July 2022. 54. Diccionario panhispánico de dudas.{{cite book}}: CS1 maint: url-status (link) 55. Ortografía de la lengua española. For the purpose of promoting a process tending towards unification, the use of the point as decimal separator is recommended.(2.2.1.2.1) 56. "Luvut ja numerot: numeroiden ryhmittely". Kielitoimiston ohjepankki (in Finnish). Retrieved 27 June 2018. 57. Taalunieversum. "10.000.000 / 10 000 000". Retrieved 1 September 2017. 58. Röll. "Union Pacific-Eisenbahn". Enzyklopädie des Eisenbahnwesens. Retrieved 26 August 2014., entry "Union Pacific-Eisenbahn", largest numbers in table 59. Röll. "Bilanz". Enzyklopädie des Eisenbahnwesens. Retrieved 26 August 2014., entry "Bilanz", sums in last table	Wikipedia
Rado graph In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of Wilhelm Ackermann (1937). The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other. Every finite or countably infinite graph is an induced subgraph of the Rado graph, and can be found as an induced subgraph by a greedy algorithm that builds up the subgraph one vertex at a time. The Rado graph is uniquely defined, among countable graphs, by an extension property that guarantees the correctness of this algorithm: no matter which vertices have already been chosen to form part of the induced subgraph, and no matter what pattern of adjacencies is needed to extend the subgraph by one more vertex, there will always exist another vertex with that pattern of adjacencies that the greedy algorithm can choose. The Rado graph is highly symmetric: any isomorphism of its finite induced subgraphs can be extended to a symmetry of the whole graph. The first-order logic sentences that are true of the Rado graph are also true of almost all random finite graphs, and the sentences that are false for the Rado graph are also false for almost all finite graphs. In model theory, the Rado graph is an example of the unique countable model of an ω-categorical theory. History The Rado graph was first constructed by Ackermann (1937) in two ways, with vertices either the hereditarily finite sets or the natural numbers. (Strictly speaking Ackermann described a directed graph, and the Rado graph is the corresponding undirected graph given by forgetting the directions on the edges.) Erdős & Rényi (1963) constructed the Rado graph as the random graph on a countable number of points. They proved that it has infinitely many automorphisms, and their argument also shows that it is unique though they did not mention this explicitly. Richard Rado (1964) rediscovered the Rado graph as a universal graph, and gave an explicit construction of it with vertex set the natural numbers. Rado's construction is essentially equivalent to one of Ackermann's constructions. Constructions Binary numbers Ackermann (1937) and Rado (1964) constructed the Rado graph using the BIT predicate as follows. They identified the vertices of the graph with the natural numbers 0, 1, 2, ... An edge connects vertices $x$ and $y$ in the graph (where $x<y$) whenever the $x$th bit of the binary representation of $y$ is nonzero. Thus, for instance, the neighbors of vertex 0 consist of all odd-numbered vertices, because the numbers whose 0th bit is nonzero are exactly the odd numbers. Vertex 1 has one smaller neighbor, vertex 0, as 1 is odd and vertex 0 is connected to all odd vertices. The larger neighbors of vertex 1 are all vertices with numbers congruent to 2 or 3 modulo 4, because those are exactly the numbers with a nonzero bit at index 1.[1] Random graph The Rado graph arises almost surely in the Erdős–Rényi model of a random graph on countably many vertices. Specifically, one may form an infinite graph by choosing, independently and with probability 1/2 for each pair of vertices, whether to connect the two vertices by an edge. With probability 1 the resulting graph is isomorphic to the Rado graph. This construction also works if any fixed probability $p$ not equal to 0 or 1 is used in place of 1/2.[2] This result, shown by Paul Erdős and Alfréd Rényi (1963), justifies the definite article in the common alternative name "the random graph" for the Rado graph. Repeatedly drawing a finite graph from the Erdős–Rényi model will in general lead to different graphs; however, when applied to a countably infinite graph, the model almost always produces the same infinite graph.[3] For any graph generated randomly in this way, the complement graph can be obtained at the same time by reversing all the choices: including an edge when the first graph did not include the same edge, and vice versa. This construction of the complement graph is an instance of the same process of choosing randomly and independently whether to include each edge, so it also (with probability 1) generates the Rado graph. Therefore, the Rado graph is a self-complementary graph.[4] Other constructions In one of Ackermann's original 1937 constructions, the vertices of the Rado graph are indexed by the hereditarily finite sets, and there is an edge between two vertices exactly when one of the corresponding finite sets is a member of the other. A similar construction can be based on Skolem's paradox, the fact that there exists a countable model for the first-order theory of sets. One can construct the Rado graph from such a model by creating a vertex for each set, with an edge connecting each pair of sets where one set in the pair is a member of the other.[5] The Rado graph may also be formed by a construction resembling that for Paley graphs, taking as the vertices of a graph all the prime numbers that are congruent to 1 modulo 4, and connecting two vertices by an edge whenever one of the two numbers is a quadratic residue modulo the other. By quadratic reciprocity and the restriction of the vertices to primes congruent to 1 mod 4, this is a symmetric relation, so it defines an undirected graph, which turns out to be isomorphic to the Rado graph.[6] Another construction of the Rado graph shows that it is an infinite circulant graph, with the integers as its vertices and with an edge between each two integers whose distance (the absolute value of their difference) belongs to a particular set $S$. To construct the Rado graph in this way, $S$ may be chosen randomly, or by choosing the indicator function of $S$ to be the concatenation of all finite binary sequences.[7] The Rado graph can also be constructed as the block intersection graph of an infinite block design in which the number of points and the size of each block are countably infinite.[8] It can also be constructed as the Fraïssé limit of the class of finite graphs.[9] Properties Extension The Rado graph satisfies the following extension property: for every two disjoint finite sets of vertices $U$ and $V$, there exists a vertex $x$ outside both sets that is connected to all vertices in $U$, but has no neighbors in $V$.[2] For instance, with the binary-number definition of the Rado graph, let $x=2^{1+\max(U\cup V)}+\sum _{u\in U}2^{u}.$ Then the nonzero bits in the binary representation of $x$ cause it to be adjacent to everything in $U$. However, $x$ has no nonzero bits in its binary representation corresponding to vertices in $V$, and $x$ is so large that the $x$th bit of every element of $V$ is zero. Thus, $x$ is not adjacent to any vertex in $V$.[10] With the random-graph definition of the Rado graph, each vertex outside the union of $U$ and $V$ has probability $1/2^{\|U\|+\|V\|}$ of fulfilling the extension property, independently of the other vertices. Because there are infinitely many vertices to choose from, each with the same finite probability of success, the probability is one that there exists a vertex that fulfils the extension property.[2] With the Paley graph definition, for any sets $U$ and $V$, by the Chinese remainder theorem, the numbers that are quadratic residues modulo every prime in $U$ and nonresidues modulo every prime in $V$ form a periodic sequence, so by Dirichlet's theorem on primes in arithmetic progressions this number-theoretic graph has the extension property.[6] Induced subgraphs The extension property can be used to build up isomorphic copies of any finite or countably infinite graph $G$ within the Rado graph, as induced subgraphs. To do so, order the vertices of $G$, and add vertices in the same order to a partial copy of $G$ within the Rado graph. At each step, the next vertex in $G$ will be adjacent to some set $U$ of vertices in $G$ that are earlier in the ordering of the vertices, and non-adjacent to the remaining set $V$ of earlier vertices in $G$. By the extension property, the Rado graph will also have a vertex $x$ that is adjacent to all the vertices in the partial copy that correspond to members of $U$, and non-adjacent to all the vertices in the partial copy that correspond to members of $V$. Adding $x$ to the partial copy of $G$ produces a larger partial copy, with one more vertex.[11] This method forms the basis for a proof by induction, with the 0-vertex subgraph as its base case, that every finite or countably infinite graph is an induced subgraph of the Rado graph.[11] Uniqueness The Rado graph is, up to graph isomorphism, the only countable graph with the extension property. For example, let $G$ and $H$ be two countable graphs with the extension property, let $G_{i}$ and $H_{i}$ be isomorphic finite induced subgraphs of $G$ and $H$ respectively, and let $g_{i}$ and $h_{i}$ be the first vertices in an enumeration of the vertices of $G$ and $H$ respectively that do not belong to $G_{i}$ and $H_{i}$. Then, by applying the extension property twice, one can find isomorphic induced subgraphs $G_{i+1}$ and $H_{i+1}$ that include $g_{i}$ and $h_{i}$ together with all the vertices of the previous subgraphs. By repeating this process, one may build up a sequence of isomorphisms between induced subgraphs that eventually includes every vertex in $G$ and $H$. Thus, by the back-and-forth method, $G$ and $H$ must be isomorphic.[12] Because the graphs constructed by the random graph construction, binary number construction, and Paley graph construction are all countable graphs with the extension property, this argument shows that they are all isomorphic to each other.[13] Symmetry group Applying the back-and-forth construction to any two isomorphic finite subgraphs of the Rado graph extends their isomorphism to an automorphism of the entire Rado graph. The fact that every isomorphism of finite subgraphs extends to an automorphism of the whole graph is expressed by saying that the Rado graph is ultrahomogeneous. In particular, there is an automorphism taking any ordered pair of adjacent vertices to any other such ordered pair, so the Rado graph is a symmetric graph.[12] The automorphism group of the Rado graph is a simple group, whose number of elements is the cardinality of the continuum. Every subgroup of this group whose index is less than the cardinality of the continuum contains the pointwise stabilizer of a finite set of vertices, and furthermore is contained within the setwise stabilizer of the same set.[14] The statement about pointwise stabilizers is called the small index property,[14] and proving it required showing that for every finite graph $X$, there is a finite graph $Z$ containing $X$ as an induced subgraph such that every isomorphism between induced subgraphs of $X$ extends to an automorphism of $Z$.[15] This is called the extension property for partial automorphisms and has since been generalized to further structures in order to show the small index property and other properties.[16] The construction of the Rado graph as an infinite circulant graph shows that its symmetry group includes automorphisms that generate a transitive infinite cyclic group. The difference set of this construction (the set of distances in the integers between adjacent vertices) can be constrained to include the difference 1, without affecting the correctness of this construction, from which it follows that the Rado graph contains an infinite Hamiltonian path whose symmetries are a subgroup of the symmetries of the whole graph.[17] Robustness against finite changes If a graph $G$ is formed from the Rado graph by deleting any finite number of edges or vertices, or adding a finite number of edges, the change does not affect the extension property of the graph. For any pair of sets $U$ and $V$ it is still possible to find a vertex in the modified graph that is adjacent to everything in $U$ and nonadjacent to everything in $V$, by adding the modified parts of $G$ to $V$ and applying the extension property in the unmodified Rado graph. Therefore, any finite modification of this type results in a graph that is isomorphic to the Rado graph.[18] Partitions For any partition of the vertices of the Rado graph into two sets $A$ and $B$, or more generally for any partition into finitely many subsets, at least one of the subgraphs induced by one of the partition sets is isomorphic to the whole Rado graph. Cameron (2001) gives the following short proof: if none of the parts induces a subgraph isomorphic to the Rado graph, they all fail to have the extension property, and one can find pairs of sets $U_{i}$ and $V_{i}$ that cannot be extended within each subgraph. But then, the union of the sets $U_{i}$ and the union of the sets $V_{i}$ would form a set that could not be extended in the whole graph, contradicting the Rado graph's extension property. This property of being isomorphic to one of the induced subgraphs of any partition is held by only three countably infinite undirected graphs: the Rado graph, the complete graph, and the empty graph.[19] Bonato, Cameron & Delić (2000) and Diestel et al. (2007) investigate infinite directed graphs with the same partition property; all are formed by choosing orientations for the edges of the complete graph or the Rado graph. A related result concerns edge partitions instead of vertex partitions: for every partition of the edges of the Rado graph into finitely many sets, there is a subgraph isomorphic to the whole Rado graph that uses at most two of the colors. However, there may not necessarily exist an isomorphic subgraph that uses only one color of edges.[20] More generally, for every finite graph $A$ there is a number $d_{A}$ (called the big Ramsey degree of $A$ in the Rado graph) such that for every partition of the copies of $A$ in the Rado graph into finitely many sets, there is an induced subgraph isomorphic to the whole Rado graph that uses at most $d_{A}$ of the colors.[21][22] Model theory and 0-1 laws Fagin (1976) used the Rado graph to prove a zero–one law for first-order statements in the logic of graphs. When a logical statement of this type is true or false for the Rado graph, it is also true or false (respectively) for almost all finite graphs. First-order properties The first-order language of graphs is the collection of well-formed sentences in mathematical logic formed from variables representing the vertices of graphs, universal and existential quantifiers, logical connectives, and predicates for equality and adjacency of vertices. For instance, the condition that a graph does not have any isolated vertices may be expressed by the sentence $\forall u:\exists v:u\sim v$ where the $\sim $ symbol indicates the adjacency relation between two vertices.[23] This sentence $S$ is true for some graphs, and false for others; a graph $G$ is said to model $S$, written $G\models S$, if $S$ is true of the vertices and adjacency relation of $G$.[24] The extension property of the Rado graph may be expressed by a collection of first-order sentences $E_{i,j}$, stating that for every choice of $i$ vertices in a set $A$ and $j$ vertices in a set $B$, all distinct, there exists a vertex adjacent to everything in $A$ and nonadjacent to everything in $B$.[25] For instance, $E_{1,1}$ can be written as $\forall a:\forall b:a\neq b\rightarrow \exists c:c\neq a\wedge c\neq b\wedge c\sim a\wedge \lnot (c\sim b).$ Completeness Gaifman (1964) proved that the sentences $E_{i,j}$, together with additional sentences stating that the adjacency relation is symmetric and antireflexive (that is, that a graph modeling these sentences is undirected and has no self-loops), are the axioms of a complete theory. This means that, for each first-order sentence $S$, exactly one of $S$ and its negation can be proven from these axioms. Because the Rado graph models the extension axioms, it models all sentences in this theory.[26] In logic, a theory that has only one model (up to isomorphism) with a given infinite cardinality $\lambda $ is called $\lambda $-categorical. The fact that the Rado graph is the unique countable graph with the extension property implies that it is also the unique countable model for its theory. This uniqueness property of the Rado graph can be expressed by saying that the theory of the Rado graph is ω-categorical. Łoś and Vaught proved in 1954 that when a theory is $\lambda $–categorical (for some infinite cardinal $\lambda $) and, in addition, has no finite models, then the theory must be complete.[27] Therefore, Gaifman's theorem that the theory of the Rado graph is complete follows from the uniqueness of the Rado graph by the Łoś–Vaught test.[28] Finite graphs and computational complexity As Fagin (1976) proved, the first-order sentences provable from the extension axioms and modeled by the Rado graph are exactly the sentences true for almost all random finite graphs. This means that if one chooses an $n$-vertex graph uniformly at random among all graphs on $n$ labeled vertices, then the probability that such a sentence will be true for the chosen graph approaches one in the limit as $n$ approaches infinity. Symmetrically, the sentences that are not modeled by the Rado graph are false for almost all random finite graphs. It follows that every first-order sentence is either almost always true or almost always false for random finite graphs, and these two possibilities can be distinguished by determining whether the Rado graph models the sentence. Fagin's proof uses the compactness theorem.[29] Based on this equivalence, the theory of sentences modeled by the Rado graph has been called "the theory of the random graph" or "the almost sure theory of graphs". Because of this 0-1 law, it is possible to test whether any particular first-order sentence is modeled by the Rado graph in a finite amount of time, by choosing a large enough value of $n$ and counting the number of $n$-vertex graphs that model the sentence. However, here, "large enough" is at least exponential in the size of the sentence. For instance the extension axiom $E_{k,0}$ implies the existence of a $(k+1)$-vertex clique, but a clique of that size exists with high probability only in random graphs of size exponential in $k$. It is unlikely that determining whether the Rado graph models a given sentence can be done more quickly than exponential time, as the problem is PSPACE-complete.[30] Other model-theoretic properties The Rado graph is ultrahomogeneous, and thus is the Fraïssé limit of its class of finite substructures, i.e. the class of finite graphs.[31] Given that it is also in a finite relational language, ultrahomogeneity is equivalent to its theory having quantifier elimination and being ω-categorical.[32] As the Rado graph is thus the countable model of a countable ω-categorical theory, it is both prime and saturated.[33][34] The theory of the Rado graph is a prototypical example of a theory with the independence property, and of a simple theory that is not stable.[35] Related concepts Although the Rado graph is universal for induced subgraphs, it is not universal for isometric embeddings of graphs, where an isometric embedding is a graph isomorphism which preserves distance. The Rado graph has diameter two, and so any graph with larger diameter does not embed isometrically into it. Moss (1989, 1991) has described a family of universal graphs for isometric embedding, one for each possible finite graph diameter; the graph in his family with diameter two is the Rado graph. The Henson graphs are countable graphs (one for each positive integer $i$) that do not contain an $i$-vertex clique, and are universal for $i$-clique-free graphs. They can be constructed as induced subgraphs of the Rado graph.[17] The Rado graph, the Henson graphs and their complements, disjoint unions of countably infinite cliques and their complements, and infinite disjoint unions of isomorphic finite cliques and their complements are the only possible countably infinite homogeneous graphs.[36] The universality property of the Rado graph can be extended to edge-colored graphs; that is, graphs in which the edges have been assigned to different color classes, but without the usual edge coloring requirement that each color class form a matching. For any finite or countably infinite number of colors $\chi $, there exists a unique countably-infinite $\chi $-edge-colored graph $G_{\chi }$ such that every partial isomorphism of a $\chi $-edge-colored finite graph can be extended to a full isomorphism. With this notation, the Rado graph is just $G_{1}$. Truss (1985) investigates the automorphism groups of this more general family of graphs. While the Rado graph is countable universal for the class of all graphs, not all graph classes have a countable universal graph. For example, there is no countable graph omitting the 4-cycle as a subgraph that contains all other such countable graphs as (not necessarily induced) subgraphs.[37] It follows from the classical model theory considerations of constructing a saturated model that under the continuum hypothesis CH, there is a universal graph with continuum many vertices. Of course, under CH, the continuum is equal to $\aleph _{1}$, the first uncountable cardinal. Shelah (1984, 1990) uses forcing to investigate universal graphs with $\aleph _{1}$ many vertices and shows that even in the absence of CH, there may exist a universal graph of size $\aleph _{1}$. He also investigates analogous questions for higher cardinalities. Notes 1. Ackermann (1937); Rado (1964). 2. See Cameron (1997), Fact 1 and its proof. 3. Erdős & Rényi (1963). 4. Cameron (1997), Proposition 5. 5. Cameron (1997), Theorem 2. 6. Cameron (1997, 2001) 7. Cameron (1997), Section 1.2. 8. Horsley, Pike & Sanaei (2011) 9. Hodges (1997), p. 350. 10. Essentially the same construction, described in set-theoretic terms rather than using binary numbers, is given as Theorem 2 of Cameron (1997). 11. Cameron (1997), Proposition 6. 12. Cameron (2001). 13. Cameron (1997), Fact 2. 14. Cameron (1997), Section 1.8: The automorphism group. 15. Hrushovski (1992) 16. Macpherson (2011), Sections 5.2 and 5.3. 17. Henson (1971). 18. Cameron (1997), Section 1.3: Indestructibility. 19. Cameron (1990); Diestel et al. (2007). 20. Pouzet & Sauer (1996). 21. Sauer (2006) 22. Dobrinen (2021) 23. Spencer (2001), Section 1.2, "What Is a First Order Theory?", pp. 15–17. 24. See, e.g., Grandjean (1983), p. 184. 25. Spencer (2001), Section 1.3, "Extension Statements and Rooted Graphs", pp. 17–18. 26. Gaifman (1964); Marker (2002), Theorem 2.4.2, p. 50. 27. Łoś (1954); Vaught (1954); Enderton (1972), p. 147. 28. Marker (2002), Theorem 2.2.6, p. 42. 29. Fagin (1976); Marker (2002), Theorem 2.4.4, pp. 51–52. 30. Grandjean (1983). 31. Macpherson (2011), Theorem 2.1.3, Example 2.2.1. 32. Macpherson (2011), Corollary 3.1.3, Proposition 3.1.6. 33. Rothmaler (2000), Theorem 13.3.1, Theorem 13.2.1. 34. McNulty (2016), p.71 and p.75. 35. Baldwin (2018) 36. Lachlan & Woodrow (1980). 37. Cherlin (2011), Fact 1.3 References • Ackermann, Wilhelm (1937), "Die Widerspruchsfreiheit der allgemeinen Mengenlehre", Mathematische Annalen, 114 (1): 305–315, doi:10.1007/BF01594179, S2CID 120576556 • Bonato, Anthony; Cameron, Peter; Delić, Dejan (2000), "Tournaments and orders with the pigeonhole property", Canadian Mathematical Bulletin, 43 (4): 397–405, doi:10.4153/CMB-2000-047-6, MR 1793941. • Baldwin, John T. (2018), Model theory and the philosophy of mathematical practice: formalization without foundationalism, Cambridge University Press, doi:10.1017/9781316987216, ISBN 978-1-107-18921-8, MR 3793636, S2CID 126311148. • Cameron, Peter J. (1990), Oligomorphic permutation groups, London Mathematical Society Lecture Note Series, vol. 152, Cambridge: Cambridge University Press, ISBN 0-521-38836-8, MR 1066691. • Cameron, Peter J. (1997), "The random graph", The mathematics of Paul Erdős, II, Algorithms and Combinatorics, vol. 14, Berlin: Springer, pp. 333–351, arXiv:1301.7544, Bibcode:2013arXiv1301.7544C, MR 1425227. • Cameron, Peter J. (2001), "The random graph revisited" (PDF), European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., vol. 201, Basel: Birkhäuser, pp. 267–274, doi:10.1007/978-3-0348-8268-2_15, MR 1905324. • Cherlin, Gregory (2011), "Forbidden substructures and combinatorial dichotomies: WQO and universality", Discrete Mathematics, 311 (15): 1543–1584, doi:10.1016/j.disc.2011.03.014, MR 2800977. • Diestel, Reinhard; Leader, Imre; Scott, Alex; Thomassé, Stéphan (2007), "Partitions and orientations of the Rado graph", Transactions of the American Mathematical Society, 359 (5): 2395–2405, doi:10.1090/S0002-9947-06-04086-4, MR 2276626. • Dobrinen, Natasha (2021), "Ramsey theory of homogeneous structures: current trends and open problems", arXiv:2110.00655v1 [math.LO]. • Enderton, Herbert B. (1972), A mathematical introduction to logic, Academic Press, New York-London, MR 0337470. • Erdős, P.; Rényi, A. (1963), "Asymmetric graphs", Acta Mathematica Academiae Scientiarum Hungaricae, 14 (3–4): 295–315, doi:10.1007/BF01895716, MR 0156334. • Fagin, Ronald (1976), "Probabilities on finite models" (PDF), The Journal of Symbolic Logic, 41 (1): 50–58, doi:10.1017/s0022481200051756, JSTOR 2272945, MR 0476480, S2CID 2563318. • Gaifman, Haim (1964), "Concerning measures in first order calculi", Israel Journal of Mathematics, 2: 1–18, doi:10.1007/BF02759729, MR 0175755. • Grandjean, Étienne (1983), "Complexity of the first-order theory of almost all finite structures", Information and Control, 57 (2–3): 180–204, doi:10.1016/S0019-9958(83)80043-6, MR 0742707. • Henson, C. Ward (1971), "A family of countable homogeneous graphs", Pacific Journal of Mathematics, 38: 69–83, doi:10.2140/pjm.1971.38.69, MR 0304242. • Hodges, Wilfrid (1997), A Shorter Model Theory, Cambridge University Press, ISBN 0-521-58713-1, OCLC 468298248 • Horsley, Daniel; Pike, David A.; Sanaei, Asiyeh (2011), "Existential closure of block intersection graphs of infinite designs having infinite block size", Journal of Combinatorial Designs, 19 (4): 317–327, doi:10.1002/jcd.20283, MR 2838911, S2CID 120707836. • Hrushovski, Ehud (1992), "Extending partial isomorphisms of graphs", Combinatorica, 12 (4): 411–416, doi:10.1007/BF01305233, MR 1194731, S2CID 19939702. • Lachlan, A. H.; Woodrow, Robert E. (1980), "Countable ultrahomogeneous undirected graphs", Transactions of the American Mathematical Society, 262 (1): 51–94, doi:10.2307/1999974, JSTOR 1999974, MR 0583847. • Łoś, J. (1954), "On the categoricity in power of elementary deductive systems and some related problems", Colloquium Math., 3: 58–62, doi:10.4064/cm-3-1-58-62, MR 0061561. • Macpherson, Dugald (2011), "A survey of homogeneous structures", Discrete Mathematics, 311 (15): 1599–1634, doi:10.1016/j.disc.2011.01.024, MR 2800979. • Marker, David (2002), Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, ISBN 0-387-98760-6, MR 1924282. • McNulty, George F. (2016), Elementary Model Theory (PDF), pp. 71–75, retrieved 5 April 2023. • Moss, Lawrence S. (1989), "Existence and nonexistence of universal graphs", Polska Akademia Nauk. Fundamenta Mathematicae, 133 (1): 25–37, doi:10.4064/fm-133-1-25-37, MR 1059159. • Moss, Lawrence S. (1991), "The universal graphs of fixed finite diameter", Graph theory, combinatorics, and applications. Vol. 2 (Kalamazoo, MI, 1988), Wiley-Intersci. Publ., New York: Wiley, pp. 923–937, MR 1170834. • Pouzet, Maurice; Sauer, Norbert (1996), "Edge partitions of the Rado graph", Combinatorica, 16 (4): 505–520, doi:10.1007/BF01271269, MR 1433638, S2CID 206793062. • Rado, Richard (1964), "Universal graphs and universal functions" (PDF), Acta Arith., 9 (4): 331–340, doi:10.4064/aa-9-4-331-340. • Rothmaler, Philipp (2000), Introduction to model theory, Algebra, logic and applications, vol. 15, Gordon and Breach Science Publishers, ISBN 90-5699-287-2, MR 1800596. • Sauer, Norbert (2006), "Coloring subgraphs of the Rado graph", Combinatorica, 26 (2): 231–253, doi:10.1007/s00493-006-0015-0, MR 2223636, S2CID 19251747. • Shelah, Saharon (1984), "On universal graphs without instances of CH", Annals of Pure and Applied Logic, 26 (1): 75–87, doi:10.1016/0168-0072(84)90042-3, MR 0739914. • Shelah, Saharon (1990), "Universal graphs without instances of CH: revisited", Israel Journal of Mathematics, 70 (1): 69–81, doi:10.1007/BF02807219, MR 1057268. • Spencer, Joel (2001), The Strange Logic of Random Graphs, Algorithms and Combinatorics, vol. 22, Springer-Verlag, Berlin, doi:10.1007/978-3-662-04538-1, ISBN 3-540-41654-4, MR 1847951. • Truss, J. K. (1985), "The group of the countable universal graph", Mathematical Proceedings of the Cambridge Philosophical Society, 98 (2): 213–245, Bibcode:1985MPCPS..98..213T, doi:10.1017/S0305004100063428, MR 0795890, S2CID 122772888. • Vaught, Robert L. (1954), "Applications to the Löwenheim-Skolem-Tarski theorem to problems of completeness and decidability", Indagationes Mathematicae, 16: 467–472, doi:10.1016/S1385-7258(54)50058-2, MR 0063993.	Wikipedia
Bochner integral In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions. Definition Let $(X,\Sigma ,\mu )$ be a measure space, and $B$ be a Banach space. The Bochner integral of a function $f:X\to B$ is defined in much the same way as the Lebesgue integral. First, define a simple function to be any finite sum of the form $s(x)=\sum _{i=1}^{n}\chi _{E_{i}}(x)b_{i}$ where the $E_{i}$ are disjoint members of the $\sigma $-algebra $\Sigma ,$ the $b_{i}$ are distinct elements of $B,$ and χE is the characteristic function of $E.$ If $\mu \left(E_{i}\right)$ is finite whenever $b_{i}\neq 0,$ then the simple function is integrable, and the integral is then defined by $\int _{X}\left[\sum _{i=1}^{n}\chi _{E_{i}}(x)b_{i}\right]\,d\mu =\sum _{i=1}^{n}\mu (E_{i})b_{i}$ exactly as it is for the ordinary Lebesgue integral. A measurable function $f:X\to B$ is Bochner integrable if there exists a sequence of integrable simple functions $s_{n}$ such that $\lim _{n\to \infty }\int _{X}\\|f-s_{n}\\|_{B}\,d\mu =0,$ where the integral on the left-hand side is an ordinary Lebesgue integral. In this case, the Bochner integral is defined by $\int _{X}f\,d\mu =\lim _{n\to \infty }\int _{X}s_{n}\,d\mu .$ It can be shown that the sequence $\left\{\int _{X}s_{n}\,d\mu \right\}_{n=1}^{\infty }$ is a Cauchy sequence in the Banach space $B,$ hence the limit on the right exists; furthermore, the limit is independent of the approximating sequence of simple functions $\{s_{n}\}_{n=1}^{\infty }.$ These remarks show that the integral is well-defined (i.e independent of any choices). It can be shown that a function is Bochner integrable if and only if it lies in the Bochner space $L^{1}.$ Properties Elementary properties Many of the familiar properties of the Lebesgue integral continue to hold for the Bochner integral. Particularly useful is Bochner's criterion for integrability, which states that if $(X,\Sigma ,\mu )$ is a measure space, then a Bochner-measurable function $f\colon X\to B$ is Bochner integrable if and only if $\int _{X}\\|f\\|_{B}\,\mathrm {d} \mu <\infty .$ Here, a function $f\colon X\to B$ is called Bochner measurable if it is equal $\mu $-almost everywhere to a function $g$ taking values in a separable subspace $B_{0}$ of $B$, and such that the inverse image $g^{-1}(U)$ of every open set $U$ in $B$ belongs to $\Sigma $. Equivalently, $f$ is the limit $\mu $-almost everywhere of a sequence of countably-valued simple functions. Linear operators If $T\colon B\to B'$ is a continuous linear operator between Banach spaces $B$ and $B'$, and $f\colon X\to B$ is Bochner integrable, then it is relatively straightforward to show that $Tf\colon X\to B'$ is Bochner integrable and integration and the application of $T$ may be interchanged: $\int _{E}Tf\,\mathrm {d} \mu =T\int _{E}f\,\mathrm {d} \mu $ for all measurable subsets $E\in \Sigma $. A non-trivially stronger form of this result, known as Hille's theorem, also holds for closed operators.[1] If $T\colon B\to B'$ is a closed linear operator between Banach spaces $B$ and $B'$ and both $f\colon X\to B$ and $Tf\colon X\to B'$ are Bochner integrable, then $\int _{E}Tf\,\mathrm {d} \mu =T\int _{E}f\,\mathrm {d} \mu $ for all measurable subsets $E\in \Sigma $. Dominated convergence theorem A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if $f_{n}\colon X\to B$ is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function $f$, and if $\\|f_{n}(x)\\|_{B}\leq g(x)$ for almost every $x\in X$, and $g\in L^{1}(\mu )$, then $\int _{E}\\|f-f_{n}\\|_{B}\,\mathrm {d} \mu \to 0$ as $n\to \infty $ and $\int _{E}f_{n}\,\mathrm {d} \mu \to \int _{E}f\,\mathrm {d} \mu $ for all $E\in \Sigma $. If $f$ is Bochner integrable, then the inequality $\left\\|\int _{E}f\,\mathrm {d} \mu \right\\|_{B}\leq \int _{E}\\|f\\|_{B}\,\mathrm {d} \mu $ holds for all $E\in \Sigma .$ In particular, the set function $E\mapsto \int _{E}f\,\mathrm {d} \mu $ defines a countably-additive $B$-valued vector measure on $X$ which is absolutely continuous with respect to $\mu $. Radon–Nikodym property An important fact about the Bochner integral is that the Radon–Nikodym theorem fails to hold in general, and instead is a property (the Radon–Nikodym property) defining an important class of nice Banach spaces. Specifically, if $\mu $ is a measure on $(X,\Sigma ),$ then $B$ has the Radon–Nikodym property with respect to $\mu $ if, for every countably-additive vector measure $\gamma $ on $(X,\Sigma )$ with values in $B$ which has bounded variation and is absolutely continuous with respect to $\mu ,$ there is a $\mu $-integrable function $g:X\to B$ such that $\gamma (E)=\int _{E}g\,d\mu $ for every measurable set $E\in \Sigma .$[2] The Banach space $B$ has the Radon–Nikodym property if $B$ has the Radon–Nikodym property with respect to every finite measure.[2] Equivalent formulations include: • Bounded discrete-time martingales in $B$ converge a.s.[3] • Functions of bounded-variation into $B$ are differentiable a.e.[4] • For every bounded $D\subseteq B$, there exists $f\in B^{}$ and $\delta \in \mathbb {R} ^{+}$ such that $\{x:f(x)+\delta >\sup {f(D)}\}\subseteq D$ has arbitrarily small diameter.[3] It is known that the space $\ell _{1}$ has the Radon–Nikodym property, but $c_{0}$ and the spaces $L^{\infty }(\Omega ),$ $L^{1}(\Omega ),$ for $\Omega $ an open bounded subset of $\mathbb {R} ^{n},$ and $C(K),$ for $K$ an infinite compact space, do not.[5] Spaces with Radon–Nikodym property include separable dual spaces (this is the Dunford–Pettis theorem) and reflexive spaces, which include, in particular, Hilbert spaces.[2] See also • Bochner space – Mathematical concept • Bochner measurable function • Pettis integral • Vector measure • Weakly measurable function References 1. Diestel, Joseph; Uhl, Jr., John Jerry (1977). Vector Measures. Mathematical Surveys. American Mathematical Society. doi:10.1090/surv/015. (See Theorem II.2.6) 2. Bárcenas, Diómedes (2003). "The Radon–Nikodym Theorem for Reflexive Banach Spaces" (PDF). Divulgaciones Matemáticas. 11 (1): 55–59 [pp. 55–56]. 3. Bourgin 1983, pp. 31, 33. Thm. 2.3.6-7, conditions (1,4,10). 4. Bourgin 1983, p. 16. "Early workers in this field were concerned with the Banach space property that each X-valued function of bounded variation on [0,1] be differentiable almost surely. It turns out that this property (known as the Gelfand-Fréchet property) is also equivalent to the RNP [Radon-Nikodym Property]." 5. Bourgin 1983, p. 14. • Bochner, Salomon (1933), "Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind" (PDF), Fundamenta Mathematicae, 20: 262–276 • Bourgin, Richard D. (1983). Geometric Aspects of Convex Sets with the Radon-Nikodým Property. Lecture Notes in Mathematics 993. Berlin: Springer-Verlag. doi:10.1007/BFb0069321. ISBN 3-540-12296-6. • Cohn, Donald (2013), Measure Theory, Birkhäuser Advanced Texts Basler Lehrbücher, Springer, doi:10.1007/978-1-4614-6956-8, ISBN 978-1-4614-6955-1 • Yosida, Kôsaku (1980), Functional Analysis, Classics in Mathematics, vol. 123, Springer, doi:10.1007/978-3-642-61859-8, ISBN 978-3-540-58654-8 • Diestel, Joseph (1984), Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, vol. 92, Springer, doi:10.1007/978-1-4612-5200-9, ISBN 978-0-387-90859-5 • Diestel; Uhl (1977), Vector measures, American Mathematical Society, ISBN 978-0-8218-1515-1 • Hille, Einar; Phillips, Ralph (1957), Functional Analysis and Semi-Groups, American Mathematical Society, ISBN 978-0-8218-1031-6 • Lang, Serge (1993), Real and Functional Analysis (3rd ed.), Springer, ISBN 978-0387940014 • Sobolev, V. I. (2001) [1994], "Bochner integral", Encyclopedia of Mathematics, EMS Press • van Dulst, D. (2001) [1994], "Vector measures", Encyclopedia of Mathematics, EMS Press Integrals Types of integrals • Riemann integral • Lebesgue integral • Burkill integral • Bochner integral • Daniell integral • Darboux integral • Henstock–Kurzweil integral • Haar integral • Hellinger integral • Khinchin integral • Kolmogorov integral • Lebesgue–Stieltjes integral • Pettis integral • Pfeffer integral • Riemann–Stieltjes integral • Regulated integral Integration techniques • Substitution • Trigonometric • Euler • Weierstrass • By parts • Partial fractions • Euler's formula • Inverse functions • Changing order • Reduction formulas • Parametric derivatives • Differentiation under the integral sign • Laplace transform • Contour integration • Laplace's method • Numerical integration • Simpson's rule • Trapezoidal rule • Risch algorithm Improper integrals • Gaussian integral • Dirichlet integral • Fermi–Dirac integral • complete • incomplete • Bose–Einstein integral • Frullani integral • Common integrals in quantum field theory Stochastic integrals • Itô integral • Russo–Vallois integral • Stratonovich integral • Skorokhod integral Miscellaneous • Basel problem • Euler–Maclaurin formula • Gabriel's horn • Integration Bee • Proof that 22/7 exceeds π • Volumes • Washers • Shells Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C*-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons Analysis in topological vector spaces Basic concepts • Abstract Wiener space • Classical Wiener space • Bochner space • Convex series • Cylinder set measure • Infinite-dimensional vector function • Matrix calculus • Vector calculus Derivatives • Differentiable vector–valued functions from Euclidean space • Differentiation in Fréchet spaces • Fréchet derivative • Total • Functional derivative • Gateaux derivative • Directional • Generalizations of the derivative • Hadamard derivative • Holomorphic • Quasi-derivative Measurability • Besov measure • Cylinder set measure • Canonical Gaussian • Classical Wiener measure • Measure like set functions • infinite-dimensional Gaussian measure • Projection-valued • Vector • Bochner / Weakly / Strongly measurable function • Radonifying function Integrals • Bochner • Direct integral • Dunford • Gelfand–Pettis/Weak • Regulated • Paley–Wiener Results • Cameron–Martin theorem • Inverse function theorem • Nash–Moser theorem • Feldman–Hájek theorem • No infinite-dimensional Lebesgue measure • Sazonov's theorem • Structure theorem for Gaussian measures Related • Crinkled arc • Covariance operator Functional calculus • Borel functional calculus • Continuous functional calculus • Holomorphic functional calculus Applications • Banach manifold (bundle) • Convenient vector space • Choquet theory • Fréchet manifold • Hilbert manifold	Wikipedia
Radon transform In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann Radon,[1] who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes (integrating over lines is known as the X-ray transform). It was later generalized to higher-dimensional Euclidean spaces and more broadly in the context of integral geometry. The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object. Explanation If a function$f$ represents an unknown density, then the Radon transform represents the projection data obtained as the output of a tomographic scan. Hence the inverse of the Radon transform can be used to reconstruct the original density from the projection data, and thus it forms the mathematical underpinning for tomographic reconstruction, also known as iterative reconstruction. The Radon transform data is often called a sinogram because the Radon transform of an off-center point source is a sinusoid. Consequently, the Radon transform of a number of small objects appears graphically as a number of blurred sine waves with different amplitudes and phases. The Radon transform is useful in computed axial tomography (CAT scan), barcode scanners, electron microscopy of macromolecular assemblies like viruses and protein complexes, reflection seismology and in the solution of hyperbolic partial differential equations. Definition Let $f({\textbf {x}})=f(x,y)$ be a function that satisfies the three regularity conditions:[2] 1. $f({\textbf {x}})$ is continuous; 2. the double integral $\iint {\dfrac {\vert f({\textbf {x}})\vert }{\sqrt {x^{2}+y^{2}}}}\,dx\,dy$, extending over the whole plane, converges; 3. for any arbitrary point $(x,y)$ on the plane it holds that $\lim _{r\to \infty }\int _{0}^{2\pi }f(x+r\cos \varphi ,y+r\sin \varphi )\,d\varphi =0.$ The Radon transform, $Rf$, is a function defined on the space of straight lines $L\subset \mathbb {R} ^{2}$ by the line integral along each such line as: $Rf(L)=\int _{L}f(\mathbf {x} )\vert d\mathbf {x} \vert .$ Concretely, the parametrization of any straight line $L$ with respect to arc length $z$ can always be written: $(x(z),y(z))={\Big (}(z\sin \alpha +s\cos \alpha ),(-z\cos \alpha +s\sin \alpha ){\Big )}\,$ where $s$ is the distance of $L$ from the origin and $\alpha $ is the angle the normal vector to $L$ makes with the $X$-axis. It follows that the quantities $(\alpha ,s)$ can be considered as coordinates on the space of all lines in $\mathbb {R} ^{2}$, and the Radon transform can be expressed in these coordinates by: ${\begin{aligned}Rf(\alpha ,s)&=\int _{-\infty }^{\infty }f(x(z),y(z))\,dz\\&=\int _{-\infty }^{\infty }f{\big (}(z\sin \alpha +s\cos \alpha ),(-z\cos \alpha +s\sin \alpha ){\big )}\,dz.\end{aligned}}$ More generally, in the $n$-dimensional Euclidean space $\mathbb {R} ^{n}$, the Radon transform of a function $f$ satisfying the regularity conditions is a function $Rf$ on the space $\Sigma _{n}$ of all hyperplanes in $\mathbb {R} ^{n}$. It is defined by: Shepp Logan phantom Radon transform Inverse Radon transform $Rf(\xi )=\int _{\xi }f(\mathbf {x} )\,d\sigma (\mathbf {x} ),\quad \forall \xi \in \Sigma _{n}$ where the integral is taken with respect to the natural hypersurface measure, $d\sigma $ (generalizing the $\vert d\mathbf {x} \vert $ term from the $2$-dimensional case). Observe that any element of $\Sigma _{n}$ is characterized as the solution locus of an equation $\mathbf {x} \cdot \alpha =s$, where $\alpha \in S^{n-1}$ is a unit vector and $s\in \mathbb {R} $. Thus the $n$-dimensional Radon transform may be rewritten as a function on $S^{n-1}\times \mathbb {R} $ via: $Rf(\alpha ,s)=\int _{\mathbf {x} \cdot \alpha =s}f(\mathbf {x} )\,d\sigma (\mathbf {x} ).$ It is also possible to generalize the Radon transform still further by integrating instead over $k$-dimensional affine subspaces of $\mathbb {R} ^{n}$. The X-ray transform is the most widely used special case of this construction, and is obtained by integrating over straight lines. Relationship with the Fourier transform The Radon transform is closely related to the Fourier transform. We define the univariate Fourier transform here as: ${\hat {f}}(\omega )=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\omega }\,dx.$ For a function of a $2$-vector $\mathbf {x} =(x,y)$, the univariate Fourier transform is: ${\hat {f}}(\mathbf {w} )=\iint _{\mathbb {R} ^{2}}f(\mathbf {x} )e^{-2\pi i\mathbf {x} \cdot \mathbf {w} }\,dx\,dy.$ For convenience, denote ${\mathcal {R}}_{\alpha }[f](s)={\mathcal {R}}[f](\alpha ,s)$. The Fourier slice theorem then states: ${\widehat {{\mathcal {R}}_{\alpha }[f]}}(\sigma )={\hat {f}}(\sigma \mathbf {n} (\alpha ))$ where $\mathbf {n} (\alpha )=(\cos \alpha ,\sin \alpha ).$ Thus the two-dimensional Fourier transform of the initial function along a line at the inclination angle $\alpha $ is the one variable Fourier transform of the Radon transform (acquired at angle $\alpha $) of that function. This fact can be used to compute both the Radon transform and its inverse. The result can be generalized into n dimensions: ${\hat {f}}(r\alpha )=\int _{\mathbb {R} }{\mathcal {R}}f(\alpha ,s)e^{-2\pi isr}\,ds.$ Dual transform The dual Radon transform is a kind of adjoint to the Radon transform. Beginning with a function g on the space $\Sigma _{n}$, the dual Radon transform is the function ${\mathcal {R}}^{}g$ on Rn defined by: ${\mathcal {R}}^{}g(\mathbf {x} )=\int _{\mathbf {x} \in \xi }g(\xi )\,d\mu (\xi ).$ The integral here is taken over the set of all hyperplanes incident with the point ${\textbf {x}}\in \mathbb {R} ^{n}$, and the measure $d\mu $ is the unique probability measure on the set $\{\xi \|\mathbf {x} \in \xi \}$ invariant under rotations about the point $\mathbf {x} $. Concretely, for the two-dimensional Radon transform, the dual transform is given by: ${\mathcal {R}}^{}g(\mathbf {x} )={\frac {1}{2\pi }}\int _{\alpha =0}^{2\pi }g(\alpha ,\mathbf {n} (\alpha )\cdot \mathbf {x} )\,d\alpha .$ In the context of image processing, the dual transform is commonly called back-projection[3] as it takes a function defined on each line in the plane and 'smears' or projects it back over the line to produce an image. Intertwining property Let $\Delta $ denote the Laplacian on $\mathbb {R} ^{n}$ defined by: $\Delta ={\frac {\partial ^{2}}{\partial x_{1}^{2}}}+\cdots +{\frac {\partial ^{2}}{\partial x_{n}^{2}}}$ This is a natural rotationally invariant second-order differential operator. On $\Sigma _{n}$, the "radial" second derivative $Lf(\alpha ,s)\equiv {\frac {\partial ^{2}}{\partial s^{2}}}f(\alpha ,s)$ is also rotationally invariant. The Radon transform and its dual are intertwining operators for these two differential operators in the sense that:[4] ${\mathcal {R}}(\Delta f)=L({\mathcal {R}}f),\quad {\mathcal {R}}^{}(Lg)=\Delta ({\mathcal {R}}^{}g).$ In analysing the solutions to the wave equation in multiple spatial dimensions, the intertwining property leads to the translational representation of Lax and Philips.[5] In imaging[6] and numerical analysis[7] this is exploited to reduce multi-dimensional problems into single-dimensional ones, as a dimensional splitting method. Reconstruction approaches The process of reconstruction produces the image (or function $f$ in the previous section) from its projection data. Reconstruction is an inverse problem. Radon inversion formula In the two-dimensional case, the most commonly used analytical formula to recover $f$ from its Radon transform is the Filtered Back-projection Formula or Radon Inversion Formula[8]: $f(\mathbf {x} )=\int _{0}^{\pi }({\mathcal {R}}f(\cdot ,\theta )h)(\left\langle \mathbf {x} ,\mathbf {n} _{\theta }\right\rangle )\,d\theta $ where $h$ is such that ${\hat {h}}(k)=\|k\|$.[8] The convolution kernel $h$ is referred to as Ramp filter in some literature. Ill-posedness Intuitively, in the filtered back-projection formula, by analogy with differentiation, for which $ \left({\widehat {{\frac {d}{dx}}f}}\right)\!(k)=ik{\widehat {f}}(k)$, we see that the filter performs an operation similar to a derivative. Roughly speaking, then, the filter makes objects more singular. A quantitive statement of the ill-posedness of Radon inversion goes as follows: ${\widehat {{\mathcal {R}}^{}{\mathcal {R}}g}}(k)={\frac {1}{\\|\mathbf {k} \\|}}{\hat {g}}(\mathbf {k} )$ where ${\mathcal {R}}^{}$ is the previously defined adjoint to the Radon Transform. Thus for $g(\mathbf {x} )=e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle }$, we have: ${\mathcal {R}}^{}{\mathcal {R}}g={\frac {1}{\\|\mathbf {k_{0}} \\|}}e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle }$ The complex exponential $e^{i\left\langle \mathbf {k} _{0},\mathbf {x} \right\rangle }$ is thus an eigenfunction of ${\mathcal {R}}^{}{\mathcal {R}}$ with eigenvalue $ {\frac {1}{\\|\mathbf {k} _{0}\\|}}$. Thus the singular values of ${\mathcal {R}}$ are $ {\frac {1}{\sqrt {\\|\mathbf {k} \\|}}}$. Since these singular values tend to $0$, ${\mathcal {R}}^{-1}$ is unbounded.[8] Iterative reconstruction methods Compared with the Filtered Back-projection method, iterative reconstruction costs large computation time, limiting its practical use. However, due to the ill-posedness of Radon Inversion, the Filtered Back-projection method may be infeasible in the presence of discontinuity or noise. Iterative reconstruction methods (e.g. iterative Sparse Asymptotic Minimum Variance[9]) could provide metal artefact reduction, noise and dose reduction for the reconstructed result that attract much research interest around the world. Inversion formulas Explicit and computationally efficient inversion formulas for the Radon transform and its dual are available. The Radon transform in $n$ dimensions can be inverted by the formula:[10] $c_{n}f=(-\Delta )^{(n-1)/2}R^{}Rf\,$ where $c_{n}=(4\pi )^{(n-1)/2}{\frac {\Gamma (n/2)}{\Gamma (1/2)}}$, and the power of the Laplacian $(-\Delta )^{(n-1)/2}$ is defined as a pseudo-differential operator if necessary by the Fourier transform: $\left[{\mathcal {F}}(-\Delta )^{(n-1)/2}\varphi \right](\xi )=\|2\pi \xi \|^{n-1}({\mathcal {F}}\varphi )(\xi ).