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P-adic Exponential Function
In mathematics, particularly ''p''-adic analysis, the ''p''-adic exponential function is a ''p''-adic analogue of the usual exponential function on the complex numbers. As in the complex case, it has an inverse function, named the ''p''-adic logarithm. Definition The usual exponential function on C is defined by the infinite series :\exp(z)=\sum_^\infty \frac. Entirely analogously, one defines the exponential function on C''p'', the completion of the algebraic closure of Q''p'', by :\exp_p(z)=\sum_^\infty\frac. However, unlike exp which converges on all of C, exp''p'' only converges on the disc :, z, _p [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius Of Convergence
In mathematics, the radius of convergence of a power series is the radius of the largest Disk (mathematics), disk at the Power series, center of the series in which the series Convergent series, converges. It is either a non-negative real number or \infty. When it is positive, the power series absolute convergence, converges absolutely and compact convergence, uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges. In case of multiple singularities of a function (singularities are those values of the argument for which the function is not defined), the radius of convergence is the shortest or minimum of all the respective distances (which are all non-negative numbers) calculated from the center of the disk of convergence to the respective singularities of the function. Definition For a power series ''f'' defined as: :f(z) = \sum_^\infty c_n (z-a)^n, where *''a'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbach (1997, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College London, University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Identity
In mathematics, Euler's identity (also known as Euler's equation) is the Equality (mathematics), equality e^ + 1 = 0 where :e is E (mathematical constant), Euler's number, the base of natural logarithms, :i is the imaginary unit, which by definition satisfies i^2 = -1, and :\pi is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula e^ = \cos x + i\sin x when evaluated for x = \pi. Euler's identity is considered an exemplar of mathematical beauty, as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in Lindemann–Weierstrass theorem#Transcendence of e and π, a proof that is Transcendental number, transcendental, which implies the impossibility of squaring the circle. Mathematical beauty Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of Unity
In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre. Roots of unity can be defined in any field (mathematics), field. If the characteristic of a field, characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, converse (logic), conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Analysis
In mathematics, ''p''-adic analysis is a branch of number theory that studies functions of ''p''-adic numbers. Along with the more classical fields of real and complex analysis, which deal, respectively, with functions on the real and complex numbers, it belongs to the discipline of mathematical analysis. The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of locally compact groups ( abstract harmonic analysis). The usual meaning taken for ''p''-adic analysis is the theory of ''p''-adic-valued functions on spaces of interest. Applications of ''p''-adic analysis have mainly been in number theory, where it has a significant role in diophantine geometry and diophantine approximation. Some applications have required the development of ''p''-adic functional analysis and spectral theory. In many ways ''p''-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E (mathematical Constant)
The number is a mathematical constant approximately equal to 2.71828 that is the base of a logarithm, base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted \gamma. Alternatively, can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest. The number is of great importance in mathematics, alongside 0, 1, Pi, , and . All five appear in one formulation of Euler's identity e^+1=0 and play important and recurring roles across mathematics. Like the constant , is Irrational number, irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is Transcendental number, transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