$ For computational purposes, the power of the Laplacian is commuted with the dual transform $R^{}$ to give:[11] $c_{n}f={\begin{cases}R^{}{\frac {d^{n-1}}{ds^{n-1}}}Rf&n{\text{ odd}}\\R^{}{\mathcal {H}}_{s}{\frac {d^{n-1}}{ds^{n-1}}}Rf&n{\text{ even}}\end{cases}}$ where ${\mathcal {H}}_{s}$ is the Hilbert transform with respect to the s variable. In two dimensions, the operator ${\mathcal {H}}_{s}{\frac {d}{ds}}$ appears in image processing as a ramp filter.[12] One can prove directly from the Fourier slice theorem and change of variables for integration that for a compactly supported continuous function $f$ of two variables: $f={\frac {1}{2}}R^{}H_{s}{\frac {d}{ds}}Rf.$ Thus in an image processing context the original image $f$ can be recovered from the 'sinogram' data $Rf$ by applying a ramp filter (in the $s$ variable) and then back-projecting. As the filtering step can be performed efficiently (for example using digital signal processing techniques) and the back projection step is simply an accumulation of values in the pixels of the image, this results in a highly efficient, and hence widely used, algorithm. Explicitly, the inversion formula obtained by the latter method is:[3] $f(x)={\begin{cases}\displaystyle -\imath 2\pi (2\pi )^{-n}(-1)^{n/2}\int _{S^{n-1}}{\frac {\partial ^{n-1}}{2\partial s^{n-1}}}Rf(\alpha ,\alpha \cdot x)\,d\alpha &n{\text{ odd}}\\\displaystyle (2\pi )^{-n}(-1)^{n/2}\iint _{\mathbb {R} \times S^{n-1}}{\frac {\partial ^{n-1}}{q\partial s^{n-1}}}Rf(\alpha ,\alpha \cdot x+q)\,d\alpha \,dq&n{\text{ even}}\\\end{cases}}$ The dual transform can also be inverted by an analogous formula: $c_{n}g=(-L)^{(n-1)/2}R(R^{}g).\,$ Radon transform in algebraic geometry In algebraic geometry, a Radon transform (also known as the Brylinski–Radon transform) is constructed as follows. Write $\mathbf {P} ^{d}\,{\stackrel {p_{1}}{\gets }}\,H\,{\stackrel {p_{2}}{\to }}\,\mathbf {P} ^{\vee ,d}$ for the universal hyperplane, i.e., H consists of pairs (x, h) where x is a point in d-dimensional projective space $\mathbf {P} ^{d}$ and h is a point in the dual projective space (in other words, x is a line through the origin in (d+1)-dimensional affine space, and h is a hyperplane in that space) such that x is contained in h. Then the Brylinksi–Radon transform is the functor between appropriate derived categories of étale sheaves $\operatorname {Rad} :=Rp_{2,}p_{1}^{}:D(\mathbf {P} ^{d})\to D(\mathbf {P} ^{\vee ,d}).$ :=Rp_{2,}p_{1}^{}:D(\mathbf {P} ^{d})\to D(\mathbf {P} ^{\vee ,d}).} The main theorem about this transform is that this transform induces an equivalence of the categories of perverse sheaves on the projective space and its dual projective space, up to constant sheaves.[13] See also • Periodogram • Matched filter • Deconvolution • X-ray transform • Funk transform • The Hough transform, when written in a continuous form, is very similar, if not equivalent, to the Radon transform.[14] • Cauchy–Crofton theorem is a closely related formula for computing the length of curves in space. • Fast Fourier transform Notes 1. Radon 1917. 2. Radon 1986. 3. Roerdink 2001. 4. Helgason 1984, Lemma I.2.1. 5. Lax, P. D.; Philips, R. S. (1964). "Scattering theory". Bull. Amer. Math. Soc. 70 (1): 130–142. doi:10.1090/s0002-9904-1964-11051-x. 6. Bonneel, N.; Rabin, J.; Peyre, G.; Pfister, H. (2015). "Sliced and Radon Wasserstein Barycenters of Measures". Journal of Mathematical Imaging and Vision. 51 (1): 22–25. doi:10.1007/s10851-014-0506-3. S2CID 1907942. 7. Rim, D. (2018). "Dimensional Splitting of Hyperbolic Partial Differential Equations Using the Radon Transform". SIAM J. Sci. Comput. 40 (6): A4184–A4207. arXiv:1705.03609. Bibcode:2018SJSC...40A4184R. doi:10.1137/17m1135633. S2CID 115193737. 8. Candès 2016b. 9. Abeida, Habti; Zhang, Qilin; Li, Jian; Merabtine, Nadjim (2013). "Iterative Sparse Asymptotic Minimum Variance Based Approaches for Array Processing" (PDF). IEEE Transactions on Signal Processing. IEEE. 61 (4): 933–944. arXiv:1802.03070. Bibcode:2013ITSP...61..933A. doi:10.1109/tsp.2012.2231676. ISSN 1053-587X. S2CID 16276001. 10. Helgason 1984, Theorem I.2.13. 11. Helgason 1984, Theorem I.2.16. 12. Nygren 1997. 13. Kiehl & Weissauer (2001, Ch. IV, Cor. 2.4) 14. van Ginkel, Hendricks & van Vliet 2004. References • Kiehl, Reinhardt; Weissauer, Rainer (2001), Weil conjectures, perverse sheaves and l'adic Fourier transform, Springer, doi:10.1007/978-3-662-04576-3, ISBN 3-540-41457-6, MR 1855066 • Radon, Johann (1917), "Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten", Berichte über die Verhandlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse [Reports on the Proceedings of the Royal Saxonian Academy of Sciences at Leipzig, Mathematical and Physical Section], Leipzig: Teubner (69): 262–277; Translation: Radon, J. (December 1986), translated by Parks, P.C., "On the determination of functions from their integral values along certain manifolds", IEEE Transactions on Medical Imaging, 5 (4): 170–176, doi:10.1109/TMI.1986.4307775, PMID 18244009, S2CID 26553287. • Roerdink, J.B.T.M. (2001) [1994], "Tomography", Encyclopedia of Mathematics, EMS Press. • Helgason, Sigurdur (1984), Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press, ISBN 0-12-338301-3. • Candès, Emmanuel (February 2, 2016a). "Applied Fourier Analysis and Elements of Modern Signal Processing – Lecture 9" (PDF). • Candès, Emmanuel (February 4, 2016b). "Applied Fourier Analysis and Elements of Modern Signal Processing – Lecture 10" (PDF). • Nygren, Anders J. (1997). "Filtered Back Projection". Tomographic Reconstruction of SPECT Data. • van Ginkel, M.; Hendricks, C.L. Luengo; van Vliet, L.J. (2004). "A short introduction to the Radon and Hough transforms and how they relate to each other" (PDF). Archived (PDF) from the original on 2016-07-29. Further reading • Lokenath Debnath; Dambaru Bhatta (19 April 2016). Integral Transforms and Their Applications. CRC Press. ISBN 978-1-4200-1091-6. • Deans, Stanley R. (1983), The Radon Transform and Some of Its Applications, New York: John Wiley & Sons • Helgason, Sigurdur (2008), Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39 (2nd ed.), Providence, R.I.: American Mathematical Society, doi:10.1090/surv/039, ISBN 978-0-8218-4530-1, MR 2463854 • Herman, Gabor T. (2009), Fundamentals of Computerized Tomography: Image Reconstruction from Projections (2nd ed.), Springer, ISBN 978-1-85233-617-2 • Minlos, R.A. (2001) [1994], "Radon transform", Encyclopedia of Mathematics, EMS Press • Natterer, Frank (June 2001), The Mathematics of Computerized Tomography, Classics in Applied Mathematics, vol. 32, Society for Industrial and Applied Mathematics, ISBN 0-89871-493-1 • Natterer, Frank; Wübbeling, Frank (2001), Mathematical Methods in Image Reconstruction, Society for Industrial and Applied Mathematics, ISBN 0-89871-472-9 External links • Weisstein, Eric W. "Radon Transform". MathWorld. • Analytical projection (the Radon transform) (video). Part of the "Computed Tomography and the ASTRA Toolbox" course. University of Antwerp. September 10, 2015.	Wikipedia
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets.[1] These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures. Motivation A common problem is to find a good notion of a measure on a topological space that is compatible with the topology in some sense. One way to do this is to define a measure on the Borel sets of the topological space. In general there are several problems with this: for example, such a measure may not have a well defined support. Another approach to measure theory is to restrict to locally compact Hausdorff spaces, and only consider the measures that correspond to positive linear functionals on the space of continuous functions with compact support (some authors use this as the definition of a Radon measure). This produces a good theory with no pathological problems, but does not apply to spaces that are not locally compact. If there is no restriction to non-negative measures and complex measures are allowed, then Radon measures can be defined as the continuous dual space on the space of continuous functions with compact support. If such a Radon measure is real then it can be decomposed into the difference of two positive measures. Furthermore, an arbitrary Radon measure can be decomposed into four positive Radon measures, where the real and imaginary parts of the functional are each the differences of two positive Radon measures. The theory of Radon measures has most of the good properties of the usual theory for locally compact spaces, but applies to all Hausdorff topological spaces. The idea of the definition of a Radon measure is to find some properties that characterize the measures on locally compact spaces corresponding to positive functionals, and use these properties as the definition of a Radon measure on an arbitrary Hausdorff space. Definitions Let m be a measure on the σ-algebra of Borel sets of a Hausdorff topological space X. • The measure m is called inner regular or tight if, for any open set U, m(U) is the supremum of m(K) over all compact subsets K of U. • The measure m is called outer regular if, for any Borel set B, m(B) is the infimum of m(U) over all open sets U containing B. • The measure m is called locally finite if every point of X has a neighborhood U for which m(U) is finite. If m is locally finite, then it follows that m is finite on compact sets, and for locally compact Hausdorff spaces, the converse holds, too. Thus, in this case, local finiteness may be equivalently replaced by finiteness on compact subsets. The measure m is called a Radon measure if it is inner regular and locally finite. In many situations, such as finite measures on locally compact spaces, this also implies outer regularity (see also Radon spaces). (It is possible to extend the theory of Radon measures to non-Hausdorff spaces, essentially by replacing the word "compact" by "closed compact" everywhere. However, there seem to be almost no applications of this extension.) Radon measures on locally compact spaces When the underlying measure space is a locally compact topological space, the definition of a Radon measure can be expressed in terms of continuous linear functionals on the space of continuous functions with compact support. This makes it possible to develop measure and integration in terms of functional analysis, an approach taken by Bourbaki and a number of other authors.[2] Measures In what follows X denotes a locally compact topological space. The continuous real-valued functions with compact support on X form a vector space K(X) = CC(X), which can be given a natural locally convex topology. Indeed, K(X) is the union of the spaces K(X, K) of continuous functions with support contained in compact sets K. Each of the spaces K(X, K) carries naturally the topology of uniform convergence, which makes it into a Banach space. But as a union of topological spaces is a special case of a direct limit of topological spaces, the space K(X) can be equipped with the direct limit locally convex topology induced by the spaces K(X, K); this topology is finer than the topology of uniform convergence. If m is a Radon measure on $X,$ then the mapping $I:f\mapsto \int f(x)\,m(dx)$ is a continuous positive linear map from K(X) to R. Positivity means that I(f) ≥ 0 whenever f is a non-negative function. Continuity with respect to the direct limit topology defined above is equivalent to the following condition: for every compact subset K of X there exists a constant MK such that, for every continuous real-valued function f on X with support contained in K, $\|I(f)\|\leq M_{K}\sup _{x\in X}\|f(x)\|.$ Conversely, by the Riesz–Markov–Kakutani representation theorem, each positive linear form on K(X) arises as integration with respect to a unique regular Borel measure. A real-valued Radon measure is defined to be any continuous linear form on K(X); they are precisely the differences of two Radon measures. This gives an identification of real-valued Radon measures with the dual space of the locally convex space K(X). These real-valued Radon measures need not be signed measures. For example, sin(x) dx is a real-valued Radon measure, but is not even an extended signed measure as it cannot be written as the difference of two measures at least one of which is finite. Some authors use the preceding approach to define (positive) Radon measures to be the positive linear forms on K(X).[3] In this set-up it is common to use a terminology in which Radon measures in the above sense are called positive measures and real-valued Radon measures as above are called (real) measures. Integration To complete the buildup of measure theory for locally compact spaces from the functional-analytic viewpoint, it is necessary to extend measure (integral) from compactly supported continuous functions. This can be done for real or complex-valued functions in several steps as follows: 1. Definition of the upper integral μ(g) of a lower semicontinuous positive (real-valued) function g as the supremum (possibly infinite) of the positive numbers μ(h) for compactly supported continuous functions h ≤ g; 2. Definition of the upper integral μ(f) for an arbitrary positive (real-valued) function f as the infimum of upper integrals μ(g) for lower semi-continuous functions g ≥ f; 3. Definition of the vector space F = F(X, μ) as the space of all functions f on X for which the upper integral μ(\|f\|) of the absolute value is finite; the upper integral of the absolute value defines a semi-norm on F, and F is a complete space with respect to the topology defined by the semi-norm; 4. Definition of the space L1(X, μ) of integrable functions as the closure inside F of the space of continuous compactly supported functions. 5. Definition of the integral for functions in L1(X, μ) as extension by continuity (after verifying that μ is continuous with respect to the topology of L1(X, μ)); 6. Definition of the measure of a set as the integral (when it exists) of the indicator function of the set. It is possible to verify that these steps produce a theory identical with the one that starts from a Radon measure defined as a function that assigns a number to each Borel set of X. The Lebesgue measure on R can be introduced by a few ways in this functional-analytic set-up. First, it is possibly to rely on an "elementary" integral such as the Daniell integral or the Riemann integral for integrals of continuous functions with compact support, as these are integrable for all the elementary definitions of integrals. The measure (in the sense defined above) defined by elementary integration is precisely the Lebesgue measure. Second, if one wants to avoid reliance on Riemann or Daniell integral or other similar theories, it is possible to develop first the general theory of Haar measures and define the Lebesgue measure as the Haar measure λ on R that satisfies the normalisation condition λ([0, 1]) = 1. Examples The following are all examples of Radon measures: • Lebesgue measure on Euclidean space (restricted to the Borel subsets); • Haar measure on any locally compact topological group; • Dirac measure on any topological space; • Gaussian measure on Euclidean space ℝn with its Borel sigma algebra; • Probability measures on the σ-algebra of Borel sets of any Polish space. This example not only generalizes the previous example, but includes many measures on non-locally compact spaces, such as Wiener measure on the space of real-valued continuous functions on the interval [0, 1]. • A measure on ℝ is a Radon measure if and only if it is a locally finite Borel measure. The following are not examples of Radon measures: • Counting measure on Euclidean space is an example of a measure that is not a Radon measure, since it is not locally finite. • The space of ordinals at most equal to Ω, the first uncountable ordinal with the order topology is a compact topological space. The measure which equals 1 on any Borel set that contains an uncountable closed subset of [1, Ω), and 0 otherwise, is Borel but not Radon, as the one-point set {Ω} has measure zero but any open neighbourhood of it has measure 1.[4] • Let X be the interval [0, 1) equipped with the topology generated by the collection of half open intervals {[a, b) : 0 ≤ a < b ≤ 1}. This topology is sometimes called Sorgenfrey line. On this topological space, standard Lebesgue measure is not Radon since it is not inner regular, since compact sets are at most countable. • Let Z be a Bernstein set in [0, 1] (or any Polish space). Then no measure which vanishes at points on Z is a Radon measure, since any compact set in Z is countable. • Standard product measure on (0, 1)κ for uncountable κ is not a Radon measure, since any compact set is contained within a product of uncountably many closed intervals, each of which is shorter than 1. We note that, intuitively, the Radon measure is useful in mathematical finance particularly for working with Lévy processes because it has the properties of both Lebesgue and Dirac measures, as unlike the Lebesgue, a Radon measure on a single point is not necessarily of measure 0.[5] Basic properties Moderated Radon measures Given a Radon measure m on a space X, we can define another measure M (on the Borel sets) by putting $M(B)=\inf\{m(V)\mid V{\text{ is an open set with }}B\subseteq V\subseteq X\}.$ The measure M is outer regular, and locally finite, and inner regular for open sets. It coincides with m on compact and open sets, and m can be reconstructed from M as the unique inner regular measure that is the same as M on compact sets. The measure m is called moderated if M is σ-finite; in this case the measures m and M are the same. (If m is σ-finite this does not imply that M is σ-finite, so being moderated is stronger than being σ-finite.) On a hereditarily Lindelöf space every Radon measure is moderated. An example of a measure m that is σ-finite but not moderated as follows.[6] The topological space X has as underlying set the subset of the real plane given by the y-axis of points (0, y) together with the points (1/n, m/n2) with m, n positive integers. The topology is given as follows. The single points (1/n, m/n2) are all open sets. A base of neighborhoods of the point (0, y) is given by wedges consisting of all points in X of the form (u, v) with \|v − y\| ≤ \|u\| ≤ 1/n for a positive integer n. This space X is locally compact. The measure m is given by letting the y-axis have measure 0 and letting the point (1/n, m/n2) have measure 1/n3. This measure is inner regular and locally finite, but is not outer regular as any open set containing the y-axis has measure infinity. In particular the y-axis has m-measure 0 but M-measure infinity. Radon spaces Main article: Radon space A topological space is called a Radon space if every finite Borel measure is a Radon measure, and strongly Radon if every locally finite Borel measure is a Radon measure. Any Suslin space is strongly Radon, and moreover every Radon measure is moderated. Duality On a locally compact Hausdorff space, Radon measures correspond to positive linear functionals on the space of continuous functions with compact support. This is not surprising as this property is the main motivation for the definition of Radon measure. Metric space structure The pointed cone M+(X) of all (positive) Radon measures on X can be given the structure of a complete metric space by defining the Radon distance between two measures m1, m2 ∈ M+(X) to be $\rho (m_{1},m_{2})=\sup \left\{\left.\int _{X}f(x)(m_{1}-m_{2})(dx)\ \right\|\mathrm {continuous\,} f:X\to [-1,1]\subset \mathbb {R} \right\}.$ This metric has some limitations. For example, the space of Radon probability measures on X, ${\mathcal {P}}(X)=\{m\in {\mathcal {M}}_{+}(X)\mid m(X)=1\},$ is not sequentially compact with respect to the Radon metric: i.e., it is not guaranteed that any sequence of probability measures will have a subsequence that is convergent with respect to the Radon metric, which presents difficulties in certain applications. On the other hand, if X is a compact metric space, then the Wasserstein metric turns P(X) into a compact metric space. Convergence in the Radon metric implies weak convergence of measures: $\rho (m_{n},m)\to 0\Rightarrow m_{n}\rightharpoonup m,$ but the converse implication is false in general. Convergence of measures in the Radon metric is sometimes known as strong convergence, as contrasted with weak convergence. See also • Radonifying function • Vague topology References 1. Folland 1999, p. 212 2. Bourbaki 2004 harvnb error: no target: CITEREFBourbaki2004 (help) 3. Bourbaki 2004 harvnb error: no target: CITEREFBourbaki2004 (help); Hewitt & Stromberg 1965; Dieudonné 1970. 4. Schwartz 1974, p. 45 5. Cont, Rama, and Peter Tankov. Financial modelling with jump processes. Chapman & Hall, 2004. 6. Bourbaki 2004, Exercise 5 of section 1 harvnb error: no target: CITEREFBourbaki2004 (help) Bibliography • Bourbaki, Nicolas (2004a), Integration I, Springer Verlag, ISBN 3-540-41129-1. Functional-analytic development of the theory of Radon measure and integral on locally compact spaces. • Bourbaki, Nicolas (2004b), Integration II, Springer Verlag, ISBN 3-540-20585-3. Haar measure; Radon measures on general Hausdorff spaces and equivalence between the definitions in terms of linear functionals and locally finite inner regular measures on the Borel sigma-algebra. • Dieudonné, Jean (1970), Treatise on analysis, vol. 2, Academic Press. Contains a simplified version of Bourbaki's approach, specialised to measures defined on separable metrizable spaces. • Folland, Gerald (1999), Real Analysis: Modern techniques and their applications, New York: John Wiley & Sons, Inc., p. 212, ISBN 0-471-31716-0 • Hewitt, Edwin; Stromberg, Karl (1965), Real and abstract analysis, Springer-Verlag • König, Heinz (1997), Measure and integration: an advanced course in basic procedures and applications, New York: Springer, ISBN 3-540-61858-9 • Schwartz, Laurent (1974), Radon measures on arbitrary topological spaces and cylindrical measures, Oxford University Press, ISBN 0-19-560516-0 External links • R. A. Minlos (2001) [1994], "Radon measure", Encyclopedia of Mathematics, EMS Press Measure theory Basic concepts • Absolute continuity of measures • Lebesgue integration • Lp spaces • Measure • Measure space • Probability space • Measurable space/function Sets • Almost everywhere • Atom • Baire set • Borel set • equivalence relation • Borel space • Carathéodory's criterion • Cylindrical σ-algebra • Cylinder set • 𝜆-system • Essential range • infimum/supremum • Locally measurable • π-system • σ-algebra • Non-measurable set • Vitali set • Null set • Support • Transverse measure • Universally measurable Types of Measures • Atomic • Baire • Banach • Besov • Borel • Brown • Complex • Complete • Content • (Logarithmically) Convex • Decomposable • Discrete • Equivalent • Finite • Inner • (Quasi-) Invariant • Locally finite • Maximising • Metric outer • Outer • Perfect • Pre-measure • (Sub-) Probability • Projection-valued • Radon • Random • Regular • Borel regular • Inner regular • Outer regular • Saturated • Set function • σ-finite • s-finite • Signed • Singular • Spectral • Strictly positive • Tight • Vector Particular measures • Counting • Dirac • Euler • Gaussian • Haar • Harmonic • Hausdorff • Intensity • Lebesgue • Infinite-dimensional • Logarithmic • Product • Projections • Pushforward • Spherical measure • Tangent • Trivial • Young Maps • Measurable function • Bochner • Strongly • Weakly • Convergence: almost everywhere • of measures • in measure • of random variables • in distribution • in probability • Cylinder set measure • Random: compact set • element • measure • process • variable • vector • Projection-valued measure Main results • Carathéodory's extension theorem • Convergence theorems • Dominated • Monotone • Vitali • Decomposition theorems • Hahn • Jordan • Maharam's • Egorov's • Fatou's lemma • Fubini's • Fubini–Tonelli • Hölder's inequality • Minkowski inequality • Radon–Nikodym • Riesz–Markov–Kakutani representation theorem Other results • Disintegration theorem • Lifting theory • Lebesgue's density theorem • Lebesgue differentiation theorem • Sard's theorem For Lebesgue measure • Isoperimetric inequality • Brunn–Minkowski theorem • Milman's reverse • Minkowski–Steiner formula • Prékopa–Leindler inequality • Vitale's random Brunn–Minkowski inequality Applications & related • Convex analysis • Descriptive set theory • Probability theory • Real analysis • Spectral theory	Wikipedia
Radon's theorem In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that: Any set of d + 2 points in Rd can be partitioned into two sets whose convex hulls intersect. A point in the intersection of these convex hulls is called a Radon point of the set. For example, in the case d = 2, any set of four points in the Euclidean plane can be partitioned in one of two ways. It may form a triple and a singleton, where the convex hull of the triple (a triangle) contains the singleton; alternatively, it may form two pairs of points that form the endpoints of two intersecting line segments. Proof and construction Consider any set $X=\{x_{1},x_{2},\dots ,x_{d+2}\}\subset \mathbf {R} ^{d}$ of d + 2 points in d-dimensional space. Then there exists a set of multipliers a1, ..., ad + 2, not all of which are zero, solving the system of linear equations $\sum _{i=1}^{d+2}a_{i}x_{i}=0,\quad \sum _{i=1}^{d+2}a_{i}=0,$ because there are d + 2 unknowns (the multipliers) but only d + 1 equations that they must satisfy (one for each coordinate of the points, together with a final equation requiring the sum of the multipliers to be zero). Fix some particular nonzero solution a1, ..., ad + 2. Let $I\subseteq X$ be the set of points with positive multipliers, and let $J=X\setminus I$ be the set of points with multipliers that are negative or zero. Then $I$ and $J$ form the required partition of the points into two subsets with intersecting convex hulls. The convex hulls of $I$ and $J$ must intersect, because they both contain the point $p=\sum _{x_{i}\in I}{\frac {a_{i}}{A}}x_{i}=\sum _{x_{j}\in J}{\frac {-a_{j}}{A}}x_{j},$ where $A=\sum _{x_{i}\in I}a_{i}=-\sum _{x_{j}\in J}a_{j}.$ The left hand side of the formula for $p$ expresses this point as a convex combination of the points in $I$, and the right hand side expresses it as a convex combination of the points in $J$. Therefore, $p$ belongs to both convex hulls, completing the proof. This proof method allows for the efficient construction of a Radon point, in an amount of time that is polynomial in the dimension, by using Gaussian elimination or other efficient algorithms to solve the system of equations for the multipliers.[1] Topological Radon theorem An equivalent formulation of Radon's theorem is: If ƒ is any affine function from a (d + 1)-dimensional simplex Δd+1 to Rd, then there are two disjoint faces of Δd+1 whose images under ƒ intersect. They are equivalent because any affine function on a simplex is uniquely determined by the images of its vertices. Formally, let ƒ be an affine function from Δd+1 to Rd. Let $v_{1},v_{2},\dots ,v_{d+2}$ be the vertices of Δd+1, and let $x_{1},x_{2},\dots ,x_{d+2}$ be their images under ƒ. By the original formulation, the $x_{1},x_{2},\dots ,x_{d+2}$ can be partitioned into two disjoint subsets, e.g. (xi)i in I and (xj)j in J, with overlapping convex hull. Because f is affine, the convex hull of (xi)i in I is the image of the face spanned by the vertices (vi)i in I, and similarly the convex hull of (xj)j in J is the image of the face spanned by the vertices (vj)j in j. These two faces are disjoint, and their images under f intersect - as claimed by the new formulation. The topological Radon theorem generalizes this formluation. It allows f to be any continuous function - not necessarily affine:[2] If ƒ is any continuous function from a (d + 1)-dimensional simplex Δd+1 to Rd, then there are two disjoint faces of Δd+1 whose images under ƒ intersect. More generally, if K is any (d + 1)-dimensional compact convex set, and ƒ is any continuous function from K to d-dimensional space, then there exists a linear function g such that some point where g achieves its maximum value and some other point where g achieves its minimum value are mapped by ƒ to the same point. In the case where K is a simplex, the two simplex faces formed by the maximum and minimum points of g must then be two disjoint faces whose images have a nonempty intersection. This same general statement, when applied to a hypersphere instead of a simplex, gives the Borsuk–Ulam theorem, that ƒ must map two opposite points of the sphere to the same point.[2] Proofs The topological Radon theorem was originally proved by Bajmoczy and Barany[2] in the following way: • Construct a continuous map g from Sd (the d-dimensional sphere) to Δd+1, such that for every point x on the sphere, g(x) and g(-x) are on two disjoint faces of Δd+1. • Apply the Borsuk–Ulam theorem to the function $f\circ g$, which is a continuous function from Sd to Rd. The theorem says that, for any such function, there exists some point y on Sd, such that f(g(y)) = f(g(-y)). • The points g(y) and g(-y) are on two disjoint faces of Δd+1, and they are mapped by f to the same point of Rd. This implies that the images of these two disjoint faces intersect. Another proof was given by Lovasz and Schrijver.[3] A third proof is given by Matousek:[4]: 115 • Let K be the simplex Δd+1, and let $K_{\Delta }^{2}$ be the deleted join of K with itself. • The geometric realization of $K_{\Delta }^{2}$ is homeomorphic to the sphere Sd+1. • Therefore, the Z2-index of $K_{\Delta }^{2}$ equals d+1. • The topological Radon theorem follows from the following more general theorem. For any simplicial complex K, if the Z2-index of $K_{\Delta }^{2}$ is larger than d, then for every continuous mapping from \|\|K\|\| to Rd, the images of two disjoint faces of K intersect. Applications The Radon point of any four points in the plane is their geometric median, the point that minimizes the sum of distances to the other points.[5][6] Radon's theorem forms a key step of a standard proof of Helly's theorem on intersections of convex sets;[7] this proof was the motivation for Radon's original discovery of Radon's theorem. Radon's theorem can also be used to calculate the VC dimension of d-dimensional points with respect to linear separations. There exist sets of d + 1 points (for instance, the points of a regular simplex) such that every two nonempty subsets can be separated from each other by a hyperplane. However, no matter which set of d + 2 points is given, the two subsets of a Radon partition cannot be linearly separated. Therefore, the VC dimension of this system is exactly d + 1.[8] A randomized algorithm that repeatedly replaces sets of d + 2 points by their Radon point can be used to compute an approximation to a centerpoint of any point set, in an amount of time that is polynomial in both the number of points and the dimension.[1] Related concepts Geometric median. The Radon point of three points in a one-dimensional space is just their median. The geometric median of a set of points is the point minimizing the sum of distances to the points in the set; it generalizes the one-dimensional median and has been studied both from the point of view of facility location and robust statistics. For sets of four points in the plane, the geometric median coincides with the Radon point. Tverberg's theorem. A generalization for partition into r sets was given by Helge Tverberg (1966) and is now known as Tverberg's theorem. It states that for any set of $(d+1)(r-1)+1\ $points in Euclidean d-space, there is a partition into r subsets whose convex hulls intersect in at least one common point. Carathéodory's theorem states that any point in the convex hull of some set of points is also within the convex hull of a subset of at most d + 1 of the points; that is, that the given point is part of a Radon partition in which it is a singleton. One proof of Carathéodory's theorem uses a technique of examining solutions to systems of linear equations, similar to the proof of Radon's theorem, to eliminate one point at a time until at most d + 1 remain. Convex geometries. Concepts related to Radon's theorem have also been considered for convex geometries, families of finite sets with the properties that the intersection of any two sets in the family remains in the family, and that the empty set and the union of all the sets belongs to the family. In this more general context, the convex hull of a set S is the intersection of the family members that contain S, and the Radon number of a space is the smallest r such that any r points have two subsets whose convex hulls intersect. Similarly, one can define the Helly number h and the Carathéodory number c by analogy to their definitions for convex sets in Euclidean spaces, and it can be shown that these numbers satisfy the inequalities h < r ≤ ch + 1.[9] Radon theorem for graphs. In an arbitrary undirected graph, one may define a convex set to be a set of vertices that includes every induced path connecting a pair of vertices in the set. With this definition, every set of ω + 1 vertices in the graph can be partitioned into two subsets whose convex hulls intersect, and ω + 1 is the minimum number for which this is possible, where ω is the clique number of the given graph.[10] For related results involving shortest paths instead of induced paths see Chepoi (1986) and Bandelt & Pesch (1989). Notes 1. Clarkson et al. (1996). 2. Bajmóczy, E. G.; Bárány, I. (1979-09-01). "On a common generalization of Borsuk's and Radon's theorem". Acta Mathematica Academiae Scientiarum Hungaricae. 34 (3): 347–350. doi:10.1007/BF01896131. ISSN 1588-2632. S2CID 12971298. 3. Lovász, László; Schrijver, Alexander (1998). "A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs". Proceedings of the American Mathematical Society. 126 (5): 1275–1285. doi:10.1090/S0002-9939-98-04244-0. ISSN 0002-9939. S2CID 7790459. 4. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3 5. Cieslik, Dietmar (2006), Shortest Connectivity: An Introduction with Applications in Phylogeny, Combinatorial Optimization, vol. 17, Springer, p. 6, ISBN 9780387235394. 6. Plastria, Frank (2006), "Four-point Fermat location problems revisited. New proofs and extensions of old results" (PDF), IMA Journal of Management Mathematics, 17 (4): 387–396, doi:10.1093/imaman/dpl007, Zbl 1126.90046. 7. Matoušek (2002), p. 11. 8. Epsilon-nets and VC-dimension, Lecture Notes by Marco Pellegrini, 2004. 9. Kay & Womble (1971). 10. Duchet (1987). References • Bajmóczy, E. G.; Bárány, I. (1979), "A common generalization of Borsuk's and Radon's theorem", Acta Mathematica Hungarica, 34 (3–4): 347–350, doi:10.1007/BF01896131. • Bandelt, H.-J.; Pesch, E. (1989), "A Radon theorem for Helly graphs", Archiv der Mathematik, 52 (1): 95–98, doi:10.1007/BF01197978. • Chepoi, V. D. (1986), "Some properties of the d-convexity in triangulated graphs", Mat. Issled. (in Russian), 87: 164–177. As cited by Bandelt & Pesch (1989). • Clarkson, Kenneth L.; Eppstein, David; Miller, Gary L.; Sturtivant, Carl; Teng, Shang-Hua (1996), "Approximating center points with iterated Radon points" (PDF), International Journal of Computational Geometry & Applications, 6 (3): 357–377, doi:10.1142/s021819599600023x, MR 1409651. • Danzer, L.; Grünbaum, B.; Klee, V. (1963), "Helly's theorem and its relatives", Convexity, Proc. Symp. Pure Math., vol. 7, American Mathematical Society, pp. 101–179. • Duchet, Pierre (1987), "Convex sets in graphs. II. Minimal path convexity", Journal of Combinatorial Theory, Series A, 44 (3): 307–316, doi:10.1016/0095-8956(88)90039-1. As cited by Bandelt & Pesch (1989). • Eckhoff, J. (1993), "Helly, Radon, and Carathéodory type theorems", Handbook of Convex Geometry, vol. A, B, Amsterdam: North-Holland, pp. 389–448. • Kay, David C.; Womble, Eugene W. (1971), "Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers", Pacific Journal of Mathematics, 38 (2): 471–485, doi:10.2140/pjm.1971.38.471, MR 0310766. • Matoušek, J. (2002), "1.3 Radon's Lemma and Helly's Theorem", Lectures on Discrete Geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, pp. 9–12, ISBN 978-0-387-95373-1. • Matoušek, J. (2003), "5.1 Nonembeddability Theorems: An Introduction", Using the Borsuk–Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry, Springer-Verlag, pp. 88–92. • Radon, J. (1921), "Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten", Mathematische Annalen, 83 (1–2): 113–115, doi:10.1007/BF01464231. • Tverberg, H. (1966), "A generalization of Radon's theorem" (PDF), Journal of the London Mathematical Society, 41: 123–128, doi:10.1112/jlms/s1-41.1.123.	Wikipedia
Radon–Nikodym theorem In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space. One way to derive a new measure from one already given is to assign a density to each point of the space, then integrate over the measurable subset of interest. This can be expressed as $\nu (A)=\int _{A}f\,d\mu ,$ where ν is the new measure being defined for any measurable subset A and the function f is the density at a given point. The integral is with respect to an existing measure μ, which may often be the canonical Lebesgue measure on the real line R or the n-dimensional Euclidean space Rn (corresponding to our standard notions of length, area and volume). For example, if f represented mass density and μ was the Lebesgue measure in three-dimensional space R3, then ν(A) would equal the total mass in a spatial region A. The Radon–Nikodym theorem essentially states that, under certain conditions, any measure ν can be expressed in this way with respect to another measure μ on the same space. The function f is then called the Radon–Nikodym derivative and is denoted by ${\tfrac {d\nu }{d\mu }}$.[1] An important application is in probability theory, leading to the probability density function of a random variable. The theorem is named after Johann Radon, who proved the theorem for the special case where the underlying space is Rn in 1913, and for Otto Nikodym who proved the general case in 1930.[2] In 1936 Hans Freudenthal generalized the Radon–Nikodym theorem by proving the Freudenthal spectral theorem, a result in Riesz space theory; this contains the Radon–Nikodym theorem as a special case.[3] A Banach space Y is said to have the Radon–Nikodym property if the generalization of the Radon–Nikodym theorem also holds, mutatis mutandis, for functions with values in Y. All Hilbert spaces have the Radon–Nikodym property. Formal description Radon–Nikodym theorem The Radon–Nikodym theorem involves a measurable space $(X,\Sigma )$ on which two σ-finite measures are defined, $\mu $ and $\nu .$ It states that, if $\nu \ll \mu $ (that is, if $\nu $ is absolutely continuous with respect to $\mu $), then there exists a $\Sigma $-measurable function $f:X\to [0,\infty ),$ such that for any measurable set $A\subseteq X,$ $\nu (A)=\int _{A}f\,d\mu .$ Radon–Nikodym derivative The function $f$ satisfying the above equality is uniquely defined up to a $\mu $-null set, that is, if $g$ is another function which satisfies the same property, then $f=g$ $\mu $-almost everywhere. The function $f$ is commonly written ${\frac {d\nu }{d\mu }$ and is called the Radon–Nikodym derivative. The choice of notation and the name of the function reflects the fact that the function is analogous to a derivative in calculus in the sense that it describes the rate of change of density of one measure with respect to another (the way the Jacobian determinant is used in multivariable integration). Extension to signed or complex measures A similar theorem can be proven for signed and complex measures: namely, that if $\mu $ is a nonnegative σ-finite measure, and $\nu $ is a finite-valued signed or complex measure such that $\nu \ll \mu ,$ that is, $\nu $ is absolutely continuous with respect to $\mu ,$ then there is a $\mu $-integrable real- or complex-valued function $g$ on $X$ such that for every measurable set $A,$ $\nu (A)=\int _{A}g\,d\mu .$ Examples In the following examples, the set X is the real interval [0,1], and $\Sigma $ is the Borel sigma-algebra on X. 1. $\mu $ is the length measure on X. $\nu $ assigns to each subset Y of X, twice the length of Y. Then, $ {\frac {d\nu }{d\mu }}=2$. 2. $\mu $ is the length measure on X. $\nu $ assigns to each subset Y of X, the number of points from the set {0.1, …, 0.9} that are contained in Y. Then, $\nu $ is not absolutely-continuous with respect to $\mu $ since it assigns non-zero measure to zero-length points. Indeed, there is no derivative $ {\frac {d\nu }{d\mu }}$: there is no finite function that, when integrated e.g. from $(0.1-\varepsilon )$ to $(0.1+\varepsilon )$, gives $1$ for all $\varepsilon >0$. 3. $\mu =\nu +\delta _{0}$, where $\nu $ is the length measure on X and $\delta _{0}$ is the Dirac measure on 0 (it assigns a measure of 1 to any set containing 0 and a measure of 0 to any other set). Then, $\nu $ is absolutely continuous with respect to $\mu $, and $ {\frac {d\nu }{d\mu }}=1_{X\setminus \{0\}}$ – the derivative is 0 at $x=0$ and 1 at $x>0$.[4] Properties • Let ν, μ, and λ be σ-finite measures on the same measurable space. If ν ≪ λ and μ ≪ λ (ν and μ are both absolutely continuous with respect to λ), then ${\frac {d(\nu +\mu )}{d\lambda }}={\frac {d\nu }{d\lambda }}+{\frac {d\mu }{d\lambda }}\quad \lambda {\text{-almost everywhere}}.$ • If ν ≪ μ ≪ λ, then ${\frac {d\nu }{d\lambda }}={\frac {d\nu }{d\mu }}{\frac {d\mu }{d\lambda }}\quad \lambda {\text{-almost everywhere}}.$ • In particular, if μ ≪ ν and ν ≪ μ, then ${\frac {d\mu }{d\nu }}=\left({\frac {d\nu }{d\mu }}\right)^{-1}\quad \nu {\text{-almost everywhere}}.$ • If μ ≪ λ and g is a μ-integrable function, then $\int _{X}g\,d\mu =\int _{X}g{\frac {d\mu }{d\lambda }}\,d\lambda .$ • If ν is a finite signed or complex measure, then ${d\|\nu \| \over d\mu }=\left\|{d\nu \over d\mu }\right\|.$ Applications Probability theory The theorem is very important in extending the ideas of probability theory from probability masses and probability densities defined over real numbers to probability measures defined over arbitrary sets. It tells if and how it is possible to change from one probability measure to another. Specifically, the probability density function of a random variable is the Radon–Nikodym derivative of the induced measure with respect to some base measure (usually the Lebesgue measure for continuous random variables). For example, it can be used to prove the existence of conditional expectation for probability measures. The latter itself is a key concept in probability theory, as conditional probability is just a special case of it. Financial mathematics Amongst other fields, financial mathematics uses the theorem extensively, in particular via the Girsanov theorem. Such changes of probability measure are the cornerstone of the rational pricing of derivatives and are used for converting actual probabilities into those of the risk neutral probabilities. Information divergences If μ and ν are measures over X, and μ ≪ ν • The Kullback–Leibler divergence from ν to μ is defined to be $D_{\text{KL}}(\mu \parallel \nu )=\int _{X}\log \left({\frac {d\mu }{d\nu }}\right)\;d\mu .$ • For α > 0, α ≠ 1 the Rényi divergence of order α from ν to μ is defined to be $D_{\alpha }(\mu \parallel \nu )={\frac {1}{\alpha -1}}\log \left(\int _{X}\left({\frac {d\mu }{d\nu }}\right)^{\alpha -1}\;d\mu \right).$ The assumption of σ-finiteness The Radon–Nikodym theorem above makes the assumption that the measure μ with respect to which one computes the rate of change of ν is σ-finite. Negative example Here is an example when μ is not σ-finite and the Radon–Nikodym theorem fails to hold. Consider the Borel σ-algebra on the real line. Let the counting measure, μ, of a Borel set A be defined as the number of elements of A if A is finite, and ∞ otherwise. One can check that μ is indeed a measure. It is not σ-finite, as not every Borel set is at most a countable union of finite sets. Let ν be the usual Lebesgue measure on this Borel algebra. Then, ν is absolutely continuous with respect to μ, since for a set A one has μ(A) = 0 only if A is the empty set, and then ν(A) is also zero. Assume that the Radon–Nikodym theorem holds, that is, for some measurable function f one has $\nu (A)=\int _{A}f\,d\mu $ for all Borel sets. Taking A to be a singleton set, A = {a}, and using the above equality, one finds $0=f(a)$ for all real numbers a. This implies that the function f , and therefore the Lebesgue measure ν, is zero, which is a contradiction. Positive result Assuming $\nu \ll \mu ,$ the Radon–Nikodym theorem also holds if $\mu $ is localizable and $\nu $ is accessible with respect to $\mu $,[5]: p. 189, Exercise 9O i.e., $\nu (A)=\sup\{\nu (B):B\in {\cal {P}}(A)\cap \mu ^{\operatorname {pre} }(\mathbb {R} _{\geq 0})\}$ for all $A\in \Sigma .$[6]: Theorem 1.111 (Radon–Nikodym, II) [5]: p. 190, Exercise 9T(ii) Proof This section gives a measure-theoretic proof of the theorem. There is also a functional-analytic proof, using Hilbert space methods, that was first given by von Neumann. For finite measures μ and ν, the idea is to consider functions f with f dμ ≤ dν. The supremum of all such functions, along with the monotone convergence theorem, then furnishes the Radon–Nikodym derivative. The fact that the remaining part of μ is singular with respect to ν follows from a technical fact about finite measures. Once the result is established for finite measures, extending to σ-finite, signed, and complex measures can be done naturally. The details are given below. For finite measures Constructing an extended-valued candidate First, suppose μ and ν are both finite-valued nonnegative measures. Let F be the set of those extended-value measurable functions f : X → [0, ∞] such that: $\forall A\in \Sigma :\qquad \int _{A}f\,d\mu \leq \nu (A)$ :\qquad \int _{A}f\,d\mu \leq \nu (A)} F ≠ ∅, since it contains at least the zero function. Now let f1, f2 ∈ F, and suppose A is an arbitrary measurable set, and define: ${\begin{aligned}A_{1}&=\left\{x\in A:f_{1}(x)>f_{2}(x)\right\},\\A_{2}&=\left\{x\in A:f_{2}(x)\geq f_{1}(x)\right\}.\end{aligned}}$ Then one has $\int _{A}\max \left\{f_{1},f_{2}\right\}\,d\mu =\int _{A_{1}}f_{1}\,d\mu +\int _{A_{2}}f_{2}\,d\mu \leq \nu \left(A_{1}\right)+\nu \left(A_{2}\right)=\nu (A),$ and therefore, max{ f 1, f 2} ∈ F. Now, let { fn } be a sequence of functions in F such that $\lim _{n\to \infty }\int _{X}f_{n}\,d\mu =\sup _{f\in F}\int _{X}f\,d\mu .$ By replacing fn with the maximum of the first n functions, one can assume that the sequence { fn } is increasing. Let g be an extended-valued function defined as $g(x):=\lim _{n\to \infty }f_{n}(x).$ By Lebesgue's monotone convergence theorem, one has $\lim _{n\to \infty }\int _{A}f_{n}\,d\mu =\int _{A}\lim _{n\to \infty }f_{n}(x)\,d\mu (x)=\int _{A}g\,d\mu \leq \nu (A)$ for each A ∈ Σ, and hence, g ∈ F. Also, by the construction of g, $\int _{X}g\,d\mu =\sup _{f\in F}\int _{X}f\,d\mu .$ Proving equality Now, since g ∈ F, $\nu _{0}(A):=\nu (A)-\int _{A}g\,d\mu $ defines a nonnegative measure on Σ. To prove equality, we show that ν0 = 0. Suppose ν0 ≠ 0; then, since μ is finite, there is an ε > 0 such that ν0(X) > ε μ(X). To derive a contradiction from ν0 ≠ 0, we look for a positive set P ∈ Σ for the signed measure ν0 − ε μ (i.e. a measurable set P, all of whose measurable subsets have non-negative ν0 − εμ measure), where also P has positive μ-measure. Conceptually, we're looking for a set P, where ν0 ≥ ε μ in every part of P. A convenient approach is to use the Hahn decomposition (P, N) for the signed measure ν0 − ε μ. Note then that for every A ∈ Σ one has ν0(A ∩ P) ≥ ε μ(A ∩ P), and hence, ${\begin{aligned}\nu (A)&=\int _{A}g\,d\mu +\nu _{0}(A)\\&\geq \int _{A}g\,d\mu +\nu _{0}(A\cap P)\\&\geq \int _{A}g\,d\mu +\varepsilon \mu (A\cap P)=\int _{A}\left(g+\varepsilon 1_{P}\right)\,d\mu ,\end{aligned}}$ where 1P is the indicator function of P. Also, note that μ(P) > 0 as desired; for if μ(P) = 0, then (since ν is absolutely continuous in relation to μ) ν0(P) ≤ ν(P) = 0, so ν0(P) = 0 and $\nu _{0}(X)-\varepsilon \mu (X)=\left(\nu _{0}-\varepsilon \mu \right)(N)\leq 0,$ contradicting the fact that ν0(X) > εμ(X). Then, since also $\int _{X}\left(g+\varepsilon 1_{P}\right)\,d\mu \leq \nu (X)<+\infty ,$ g + ε 1P ∈ F and satisfies $\int _{X}\left(g+\varepsilon 1_{P}\right)\,d\mu >\int _{X}g\,d\mu =\sup _{f\in F}\int _{X}f\,d\mu .$ This is impossible because it violates the definition of a supremum; therefore, the initial assumption that ν0 ≠ 0 must be false. Hence, ν0 = 0, as desired. Restricting to finite values Now, since g is μ-integrable, the set {x ∈ X : g(x) = ∞} is μ-null. Therefore, if a f is defined as $f(x)={\begin{cases}g(x)&{\text{if }}g(x)<\infty \\0&{\text{otherwise,}}\end{cases}}$ then f has the desired properties. Uniqueness As for the uniqueness, let f, g : X → [0, ∞) be measurable functions satisfying $\nu (A)=\int _{A}f\,d\mu =\int _{A}g\,d\mu $ for every measurable set A. Then, g − f is μ-integrable, and $\int _{A}(g-f)\,d\mu =0.$ In particular, for A = {x ∈ X : f(x) > g(x)}, or {x ∈ X : f(x) < g(x)}. It follows that $\int _{X}(g-f)^{+}\,d\mu =0=\int _{X}(g-f)^{-}\,d\mu ,$ and so, that (g − f )+ = 0 μ-almost everywhere; the same is true for (g − f )−, and thus, f = g μ-almost everywhere, as desired. For σ-finite positive measures If μ and ν are σ-finite, then X can be written as the union of a sequence {Bn}n of disjoint sets in Σ, each of which has finite measure under both μ and ν. For each n, by the finite case, there is a Σ-measurable function fn : Bn → [0, ∞) such that $\nu _{n}(A)=\int _{A}f_{n}\,d\mu $ for each Σ-measurable subset A of Bn. The sum $ \left(\sum _{n}f_{n}1_{B_{n}}\right):=f$ of those functions is then the required function such that $ \nu (A)=\int _{A}f\,d\mu $. As for the uniqueness, since each of the fn is μ-almost everywhere unique, so is f. For signed and complex measures If ν is a σ-finite signed measure, then it can be Hahn–Jordan decomposed as ν = ν+ − ν− where one of the measures is finite. Applying the previous result to those two measures, one obtains two functions, g, h : X → [0, ∞), satisfying the Radon–Nikodym theorem for ν+ and ν− respectively, at least one of which is μ-integrable (i.e., its integral with respect to μ is finite). It is clear then that f = g − h satisfies the required properties, including uniqueness, since both g and h are unique up to μ-almost everywhere equality. If ν is a complex measure, it can be decomposed as ν = ν1 + iν2, where both ν1 and ν2 are finite-valued signed measures. Applying the above argument, one obtains two functions, g, h : X → [0, ∞), satisfying the required properties for ν1 and ν2, respectively. Clearly, f = g + ih is the required function. The Lebesgue decomposition theorem Lebesgue's decomposition theorem shows that the assumptions of the Radon–Nikodym theorem can be found even in a situation which is seemingly more general. Consider a σ-finite positive measure $\mu $ on the measure space $(X,\Sigma )$ and a σ-finite signed measure $\nu $ on $\Sigma $, without assuming any absolute continuity. Then there exist unique signed measures $\nu _{a}$ and $\nu _{s}$ on $\Sigma $ such that $\nu =\nu _{a}+\nu _{s}$, $\nu _{a}\ll \mu $, and $\nu _{s}\perp \mu $. The Radon–Nikodym theorem can then be applied to the pair $\nu _{a},\mu $. See also • Girsanov theorem • Radon–Nikodym set Notes 1. Billingsley, Patrick (1995). Probability and Measure (Third ed.). New York: John Wiley & Sons. pp. 419–427. ISBN 0-471-00710-2. 2. Nikodym, O. (1930). "Sur une généralisation des intégrales de M. J. Radon" (PDF). Fundamenta Mathematicae (in French). 15: 131–179. doi:10.4064/fm-15-1-131-179. JFM 56.0922.02. Retrieved 2018-01-30. 3. Zaanen, Adriaan C. (1996). Introduction to Operator Theory in Riesz Spaces. Springer. ISBN 3-540-61989-5. 4. "Calculating Radon Nikodym derivative". Stack Exchange. April 7, 2018. 5. Brown, Arlen; Pearcy, Carl (1977). Introduction to Operator Theory I: Elements of Functional Analysis. ISBN 978-1461299288. 6. Fonseca, Irene; Leoni, Giovanni. Modern Methods in the Calculus of Variations: Lp Spaces. Springer. p. 68. ISBN 978-0-387-35784-3. References • Lang, Serge (1969). Analysis II: Real analysis. Addison-Wesley. Contains a proof for vector measures assuming values in a Banach space. • Royden, H. L.; Fitzpatrick, P. M. (2010). Real Analysis (4th ed.). Pearson. Contains a lucid proof in case the measure ν is not σ-finite. • Shilov, G. E.; Gurevich, B. L. (1978). Integral, Measure, and Derivative: A Unified Approach. Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8. • Stein, Elias M.; Shakarchi, Rami (2005). Real analysis: measure theory, integration, and Hilbert spaces. Princeton lectures in analysis. Princeton, N.J: Princeton University Press. ISBN 978-0-691-11386-9. Contains a proof of the generalisation. • Teschl, Gerald. "Topics in Real and Functional Analysis". (lecture notes). This article incorporates material from Radon–Nikodym theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Measure theory Basic concepts • Absolute continuity of measures • Lebesgue integration • Lp spaces • Measure • Measure space • Probability space • Measurable space/function Sets • Almost everywhere • Atom • Baire set • Borel set • equivalence relation • Borel space • Carathéodory's criterion • Cylindrical σ-algebra • Cylinder set • 𝜆-system • Essential range • infimum/supremum • Locally measurable • π-system • σ-algebra • Non-measurable set • Vitali set • Null set • Support • Transverse measure • Universally measurable Types of Measures • Atomic • Baire • Banach • Besov • Borel • Brown • Complex • Complete • Content • (Logarithmically) Convex • Decomposable • Discrete • Equivalent • Finite • Inner • (Quasi-) Invariant • Locally finite • Maximising • Metric outer • Outer • Perfect • Pre-measure • (Sub-) Probability • Projection-valued • Radon • Random • Regular • Borel regular • Inner regular • Outer regular • Saturated • Set function • σ-finite • s-finite • Signed • Singular • Spectral • Strictly positive • Tight • Vector Particular measures • Counting • Dirac • Euler • Gaussian • Haar • Harmonic • Hausdorff • Intensity • Lebesgue • Infinite-dimensional • Logarithmic • Product • Projections • Pushforward • Spherical measure • Tangent • Trivial • Young Maps • Measurable function • Bochner • Strongly • Weakly • Convergence: almost everywhere • of measures • in measure • of random variables • in distribution • in probability • Cylinder set measure • Random: compact set • element • measure • process • variable • vector • Projection-valued measure Main results • Carathéodory's extension theorem • Convergence theorems • Dominated • Monotone • Vitali • Decomposition theorems • Hahn • Jordan • Maharam's • Egorov's • Fatou's lemma • Fubini's • Fubini–Tonelli • Hölder's inequality • Minkowski inequality • Radon–Nikodym • Riesz–Markov–Kakutani representation theorem Other results • Disintegration theorem • Lifting theory • Lebesgue's density theorem • Lebesgue differentiation theorem • Sard's theorem For Lebesgue measure • Isoperimetric inequality • Brunn–Minkowski theorem • Milman's reverse • Minkowski–Steiner formula • Prékopa–Leindler inequality • Vitale's random Brunn–Minkowski inequality Applications & related • Convex analysis • Descriptive set theory • Probability theory • Real analysis • Spectral theory	Wikipedia
Vector fields on spheres In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras. Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed on a sphere in $n$-dimensional Euclidean space. A definitive answer was provided in 1962 by Frank Adams. It was already known,[1] by direct construction using Clifford algebras, that there were at least $\rho (n)-1$ such fields (see definition below). Adams applied homotopy theory and topological K-theory[2] to prove that no more independent vector fields could be found. Hence $\rho (n)-1$ is the exact number of pointwise linearly independent vector fields that exist on an ($n-1$)-dimensional sphere. Technical details In detail, the question applies to the 'round spheres' and to their tangent bundles: in fact since all exotic spheres have isomorphic tangent bundles, the Radon–Hurwitz numbers $\rho (n)$ determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere. The case of $n$ odd is taken care of by the Poincaré–Hopf index theorem (see hairy ball theorem), so the case $n$ even is an extension of that. Adams showed that the maximum number of continuous (smooth would be no different here) pointwise linearly-independent vector fields on the ($n-1$)-sphere is exactly $\rho (n)-1$. The construction of the fields is related to the real Clifford algebras, which is a theory with a periodicity modulo 8 that also shows up here. By the Gram–Schmidt process, it is the same to ask for (pointwise) linear independence or fields that give an orthonormal basis at each point. Radon–Hurwitz numbers The Radon–Hurwitz numbers $\rho (n)$ occur in earlier work of Johann Radon (1922) and Adolf Hurwitz (1923) on the Hurwitz problem on quadratic forms.[3] For $n$ written as the product of an odd number $A$ and a power of two $2^{B}$, write $B=c+4d,0\leq c<4$. Then[3] $\rho (n)=2^{c}+8d$. The first few values of $\rho (2n)$ are (from (sequence A053381 in the OEIS)): 2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, ... For odd $n$, the value of the function $\rho (n)$ is one. These numbers occur also in other, related areas. In matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real $n\times n$ matrices, for which each non-zero matrix is a similarity transformation, i.e. a product of an orthogonal matrix and a scalar matrix. In quadratic forms, the Hurwitz problem asks for multiplicative identities between quadratic forms. The classical results were revisited in 1952 by Beno Eckmann. They are now applied in areas including coding theory and theoretical physics. References 1. James, I. M. (1957). "Whitehead products and vector-fields on spheres". Proceedings of the Cambridge Philosophical Society. 53 (4): 817–820. doi:10.1017/S0305004100032928. S2CID 119646042. 2. Adams, J. F. (1962). "Vector Fields on Spheres". Annals of Mathematics. 75 (3): 603–632. doi:10.2307/1970213. JSTOR 1970213. Zbl 0112.38102. 3. Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. p. 127. ISBN 0-521-42668-5. Zbl 0785.11022. • Porteous, I.R. (1969). Topological Geometry. Van Nostrand Reinhold. pp. 336–352. ISBN 0-442-06606-6. Zbl 0186.06304. • Miller, H.R. "Vector fields on spheres, etc. (course notes)" (PDF). Retrieved 10 November 2018.	Wikipedia
Radon–Nikodym set In the theory of fair cake-cutting, the Radon–Nikodym set (RNS) is a geometric object that represents a cake, based on how different people evaluate the different parts of the cake. Example Suppose we have a cake made of four parts. There are two people, Alice and George, with different tastes: each person values the different parts of the cake differently. The table below describes the parts and their values; the last row, "RNS Point", is explained afterwards. ChocolateLemonVanillaCherries Alice's value18912 George's value18048 RNS point(0.5,0.5)(1,0)(0.2,0.8)(0.2,0.8) The "RNS point" of a piece of cake describes the relative values of the partners to that piece. It has two coordinates – one for Alice and one for George. For example: • The partners agree on the values for the chocolate part, so the coordinates of its RNS point are also equal (they are normalized such that their sum is 1). • The lemon part is only valuable for Alice, so in its RNS point, only Alice's coordinate is 1 while George's coordinate is 0. • In both the vanilla and the cherries part, the ratio between Alice's value to George's value is 1:4. Hence, this is also the ratio between the coordinates of their RNS points. Note that both the vanilla and the cherries are mapped to the same RNS point. The RNS of a cake is just the set of all its RNS points; in the above cake this set contains three points: {(0.5,0.5), (1,0), (0.2,0.8)}. It can be represented by the segment (1,0)-(0,1): (1.0, 0.0)(0.9, 0.1)(0.8, 0.2)(0.7, 0.3)(0.6, 0.4)(0.5, 0.5)(0.4, 0.6)(0.3, 0.7)(0.2, 0.8)(0.1, 0.9)(0.0, 1.0) Lemon––––Chocolate––Vanilla, Cherries–– In effect, the cake is decomposed and re-constructed on the segment (1,0)-(0,1). Definitions There is a set $C$ ("the cake"), and a set $\mathbb {C} $ which is a sigma-algebra of subsets of $C$. There are $n$ partners. Every partner $i$ has a personal value measure $V_{i}:\mathbb {C} \to \mathbb {R} $. This measure determines how much each subset of $C$ is worth to that partner. Define the following measure: $V=\sum _{i=1}^{n}V_{i}$ Note that each $V_{i}$ is an absolutely continuous measure with respect to $V$. Therefore, by the Radon–Nikodym theorem, it has a Radon–Nikodym derivative, which is a function $v_{i}:C\to [0,\infty )$ such that for every measurable subset $X\in \mathbb {C} $: $V_{i}(X)=\int _{X}v_{i}\,dV$ The $v_{i}$ are called value-density functions. They have the following properties, for almost all points of the cake $x\in C$:[1]: 222 • $\sum _{i=1}^{n}v_{i}(x)=1$ • $\forall i:0\leq v_{i}(x)\leq 1$ For every point $x\in C$, the RNS point of $x$ is defined by: $v(x)=(v_{1}(x),\dots ,v_{n}(x))$ Note that $v(x)$ is always a point in the $(n-1)$-dimensional unit simplex in $\mathbb {R} ^{n}$, denoted by $\Delta ^{n-1}$ (or just $\Delta $ when $n$ is clear from the context). The RNS of a cake is the set of all its RNS points: $RNS(C)=\{v(x)\mid x\in C\}$ The cake is decomposed and then re-constructed inside $\Delta $. Each vertex of $\Delta $ is associated with one of the n partners. Each fraction of the cake is mapped to a point in $\Delta $ according to the valuations: the more valuable a piece is to a partner, the closer it is to that partner's vertex. This is shown in the above example for $n=2$ partners (where $\Delta $ is just the segment between (1,0) and (0,1)). Akin[2] describes the meaning of the RNS for $n=3$ partners: We imagine a table shaped like an equilateral triangle with each consumer seated at a vertex... the desirability to consumer $i$ of a fragment of cake at a point $v\in \Delta $ is given by the barycentric coordinate $v_{i}$ measuring its closeness to vertex $i$. Thus, $v_{i}$ is 1 at the vertex and declines linearly to value 0 at the opposite face. Efficient RNS partitions The unit simplex $\Delta $ can be partitioned among the partners, giving each partner $i$ a subset $\Delta _{i}$. Each such partition induces a partition of the cake $C$, in which partner $i$ receives the bits of $C$ whose RNS-points fall within $\Delta _{i}$. Here are two example partitions for the two-partner example, where $\Delta $ is the segment between (1,0) and (0,1) • Cut $\Delta $ in the point (0.4,0.6). Give the segment (1,0)-(0.4,0.6) to Alice and the segment (0.4,0.6)-(0,1) to George. This corresponds to giving the Lemon and Chocolate to Alice (total value 27) and the rest to George (total value 12). • Cut in the same point (0.4,0.6), but give the segment (1,0)-(0.4,0.6) to George (total value 18) and the segment (0.4,0.6)-(0,1) to Alice (total value 3). The first partition looks much more efficient than the second one: in the first partition, each partner is given the pieces that are more valuable to him/her (closer to his/her vertex of the simplex), while in the second partition the opposite is true. In fact, the first partition is Pareto efficient while the second partition is not. For example, in the second partition, Alice can give the cherries to George in exchange for 2/9 of the chocolate; this will improve Alice's utility by 2 and George's utility by 4. This example illustrates a general fact that we define below. For every point $w=(w_{1},\dots ,w_{n})\in \Delta $: • Say that a partition of $\Delta =\Delta _{1}\cup \cdots \cup \Delta _{n}$ belongs to $w$, if: For all $i,j$ and for all $(v_{1},\dots ,v_{n})\in \Delta _{i}$: ${\frac {v_{i}}{v_{j}}}\geq {\frac {w_{i}}{w_{j}}}$ • Say that a partition of $C=X_{1}\cup \cdots \cup X_{n}$ belongs to $w$, if it is induced by a partition of $\Delta $ that belongs to $w$. I.e: For all $i,j$ and for all $x\in X_{i}$: ${\frac {v_{i}(x)}{v_{j}(x)}}\geq {\frac {w_{i}}{w_{j}}}$ It is possible to prove that:[1]: 241–244 A partition $C=X_{1}\cup \cdots \cup X_{n}$ belongs to a positive point $w=(w_{1},\dots ,w_{n})\in \Delta ^{+}$, if-and-only-if it maximizes the sum: ${\frac {V_{1}(X_{1})}{w_{1}}}+\cdots +{\frac {V_{1}(X_{n})}{w_{n}}}$ I.e, iff it is a weighted-utilitarian-maximal division with weight vector $w$. Since every Pareto-efficient division is weighetd-utilitarian-maximal for some selection of weights,[3] the following theorem is also true:[1]: 246 A positive partition $C=X_{1}\cup \cdots \cup X_{n}$ belongs to some positive point in $\Delta ^{+}$, if-and-only-if it is Pareto-efficient. So there is a mapping between the set of Pareto-efficient partitions and the points in $\Delta $. Returning to the above example: • The first partition (giving the Lemon and Chocolate to Alice and the rest to George) belongs to the point $(0.4,0.6)$, as well as to other points such as $(0.3,0.7)$ (some partitions belong to more than one point). Indeed, it is a utilitarian cake-cutting that maximizes the sum ${\frac {V_{\text{Alice}}}{0.4}}+{\frac {V_{\text{George}}}{0.6}}$, and it is also Pareto-efficient. • In contrast, the second partition does not belong to any point, and indeed it is not Pareto-efficient. • There are some points to which many different partitions belong. For example, the point $(0.5,0.5)$. This is a point of the RNS and there is a positive mass of cake associated with it, so any partition of that mass leads to a partition that belongs to $(0.5,0.5)$. For example, giving the Lemon and Chocolate to Alice (value 27) and the rest to George (value 12) belongs to $(0.5,0.5)$; giving only the Lemon to Alice (value 9) and the rest to George (value 30) also belongs to it; giving the Lemon and half the chocolate to Alice (value 18) and the rest to George (value 21) also belongs to it; etc. All these partitions maximize the sum ${\frac {V_{\text{Alice}}}{0.5}}+{\frac {V_{\text{George}}}{0.5}}$; indeed, this sum is 78 in all these partitions. They are all Pareto-efficient. History The RNS was introduced as part of the Dubins–Spanier theorems and used in the proof of Weller's theorem and later results by Ethan Akin.[2] The term "Radon–Nikodym set" was coined by Julius Barbanel.[1] See also • Individual pieces set References 1. Barbanel, Julius B.; with an introduction by Alan D. Taylor (2005). The geometry of efficient fair division. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511546679. ISBN 0-521-84248-4. MR 2132232.{{cite book}}: CS1 maint: multiple names: authors list (link) Short summary is available at: Barbanel, J. (2010). "A Geometric Approach to Fair Division". The College Mathematics Journal. 41 (4): 268. doi:10.4169/074683410x510263. 2. Akin, Ethan (1995). "Vilfredo Pareto cuts the cake". Journal of Mathematical Economics. 24: 23. doi:10.1016/0304-4068(94)00674-y. 3. Barbanel, Julius B.; Zwicker, William S. (1997). "Two applications of a theorem of Dvoretsky, Wald, and Wolfovitz to cake division". Theory and Decision. 43 (2): 203. doi:10.1023/a:1004966624893.	Wikipedia
Radon–Riesz property The Radon–Riesz property is a mathematical property for normed spaces that helps ensure convergence in norm. Given two assumptions (essentially weak convergence and continuity of norm), we would like to ensure convergence in the norm topology. Definition Suppose that (X, \|\|·\|\|) is a normed space. We say that X has the Radon–Riesz property (or that X is a Radon–Riesz space) if whenever $(x_{n})$ is a sequence in the space and $x$ is a member of X such that $(x_{n})$ converges weakly to $x$ and $\lim _{n\to \infty }\Vert x_{n}\Vert =\Vert x\Vert $, then $(x_{n})$ converges to $x$ in norm; that is, $\lim _{n\to \infty }\Vert x_{n}-x\Vert =0$. Other names Although it would appear that Johann Radon was one of the first to make significant use of this property in 1913, M. I. Kadets and V. L. Klee also used versions of the Radon–Riesz property to make advancements in Banach space theory in the late 1920s. It is common for the Radon–Riesz property to also be referred to as the Kadets–Klee property or property (H). According to Robert Megginson, the letter H does not stand for anything. It was simply referred to as property (H) in a list of properties for normed spaces that starts with (A) and ends with (H). This list was given by K. Fan and I. Glicksberg (Observe that the definition of (H) given by Fan and Glicksberg includes additionally the rotundity of the norm, so it does not coincide with the Radon-Riesz property itself). The "Riesz" part of the name refers to Frigyes Riesz. He also made some use of this property in the 1920s. It is important to know that the name "Kadets-Klee property" is used sometimes to speak about the coincidence of the weak topologies and norm topologies in the unit sphere of the normed space. Examples 1. Every real Hilbert space is a Radon–Riesz space. Indeed, suppose that H is a real Hilbert space and that $(x_{n})$ is a sequence in H converging weakly to a member $x$ of H. Using the two assumptions on the sequence and the fact that $\langle x_{n}-x,x_{n}-x\rangle =\langle x_{n},x_{n}\rangle -\langle x_{n},x\rangle -\langle x,x_{n}\rangle +\langle x,x\rangle ,$ and letting n tend to infinity, we see that $\lim _{n\to \infty }{\langle x_{n}-x,x_{n}-x\rangle }=0.$ Thus H is a Radon–Riesz space. 2. Every uniformly convex Banach space is a Radon-Riesz space. See Section 3.7 of Haim Brezis' Functional analysis. See also • Johann Radon • Frigyes Riesz • Hilbert space or Banach space theory • Weak topology • Normed space • Functional analysis • Schur's property References • Megginson, Robert E. (1998), An Introduction to Banach Space Theory, New York Berlin Heidelberg: Springer-Verlag, ISBN 0-387-98431-3	Wikipedia
Partition regularity In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets. Given a set $X$, a collection of subsets $\mathbb {S} \subset {\mathcal {P}}(X)$ is called partition regular if every set A in the collection has the property that, no matter how A is partitioned into finitely many subsets, at least one of the subsets will also belong to the collection. That is, for any $A\in \mathbb {S} $, and any finite partition $A=C_{1}\cup C_{2}\cup \cdots \cup C_{n}$, there exists an i ≤ n such that $C_{i}$ belongs to $\mathbb {S} $. Ramsey theory is sometimes characterized as the study of which collections $\mathbb {S} $ are partition regular. Examples • The collection of all infinite subsets of an infinite set X is a prototypical example. In this case partition regularity asserts that every finite partition of an infinite set has an infinite cell (i.e. the infinite pigeonhole principle.) • Sets with positive upper density in $\mathbb {N} $: the upper density ${\overline {d}}(A)$ of $A\subset \mathbb {N} $ is defined as ${\overline {d}}(A)=\limsup _{n\rightarrow \infty }{\frac {\|\{1,2,\ldots ,n\}\cap A\|}{n}}.$ (Szemerédi's theorem) • For any ultrafilter $\mathbb {U} $ on a set $X$, $\mathbb {U} $ is partition regular: for any $A\in \mathbb {U} $, if $A=C_{1}\sqcup \cdots \sqcup C_{n}$, then exactly one $C_{i}\in \mathbb {U} $. • Sets of recurrence: a set R of integers is called a set of recurrence if for any measure-preserving transformation $T$ of the probability space (Ω, β, μ) and $A\in \beta $ of positive measure there is a nonzero $n\in R$ so that $\mu (A\cap T^{n}A)>0$. • Call a subset of natural numbers a.p.-rich if it contains arbitrarily long arithmetic progressions. Then the collection of a.p.-rich subsets is partition regular (Van der Waerden, 1927). • Let $[A]^{n}$ be the set of all n-subsets of $A\subset \mathbb {N} $. Let $\mathbb {S} ^{n}=\bigcup _{A\subset \mathbb {N} }^{}[A]^{n}$. For each n, $\mathbb {S} ^{n}$ is partition regular. (Ramsey, 1930). • For each infinite cardinal $\kappa $, the collection of stationary sets of $\kappa $ is partition regular. More is true: if $S$ is stationary and $S=\bigcup _{\alpha <\lambda }S_{\alpha }$ for some $\lambda <\kappa $, then some $S_{\alpha }$ is stationary. • The collection of $\Delta $-sets: $A\subset \mathbb {N} $ is a $\Delta $-set if $A$ contains the set of differences $\{s_{m}-s_{n}:m,n\in \mathbb {N} ,\,n<m\}$ for some sequence $\langle s_{n}\rangle _{n=1}^{\infty }$. • The set of barriers on $\mathbb {N} $: call a collection $\mathbb {B} $ of finite subsets of $\mathbb {N} $ a barrier if: • $\forall X,Y\in \mathbb {B} ,X\not \subset Y$ and • for all infinite $I\subset \cup \mathbb {B} $, there is some $X\in \mathbb {B} $ such that the elements of X are the smallest elements of I; i.e. $X\subset I$ and $\forall i\in I\setminus X,\forall x\in X,x<i$. This generalizes Ramsey's theorem, as each $[A]^{n}$ is a barrier. (Nash-Williams, 1965) • Finite products of infinite trees (Halpern–Läuchli, 1966) • Piecewise syndetic sets (Brown, 1968) • Call a subset of natural numbers i.p.-rich if it contains arbitrarily large finite sets together with all their finite sums. Then the collection of i.p.-rich subsets is partition regular (Folkman–Rado–Sanders, 1968). • (m, p, c)-sets (Deuber, 1973) • IP sets (Hindman, 1974, see also Hindman, Strauss, 1998) • MTk sets for each k, i.e. k-tuples of finite sums (Milliken–Taylor, 1975) • Central sets; i.e. the members of any minimal idempotent in $\beta \mathbb {N} $, the Stone–Čech compactification of the integers. (Furstenberg, 1981, see also Hindman, Strauss, 1998) Diophantine equations A Diophantine equation $P(\mathbf {x} )=0$ is called partition regular if the collection of all infinite subsets of $\mathbb {N} $ containing a solution is partition regular. Rado's theorem characterises exactly which systems of linear Diophantine equations $\mathbf {A} \mathbf {x} =\mathbf {0} $ are partition regular. Much progress has been made recently on classifying nonlinear Diophantine equations.[1][2] References 1. Di Nasso, Mauro; Luperi Baglini, Lorenzo (January 2018). "Ramsey properties of nonlinear Diophantine equations". Advances in Mathematics. 324: 84–117. arXiv:1606.02056. doi:10.1016/j.aim.2017.11.003. ISSN 0001-8708. 2. Barrett, Jordan Mitchell; Lupini, Martino; Moreira, Joel (May 2021). "On Rado conditions for nonlinear Diophantine equations". European Journal of Combinatorics. 94: 103277. arXiv:1907.06163. doi:10.1016/j.ejc.2020.103277. ISSN 0195-6698. Sources 1. Vitaly Bergelson, N. Hindman Partition regular structures contained in large sets are abundant J. Comb. Theory A 93 (2001), 18–36. 2. T. Brown, An interesting combinatorial method in the theory of locally finite semigroups, Pacific J. Math. 36, no. 2 (1971), 285–289. 3. W. Deuber, Mathematische Zeitschrift 133, (1973) 109–123 4. N. Hindman, Finite sums from sequences within cells of a partition of N, J. Comb. Theory A 17 (1974) 1–11. 5. C.St.J.A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965), 33–39. 6. N. Hindman, D. Strauss, Algebra in the Stone–Čech compactification, De Gruyter, 1998 7. J.Sanders, A Generalization of Schur's Theorem, Doctoral Dissertation, Yale University, 1968.	Wikipedia
Radó's theorem (Riemann surfaces) In mathematical complex analysis, Radó's theorem, proved by Tibor Radó (1925), states that every connected Riemann surface is second-countable (has a countable base for its topology). The Prüfer surface is an example of a surface with no countable base for the topology, so cannot have the structure of a Riemann surface. The obvious analogue of Radó's theorem in higher dimensions is false: there are 2-dimensional connected complex manifolds that are not second-countable. References • Hubbard, John Hamal (2006), Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, ISBN 978-0-9715766-2-9, MR 2245223 • Radó, Tibor (1925), "Über den Begriff der Riemannschen Fläche", Acta Szeged, 2 (2): 101–121, JFM 51.0273.01	Wikipedia
Radó's theorem (harmonic functions) In mathematics, Radó's theorem is a result about harmonic functions, named after Tibor Radó. Informally, it says that any "nice looking" shape without holes can be smoothly deformed into a disk. See also Rado's theorem (Ramsey theory) Suppose Ω is an open, connected and convex subset of the Euclidean space R2 with smooth boundary ∂Ω and suppose that D is the unit disk. Then, given any homeomorphism μ : ∂D → ∂Ω, there exists a unique harmonic function u : D → Ω such that u = μ on ∂D and u is a diffeomorphism. References • R. Schoen, S. T. Yau. (1997) Lectures on Harmonic Maps. International Press, Inc., Boston, Massachusetts. ISBN 1-57146-002-0, page 4. This article incorporates material from Rado's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.	Wikipedia
Raegan Higgins Raegan J. Higgins is an American mathematician and co-director of the EDGE program for Women. She is also one of the co-founders of the website Mathematically Gifted & Black, which highlights the accomplishments of Black mathematicians.[1] Raegan J. Higgins BornBaton Rouge, LA Known forWork with the EDGE program Scientific career Doctoral advisorAllan Clemens Peterson and Lynn Erbe Research Higgins studies time scales and its application to mathematical biology.[2] Education Higgins went to Xavier University of Louisiana in New Orleans, Louisiana.[3] She attended University of Nebraska-Lincoln for her graduate studies studying under the advisement of Lynn Erbe and Allan Clemens Peterson.[4] She graduated in 2008 and was one of the first two African-American women to earn a doctoral degree in Mathematics from University of Nebraska-Lincoln.[3] Career Higgins, along with Ami Radunskaya, is co-director of the EDGE program which supports women who are pursuing graduate degrees and ultimately careers in the mathematical sciences. She participated in the EDGE program in 2002 as a graduate student. She was also a workshop facilitator from 2014 to 2017.[5] Professor Higgins became Co-Director of the program in 2017.[6] In 2008, Higgins joined the faculty at Texas Tech University in the Department of Mathematics & Statistics.[7] She earned tenure and was promoted to Associate Professor, becoming the first African American to receive Tenure and Promotion in the Mathematics & Statistics Department at Texas Tech.[3] Honors and Grant Awards In 2020, Higgins received the Association for Women in Mathematics (AWM) service award.[8] She has won several National Science Foundation grants for various programs in mathematical education.[9][10][11][12] She also earned the 2021 AWM Gweneth Humphreys Award.[13] She co-delivered an invited plenary address at the 2021 National Math Festival.[14] Higgins' accomplishments earned her recognition by Mathematically Gifted & Black as a Black History Month 2018 Honoree.[15] References 1. "About Us". Mathematically Gifted & Black. 2. "TTU Raegan Higgins Research Spotlight". Retrieved June 12, 2020. 3. "Raegan Higgins". Mathematically Gifted & Black. Retrieved June 10, 2020. 4. "Raegan Higgins – The Mathematics Genealogy Project". genealogy.math.ndsu.nodak.edu. Retrieved June 10, 2020. 5. "EDGE for Women Past Programs". Retrieved June 10, 2020. 6. "EDGE for Women Directors". Retrieved June 10, 2020. 7. "Texas Tech Faculty Page". Retrieved June 10, 2020. 8. "AWM Service Awards 2020". Retrieved June 10, 2020. 9. "NSF Award Search: Award#1852944 – Leveraging Learning Assistantships, Mentoring, and Scholarships to Develop Self-Determined Mathematics Teachers for West Texas". nsf.gov. Retrieved June 10, 2020. 10. "NSF Award Search: Award#0831420 – The West Texas Middle School Math Partnership". nsf.gov. Retrieved June 10, 2020. 11. "NSF Award Search: Award#1356604 – South Plains Mathematics Fellows". nsf.gov. Retrieved June 10, 2020. 12. "NSF Award Search: Award#1701664 – Pre-Alliance Planning: The Bridges Across Texas Louis Stokes Alliances for Minority Participation". nsf.gov. Retrieved June 10, 2020. 13. "Raegan Higgins Wins the Humphreys Award" (PDF). AWM Newsletter. January–February 2021. 14. "2021 Festival Online". National Math Festival. November 18, 2020. Retrieved June 7, 2021. 15. "Raegan Higgins". Mathematically Gifted & Black. External links • Home page • Video interview at Meet a Mathematician! Authority control: Academics • MathSciNet • Mathematics Genealogy Project • ORCID • zbMATH	Wikipedia
Rafe Mazzeo Rafe Roys Mazzeo (born 1961) is an American mathematician working in differential geometry, microlocal analysis, and partial differential equations.[2] He is currently a professor of mathematics. He served as the department chair at Stanford University from 2007 to 2010 and 2019–2022.[2] Rafe Mazzeo Born1961 (age 61–62) Boston, Massachusetts, US NationalityAmerican Alma materMIT AwardsSloan Research Fellowship (1991-1995)[1] Scientific career FieldsMathematics InstitutionsStanford University ThesisHodge cohomology of negatively curved manifolds (1986) Doctoral advisorRichard Burt Melrose Education and career Mazzeo obtained his B.S. degree from MIT in 1982. He completed his Ph.D. in mathematics at MIT under the supervision of Richard Burt Melrose in 1986. His Ph.D. thesis was titled "Hodge cohomology of negatively curved manifolds."[1] After obtaining his Ph.D. degree, Mazzeo joined Stanford University, where he became a full professor in 1997.[1] Contributions Mazzeo has published more than 150 mathematics papers,[3][4] and his work has been cited more than 5000 times.[4] His work has been published in many prestigious mathematics journals, including Annals of Mathematics,[5] Inventiones Mathematicae,[6] and Duke Mathematical Journal.[7] He has had 11 doctoral students.[8] He is one of the founders of the Stanford University Mathematics Camp. He is Faculty Director of the Stanford Online High School, and has been Director of the IAS/Park City Mathematics Institute since 2015. Awards and fellowships Mazzeo has received many awards, including a National Science Foundation Postdoctoral Fellowship, Alfred P. Sloan Foundation Research Fellowship, National Science Foundation Young Investigator Fellowship, and Louis and Claude Rosenberg Jr. University Fellowship in Undergraduate Education.[1] In 2013, he became a fellow of the American Mathematical Society.[9] He was elected as a Member of the American Academy of Arts and Sciences in 2022.[10] Selected publications • Mazzeo, Rafe; Rubinstein, Yanir; Jeffres, Thalia (2016). "Kähler–Einstein metrics with edge singularities". Annals of Mathematics. 183: 95–176. arXiv:1105.5216. doi:10.4007/annals.2016.183.1.3. S2CID 119306806. Retrieved 4 September 2021. • Mazzeo, Rafe; Pacard, Frank; Schoen, Richard; Korevaar, Nick (1999). "Refined asymptotics for constant scalar curvature metrics with isolated singularities". Inventiones Mathematicae. 135 (2): 233–272. arXiv:math/9807038. Bibcode:1999InMat.135..233K. doi:10.1007/s002220050285. S2CID 5536612. Retrieved 5 September 2021. • Mazzeo, Rafe; Hausel, Tamás; Hunsicker, Eugenie (2004). "Hodge cohomology of gravitational instantons". Duke Mathematical Journal. 122 (3): 485–548. doi:10.1215/S0012-7094-04-12233-X. S2CID 14883967. Retrieved 5 September 2021. • Mazzeo, Rafe (1991). "Elliptic theory of differential edge operators I". Communications in Partial Differential Equations. 16 (10): 1615–1664. doi:10.1080/03605309108820815. • Mazzeo, Rafe; Melrose, Richard (1987). "Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature". Journal of Functional Analysis. 75 (2): 260–310. doi:10.1016/0022-1236(87)90097-8. • Mazzeo, Rafe; Epstein, Charles (2013). Degenerate Diffusion Operators Arising in Population Biology. Princeton University Press. ISBN 978-0691157122. References 1. "Curriculum VitÆ \| Rafe Mazzeo". Web.stanford.edu. Retrieved 7 November 2021. 2. "Rafe Mazzeo \| Mathematics". Mathematics.stanford.edu. Retrieved 7 November 2021. 3. "Rafe Mazzeo". Researchgate.net. Retrieved 7 November 2021. 4. "Rafe Mazzeo". Scholar.google.com. Retrieved 7 November 2021. 5. Mazzeo, Rafe; Yanir A. Rubinstein; Thalia Jeffres (2016). "Kähler–Einstein metrics with edge singularities". Annals of Mathematics. 183: 95–176. arXiv:1105.5216. doi:10.4007/annals.2016.183.1.3. S2CID 119306806. Retrieved 4 September 2021. 6. Mazzeo, Rafe; Frank Pacard; Richard Schoen; Nick Korevaar (1999). "Refined asymptotics for constant scalar curvature metrics with isolated singularities". Inventiones Mathematicae. 135 (2): 233–272. arXiv:math/9807038. Bibcode:1999InMat.135..233K. doi:10.1007/s002220050285. S2CID 5536612. Retrieved 5 September 2021. 7. Mazzeo, Rafe; Tamás Hausel; Eugenie Hunsicker (2004). "Hodge cohomology of gravitational instantons". Duke Mathematical Journal. 122 (3): 485–548. doi:10.1215/S0012-7094-04-12233-X. S2CID 14883967. Retrieved 5 September 2021. 8. "Rafe Mazzeo - The Mathematics Genealogy Project". Genealogy.math.ndsu.nodak.edu. Retrieved 7 November 2021. 9. "Fellows of the American Mathematical Society". American Mathematical Society. Retrieved 7 November 2021. 10. "New Members American Academy of Arts and Sciences". amacad.org. Retrieved 1 May 2022. External links • Rafe Mazzeo's Home Page Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States Academics • CiNii • Google Scholar • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef {[Category:Fellow of the American Academy of Arts and Sciences]}	Wikipedia
Ragsdale conjecture The Ragsdale conjecture is a mathematical conjecture that concerns the possible arrangements of real algebraic curves embedded in the projective plane. It was proposed by Virginia Ragsdale in her dissertation in 1906 and was disproved in 1979. It has been called "the oldest and most famous conjecture on the topology of real algebraic curves".[1] Formulation of the conjecture Ragsdale's dissertation, "On the Arrangement of the Real Branches of Plane Algebraic Curves," was published by the American Journal of Mathematics in 1906. The dissertation was a treatment of Hilbert's sixteenth problem, which had been proposed by Hilbert in 1900, along with 22 other unsolved problems of the 19th century; it is one of the handful of Hilbert's problems that remains wholly unresolved. Ragsdale formulated a conjecture that provided an upper bound on the number of topological circles of a certain type,[2] along with the basis of evidence. Conjecture Ragsdale's main conjecture is as follows. Assume that an algebraic curve of degree 2k contains p even and n odd ovals. Ragsdale conjectured that $p\leq {\tfrac {3}{2}}k(k-1)+1\quad {\text{and}}\quad n\leq {\tfrac {3}{2}}k(k-1).$ She also posed the inequality $\|2(p-n)-1\|\leq 3k^{2}-3k+1,$ and showed that the inequality could not be further improved. This inequality was later proved by Petrovsky. Disproving the conjecture The conjecture was held of very high importance in the field of real algebraic geometry for most of the twentieth century. Later, in 1980, Oleg Viro[3] introduced a technique known as "patchworking algebraic curves"[1] and used to generate a counterexample to the conjecture. In 1993, Ilia Itenberg[4] produced additional counterexamples to the Ragsdale conjecture, so Viro and Itenberg wrote a paper in 1996 discussing their work on disproving the conjecture using the "patchworking" technique.[1] The problem of finding a sharp upper bound remains unsolved. References 1. Itenberg, Ilya; Oleg, Viro (1996). "Patchworking algebraic curves disproves the ragsdale conjecture". The Mathematical Intelligencer. Springer-Verlag. 18 (4): 19–28. doi:10.1007/BF03026748. 2. De Loera, Jesús; Wicklin, Frederick J. "Biographies of Women in Mathematics: Virginia Ragsdale". Anges Scott College. Retrieved 22 March 2019. 3. Viro, Oleg Ya. (1980). "Кривые степени 7, кривые степени 8 и гипотеза Рэгсдейл" [Curves of degree 7, curves of degree 8 and the hypothesis of Ragsdale]. Doklady Akademii Nauk SSSR. 254 (6): 1306–1309. Translated in "Кривые степени 7, кривые степени 8 и гипотеза Рэгсдейл" [Curves of degree 7, curves of degree 8 and the hypothesis of Ragsdale]. Soviet Mathematics - Doklady. 22: 566–570. 1980. Zbl 0422.14032. 4. Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii (2007). Tropical algebraic geometry. Oberwolfach Seminars. Vol. 35. Basel: Birkhäuser. pp. 34–35. ISBN 978-3-7643-8309-1. Zbl 1162.14300. Disproved conjectures • Borsuk's • Chinese hypothesis • Connes • Euler's sum of powers • Ganea • Hedetniemi's • Hauptvermutung • Hirsch • Kalman's • Keller's • Mertens • Ono's inequality • Pólya • Ragsdale • Schoen–Yau • Seifert • Tait's • Von Neumann • Weyl–Berry • Williamson	Wikipedia
Rainbow coloring In graph theory, a path in an edge-colored graph is said to be rainbow if no color repeats on it. A graph is said to be rainbow-connected (or rainbow colored) if there is a rainbow path between each pair of its vertices. If there is a rainbow shortest path between each pair of vertices, the graph is said to be strongly rainbow-connected (or strongly rainbow colored).[1] Definitions and bounds The rainbow connection number of a graph $G$ is the minimum number of colors needed to rainbow-connect $G$, and is denoted by ${\text{rc}}(G)$. Similarly, the strong rainbow connection number of a graph $G$ is the minimum number of colors needed to strongly rainbow-connect $G$, and is denoted by ${\text{src}}(G)$. Clearly, each strong rainbow coloring is also a rainbow coloring, while the converse is not true in general. It is easy to observe that to rainbow-connect any connected graph $G$, we need at least ${\text{diam}}(G)$ colors, where ${\text{diam}}(G)$ is the diameter of $G$ (i.e. the length of the longest shortest path). On the other hand, we can never use more than $m$ colors, where $m$ denotes the number of edges in $G$. Finally, because each strongly rainbow-connected graph is rainbow-connected, we have that ${\text{diam}}(G)\leq {\text{rc}}(G)\leq {\text{src}}(G)\leq m$. The following are the extremal cases:[1] • ${\text{rc}}(G)={\text{src}}(G)=1$ if and only if $G$ is a complete graph. • ${\text{rc}}(G)={\text{src}}(G)=m$ if and only if $G$ is a tree. The above shows that in terms of the number of vertices, the upper bound ${\text{rc}}(G)\leq n-1$ is the best possible in general. In fact, a rainbow coloring using $n-1$ colors can be constructed by coloring the edges of a spanning tree of $G$ in distinct colors. The remaining uncolored edges are colored arbitrarily, without introducing new colors. When $G$ is 2-connected, we have that ${\text{rc}}(G)\leq \lceil n/2\rceil $.[2] Moreover, this is tight as witnessed by e.g. odd cycles. Exact rainbow or strong rainbow connection numbers The rainbow or the strong rainbow connection number has been determined for some structured graph classes: • ${\text{rc}}(C_{n})={\text{src}}(C_{n})=\lceil n/2\rceil $, for each integer $n\geq 4$, where $C_{n}$ is the cycle graph.[1] • ${\text{rc}}(W_{n})=3$, for each integer $n\geq 7$, and ${\text{src}}(W_{n})=\lceil n/3\rceil $, for $n\geq 3$, where $W_{n}$ is the wheel graph.[1] Complexity The problem of deciding whether ${\text{rc}}(G)=2$ for a given graph $G$ is NP-complete.[3] Because ${\text{rc}}(G)=2$ if and only if ${\text{src}}(G)=2$,[1] it follows that deciding if ${\text{src}}(G)=2$ is NP-complete for a given graph $G$. Variants and generalizations Chartrand, Okamoto and Zhang[4] generalized the rainbow connection number as follows. Let $G$ be an edge-colored nontrivial connected graph of order $n$. A tree $T$ is a rainbow tree if no two edges of $T$ are assigned the same color. Let $k$ be a fixed integer with $2\leq k\leq n$. An edge coloring of $G$ is called a $k$-rainbow coloring if for every set $S$ of $k$ vertices of $G$, there is a rainbow tree in $G$ containing the vertices of $S$. The $k$-rainbow index ${\text{rx}}_{k}(G)$ of $G$ is the minimum number of colors needed in a $k$-rainbow coloring of $G$. A $k$-rainbow coloring using ${\text{rx}}_{k}(G)$ colors is called a minimum $k$-rainbow coloring. Thus ${\text{rx}}_{2}(G)$ is the rainbow connection number of $G$. Rainbow connection has also been studied in vertex-colored graphs. This concept was introduced by Krivelevich and Yuster.[5] Here, the rainbow vertex-connection number of a graph $G$, denoted by ${\text{rvc}}(G)$, is the minimum number of colors needed to color $G$ such that for each pair of vertices, there is a path connecting them whose internal vertices are assigned distinct colors. See also • Rainbow matching Notes 1. Chartrand et al. (2008). 2. Ekstein et al. (2013). 3. Chakraborty et al. (2011). 4. Chartrand, Okamoto & Zhang (2010). 5. Krivelevich & Yuster (2010). References • Chartrand, Gary; Johns, Garry L.; McKeon, Kathleen A.; Zhang, Ping (2008), "Rainbow connection in graphs", Mathematica Bohemica, 133 (1): 85–98, doi:10.21136/MB.2008.133947. • Chartrand, Gary; Okamoto, Futaba; Zhang, Ping (2010), "Rainbow trees in graphs and generalized connectivity", Networks, 55 (4): NA, doi:10.1002/net.20339, S2CID 7505197. • Chakraborty, Sourav; Fischer, Eldar; Matsliah, Arie; Yuster, Raphael (2011), "Hardness and algorithms for rainbow connection", Journal of Combinatorial Optimization, 21 (3): 330–347, arXiv:0809.2493, doi:10.1007/s10878-009-9250-9, S2CID 10874392. • Krivelevich, Michael; Yuster, Raphael (2010), "The Rainbow Connection of a Graph Is (at Most) Reciprocal to Its Minimum Degree", Journal of Graph Theory, 63 (3): 185–191, doi:10.1002/jgt.20418. • Li, Xueliang; Shi, Yongtang; Sun, Yuefang (2013), "Rainbow Connections of Graphs: A Survey", Graphs and Combinatorics, 29 (1): 1–38, arXiv:1101.5747, doi:10.1007/s00373-012-1243-2, S2CID 253898232. • Li, Xueliang; Sun, Yuefang (2012), Rainbow connections of graphs, Springer, p. 103, ISBN 978-1-4614-3119-0. • Ekstein, Jan; Holub, Přemysl; Kaiser, Tomáš; Koch, Maria; Camacho, Stephan Matos; Ryjáček, Zdeněk; Schiermeyer, Ingo (2013), "The rainbow connection number of 2-connected graphs", Discrete Mathematics, 313 (19): 1884–1892, arXiv:1110.5736, doi:10.1016/j.disc.2012.04.022, S2CID 16596310.	Wikipedia
Covering problems In combinatorics and computer science, covering problems are computational problems that ask whether a certain combinatorial structure 'covers' another, or how large the structure has to be to do that. Covering problems are minimization problems and usually integer linear programs, whose dual problems are called packing problems. The most prominent examples of covering problems are the set cover problem, which is equivalent to the hitting set problem, and its special cases, the vertex cover problem and the edge cover problem. Covering/packing-problem pairs Covering problems Packing problems Minimum set cover Maximum set packing Minimum edge cover Maximum matching Minimum vertex cover Maximum independent set Bin covering Bin packing Polygon covering Rectangle packing General linear programming formulation In the context of linear programming, one can think of any linear program as a covering problem if the coefficients in the constraint matrix, the objective function, and right-hand side are nonnegative.[1] More precisely, consider the following general integer linear program: minimize $\sum _{i=1}^{n}c_{i}x_{i}$ subject to $\sum _{i=1}^{n}a_{ji}x_{i}\geq b_{j}{\text{ for }}j=1,\dots ,m$ $x_{i}\in \left\{0,1,2,\ldots \right\}{\text{ for }}i=1,\dots ,n$. Such an integer linear program is called a covering problem if $a_{ji},b_{j},c_{i}\geq 0$ for all $i=1,\dots ,n$ and $j=1,\dots ,m$. Intuition: Assume having $n$ types of object and each object of type $i$ has an associated cost of $c_{i}$. The number $x_{i}$ indicates how many objects of type $i$ we buy. If the constraints $A\mathbf {x} \geq \mathbf {b} $ are satisfied, it is said that $\mathbf {x} $ is a covering (the structures that are covered depend on the combinatorial context). Finally, an optimal solution to the above integer linear program is a covering of minimal cost. Kinds of covering problems There are various kinds of covering problems in graph theory, computational geometry and more; see Category:Covering problems. Other stochastic related versions of the problem can be found. [2] For Petri nets, for example, the covering problem is defined as the question if for a given marking, there exists a run of the net, such that some larger (or equal) marking can be reached. Larger means here that all components are at least as large as the ones of the given marking and at least one is properly larger. Rainbow covering and conflict-free covering In some covering problems, the covering should satisfy some additional requirements. In particular, in the rainbow covering problem, each of the original objects has a "color", and it is required that the covering contains exactly one (or at most one) object of each color. Rainbow covering was studied e.g. for covering points by intervals:[3] • There is a set J of n colored intervals on the real line, and a set P of points on the real line. • A subset Q of J is called a rainbow set if it contains at most a single interval of each color. • A set of intervals J is called a covering of P if each point in P is contained in at least one interval of Q. • The Rainbow covering problem is the problem of finding a rainbow set Q that is a covering of P. The problem is NP-hard (by reduction from linear SAT). A more general notion is conflict-free covering.[4] In this problem: • There is a set O of m objects, and a conflict-graph GO on O. • A subset Q of O is called conflict-free if it is an independent set in GO, that is, no two objects in Q are connected by an edge in GO. • A rainbow set is a conflict-free set in the special case in which GO is made of disjoint cliques, where each clique represents a color. Conflict-free set cover is the problem of finding a conflict-free subset of O that is a covering of P. Banik, Panolan, Raman, Sahlot and Saurabh[5] prove the following for the special case in which the conflict-graph has bounded arboricity: • If the geometric cover problem is fixed-parameter tractable (FPT), then the conflict-free geometric cover problem is FPT. • If the geometric cover problem admits an r-approximation algorithm, then the conflict-free geometric cover problem admits a similar approximation algorithm in FPT time. References 1. Vazirani, Vijay V. (2001). Approximation Algorithms. Springer-Verlag. ISBN 3-540-65367-8.: 112 2. Douek-Pinkovich, Y., Ben-Gal, I., & Raviv, T. (2022). "The Stochastic Test Collection Problem: Models, Exact and Heuristic Solution Approaches" (PDF). European Journal of Operational Research, 299 (2022), 945–959}.{{cite web}}: CS1 maint: multiple names: authors list (link) 3. Arkin, Esther M.; Banik, Aritra; Carmi, Paz; Citovsky, Gui; Katz, Matthew J.; Mitchell, Joseph S. B.; Simakov, Marina (2018-12-11). "Selecting and covering colored points". Discrete Applied Mathematics. 250: 75–86. doi:10.1016/j.dam.2018.05.011. ISSN 0166-218X. 4. Banik, Aritra; Sahlot, Vibha; Saurabh, Saket (2020-08-01). "Approximation algorithms for geometric conflict free covering problems". Computational Geometry. 89: 101591. doi:10.1016/j.comgeo.2019.101591. ISSN 0925-7721. S2CID 209959954. 5. Banik, Aritra; Panolan, Fahad; Raman, Venkatesh; Sahlot, Vibha; Saurabh, Saket (2020-01-01). "Parameterized Complexity of Geometric Covering Problems Having Conflicts". Algorithmica. 82 (1): 1–19. doi:10.1007/s00453-019-00600-w. ISSN 1432-0541. S2CID 254027914.	Wikipedia
Rainbow matching In the mathematical discipline of graph theory, a rainbow matching in an edge-colored graph is a matching in which all the edges have distinct colors. Definition Given an edge-colored graph G = (V,E), a rainbow matching M in G is a set of pairwise non-adjacent edges, that is, no two edges share a common vertex, such that all the edges in the set have distinct colors. A maximum rainbow matching is a rainbow matching that contains the largest possible number of edges. History Rainbow matchings are of particular interest given their connection to transversals of Latin squares. Denote by Kn,n the complete bipartite graph on n + n vertices. Every proper n-edge coloring of Kn,n corresponds to a Latin square of order n. A rainbow matching then corresponds to a transversal of the Latin square, meaning a selection of n positions, one in each row and each column, containing distinct entries. This connection between transversals of Latin squares and rainbow matchings in Kn,n has inspired additional interest in the study of rainbow matchings in triangle-free graphs.[1] Existence when each edge has a single color An edge-coloring is called proper if each edge has a single color, and each two edges of the same color have no vertex in common. A proper edge-coloring does not guarantee the existence of a perfect rainbow matching. For example, consider the graph K2,2: the complete bipartite graph on 2+2 vertices. Suppose the edges (x1,y1) and (x2,y2) are colored green, and the edges (x1,y2) and (x2,y1) are colored blue. This is a proper coloring, but there are only two perfect matchings, and each of them is colored by a single color. This invokes the question: when does a large rainbow matching is guaranteed to exist? Bounds depending only on the number of vertices Much of the research on this question was published using the terminology of Latin transversals in Latin squares. Translated into the rainbow matching terminology: • In 1967, H. J. Ryser conjectured that, when n is odd, every proper edge-coloring of Kn,n has a rainbow matching of size n.[2] • In 1975, S. K. Stein and Brualdi conjectured that, when n is even, every proper edge-coloring of Kn,n has a rainbow matching of size n – 1.[3] (it is known that a rainbow matching of size n need not exist in this case). A more general conjecture of Stein is that a rainbow matching of size n – 1 exists not only for a proper edge-coloring, but for any coloring in which each color appears on exactly n edges.[2] Some weaker versions of these conjectures have been proved: • Every proper edge-coloring of Kn,n has a rainbow matching of size 2n/3.[4] • Every proper edge-coloring of Kn,n has a rainbow matching of size $n-{\sqrt {n}}.$[5] • Every proper edge-coloring of Kn,n has a rainbow matching of size n – 11 log22(n).[6] Bounds depending on the minimum degree Wang asked if there is a function f(d) such that every properly edge-colored graph G with minimum degree d and at least f(d) vertices must have a rainbow matching of size d.[7] Obviously at least 2d vertices are necessary, but how many are sufficient? • Diemunsch, et al. answered this question in the affirmative and showed that given a properly edge-colored graph G with minimum degree d and order at least f(d) = 98δ/23, there exists a rainbow matching of size d in G.[8] • This bound was later improved to f(d) = 4d – 3 by Andras Gyarfas and Gabor N. Sarkozy.[9] They also show that any graph with at least 2d vertices has a rainbow matching of size at least d – 2d2/3. These are the best known estimate to date. Existence when the same edge may have different colors Suppose that each edge may have several different colors, while each two edges of the same color must still have no vertex in common. In other words, each color is a matching. How many colors are needed in order to guarantee the existence of a rainbow matching? In complete bipartite graphs Drisko[10] studied this question using the terminology of Latin rectangles. He proved that, for any n ≤ k, in the complete bipartite graph Kn,k, any family of 2n – 1 matchings (=colors) of size n has a perfect rainbow matching (of size n). He applied this theorem to questions about group actions and difference sets. Drisko also showed that 2n – 1 matchings may be necessary: consider a family of 2n – 2 matchings, of which n – 1 are { (x1, y1), (x2, y2), ..., (xn, yn)} and the other n – 1 are {(x1, y2), (x2, y3), …, (xn, y1) }. Then the largest rainbow matching is of size n – 1 (e.g. take one edge from each of the first n – 1 matchings). Alon[11] showed that Drisko's theorem implies an older result[12] in additive number theory. In general bipartite graphs Aharoni and Berger[13] generalized Drisko's theorem to any bipartite graph, namely: any family of 2n – 1 matchings of size n in a bipartite graph has a rainbow matching of size n. Aharoni, Kotlar and Ziv[14] showed that Drisko's extremal example is unique in any bipartite graph. In general graphs In general graphs, 2n – 1 matchings are no longer sufficient. When n is even, one can add to Drisko's example the matching { (x1, x2), (y1, y2), (x2, x3), (y2, y3), … } and get a family of 2n – 1 matchings without any rainbow matching. Aharoni, Berger, Chudnovsky, Howard and Seymour[15] proved that, in a general graph, 3n – 2 matchings (=colors) are always sufficient. It is not known whether this is tight: currently the best lower bound for even n is 2n and for odd n it is 2n – 1.[16] Rainbow fractional matchings A fractional matching is a set of edges with a non-negative weight assigned to each edge, such that the sum of weights adjacent to each vertex is at most 1. The size of a fractional matching is the sum of weights of all edges. It is a generalization of a matching, and can be used to generalize both the colors and the rainbow matching: • Instead of requiring that each color be a matching of size n, the requirement is weakened: each "color" can be an arbitrary set of edges, but it should admit a fractional matching of size at least n. • Instead of looking for a rainbow matching, we look for a rainbow fractional matching - a fractional matching in which each edge with a positive weight has a different color. It is known that, in a bipartite graph, the maximum fractional matching size equals the maximum matching size. Therefore, the theorem of Aharoni and Berger[13] is equivalent to the following. Let n be any positive integer. Given any family of 2n – 1 fractional-matchings (=colors) of size n in a bipartite graph, there exists a rainbow-fractional-matching of size n. Aharoni, Holzman and Jiang extend this theorem to arbitrary graphs as follows. Let n be any positive integer or half-integer. Any family of 2n fractional-matchings (=colors) of size at least n in an arbitrary graph has a rainbow-fractional-matching of size n.[16]: Thm.1.5 The 2n is the smallest possible for fractional matchings in arbitrary graphs: the extremal case is constructed using an odd-length cycle. Partial proof For the case of perfect fractional matchings, both the above theorems can derived from the colorful Caratheodory theorem. For every edge e in E, let 1e be a vector of size \|V\|, where for each vertex v in V, element v in 1e equals 1 if e is adjacent to v, and 0 otherwise (so each vector 1e has 2 ones and \|V\|-2 zeros). Every fractional matching corresponds to a conical combination of edges, in which each element is at most 1. A conical combination in which each element is exactly 1 corresponds to a perfect fractional matching. In other words, a collection F of edges admits a perfect fractional matching, if and only if 1v (the vector of \|V\| ones) is contained in the conical hull of the vectors 1e for e in F. Consider a graph with 2n vertices, and suppose there are 2n subsets of edges, each of which admits a perfect fractional matching (of size n). This means that the vector 1v is in the conical hull of each of these n subsets. By the colorful Caratheodory theorem, there exists a selection of 2n edges, one from each subset, that their conical hull contains 1v. This corresponds to a rainbow perfect fractional matching. The expression 2n is the dimension of the vectors 1e - each vector has 2n elements. Now, suppose that the graph is bipartite. In a bipartite graph, there is a constraint on the vectors 1e: the sum of elements corresponding to each part of the graph must be 1. Therefore, the vectors 1e live in a (2n – 1)-dimensional space. Therefore, the same argument as above holds when there are only 2n – 1 subsets of edges. Rainbow matching in hypergraphs An r-uniform hypergraph is a set of hyperedges each of which contains exactly r vertices (so a 2-uniform hypergraph is a just a graph without self-loops). Aharoni, Holzman and Jiang extend their theorem to such hypergraphs as follows. Let n be any positive rational number. Any family of ⌈r⋅n⌉ fractional-matchings (=colors) of size at least n in an r-uniform hypergraph has a rainbow-fractional-matching of size n.[16]: Thm.1.6 The ⌈r⋅n⌉ is the smallest possible when n is an integer. An r-partite hypergraph is an r-uniform hypergraph in which the vertices are partitioned into r disjoint sets and each hyperedge contains exactly one vertex of each set (so a 2-partite hypergraph is a just bipartite graph). Let n be any positive integer. Any family of rn – r + 1 fractional-matchings (=colors) of size at least n in an r-partite hypergraph has a rainbow-fractional-matching of size n.[16]: Thm.1.7 The rn – r + 1 is the smallest possible: the extremal case is when n = r – 1 is a prime power, and all colors are edges of the truncated projective plane of order n. So each color has n2 = rn – r + 1 edges and a fractional matching of size n, but any fractional matching of that size requires all rn – r + 1 edges.[17] Partial proof For the case of perfect fractional matchings, both the above theorems can derived from the colorful caratheodory theorem in the previous section. For a general r-uniform hypergraph (admitting a perfect matching of size n), the vectors 1e live in a (rn)-dimensional space. For an r-uniform r-partite hypergraph, the r-partiteness constraints imply that the vectors 1e live in a (rn – r + 1)-dimensional space. Notes The above results hold only for rainbow fractional matchings. In contrast, the case of rainbow integral matchings in r-uniform hypergraphs is much less understood. The number of required matchings for a rainbow matching of size n grows at least exponentially with n. See also: matching in hypergraphs Computation Garey and Johnson have shown that computing a maximum rainbow matching is NP-complete even for edge-colored bipartite graphs.[18] Applications Rainbow matchings have been applied for solving packing problems.[19] See also • Rainbow coloring • Rainbow-colorable hypergraph • Rainbow-independent set • Rainbow covering • Hall-type theorems for hypergraphs References 1. West, D.B. (2009), Rainbow Matchings 2. Aharoni, Ron; Berger, Eli; Kotlar, Dani; Ziv, Ran (2017-01-04). "On a conjecture of Stein". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 87 (2): 203–211. doi:10.1007/s12188-016-0160-3. ISSN 0025-5858. S2CID 119139740. 3. Stein, Sherman (1975-08-01). "Transversals of Latin squares and their generalizations". Pacific Journal of Mathematics. 59 (2): 567–575. doi:10.2140/pjm.1975.59.567. ISSN 0030-8730. 4. Koksma, Klaas K. (1969-07-01). "A lower bound for the order of a partial transversal in a latin square". Journal of Combinatorial Theory. 7 (1): 94–95. doi:10.1016/s0021-9800(69)80009-8. ISSN 0021-9800. 5. Woolbright, David E (1978-03-01). "An n × n Latin square has a transversal with at least n−n distinct symbols". Journal of Combinatorial Theory, Series A. 24 (2): 235–237. doi:10.1016/0097-3165(78)90009-2. ISSN 0097-3165. 6. Hatami, Pooya; Shor, Peter W. (2008-10-01). "A lower bound for the length of a partial transversal in a Latin square". Journal of Combinatorial Theory, Series A. 115 (7): 1103–1113. doi:10.1016/j.jcta.2008.01.002. ISSN 0097-3165. 7. Wang, Guanghui (2009), "Rainbow Matchings in Properly Edge Colored Graphs", The Electronic Journal of Combinatorics, 18 (1): 162 8. Diemunsch, Jennifer; Ferrara, Michael; Lo, Allan; Moffatt, Casey; Pfender, Florian; Wenger, Paul S. (2012), "Rainbow Matchings of Size δ(G) in Properly Edge-Colored Graphs", The Electronic Journal of Combinatorics, 19 (2): 52, doi:10.37236/2443, S2CID 119177198 9. Gyarfas, Andras; Sarkozy, Gabor N. (2012). "Rainbow matchings and partial transversals of Latin squares". arXiv:1208.5670 [CO math. CO]. 10. Drisko, Arthur A. (1998-11-01). "Transversals in Row-Latin Rectangles". Journal of Combinatorial Theory, Series A. 84 (2): 181–195. doi:10.1006/jcta.1998.2894. ISSN 0097-3165. 11. Alon, Noga (2011). "Multicolored Matchings in Hypergraphs". Moscow Journal of Combinatorics and Number Theory. 1: 3–10. 12. Flores, Carlos; Ordaz, Oscar (1996-05-01). "On the Erdös-Ginzburg-Ziv theorem". Discrete Mathematics. 152 (1–3): 321–324. doi:10.1016/0012-365x(94)00328-g. ISSN 0012-365X. 13. Aharoni, Ron; Berger, Eli (2009-09-25). "Rainbow Matchings in $r$-Partite $r$-Graphs". The Electronic Journal of Combinatorics. 16 (1). doi:10.37236/208. ISSN 1077-8926. 14. Aharoni, Ron; Kotlar, Dani; Ziv, Ran (2018-01-01). "Uniqueness of the extreme cases in theorems of Drisko and Erdős–Ginzburg–Ziv". European Journal of Combinatorics. 67: 222–229. doi:10.1016/j.ejc.2017.08.008. ISSN 0195-6698. S2CID 38268762. 15. Aharoni, Ron; Berger, Eli; Chudnovsky, Maria; Howard, David; Seymour, Paul (2019-06-01). "Large rainbow matchings in general graphs". European Journal of Combinatorics. 79: 222–227. arXiv:1611.03648. doi:10.1016/j.ejc.2019.01.012. ISSN 0195-6698. S2CID 42126880. 16. Aharoni, Ron; Holzman, Ron; Jiang, Zilin (2019-10-29). "Rainbow Fractional Matchings". Combinatorica. 39 (6): 1191–1202. arXiv:1805.09732. doi:10.1007/s00493-019-4019-y. ISSN 0209-9683. S2CID 119173114. 17. Füredi, Zoltán (1989-05-01). "Covering the complete graph by partitions". Discrete Mathematics. 75 (1–3): 217–226. doi:10.1016/0012-365x(89)90088-5. ISSN 0012-365X. 18. Garey, M. R.; Johnson, D. S. (1979). Victor Klee (ed.). Computers and Intractability: A Guide to the Theory of NP-Completeness. A Series of Books in the Mathematical Sciences. San Francisco, Calif.: W. H. Freeman and Co. pp. x+338. ISBN 0-7167-1045-5. MR 0519066. 19. Bannach, Max; Berndt, Sebastian; Maack, Marten; Mnich, Matthias; Lassota, Alexandra; Rau, Malin; Skambath, Malte (2020-07-06). "Solving Packing Problems with Few Small Items Using Rainbow Matchings". arXiv:2007.02660 [cs.DS].	Wikipedia
John Rainwater The fictitious mathematician John Rainwater was created as a student prank but has become known as the author of important results in functional analysis. John Rainwater Known forRainwater's theorem, Rainwater seminar (functional analysis) Scientific career InfluencesJohn R. Isbell, Robert R. Phelps InfluencedJoseph Diestel At the University of Washington in 1952, John Rainwater was invented and enrolled in a mathematics course by graduate students who were in possession of a duplicate student-registration form. Later, mathematicians published under the pseudonym of John Rainwater. Papers were published under the name Rainwater mainly in functional analysis, particularly in the geometric theory of Banach spaces and in convex functions. Rainwater's theorem is an important result in summability theory and functional analysis. The University of Washington's seminar in functional analysis is called the Rainwater seminar, and the associated Rainwater notes have influenced Banach-space theory and convex analysis.[1] The concept of a fictional pseudonym used by multiple people creating valuable mathematics is not unique. Most notably, Nicolas Bourbaki has been the collective pseudonym for a number of leading mathematicians writing in French for many decades. Creation John Rainwater was invented by graduate students at the University of Washington in 1952, when students used an extra registration form to enroll Rainwater in a course on real functions. Students submitted homework for Rainwater throughout the semester. The professor caught on to the prank around the middle of the term. Other students in the class were made aware of the situation from the professor's enigmatic remarks after he became the victim of a novelty "exploding" fountain pen bearing Rainwater's name.[2] Research Early on, Rainwater distinguished himself by solving problems in the American Mathematical Monthly, whose sponsoring society, the Mathematical Association of America, invited him to join. John R. Isbell published the first paper in Rainwater's name. Other mathematicians have published papers using the name "Rainwater", and acknowledged "Rainwater's assistance" in articles. The seminar on functional analysis at the University of Washington has been called the "Rainwater seminar".[1] Rainwater's theorem is an important result in summability theory and Banach-space theory.[3][4] Evaluation In 2002 Robert Phelps summarized the impact of Rainwater's research. The first Rainwater paper (by Isbell) was in topology and had had 19 citations. While only one page, Rainwater's note in the 1963 Proceedings of the American Mathematical Society had had eight citations in papers; its main result has been called "Rainwater's theorem" in books on convex functions and functional analysis. "There is even one citation to number 13, his unpublished 1967 Rainwater Seminar note on Lindenstrauss spaces," which are named after a construction by Joram Lindenstrauss. "In summary, it appears that most of John Rainwater's published work has been reasonably well received."[1] While Rainwater is lesser known and younger than Nicolas Bourbaki, the collective pseudonym for a number of leading mathematicians writing in French, he is more senior and has more publications than the combination of research by three other pseudonymous mathematicians—Peter Orno, M. G. Stanley, and H. C. Enos.[5] Mathematicians publishing as Rainwater Many internationally renowned mathematicians have published under the name of John Rainwater. John Isbell wrote Rainwater's first, second, and tenth papers; by 2002, Isbell had also written or coauthored six other pseudonymous papers under two other names. Functional analyst Robert R. Phelps wrote the third, ninth, eleventh (an unpublished note for the Rainwater seminar), twelfth, and thirteenth (with Peter D. Morris), fifteenth (with Isaac Namioka), and sixteenth (with David Preiss) papers. Irving Glicksberg wrote the fourth and eighth papers. Edgar Asplund wrote the seventh. "Paper 14 is a departure for John Rainwater. Not only is it in algebra, but he doesn't thank anyone for helpful conversations. He notes, however, that his work was supported by four different grants. (Culprits this time were Ken Brown, Ken Goodearl, Toby Stafford and Bob Warfield.)" John Rainwater's c.v. lists an incomplete collection of problems or solutions that he contributed to the American Mathematical Monthly, the earliest in 1959 (by John Isbell).[1] Notes 1. Phelps (2002) 2. Phelps (2002) wrote that John Rainwater came into existence at the University of Washington in 1952 when Nick Massey, a mathematics graduate student in Prof. Maynard Arsove's beginning real variables class, erroneously received a blank registration card. (In those years, each student filled out a card for every class, which first circulated among various tabulating clerks in the registrar's office before being sent to the professor.) He and a fellow graduate student, Sam Saunders, decided to use the card to enroll a fictional student, and since it was raining at the time, decided to call him "John Rainwater". They handed in John Rainwater's homework regularly, so it wasn't until after the first midterm exam that Prof. Arsove became aware of the deception. He took it well, even when he later opened an "exploding" fountain pen with John Rainwater's name engraved on it which had been left on the classroom table. After remarks by Arsove, such as "I guess I'll never see Rainwater except in a barrel," virtually all the students learned of the Rainwater prank. (Phelps 2002) 3. Diestel, Joseph (1975), Geometry of Banach spaces: Selected topics, Lecture Notes in Mathematics, vol. 485, Springer-Verlag, pp. xi+282, ISBN 978-3-540-07402-1, MR 0461094, Pages 94 and 125 discuss Rainwater's theorem 4. Diestel, Joseph (1984), "IX Extremal tests for weak convergence of sequences and series (Rainwater's theorem, pp. 147–157)", Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, New York: Springer-Verlag, pp. xii+261, ISBN 0-387-90859-5, MR 0737004, Pages 147, 154, 157, 167, and 170–173 discuss Rainwater's theorem 5. (Phelps 2002) References • Diestel, Joseph (1975), Geometry of Banach spaces: Selected topics, Lecture Notes in Mathematics, vol. 485, Springer-Verlag, pp. xi+282, ISBN 978-3-540-07402-1, MR 0461094, Pages 94 and 125 discuss Rainwater's theorem • Diestel, Joseph (1984), "IX Extremal tests for weak convergence of sequences and series (Rainwater's theorem, pp. 147–157)", Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, New York: Springer-Verlag, pp. xii+261, ISBN 0-387-90859-5, MR 0737004, Pages 147, 154, 157, 167, and 170–173 discuss Rainwater's theorem • Phelps, Robert R. (2002). Melvin Henriksen (ed.). "Biography of John Rainwater". Topological Commentary. Topology Atlas (York University, Canada). 7 (2). ISSN 1499-9226. {{cite journal}}: External link in \|journal= (help) Further reading The Rainwater seminar and Rainwater notes are listed as influences by the following books: • Asimow, L.; Ellis, A. J. (1980). Convexity theory and its applications in functional analysis. London Mathematical Society Monographs. Vol. 16. London-New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. pp. x+266. ISBN 0-12-065340-0. MR 0623459. • Bourgin, Richard D. (1983). Geometric aspects of convex sets with the Radon-Nikodým property. Lecture Notes in Mathematics. Vol. 993. Berlin: Springer-Verlag. pp. xii+474. ISBN 3-540-12296-6. MR 0704815. • Giles, John R. (1982). Convex analysis with application in the differentiation of convex functions. Research Notes in Mathematics. Vol. 58. Boston, Mass.-London: Pitman (Advanced Publishing Program). pp. x+278. ISBN 0-273-08537-9. MR 0650456. See also • Monsieur LeBlanc (pseudonym of Sophie Germain) Collaborative pseudonyms: • Nicolas Bourbaki • Arthur Besse • Blanche Descartes • G. W. Peck External resources • Mathematical Reviews. "John Rainwater". Retrieved 2011-04-02. Authority control International • VIAF National • United States • Czech Republic Academics • MathSciNet • Scopus • zbMATH	Wikipedia
Raised cosine distribution In probability theory and statistics, the raised cosine distribution is a continuous probability distribution supported on the interval $[\mu -s,\mu +s]$. The probability density function (PDF) is $f(x;\mu ,s)={\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\,\pi \right)\right]\,={\frac {1}{s}}\operatorname {hvc} \left({\frac {x-\mu }{s}}\,\pi \right)\,$ Raised cosine Probability density function Cumulative distribution function Parameters $\mu \,$(real) $s>0\,$(real) Support $x\in [\mu -s,\mu +s]\,$ PDF ${\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\,\pi \right)\right]\,={\frac {1}{s}}\operatorname {hvc} \left({\frac {x-\mu }{s}}\,\pi \right)\,$ CDF ${\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]$ Mean $\mu \,$ Median $\mu \,$ Mode $\mu \,$ Variance $s^{2}\left({\frac {1}{3}}-{\frac {2}{\pi ^{2}}}\right)\,$ Skewness $0\,$ Ex. kurtosis ${\frac {6(90-\pi ^{4})}{5(\pi ^{2}-6)^{2}}}=-0.59376\ldots \,$ MGF ${\frac {\pi ^{2}\sinh(st)}{st(\pi ^{2}+s^{2}t^{2})}}\,e^{\mu t}$ CF ${\frac {\pi ^{2}\sin(st)}{st(\pi ^{2}-s^{2}t^{2})}}\,e^{i\mu t}$ for $\mu -s\leq x\leq \mu +s$ and zero otherwise. The cumulative distribution function (CDF) is $F(x;\mu ,s)={\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]$ for $\mu -s\leq x\leq \mu +s$ and zero for $x<\mu -s$ and unity for $x>\mu +s$. The moments of the raised cosine distribution are somewhat complicated in the general case, but are considerably simplified for the standard raised cosine distribution. The standard raised cosine distribution is just the raised cosine distribution with $\mu =0$ and $s=1$. Because the standard raised cosine distribution is an even function, the odd moments are zero. The even moments are given by: ${\begin{aligned}\operatorname {E} (x^{2n})&={\frac {1}{2}}\int _{-1}^{1}[1+\cos(x\pi )]x^{2n}\,dx=\int _{-1}^{1}x^{2n}\operatorname {hvc} (x\pi )\,dx\\[5pt]&={\frac {1}{n+1}}+{\frac {1}{1+2n}}\,_{1}F_{2}\left(n+{\frac {1}{2}};{\frac {1}{2}},n+{\frac {3}{2}};{\frac {-\pi ^{2}}{4}}\right)\end{aligned}}$ where $\,_{1}F_{2}$ is a generalized hypergeometric function. See also • Hann function • Havercosine (hvc) References • Horst Rinne (2010). "Location-Scale Distributions – Linear Estimation and Probability Plotting Using MATLAB" (PDF). p. 116. Retrieved 2012-11-16. Probability distributions (list) Discrete univariate with finite support • Benford • Bernoulli • beta-binomial • binomial • categorical • hypergeometric • negative • Poisson binomial • Rademacher • soliton • discrete uniform • Zipf • Zipf–Mandelbrot with infinite support • beta negative binomial • Borel • Conway–Maxwell–Poisson • discrete phase-type • Delaporte • extended negative binomial • Flory–Schulz • Gauss–Kuzmin • geometric • logarithmic • mixed Poisson • negative binomial • Panjer • parabolic fractal • Poisson • Skellam • Yule–Simon • zeta Continuous univariate supported on a bounded interval • arcsine • ARGUS • Balding–Nichols • Bates • beta • beta rectangular • continuous Bernoulli • Irwin–Hall • Kumaraswamy • logit-normal • noncentral beta • PERT • raised cosine • reciprocal • triangular • U-quadratic • uniform • Wigner semicircle supported on a semi-infinite interval • Benini • Benktander 1st kind • Benktander 2nd kind • beta prime • Burr • chi • chi-squared • noncentral • inverse • scaled • Dagum • Davis • Erlang • hyper • exponential • hyperexponential • hypoexponential • logarithmic • F • noncentral • folded normal • Fréchet • gamma • generalized • inverse • gamma/Gompertz • Gompertz • shifted • half-logistic • half-normal • Hotelling's T-squared • inverse Gaussian • generalized • Kolmogorov • Lévy • log-Cauchy • log-Laplace • log-logistic • log-normal • log-t • Lomax • matrix-exponential • Maxwell–Boltzmann • Maxwell–Jüttner • Mittag-Leffler • Nakagami • Pareto • phase-type • Poly-Weibull • Rayleigh • relativistic Breit–Wigner • Rice • truncated normal • type-2 Gumbel • Weibull • discrete • Wilks's lambda supported on the whole real line • Cauchy • exponential power • Fisher's z • Kaniadakis κ-Gaussian • Gaussian q • generalized normal • generalized hyperbolic • geometric stable • Gumbel • Holtsmark • hyperbolic secant • Johnson's SU • Landau • Laplace • asymmetric • logistic • noncentral t • normal (Gaussian) • normal-inverse Gaussian • skew normal • slash • stable • Student's t • Tracy–Widom • variance-gamma • Voigt with support whose type varies • generalized chi-squared • generalized extreme value • generalized Pareto • Marchenko–Pastur • Kaniadakis κ-exponential • Kaniadakis κ-Gamma • Kaniadakis κ-Weibull • Kaniadakis κ-Logistic • Kaniadakis κ-Erlang • q-exponential • q-Gaussian • q-Weibull • shifted log-logistic • Tukey lambda Mixed univariate continuous- discrete • Rectified Gaussian Multivariate (joint) • Discrete: • Ewens • multinomial • Dirichlet • negative • Continuous: • Dirichlet • generalized • multivariate Laplace • multivariate normal • multivariate stable • multivariate t • normal-gamma • inverse • Matrix-valued: • LKJ • matrix normal • matrix t • matrix gamma • inverse • Wishart • normal • inverse • normal-inverse • complex Directional Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham Degenerate and singular Degenerate Dirac delta function Singular Cantor Families • Circular • compound Poisson • elliptical • exponential • natural exponential • location–scale • maximum entropy • mixture • Pearson • Tweedie • wrapped • Category • Commons	Wikipedia
Raj Chandra Bose Raj Chandra Bose (or Basu) (19 June 1901 – 31 October 1987) was an Indian American mathematician and statistician best known for his work in design theory, finite geometry and the theory of error-correcting codes in which the class of BCH codes is partly named after him. He also invented the notions of partial geometry, association scheme, and strongly regular graph and started a systematic study of difference sets to construct symmetric block designs. He was notable for his work along with S. S. Shrikhande and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that for no n do there exist two mutually orthogonal Latin squares of order 4n + 2. Raj Chandra Bose Raj Chandra Bose Born(1901-06-19)19 June 1901 Narmadapuram, Madhya Pradesh, India Died31 October 1987(1987-10-31) (aged 86) Fort Collins, Colorado CitizenshipIndia Alma materRajabazar Science College (University of Calcutta) Known forAssociation scheme Bose–Mesner algebra Euler's conjecture on Latin squares Strongly regular graphs Partial Geometries Morse Code Notable Awards Elected Fellow of the US Academy of Sciences AwardsElected Member of the US National Academy of Sciences Scientific career FieldsMathematics and Statistics InstitutionsIndian Statistical Institute Colorado State University University of North Carolina at Chapel Hill Doctoral studentsDijen K. Ray-Chaudhuri Sharadchandra Shankar Shrikhande J. N. Srivastava Early life Bose was born in Hoshangabad, India; he was the first of five children. His father was a physician and life was good until 1918 when his mother died in the influenza pandemic. His father died of a stroke the following year. Despite difficult circumstances, Bose continued to study securing first class in both the Masters examinations in Pure and Applied mathematics in 1925 and 1927 respectively at the Rajabazar Science College campus of University of Calcutta. His research was performed under the supervision of the geometry Professor Syamadas Mukhopadhyaya from Calcutta. Bose worked as a lecturer at Asutosh College, Calcutta. He published his work on the differential geometry of convex curves. Academic life Bose's course changed in December 1932 when P. C. Mahalanobis, director of the new (1931) Indian Statistical Institute, offered Bose a part-time job. Mahalanobis had seen Bose's geometrical work and wanted him to work on statistics. The day after Bose moved in, the secretary brought him all the volumes of Biometrika with a list of 50 papers to read and also Ronald Fisher's Statistical Methods for Research Workers. Mahalanobis told him, "You were saying that you do not know much statistics. You master the 50 papers ... and Fisher's book. This will suffice for your statistical education for the present." With Samarendra Nath Roy, who joined the ISI a little later, Bose was the chief mathematician at the Institute. He first worked with multivariate analysis where he collaborated with Mahalanobis and Roy. In 1938–9 Fisher visited India and talked about the design of experiments. Roy had the idea of using the theory of finite fields and finite geometry to solve problems in design. The development of a mathematical theory of design would be Bose's main preoccupation until the mid-1950s. In 1935 Bose had become full-time at the Institute. In 1940 joined the University of Calcutta where C. R. Rao and H. K. Nandi were in the first group of students he taught. In 1945 Bose became Head of the Department of Statistics. University authorities in the United States told him he needed to have a doctorate. So he submitted his published papers on multivariate analysis and the design of experiments and was awarded a D. Litt. in 1947. In 1947 Bose went to the United States as a visiting professor at Columbia University and the University of North Carolina at Chapel Hill. He received offers from American universities and he was also offered positions in India. The Indian jobs involved very heavy administration, which he saw as the end of his research work and in March 1949 he joined the University of North Carolina at Chapel Hill as Professor of Statistics. In the years at Chapel Hill Bose made important discoveries on coding theory (with D.K. Ray-Chaudhuri) and constructed (with S. S. Shrikhande and E. T. Parker) a Graeco-Latin square of size 10, a counterexample to Euler's conjecture that no Graeco-Latin square of size 4k + 2 exists. In 1971, he retired at the age of 70. He then accepted a chair at Colorado State University of Fort Collins from which he retired in 1980. His final doctoral student finished after this second retirement. Bose died in Colorado, aged 86, in 1987. He is survived by two daughters. The elder, Purabi Schur, is retired from the Library of Congress and the younger, Sipra Bose Johnson, is retired as a professor of anthropology from the State University of New York at New Paltz. Some articles by R. C. Bose • R. C. Bose, On the construction of balanced incomplete block designs, Annals of Eugenics. 9 (1939), 358–399. • R. C. Bose and K. R. Nair, Partially balanced incomplete block designs, Sankhya 4 (1939), 337–372. • Bose, Raj Chandra; Mesner, D. M. (1959). "On linear associative algebras corresponding to association schemes of partially balanced designs". Annals of Mathematical Statistics. 30 (1): 21–38. doi:10.1214/aoms/1177706356. JSTOR 2237117. MR 0102157. • R. C. Bose and S. S. Shrikhande, On the falsity of Euler's conjecture about the non-existence of two orthogonal Latin squares of order 4t + 2, Proceedings of the National Academy of Sciences USA, 45, (1959), 734–737. • R. C. Bose and D.K. Ray-Chaudhuri On a class of error-correcting binary codes, Information and control, 3, (1960), 68–79. Autobiography • J. Gani (ed) (1982) The Making of Statisticians, New York: Springer-Verlag. This has a chapter in which Bose tells the story of his life. Discussions • Norman R. Draper (1990) Obituary: Raj Chandra Bose, Journal of the Royal Statistical Society Series A, Vol. 153, No. 1. pp. 98–99. • "Bose, Raj Chandra", pp. 183–184 in Leading Personalities in Statistical Sciences from the Seventeenth Century to the Present, (ed. N. L. Johnson and S. Kotz) 1997. New York: Wiley. Originally p See also • Association scheme • Block design • Bose–Mesner algebra • Combinatorial design • Design of experiments External links • R. C. Bose:another photograph on the Portraits of Statisticians page. • Indian Statistical Institute: useful background information • Raj Chandra Bose at the Mathematics Genealogy Project • Peter Cameron's Quotes on Mathematics: where the story about fields comes from • Weisstein, Eric W. "Euler's Graeco-Roman Squares Conjecture". MathWorld. • O'Connor, John J.; Robertson, Edmund F., "Raj Chandra Bose", MacTutor History of Mathematics Archive, University of St Andrews Authority control International • FAST • ISNI • VIAF National • Catalonia • Germany • Israel • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • SNAC • IdRef	Wikipedia
Rajchman measure In mathematics, a Rajchman measure, studied by Rajchman (1928), is a regular Borel measure on a locally compact group such as the circle, whose Fourier transform vanishes at infinity. References • Lyons, Russell (1995), "Seventy years of Rajchman measures" (PDF), Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), The Journal of Fourier Analysis and Applications: 363–377, ISSN 1069-5869, MR 1364897 • Rajchman, A. (1928), "Sur une classe de fonctions à variation bornée", C. R. Acad. Sci. Paris, 187: 1026–1028	Wikipedia
Rajeev Motwani Rajeev Motwani (Hindi: राजीव मोटवानी , 24 March 1962 – 5 June 2009) was an Indian American professor of Computer Science at Stanford University whose research focused on theoretical computer science. He was a special advisor to Sequoia Capital. He was a winner of the Gödel Prize in 2001.[2][3][4] Rajeev Motwani Rajeev Motwani in 2006 Born Rajeev Motwani (1962-03-24)24 March 1962 Jammu, J&K, India Died5 June 2009(2009-06-05) (aged 47) Atherton, California, United States NationalityAmerican CitizenshipAmerican EducationSt. Columba's School, Delhi Alma materIIT Kanpur (B.Tech C.S.) UC Berkeley (Ph.D. C.S.) SpouseAsha Jadeja Motwani AwardsGödel Prize Scientific career Fieldstheoretical computer science data privacy web search robotics computational drug design ThesisProbabilistic Analysis of Matching and network flow Algorithms (1988) Doctoral advisorRichard M. Karp[1] Doctoral students • Moses Charikar • Piotr Indyk • David Karger • Sanjeev Khanna • Suresh Venkatasubramanian Websitetheory.stanford.edu/~rajeev Education Rajeev Motwani was born in Jammu, Jammu and Kashmir, India on 24 March 1962, and grew up in New Delhi.[5] His father was in the Indian Army. He had two brothers. As a child, inspired by luminaries like Gauss, he wanted to become a mathematician. Motwani went to St Columba's School, New Delhi. He completed his B.Tech. in Computer Science from the Indian Institute of Technology Kanpur in Kanpur, Uttar Pradesh in 1983 and got his Ph.D. in Computer Science from the University of California, Berkeley in Berkeley, California, United States in 1988, under the supervision of Richard M. Karp.[1] Career Motwani joined Stanford soon after U.C. Berkeley. He founded the Mining Data at Stanford project (MIDAS), an umbrella organization for several groups looking into new and innovative data management concepts. His research included data privacy, web search, robotics, and computational drug design. He is also one of the originators of the Locality-sensitive hashing algorithm. Motwani was one of the co-authors (with Larry Page and Sergey Brin, and Terry Winograd) of an influential early paper on the PageRank algorithm. He also co-authored another seminal search paper What Can You Do With A Web In Your Pocket with those same authors.[6] PageRank was the basis for search techniques of Google (founded by Page and Brin), and Motwani advised or taught many of Google's developers and researchers,[7] including the first employee, Craig Silverstein.[8] He was an author of two widely used theoretical computer science textbooks: Randomized Algorithms with Prabhakar Raghavan[9] and Introduction to Automata Theory, Languages, and Computation with John Hopcroft and Jeffrey Ullman.[10] He was an avid angel investor and helped fund a number of startups to emerge from Stanford. He sat on boards including Google, Kaboodle, Mimosa Systems (acquired by Iron Mountain Incorporated), Adchemy, Baynote, Vuclip, NeoPath Networks (acquired by Cisco Systems in 2007), Tapulous and Stanford Student Enterprises. He was active in the Business Association of Stanford Entrepreneurial Students (BASES).[11][12][13] He was a winner of the Gödel Prize in 2001 for his work on the PCP theorem and its applications to hardness of approximation.[14][15] He served on the editorial boards of SIAM Journal on Computing, Journal of Computer and System Sciences, ACM Transactions on Knowledge Discovery from Data, and IEEE Transactions on Knowledge and Data Engineering. Death Motwani was found dead in his pool in the backyard of his Atherton, San Mateo County, California home on 5 June 2009. The San Mateo County coroner, Robert Foucrault, ruled the death an accidental drowning. Toxicology tests showed that Motwani's blood alcohol content was 0.26 percent.[16] He could not swim, but was planning on taking lessons, according to his friends.[17] Personal life Motwani, and his wife Asha Jadeja Motwani, had two daughters named Naitri and Anya.[18] After his death, his family donated US$1.5 million in 2011 and a building was named in his honor at IIT Kanpur.[19] Awards • Gödel Prize in 2001 • Okawa Foundation Research Award[20] • Arthur Sloan Research Fellowship[20] • National Young Investigator Award from the National Science Foundation • Distinguished Alumnus Award from IIT Kanpur in 2006[19] • Bergmann Memorial Award from the US-Israel Bi-National Science Foundation • IBM Faculty Award References 1. Rajeev Motwani at the Mathematics Genealogy Project 2. Rajeev Motwani at DBLP Bibliography Server 3. Rajeev Motwani author profile page at the ACM Digital Library 4. Raghavan, Prabhakar (2012). "Rajeev Motwani (1962-2009)" (PDF). Theory of Computing. 8: 55–57. doi:10.4086/toc.2012.v008a003. 5. Rajeev Motwani, computer scientist at Stanford; adviser, investor in Silicon Valley, dead at 47 6. Brin, Sergey; Motwani, Rajeev; Page, Lawrence; Winograd, Terry (1998). "What can you do with a Web in your Pocket?". IEEE Data Engineering Bulletin. 21 (2): 37–47. Archived from the original on 10 June 2009. 7. Alfred Spector, VP of Research (8 June 2009). "Remembering Rajeev Motwani". Retrieved 11 September 2013. 8. "Craig Silverstein's website". Stanford University. Archived from the original on 2 October 1999. Retrieved 12 October 2010. 9. Raghavan, Prabhakar; Motwani, Rajeev (1995). Randomized algorithms. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-47465-8. 10. Ullman, Jeffrey D.; Hopcroft, John E.; Motwani, Rajeev (2007). Introduction to automata theory, languages, and computation. Boston: Pearson/Addison Wesley. ISBN 978-0-321-45536-9. 11. NeoPath Networks Locks Up $6M Equity Financing; August Capital and DCM-Doll Capital Management Lead the Investment 2004-03-08 12. "Cisco kisses NeoPath products goodbye" Archived 2009-06-10 at the Wayback Machine by Deni Connor, Network World, 2007-04-04. Retrieved 2009-06-06. 13. Rajeev Motwani, Google founders’ professor and early investor, dies 2009-06-05 14. 2001 Gödel Prize citation 15. Arora, S.; Lund, C.; Motwani, R.; Sudan, M.; Szegedy, M. (1998). "Proof verification and the hardness of approximation problems". Journal of the ACM. 45 (3): 501–555. doi:10.1145/278298.278306. S2CID 8561542. 16. Lee, Henry K. (16 July 2009). "Stanford tech mentor was drunk when he drowned". San Francisco Chronicle. Hearst Communications, Inc. pp. D–4. Retrieved 17 July 2009. 17. Weaver, Matthew (7 June 2009). "Google founders' mentor found dead in swimming pool". guardian.co.uk. Guardian News and Media Limited. 18. Google mentor Rajeev Motwani dies in freak accident Archived 2009-06-10 at the Wayback Machine 2009-06-07 19. "The Rajeev Motwani Building: Department of Computer Science and Engineering". Archived from the original on 22 May 2013. Retrieved 11 September 2013. 20. "Rajeev Motwani passes away". Thaindian. 6 June 2009. Retrieved 11 September 2013. External links • Mathematician at heart • Professor Rajeev Motwani at The Telegraph Gödel Prize laureates 1990s • Babai / Goldwasser / Micali / Moran / Rackoff (1993) • Håstad (1994) • Immerman / Szelepcsényi (1995) • Jerrum / Sinclair (1996) • Halpern / Moses (1997) • Toda (1998) • Shor (1999) 2000s • Vardi / Wolper (2000) • Arora / Feige / Goldwasser / Lund / Lovász / Motwani / Safra / Sudan / Szegedy (2001) • Sénizergues (2002) • Freund / Schapire (2003) • Herlihy / Saks / Shavit / Zaharoglou (2004) • Alon / Matias / Szegedy (2005) • Agrawal / Kayal / Saxena (2006) • Razborov / Rudich (2007) • Teng / Spielman (2008) • Reingold / Vadhan / Wigderson (2009) 2010s • Arora / Mitchell (2010) • Håstad (2011) • Koutsoupias / Papadimitriou / Roughgarden / É. Tardos / Nisan / Ronen (2012) • Boneh / Franklin / Joux (2013) • Fagin / Lotem / Naor (2014) • Spielman / Teng (2015) • Brookes / O'Hearn (2016) • Dwork / McSherry / Nissim / Smith (2017) • Regev (2018) • Dinur (2019) 2020s • Moser / G. Tardos (2020) • Bulatov / Cai / Chen / Dyer / Richerby (2021) • Brakerski / Gentry / Vaikuntanathan (2022) Authority control International • ISNI • VIAF National • Norway • Germany • Israel • United States • Japan • Czech Republic • Netherlands • Poland Academics • Association for Computing Machinery • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef scientists]]	Wikipedia
Rake (angle) A rake is an angle of slope[1] measured from horizontal, or in some contexts from a vertical line 90° perpendicular to horizontal. A 60° rake would mean that the line is pointing 60° up from horizontal, either forwards or backwards relative to the object. Usage Though the term may be used in a general manner, it is commonly applied in several specific contexts. The rake of a ship's prow is the angle at which the prow rises from the water (the rake below water being called the bow rake). A motorcycle or bicycle fork rake is the angle at which the forks are angled down towards the ground. See also caster angle, which is the angular displacement of the steering axis from the vertical axis of a steered wheel. In machining and sawing the rake angle is the angle from the cutting head to the object being worked on (with a perpendicular angle conventionally being a 0° rake). In geology the rake is the angle at which one rock moves against another in a geological fault. In a theatre or opera house the stage can be raked to slope up towards the back of the stage to allow better viewing for the audience. See also • Pitch angle, one of the angular degrees of freedom of any stiff body (for example a vehicle), describing rotation about the side-to-side axis References 1. "rake(3)". Oxford Dictionaries. Retrieved 22 February 2011.	Wikipedia
Raking Raking (also called "raking ratio estimation" or "iterative proportional fitting") is the statistical process of adjusting data sample weights of a contingency table to match desired marginal totals.[1][2][3] References 1. "1. How different weighting methods work". 26 January 2018. 2. Kalton, Graham; Flores-Cervantes, Ismael (2003). "Weighting Methods" (PDF). Journal of Official Statistics. 19 (2): 81–97. 3. Battaglia, Michael; Izrael, David (2009). "Practical Considerations in Raking Survey Data". Survey Practice. 2 (5): 1–10. doi:10.29115/SP-2009-0019.	Wikipedia
Ralph Greenberg Ralph Greenberg (born 1944) is an American mathematician who has made contributions to number theory, in particular Iwasawa theory. Ralph Greenberg Born1944 (age 78–79) Chester, Pennsylvania NationalityAmerican Alma materUniversity of Pennsylvania Princeton University Scientific career FieldsMathematics InstitutionsUniversity of Washington Doctoral advisorKenkichi Iwasawa He was born in Chester, Pennsylvania[1] and studied at the University of Pennsylvania, earning a B.A. in 1966,[1] after which he attended Princeton University, earning his doctorate in 1971 under the supervision of Kenkichi Iwasawa.[2] Greenberg's results include a proof (joint with Glenn Stevens) of the Mazur–Tate–Teitelbaum conjecture as well as a formula for the derivative of a p-adic Dirichlet L-function at $s=0$ (joint with Bruce Ferrero). Greenberg is also well known for his many conjectures. In his PhD thesis, he conjectured that the Iwasawa μ- and λ-invariants of the cyclotomic $\mathbb {Z} _{p}$-extension of a totally real field are zero, a conjecture that remains open as of September 2012. In the 1980s, he introduced the notion of a Selmer group for a p-adic Galois representation and generalized the "main conjectures" of Iwasawa and Barry Mazur to this setting. He has since generalized this setup to present Iwasawa theory as the theory of p-adic deformations of motives. He also provided an arithmetic theory of L-invariants generalizing his aforementioned work with Stevens. Greenberg was an invited speaker in International Congress of Mathematicians 2010, Hyderabad on the topic of "Number Theory."[3] In 2012, he became a fellow of the American Mathematical Society.[4] In the late 1990s and early 2000s, Greenberg publicly disputed NASA conspiracy theorist and pseudoscientist Richard C. Hoagland's mathematical interpretations of the so-called "D&M Pyramid" and surrounding features found on the Cydonia Planitia region of Mars as being conclusive signs of extraterrestrial intelligence and challenged him to a public debate. Hoagland has yet to respond.[5] References 1. "The Institute for Advanced Studies, Annual Report 1981/82" (PDF). Institute for Advanced Study. Retrieved January 9, 2020. 2. "Curriculum Vita". 3. "ICM Plenary and Invited Speakers since 1897". International Congress of Mathematicians. 4. "List of Fellows of the American Mathematical Society". Retrieved 2013-01-19. 5. "Letter to Richard Hoagland". External links • Ralph Greenberg at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Catalonia • Germany • Israel • United States Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • SNAC • IdRef	Wikipedia
Ralph Kaufmann Ralph Martin Kaufmann (born August 4, 1969) is a German mathematician working in the United States. Ralph Kaufmann Kaufmann at Oberwolfach, 2010 Born (1969-08-04) August 4, 1969 Bendorf, Rheinland-Pfalz, Germany NationalityGerman Alma materUniversity of Bonn Scientific career FieldsMathematics, Physics, Philosophy InstitutionsMax Planck Institute for Mathematics, Institut des Hautes Études Scientifiques, Purdue University Doctoral advisorYuri Manin Career Kaufmann studied mathematics, physics and philosophy at the University of Bonn. He obtained a master's degree in Physics in 1994 under the supervision of Werner Nahm and a master's degree in philosophy under the supervision of Rainer Stuhlmann-Laeisz in 1996. His doctoral studies were carried out at the Max Planck Institute for Mathematics under the supervision of Yuri Manin, and he graduated summa cum laude from University of Bonn in 1997 with a thesis entitled "The geometry of the moduli space of pointed curves, the tensor product in the theory of Frobenius manifolds and the explicit Künneth formula in quantum cohomology".[1] He remained at the Max Planck Institute for one year after his graduation as a researcher before moving to the Institut des Hautes Études Scientifiques for a year with a Marie Curie Fellowship from the European Union. In 1999 Kaufmann moved to the United States, where he has held several positions. He arrived at his current institution, Purdue University, in 2007 as an associate professor, being promoted to full professor in 2012. Kaufmann has been a member of the Institute for Advanced Study in Princeton,[2] and has held visiting positions at the Max Planck Institute for Mathematics in Bonn, the Institut des Hautes Études Scientifiques, the Mittag-Leffler Institute in Stockholm, the Isaac Newton Institute in Cambridge, and the Mathematical Sciences Research Institute in Berkeley, California.[3] Research Kaufmann's research has covered different areas of mathematics and theoretical physics. After briefly working on the Virasoro algebra,[4] he started to work on the quantum cohomology of a product[5] and the Künneth formula, obtaining results on explicit formulas[6] and global results.[7] In collaboration with Yuri Manin and Don Zagier he started the study of higher Weil–Peterson volumes,[8] later continued by Losef–Manin and Maryam Mirzakhani. Kaufmann then started to study stringy and mirror phenomena for orbifolds[9] and singularities.[10] This led to the invention of stringy K-theory.[11][12] Kaufmann has also worked on string topology, invented by Moira Chas and Dennis Sullivan, and operad theory.[13][14][15][16][17] Here he first proved a cyclic version of Pierre Deligne's conjecture in deformation theory[18] and provided an extension of string topology to moduli spaces.[19][20][21][22] More recently Kaufmann has introduced the notion of Feynman categories[23] to give a common framework for various aspects of algebra, geometry, topology, and category theory.[24][25][26] In mathematical physics, he has also studied the geometry of wire networks[27] as well as periodic systems[28][29] and topological insulators.[30] In philosophy, he has recently worked on Hegel's theory of mathematics[31][32] and on Friedrich Hölderlin.[33] As of January 2018, eight graduate students have obtained their PhD under Kaufmann's supervision.[34] Honors and awards • Heinrich Hörlein Gedächtnis Preis for his dissertation. • Simons Fellow[35] Publications (books authored) • Kaufmann, Ralph M.; Ward, Benjamin C. (2017). Feynman categories. Paris. ISBN 978-2-85629-852-7. OCLC 990103099.{{cite book}}: CS1 maint: location missing publisher (link) • Pastior, Oskar; Kaufmann, Ralph; Egger, Oswald (2006). Gewichtete Gedichte : Chronologie der Materialien (in German). Wien-Hombroich: Das böhmische Dorf. ISBN 3-902024-08-9. OCLC 69983395. References 1. Bonner Mathematische Schriften 312, 95 p., Bonn 1997. 2. "Ralph Martin Kaufmann". Institute for Advanced Study. Retrieved 2018-01-03. 3. "Ralph M. Kaufmann". Purdue University, Department of Mathematics. Retrieved 2018-01-03. 4. Kaufmann, Ralph M. "Path Space Decompositions for the Virasoro Algebra and its Verma Modules". Internat. J. of Modern Phys. A 10 (1995), 943-961. 5. Maxim Kontsevich and Yuri Manin with appendix by R. Kaufmann. "Quantum cohomology of a product". Invent. Math. 124 (1996), 313-339. 6. Kaufmann, Ralph M. "The intersection form in H*(M0,n) and the explicit Künneth formula in quantum cohomology". Internat. Math. Res. Notices 19 (1996), 929-954. 7. Kaufmann, Ralph M. "The tensor product in the theory of Frobenius manifolds". Internat. J. of Math. 10 (1999), 159-206 8. Kaufmann, R.; Manin, Yu.; and Zagier, D. "Higher Weil-Petersson Volumes of Moduli Spaces of Stable n-pointed Curves". Comm. Math. Phys. 181 (1996), 763-787. 9. Kaufmann, Ralph M. "Orbifolding Frobenius Algebras". Internat. J. of Math. 14 (2003), 573-619 10. Kaufmann, Ralph M. "Singularities with Symmetries, Orbifold Frobenius algebras and Mirror Symmetry" Contemp. Math., 403 (2006), 67-116 11. Jarvis, T.; Kaufmann, R. and Kimura, T. "Stringy K-theory and the Chern character". Inventiones Math. 168, 1 (2007), 23-81 12. Kaufmann, Ralph M. "Global Stringy Orbifold Cohomology, K-Theory and de Rham Theory". Letters in Mathematical Physics, 94, 2 (2010) 165-195. 13. Kaufmann, Ralph M.; Livernet, Muriel and Penner, Robert C. "Arc Operads and Arc Algebras". Geometry and Topology 7 (2003), 511-568. 14. Kaufmann, Ralph M (15 April 2005). "On several varieties of cacti and their relations". Algebraic & Geometric Topology. Mathematical Sciences Publishers. 5 (1): 237–300. doi:10.2140/agt.2005.5.237. ISSN 1472-2739. S2CID 2067462. 15. Kaufmann, Ralph M. "On Spineless Cacti, Deligne's Conjecture and Connes-Kreimer's Hopf Algebra." Topology 46, 1 (2007), 39-88. 16. Kaufmann, Ralph M. "Operads, Moduli of Surfaces and Quantum Algebras", in: N. Tongring and R. C. Penner "Woods Hole Mathematics. Perspectives in Mathematics and Physics", Series on Knots and Everything - Vol. 34, World Scientific 2004. 17. Kaufmann, Ralph M.; Schwell, Rachel (2010). "Associahedra, cyclohedra and a topological solution to the $A_{\infty }$-Deligne conjecture". Advances in Mathematics. 223 (6): 2166–2199. doi:10.1016/j.aim.2009.11.001. 18. Kaufmann, Ralph M. "A proof of a cyclic version of Deligne's conjecture via Cacti" Math. Res. Letters 15, 5 (2008), 901--921. 19. Kaufmann, Ralph M. "Moduli space actions on the Hochschild cochain complex I: cell models". Journal of Noncommutative Geometry 1, 3 (2007) 333-384., Kaufmann, Ralph M. "Moduli space actions on the Hochschild cochain complex II: correlators". Journal of Noncommutative Geometry 2, 3 (2008), 283-332. 20. Kaufmann, Ralph M. "Open/Closed String Topology and Moduli Space Actions via Open/Closed Hochschild Actions". SIGMA 6 (2010) 036, 33 pages. Kaufmann, Ralph M. 21. "Graphs, strings and actions" in: Algebra, Arithmetic and Geometry Volume II: In Honor of Yu. I. Manin. Progress in Mathematics 270, 127--178. Birhauser, Boston (2010). 22. Kaufmann, Ralph M. "Arc Geometry and Algebra: Foliations, Moduli Spaces, String Topology and Field Theory". Chapter in Handbook of Teichmüller Theory IV. Ed: Athanase Papadopoulos. European Mathematical Society 2014. 23. Kaufmann, Ralph M. and Ward, Benjamin C. "Feynman categories". Astérisque 387 (2017), x+161 pages. 24. Kaufmann, Ralph M., Ward, Benjamin C. and Zuniga, J. Javier "The odd origin of Gerstenhaber brackets, Batalin-Vilkovisky operators and the master equations". Journal of Math. Phys. 56 (2015), 103504. 25. Kaufmann, Ralph M., Lucas, Jason. "Decorated Feynman categories". arXiv:1602.00823 J. of Noncommutative Geometry to appear. 26. Berger, Clemens and Kaufmann, Ralph M. "Comprehensive Factorization Systems". arXiv:1710.09438. Special Issue of Tbilisi Mathematical Journal in honor of Peter J. Freyd and F.W. Lawvere on the occasion of their 80th birthdays to appear 27. Kaufmann, Ralph M., Khlebnikov, Sergei and Wehefritz-Kaufmann, Birgit. "The geometry of the Double Gyroid wire network: Quantum and Classical". Journal of Noncommutative Geometry, 6 (2012), 623-664. 28. Kaufmann, Ralph M., Khlebnikov, Sergei and Wehefritz-Kaufmann, Birgit. "Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid". Annals of Physics, 327 (2012), 2865-2884. 29. Kaufmann, Ralph M., Khlebnikov, Sergei and Wehefritz-Kaufmann, Birgit. "Re-gauging groupoid, symmetries and degeneracies for Graph Hamiltonians and applications to the Gyroid wire network". Ann. H. Poincaré 7, 6 (2016), 1383-1414. 30. Kaufmann, Ralph M., Li, Dan, and Wehefritz-Kaufmann, Birgit. "Notes on topological insulators". Reviews in Math. Physics, Vol. 28, No. 10 (2016) 1630003 31. Kaufmann, Ralph M. and Yeomans, Christopher L. "Math by Pure Thinking: R First and the Divergence Of Measures in Hegel's Philosophy of Mathematics". European Journal of Philosophy, ahead of print. 32. Kaufmann, Ralph M. and Yeomans, Christopher L. "Hegel on Calculus". History of Philosophy Quarterly, to appear. 33. Kaufmann, Ralph M. Der Dichter spricht: Eine Rezeption Hölderlins 'Verfahrensweise des Poetischen Geistes'", Zeitschrift für Kulturphilosophie, 2017/1 "Sprache und Gestalt", Meiner Verlag, Hamburg. 34. Ralph Kaufmann at the Mathematics Genealogy Project 35. "Simons Fellows Awardees: Mathematics". Simons Foundation. Retrieved 2018-01-03. External links • Homepage at Purdue • Papers on the eprints ArXiv Authority control National • Germany • United States Academics • Google Scholar • MathSciNet • Mathematics Genealogy Project • ORCID • zbMATH	Wikipedia
Ralph Kenna Professor Ralph Kenna (born 27 August 1964) is an Irish mathematician and theoretical physicist who is head of the statistical physics research group at Coventry University. He is a specialist in statistical physics, complex systems and Irish mythology.[1][2] Professor Ralph Kenna Born Ralph Jude Kenna (1964-08-27)27 August 1964 Athlone, Ireland NationalityIreland CitizenshipIreland Alma mater • University of Graz[3] • Trinity College Dublin[4] Scientific career Fields • Statistical physics • complex systems • Comparative mythology Institutions • Coventry University • Trinity College Dublin • University of Liverpool • University of Graz Early life and education Kenna was born in Athlone, on the border between counties Roscommon and Westmeath, Ireland, on 27 August 1964.[5] He attended Marist College, Athlone (best leaving certificate, class of 1981) and obtained his B.A. (mod) degree in Theoretical Physics from Trinity College Dublin in 1985. He also obtained his M.Sc. (1988) from Trinity College Dublin. He completed his PhD (Dr. rer. nat.) at the University of Graz[3] under Professor Christian Lang in 1993.[6] Career Kenna was an EU Marie Curie Research Fellow at the University of Liverpool from 1994 to 1997 and at Trinity College Dublin from 1997 to 1999. He lectured at Trinity from 1998 to 2002, when he moved to Coventry University. In 2005, he co-founded the Applied Mathematics Research Centre at Coventry and founded the Statistical Physics Group there. In 2018 these joined the Fluid and Complex Systems Research Centre, Coventry University, of which Kenna is Deputy Director.[7] Research Kenna's research interests relate to field theory, statistical physics (especially phase transitions and critical phenomena) and complex systems (especially applied to Irish mythology and other epic narratives). Statistical physics In statistical physics, Kenna is noted for his development of scaling relations for logarithmic corrections.[8][9] Already in his PhD Thesis[10] he introduced a renormalization group basis for finite-size scaling (FSS) for logarithmic corrections at the upper critical dimension and, with Bertrand Berche, he extended this to higher dimensions in 2012.[11] They proposed that universality lives at the pseudocritical point instead of at the critical point and a new form for hyperscaling, valid in high dimensions.[11] This led to the introduction of the new critical exponent ϙ (koppa) and its logarithmic counterpart ϙ̂ (ϙ-hat or koppa-hat)[12] to govern the finite-size dependence of the correlation length and a new form for FSS, called QFSS,[13] to replace standard prescription above the upper critical dimension. Fourier analysis showed two regimes that control finite systems in high dimensions instead of Landau scaling.[14] The Q sector is affected by dangerous irrelevant variables and the Gaussian, or G, sector (G) is not but both are physical. The pseudocritical point resides in the Q sector while the critical point itself may be either Q or G.[15] Formal similarities between spin systems on lattices and on scale-free networks mean an analogy between dimensionality and power-law decay of the node degree distribution there. Logarithmic corrections at critical values[16][17] also obey the scaling relations developed by Kenna in 2006.[8][9] Kenna's work on percolation theory in high dimensions and spin models on annealed scale-free networks has featured in the Institute of Physics News.[18][19] Complex Systems In 2010, with Bertrand Berche, Kenna quantified the notion of critical mass of academic research groups.[20] Using data from the UK's Research Assessment Exercise 2008 and the French counterpart (AERES) they tracked how research group quality depends on the size of the group.[21] [22][23] They found quality rises linearly with group size up to a point which they later identified as akin to the Dunbar number in anthropology. Critical mass, defined as the minimum size a group needs to achieve to be sustainable, is half that size.[24] Subsequently, with Olesya Mryglod and Yurij Holovatch, Kenna and Berche used scientometrics to predict the outcome of the UK's Research Excellence Framework 2014.[25][26] They found that correlations between metrics and peer review are poor and the former cannot reliably be used to replace the latter.[27] This went some way to halting the overuse of metrics at the Research Excellence Framework 2021.[28][29][30][31] Comparative Mythology In comparative mythology Kenna is noted for pioneering the usage of complex networks in the study of Irish and other mythologies. His first paper[32] on the topic was downloaded over 30,000 times in 10 years, a record [33] for Europe's flagship letters journal in physics, and resulted in considerable media coverage in international press.[34][35][36][37][38][39][40] Other major works include investigations into the Sagas of Icelanders.[41]Kenna's team found that whether the sagas are historically accurate or not, the properties of the social worlds they record are similar to those of real social networks.[42][43][44][45][46][47] The epic poems of Ossian[48] were the focus of the next subject of study with conclusions broadly in line with the view they were misappropriated from Irish sources. [49][50] The Viking Age in Ireland as portrayed in Cogadh Gaedhel re Gallaibh was next tackled by Kenna's team.[51] They developed a measure to place hostility on a spectrum between civil war and international conflict. Their findings[52][53][54][55][56][57] quantified and supported the traditional view of the Viking age in Ireland as one of international conflict and challenged recent revisionist claims. A study of the character Fraoch identified quantifiable differences between the two parts of his story, supporting the suggestion it was set to writing by two different scribes, one of whom embellished the tale.[58] Kenna and co-workers also studied Ukrainian mythology. They compared the Kyiv bylyny cycle to other prominent European epics to identify universal and distinguishing properties of its social networks.[59] Kenna's team developed mathematical and statistical methods to probe how a modern complex narrative - namely George R. R. Martin's epic novels, A Song of Ice and Fire - achieved broad acclaim without surrendering to the need for reductionist simplifications.[60] This and other works on narratology led to sustained media interest.[61][62][63][64] Awards, grants and honours Kenna was awarded a Council of Europe scholarship to study in Austria where his PhD was then fully funded by a grant from the Austrian Science Fund (1990-1993). He twice held EU Marie Curie Research Fellowships (1994–1997 at the University of Liverpool and in 1997 at Trinity College Dublin). He also held an Enterprise Ireland Fellowship while in Trinity (1997-1999). In 2004 Kenna became Member of the Institute of Physics and in 2011 he was elected Fellow of the Institute of Mathematics and its Applications. He held a number of grants throughout the years, including from the Leverhulme Trust[65] (2010-2013) and European Science Foundation[66] (2014), to support his work on comparative mythology. Kenna held or supported a number of Marie Skłodowska-Curie Actions including an FP7 International Research Staff Exchange Scheme [67][68] with Christian von Ferber (2012-2016; 2014-2018) and an FP7-Marie Curie International Incoming Fellowship[69] for Dr Nikolay Izmailyan to visit him in Coventry (2013-2015). Kenna has been supported by Armenian State Committee of Science Thematic Grants to collaborate with Nerses Izmailyan (2021-2025). In 2019 Kenna was awarded Doctor honoris causa by the Academic Council of the Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine. His certificate of election reads:[70] By the decision of the Academic Council of the ICMP of the National Acad. Sci. of Ukraine of February 11, 2019 the title of Doctor honoris causa was conferred on Ralph Kenna (Coventry, England) for the studies which provided explanation of scaling and universality in complex systems of many interacting agents, pioneering interdisciplinary applications of statistical physics as well as for his personal initiative in different forms of collaboration with Ukraine and his engagement in preparation of young scientists.[70] In 2023 Kenna was selected as Member of the London Mathematical Society Research Policy Committee [71] Personal life Ralph is first child of Pat Kenna[72] and Irene Kenna (nee Benson). Ralph had an out-of-hospital cardiac arrest in 2017.[73] Despite being 30 minutes gone, he made a full recovery.[74] He became fund raiser for the air ambulance who brought him back.[75] He married Claire Dutton in 2015. Claire has an M.Sc. in medical statistics and in 2018 published two papers about out-of-hospital cardiac arrests.[76] They have one child. Selected publications • Universal scaling relations for logarithmic-correction exponents, in "Order, Disorder, and Criticality: Advanced Problems of Phase Transition Theory". World Scientific, 2012 pp 1–46. Yu. Holovatch (editor) https://doi.org/10.1142/9789814417891_0001 • A new critical exponent ϙ and its logarithmic counterpart. Condensed Matter Physics, Vol. 16, 23601 March 11, 2013. (With Bertrand Berche) http://www.icmp.lviv.ua/journal/zbirnyk.74/23601/art23601.pdf • Maths meets myths: Quantitative approaches to ancient narratives. Springer, 2016. (Editor with Máirín MacCarron & Pádraig MacCarron) ISBN 978-3-319-39445-9 • Narrative structure of A Song of Ice and Fire creates a fictional world with realistic measures of social complexity. Proceedings of the National Academy of Sciences, Vol. 117 No. 46 November 17, 2020. (With Thomas Gessey-Jones, Robin Dunbar, Pádraig MacCarron, Cathal O’Conchobhair & Joseph Yose) https://www.pnas.org/doi/full/10.1073/pnas.2006465117 References 1. Kenna, Ralph. "AMRC, Coventry University: Single view". Complexity-coventry.org. Retrieved 15 November 2016. 2. Kenna, Ralph. "Physics Tree: Single view". academictree.org. Retrieved 2 July 2023. 3. "Physics - Ralph Kenna". Physics.aps.org. Retrieved 15 November 2016. 4. "The Atlas of Irish Mathematics: Westmeath (Feb 2018)". 5. "50th birthday of Ralph Kenna" (PDF). Icmp.lviv.ua. Retrieved 15 November 2016. 6. "Mathematics genealogy- Ralph Kenna". /www.mathgenealogy.org. Retrieved 15 November 2016. 7. "Deputy Director - Ralph Kenna". Fluid and Complex Systems Research Centre, Coventry University. Retrieved 30 June 2023. 8. Kenna, R.; Johnston, D.; Janke, W. (2006). "Scaling Relations for Logarithmic Corrections". Physical Review Letters. 96 (11): 115701. arXiv:cond-mat/0605162. Bibcode:2006PhRvL..96k5701K. doi:10.1103/PhysRevLett.96.115701. PMID 16605842. S2CID 5844607. 9. Kenna, R.; Johnston, D.; Janke, W. (2006). "Self-Consistent Scaling Theory for Logarithmic-Correction Exponents". Physical Review Letters. 97 (15): 155702. arXiv:cond-mat/0608127. Bibcode:2006PhRvL..97o5702K. doi:10.1103/PhysRevLett.97.155702. PMID 17155337. S2CID 7648192. 10. Kenna, R.; Lang, C.B. (1991). "Finite-Size Scaling and the Zeroes of the Partition Function in the $\varphi _{4}^{4}$ Model". Physics Letters B. 264 (3–4): 396. Bibcode:1991PhLB..264..396K. doi:10.1016/0370-2693(91)90367-Y. 11. Berche, B.; Kenna, R.; Walter, J.-C. (2012). "Hyperscaling above the upper critical dimension". Nuclear Physics B. 865 (1): 115–132. arXiv:1402.1657. Bibcode:2012NuPhB.865..115B. doi:10.1016/j.nuclphysb.2012.07.021. S2CID 119216957. 12. Kenna, R.; Berche, B. (2013). "A new critical exponent ϙ and its logarithmic counterpart" (PDF). Condensed Matter Physics. 16: 23601. doi:10.5488/CMP.16.23601. S2CID 118152535. 13. Kenna, R.; Berche, B. (2013). "A new critical exponent ϙ and its logarithmic counterpart" (PDF). Condensed Matter Physics. 16: 23601. doi:10.5488/CMP.16.23601. S2CID 118152535. 14. Flores-Sola, E.; Berche, B.; Kenna, R.; Weigl, M. (2016). "Role of Fourier modes in finite-size scaling above the upper critical dimension". Physical Review Letters. 116 (11): 115701. arXiv:1511.04321. Bibcode:2016PhRvL.116k5701F. doi:10.1103/PhysRevLett.116.115701. PMID 27035310. S2CID 206271579. 15. R. Kenna and B. Berche, Scaling and Finite-Size Scaling above the Upper Critical Dimension, invited contribution to "Order, Disorder, and Criticality: Advanced Problems of Phase Transition Theory", Yu. Holovatch (editor). vol. 4 World Scientific, Singapore, 2015 (ISBN 978-981-4632-67-6 [hardcover)] 16. Palchykov, C.; von Ferber, C.; Folk, R.; Holovatch, Yu.; Kenna, R. (2010). "Critical phenomena on scale-free networks: logarithmic corrections and scaling functions". Physical Review E. 82 (1 Pt 1): 011145. arXiv:1004.0097. Bibcode:2010PhRvE..82a1145P. doi:10.1103/PhysRevE.82.011145. PMID 20866603. S2CID 14311802. 17. Krasnytska, M.; Berche, B.; Holovatch, Yu.; Kenna, R. (2016). "Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks". J. Phys. A. 49 (13): 135001. arXiv:1510.00534. Bibcode:2016JPhA...49m5001K. doi:10.1088/1751-8113/49/13/135001. S2CID 119280739. 18. Brown, Phil (21 July 2017). "Where physics and mathematics meet for coffee". Retrieved 8 July 2023. 19. Journal of Physics A: Mathematical and Theoretical (17 February 2016). "Interview with Mariana Krasnytska, Bertrand Berche, Ralph Kenna and Yurij Holovatch". Retrieved 8 July 2013. 20. Kenna, MR; Berche, B. (2010). "The extensive nature of group quality". EPL. 90 (5): 58002. arXiv:1004.3155. Bibcode:2010EL.....9058002K. doi:10.1209/0295-5075/90/58002. S2CID 53622739. 21. Jump, Paul (8 July 2010). "Truth in numbers: study pinpoints 'critical mass' for research success". Retrieved 6 July 2023. 22. Hendy, Shaun (20 October 2010). "Critical mass or is mass critical?". Retrieved 6 July 2023. 23. Kennna, Ralph; Berche, Bertrand. "Masse critique et Groupes de Recherche". Retrieved 6 July 2023. 24. Kenna, R.; Berche, B. (2011). "Critical mass and the dependency of research quality on group size". Scientometrics. 86 (2): 527–540. arXiv:1006.0928. doi:10.1007/s11192-010-0282-9. S2CID 37876062. 25. Mryglod, O.; Kenna, R.; Holovatch, Y.; Berche, B. (2015). "Predicting Results of the Research Excellence Framework using departmental h-Index". Scientometrics. 102 (3): 2165–2180. arXiv:1411.1996. doi:10.1007/s11192-014-1512-3. S2CID 255006888. 26. Mryglod, O.; Kenna, R.; Holovatch, Y.; Berche, B. (2015). "Predicting Results of the Research Excellence Framework using departmental h-Index - revisited". Scientometrics. 104 (3): 1013–1017. arXiv:1501.07857. doi:10.1007/s11192-015-1567-9. S2CID 255014082. 27. "The predicted results for the 2014 ref aret in". www.timeshighereducation.com. 27 November 2014. Retrieved 15 November 2016. 28. "Counting what Counts Confidently (CCC)". ref.ac.uk. 31 March 2021. Retrieved 31 March 2021.. 29. Jump, Paul (27 November 2014). "The (predicted) results for the 2014 REF are in". Retrieved 6 July 2023. 30. Jump, Paul (5 September 2015). "Can the research excellence framework run on metrics?". Retrieved 6 July 2023. 31. Burke, Maria (15 December 2014). "Can research quality be predicted by metrics?". Retrieved 6 July 2023. 32. Kenna, R.; MacCarron, P. (2012). "Universal properties of mythological networks". EPL. 99 (2): 28002. arXiv:1205.4324. Bibcode:2012EL.....9928002M. doi:10.1209/0295-5075/99/28002. S2CID 751459. 33. Mryglod, O.; Kenna, R.; Holovatch, Y. (2014). "Is your EPL attractive? Classification of publications through download statistics". Europhysics Letters. 108 (5): 50011. arXiv:1409.7889. Bibcode:2014EL....10850011M. doi:10.1209/0295-5075/108/50011. S2CID 17216146. 34. "Publications - European Physical Society (EPS)". 35. Colins, Nick (25 July 2012). "Beowulf and Iliad 'more plausible than Shakespeare'". Retrieved 6 July 2023. 36. "Gods and dragon myths 'more realistic than Harry'". 25 July 2012. Retrieved 7 July 2023. 37. Carbery, Genevieve (26 July 2012). "Jul 26 2012". Retrieved 7 July 2023. 38. Czepel, Robert (27 July 2012). "Heldenreise trifft Netzwerktheorie". Retrieved 7 July 2023. 39. Langenbach, Jürgen (31 July 2012). "Ist Beowulf klarer als Shakespeare?". Retrieved 7 July 2023. 40. "Sociální Sítě v eposech naznačují: Achilles možná Existoval". 1 August 2012. Retrieved 7 July 2023. 41. MacCarron, P.; Kenna, R. (2013). "Network analysis of the Íslendinga sögur – the Sagas of Icelanders". Eur. Phys. J. B. 86: 407. arXiv:1309.6134. doi:10.1140/epjb/e2013-40583-3. 42. "Research suggests the Vikings may have been more social than savage". 1 October 2013. Retrieved 7 July 2023. 43. Greenwood, Veronique (15 July 2014). "The Viking Facebook". Retrieved 7 July 2023. 44. Gunaratnam, Gina (9 February 2015). "Physics and society: Network analysis of the Íslendinga sögur - the Sagas of Icelanders". Retrieved 7 July 2023. 45. "Vikings may Have Been More Social than Savage". Retrieved 7 July 2023. 46. Skett, Philippa (2 June 2016). "Using network science to uncover the truth behind bloodshed, battles and conflict in the Sagas of Icelanders". Retrieved 7 July 2023. 47. "Viking sagas: Six degrees of Icelandic separation Social networks from the Viking era". 18 December 2013. Retrieved 7 July 2023. 48. Yose, J.; Kenna, R.; MacCarron, P.; Platini, T.; Tonra, T. (2016). "A Networks-Science Investigation into the Epic Poems of Ossian". Advances in Complex Systems. 19 (4n05): 1650008. arXiv:1610.00142. doi:10.1142/S0219525916500089. S2CID 19540179. 49. "Ossian poems a hoax". 21 October 2016. 50. Siggins, Lorna (22 October 2016). "Scottish bard Ossian 'fabricated' from Irish mythology". Retrieved 7 July 2023. 51. Yose, J.; Kenna, R.; MacCarron, M.; MacCarron, P. (2018). "Network Analysis of the Viking Age in Ireland as portrayed in Cogadh Gaedhel re Gallaibh". Royal Society Open Science. 5 (1): 171024. doi:10.1098/rsos.171024. PMC 5792891. PMID 29410814. 52. Enright, Seamus (27 January 2018). "Breifne king's role in battle doesn't add up". 53. "The Battle of Clontarf in 1014: an analysis by a theoretical physicist and mathematician". Politics.ie. 27 January 2018. 54. "Battle of Clontarf: It's a mathematical question". 55. "Who was actually fighting whom at Clontarf in 1014?". 24 January 2018. 56. "Network science shines new light on Battle of Clontarf". 24 January 2018. 57. O'Sullivan, Kevin (24 January 2018). "Battle of Clontarf: It's a mathematical question" (PDF). Retrieved 7 July 2023. 58. Janickyj, M.; Curley, D.; Maccarron, P.; McCarthy, M.; Yose, J.; Kenna, R. (2022). "The enigmatic Fráoch and his eponymous tale — network analysis of an early irish heroic romance". Advances in Complex Systems. 25 (5n06): 2240006. doi:10.1142/S0219525922400069. S2CID 250962477. 59. Sarkanych, P.; Fedorak, F.; Holovatch, Yu.; MacCarron, P.; Yose, J.; Kenna, R. (2023). "Network analysis of the Kyiv bylyny cycle -- east Slavic epic narratives". Advances in Complex Systems. 25 (5n06): 2240007. doi:10.1142/S0219525922400070. 60. Gessey-Jones, T.; Connaughton, C.; Dunbar, R.; Kenna, R.; MacCarron, P.; O'Conchobhair, C.; Yose, J. (2020). "Narrative structure of A Song of Ice and Fire creates a fictional world with realistic measures of social complexity". PNAS. 117: 28582–28588. doi:10.1073/pnas.2006465117. 61. "Padraig Mac Carron and Ralph Kenna: If Achilles Used Facebook..." Historynewsnetwork.org. Retrieved 15 November 2016. 62. Maths Meets Myths: Quantitative Approaches to Ancient Narratives - Ralph Kenna. Retrieved 15 November 2016. {{cite book}}: \|website= ignored (help) 63. "Physicists study Homer's Iliad and other classics for hidden truths". Science Daily. Archived from the original on 14 March 2016. 64. "The truth behind the world's greatest literary hoax". Independent.co.uk. 20 October 2016. Retrieved 15 November 2016. 65. "Grant listing". leverhulme.ac.uk. 17 September 2010. Retrieved 2 July 2023. 66. "MMM2014". complexity-coventry.org. 1 February 2012. Retrieved 2 July 2023. 67. "SPIDER". cordis.europa.eu. 1 February 2012. Retrieved 2 July 2023. 68. "Success story". ec.europa.eu. 3 May 2019. Retrieved 2 July 2023. 69. "RAVEN". cordis.europa.eu. 1 April 2013. Retrieved 2 July 2023. 70. "Ralph Kenna \| Institute for Condensed Matter Physics". www.icmp.lviv.ua. Retrieved 23 June 2023. 71. https://www.lms.ac.uk/about/committees/research-policy-committee. 72. "Patrick Kenna: Death Notice". RIP.IE. 7 November 2009. Retrieved 2 July 2023. 73. Ralph Kenna (12 August 2017). "Ralph Kenna: Ralph's story". Web homepage. Air Ambulance. Retrieved 2 July 2023. 74. Ralph Kenna (12 August 2017). "Ralph Kenna: Second chance of life". Web homepage. Air Ambulance. Retrieved 2 July 2023. 75. Ralph Kenna (12 August 2017). "Ralph Kenna: Fund raising". Web homepage. Rugby Observer. Retrieved 2 July 2023. 76. "Claire Kenna: Cardiac arrest paper". pubmed.ncbi.nlm.nih.gov. 7 November 2009. Retrieved 2 July 2023. Authority control International • VIAF National • Germany Academics • DBLP • Google Scholar • ORCID • Scopus • zbMATH Other • IdRef	Wikipedia
Ralph Lent Jeffery Ralph Lent Jeffery (3 October 1889 Overton, Yarmouth County, Nova Scotia, Canada – 1975 Wolfville, Nova Scotia) was a Canadian mathematician working on analysis. He taught at several institutions including Acadia University, the University of Saskatchewan and Queen's University. Jeffery Hall at Queen's was named for him. In 1937 he was elected a Fellow of the Royal Society of Canada. In 1925 Jeffery exhibited a bounded function of two real variables, continuous in each, yet fails to have an integral. In 1951 Jeffery published Theory of Functions of a Real Variable which was noted for its coverage of integration theory.[1][2] Selected papers • 1925: "Definite integrals containing a parameter", Annals of Mathematics 26(3): 173 to 80 doi:10.2307/1967895 • 1926: "Functions of two variables for which the double integral does not exist", American Mathematical Monthly 33(3): 142,3 • 1931: "The uniform approximation of a sequence of integrals", American Journal of Mathematics 53(1): 61 to 71 • 1933: "Sets of k-extent in n-dimensional space", Transactions of the American Mathematical Society 35(3): 629 to 47 doi:10.2307/1989852 References 1. A Rosenthal (1952) "Review: Functions of a Real Variable", Bulletin of the American Mathematical Society 58(4): 517–9 via Project Euclid 2. W.W. Rogosinski (1952) "Review: Functions of a Real Variable, The Mathematical Gazette doi:10.2307/3608234 • O'Connor, John J.; Robertson, Edmund F., "Ralph Lent Jeffery", MacTutor History of Mathematics Archive, University of St Andrews • Ralph Lent Jeffery at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Norway • Germany • Israel • United States • Australia • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Ralph Louis Cohen Ralph Louis Cohen (born 1952) is an American mathematician, specializing in algebraic topology and differential topology.[1] Ralph Louis Cohen OccupationMathematician, university teacher, academic Employer • Stanford University Awards • Fellow of the American Mathematical Society (2012, 2013) • Continuing Bass University Fellow in Undergraduate Education (2005) Websitehttps://profiles.stanford.edu/ralph-cohen Career Cohen received his bachelor's degree from the University of Michigan in 1973 and his Ph.D. in 1978 from Brandeis University where he worked under the supervision of Edgar H. Brown, Jr. His thesis was titled On Odd Primary Stable Homotopy Theory. He did his postdoctoral training as an L.E. Dickson Instructor at the University Chicago, and then became an Assistant Professor of Mathematics at Stanford University in 1980. In 1983, he became an Associate Professor and was promoted to Full Professor in 1987. Cohen is now the Barbara Kimball Browning Professor of Mathematics at Stanford. He was Chair of the Mathematics Department from 1992 to 1995, from 1999 to 2009 he was the Director of the Mathematics Research Center at Stanford, and from 2010 to 2016 was the Senior Associate Dean for the Natural Sciences in the School of Humanities and Sciences. Cohen has been a visiting professor at Princeton University, the University of Oxford, the University of Cambridge, Paris Diderot University, Paris 13 University, the University of Lille, and the University of Copenhagen. He was a founding editor of both the Journal of Topology and Geometry & Topology. Cohen has been the Ph.D. supervisor to over 30 doctoral students, including Ulrike Tillmann and Ernesto Lupercio. Research In 1985, Cohen proved the Immersion Conjecture, which says that each smooth, compact n-manifold has an immersion in Euclidean space of dimension $2n-\alpha (n)$, where $\alpha (n)$ is the number of ones in the binary expansion of $n$. In 1991, Cohen, together with Frederick Cohen, Benjamin Mann, and R. James Milgram gave a complete description of the algebraic topology of the space of rational functions, and in the following years he made several contributions to the study of related moduli spaces. In 1995 Cohen, John D. S. Jones, and Graeme Segal introduced an approach for understanding the homotopy theory underlying Floer homology theory in Symplectic geometry. Since 2002 Cohen has been one of the developers and contributors to the theory of String topology, which was introduced originally by Moira Chas and Dennis Sullivan. In 1995, Cohen was a founder of the Stanford University Math Camp (SUMaC), a summer camp for mathematically talented high school students. In 2002 Cohen received the Distinguished Teaching Award from Stanford University, and in 2005 he became a Bass Fellow in Undergraduate Education at Stanford. Awards In 1982 Cohen was a Sloan Research Fellow. In 1983 he was an invited speaker at the International Congress of Mathematicians in Warsaw. In 1984 he received the Presidential Young Investigator Award. In 1988 he received an NSF International Award, in 2010 he served on the Executive Committee of the American Mathematical Society, and in 2012 he was elected a Fellow of the American Mathematical Society. Selected publications • Cohen, Ralph L. (1985). "The immersion conjecture for differentiable manifolds". Annals of Mathematics. 122 (2): 237–328. doi:10.2307/1971304. JSTOR 1971304. MR 0808220. • Cohen, Frederick R.; Cohen, Ralph L.; Mann, Benjamin M.; Milgram, R. James (1991). "The topology of rational functions and divisors of surfaces". Acta Mathematica. 166 (3–4): 163–221. doi:10.1007/bf02398886. MR 1097023. • Cohen, Ralph L.; Jones, John D. S.; Segal, Graeme (1995), "Floer's infinite dimensional Morse theory and homotopy theory", The Floer Memorial Volume, Progress in Mathematics, vol. 133, Birkhäuser Verlag, pp. 297–325, doi:10.1007/978-3-0348-9217-9_13, MR 1362832 • Cohen, Ralph L.; Lupercio, Ernesto; Segal, Graeme B. (1999). "Holomorphic spheres in loop groups and Bott periodicity". Asian Journal of Mathematics. 3 (4): 801–818. doi:10.4310/ajm.1999.v3.n4.a5. MR 1797579. • Cohen, Ralph L.; Jones, John D.S. (2002). "A homotopy theoretic realization of string topology". Mathematische Annalen. 324 (4): 773–798. arXiv:math/0107187. doi:10.1007/s00208-002-0362-0. MR 1942249. S2CID 16916132. • A homotopy theoretic realization of string topology, (with John D.S Jones) Mathematische Annalen, vol. 324, 773-798 (2002) • Cohen, Ralph L.; Jones, John D. S. (2016). "Gauge theory and string topology". Boletín de la Sociedad Matemática Mexicana. 23 (1): 233–255. arXiv:1304.0613. doi:10.1007/s40590-016-0134-6. MR 3633134. S2CID 118378937. • with Kathryn Hess, Alexander A. Voronov: String Topology and Cyclic Homology, Birkhäuser 2006[2] • with Gunnar Carlsson: The What, Where and Why of Mathematics. A handbook for Teachers. 1991. • with Gunnar Carlsson: Topics in Algebra. 1999. References 1. "Ralph L. Cohen". stanford.edu. 2. Latschev, Janko (2010). "Review: String topology and cyclic homology by Ralph L. Cohen, Kathryn Hess, and Alexander A. Voronov" (PDF). Bull. Amer. Math. Soc. (N.S.). 47 (4): 705–712. doi:10.1090/s0273-0979-09-01265-8. External links • Ralph Louis Cohen at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Germany • Israel • United States • Czech Republic • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Ralph S. Phillips Ralph Saul Phillips (23 June 1913 – 23 November 1998) was an American mathematician and academic known for his contributions to functional analysis, scattering theory, and servomechanisms. He served as a Professor of mathematics at Stanford University. He made major contributions to acoustical scattering theory in collaboration with Peter Lax, proving remarkable results on local energy decay and the connections between poles of the scattering matrix and the analytic properties of the resolvent. With Lax, he coauthored the widely referred book on scattering theory titled Scattering Theory for Automorphic Functions. Phillips received the 1997 Leroy P. Steele Prize for Lifetime Achievement.[2] Ralph S. Phillips Born(1913-06-23)June 23, 1913 Oakland, California, US DiedNovember 23, 1998(1998-11-23) (aged 85) Monte Sereno, California, US Alma materUniversity of California University of Michigan Known forScattering theory AwardsLeroy P. Steele Prize (1997) Scientific career FieldsFunctional analysis InstitutionsUniversity of California Stanford University ThesisIntegration in a Convex Linear Topological Space[1] Doctoral advisorTheophil Henry Hildebrandt[1] Doctoral studentsA.V. Balakrishnan J. Thomas Beale Andrew Majda Michael C. Reed Education and career Phillips was born in Oakland on 23 June 1913. He received his bachelor's degree from the University of California at Los Angeles (UCLA) in 1935 and his Ph.D. from the University of Michigan in 1939 under the direction of Theophil H. Hildebrandt. From 1939 until 1942 he was a member of the Institute for Advanced Study in Princeton, an instructor at the University of Washington, and an instructor at Harvard University. It was at the University of Washington he met his future wife, Jean.[3] During the war he led a research group at the Radiation Laboratory at the Massachusetts Institute of Technology, the facility where much of the theoretical and practical work on radar technology was done. This work led to his book Theory of Servomechanisms,[4] which for many years was the standard text in the subject. After the war he returned to mathematics, joining as an assistant professor at the Courant Institute of Mathematical Sciences. He moved to the University of Southern California the next year and returned to UCLA in 1958. In 1960 he joined Stanford University and remained there until his death in 1998. He was the Robert Grimmett Professor of Mathematics at Stanford. Philips's work (with A. Lubotzky and P. Sarnak) on Ramanujan graphs had a huge impact on combinatorics and computer science. Books • Scattering Theory for Automorphic Functions, with P. D. Lax. Princeton University Press (1977). ISBN 0-691-08184-0 References 1. Ralph S. Phillips at the Mathematics Genealogy Project 2. "1997 Steele Prizes" (PDF). American Mathematical Society. Retrieved 26 June 2010. 3. Peter Sarnak. "Ralph Phillips" (PDF). American Mathematical Society. Retrieved 26 June 2010. 4. H. M. James; N. B. Nichols; R. S. Phillips, eds. (1947). Theory of Servomechanisms (PDF). McGraw-Hill. Archived from the original (PDF) on 2010-07-05. Retrieved 2010-06-26. External links • Ralph S. Phillips at the Mathematics Genealogy Project • Ralph S. Phillips, editorial in Journal of Evolution Equations Authority control International • FAST • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States • Latvia • Czech Republic • Netherlands • Poland Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef	Wikipedia
Ralph Gordon Stanton Ralph Gordon Stanton (21 October 1923 – 21 April 2010) was a Canadian mathematician, teacher, scholar, and pioneer in mathematics and computing education.[1] As a researcher, he made important contributions in the area of discrete mathematics; and as an educator and administrator, was also instrumental in founding the Faculty of Mathematics at the University of Waterloo, and for establishing its unofficial mascot of the pink tie. Ralph Gordon Stanton Born(1923-10-21)October 21, 1923 Lambeth, Ontario, Canada DiedApril 21, 2010(2010-04-21) (aged 86) Alma mater • University of Western Ontario • University of Toronto Scientific career FieldsMathematics Life and education Stanton was born in Lambeth, Ontario, Canada on 21 October 1923. He was the eldest of four children.[2] Stanton received his BA in Mathematics and Physics in 1944 from the University of Western Ontario.[1] He went on to receive his MA in 1945 and PhD in 1948, both from the University of Toronto.[1] His PhD dissertation was on the topic "On The Mathiew Group M(Sub 24)", under advisor Richard Dagobert Brauer.[3] He received honorary Doctor of Science degrees from the University of Queensland in 1989, and from the University of Natal in 1997. He also received an honorary D. Math from the University of Waterloo in 1997.[2] Career Faculty positions From 1946 to 1957 Stanton taught at the University of Toronto. In 1957, he moved to Kitchener-Waterloo to work at what was then Waterloo College, which was undergoing expansion, and became what is currently the University of Waterloo. At the time of his arrival he constituted the entirety of the Mathematics Department.[4] Stanton became the university's first Dean of Graduate Studies in 1960.[5] He turned the Department of Mathematics into the Faculty of Mathematics, which when it opened on January 1, 1967 was the first of its kind in North America.[6] In 1967 he moved to York University to found their Graduate program in Mathematics. In 1970 he moved to the University of Manitoba's Department of Computer Science, serving successively as Head, Professor, and Distinguished Professor. Research Stanton's main areas of research were in statistics and applied statistics; algebra; mathematical biology; combinatorial design theory, including pair-wise balanced designs, difference sets, covering and packing designs, and room squares; graph theory, including graph models of networks; and algorithms.[2][4] Teaching and other influences Stanton's influence on the young University of Waterloo extended to many areas. He hired Wes Graham, who Stanton had taught as an undergraduate. Graham became one of the first professors of computer science at the university, and the first Director of its Computing Centre in 1962. Stanton was one of five members of the Academic Advisory Committee that, in 1958, urged the board of governors to buy the Schweitzer farm on the outskirts of Waterloo that today houses the main campus. He introduced computers to classroom teaching in 1960, and introduced co-op programs in applied mathematics and computer science. [2] His interest in teaching extended to the secondary school level. He encouraged teaching of computing science and mathematics in high schools, serving as editor of two high school mathematical journals, member of Ontario provincial curriculum committees, and was actively involved in developing the Canadian Junior Mathematics Contest and the Descartes Senior Mathematics Competition, now administered by the University of Waterloo's Centre for Education in Mathematics and Computing. Stanton's gaudy neckties were the inspiration for the University of Waterloo's Faculty of Mathematics mascot, a giant pink tie that was hung by students over the Math and Computer Building when it opened in 1968. The pink tie remains the unofficial symbol of the Mathematics Faculty, with the Mathematics Society distributing more than 1000 pink ties to new students in the first week of the school year.[7] Stanton founded and administered three not-for-profit corporations dedicated to mathematical research and communication.[2] "Utilitas Mathematica Publishing" started in the 1970s and published conference proceedings in mathematics and scientific computing. He founded the Charles Babbage Research Centre (CBRC), a registered charitable organization, to promote conferences and encourage the publication of research. The CBRC currently has published the Canadian journal of combinatorics, Ars Combinatoria, since its inception in 1976, and continues to publish six volumes per year. [8] In 1990 he began his final project, the Institute of Combinatorics and its Applications (ICA). Although the institute was minimally active after Stanton's death in 2010, in March 2016 it resumed its full activities. Stanton also helped organize the first Southeastern Conference on Combinatorics, Graph Theory, and Computing. He continued as one of the organizers until at least 1991, at which point it was the largest combinatorial meeting in the world. [4] Awards In 1985 he was awarded the Killam Prize in Mathematics for Natural Sciences from the Canada Council for the Arts. (As of 2017, five prizes of $100,000 Cdn are awarded annually.)[9] Literature collection Stanton's collection of French and Portuguese literature was described as one of the world's largest private collections of classical Portuguese literature.[10] He donated his collection to the Fisher Rare Book Library at the University of Toronto.[11] His donations spanned from 1987 to his death in 2010, with the bulk being donated in the 1990s. Thanks to his generosity, the Fisher Rare Book Library now "boasts comprehensive collections of most of the significant French playwrights of the Classical period". An example of his donations is a two-volume set of the 1587 edition of Holinshed's Chronicles. References 1. "Stanton, Ralph G. fonds". Retrieved 26 March 2017. 2. "Passages: Ralph Gordon Stanton". Winnipeg Free Press. May 1, 2010. Retrieved 25 March 2017. 3. "Mathematics Genealogy Project". Retrieved 26 March 2017. 4. McKay, Brendan D.; Seberry, Jennifer; Vanstone, Scott A. (1991). "Ralph Gordon Stanton". Discrete Mathematics. 92 (1–3): 1–8. doi:10.1016/0012-365x(91)90261-y. 5. "Dr. Stanton Heads Graduate Program". University of Waterloo Quarterly Report. University of Waterloo Information Services. 2 (2): 1. Retrieved 26 January 2018. 6. "Unbundling Computing History". Retrieved 26 March 2017. 7. "Legend of the Pink Tie". 5 December 2014. Retrieved 10 October 2020. 8. "Ars Combinatoria". Retrieved 26 March 2017. 9. "Killam program Archives". Canada Council for the Arts. Retrieved 26 March 2017. 10. "UW Daily Bulletin, Apr 28, 2010". Retrieved 26 March 2017. 11. "In Memoriam: Ralph Stanton". Retrieved 26 March 2017. Authority control International • ISNI • VIAF National • Norway • Germany • Israel • United States • Japan • Australia • Netherlands Academics • CiNii • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Ralucca Gera Ralucca Michelle Gera (née Muntean) is an American mathematician specializing in graph theory, including graph coloring, dominating sets, and spectral graph theory. Her interests also include personalized learning in mathematics education. She is a professor of mathematics at the Naval Postgraduate School.[1] Education and career Gera was an undergraduate at Western Michigan University, graduating Phi Beta Kappa and with honors in mathematics in 1999.[2][3] She remained at Western Michigan University for her doctoral studies, completing a PhD in 2005 under the supervision of Ping Zhang; her dissertation was Stratification and Domination in Graphs and Digraphs.[4] She has been a faculty member at the Naval Postgraduate School since 2005,[1] and became a full professor there in 2018. She served as Associate Provost For Graduate Education from 2018 to 2021, and at the same time served as the founding director of the Teaching and Learning Commons at the Naval Postgraduate School[5] Recognition In 2016, the Naval Postgraduate School gave Gera their Richard W. Hamming Excellence in Teaching Award.[2][6] References 1. "Ralucca Gera, PhD, Professor of Mathematics, Principal Investigator, Campus of the Future", Faces of NPS, Naval Postgraduate School Foundation & Alumni Association, retrieved 2023-01-06 2. Ralucca Gera: Awards and Grants, Naval Postgraduate School, retrieved 2023-01-06 3. "Students inducted into WMU's Phi Beta Kappa chapter", WMU News, Western Michigan University, 22 July 1999, retrieved 2023-01-06 4. Ralucca Gera at the Mathematics Genealogy Project 5. Ralucca Gera: Biography, Naval Postgraduate School, retrieved 2023-01-06 6. Chagoya, Javier (27 September 2016), "Applied Mathematics Professor Earns Hamming Teaching Award", Today@NPS, Naval Postgraduate School External links • Home page • Ralucca Gera publications indexed by Google Scholar Authority control International • VIAF National • Germany Academics • Association for Computing Machinery • DBLP • MathSciNet • Mathematics Genealogy Project • ORCID • zbMATH Other • IdRef	Wikipedia
Ram Kishore Saxena Ram Kishore Saxena D.Sc , FNASc (11 November 1936) is an Indian mathematician and Emeritus professor, UGC Jai Narain Vyas University and former Professor and Head, Department of Mathematics.[1][2][3][4] Ram Kishore Saxena DSc,FNASc Born (1936-11-11) 11 November 1936 Jaipur Rajasthan NationalityIndian EducationUniversity of Rajasthan BSc,MSc 1956,PhD1962 University of Jodhpur DSc 1971 Alma materMcGill University , Research Associate University of Jodhpur University of Rajasthan Organization(s)University of Sulaymaniyah Associate Professor Al-Fateh University Professor University of Jodhpur Jaswant College, Jodhpur Lohia College,Churu Maharana Bhupal College, Udaipur Mohanlal Sukhadia University Jai Narain Vyas University Known forMathematics Notable workFox H-function Meijer G-function SpouseBimlesh Saxena AwardsPost-Doctoral Fellowship of National Research Council Canada Fellow of National Academy of Sciences, India Published work Saxena has published 356 research papers;[4][5][6] under his supervision many scholars has done PhD and post-doctoral research.[1] Saxena has published books.[7][8][9][10][11] References 1. "Curriculum Vitae 1. Name : Saxena Ram Kshore 2. Date of - [PDF Document]". documents.pub. Retrieved 2022-09-16. 2. "Ram Saxena - The Mathematics Genealogy Project". www.mathgenealogy.org. Retrieved 2022-09-16. 3. "Saxena, Ram Kishore [WorldCat Identities]". 4. "MR: Saxena, Ram Kishore - 193316". mathscinet.ams.org. Retrieved 2022-09-16. 5. "Ram Kishore Saxena". 6. "Ram Kishore Saxena". 7. Mathai, A. M.; Saxena, R. K.; Saxena, Ram Kishore (1973). Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Springer. ISBN 978-0-387-06482-6. 8. "H-function". 9. "The H-Function: Theory And Applications - A.M. Mathai, Ram Kishore Saxena, Hans J. Haubold". www.iri.upc.edu. Retrieved 2022-09-16. 10. Mathai, A. M.; Saxena, Rajendra Kumar (1978). The H-function with Applications in Statistics and Other Disciplines. Wiley. ISBN 978-0-470-26380-8. 11. "Journal of Ramanujan Society of Mathematics and Mathematical Sciences". www.rsmams.org. Retrieved 2022-09-16. Authority control International • VIAF • 2 National • Israel • United States • Czech Republic Academics • MathSciNet • Mathematics Genealogy Project	Wikipedia
Rama Cont Rama Cont is the Professor of Mathematical Finance at the University of Oxford.[7] [8] He is known for contributions to probability theory, stochastic analysis and mathematical modelling in finance, in particular mathematical models of systemic risk.[3] He was awarded the Louis Bachelier Prize by the French Academy of Sciences in 2010. Rama Cont Born Rama Cont (1972-06-30) 30 June 1972 Tehran, Iran Nationality Iran Alma materÉcole Polytechnique Known forSystemic risk modelling, Functional Ito calculus, Pathwise Ito calculus, Model risk, Liquidity at risk Awards • Louis Bachelier Prize (2010)[1] • Royal Society Award for Excellence in Interdisciplinary Research(2017)[2] • SIAM Fellow (2017)[3] Scientific career Fields • Probability • Mathematical Finance Institutions • University of Oxford • Imperial College London • Sorbonne University • Columbia University • École Polytechnique ThesisDes marches aléatoires aux marchés aléatoires. Modélisation statistique des marchés financiers: études empiriques et approches théoriques.[4] (1998) Doctoral advisorJean-Philippe Bouchaud[5] InfluencesBenoit Mandelbrot[6] Websitepeople.maths.ox.ac.uk/rama.cont/ Biography Born in Tehran (Iran), Cont obtained his undergraduate degree from Ecole Polytechnique (France),[8] a master's degree in theoretical physics from Ecole Normale Superieure and a degree in Chinese Language from Institut national des langues et civilisations orientales.[9] His doctoral thesis focused on the application of Lévy processes in financial modelling. Career and achievements Cont started his career as a CNRS researcher in applied mathematics at Ecole Polytechnique (France) in 1998 and held academic positions at Ecole Polytechnique, Columbia University and Imperial College London.[9] He was appointed 'Directeur de Recherche CNRS' (CNRS Senior Research Scientist) in 2008 and was chair of mathematical finance at Imperial College London[10] from 2012 to 2018. He was elected Statutory Professor in Mathematical Finance at the Oxford Mathematical Institute and professorial fellow of St Hugh's College, Oxford in 2018.[11][12] Cont's research focuses on probability theory, stochastic analysis and mathematical modelling in finance.[13] His mathematical work focuses on pathwise methods in stochastic analysis [14] and the Functional Ito calculus.[15] In quantitative finance he is known in particular for his work on models based on jump processes,[16] the stochastic modelling of limit order books as queueing systems [17] ,[18] machine learning methods in finance [19] and the mathematical modelling of systemic risk.[20] [21] He was editor in chief of the Encyclopedia of Quantitative Finance.[22] Cont has served as advisor to central banks and international organizations such as the International Monetary Fund and the Bank for International Settlements on stress testing and systemic risk monitoring. His work on network models, financial stability and central clearing [23] has influenced central banks and regulators .[24] He has given numerous media interviews[25] [26][27][6] [28] on issues related to systemic risk and financial regulation. Scientific contributions Systemic risk modeling Work by Cont and his collaborators on mathematical modeling of systemic risk and financial stability, in particular on network models of financial contagion and the modeling of indirect contagion via 'fire sales', has influenced academic research and policy in this area.[24][29] Central clearing Cont's research on central clearing in over-the-counter (OTC) markets has influenced risk management practices of central counterparties and regulatory thinking on central clearing.[30] Cont has argued that central clearing does not eliminate counterparty risk but transforms it into liquidity risk, therefore risk management and stress testing of central counterparties should focus on liquidity risk and liquidity resources, not capital.[31] Risk measurement and Model risk Cont introduced a rigorous approach for the assessment of model risk [32] which has been influential in the design of model risk management frameworks in financial institutions. [33][34][35] Cont, Deguest and Scandolo[36] introduced the concept of 'risk measurement procedure', an empirical counterpart of the notion of risk measure, and defined a robust class of risk measurement procedures known as 'Range Value-at-risk' (RVaR), a robust alternative to Expected shortfall.[37] Cont, Kotlicki and Valderrama define the concept of Liquidity at risk,[38] as the amount of liquid assets needed by a financial institution to face liquidity outflows in this scenario. Awards and honours Cont was awarded the Louis Bachelier Prize by the French Academy of Sciences in 2010 for his work on mathematical modelling of financial markets.[1] He was elected Fellow of the Society for Industrial and Applied Mathematics (SIAM) in 2017 for "contributions to stochastic analysis and mathematical finance".[3] He received the Award for Excellence in Interdisciplinary Research (APEX) from the Royal Society in 2017 for his research on mathematical modelling of systemic risk.[2] [39] Publications • Ananova, Anna; Cont, Rama (2017). "Pathwise integration with respect to paths of finite quadratic variation". Journal de Mathématiques Pures et Appliquées. 107 (6): 737–757. doi:10.1016/j.matpur.2016.10.004. S2CID 16318176. • Cont, R. (2006). "Model Uncertainty and Its Impact on the Pricing of Derivative Instruments" (PDF). Mathematical Finance. 16 (3): 519–547. doi:10.1111/j.1467-9965.2006.00281.x. S2CID 16075069. • Cont, Rama; Fournie, David-Antoine (2013). "Functional Ito calculus and stochastic integral representation of martingales". The Annals of Probability. 41 (1): 109–133. doi:10.1214/11-AOP721. S2CID 8840440. • Cont, Rama; Moussa, Amal; Santos, Edson Bastos (2013). "Network structure and systemic risk in banking systems". In Fouque, Jean-Pierre; Langsam, Joseph (eds.). Handbook of Systemic Risk (PDF). Cambridge University Press. CiteSeerX 10.1.1.637.587. doi:10.1017/CBO9781139151184.018. ISBN 9781107023437. Archived from the original (PDF) on 1 Aug 2013. Retrieved 5 August 2018. Alt URL • Bally, Vlad; Caramellino, Lucia; Cont, Rama (2016). Stochastic integration by parts and Functional Ito calculus. Springer. doi:10.1007/978-3-319-27128-6. ISBN 9783319271286. • Cont, Rama; Deguest, Romain; Giacomo, Giacomo (2010). "Robustness and Sensitivity Analysis of Risk Measurement Procedures" (PDF). Quantitative Finance. 10 (6): 593–606. doi:10.1080/14697681003685597. S2CID 158678050. • Cont, Rama; De Larrard, Adrien (2013). "Price Dynamics in a Markovian Limit Order Market". SIAM Journal on Financial Mathematics. 4 (1): 1–25. arXiv:1104.4596. doi:10.1137/110856605. S2CID 1238587. • Cont, Rama; Tankov, Peter (2004). Financial Modelling with Jump Processes. CRC Press. ISBN 9781584884132. • Cont, Rama; Kotlicki, Artur; Valderrama, Laura (2020). "Liquidity at Risk: Joint Stress Testing of Solvency and Liquidity". Journal of Banking and Finance. 118: 105871. doi:10.1016/j.jbankfin.2020.105871. • Cont, Rama; Stoikov, Sasha; Talreja, Rishi (2010). "A Stochastic Model for Order Book Dynamics". Operations Research. 58 (3): 549–563. doi:10.1287/opre.1090.0780. References 1. London Mathematical Society. "Louis Bachelier Prize". LMS. Retrieved 5 August 2018. 2. "APEX Awards \| Royal Society". 3. Society for Industrial and Applied Mathematics. "SIAM Fellows: Class of 2017". SIAM. Retrieved 5 August 2018. 4. "Rama Cont - the Mathematics Genealogy Project". 5. Rama Cont at the Mathematics Genealogy Project 6. Sorman, Guy. "Wild Randomness". Forbes (August 2009). Retrieved 5 August 2018. 7. "Prof. Rama Cont \| Mathematical Institute". 8. "Rama Cont - Who's Who". 9. "Rama Cont's CV" (PDF). Deutsche GesellSchaft fuer Versicherungs und Finanzmathematik. Retrieved 2018-08-03. 10. "Home - Professor Rama Cont". 11. "Appointments". Oxford Gazette. Oxford University. 2018. Retrieved 22 January 2018. 12. "Rama Cont appointed to the Professorship of Mathematical Finance in Oxford". Oxford University. 2018. Retrieved 1 February 2018. 13. http://rama.cont.perso.math.cnrs.fr/ 14. Ananova, Anna; Cont, Rama (2017). "Pathwise integration with respect to paths of finite quadratic variation". Journal de Mathématiques Pures et Appliquées. 107 (6): 737–757. doi:10.1016/j.matpur.2016.10.004. S2CID 16318176. 15. Bally, Vlad; Caramellino, Lucia; Cont, Rama (2016). Stochastic Integration by Parts and Functional Itô Calculus. Advanced Courses in Mathematics - CRM Barcelona. doi:10.1007/978-3-319-27128-6. ISBN 978-3-319-27127-9. 16. Cont, Rama; Tankov, Peter (2004). Financial Modelling with Jump Processes. CRC Press. ISBN 9781584884132. 17. Cont, Rama; De Larrard, Adrien (2013). "Price Dynamics in a Markovian Limit Order Market". SIAM Journal on Financial Mathematics. 4 (1): 1–25. arXiv:1104.4596. doi:10.1137/110856605. S2CID 1238587. 18. Cont, Rama; Stoikov, Sasha; Talreja, Rishi (2008). "A Stochastic Model for Order Book Dynamics". Operations Research. 58 (7176): 340–344. doi:10.1287/opre.1090.0780. 19. Mannix, Rob (2018). "Neural network learns 'universal model' for stock-price moves". RISK. 20. Systemic Risk: a challenge for Mathematical Modelling on YouTube 21. Cont, Rama; Moussa, Amal; Santos, Edson Bastos (2013). "Network structure and systemic risk in banking systems". In Fouque, Jean-Pierre; Langsam, Joseph (eds.). Handbook of Systemic Risk (PDF). Cambridge University Press. CiteSeerX 10.1.1.637.587. doi:10.1017/CBO9781139151184.018. ISBN 9781107023437. Archived from the original (PDF) on 1 Aug 2013. Retrieved 5 August 2018. Alt URL 22. Cont, Rama (2010). Encyclopedia of Quantitative Finance. Chichester: Wiley. ISBN 9780470057568. 23. "Rama Cont - Central bank research hub". BIS. Retrieved 5 August 2018. 24. Yellen, Janet. "Interconnectedness and Systemic Risk: Lessons from the Financial Crisis and Policy Implications". Board of Governors of the Federal Reserve System. Retrieved 5 August 2018. 25. Cypel, Sylvain (17 September 2008). "Si AIG s'écroule, toute l'économie américaine est affectée". Le Monde.fr. Le Monde. Retrieved 5 August 2018. 26. ISDA AGM 2018: Rama Cont - Imperial College London. YouTube. Archived from the original on 2021-12-09. 27. Cypel, Sylvain (3 May 2010). "Les "conflits d'intérêts" d'Abacus". Le Monde.fr. Le Monde. Retrieved 5 August 2018. 28. "Les chambres de compensation transforment le risque de contrepartie en risque de liquidité". 27 February 2017. 29. [John Fell, Francesco Mazzaferro, Richard Portes, Eric Schaanning (2020) Fallen angels and indirect contagion: Rationale for and lessons from a system-wide analysis, 11 September 2020 https://voxeu.org/article/fallen-angels-and-indirect-contagion] 30. Cont, Rama (2015), "The end of the waterfall: default resources of central counterparties", Journal of Risk Management in Financial Institutions, Henry Stewart Publications, 8, SSRN 2588986, retrieved 2020-12-09 31. Cont, Rama (2017), "Central clearing and risk transformation", Financial Stability Review, Banque de France, 21, SSRN 2919260, retrieved 2020-12-09 • Cont, Rama (2006). "Model Uncertainty and Its Impact on the Pricing of Derivative Instruments" (PDF). Mathematical Finance. 16 (3): 519–547. doi:10.1111/j.1467-9965.2006.00281.x. S2CID 16075069. 32. Morini, Massimo (2012). Understanding and Managing Model Risk: A Practical Guide for Quants, Traders and Validators. Wiley. doi:10.1002/9781118467312. ISBN 9781118467312. 33. http://nx.numerix.com/rs/numerix2/images/ModelRiskManagement_2015SlidesMerged.pdf 34. "Model risk". 17 September 2017. 35. Cont, Rama; Deguest, Romain; Giacomo, Giacomo (2010). "Robustness and Sensitivity Analysis of Risk Measurement Procedures" (PDF). Quantitative Finance. 10 (6): 593–606. doi:10.1080/14697681003685597. S2CID 158678050. 36. Fissler, Tobias; Ziegel, Johanna F. (2021). "On the elicitability of range value at risk". Statistics & Risk Modeling. 38 (1–2): 25–46. arXiv:1902.04489. doi:10.1515/strm-2020-0037. S2CID 85517050. 37. Cont, Rama; Kotlicki, Artur; Valderrama, Laura (2020). "Liquidity at Risk: Joint Stress Testing of Solvency and Liquidity". Journal of Banking and Finance. 118: 105871. doi:10.1016/j.jbankfin.2020.105871. 38. Dunning, Hayley. "Maths researcher awarded funding for interdisciplinary risk project". Imperial College London. Retrieved 5 August 2018. External links • Professional Homepage • Rama Cont publications indexed by Google Scholar • Rama Cont at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States • Czech Republic • Netherlands Academics • DBLP • Google Scholar • MathSciNet • Mathematics Genealogy Project • ORCID • Scopus • zbMATH Other • IdRef	Wikipedia
Srinivasa Ramanujan Srinivasa Ramanujan FRS (/ˈsriːnɪvɑːsə rɑːˈmɑːnʊdʒən/ SREE-nih-vah-sə rah-MAH-nuuj-ən;[1] born Srinivasa Ramanujan Aiyangar, IPA: [sriːniʋaːsa ɾaːmaːnud͡ʑan ajːaŋgar]; 22 December 1887 – 26 April 1920)[2][3] was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Srinivasa Ramanujan FRS Born Srinivasa Ramanujan Aiyangar (1887-12-22)22 December 1887 Erode, Mysore State, British India (now in Tamil Nadu, India) Died26 April 1920(1920-04-26) (aged 32) Kumbakonam, Tanjore District, Madras Presidency, British India (now Thanjavur district, Tamil Nadu, India) CitizenshipBritish Indian Education • Government Arts College (no degree) • Pachaiyappa's College (no degree) • Trinity College, Cambridge (BA) Known for • Ramanujan's sum • Landau–Ramanujan constant • Mock theta functions • Ramanujan conjecture • Ramanujan prime • Ramanujan–Soldner constant • Ramanujan theta function • Rogers–Ramanujan identities • Ramanujan's master theorem • Hardy–Ramanujan asymptotic formula • Ramanujan–Sato series AwardsFellow of the Royal Society (1918) Scientific career FieldsMathematics InstitutionsTrinity College, Cambridge ThesisHighly Composite Numbers (1916) Academic advisors • G. H. Hardy • J. E. Littlewood InfluencesG. S. Carr Signature Ramanujan initially developed his own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered".[4] Seeking mathematicians who could better understand his work, in 1913 he began a postal correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged for him to travel to Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some that "defeated me completely; I had never seen anything in the least like them before",[5] and some recently proven but highly advanced results. During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations).[6] Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research.[7] Of his thousands of results, all but a dozen or two have now been proven correct.[8] The Ramanujan Journal, a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan,[9] and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death as a source of new mathematical ideas. As late as 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death.[10][11] He became one of the youngest Fellows of the Royal Society and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge. Of his original letters, Hardy stated that a single look was enough to show they could have been written only by a mathematician of the highest calibre, comparing Ramanujan to mathematical geniuses such as Euler and Jacobi. In 1919, ill health—now believed to have been hepatic amoebiasis (a complication from episodes of dysentery many years previously)—compelled Ramanujan's return to India, where he died in 1920 at the age of 32. His last letters to Hardy, written in January 1920, show that he was still continuing to produce new mathematical ideas and theorems. His "lost notebook", containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976. Early life Ramanujan (literally, "younger brother of Rama", a Hindu deity)[12] was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, in present-day Tamil Nadu.[13] His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a clerk in a sari shop.[14][2] His mother, Komalatammal, was a housewife and sang at a local temple.[15] They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam.[16] The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted smallpox, but recovered, unlike the 4,000 others who died in a bad year in the Thanjavur district around this time. He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). His mother gave birth to two more children, in 1891 and 1894, both of whom died before their first birthdays.[12] On 1 October 1892, Ramanujan was enrolled at the local school.[17] After his maternal grandfather lost his job as a court official in Kanchipuram,[18] Ramanujan and his mother moved back to Kumbakonam, and he was enrolled in Kangayan Primary School.[19] When his paternal grandfather died, he was sent back to his maternal grandparents, then living in Madras. He did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.[19] Since Ramanujan's father was at work most of the day, his mother took care of the boy, and they had a close relationship. From her, he learned about tradition and puranas, to sing religious songs, to attend pujas at the temple, and to maintain particular eating habits—all part of Brahmin culture.[20] At Kangayan Primary School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, and arithmetic with the best scores in the district.[21] That year, Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time.[21] A child prodigy by age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book written by S. L. Loney on advanced trigonometry.[22][23] He mastered this by the age of 13 while discovering sophisticated theorems on his own. By 14, he received merit certificates and academic awards that continued throughout his school career, and he assisted the school in the logistics of assigning its 1,200 students (each with differing needs) to its approximately 35 teachers.[24] He completed mathematical exams in half the allotted time, and showed a familiarity with geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902. He would later develop his own method to solve the quartic. In 1903, he tried to solve the quintic, not knowing that it was impossible to solve with radicals.[25] In 1903, when he was 16, Ramanujan obtained from a friend a library copy of A Synopsis of Elementary Results in Pure and Applied Mathematics, G. S. Carr's collection of 5,000 theorems.[26][27] Ramanujan reportedly studied the contents of the book in detail.[28] The next year, Ramanujan independently developed and investigated the Bernoulli numbers and calculated the Euler–Mascheroni constant up to 15 decimal places.[29] His peers at the time said they "rarely understood him" and "stood in respectful awe" of him.[24] When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum.[30] He received a scholarship to study at Government Arts College, Kumbakonam,[31][32] but was so intent on mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.[33] In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam, and stayed in Rajahmundry[34] for about a month.[33] He later enrolled at Pachaiyappa's College in Madras. There, he passed in mathematics, choosing only to attempt questions that appealed to him and leaving the rest unanswered, but performed poorly in other subjects, such as English, physiology, and Sanskrit.[35] Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later. Without an FA degree, he left college and continued to pursue independent research in mathematics, living in extreme poverty and often on the brink of starvation.[36] In 1910, after a meeting between the 23-year-old Ramanujan and the founder of the Indian Mathematical Society, V. Ramaswamy Aiyer, Ramanujan began to get recognition in Madras's mathematical circles, leading to his inclusion as a researcher at the University of Madras.[37] Adulthood in India On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994),[38] a girl his mother had selected for him a year earlier and who was ten years old when they married.[39][40][41] It was not unusual then for marriages to be arranged with girls at a young age. Janaki was from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate in the marriage ceremony.[42] As was common at that time, Janaki continued to stay at her maternal home for three years after marriage, until she reached puberty. In 1912, she and Ramanujan's mother joined Ramanujan in Madras.[43] After the marriage, Ramanujan developed a hydrocele testis.[44] The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac, but his family could not afford the operation. In January 1910, a doctor volunteered to do the surgery at no cost.[45] After his successful surgery, Ramanujan searched for a job. He stayed at a friend's house while he went from door to door around Madras looking for a clerical position. To make money, he tutored students at Presidency College who were preparing for their Fellow of Arts exam.[46] In late 1910, Ramanujan was sick again. He feared for his health, and told his friend R. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras Christian College."[47] After Ramanujan recovered and retrieved his notebooks from Iyer, he took a train from Kumbakonam to Villupuram, a city under French control.[48][49] In 1912, Ramanujan moved with his wife and mother to a house in Saiva Muthaiah Mudali street, George Town, Madras, where they lived for a few months.[50] In May 1913, upon securing a research position at Madras University, Ramanujan moved with his family to Triplicane.[51] Pursuit of career in mathematics In 1910, Ramanujan met deputy collector V. Ramaswamy Aiyer, who founded the Indian Mathematical Society.[52] Wishing for a job at the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled: I was struck by the extraordinary mathematical results contained in [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department.[53] Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.[52] Some of them looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society.[54][55][56] Rao was impressed by Ramanujan's research but doubted that it was his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding of his work but concluded that he was not a fraud.[57] Ramanujan's friend C. V. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed to give him another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately convinced him of Ramanujan's brilliance.[57] When Rao asked him what he wanted, Ramanujan replied that he needed work and financial support. Rao consented and sent him to Madras. He continued his research with Rao's financial aid. With Aiyer's help, Ramanujan had his work published in the Journal of the Indian Mathematical Society.[58] One of the first problems he posed in the journal[30] was to find the value of: ${\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}$ He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied an incomplete[59] solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem. $x+n+a={\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\cdots }}}}}}$ Using this equation, the answer to the question posed in the Journal was simply 3, obtained by setting x = 2, n = 1, and a = 0.[60] Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers. One property he discovered was that the denominators (sequence A027642 in the OEIS) of the fractions of Bernoulli numbers are always divisible by six. He also devised a method of calculating Bn based on previous Bernoulli numbers. One of these methods follows: It will be observed that if n is even but not equal to zero, 1. Bn is a fraction and the numerator of Bn/n in its lowest terms is a prime number, 2. the denominator of Bn contains each of the factors 2 and 3 once and only once, 3. 2n(2n − 1)Bn/n is an integer and 2(2n − 1)Bn consequently is an odd integer. In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures.[61] His writing initially had many flaws. As Journal editor M. T. Narayana Iyengar noted: Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.[62] Ramanujan later wrote another paper and also continued to provide problems in the Journal.[63] In early 1912, he got a temporary job in the Madras Accountant General's office, with a monthly salary of 20 rupees. He lasted only a few weeks.[64] Toward the end of that assignment, he applied for a position under the Chief Accountant of the Madras Port Trust. In a letter dated 9 February 1912, Ramanujan wrote: Sir, I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.[65] Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".[66] Three weeks after he applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month.[67] At his office, Ramanujan easily and quickly completed the work he was given and spent his spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits. Contacting British mathematicians In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. M. J. M. Hill of University College London commented that Ramanujan's papers were riddled with holes.[68] He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the necessary educational background and foundation to be accepted by mathematicians.[69] Although Hill did not offer to take Ramanujan on as a student, he gave thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.[70] The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment.[71] On 16 January 1913, Ramanujan wrote to G. H. Hardy.[72] Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible fraud.[73] Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".[74]: 494 One of the theorems Hardy found amazing was on the bottom of page three (valid for 0 < a < b + 1/2): $\int \limits _{0}^{\infty }{\frac {1+{\dfrac {x^{2}}{(b+1)^{2}}}}{1+{\dfrac {x^{2}}{a^{2}}}}}\times {\frac {1+{\dfrac {x^{2}}{(b+2)^{2}}}}{1+{\dfrac {x^{2}}{(a+1)^{2}}}}}\times \cdots \,dx={\frac {\sqrt {\pi }}{2}}\times {\frac {\Gamma \left(a+{\frac {1}{2}}\right)\Gamma (b+1)\Gamma (b-a+1)}{\Gamma (a)\Gamma \left(b+{\frac {1}{2}}\right)\Gamma \left(b-a+{\frac {1}{2}}\right)}}.$ Hardy was also impressed by some of Ramanujan's other work relating to infinite series: $1-5\left({\frac {1}{2}}\right)^{3}+9\left({\frac {1\times 3}{2\times 4}}\right)^{3}-13\left({\frac {1\times 3\times 5}{2\times 4\times 6}}\right)^{3}+\cdots ={\frac {2}{\pi }}$ $1+9\left({\frac {1}{4}}\right)^{4}+17\left({\frac {1\times 5}{4\times 8}}\right)^{4}+25\left({\frac {1\times 5\times 9}{4\times 8\times 12}}\right)^{4}+\cdots ={\frac {2{\sqrt {2}}}{{\sqrt {\pi }}\,\Gamma ^{2}\left({\frac {3}{4}}\right)}}.$ The first result had already been determined by G. Bauer in 1859. The second was new to Hardy, and was derived from a class of functions called hypergeometric series, which had first been researched by Euler and Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals.[75] After seeing Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything in the least like them before",[76] and that they "must be true, because, if they were not true, no one would have the imagination to invent them".[76] Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power".[74]: 494–495 One colleague, E. H. Neville, later remarked that "not one [theorem] could have been set in the most advanced mathematical examination in the world".[63] On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest in his work, adding that it was "essential that I should see proofs of some of your assertions".[77] Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip.[78] In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land".[79] Meanwhile, he sent Hardy a letter packed with theorems, writing, "I have found a friend in you who views my labour sympathetically."[80] To supplement Hardy's endorsement, Gilbert Walker, a former mathematical lecturer at Trinity College, Cambridge, looked at Ramanujan's work and expressed amazement, urging the young man to spend time at Cambridge.[81] As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan".[82] The board agreed to grant Ramanujan a monthly research scholarship of 75 rupees for the next two years at the University of Madras.[83] While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one instance, Iyer submitted some of Ramanujan's theorems on summation of series to the journal, adding, "The following theorem is due to S. Ramanujan, the mathematics student of Madras University." Later in November, British Professor Edward B. Ross of Madras Christian College, whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived in the day's mail.[84] In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.[85] Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England.[86] Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted the proposal; Neville said, "Ramanujan needed no converting" and "his parents' opposition had been withdrawn".[63] Apparently, Ramanujan's mother had a vivid dream in which the family goddess, the deity of Namagiri, commanded her "to stand no longer between her son and the fulfilment of his life's purpose".[63] On 17 March 1914, Ramanujan traveled to England by ship,[87] leaving his wife to stay with his parents in India. Life in England Ramanujan departed from Madras aboard the S.S. Nevasa on 17 March 1914.[88] When he disembarked in London on 14 April, Neville was waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Court, a five-minute walk from Hardy's room.[89] Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems in the notebooks. Hardy saw that some were wrong, others had already been discovered, and the rest were new breakthroughs.[90] Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a Jacobi",[91] while Hardy said he "can compare him only with Euler or Jacobi."[92] Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood, and published part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. In the previous few decades, the foundations of mathematics had come into question and the need for mathematically rigorous proofs recognised. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his intuition and insights. Hardy tried his best to fill the gaps in Ramanujan's education and to mentor him in the need for formal proofs to support his results, without hindering his inspiration—a conflict that neither found easy. Ramanujan was awarded a Bachelor of Arts by Research degree[93][94] (the predecessor of the PhD degree) in March 1916 for his work on highly composite numbers, sections of the first part of which had been published the preceding year in the Proceedings of the London Mathematical Society. The paper was more than 50 pages long and proved various properties of such numbers. Hardy disliked this topic area but remarked that though it engaged with what he called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra of inequalities'.[95] On 6 December 1917, Ramanujan was elected to the London Mathematical Society. On 2 May 1918, he was elected a Fellow of the Royal Society,[96] the second Indian admitted, after Ardaseer Cursetjee in 1841. At age 31, Ramanujan was one of the youngest Fellows in the Royal Society's history. He was elected "for his investigation in elliptic functions and the Theory of Numbers." On 13 October 1918, he was the first Indian to be elected a Fellow of Trinity College, Cambridge.[97] Illness and death Ramanujan had numerous health problems throughout his life. His health worsened in England; possibly he was also less resilient due to the difficulty of keeping to the strict dietary requirements of his religion there and because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and a severe vitamin deficiency, and confined to a sanatorium. In 1919, he returned to Kumbakonam, Madras Presidency, and in 1920 he died at the age of 32. After his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series and continued fractions.[43] Ramanujan's widow, Smt. Janaki Ammal, moved to Bombay. In 1931, she returned to Madras and settled in Triplicane, where she supported herself on a pension from Madras University and income from tailoring. In 1950, she adopted a son, W. Narayanan, who eventually became an officer of the State Bank of India and raised a family. In her later years, she was granted a lifetime pension from Ramanujan's former employer, the Madras Port Trust, and pensions from, among others, the Indian National Science Academy and the state governments of Tamil Nadu, Andhra Pradesh and West Bengal. She continued to cherish Ramanujan's memory, and was active in efforts to increase his public recognition; prominent mathematicians, including George Andrews, Bruce C. Berndt and Béla Bollobás made it a point to visit her while in India. She died at her Triplicane residence in 1994.[42][43] A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D. A. B. Young[98] concluded that his medical symptoms—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from hepatic amoebiasis, an illness then widespread in Madras, than tuberculosis. He had two episodes of dysentery before he left India. When not properly treated, amoebic dysentery can lie dormant for years and lead to hepatic amoebiasis, whose diagnosis was not then well established.[99] At the time, if properly diagnosed, amoebiasis was a treatable and often curable disease;[99][100] British soldiers who contracted it during the First World War were being successfully cured of amoebiasis around the time Ramanujan left England.[101] Personality and spiritual life While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing. —Srinivasa Ramanujan[102] Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners.[103] He lived a simple life at Cambridge.[104] Ramanujan's first Indian biographers describe him as a rigorously orthodox Hindu. He credited his acumen to his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to her for inspiration in his work[105] and said he dreamed of blood drops that symbolised her consort, Narasimha. Later he had visions of scrolls of complex mathematical content unfolding before his eyes.[106] He often said, "An equation for me has no meaning unless it expresses a thought of God."[107] Hardy cites Ramanujan as remarking that all religions seemed equally true to him.[108] Hardy further argued that Ramanujan's religious belief had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict vegetarianism.[109] Similarly, in an interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers to Ramanujan's mathematical thinking. It is not true. He has meticulously recorded every result in his three notebooks," further speculating that Ramanujan worked out intermediate results on slate that he could not afford the paper to record more permanently.[8] Mathematical achievements In mathematics, there is a distinction between insight and formulating or working through a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct of his work, new directions of research were opened up. Examples of the most intriguing of these formulae include infinite series for π, one of which is given below: ${\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.$ This result is based on the negative fundamental discriminant d = −4 × 58 = −232 with class number h(d) = 2. Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 3962, which is related to the fact that $ e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .$ This might be compared to Heegner numbers, which have class number 1 and yield similar formulae. Ramanujan's series for π converges extraordinarily rapidly and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation 9801√2/4412 for π, which is correct to six decimal places; truncating it to the first two terms gives a value correct to 14 decimal places. See also the more general Ramanujan–Sato series. One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which P. C. Mahalanobis posed a problem: Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x?' This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. 'It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind', Ramanujan replied."[110][111] His intuition also led him to derive some previously unknown identities, such as ${\begin{aligned}&\left(1+2\sum _{n=1}^{\infty }{\frac {\cos(n\theta )}{\cosh(n\pi )}}\right)^{-2}+\left(1+2\sum _{n=1}^{\infty }{\frac {\cosh(n\theta )}{\cosh(n\pi )}}\right)^{-2}\\[6pt]={}&{\frac {2\Gamma ^{4}{\bigl (}{\frac {3}{4}}{\bigr )}}{\pi }}={\frac {8\pi ^{3}}{\Gamma ^{4}{\bigl (}{\frac {1}{4}}{\bigr )}}}\end{aligned}}$ for all θ such that $\|\Re (\theta )\|<\pi $ and $\|\Im (\theta )\|<\pi $, where Γ(z) is the gamma function, and related to a special value of the Dedekind eta function. Expanding into series of powers and equating coefficients of θ0, θ4, and θ8 gives some deep identities for the hyperbolic secant. In 1918, Hardy and Ramanujan studied the partition function P(n) extensively. They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. In 1937, Hans Rademacher refined their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae called the circle method.[112] In the last year of his life, Ramanujan discovered mock theta functions.[113] For many years, these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms. The Ramanujan conjecture Main article: Ramanujan–Petersson conjecture Although there are numerous statements that could have borne the name Ramanujan conjecture, one was highly influential on later work. In particular, the connection of this conjecture with conjectures of André Weil in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the tau-function, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal in 1978 for that work.[7][114] In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called τ(n) (the Ramanujan tau function).[115] He proved many congruences for these numbers, such as τ(p) ≡ 1 + p11 mod 691 for primes p. This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture that there is a theory of Galois representations that "explains" these congruences and more generally all modular forms. Δ(z) is the first example of a modular form to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The proof of Fermat's Last Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this theory, there would be no proof of Fermat's Last Theorem.[116] Ramanujan's notebooks Further information: Ramanujan's lost notebook While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of looseleaf paper. They were mostly written up without any derivations. This is probably the origin of the misapprehension that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to prove most of his results, but chose not to record the proofs in his notes. This may have been for any number of reasons. Since paper was very expensive, Ramanujan did most of his work and perhaps his proofs on slate, after which he transferred the final results to paper. At the time, slates were commonly used by mathematics students in the Madras Presidency. He was also quite likely to have been influenced by the style of G. S. Carr's book, which stated results without proofs. It is also possible that Ramanujan considered his work to be for his personal interest alone and therefore recorded only the results.[117] The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material. The second has 256 pages in 21 chapters and 100 unorganised pages, and the third 33 unorganised pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself wrote papers exploring material from Ramanujan's work, as did G. N. Watson, B. M. Wilson, and Bruce Berndt.[117] In 1976, George Andrews rediscovered a fourth notebook with 87 unorganised pages, the so-called "lost notebook".[99] Hardy–Ramanujan number 1729 Main article: 1729 (number) The number 1729 is known as the Hardy–Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. In Hardy's words:[118] I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." Immediately before this anecdote, Hardy quoted Littlewood as saying, "Every positive integer was one of [Ramanujan's] personal friends."[119] The two different ways are: $1729=1^{3}+12^{3}=9^{3}+10^{3}.$ Generalisations of this idea have created the notion of "taxicab numbers". Mathematicians' views of Ramanujan In his obituary of Ramanujan, written for Nature in 1920, Hardy observed that Ramanujan's work primarily involved fields less known even among other pure mathematicians, concluding: His insight into formulae was quite amazing, and altogether beyond anything I have met with in any European mathematician. It is perhaps useless to speculate as to his history had he been introduced to modern ideas and methods at sixteen instead of at twenty-six. It is not extravagant to suppose that he might have become the greatest mathematician of his time. What he actually did is wonderful enough… when the researches which his work has suggested have been completed, it will probably seem a good deal more wonderful than it does to-day.[74] Hardy further said:[120] He combined a power of generalisation, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was..." When asked about the methods Ramanujan employed to arrive at his solutions, Hardy said they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account."[121] He also said that he had "never met his equal, and can compare him only with Euler or Jacobi".[121] Littlewood reportedly said that helping Ramanujan catch up with European mathematics beyond what was available in India was very difficult, because each new point mentioned to Ramanujan caused him to produce original ideas that prevented Littlewood from continuing the lesson.[122] K. Srinivasa Rao has said,[123] "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdős has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100. Hardy gave himself a score of 25, J. E. Littlewood 30, David Hilbert 80 and Ramanujan 100.'" During a May 2011 lecture at IIT Madras, Berndt said that over the last 40 years, as nearly all of Ramanujan's conjectures had been proven, there had been greater appreciation of Ramanujan's work and brilliance, and that Ramanujan's work was now pervading many areas of modern mathematics and physics.[113][124] Posthumous recognition Further information: List of things named after Srinivasa Ramanujan The year after his death, Nature listed Ramanujan among other distinguished scientists and mathematicians on a "Calendar of Scientific Pioneers" who had achieved eminence.[125] Ramanujan's home state of Tamil Nadu celebrates 22 December (Ramanujan's birthday) as 'State IT Day'. Stamps picturing Ramanujan were issued by the government of India in 1962, 2011, 2012 and 2016.[126] Since Ramanujan's centennial year, his birthday, 22 December, has been annually celebrated as Ramanujan Day by the Government Arts College, Kumbakonam, where he studied, and at the IIT Madras in Chennai. The International Centre for Theoretical Physics (ICTP) has created a prize in Ramanujan's name for young mathematicians from developing countries in cooperation with the International Mathematical Union, which nominates members of the prize committee. SASTRA University, a private university based in Tamil Nadu, has instituted the SASTRA Ramanujan Prize of US$10,000 to be given annually to a mathematician not exceeding age 32 for outstanding contributions in an area of mathematics influenced by Ramanujan.[127] Based on the recommendations of a committee appointed by the University Grants Commission (UGC), Government of India, the Srinivasa Ramanujan Centre, established by SASTRA, has been declared an off-campus centre under the ambit of SASTRA University. House of Ramanujan Mathematics, a museum of Ramanujan's life and work, is also on this campus. SASTRA purchased and renovated the house where Ramanujan lived at Kumabakonam.[127] In 2011, on the 125th anniversary of his birth, the Indian government declared that 22 December will be celebrated every year as National Mathematics Day.[128] Then Indian Prime Minister Manmohan Singh also declared that 2012 would be celebrated as National Mathematics Year and 22 December as National Mathematics Day of India.[129] Ramanujan IT City is an information technology (IT) special economic zone (SEZ) in Chennai that was built in 2011. Situated next to the Tidel Park, it includes 25 acres (10 ha) with two zones, with a total area of 5.7 million square feet (530,000 m2), including 4.5 million square feet (420,000 m2) of office space.[130] Commemorative postal stamps Commemorative stamps released by India Post (by year): 1962 2011 2012 2016 In popular culture • The Man Who Loved Numbers is a 1988 PBS NOVA documentary about Ramanujan (S15, E9).[131] • The Man Who Knew Infinity is a 2015 film based on Kanigel's book of the same name. British actor Dev Patel portrays Ramanujan.[132][133][134] • Ramanujan, an Indo-British collaboration film chronicling Ramanujan's life, was released in 2014 by the independent film company Camphor Cinema.[135] The cast and crew include director Gnana Rajasekaran, cinematographer Sunny Joseph and editor B. Lenin.[136][137] Indian and English stars Abhinay Vaddi, Suhasini Maniratnam, Bhama, Kevin McGowan and Michael Lieber star in pivotal roles.[138] • Nandan Kudhyadi directed the Indian documentary films The Genius of Srinivasa Ramanujan (2013) and Srinivasa Ramanujan: The Mathematician and His Legacy (2016) about the mathematician.[139] • Ramanujan (The Man Who Reshaped 20th Century Mathematics), an Indian docudrama film directed by Akashdeep released in 2018.[140] • M. N. Krish's thriller novel The Steradian Trail weaves Ramanujan and his accidental discovery into its plot connecting religion, mathematics, finance and economics.[141][142] • Partition, a play by Ira Hauptman about Hardy and Ramanujan, was first performed in 2013.[143][144][145][146] • The play First Class Man by Alter Ego Productions[147] was based on David Freeman's First Class Man. The play centres around Ramanujan and his complex and dysfunctional relationship with Hardy. On 16 October 2011 it was announced that Roger Spottiswoode, best known for his James Bond film Tomorrow Never Dies, is working on the film version, starring Siddharth.[148] • A Disappearing Number is a British stage production by the company Complicite that explores the relationship between Hardy and Ramanujan.[149] • David Leavitt's novel The Indian Clerk explores the events following Ramanujan's letter to Hardy.[150][151] • Google honoured Ramanujan on his 125th birth anniversary by replacing its logo with a doodle on its home page.[152][153] • Ramanujan was mentioned in the 1997 film Good Will Hunting, in a scene where professor Gerald Lambeau (Stellan Skarsgård) explains to Sean Maguire (Robin Williams) the genius of Will Hunting (Matt Damon) by comparing him to Ramanujan.[154] Selected papers • Ramanujan, S. (1914). "Some definite integrals". Messenger Math. 44: 10–18. • Ramanujan, S. (1914). "Some definite integrals connected with Gauss's sums". Messenger Math. 44: 75–85. • Ramanujan, S. (1915). "On certain infinite series". Messenger Math. 45: 11–15. • Ramanujan, S. (1915). "Highly Composite Numbers". Proceedings of the London Mathematical Society. 14 (1): 347–409. doi:10.1112/plms/s2_14.1.347. • Ramanujan, S. (1915). "On the number of divisors of a number". The Journal of the Indian Mathematical Society. 7 (4): 131–133. • Ramanujan, S. (1915). "Short Note: On the sum of the square roots of the first n natural numbers". The Journal of the Indian Mathematical Society. 7 (5): 173–175. • Ramanujan, S. (1916). "Some formulae in the analytical theory of numbers". Messenger Math. 45: 81–84. • Ramanujan, S. (1916). "A Series for Euler's Constant γ". Messenger Math. 46: 73–80. • Ramanujan, S. (1917). "On the expression of numbers in the form ax2 + by2 + cz2 + du2". Mathematical Proceedings of the Cambridge Philosophical Society. 19: 11–21. • Hardy, G. H.; Ramanujan, S. (1917). "Asymptotic Formulae for the Distribution of Integers of Various Types". Proceedings of the London Mathematical Society. 16 (1): 112–132. doi:10.1112/plms/s2-16.1.112. • Hardy, G. H.; Ramanujan, Srinivasa (1918). "Asymptotic Formulae in Combinatory Analysis". Proceedings of the London Mathematical Society. 17 (1): 75–115. doi:10.1112/plms/s2-17.1.75. • Hardy, G. H.; Ramanujan, Srinivasa (1918). "On the coefficients in the expansions of certain modular functions". Proc. R. Soc. A. 95 (667): 144–155. Bibcode:1918RSPSA..95..144H. doi:10.1098/rspa.1918.0056. • Ramanujan, Srinivasa (1919). "Some definite integrals". The Journal of the Indian Mathematical Society. 11 (2): 81–88. • Ramanujan, S. (1919). "A proof of Bertrand's postulate". The Journal of the Indian Mathematical Society. 11 (5): 181–183. • Ramanujan, S. (1920). "A class of definite integrals". Quart. J. Pure. Appl. Math. 48: 294–309. hdl:2027/uc1.$b417568. • Ramanujan, S. (1921). "Congruence properties of partitions". Math. Z. 9 (1–2): 147–153. doi:10.1007/BF01378341. S2CID 121753215. Posthumously published extract of a longer, unpublished manuscript. Further works of Ramanujan's mathematics • George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part I (Springer, 2005, ISBN 0-387-25529-X)[155] • George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part II, (Springer, 2008, ISBN 978-0-387-77765-8) • George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part III, (Springer, 2012, ISBN 978-1-4614-3809-0) • George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part IV, (Springer, 2013, ISBN 978-1-4614-4080-2) • George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part V, (Springer, 2018, ISBN 978-3-319-77832-7) • M. P. Chaudhary, A simple solution of some integrals given by Srinivasa Ramanujan, (Resonance: J. Sci. Education – publication of Indian Academy of Science, 2008)[156] • M.P. Chaudhary, Mock theta functions to mock theta conjectures, SCIENTIA, Series A : Math. Sci., (22)(2012) 33–46. • M.P. Chaudhary, On modular relations for the Roger-Ramanujan type identities, Pacific J. Appl. Math., 7(3)(2016) 177–184. Selected publications on Ramanujan and his work • Berndt, Bruce C. (1998). Butzer, P. L.; Oberschelp, W.; Jongen, H. Th. (eds.). Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe (PDF). Turnhout, Belgium: Brepols Verlag. pp. 119–146. ISBN 978-2-503-50673-9. Archived (PDF) from the original on 9 September 2004. • Berndt, Bruce C.; Rankin, Robert A. (1995). Ramanujan: Letters and Commentary. Vol. 9. Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-0287-8. • Berndt, Bruce C.; Rankin, Robert A. (2001). Ramanujan: Essays and Surveys. Vol. 22. Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-2624-9. • Berndt, Bruce C. (2006). Number Theory in the Spirit of Ramanujan. Vol. 9. Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-4178-5. • Berndt, Bruce C. (1985). Ramanujan's Notebooks: Part I. New York: Springer. ISBN 978-0-387-96110-1. • Berndt, Bruce C. (1999). Ramanujan's Notebooks: Part II. New York: Springer. ISBN 978-0-387-96794-3. • Berndt, Bruce C. (2004). Ramanujan's Notebooks: Part III. New York: Springer. ISBN 978-0-387-97503-0. • Berndt, Bruce C. (1993). Ramanujan's Notebooks: Part IV. New York: Springer. ISBN 978-0-387-94109-7. • Berndt, Bruce C. (2005). Ramanujan's Notebooks: Part V. New York: Springer. ISBN 978-0-387-94941-3. • Hardy, G. H. (March 1937). "The Indian Mathematician Ramanujan". The American Mathematical Monthly. 44 (3): 137–155. doi:10.2307/2301659. JSTOR 2301659. • Hardy, G. H. (1978). Ramanujan. New York: Chelsea Pub. Co. ISBN 978-0-8284-0136-4. • Hardy, G. H. (1999). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-2023-0. • Henderson, Harry (1995). Modern Mathematicians. New York: Facts on File Inc. ISBN 978-0-8160-3235-8. • Kanigel, Robert (1991). The Man Who Knew Infinity: a Life of the Genius Ramanujan. New York: Charles Scribner's Sons. ISBN 978-0-684-19259-8. • Leavitt, David (2007). The Indian Clerk (paperback ed.). London: Bloomsbury. ISBN 978-0-7475-9370-6. • Narlikar, Jayant V. (2003). Scientific Edge: the Indian Scientist From Vedic to Modern Times. New Delhi, India: Penguin Books. ISBN 978-0-14-303028-7. • Ono, Ken; Aczel, Amir D. (13 April 2016). My Search for Ramanujan: How I Learned to Count. Springer. ISBN 978-3319255668. • Sankaran, T. M. (2005). "Srinivasa Ramanujan- Ganitha lokathile Mahaprathibha" (in Malayalam). Kochi, India: Kerala Sastra Sahithya Parishath. {{cite journal}}: Cite journal requires \|journal= (help) Selected publications on works of Ramanujan • Ramanujan, Srinivasa; Hardy, G. H.; Seshu Aiyar, P. V.; Wilson, B. M.; Berndt, Bruce C. (2000). Collected Papers of Srinivasa Ramanujan. AMS. ISBN 978-0-8218-2076-6. This book was originally published in 1927[157] after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third reprint contains additional commentary by Bruce C. Berndt. • S. Ramanujan (1957). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research. These books contain photocopies of the original notebooks as written by Ramanujan. • S. Ramanujan (1988). The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa. ISBN 978-3-540-18726-4. This book contains photo copies of the pages of the "Lost Notebook". • Problems posed by Ramanujan, Journal of the Indian Mathematical Society. • S. Ramanujan (2012). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research. This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai. See also • 1729 (number) • Brown numbers • List of amateur mathematicians • List of Indian mathematicians • Ramanujan graph • Ramanujan summation • Ramanujan's constant • Ramanujan's ternary quadratic form • Rank of a partition References 1. Olausson, Lena; Sangster, Catherine (2006). Oxford BBC Guide to Pronunciation. Oxford University Press. p. 322. ISBN 978-0-19-280710-6. 2. Kanigel, Robert (2004). "Ramanujan, Srinivasa". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/51582. (Subscription or UK public library membership required.) 3. "Ramanujan Aiyangar, Srinivasa (1887-1920)". trove.nla.gov.au. 4. Hans Eysenck (1995). Genius, p. 197. Cambridge University Press, ISBN 0-521-48508-8. 5. Hardy, Godfrey Harold (1940). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge University Press. p. 9. ISBN 0-8218-2023-0. 6. Berndt, Bruce C. (12 December 1997). Ramanujan's Notebooks. Vol. Part 5. Springer Science & Business. p. 4. ISBN 978-0-38794941-3. 7. Ono, Ken (June–July 2006). "Honoring a Gift from Kumbakonam" (PDF). Notices of the American Mathematical Society. 53 (6): 640–51 [649–50]. Archived (PDF) from the original on 21 June 2007. Retrieved 23 June 2007. 8. "Rediscovering Ramanujan". Frontline. 16 (17): 650. August 1999. Archived from the original on 25 September 2013. Retrieved 20 December 2012. 9. Alladi, Krishnaswami; Elliott, P. D. T. A.; Granville, A. (30 September 1998). Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdos. Springer Science & Business. p. 6. ISBN 978-0-79238273-7. 10. Deep meaning in Ramanujan's 'simple' pattern Archived 3 August 2017 at the Wayback Machine 11. "Mathematical proof reveals magic of Ramanujan's genius" Archived 9 July 2017 at the Wayback Machine. New Scientist. 12. Kanigel 1991, p. 12 13. Kanigel 1991, p. 11 14. Kanigel 1991, pp. 17–18 15. Berndt & Rankin 2001a, p. 89 harvnb error: no target: CITEREFBerndtRankin2001a (help) 16. Srinivasan, Pankaja (19 October 2012). "The Nostalgia Formula". The Hindu. Retrieved 7 September 2016. 17. Kanigel 1991, p. 13 18. Kanigel 1991, p. 19 19. Kanigel 1991, p. 14 20. Kanigel 1991, p. 20 21. Kanigel 1991, p. 25 22. Berndt & Rankin 2001b, p. 9 harvnb error: no target: CITEREFBerndtRankin2001b (help) 23. Hardy, G. H. (1999). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Providence, Rhode Island: American Mathematical Society. p. 2. ISBN 978-0-8218-2023-0. 24. Kanigel 1991, p. 27 25. "Srinivasa Ramanujan - Biography". Maths History. Retrieved 29 October 2022. 26. Kanigel 1991, p. 39 27. McElroy, Tucker (2005). A to Z of mathematicians. Facts on File. p. 221. ISBN 0-8160-5338-3- 28. Ramanujan Aiyangar, Srinivasa; Hardy, Godfrey Harold; Aiyar, P. Veṅkatesvara Seshu (2000), "Collected papers of Srinivasa Ramanujan", Nature, 123 (3104): xii, Bibcode:1929Natur.123..631L, doi:10.1038/123631a0, ISBN 978-0-8218-2076-6, S2CID 44812911 29. Kanigel 1991, p. 90 30. Kanigel 1991, p. ?? 31. Kanigel 1991, p. 28 32. Kanigel 1991, p. 45 33. Kanigel 1991, pp. 47–48 34. "Ramanujan lost and found: a 1905 letter from The Hindu". The Hindu. Chennai, India. 25 December 2011. 35. Krishnamachari, Suganthi (27 June 2013). "Travails of a Genius". The Hindu. Archived from the original on 26 August 2017. Retrieved 7 September 2016. 36. Kanigel 1991, p. 55–56 37. Krishnamurthy, V. "Srinivasa Ramanujan – His life and his genius". www.krishnamurthys.com. (Expository address delivered on Sep.16, 1987 at Visvesvarayya Auditorium as part of the celebrations of Ramanujan Centenary by the IISC, Bangalore). Archived from the original on 21 September 2016. Retrieved 7 September 2016. 38. "The seamstress and the mathematician". Live mint. 20 April 2018. 39. >Kanigel 1991, p. 71 40. Bullough, V.L. (1990). "2. History in adult human sexual behavior with children and adolescents in Western societies". Pedophilia: Biosocial Dimensions. New York: Springer-Verlag. p. 71. ISBN 978-1-46139684-0. 41. Kolata, Gina (19 June 1987). "Remembering a 'Magical Genius'". Science. New Series. 236 (4808): 1519–21. Bibcode:1987Sci...236.1519K. doi:10.1126/science.236.4808.1519. PMID 17835731. 42. "Ramanujan's wife: Janakiammal (Janaki)" (PDF). Chennai: Institute of Mathematical Sciences. Archived from the original (PDF) on 24 December 2012. Retrieved 10 November 2012. 43. Janardhanan, Arun (6 December 2015). "A passage to infinity". Indian Express. Archived from the original on 5 September 2016. Retrieved 7 September 2016. 44. >Kanigel 1991, p. 72 45. Ramanujan, Srinivasa (1968). P. K. Srinivasan (ed.). Ramanujan Memorial Number: Letters and Reminiscences. Vol. 1. Madras: Muthialpet High School. 100. 46. Kanigel 1991, p. 73 47. Kanigel 1991, pp. 74–75 48. Ranganathan, Shiyali Ramamrita (1967). Ramanujan: The Man and the Mathematician. Bombay: Asia Publishing House. p. 23. ISBN 9788185273372. 49. Srinivasan (1968), Vol. 1, p. 99. 50. Rao, K. Srinivasa. "Ramanujan's wife Janakiammal (Janaki)" (PDF). IMSC. Institute of Mathematical Sciences, Chennai. Archived from the original (PDF) on 10 January 2017. Retrieved 7 September 2016. 51. "About Ramanujan". The Ramanujan Institute. Archived from the original on 6 October 2016. Retrieved 7 September 2016. 52. Kanigel 1991, p. 77 53. Srinivasan (1968), Vol. 1, p. 129. 54. Srinivasan (1968), Vol. 1, p. 86. 55. Neville, Eric Harold (January 1921). "The Late Srinivasa Ramanujan". Nature. 106 (2673): 661–662. Bibcode:1921Natur.106..661N. doi:10.1038/106661b0. S2CID 4185656. 56. Ranganathan 1967, p. 24 57. Kanigel 1991, p. 80 58. Kanigel 1991, p. 86 59. Herschfeld, Aaron (August 1935). "On Infinite Radicals". The American Mathematical Monthly. 42 (7): 419–429. doi:10.1080/00029890.1935.11987745. ISSN 0002-9890. 60. Kanigel 1991, p. 87 61. Kanigel 1991, p. 91 62. Seshu Iyer, P. V. (June 1920). "The Late Mr. S. Ramanujan, B.A., F.R.S.". Journal of the Indian Mathematical Society. 12 (3): 83. 63. Neville, Eric Harold (1942). "Srinivasa Ramanujan". Nature. 149 (3776): 292–293. Bibcode:1942Natur.149..292N. doi:10.1038/149292a0. 64. Srinivasan (1968), p. 176. 65. Srinivasan (1968), p. 31. 66. Srinivasan (1968), p. 49. 67. Kanigel 1991, p. 96 68. Kanigel 1991, p. 105 69. Letter from M. J. M. Hill to a C. L. T. Griffith (a former student who sent the request to Hill on Ramanujan's behalf), 28 November 1912. 70. Kanigel 1991, p. 106 71. Kanigel 1991, pp. 170–171 72. "The letter that revealed Ramanujan's genius". YouTube. 73. Snow, C. P. (1966). Variety of Men. New York: Charles Scribner's Sons. pp. 30–31. 74. Hardy, G. H. (1920). "Obituary, S. Ramanujan". Nature. 105 (7): 494–495. Bibcode:1920Natur.105..494H. doi:10.1038/105494a0. S2CID 4174904. 75. Kanigel 1991, p. 167 76. Kanigel 1991, p. 168 77. Letter, Hardy to Ramanujan, 8 February 1913. 78. Letter, Ramanujan to Hardy, 22 January 1914. 79. Kanigel 1991, p. 185 80. Letter, Ramanujan to Hardy, 27 February 1913, Cambridge University Library. 81. Kanigel 1991, p. 175 82. Ram, Suresh (1972). Srinivasa Ramanujan. New Delhi: National Book Trust. p. 29. 83. Ranganathan 1967, pp. 30–31 84. Ranganathan 1967, p. 12 85. Kanigel 1991, p. 183 86. Kanigel 1991, p. 184 87. "A (very) Brief History of Srinivasa Ramanujan". YouTube. Archived from the original on 11 December 2021. 88. Kanigel 1991, p. 196 89. Kanigel 1991, p. 202 90. Hardy, G. H. (1940). Ramanujan. Cambridge: Cambridge University Press. p. 10. 91. Letter, Littlewood to Hardy, early March 1913. 92. Hardy, G. H. (1979). Collected Papers of G. H. Hardy. Vol. 7. Oxford, England: Clarendon Press. 720. 93. The Cambridge University Reporter, of 18 March 1916, reports: Bachelors designate in Arts, Srinivasa Ramanujan (Research Student), Trin. A clear photographic image of said document can be viewed on the following YouTube video at the specified timestamp: https://www.youtube.com/watch?v=uhNGCn_3hmc&t=1636 94. "The Maths PhD in the UK: Notes on its History". www.economics.soton.ac.uk. Retrieved 9 August 2020. 95. Jean-Louis Nicolas, Guy Robin (eds.), Highly Composite Numbers by Srinivasa Ramanujan, The Ramanujan Journal 1997 1, 119–153, p.121 96. Embleton, Ellen (2 October 2018). "Revisiting Ramanujan". The Royal Society. The Royal Society. Archived from the original on 16 February 2020. Retrieved 16 February 2020. 97. Kanigel 1991, pp. 299–300 98. Young, D. A. B. (1994). "Ramanujan's illness". Notes and Records of the Royal Society of London. 48 (1): 107–119. doi:10.1098/rsnr.1994.0009. PMID 11615274. S2CID 33416179. 99. Peterson, Doug. "Raiders of the Lost Notebook". UIUC College of Liberal Arts and Sciences. Archived from the original on 12 January 2014. Retrieved 11 January 2014. 100. Gunn, J. W. C.; Savage, B. (1919). "Report on the treatment of Entamoeba histolytica infections". Journal of the Royal Army Medical Corps. 33 (5): 418–426. 101. Langley, George J. (24 December 1921). "The Difficulties in Diagnosis And Treatment of Hepatic Abscess". British Medical Journal. 2 (3182): 1073–1074. doi:10.1136/bmj.2.3182.1073. JSTOR 20429465. PMC 2339657. PMID 20770524. 102. Baaquie, Belal E.; Willeboordse, Frederick H. (1 December 2009). Exploring Integrated Science. CRC Press. p. 38. ISBN 978-1-4200-8794-9. 103. "Ramanujan's Personality". Archived from the original on 27 September 2007. Retrieved 23 June 2018. 104. Kanigel 1991, pp. 234, 241 105. Kanigel 1991, p. 36 106. Kanigel 1991, p. 281 107. Chaitin, Gregory (28 July 2007). "Less Proof, More Truth". New Scientist (2614): 49. doi:10.1016/S0262-4079(07)61908-3. 108. Kanigel 1991, p. 283 109. Berndt, Bruce C.; Rankin, Robert Alexander (2001). Ramanujan: Essays and Surveys. American Mathematical Society. p. 47. ISBN 978-0-82182624-9. Retrieved 8 June 2015. 110. Ranganathan 1967, p. 82 111. Calyampudi Radhakrishna Rao (1997). Statistics and truth: putting chance to work. World Scientific. p. 185. ISBN 978-981-02-3111-8. Retrieved 7 June 2010. 112. Iwaniec, Henryk (1989). "The circle method and the Fourier coefficients of modular forms". Number theory and related topics (Bombay, 1988). Tata Inst. Fund. Res. Stud. Math. Vol. 12. Tata Inst. Fund. Res., Bombay. pp. 47–55. MR 1441326. 113. "100-Year-Old Deathbed Dreams of Mathematician Proved True". Fox News. 28 December 2012. Archived from the original on 7 January 2013. 114. Winnie, Wen-Ching (9 December 2019). "The Ramanujan conjecture and its applications". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 378 (2163). doi:10.1098/rsta.2018.0441. PMC 6939229. PMID 31813366. 115. Ramanujan, Srinivasa (1916). "On certain arithmetical functions" (PDF). Transactions of the Cambridge Philosophical Society. XXII (9). Archived from the original (PDF) on 11 June 2016. Retrieved 15 May 2016. The tau function is discussed in pages 194–197. 116. Ono, Ken; Aczel, Amir D. (13 April 2016). My Search for Ramanujan: How I Learned to Count. Springer. pp. 236–237. ISBN 978-3319255668. ideas that were critical to the proof of Fermat's last theorem 117. Berndt, Bruce C. (12 December 1997). Ramanujans Notebooks. ISBN 978-0387949413. 118. "Quotations by Hardy". Gap.dcs.st-and.ac.uk. Archived from the original on 16 July 2012. Retrieved 20 November 2012. 119. Hardy, G. H. "Obituary Notices: Srinivasa Ramanujan". Proceedings of the London Mathematical Society. 19: lvii. Archived from the original on 5 March 2016. 120. James R. Newman, ed. (2000). The world of mathematics. Mineola, NY: Dover Publications. pp. 373–4. ISBN 978-0-486-41153-8. OCLC 43555029. 121. Srinivasa Ramanujan Archived 25 March 2005 at the Wayback Machine. Retrieved 2 December 2010. 122. Alyangar, Srinivasa Ramanujan (1995). Berndt, Bruce C.; Rankin, Robert Alexander (eds.). Ramanujan: Letters and Commentary. History of Mathematics. Vol. 9. American Mathematical Society. p. 51. ISBN 0821891251. Retrieved 22 February 2023. 123. Rao, K Srinivasa. "Srinivasa Ramanujan (22 December 1887 – 26 April 1920)". Archived from the original on 16 April 2012. Retrieved 23 June 2018. 124. "Bruce Berndt on "Ramanujan's Lost Notebook", IIT Madras, 24th May 2011". youtube.com. Archived from the original on 6 December 2015. 125. "Calendar of Scientific Pioneers". Nature. 107 (2686): 252–254. 21 April 1921. Bibcode:1921Natur.107..252.. doi:10.1038/107252b0. 126. Srinivasa Ramanujan on stamps. commons.wikimedia.org 127. "Sastra University – Srinivasa Ramanujan Center – About Us". Archived from the original on 15 June 2017. Retrieved 23 June 2018. 128. "Singh's first visit to the state". CNN IBN. India. 26 December 2011. Archived from the original on 15 July 2012. Retrieved 12 April 2016. 129. "Welcome 2012 – The National Mathematical Year in India". India. 28 December 2011. Archived from the original on 6 December 2017. Retrieved 6 December 2017. 130. "Ramanujan IT City - Hardy Tower". JLL Property India. 131. "The Man Who Loved Numbers". IMDB. 22 March 1988. Retrieved 24 September 2021. 132. "Cannes: Dev Patel to Star as Famed Indian Mathematician". hollywoodreporter.com. Archived from the original on 9 January 2014. 133. Barraclough, Leo (5 December 2013). "Jeremy Irons to Co-star in 'The Man Who Knew Infinity'". variety.com. Archived from the original on 12 October 2017. 134. McNary, Dave (15 July 2014). "Dev Patel's 'The Man Who Knew Infinity' Moves to Production After 8 Years in Development". variety.com. Archived from the original on 4 July 2017. 135. "'Ramanujan' Makers Shoot in His House". Indiatimes. Times Internet Limited. Archived from the original on 11 July 2013. Retrieved 12 July 2013. 136. "Camphor Cinema Presents Their First Film Ramanujan". Box Office India. Select Publishing Company. 11 June 2013. Archived from the original on 20 August 2013. Retrieved 12 July 2013. 137. "Makers of 'Ramanujan' shoot in genius' house". Z News. Zee Media Corporation Ltd. Archived from the original on 8 July 2013. Retrieved 12 July 2013. 138. Krishnamachari, Suganthy (27 June 2013). "Travails of a genius". The Hindu. Chennai, India. Archived from the original on 1 July 2013. Retrieved 12 July 2013. 139. "Pune-based filmmaker wins 3 awards at National Science Film festival". The Indian Express. 27 February 2017. 140. "Ramanujan (The Man who reshaped 20th Century Mathematics) (2018)". Indiancine.ma. 141. Basu, Kankana (7 December 2014). "Racy read". The Hindu. Retrieved 30 April 2016. 142. "Crime in a World of High Science". 16 September 2014. Archived from the original on 15 April 2016. Retrieved 30 April 2016. 143. Ribet, Kenneth A. (December 2003). "Theater Review. Partition" (PDF). Notices of the AMS. 50 (1): 1407–1408. Archived (PDF) from the original on 6 October 2016. Retrieved 27 September 2016. 144. Harvey, Dennis (18 May 2003). "Review: 'Partition'". Archived from the original on 6 October 2016. Retrieved 23 March 2017. 145. "Partitions – a play on Ramanujan". The Hindu. 26 May 2003. Archived from the original on 20 July 2008. 146. DATTA, SRAVASTI (19 December 2014). "An ode to a genius". The Hindu. Retrieved 23 March 2017. 147. "First Class Man". Alteregoproductions.org. Archived from the original on 29 June 2007. Retrieved 20 November 2012. 148. "News / National: James Bond director to make film on Ramanujan". The Hindu. India. 16 October 2011. Archived from the original on 17 October 2011. Retrieved 18 October 2011. 149. Lunden, Jeff (15 July 2010). "'Disappearing Number': A Vivid Theatrical Equation". Morning Edition. National Public Radio. Retrieved 24 April 2018. 150. Freudenberger, Nell (16 September 2007). "Lust for Numbers". The New York Times. Archived from the original on 10 January 2012. Retrieved 4 September 2011. 151. Taylor, D. J. (26 January 2008). "Adding up to a life". The Guardian. UK. Archived from the original on 6 October 2014. Retrieved 4 September 2011. 152. "Google doodles for Ramanujan's 125th birthday". Times of India. 22 December 2012. Archived from the original on 22 December 2012. Retrieved 22 December 2012. 153. "Srinivasa Ramanujan's 125th Birthday". www.google.com. Archived from the original on 10 May 2016. Retrieved 30 April 2016. 154. Kumar, V. Krishna (2 February 2018). "A Legendary Creative Math Genius: Srinivasa Ramanujan". Psychology Today. Retrieved 24 April 2018. 155. Bressoud, David (2006). "Review: Ramanujan's Lost Notebook, Part I, by George Andrews and Bruce C. Berndt" (PDF). Bulletin of the American Mathematical Society. New Series. 43 (4): 585–591. doi:10.1090/s0273-0979-06-01110-4. Archived (PDF) from the original on 23 June 2015. 156. "A simple solution of some integrals given by Srinivasa Ramanujan" (PDF). Resonance. 13 (9): 882–884. Archived (PDF) from the original on 6 October 2018. 157. Bell, E. T. (1928). "Collected Papers of Srinivasa Ramanujan edited by G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson". Bulletin of the American Mathematical Society. 34 (6): 783–784. doi:10.1090/S0002-9904-1928-04651-7. External links Media links • Biswas, Soutik (16 March 2006). "Film to celebrate mathematics genius". BBC. Retrieved 24 August 2006. • Feature Film on Mathematics Genius Ramanujan by Dev Benegal and Stephen Fry • BBC radio programme about Ramanujan – episode 5 • A biographical song about Ramanujan's life • "Why Did This Mathematician's Equations Make Everyone So Angry?". Youtube.com. Thoughty2. 11 April 2022. Retrieved 29 June 2022. Biographical links • Srinivasa Ramanujan at the Mathematics Genealogy Project • O'Connor, John J.; Robertson, Edmund F., "Srinivasa Ramanujan", MacTutor History of Mathematics Archive, University of St Andrews • Weisstein, Eric Wolfgang (ed.). "Ramanujan, Srinivasa (1887–1920)". ScienceWorld. • A short biography of Ramanujan • "Our Devoted Site for Great Mathematical Genius" Other links • Wolfram, Stephen (27 April 2016). "Who Was Ramanujan?". • A Study Group For Mathematics: Srinivasa Ramanujan Iyengar • The Ramanujan Journal – An international journal devoted to Ramanujan • International Math Union Prizes, including a Ramanujan Prize • Hindu.com: Norwegian and Indian mathematical geniuses, Ramanujan – Essays and Surveys Archived 6 November 2012 at the Wayback Machine, Ramanujan's growing influence, Ramanujan's mentor • Hindu.com: The sponsor of Ramanujan • Bruce C. Berndt; Robert A. Rankin (2000). "The Books Studied by Ramanujan in India". American Mathematical Monthly. 107 (7): 595–601. doi:10.2307/2589114. JSTOR 2589114. MR 1786233. • "Ramanujan's mock theta function puzzle solved" • Ramanujan's papers and notebooks • Sample page from the second notebook • Ramanujan on Fried Eye • Clark, Alex. "163 and Ramanujan Constant". Numberphile. Brady Haran. Archived from the original on 4 February 2018. Retrieved 23 June 2018. Indian mathematics Mathematicians Ancient • Apastamba • Baudhayana • Katyayana • Manava • Pāṇini • Pingala • Yajnavalkya Classical • Āryabhaṭa I • Āryabhaṭa II • Bhāskara I • Bhāskara II • Melpathur Narayana Bhattathiri • Brahmadeva • Brahmagupta • Govindasvāmi • Halayudha • Jyeṣṭhadeva • Kamalakara • Mādhava of Saṅgamagrāma • Mahāvīra • Mahendra Sūri • Munishvara • Narayana • Parameshvara • Achyuta Pisharati • Jagannatha Samrat • Nilakantha Somayaji • Śrīpati • Sridhara • Gangesha Upadhyaya • Varāhamihira • Sankara Variar • Virasena Modern • Shanti Swarup Bhatnagar Prize recipients in Mathematical Science Treatises • Āryabhaṭīya • Bakhshali manuscript • Bijaganita • Brāhmasphuṭasiddhānta • Ganita Kaumudi • Karanapaddhati • Līlāvatī • Lokavibhaga • Paulisa Siddhanta • Paitamaha Siddhanta • Romaka Siddhanta • Sadratnamala • Siddhānta Shiromani • Śulba Sūtras • Surya Siddhanta • Tantrasamgraha • Vasishtha Siddhanta • Veṇvāroha • Yuktibhāṣā • Yavanajataka Pioneering innovations • Brahmi numerals • Hindu–Arabic numeral system • Symbol for zero (0) • Infinite series expansions for the trigonometric functions Centres • Kerala school of astronomy and mathematics • Jantar Mantar (Jaipur, New Delhi, Ujjain, Varanasi) Historians of mathematics • Bapudeva Sastri (1821–1900) • Shankar Balakrishna Dikshit (1853–1898) • Sudhakara Dvivedi (1855–1910) • M. Rangacarya (1861–1916) • P. C. Sengupta (1876–1962) • B. B. Datta (1888–1958) • T. Hayashi • A. A. Krishnaswamy Ayyangar (1892– 1953) • A. N. Singh (1901–1954) • C. T. Rajagopal (1903–1978) • T. A. Saraswati Amma (1918–2000) • S. N. Sen (1918–1992) • K. S. Shukla (1918–2007) • K. V. Sarma (1919–2005) Translators • Walter Eugene Clark • David Pingree Other regions • Babylon • China • Greece • Islamic mathematics • Europe Modern institutions • Indian Statistical Institute • Bhaskaracharya Pratishthana • Chennai Mathematical Institute • Institute of Mathematical Sciences • Indian Institute of Science • Harish-Chandra Research Institute • Homi Bhabha Centre for Science Education • Ramanujan Institute for Advanced Study in Mathematics • TIFR Authority control International • FAST • ISNI • VIAF National • Norway • France • BnF data • Catalonia • Germany • Israel • United States • Sweden • Japan • Czech Republic • Australia • Greece • Korea • Netherlands • Poland Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH People • Trove • 2 Other • SNAC • IdRef	Wikipedia
Ramanujan's congruences In mathematics, Ramanujan's congruences are some remarkable congruences for the partition function p(n). The mathematician Srinivasa Ramanujan discovered the congruences ${\begin{aligned}p(5k+4)&\equiv 0{\pmod {5}},\\p(7k+5)&\equiv 0{\pmod {7}},\\p(11k+6)&\equiv 0{\pmod {11}}.\end{aligned}}$ This means that: • If a number is 4 more than a multiple of 5, i.e. it is in the sequence 4, 9, 14, 19, 24, 29, . . . then the number of its partitions is a multiple of 5. • If a number is 5 more than a multiple of 7, i.e. it is in the sequence 5, 12, 19, 26, 33, 40, . . . then the number of its partitions is a multiple of 7. • If a number is 6 more than a multiple of 11, i.e. it is in the sequence 6, 17, 28, 39, 50, 61, . . . then the number of its partitions is a multiple of 11. Background In his 1919 paper,[1] he proved the first two congruences using the following identities (using q-Pochhammer symbol notation): ${\begin{aligned}&\sum _{k=0}^{\infty }p(5k+4)q^{k}=5{\frac {(q^{5})_{\infty }^{5}}{(q)_{\infty }^{6}}},\\[4pt]&\sum _{k=0}^{\infty }p(7k+5)q^{k}=7{\frac {(q^{7})_{\infty }^{3}}{(q)_{\infty }^{4}}}+49q{\frac {(q^{7})_{\infty }^{7}}{(q)_{\infty }^{8}}}.\end{aligned}}$ He then stated that "It appears there are no equally simple properties for any moduli involving primes other than these". After Ramanujan died in 1920, G. H. Hardy extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on p(n) (Ramanujan, 1921). The proof in this manuscript employs the Eisenstein series. In 1944, Freeman Dyson defined the rank function and conjectured the existence of a crank function for partitions that would provide a combinatorial proof of Ramanujan's congruences modulo 11. Forty years later, George Andrews and Frank Garvan found such a function, and proved the celebrated result that the crank simultaneously "explains" the three Ramanujan congruences modulo 5, 7 and 11. In the 1960s, A. O. L. Atkin of the University of Illinois at Chicago discovered additional congruences for small prime moduli. For example: $p(11^{3}\cdot 13k+237)\equiv 0{\pmod {13}}.$ Extending the results of A. Atkin, Ken Ono in 2000 proved that there are such Ramanujan congruences modulo every integer coprime to 6. For example, his results give $p(107^{4}\cdot 31k+30064597)\equiv 0{\pmod {31}}.$ Later Ken Ono conjectured that the elusive crank also satisfies exactly the same types of general congruences. This was proved by his Ph.D. student Karl Mahlburg in his 2005 paper Partition Congruences and the Andrews–Garvan–Dyson Crank, linked below. This paper won the first Proceedings of the National Academy of Sciences Paper of the Year prize.[2] A conceptual explanation for Ramanujan's observation was finally discovered in January 2011 [3] by considering the Hausdorff dimension of the following $P$ function in the l-adic topology: $P_{\ell }(b;z):=\sum _{n=0}^{\infty }p\left({\frac {\ell ^{b}n+1}{24}}\right)q^{n/24}.$ It is seen to have dimension 0 only in the cases where ℓ = 5, 7 or 11 and since the partition function can be written as a linear combination of these functions[4] this can be considered a formalization and proof of Ramanujan's observation. In 2001, R.L. Weaver gave an effective algorithm for finding congruences of the partition function, and tabulated 76,065 congruences.[5] This was extended in 2012 by F. Johansson to 22,474,608,014 congruences,[6] one large example being $p(999959^{4}\cdot 29k+28995221336976431135321047)\equiv 0{\pmod {29}}.$ See also • Tau-function, for which there are other so-called Ramanujan congruences • Rank of a partition • Crank of a partition References 1. Ramanujan, S. (1921). "Congruence properties of partitions". Mathematische Zeitschrift. 9 (1–2): 147–153. doi:10.1007/bf01378341. S2CID 121753215. 2. "Cozzarelli Prize". National Academy of Sciences. June 2014. Retrieved 2014-08-06. 3. Folsom, Amanda; Kent, Zachary A.; Ono, Ken (2012). "ℓ-Adic properties of the partition function". Advances in Mathematics. 229 (3): 1586. doi:10.1016/j.aim.2011.11.013. 4. Bruinier, J. H.; Ono, K. (2011). "Algebraic Formulas for the Coefficients of Half-Integral Weight Harmonic Weak Maas Forms" (PDF). arXiv:1104.1182. Bibcode:2011arXiv1104.1182H. {{cite journal}}: Cite journal requires \|journal= (help) 5. Weaver, Rhiannon L. (2001). "New congruences for the partition function". The Ramanujan Journal. 5: 53–63. doi:10.1023/A:1011493128408. S2CID 119699656. 6. Johansson, Fredrik (2012). "Efficient implementation of the Hardy–Ramanujan–Rademacher formula". LMS Journal of Computation and Mathematics. 15: 341–359. arXiv:1205.5991. doi:10.1112/S1461157012001088. S2CID 16580723. • Ono, Ken (2000). "Distribution of the partition function modulo m". Annals of Mathematics. Second Series. 151 (1): 293–307. arXiv:math/0008140. Bibcode:2000math......8140O. doi:10.2307/121118. JSTOR 121118. S2CID 119750203. Zbl 0984.11050. • Ono, Ken (2004). The web of modularity: arithmetic of the coefficients of modular forms and q-series. CBMS Regional Conference Series in Mathematics. Vol. 102. Providence, RI: American Mathematical Society. ISBN 978-0-8218-3368-1. Zbl 1119.11026. • Ramanujan, S. (1919). "Some properties of p(n), the number of partitions of n". Proceedings of the Cambridge Philosophical Society. 19: 207–210. JFM 47.0885.01. External links • Mahlburg, K. (2005). "Partition Congruences and the Andrews–Garvan–Dyson Crank" (PDF). Proceedings of the National Academy of Sciences. 102 (43): 15373–76. Bibcode:2005PNAS..10215373M. doi:10.1073/pnas.0506702102. PMC 1266116. PMID 16217020. • Dyson's rank, crank and adjoint. A list of references.	Wikipedia
Heegner number In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field $\mathbb {Q} \left[{\sqrt {-d}}\right]$ has class number 1. Equivalently, the ring of algebraic integers of $\mathbb {Q} \left[{\sqrt {-d}}\right]$ has unique factorization.[1] The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory. According to the (Baker–)Stark–Heegner theorem there are precisely nine Heegner numbers: 1, 2, 3, 7, 11, 19, 43, 67, and 163. (sequence A003173 in the OEIS) This result was conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966, and Stark further indicated the gap in Heegner's proof was minor.[2] Euler's prime-generating polynomial Euler's prime-generating polynomial $n^{2}+n+41,$ which gives (distinct) primes for n = 0, ..., 39, is related to the Heegner number 163 = 4 · 41 − 1. Rabinowitz[3] proved that $n^{2}+n+p$ gives primes for $n=0,\dots ,p-2$ if and only if this quadratic's discriminant $1-4p$ is the negative of a Heegner number. (Note that $p-1$ yields $p^{2}$, so $p-2$ is maximal.) 1, 2, and 3 are not of the required form, so the Heegner numbers that work are 7, 11, 19, 43, 67, 163, yielding prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.[4] Almost integers and Ramanujan's constant Ramanujan's constant is the transcendental number[5] $e^{\pi {\sqrt {163}}}$, which is an almost integer, in that it is very close to an integer:[6] $e^{\pi {\sqrt {163}}}=262\,537\,412\,640\,768\,743.999\,999\,999\,999\,25\ldots \approx 640\,320^{3}+744.$ This number was discovered in 1859 by the mathematician Charles Hermite.[7] In a 1975 April Fool article in Scientific American magazine,[8] "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name. This coincidence is explained by complex multiplication and the q-expansion of the j-invariant. Detail In what follows, j(z) denotes the j-invariant of the complex number z. Briefly, $\textstyle j\left({\frac {1+{\sqrt {-d}}}{2}}\right)$ is an integer for d a Heegner number, and $e^{\pi {\sqrt {d}}}\approx -j\left({\frac {1+{\sqrt {-d}}}{2}}\right)+744$ via the q-expansion. If $\tau $ is a quadratic irrational, then the j-invariant is an algebraic integer of degree $\left\|\mathrm {Cl} {\bigl (}\mathbf {Q} (\tau ){\bigr )}\right\|$, the class number of $\mathbf {Q} (\tau )$ and the minimal (monic integral) polynomial it satisfies is called the 'Hilbert class polynomial'. Thus if the imaginary quadratic extension $\mathbf {Q} (\tau )$ has class number 1 (so d is a Heegner number), the j-invariant is an integer. The q-expansion of j, with its Fourier series expansion written as a Laurent series in terms of $q=e^{2\pi i\tau }$, begins as: $j(\tau )={\frac {1}{q}}+744+196\,884q+\cdots .$ The coefficients $c_{n}$ asymptotically grow as $\ln(c_{n})\sim 4\pi {\sqrt {n}}+O{\bigl (}\ln(n){\bigr )},$ and the low order coefficients grow more slowly than $200\,000^{n}$, so for $\textstyle q\ll {\frac {1}{200\,000}}$, j is very well approximated by its first two terms. Setting $\textstyle \tau ={\frac {1+{\sqrt {-163}}}{2}}$ yields $q=-e^{-\pi {\sqrt {163}}}\quad \therefore \quad {\frac {1}{q}}=-e^{\pi {\sqrt {163}}}.$ Now $j\left({\frac {1+{\sqrt {-163}}}{2}}\right)=\left(-640\,320\right)^{3},$ so, $\left(-640\,320\right)^{3}=-e^{\pi {\sqrt {163}}}+744+O\left(e^{-\pi {\sqrt {163}}}\right).$ Or, $e^{\pi {\sqrt {163}}}=640\,320^{3}+744+O\left(e^{-\pi {\sqrt {163}}}\right)$ where the linear term of the error is, ${\frac {-196\,884}{e^{\pi {\sqrt {163}}}}}\approx {\frac {-196\,884}{640\,320^{3}+744}}\approx -0.000\,000\,000\,000\,75$ explaining why $e^{\pi {\sqrt {163}}}$ is within approximately the above of being an integer. Pi formulas The Chudnovsky brothers found in 1987 that ${\frac {1}{\pi }}={\frac {12}{640\,320^{\frac {3}{2}}}}\sum _{k=0}^{\infty }{\frac {(6k)!(163\cdot 3\,344\,418k+13\,591\,409)}{(3k)!(k!)^{3}(-640\,320)^{3k}}},$ a proof of which uses the fact that $j\left({\frac {1+{\sqrt {-163}}}{2}}\right)=-640\,320^{3}.$ For similar formulas, see the Ramanujan–Sato series. Other Heegner numbers For the four largest Heegner numbers, the approximations one obtains[9] are as follows. ${\begin{aligned}e^{\pi {\sqrt {19}}}&\approx {\color {white}000\,0}96^{3}+744-0.22\\e^{\pi {\sqrt {43}}}&\approx {\color {white}000\,}960^{3}+744-0.000\,22\\e^{\pi {\sqrt {67}}}&\approx {\color {white}00}5\,280^{3}+744-0.000\,0013\\e^{\pi {\sqrt {163}}}&\approx 640\,320^{3}+744-0.000\,000\,000\,000\,75\end{aligned}}$ Alternatively,[10] ${\begin{aligned}e^{\pi {\sqrt {19}}}&\approx 12^{3}\left(3^{2}-1\right)^{3}{\color {white}00}+744-0.22\\e^{\pi {\sqrt {43}}}&\approx 12^{3}\left(9^{2}-1\right)^{3}{\color {white}00}+744-0.000\,22\\e^{\pi {\sqrt {67}}}&\approx 12^{3}\left(21^{2}-1\right)^{3}{\color {white}0}+744-0.000\,0013\\e^{\pi {\sqrt {163}}}&\approx 12^{3}\left(231^{2}-1\right)^{3}+744-0.000\,000\,000\,000\,75\end{aligned}}$ where the reason for the squares is due to certain Eisenstein series. For Heegner numbers $d<19$, one does not obtain an almost integer; even $d=19$ is not noteworthy.[11] The integer j-invariants are highly factorisable, which follows from the form $12^{3}\left(n^{2}-1\right)^{3}=\left(2^{2}\cdot 3\cdot (n-1)\cdot (n+1)\right)^{3},$ and factor as, ${\begin{aligned}j\left({\frac {1+{\sqrt {-19}}}{2}}\right)&={\color {white}000\,0}96^{3}=\left(2^{5}\cdot 3\right)^{3}\\j\left({\frac {1+{\sqrt {-43}}}{2}}\right)&={\color {white}000\,}960^{3}=\left(2^{6}\cdot 3\cdot 5\right)^{3}\\j\left({\frac {1+{\sqrt {-67}}}{2}}\right)&={\color {white}00}5\,280^{3}=\left(2^{5}\cdot 3\cdot 5\cdot 11\right)^{3}\\j\left({\frac {1+{\sqrt {-163}}}{2}}\right)&=640\,320^{3}=\left(2^{6}\cdot 3\cdot 5\cdot 23\cdot 29\right)^{3}.\end{aligned}}$ These transcendental numbers, in addition to being closely approximated by integers (which are simply algebraic numbers of degree 1), can be closely approximated by algebraic numbers of degree 3,[12] ${\begin{aligned}e^{\pi {\sqrt {19}}}&\approx x^{24}-24.000\,31;&x^{3}-2x-2&=0\\e^{\pi {\sqrt {43}}}&\approx x^{24}-24.000\,000\,31;&x^{3}-2x^{2}-2&=0\\e^{\pi {\sqrt {67}}}&\approx x^{24}-24.000\,000\,0019;&x^{3}-2x^{2}-2x-2&=0\\e^{\pi {\sqrt {163}}}&\approx x^{24}-24.000\,000\,000\,000\,0011;&\quad x^{3}-6x^{2}+4x-2&=0\end{aligned}}$ The roots of the cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 24th root, and which explains the 24 in the approximation. They can also be closely approximated by algebraic numbers of degree 4,[13] ${\begin{aligned}e^{\pi {\sqrt {19}}}&\approx 3^{5}\left(3-{\sqrt {2\left(1-{\tfrac {96}{24}}+1{\sqrt {3\cdot 19}}\right)}}\right)^{-2}-12.000\,06\dots \\e^{\pi {\sqrt {43}}}&\approx 3^{5}\left(9-{\sqrt {2\left(1-{\tfrac {960}{24}}+7{\sqrt {3\cdot 43}}\right)}}\right)^{-2}-12.000\,000\,061\dots \\e^{\pi {\sqrt {67}}}&\approx 3^{5}\left(21-{\sqrt {2\left(1-{\tfrac {5\,280}{24}}+31{\sqrt {3\cdot 67}}\right)}}\right)^{-2}-12.000\,000\,000\,36\dots \\e^{\pi {\sqrt {163}}}&\approx 3^{5}\left(231-{\sqrt {2\left(1-{\tfrac {640\,320}{24}}+2\,413{\sqrt {3\cdot 163}}\right)}}\right)^{-2}-12.000\,000\,000\,000\,000\,21\dots \end{aligned}}$ If $x$ denotes the expression within the parenthesis (e.g. $x=3-{\sqrt {2\left(1-{\tfrac {96}{24}}+1{\sqrt {3\cdot 19}}\right)}}$), it satisfies respectively the quartic equations ${\begin{aligned}x^{4}-{\color {white}00}4\cdot 3x^{3}+{\color {white}000\,0}{\tfrac {2}{3}}(96+3)x^{2}-{\color {white}000\,000}{\tfrac {2}{3}}\cdot 3(96-6)x-3&=0\\x^{4}-{\color {white}00}4\cdot 9x^{3}+{\color {white}000\,}{\tfrac {2}{3}}(960+3)x^{2}-{\color {white}000\,00}{\tfrac {2}{3}}\cdot 9(960-6)x-3&=0\\x^{4}-{\color {white}0}4\cdot 21x^{3}+{\color {white}00}{\tfrac {2}{3}}(5\,280+3)x^{2}-{\color {white}000}{\tfrac {2}{3}}\cdot 21(5\,280-6)x-3&=0\\x^{4}-4\cdot 231x^{3}+{\tfrac {2}{3}}(640\,320+3)x^{2}-{\tfrac {2}{3}}\cdot 231(640\,320-6)x-3&=0\\\end{aligned}}$ Note the reappearance of the integers $n=3,9,21,231$ as well as the fact that ${\begin{aligned}2^{6}\cdot 3\left(-\left(1-{\tfrac {96}{24}}\right)^{2}+1^{2}\cdot 3\cdot 19\right)&=96^{2}\\2^{6}\cdot 3\left(-\left(1-{\tfrac {960}{24}}\right)^{2}+7^{2}\cdot 3\cdot 43\right)&=960^{2}\\2^{6}\cdot 3\left(-\left(1-{\tfrac {5\,280}{24}}\right)^{2}+31^{2}\cdot 3\cdot 67\right)&=5\,280^{2}\\2^{6}\cdot 3\left(-\left(1-{\tfrac {640\,320}{24}}\right)^{2}+2413^{2}\cdot 3\cdot 163\right)&=640\,320^{2}\end{aligned}}$ which, with the appropriate fractional power, are precisely the j-invariants. Similarly for algebraic numbers of degree 6, ${\begin{aligned}e^{\pi {\sqrt {19}}}&\approx \left(5x\right)^{3}-6.000\,010\dots \\e^{\pi {\sqrt {43}}}&\approx \left(5x\right)^{3}-6.000\,000\,010\dots \\e^{\pi {\sqrt {67}}}&\approx \left(5x\right)^{3}-6.000\,000\,000\,061\dots \\e^{\pi {\sqrt {163}}}&\approx \left(5x\right)^{3}-6.000\,000\,000\,000\,000\,034\dots \end{aligned}}$ where the xs are given respectively by the appropriate root of the sextic equations, ${\begin{aligned}5x^{6}-{\color {white}000\,0}96x^{5}-10x^{3}+1&=0\\5x^{6}-{\color {white}000\,}960x^{5}-10x^{3}+1&=0\\5x^{6}-{\color {white}00}5\,280x^{5}-10x^{3}+1&=0\\5x^{6}-640\,320x^{5}-10x^{3}+1&=0\end{aligned}}$ with the j-invariants appearing again. These sextics are not only algebraic, they are also solvable in radicals as they factor into two cubics over the extension $\mathbb {Q} {\sqrt {5}}$ (with the first factoring further into two quadratics). These algebraic approximations can be exactly expressed in terms of Dedekind eta quotients. As an example, let $\textstyle \tau ={\frac {1+{\sqrt {-163}}}{2}}$, then, ${\begin{aligned}e^{\pi {\sqrt {163}}}&=\left({\frac {e^{\frac {\pi i}{24}}\eta (\tau )}{\eta (2\tau )}}\right)^{24}-24.000\,000\,000\,000\,001\,05\dots \\e^{\pi {\sqrt {163}}}&=\left({\frac {e^{\frac {\pi i}{12}}\eta (\tau )}{\eta (3\tau )}}\right)^{12}-12.000\,000\,000\,000\,000\,21\dots \\e^{\pi {\sqrt {163}}}&=\left({\frac {e^{\frac {\pi i}{6}}\eta (\tau )}{\eta (5\tau )}}\right)^{6}-6.000\,000\,000\,000\,000\,034\dots \end{aligned}}$ where the eta quotients are the algebraic numbers given above. Class 2 numbers The three numbers 88, 148, 232, for which the imaginary quadratic field $\mathbb {Q} \left[{\sqrt {-d}}\right]$ has class number 2, are not Heegner numbers but have certain similar properties in terms of almost integers. For instance, ${\begin{aligned}e^{\pi {\sqrt {88}}}+8\,744&\approx {\color {white}00\,00}2\,508\,952^{2}-0.077\dots \\e^{\pi {\sqrt {148}}}+8\,744&\approx {\color {white}00\,}199\,148\,648^{2}-0.000\,97\dots \\e^{\pi {\sqrt {232}}}+8\,744&\approx 24\,591\,257\,752^{2}-0.000\,0078\dots \\\end{aligned}}$ and ${\begin{aligned}e^{\pi {\sqrt {22}}}-24&\approx {\color {white}00}\left(6+4{\sqrt {2}}\right)^{6}+0.000\,11\dots \\e^{\pi {\sqrt {37}}}+24&\approx \left(12+2{\sqrt {37}}\right)^{6}-0.000\,0014\dots \\e^{\pi {\sqrt {58}}}-24&\approx \left(27+5{\sqrt {29}}\right)^{6}-0.000\,000\,0011\dots \\\end{aligned}}$ Consecutive primes Given an odd prime p, if one computes $k^{2}\mod p$ for $\textstyle k=0,1,\dots ,{\frac {p-1}{2}}$ (this is sufficient because $\left(p-k\right)^{2}\equiv k^{2}\mod p$), one gets consecutive composites, followed by consecutive primes, if and only if p is a Heegner number.[14] For details, see "Quadratic Polynomials Producing Consecutive Distinct Primes and Class Groups of Complex Quadratic Fields" by Richard Mollin.[15] Notes and references 1. Conway, John Horton; Guy, Richard K. (1996). The Book of Numbers. Springer. p. 224. ISBN 0-387-97993-X. 2. Stark, H. M. (1969), "On the gap in the theorem of Heegner" (PDF), Journal of Number Theory, 1 (1): 16–27, Bibcode:1969JNT.....1...16S, doi:10.1016/0022-314X(69)90023-7, hdl:2027.42/33039 3. Rabinovitch, Georg "Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern." Proc. Fifth Internat. Congress Math. ( Cambridge) 1, 418–421, 1913. 4. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 88 and 144, 1983. 5. Weisstein, Eric W. "Transcendental Number". MathWorld. gives $e^{\pi {\sqrt {d}}},d\in Z^{}$, based on Nesterenko, Yu. V. "On Algebraic Independence of the Components of Solutions of a System of Linear Differential Equations." Izv. Akad. Nauk SSSR, Ser. Mat. 38, 495–512, 1974. English translation in Math. USSR 8, 501–518, 1974. 6. Ramanujan Constant – from Wolfram MathWorld 7. Barrow, John D (2002). The Constants of Nature. London: Jonathan Cape. p. 72. ISBN 0-224-06135-6. 8. Gardner, Martin (April 1975). "Mathematical Games". Scientific American. Scientific American, Inc. 232 (4): 127. Bibcode:1975SciAm.232e.102G. doi:10.1038/scientificamerican0575-102. 9. These can be checked by computing ${\sqrt[{3}]{e^{\pi {\sqrt {d}}}-744}}$ on a calculator, and ${\frac {196\,884}{e^{\pi {\sqrt {d}}}}}$ for the linear term of the error. 10. "More on e^(piSQRT(163))". 11. The absolute deviation of a random real number (picked uniformly from [0,1], say) is a uniformly distributed variable on [0, 0.5], so it has absolute average deviation and median absolute deviation of 0.25, and a deviation of 0.22 is not exceptional. 12. "Pi Formulas". 13. "Extending Ramanujan's Dedekind Eta Quotients". 14. "Simple Complex Quadratic Fields". 15. Mollin, R. A. (1996). "Quadratic polynomials producing consecutive, distinct primes and class groups of complex quadratic fields" (PDF). Acta Arithmetica. 74: 17–30. doi:10.4064/aa-74-1-17-30. External links • Weisstein, Eric W. "Heegner Number". MathWorld. • OEIS sequence A003173 (Heegner numbers: imaginary quadratic fields with unique factorization) • Gauss' Class Number Problem for Imaginary Quadratic Fields, by Dorian Goldfeld: Detailed history of problem. • Clark, Alex. "163 and Ramanujan Constant". Numberphile. Brady Haran. Archived from the original on 2013-05-16. Retrieved 2013-04-02.	Wikipedia
Ramanujan's master theorem In mathematics, Ramanujan's Master Theorem, named after Srinivasa Ramanujan,[1] is a technique that provides an analytic expression for the Mellin transform of an analytic function. The result is stated as follows: If a complex-valued function $ f(x)$ has an expansion of the form $f(x)=\sum _{k=0}^{\infty }{\frac {\,\varphi (k)\,}{k!}}(-x)^{k}$ then the Mellin transform of $ f(x)$ is given by $\int _{0}^{\infty }x^{s-1}f(x)\,dx=\Gamma (s)\,\varphi (-s)$ where $ \Gamma (s)$ is the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series. Higher-dimensional versions of this theorem also appear in quantum physics (through Feynman diagrams).[2] A similar result was also obtained by Glaisher.[3] Alternative formalism An alternative formulation of Ramanujan's Master Theorem is as follows: $\int _{0}^{\infty }x^{s-1}\left(\,\lambda (0)-x\,\lambda (1)+x^{2}\,\lambda (2)-\,\cdots \,\right)dx={\frac {\pi }{\,\sin(\pi s)\,}}\,\lambda (-s)$ which gets converted to the above form after substituting $ \lambda (n)\equiv {\frac {\varphi (n)}{\,\Gamma (1+n)\,}}$ and using the functional equation for the gamma function. The integral above is convergent for $ 0<\operatorname {\mathcal {Re}} (s)<1$ subject to growth conditions on $ \varphi $.[4] Proof A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by G. H. Hardy[5] (chapter XI) employing the residue theorem and the well-known Mellin inversion theorem. Application to Bernoulli polynomials The generating function of the Bernoulli polynomials $ B_{k}(x)$ is given by: ${\frac {z\,e^{x\,z}}{\,e^{z}-1\,}}=\sum _{k=0}^{\infty }B_{k}(x)\,{\frac {z^{k}}{k!}}$ These polynomials are given in terms of the Hurwitz zeta function: $\zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{\,(n+a)^{s}\,}}$ by $ \zeta (1-n,a)=-{\frac {B_{n}(a)}{n}}$ for $ ~n\geq 1$. Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation:[6] $\int _{0}^{\infty }x^{s-1}\left({\frac {e^{-ax}}{\,1-e^{-x}\,}}-{\frac {1}{x}}\right)dx=\Gamma (s)\,\zeta (s,a)\!$ which is valid for $ 0<\operatorname {\mathcal {Re}} (s)<1$. Application to the gamma function Weierstrass's definition of the gamma function $\Gamma (x)={\frac {\,e^{-\gamma \,x\,}}{x}}\,\prod _{n=1}^{\infty }\left(\,1+{\frac {x}{n}}\,\right)^{-1}e^{x/n}\!$ is equivalent to expression $\log \Gamma (1+x)=-\gamma \,x+\sum _{k=2}^{\infty }{\frac {\,\zeta (k)\,}{k}}\,(-x)^{k}$ where $ \zeta (k)$ is the Riemann zeta function. Then applying Ramanujan master theorem we have: $\int _{0}^{\infty }x^{s-1}{\frac {\,\gamma \,x+\log \Gamma (1+x)\,}{x^{2}}}\mathrm {d} x={\frac {\pi }{\sin(\pi s)}}{\frac {\zeta (2-s)}{2-s}}\!$ valid for $ 0<\operatorname {\mathcal {Re}} (s)<1$. Special cases of $ s={\frac {1}{2}}$ and $ s={\frac {3}{4}}$ are $\int _{0}^{\infty }{\frac {\,\gamma x+\log \Gamma (1+x)\,}{x^{5/2}}}\,\mathrm {d} x={\frac {2\pi }{3}}\,\zeta \left({\frac {3}{2}}\right)$ $\int _{0}^{\infty }{\frac {\,\gamma \,x+\log \Gamma (1+x)\,}{x^{9/4}}}\,\mathrm {d} x={\sqrt {2}}{\frac {4\pi }{5}}\zeta \left({\frac {5}{4}}\right)$ Application to Bessel functions The Bessel function of the first kind has the power series $J_{\nu }(z)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{\Gamma (k+\nu +1)k!}}{\bigg (}{\frac {z}{2}}{\bigg )}^{2k+\nu }$ By Ramanujan's Master Theorem, together with some identities for the gamma function and rearranging, we can evaluate the integral ${\frac {2^{\nu -2s}\pi }{\sin {(\pi (s-\nu ))}}}\int _{0}^{\infty }z^{s-1-\nu /2}J_{\nu }({\sqrt {z}})\,dz=\Gamma (s)\Gamma (s-\nu )$ valid for $ 0<2\operatorname {\mathcal {Re}} (s)<\operatorname {\mathcal {Re}} (\nu )+{\tfrac {3}{2}}$. Equivalently, if the spherical Bessel function $ j_{\nu }(z)$ is preferred, the formula becomes ${\frac {2^{\nu -2s}{\sqrt {\pi }}(1-2s+2\nu )}{\cos {(\pi (s-\nu ))}}}\int _{0}^{\infty }z^{s-1-\nu /2}j_{\nu }({\sqrt {z}})\,dz=\Gamma (s)\Gamma {\bigg (}{\frac {1}{2}}+s-\nu {\bigg )}$ valid for $ 0<2\operatorname {\mathcal {Re}} (s)<\operatorname {\mathcal {Re}} (\nu )+2$. The solution is remarkable in that it is able to interpolate across the major identities for the gamma function. In particular, the choice of $ J_{0}({\sqrt {z}})$ gives the square of the gamma function, $ j_{0}({\sqrt {z}})$ gives the duplication formula, $ z^{-1/2}J_{1}({\sqrt {z}})$ gives the reflection formula, and fixing to the evaluable $ s={\tfrac {1}{2}}$ or $ s=1$ gives the gamma function by itself, up to reflection and scaling. Bracket integration method The bracket integration method (method of brackets) applies Ramanujan's Master Theorem to a broad range of integrals.[7] [8] The bracket integration method generates an integral of a series expansion, introduces simplifying notations, solves linear equations, and completes the integration using formulas arising from Ramanujan's Master Theorem.[8] Generate an integral of a series expansion This method transforms the integral to an integral of a series expansion involving M variables, $ x_{1},\ldots x_{M}$, and S summation parameters, $ n_{1},\dots n_{S}$. A multivariate integral may assume this form.[2]: 8 $\int _{0}^{\infty }\cdots \int _{0}^{\infty }\sum _{n_{1},\ldots ,n_{S}=0}^{\infty }\varphi (n_{1}\cdots n_{S})\ \prod _{j=1}^{S}\left({\frac {(-1)^{n_{j}}}{n_{j}!}}\right)\prod _{j=1}^{M}(x_{j})^{(-c_{j}+a_{j1}\cdot n_{1}+\cdots +a_{jS}\cdot n_{S}-1)}\ dx_{1}\cdots dx_{M}$ (B.0) Apply special notations • The bracket ($ \langle \cdots \rangle $), indicator ($ \phi $), and monomial power notations replace terms in the series expansion.[2]: 8 $\int _{0}^{\infty }x^{c+b\cdot n-1}\ dx\ \to \ <c+b\cdot n>$ (B.1) ${\frac {(-1)^{n}}{n!}}\ \to \ \phi _{n}$ (B.2) $\prod _{j=1}^{S}\left({\frac {(-1)^{n_{j}}}{n_{j}!}}\right)\ \to \ \phi _{n_{1},\ldots ,n_{S}}$ (B.3) $\left(\sum _{k=1}^{P}u_{k}\right)^{\mp d}\ \to \ \sum _{n_{1},\ldots ,n_{P}=0}^{\infty }\varphi _{n_{1},\ldots ,n_{P}}\prod _{k=1}^{P}u_{k}^{n_{k}}{\frac {\langle \pm d+\sum _{j=1}^{P}n_{j}\rangle }{\Gamma (\pm d)}}$ (B.4) • Application of these notations transforms the integral to a bracket series containing B brackets.[7]: 56 $\sum _{n_{1},\ldots ,n_{S}=0}^{\infty }\varphi (n_{1}\cdots n_{S})\ \phi (n_{1}\cdots n_{S})\prod _{j=1}^{B}\left\langle -c_{j}+\sum _{k=1}^{S}a_{jk}\cdot n_{k}\right\rangle $ (B.5) • Each bracket series has an index defined as index = number of sums − number of brackets. • Among all bracket series representations of an integral, the representation with a minimal index is preferred.[8]: 984 Solve linear equations • The array of coefficients $ a_{jk}$ must have maximum rank, linearly independent leading columns to solve the following set of linear equations.[2]: 8 [8]: 985 • If the index is non-negative, solve this equation set for each $ n_{j}^{\ast }$. The terms $ n_{j}^{\ast }$ may be linear functions of $ \{n_{B+1}\dots n_{S}\}$. $-c_{j}+\sum _{k=1}^{B}a_{jk}\cdot n_{k}^{\ast }+\sum _{k=B+1}^{S}a_{jk}\cdot n_{k}=0$ (B.6) • If the index is zero, equation (B.6) simplifies to solving this equation set for each $ n_{j}^{\ast }$ $-c_{j}+\sum _{k=1}^{B}a_{jk}\cdot n_{k}^{\ast }=0$ (B.7) • If the index is negative, the integral cannot be determined. Apply formulas • If the index is non-negative, the formula for the integral is this form.[7]: 54 $\sum _{n_{B+1}\dots n_{S}=0}^{\infty }{\frac {\varphi (n_{1}^{\ast }\dots n_{B}^{\ast },n_{B+1}\dots n_{S})\cdot \prod _{j=1}^{B}\Gamma (-n_{j}^{\ast })}{\det \|A\|}}$ (B.8) • These rules apply.[8]: 985 • A series is generated for each choice of free summation parameters, $ \{n_{B}+1,\dots N_{S}\}$. • Series converging in a common region are added. • If a choice generates a divergent series or null series (a series with zero valued terms), the series is rejected. • A bracket series of negative index is assigned no value. • If all series are rejected, then the method cannot be applied. • If the index is zero, the formula B.8 simplifies to this formula and no sum occurs. $\ {\frac {\varphi (n_{1}^{\ast }\dots n_{S}^{\ast })\cdot \prod _{j=1}^{S}\Gamma (-n_{j}^{\ast })}{\det \|A\|}}$ (B.9) Mathematical basis • Apply this variable transformation to the general integral form (B.0).[4]: 14 $y_{k}=x_{1}^{a_{1k}}\cdot \ldots \cdot x_{M}^{a_{Mk}}$ (B.10) . • This is the transformed integral (B.11) and the result from applying Ramanujan's Master Theorem (B.12). $(\det \|A\|^{-1})\cdot \int _{0}^{\infty }\cdots \int _{0}^{\infty }\sum _{n_{1},\ldots ,n_{S}=0}^{\infty }\varphi (n_{1}\cdots n_{S})\prod _{j=1}^{S}\left({\frac {(-1)^{n_{j}}}{n_{j}!}}\right)\prod _{j=1}^{M}(y_{j})^{-n_{j}^{\ast }+n_{j}-1}\ dy_{1}\cdots dy_{M}$ (B.11) $=\sum _{n_{M+1}\cdots n_{S}=0}^{\infty }{\frac {\varphi (n_{1}^{\ast }\dots n_{M}^{\ast },n_{M+1}\dots n_{S})\cdot \prod _{j=1}^{M}\Gamma (-n_{j}^{\ast })}{\det \|A\|}}$ (B.12) • The number of brackets (B) equals the number of integrals (M) (B.1). In addition to generating the algorithm's formulas (B.8,B.9), the variable transformation also generates the algorithm's linear equations (B.6,B.7).[4]: 14 Example • The bracket integration method is applied to this integral. $\int _{0}^{\infty }x^{3/2}\cdot e^{-x^{3}/2}\ dx$ • Generate the integral of a series expansion (B.0). $\int _{0}^{\infty }\sum _{n=0}^{\infty }2^{-n}\cdot {\frac {(-1)^{n}}{n!}}\cdot x^{(3\cdot n+5/2)-1}\ dx$ • Apply special notations (B.1, B.2). $\sum _{n=0}^{\infty }2^{-n}\cdot \phi (n)\cdot \langle 3\cdot n+{\frac {5}{2}}\rangle $ • Solve the linear equation (B.7). $3\cdot n^{\ast }+{\frac {5}{2}}=0,\ n^{\ast }={\frac {-5}{6}}$ • Apply the formula (B.9). ${\frac {2^{5/6}\cdot \Gamma ({\frac {5}{6}})}{3}}$ References 1. Berndt, B. (1985). Ramanujan's Notebooks, Part I. New York: Springer-Verlag. 2. González, Iván; Moll, V.H.; Schmidt, Iván (2011). "A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams". arXiv:1103.0588 [math-ph]. 3. Glaisher, J.W.L. (1874). "A new formula in definite integrals". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 48 (315): 53–55. doi:10.1080/14786447408641072. 4. Amdeberhan, Tewodros; Gonzalez, Ivan; Harrison, Marshall; Moll, Victor H.; Straub, Armin (2012). "Ramanujan's Master Theorem". The Ramanujan Journal. 29 (1–3): 103–120. CiteSeerX 10.1.1.232.8448. doi:10.1007/s11139-011-9333-y. S2CID 8886049. 5. Hardy, G.H. (1978). Ramanujan: Twelve lectures on subjects suggested by his life and work (3rd ed.). New York, NY: Chelsea. ISBN 978-0-8284-0136-4. 6. Espinosa, O.; Moll, V. (2002). "On some definite integrals involving the Hurwitz zeta function. Part 2". The Ramanujan Journal. 6 (4): 449–468. arXiv:math/0107082. doi:10.1023/A:1021171500736. S2CID 970603. 7. Gonzalez, Ivan; Moll, Victor H. (July 2010). "Definite integrals by the method of brackets. Part 1,". Advances in Applied Mathematics. 45 (1): 50–73. doi:10.1016/j.aam.2009.11.003. 8. Gonzalez, Ivan; Jiu, Lin; Moll, Victor H. (1 January 2020). "An extension of the method of brackets. Part 2". Open Mathematics. 18 (1): 983–995. doi:10.1515/math-2020-0062. ISSN 2391-5455. External links • "Ramanujan's Master Theorem". mathworld.wolfram.com. • "rmt" (PDF). ArminStraub. publications.	Wikipedia
1729 (number) 1729 is the natural number following 1728 and preceding 1730. It is notably the first taxicab number. ← 1728 1729 1730 → • List of numbers • Integers ← 0 1k 2k 3k 4k 5k 6k 7k 8k 9k → Cardinalone thousand seven hundred twenty-nine Ordinal1729th (one thousand seven hundred twenty-ninth) Factorization7 × 13 × 19 Divisors1, 7, 13, 19, 91, 133, 247, 1729 Greek numeral,ΑΨΚΘ´ Roman numeralMDCCXXIX Binary110110000012 Ternary21010013 Senary120016 Octal33018 Duodecimal100112 Hexadecimal6C116 In mathematics 1729 is the smallest taxicab number,[1] and is variously known as Ramanujan's number or the Ramanujan–Hardy number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation:[2][3][4][5] I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." The two different ways are: 1729 = 13 + 123 = 93 + 103 The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729; 19 × 91 = 1729). 91 = 63 + (−5)3 = 43 + 33 1729 was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessy in 1657. A commemorative plaque now appears at the site of the Ramanujan-Hardy incident, at 2 Colinette Road in Putney.[6] The same expression defines 1729 as the first in the sequence of "Fermat near misses" defined, in reference to Fermat's Last Theorem, as numbers of the form 1 + z3 which are also expressible as the sum of two other cubes (sequence A050794 in the OEIS). Other properties 1729 is a sphenic number. It is the third Carmichael number, the first Chernick–Carmichael number (sequence A033502 in the OEIS), the first absolute Euler pseudoprime, and the third Zeisel number.[7] It is a centered cube number,[8] as well as a dodecagonal number,[9] a 24-gonal[10] and 84-gonal number. Investigating pairs of distinct integer-valued quadratic forms that represent every integer the same number of times, Schiemann found that such quadratic forms must be in four or more variables, and the least possible discriminant of a four-variable pair is 1729.[11] 1729 is the lowest number which can be represented by a Loeschian quadratic form $a^{2}+ab+b^{2}$ in four different ways with a and b positive integers. The integer pairs $(a,b)$ are (25,23), (32,15), (37,8) and (40,3).[12] 1729 is also the smallest integer side $d$ of an equilateral triangle for which there are three sets of non-equivalent interior points at integer distances from their vertices: {211, 1541, 1560}, {195, 1544, 1591}, and {824, 915, 1591}.[13] 1729 is the dimension of the Fourier transform on which the fastest known algorithm for multiplying two numbers is based.[14] This is an example of a galactic algorithm. See also • A Disappearing Number, a March 2007 play about Ramanujan in England during World War I. • Interesting number paradox • 4104, the second positive integer which can be expressed as the sum of two positive cubes in two different ways. References 1. Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 13. ISBN 978-1-84800-000-1. 2. Quotations by Hardy Archived 2012-07-16 at the Wayback Machine 3. Singh, Simon (15 October 2013). "Why is the number 1,729 hidden in Futurama episodes?". BBC News Online. Retrieved 15 October 2013. 4. Hardy, G H (1940). Ramanujan. New York: Cambridge University Press (original). p. 12. 5. Hardy, G. H. (1921), "Srinivasa Ramanujan", Proc. London Math. Soc., s2-19 (1): xl–lviii, doi:10.1112/plms/s2-19.1.1-u The anecdote about 1729 occurs on pages lvii and lviii 6. Marshall, Michael (24 February 2017). "A black plaque for Ramanujan, Hardy and 1,729". Good Thinking. Retrieved 7 March 2019. 7. Sloane, N. J. A. (ed.). "Sequence A051015 (Zeisel numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02. 8. Sloane, N. J. A. (ed.). "Sequence A005898 (Centered cube numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02. 9. Sloane, N. J. A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02. 10. Sloane, N. J. A. (ed.). "Sequence A051876 (24-gonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02. 11. Guy, Richard K. (2004), Unsolved Problems in Number Theory, Problem Books in Mathematics, Volume 1 (3rd ed.), Springer, ISBN 0-387-20860-7 - D1 mentions the Ramanujan-Hardy number. 12. David Mitchell (25 February 2017). "Tessellating the Ramanujan-Hardy Taxicab Number, 1729, Bedrock of Integer Sequence A198775". Retrieved 19 July 2018. 13. Ignacio Larrosa Cañestro (June 2016). "Distancias a los vértices de un triángulo equilátero (Distances to vertices of an equilateral triangle)" (PDF) (in Spanish). p. 5. 14. Harvey, David; Conversation, The. "We've found a quicker way to multiply really big numbers". phys.org. Retrieved 2021-11-01. External links • Weisstein, Eric W. "Hardy–Ramanujan Number". MathWorld. • Grime, James; Bowley, Roger. "1729: Taxi Cab Number or Hardy-Ramanujan Number". Numberphile. Brady Haran. Archived from the original on 2017-03-06. Retrieved 2013-04-02. • Why does the number 1729 show up in so many Futurama episodes?, io9.com	Wikipedia
Ramanujan Institute for Advanced Study in Mathematics Ramanujan Institute for Advanced Study in Mathematics (RIASM) is the Department of Mathematics of University of Madras. This name was adopted in 1967.[1] Ramanujan Institute for Advanced Study in Mathematics Established • 1927 as Department of Mathematics, University of Madras • 1967 as RIASM Research typePublic Field of research Mathematics DirectorProf Sushama Agarwal Narayandas AddressChepauk LocationChennai, India CampusUrban AffiliationsUniversity of Madras Operating agency NBHM, UGC Websiteriasm.unom.ac.in History The University of Madras was incorporated in 1857 and the Department of Mathematics was an integral part of the university from its beginning. The department developed from its early years to become a centre of research in mathematics with the appointment of R. Vaidyanathaswamy as a Reader in Mathematics in 1927.[2] The seeds of the Ramanujan Institute for Advanced Study in Mathematics were sown when the "Ramanujan Institute of Mathematics" was established by Alagappa Chettiar on 26 January 1950 as a memorial to the mathematician Srinivasa Ramanujan. It was governed by the Asoka Charitable Trust, Karaikudi, and was located at Krishna Vilas, Vepery, Chennai. The Ramanujan Institute of Mathematics was inaugurated by A. Lakshmanaswamy Mudaliar, Vice Chancellor of University of Madras, with T. Vijayaraghavan, a student of G.H. Hardy, as Director of the Institute. The institute faced a financial crisis when, in 1956, Asoka Charitable Trust expressed its inability to run the institute. However, due to the request from Subrahmanyan Chandrasekhar Jawaharlal Nehru took an initiative in such a way the management of the institute came to be vested with the University of Madras and the institute was taken over by the university in May 1957. The Asoka charitable Trust Started Ramanujan Institute of Mathematics in January, 1950 and as the Institute found itself in financial difficulties, Government of India agreed in 1953-54 to meet the expenses of a chair for Mathematics at the Institute Object to the condition that the Trust Would continue to spend the amount Previously spent by it for the activity of the Institute. Grants of 18,000 for each year were given the Institute for the years 1953-54 and 1994-55, but the audited accounts of the Institute revealed that the Trust was not fulfilling the condition of the Government grant and had instead built up a Reserve Fund. No Government grant was therefore paid in 1965-56 Towards the close of 1956, the Trust decided to close down the Institute, but as a result of discussions with the Government of India, Initiated by the founder of the Trust and the Vice-Chancellor of the Madras University, and later earned on by the Vice-Chancellor, it has been agreed that in view of the difficulty of manning an Institute with the limited number of available Professors of the requisite quality, the activities of the Institute may be continued in the Department of Mathematics in the University of Madras It has been decided to create a Ramanujam Professorship of Mathematics on a scale of Rs 1,000—1,500 with selection grade up to Rs 1,700 and attach the existing permanent research staff of the Institute to the said Professor The Government of India will give the necessary grants to the University to enable them to carry out this arrangement until such time as the University Grants Commission sanctions an appropriate grant.[3] During the short period of existence, the institute had a string of prominent mathematicians as visitors, including S.S. Pillai, noted number theorist, V. Ganapathy Iyer, analyst and Norbert Wiener. After the demise of T. Vijayaraghavan in 1955, C.T. Rajagopal took over as the Director of the Institute. From 1957 to 1966, the Department of Mathematics and the Ramanujan Institute of Mathematics functioned as independent bodies under the University of Madras.[4] In 1967 the University Grants Commission (India) proposed to make the Department of Mathematics of the University of Madras into one of its Centres of Advanced Study. In the same year these two institutions were amalgamated to form a UGC Centre for Advanced Study in Mathematics and named the center as the "Ramanujan Institute for Advanced Study in Mathematics" (RIASM). C.T. Rajagopal was appointed the first Director of the Ramanujan Institute for Advanced Study in Mathematics, and when he retired in 1969, the reins were taken over by T.S. Bhanumurthy. Ramanujan Museum Utilising a grant of Rs. 1 lakh received as UGC Special Assistance for Equipment and with the help of the Vikram A. Sarabhai Community Science Centre, Ahmedabad a Mathematical Laboratory was established in the institute.[5] About 65 mathematical models were acquired under the scheme. These models were exhibited • for the participants of several Refresher Courses conducted through the Academic Staff College of the University of Madras, • at the Silver Jubilee conference of the Association of Mathematics Teachers of India held at Madras during 10–13 January 1991, • in the Science Exhibition held at National Institute of Technology, Tiruchirapalli during 10–14 July 1991, and • are being lent to be exhibited in several schools in and around Chennai. Later the institute received an amount of Rs. 2 lakh from the National Board for Higher Mathematics, Rs. 1 lakh from The Hindu newspaper and Rs. 9 lakh from the Ministry of Culture and Tourism, Government of India, and a matching grant of Rs. 9 lakh from the University of Madras. The amount was used to establish the Ramanujan Museum in the premises of the Ramanujan Institute for Advanced Study in Mathematics. A grant of rupees one crore was later sanctioned by the Ministry of Human Resource Development, Government of India, towards the establishment of Ramanujan Museum and Research Centre. Courses offered • M.Sc. (Mathematics) full-time • MPhil (Mathematics) full-time • PhD (Mathematics) full-time and part-time Current research areas • Algebra • Functional analysis • Harmonic analysis • Potential theory • Differential equations • Mathematical biology • Stochastic processes • Graph theory External links • Official website References 1. "About the Department". University of Madras. Retrieved 1 March 2014. 2. "Life and Work". www.imsc.res.in. Retrieved 22 January 2023. 3. https://eparlib.nic.in/bitstream/123456789/1766/1/lsd_02_07_11-04-1959.pdf page 16 4. http://riasm.unom.ac.in/index.html 5. "Home". ramanujanmuseum.org. Authority control International • ISNI Other • IdRef	Wikipedia
Ramanujan Math Park The Ramanujan Math Park is an Indian museum and activity center dedicated to mathematics education inside the Agastya Campus Creativity Lab located in Kuppam, in Chittoor, Andhra Pradesh.[1] It is named after the Indian mathematician Srinivasa Ramanujan (1887-1920) who was from nearby Madras State.[2] It is a joint project of Agastya International Foundation and the non-profit organization Gyanome.[3] Agastya is known for its hands-on teaching methods and the Math Park follows this tradition. The park features both indoor and outdoor exhibits as well as interactive touch screen stations, all designed to enhance the mathematical experience.[4] There are plans to replicate this math park experience at other Government run schools elsewhere in India.[2] History Ramanujan Math Park was conceived, partially funded and executed by Sujatha Ramdorai and her husband Srinivasan Ramdorai along with V.S.S Sastry, an Indian mathematics communicator based in nearby Kolar. It was inaugurated in 2017, on 22 December, Ramanujan's birthday and the day celebrated in India as National Math Day.[5] References 1. Math park set up government school to innovate teaching techniques by Neel Kamal, The Times of India, October 24, 2017 2. DC inaugurates “Srinivasa Ramanuja Math Park” at Tapa by Amrit Pal Singh, Babushahi.com, October 24, 2017 3. Ramanujan Maths Park gyanome.org: News/Announcements 4. New math museums: Ramanujan Math Park New math museums 5. "Ramanujan's birthday will be National Mathematics Day" by C. Jaishankar, The Hindu, December 27, 2011 External links • Agastya International Foundation: official website • Ramanujan Math Park - Agastya Campus Creativity Lab Video	Wikipedia
Ramanujan graph In the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders. As Murty's survey paper[1] notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic geometry". These graphs are indirectly named after Srinivasa Ramanujan; their name comes from the Ramanujan–Petersson conjecture, which was used in a construction of some of these graphs. Definition Let $G$ be a connected $d$-regular graph with $n$ vertices, and let $\lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}$ be the eigenvalues of the adjacency matrix of $G$ (or the spectrum of $G$). Because $G$ is connected and $d$-regular, its eigenvalues satisfy $d=\lambda _{1}>\lambda _{2}$ $\geq \cdots \geq \lambda _{n}\geq -d$. Define $\lambda (G)=\max _{i\neq 1}\|\lambda _{i}\|=\max(\|\lambda _{2}\|,\ldots ,\|\lambda _{n}\|)$. A connected $d$-regular graph $G$ is a Ramanujan graph if $\lambda (G)\leq 2{\sqrt {d-1}}$. Many sources uses an alternative definition $\lambda '(G)=\max _{\|\lambda _{i}\|<d}\|\lambda _{i}\|$ (whenever there exists $\lambda _{i}$ with $\|\lambda _{i}\|<d$) to define Ramanujan graphs.[2] In other words, we allow $-d$ in addition to the "small" eigenvalues. Since $\lambda _{n}=-d$ if and only if the graph is bipartite, we will refer to the graphs that satisfy this alternative definition but not the first definition bipartite Ramanujan graphs. If $G$ is a Ramanujan graph, then $G\times K_{2}$ is a bipartite Ramanujan graph, so the existence of Ramanujan graphs is stronger. As observed by Toshikazu Sunada, a regular graph is Ramanujan if and only if its Ihara zeta function satisfies an analog of the Riemann hypothesis.[3] Examples and constructions Explicit examples • The complete graph $K_{d+1}$ has spectrum $d,-1,-1,\dots ,-1$, and thus $\lambda (K_{d+1})=1$ and the graph is a Ramanujan graph for every $d>1$. The complete bipartite graph $K_{d,d}$ has spectrum $d,0,0,\dots ,0,-d$ and hence is a bipartite Ramanujan graph for every $d$. • The Petersen graph has spectrum $3,1,1,1,1,1,-2,-2,-2,-2$, so it is a 3-regular Ramanujan graph. The icosahedral graph is a 5-regular Ramanujan graph.[4] • A Paley graph of order $q$ is ${\frac {q-1}{2}}$-regular with all other eigenvalues being ${\frac {-1\pm {\sqrt {q}}}{2}}$, making Paley graphs an infinite family of Ramanujan graphs. • More generally, let $f(x)$ be a degree 2 or 3 polynomial over $\mathbb {F} _{q}$. Let $S=\{f(x)\,:\,x\in \mathbb {F} _{q}\}$ be the image of $f(x)$ as a multiset, and suppose $S=-S$. Then the Cayley graph for $\mathbb {F} _{q}$ with generators from $S$ is a Ramanujan graph. Mathematicians are often interested in constructing infinite families of $d$-regular Ramanujan graphs for every fixed $d$. Such families are useful in applications. Algebraic constructions Several explicit constructions of Ramanujan graphs arise as Cayley graphs and are algebraic in nature. See Winnie Li's survey on Ramanujan's conjecture and other aspects of number theory relevant to these results.[5] Lubotzky, Phillips and Sarnak[2] and independently Margulis[6] showed how to construct an infinite family of $(p+1)$-regular Ramanujan graphs, whenever $p$ is a prime number and $p\equiv 1{\pmod {4}}$. Both proofs use the Ramanujan conjecture, which led to the name of Ramanujan graphs. Besides being Ramanujan graphs, these constructions satisfies some other properties, for example, their girth is $\Omega (\log _{p}(n))$ where $n$ is the number of nodes. Let us sketch the Lubotzky-Phillips-Sarnak construction. Let $q\equiv 1{\bmod {4}}$ be a prime not equal to $p$. By Jacobi's four-square theorem, there are $p+1$ solutions to the equation $p=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}$ where $a_{0}>0$ is odd and $a_{1},a_{2},a_{3}$ are even. To each such solution associate the $\operatorname {PGL} (2,\mathbb {Z} /q\mathbb {Z} )$ matrix ${\tilde {\alpha }}={\begin{pmatrix}a_{0}+ia_{1}&a_{2}+ia_{3}\\-a_{2}+ia_{3}&a_{0}-ia_{1}\end{pmatrix}},\qquad i{\text{ a fixed solution to }}i^{2}=-1{\bmod {q}}.$ If $p$ is not a quadratic residue modulo $q$ let $X^{p,q}$ be the Cayley graph of $\operatorname {PGL} (2,\mathbb {Z} /q\mathbb {Z} )$ with these $p+1$ generators, and otherwise, let $X^{p,q}$ be the Cayley graph of $\operatorname {PSL} (2,\mathbb {Z} /q\mathbb {Z} )$ with the same generators. Then $X^{p,q}$ is a $(p+1)$-regular graph on $n=q(q^{2}-1)$ or $q(q^{2}-1)/2$ vertices depending on whether or not $p$ is a quadratic residue modulo $q$. It is proved that $X^{p,q}$ is a Ramanujan graph. Morgenstern[7] later extended the construction of Lubotzky, Phillips and Sarnak. His extended construction holds whenever $p$ is a prime power. Arnold Pizer proved that the supersingular isogeny graphs are Ramanujan, although they tend to have lower girth than the graphs of Lubotzky, Phillips, and Sarnak.[8] Like the graphs of Lubotzky, Phillips, and Sarnak, the degrees of these graphs are always a prime number plus one. Probabilistic examples Adam Marcus, Daniel Spielman and Nikhil Srivastava[9] proved the existence of infinitely many $d$-regular bipartite Ramanujan graphs for any $d\geq 3$. Later[10] they proved that there exist bipartite Ramanujan graphs of every degree and every number of vertices. Michael B. Cohen[11] showed how to construct these graphs in polynomial time. The initial work followed an approach of Bilu and Linial. They considered an operation called a 2-lift that takes a $d$-regular graph $G$ with $n$ vertices and a sign on each edge, and produces a new $d$-regular graph $G'$ on $2n$ vertices. Bilu & Linial conjectured that there always exists a signing so that every new eigenvalue of $G'$ has magnitude at most $2{\sqrt {d-1}}$. This conjecture guarantees the existence of Ramanujan graphs with degree $d$ and $2^{k}(d+1)$ vertices for any $k$—simply start with the complete graph $K_{d+1}$, and iteratively take 2-lifts that retain the Ramanujan property. Using the method of interlacing polynomials, Marcus, Spielman, and Srivastava[9] proved Bilu & Linial's conjecture holds when $G$ is already a bipartite Ramanujan graph, which is enough to conclude the existence result. The sequel[10] proved the stronger statement that a sum of $d$ random bipartite matchings is Ramanujan with non-vanishing probability. It is still an open problem whether there are infinitely many $d$-regular (non-bipartite) Ramanujan graphs for any $d\geq 3$. In particular, the problem is open for $d=7$, the smallest case for which $d-1$ is not a prime power and hence not covered by Morgenstern's construction. Ramanujan graphs as expander graphs The constant $2{\sqrt {d-1}}$ in the definition of Ramanujan graphs is asymptotically sharp. More precisely, the Alon-Boppana bound states that for every $d$ and $\epsilon >0$, there exists $n$ such that all $d$-regular graphs $G$ with at least $n$ vertices satisfy $\lambda (G)>2{\sqrt {d-1}}-\epsilon $. This means that Ramanujan graphs are essentially the best possible expander graphs. Due to achieving the tight bound on $\lambda (G)$, the expander mixing lemma gives excellent bounds on the uniformity of the distribution of the edges in Ramanujan graphs, and any random walks on the graphs has a logarithmic mixing time (in terms of the number of vertices): in other words, the random walk converges to the (uniform) stationary distribution very quickly. Therefore, the diameter of Ramanujan graphs are also bounded logarithmically in terms of the number of vertices. Random graphs Confirming a conjecture of Alon, Friedman[12] showed that many families of random graphs are weakly-Ramanujan. This means that for every $d$ and $\epsilon >0$ and for sufficiently large $n$, a random $d$-regular $n$-vertex graph $G$ satisfies $\lambda (G)<2{\sqrt {d-1}}+\epsilon $ with high probability. While this result shows that random graphs are close to being Ramanujan, it cannot be used to prove the existence of Ramanujan graphs. It is conjectured,[13] though, that random graphs are Ramanujan with substantial probability (roughly 52%). In addition to direct numerical evidence, there is some theoretical support for this conjecture: the spectral gap of a $d$-regular graph seems to behave according to a Tracy-Widom distribution from random matrix theory, which would predict the same asymptotic. Applications of Ramanujan graphs Expander graphs have many applications to computer science, number theory, and group theory, see e.g Lubotzky's survey on applications to pure and applied math and Hoory, Linial, and Wigderson's survey which focuses on computer science.. Ramanujan graphs are in some sense the best expanders, and so they are especially useful in applications where expanders are needed. Importantly, the Lubotzky, Phillips, and Sarnak graphs can be traversed extremely quickly in practice, so they are practical for applications. Some example applications include • In an application to fast solvers for Laplacian linear systems, Lee, Peng, and Spielman[14] relied on the existence of bipartite Ramanujan graphs of every degree in order to quickly approximate the complete graph. • Lubetzky and Peres proved that the simple random walk exhibits cutoff phenomenon on all Ramanujan graphs.[15] This means that the random walk undergoes a phase transition from being completely unmixed to completely mixed in the total variation norm. This result strongly relies on the graph being Ramanujan, not just an expander—some good expanders are known to not exhibit cutoff.[16] • Ramanujan graphs of Pizer have been proposed as the basis for post-quantum elliptic-curve cryptography.[17] • Ramanujan graphs can be used to construct expander codes, which are good error correcting codes. See also • Expander graph • Alon-Boppana bound • Expander mixing lemma • Spectral graph theory References 1. Survey paper by M. Ram Murty 2. Alexander Lubotzky; Ralph Phillips; Peter Sarnak (1988). "Ramanujan graphs". Combinatorica. 8 (3): 261–277. doi:10.1007/BF02126799. S2CID 206812625. 3. Terras, Audrey (2011), Zeta functions of graphs: A stroll through the garden, Cambridge Studies in Advanced Mathematics, vol. 128, Cambridge University Press, ISBN 978-0-521-11367-0, MR 2768284 4. Weisstein, Eric W. "Icosahedral Graph". mathworld.wolfram.com. Retrieved 2019-11-29. 5. Li, Winnie (2020). "The Ramanujan conjecture and its applications". Philosophical Transactions of the Royal Society A. 378–2163 (2163). Bibcode:2020RSPTA.37880441W. doi:10.1098/rsta.2018.0441. PMC 6939229. PMID 31813366. 6. Margulis, G. A. (1988). "Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators". Probl. Peredachi Inf. 24–1: 51–60. 7. Moshe Morgenstern (1994). "Existence and Explicit Constructions of q+1 Regular Ramanujan Graphs for Every Prime Power q". Journal of Combinatorial Theory, Series B. 62: 44–62. doi:10.1006/jctb.1994.1054. 8. Pizer, Arnold K. (1990), "Ramanujan graphs and Hecke operators", Bulletin of the American Mathematical Society, New Series, 23 (1): 127–137, doi:10.1090/S0273-0979-1990-15918-X, MR 1027904 9. Adam Marcus; Daniel Spielman; Nikhil Srivastava (2013). Interlacing families I: Bipartite Ramanujan graphs of all degrees (PDF). Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium. 10. Adam Marcus; Daniel Spielman; Nikhil Srivastava (2015). Interlacing families IV: Bipartite Ramanujan graphs of all sizes (PDF). Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium. 11. Michael B. Cohen (2016). Ramanujan Graphs in Polynomial Time. Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium. arXiv:1604.03544. doi:10.1109/FOCS.2016.37. 12. Friedman, Joel (2003). "Relative expanders or weakly relatively Ramanujan graphs". Duke Math. J. 118 (1): 19–35. doi:10.1215/S0012-7094-03-11812-8. MR 1978881. 13. Miller, Steven J.; Novikoff, Tim; Sabelli, Anthony (2006-11-21). "The Distribution of the Largest Non-trivial Eigenvalues in Families of Random Regular Graphs". arXiv:math/0611649. 14. Lee, Yin Tat; Peng, Richard; Spielman, Daniel A. (2015-08-13). "Sparsified Cholesky Solvers for SDD linear systems". arXiv:1506.08204 [cs.DS]. 15. Lubetzky, Eyal; Peres, Yuval (2016-07-01). "Cutoff on all Ramanujan graphs". Geometric and Functional Analysis. 26 (4): 1190–1216. arXiv:1507.04725. doi:10.1007/s00039-016-0382-7. ISSN 1420-8970. S2CID 13803649. 16. Lubetzky, Eyal; Sly, Allan (2011-01-01). "Explicit Expanders with Cutoff Phenomena". Electronic Journal of Probability. 16 (none). doi:10.1214/EJP.v16-869. ISSN 1083-6489. S2CID 9121682. 17. Eisenträger, Kirsten; Hallgren, Sean; Lauter, Kristin; Morrison, Travis; Petit, Christophe (2018), "Supersingular isogeny graphs and endomorphism rings: Reductions and solutions" (PDF), in Nielsen, Jesper Buus; Rijmen, Vincent (eds.), Advances in Cryptology – EUROCRYPT 2018: 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tel Aviv, Israel, April 29 - May 3, 2018, Proceedings, Part III (PDF), Lecture Notes in Computer Science, vol. 10822, Cham: Springer, pp. 329–368, doi:10.1007/978-3-319-78372-7_11, MR 3794837 Further reading • Giuliana Davidoff; Peter Sarnak; Alain Valette (2003). Elementary number theory, group theory and Ramanujan graphs. LMS student texts. Vol. 55. Cambridge University Press. ISBN 0-521-53143-8. OCLC 50253269. • Sunada, Toshikazu (1986). "L-functions in geometry and some applications". In Shiohama, Katsuhiro; Sakai, Takashi; Sunada, Toshikazu (eds.). Curvature and Topology of Riemannian Manifolds: Proceedings of the 17th International Taniguchi Symposium held in Katata, Japan, August 26–31, 1985. Lecture Notes in Mathematics. Vol. 1201. Berlin: Springer. pp. 266–284. doi:10.1007/BFb0075662. ISBN 978-3-540-16770-9. MR 0859591. External links • Survey paper by M. Ram Murty • Survey paper by Alexander Lubotzky • Survey paper by Hoory, Linial, and Wigderson	Wikipedia
Ramanujan–Petersson conjecture In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan (1916, p. 176), states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ(z) of weight 12 $\Delta (z)=\sum _{n>0}\tau (n)q^{n}=q\prod _{n>0}\left(1-q^{n}\right)^{24}=q-24q^{2}+252q^{3}-1472q^{4}+4830q^{5}-\cdots ,$ where $q=e^{2\pi iz}$, satisfies $\|\tau (p)\|\leq 2p^{11/2},$ when p is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by Petersson (1930), is a generalization to other modular forms or automorphic forms. Ramanujan L-function The Riemann zeta function and the Dirichlet L-function satisfy the Euler product, $L(s,a)=\prod _{p}\left(1+{\frac {a(p)}{p^{s}}}+{\frac {a(p^{2})}{p^{2s}}}+\dots \right)$ (1) and due to their completely multiplicative property $L(s,a)=\prod _{p}\left(1-{\frac {a(p)}{p^{s}}}\right)^{-1}.$ (2) Are there L-functions other than the Riemann zeta function and the Dirichlet L-functions satisfying the above relations? Indeed, the L-functions of automorphic forms satisfy the Euler product (1) but they do not satisfy (2) because they do not have the completely multiplicative property. However, Ramanujan discovered that the L-function of the modular discriminant satisfies the modified relation $L(s,\tau )=\prod _{p}\left(1-{\frac {\tau (p)}{p^{s}}}+{\frac {1}{p^{2s-11}}}\right)^{-1},$ (3) where τ(p) is Ramanujan's tau function. The term ${\frac {1}{p^{2s-11}}}$ is thought of as the difference from the completely multiplicative property. The above L-function is called Ramanujan's L-function. Ramanujan conjecture Ramanujan conjectured the following: 1. τ is multiplicative, 2. τ is not completely multiplicative but for prime p and j in N we have: τ(p j+1) = τ(p)τ(p j ) − p11τ(p j−1 ), and 3. \|τ(p)\| ≤ 2p11/2. Ramanujan observed that the quadratic equation of u = p−s in the denominator of RHS of (3), $1-\tau (p)u+p^{11}u^{2}$ would have always imaginary roots from many examples. The relationship between roots and coefficients of quadratic equations leads the third relation, called Ramanujan's conjecture. Moreover, for the Ramanujan tau function, let the roots of the above quadratic equation be α and β, then $\operatorname {Re} (\alpha )=\operatorname {Re} (\beta )=p^{11/2},$ which looks like the Riemann Hypothesis. It implies an estimate that is only slightly weaker for all the τ(n), namely for any ε > 0: $O\left(n^{11/2+\varepsilon }\right).$ In 1917, L. Mordell proved the first two relations using techniques from complex analysis, specifically what are now known as Hecke operators. The third statement followed from the proof of the Weil conjectures by Deligne (1974). The formulations required to show that it was a consequence were delicate, and not at all obvious. It was the work of Michio Kuga with contributions also by Mikio Sato, Goro Shimura, and Yasutaka Ihara, followed by Deligne (1968) harvtxt error: no target: CITEREFDeligne1968 (help). The existence of the connection inspired some of the deep work in the late 1960s when the consequences of the étale cohomology theory were being worked out. Ramanujan–Petersson conjecture for modular forms In 1937, Erich Hecke used Hecke operators to generalize the method of Mordell's proof of the first two conjectures to the automorphic L-function of the discrete subgroups Γ of SL(2, Z). For any modular form $f(z)=\sum _{n=0}^{\infty }a_{n}q^{n}\qquad q=e^{2\pi iz},$ one can form the Dirichlet series $\varphi (s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.$ For a modular form f (z) of weight k ≥ 2 for Γ, φ(s) absolutely converges in Re(s) > k, because an = O(nk−1+ε). Since f is a modular form of weight k, (s − k)φ(s) turns out to be an entire and R(s) = (2π)−sΓ(s)φ(s) satisfies the functional equation: $R(k-s)=(-1)^{k/2}R(s);$ this was proved by Wilton in 1929. This correspondence between f and φ is one to one (a0 = (−1)k/2 Ress=k R(s)). Let g(x) = f (ix) −a0 for x > 0, then g(x) is related with R(s) via the Mellin transformation $R(s)=\int _{0}^{\infty }g(x)x^{s-1}\,dx\Leftrightarrow g(x)={\frac {1}{2\pi i}}\int _{\operatorname {Re} (s)=\sigma _{0}}R(s)x^{-s}\,ds.$ This correspondence relates the Dirichlet series that satisfy the above functional equation with the automorphic form of a discrete subgroup of SL(2, Z). In the case k ≥ 3 Hans Petersson introduced a metric on the space of modular forms, called the Petersson metric (also see Weil–Petersson metric). This conjecture was named after him. Under the Petersson metric it is shown that we can define the orthogonality on the space of modular forms as the space of cusp forms and its orthogonal space and they have finite dimensions. Furthermore, we can concretely calculate the dimension of the space of holomorphic modular forms, using the Riemann–Roch theorem (see the dimensions of modular forms). Deligne (1971) used the Eichler–Shimura isomorphism to reduce the Ramanujan conjecture to the Weil conjectures that he later proved.　The more general Ramanujan–Petersson conjecture for holomorphic cusp forms in the theory of elliptic modular forms for congruence subgroups has a similar formulation, with exponent (k − 1)/2 where k is the weight of the form. These results also follow from the Weil conjectures, except for the case k = 1, where it is a result of Deligne & Serre (1974). The Ramanujan–Petersson conjecture for Maass forms is still open (as of 2022) because Deligne's method, which works well in the holomorphic case, does not work in the real analytic case. Ramanujan–Petersson conjecture for automorphic forms Satake (1966) reformulated the Ramanujan–Petersson conjecture in terms of automorphic representations for GL(2) as saying that the local components of automorphic representations lie in the principal series, and suggested this condition as a generalization of the Ramanujan–Petersson conjecture to automorphic forms on other groups. Another way of saying this is that the local components of cusp forms should be tempered. However, several authors found counter-examples for anisotropic groups where the component at infinity was not tempered. Kurokawa (1978) and Howe & Piatetski-Shapiro (1979) showed that the conjecture was also false even for some quasi-split and split groups, by constructing automorphic forms for the unitary group U(2, 1) and the symplectic group Sp(4) that are non-tempered almost everywhere, related to the representation θ10. After the counterexamples were found, Piatetski-Shapiro (1979) suggested that a reformulation of the conjecture should still hold. The current formulation of the generalized Ramanujan conjecture is for a globally generic cuspidal automorphic representation of a connected reductive group, where the generic assumption means that the representation admits a Whittaker model. It states that each local component of such a representation should be tempered. It is an observation due to Langlands that establishing functoriality of symmetric powers of automorphic representations of GL(n) will give a proof of the Ramanujan–Petersson conjecture. Bounds towards Ramanujan over number fields Obtaining the best possible bounds towards the generalized Ramanujan conjecture in the case of number fields has caught the attention of many mathematicians. Each improvement is considered a milestone in the world of modern number theory. In order to understand the Ramanujan bounds for GL(n), consider a unitary cuspidal automorphic representation: $\pi =\bigotimes \pi _{v}.$ The Bernstein–Zelevinsky classification tells us that each p-adic πv can be obtained via unitary parabolic induction from a representation $\tau _{1,v}\otimes \cdots \otimes \tau _{d,v}.$ Here each $\tau _{i,v}$ is a representation of GL(ni), over the place v, of the form $\tau _{i_{0},v}\otimes \left\|\det \right\|_{v}^{\sigma _{i,v}}$ with $\tau _{i_{0},v}$ tempered. Given n ≥ 2, a Ramanujan bound is a number δ ≥ 0 such that $\max _{i}\left\|\sigma _{i,v}\right\|\leq \delta .$ Langlands classification can be used for the archimedean places. The generalized Ramanujan conjecture is equivalent to the bound δ = 0. Jacquet, Piatetski-Shapiro & Shalika (1981) harvtxt error: no target: CITEREFJacquetPiatetski-ShapiroShalika1981 (help) obtain a first bound of δ ≤ 1/2 for the general linear group GL(n), known as the trivial bound. An important breakthrough was made by Luo, Rudnick & Sarnak (1999), who currently hold the best general bound of δ ≡ 1/2 − (n2+1)−1 for arbitrary n and any number field. In the case of GL(2), Kim and Sarnak established the breakthrough bound of δ = 7/64 when the number field is the field of rational numbers, which is obtained as a consequence of the functoriality result of Kim (2002) on the symmetric fourth obtained via the Langlands–Shahidi method. Generalizing the Kim-Sarnak bounds to an arbitrary number field is possible by the results of Blomer & Brumley (2011). For reductive groups other than GL(n), the generalized Ramanujan conjecture would follow from principle of Langlands functoriality. An important example are the classical groups, where the best possible bounds were obtained by Cogdell et al. (2004) as a consequence of their Langlands functorial lift. The Ramanujan–Petersson conjecture over global function fields Drinfeld's proof of the global Langlands correspondence for GL(2) over a global function field leads towards a proof of the Ramanujan–Petersson conjecture. Lafforgue (2002) successfully extended Drinfeld's shtuka technique to the case of GL(n) in positive characteristic. Via a different technique that extends the Langlands–Shahidi method to include global function fields, Lomelí (2009) proves the Ramanujan conjecture for the classical groups. Applications An application of the Ramanujan conjecture is the explicit construction of Ramanujan graphs by Lubotzky, Phillips and Sarnak. Indeed, the name "Ramanujan graph" was derived from this connection. Another application is that the Ramanujan–Petersson conjecture for the general linear group GL(n) implies Selberg's conjecture about eigenvalues of the Laplacian for some discrete groups. References • Blomer, V.; Brumley, F. (2011), "On the Ramanujan conjecture over number fields", Annals of Mathematics, 174: 581–605, arXiv:1003.0559, doi:10.4007/annals.2011.174.1.18, MR 2811610, S2CID 54686173 • Cogdell, J. W.; Kim, H. H.; Piatetski-Shapiro, I. I.; Shahidi, F. (2004), "Functoriality for the classical groups", Publications Mathématiques de l'IHÉS, 99: 163–233, CiteSeerX 10.1.1.495.6662, doi:10.1007/s10240-004-0020-z, S2CID 7731057 • Deligne, Pierre (1971), "Formes modulaires et représentations l-adiques", Séminaire Bourbaki vol. 1968/69 Exposés 347–363, Lecture Notes in Mathematics, vol. 179, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058801, ISBN 978-3-540-05356-9 • Deligne, Pierre (1974), "La conjecture de Weil. I.", Publications Mathématiques de l'IHÉS, 43: 273–307, doi:10.1007/BF02684373, ISSN 1618-1913, MR 0340258, S2CID 123139343 • Deligne, Pierre; Serre, Jean-Pierre (1974), "Formes modulaires de poids 1", Annales Scientifiques de l'École Normale Supérieure, Série 4, 7 (4): 507–530, doi:10.24033/asens.1277, ISSN 0012-9593, MR 0379379 • Howe, Roger; Piatetski-Shapiro, I. I. (1979), "A counterexample to the "generalized Ramanujan conjecture" for (quasi-) split groups", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Providence, R.I., pp. 315–322, ISBN 978-0-8218-1435-2, MR 0546605{{citation}}: CS1 maint: location missing publisher (link) • Jacquet, H.; Piatetski-Shapiro, I. I.; Shalika, J. A. (1983), "Rankin-Selberg Convolutions", Amer. J. Math., 105 (2): 367–464, doi:10.2307/2374264, JSTOR 2374264, S2CID 124304599 • Kim, H. H. (2002), "Functoriality for the exterior square of GL(4) and symmetric fourth of GL(2)", Journal of the AMS, 16: 139–183 • Kurokawa, Nobushige (1978), "Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two", Inventiones Mathematicae, 49 (2): 149–165, Bibcode:1978InMat..49..149K, doi:10.1007/BF01403084, ISSN 0020-9910, MR 0511188, S2CID 120041528 • Langlands, R. P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math, vol. 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR 0302614 • Lomelí, L. (2009), "Functoriality for the classical groups over function fields", International Mathematics Research Notices: 4271–4335, doi:10.1093/imrn/rnp089, MR 2552304, archived from the original on 2007-05-27 • Luo, W.; Rudnick, Z.; Sarnak, P. (1999), "On the Generalized Ramanujan Conjecture for GL(n)", Proc. Sympos. Pure Math., Proceedings of Symposia in Pure Mathematics, 66: 301–310, doi:10.1090/pspum/066.2/1703764, ISBN 9780821810514 • Petersson, H. (1930), "Theorie der automorphen Formen beliebiger reeller Dimension und ihre Darstellung durch eine neue Art Poincaréscher Reihen.", Mathematische Annalen (in German), 103 (1): 369–436, doi:10.1007/BF01455702, ISSN 0025-5831, S2CID 122378161 • Piatetski-Shapiro, I. I. (1979), "Multiplicity one theorems", in Borel, Armand; Casselman., W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 209–212, ISBN 978-0-8218-1435-2, MR 0546599 • Ramanujan, Srinivasa (1916), "On certain arithmetical functions", Transactions of the Cambridge Philosophical Society, XXII (9): 159–184 Reprinted in Ramanujan, Srinivasa (2000), "Paper 18", Collected papers of Srinivasa Ramanujan, AMS Chelsea Publishing, Providence, RI, pp. 136–162, ISBN 978-0-8218-2076-6, MR 2280843 • Sarnak, Peter (2005), "Notes on the generalized Ramanujan conjectures" (PDF), in Arthur, James; Ellwood, David; Kottwitz, Robert (eds.), Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Providence, R.I.: American Mathematical Society, pp. 659–685, ISBN 978-0-8218-3844-0, MR 2192019 • Satake, Ichirô (1966), "Spherical functions and Ramanujan conjecture", in Borel, Armand; Mostow, George D. (eds.), Algebraic Groups and Discontinuous Subgroups (Boulder, Colo., 1965), Proc. Sympos. Pure Math., vol. IX, Providence, R.I., pp. 258–264, ISBN 978-0-8218-3213-4, MR 0211955{{citation}}: CS1 maint: location missing publisher (link) L-functions in number theory Analytic examples • Riemann zeta function • Dirichlet L-functions • L-functions of Hecke characters • Automorphic L-functions • Selberg class Algebraic examples • Dedekind zeta functions • Artin L-functions • Hasse–Weil L-functions • Motivic L-functions Theorems • Analytic class number formula • Riemann–von Mangoldt formula • Weil conjectures Analytic conjectures • Riemann hypothesis • Generalized Riemann hypothesis • Lindelöf hypothesis • Ramanujan–Petersson conjecture • Artin conjecture Algebraic conjectures • Birch and Swinnerton-Dyer conjecture • Deligne's conjecture • Beilinson conjectures • Bloch–Kato conjecture • Langlands conjecture p-adic L-functions • Main conjecture of Iwasawa theory • Selmer group • Euler system	Wikipedia

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.