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Branch of number theory
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
P-adic_analysis
Number system extending the rational numbers
p, the p-adic numbers form an extension of the rational numbers that is distinct from the real numbers, though with some similar properties; p-adic numbers
P-adic_number
on 2012-03-11, retrieved 2011-05-12 Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58
P-adic_distribution
In mathematics, the p-adic gamma function Γp is a function of a p-adic variable analogous to the gamma function. It was first explicitly defined by Morita
P-adic_gamma_function
Research program
a constant field, and the harmonic oscillator. p-adic analysis Volovich, I. V. (1987-06-01). "p-adic space-time and string theory". Theoretical and Mathematical
P-adic_quantum_mechanics
Study of objects of arithmetic interest over infinite towers of number fields
{\displaystyle \Gamma } isomorphic to the additive group of p-adic integers for some prime p. (These were called Γ {\displaystyle \Gamma } -extensions in
Iwasawa_theory
Mathematical 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
P-adic_exponential_function
notion of differentiability of functions that is particularly suited to p-adic analysis. In short, the definition is made more restrictive by allowing both
Strict_differentiability
Natural number
Knuth & Patashnik 1994, p. 111. Kennedy 1974, pp. 389. Peano 1889, p. 1. Peano 1908, p. 27. Halmos 1974, p. 32. Hodges 2009, p. 14. Hext 1990. Graham,
1
French mathematician (1936–1993)
theory and p-adic analysis. She was the second woman president of the Société mathématique de France. She wrote a textbook on the p-adic number system
Yvette_Amice
In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions
P-adic_L-function
French mathematician (born 1962)
recherche at the CNRS (IMJ-PRG) known for his work in number theory and p-adic analysis. Colmez studied at École Normale Supérieure and obtained his doctorate
Pierre_Colmez
Mathematical theory
In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields
P-adic_Hodge_theory
Relates the topology of a complete non-archimedean field to its algebraic extensions
In number theory, more specifically in p-adic analysis, Krasner's lemma is a basic result relating the topology of a complete non-archimedean field to
Krasner's_lemma
Result in modular arithmetic
power of p tends to infinity, it follows that a root or a factorization modulo p can be lifted to a root or a factorization over the p-adic integers.
Hensel's_lemma
theory p-adic analysis a branch of number theory that deals with the analysis of functions of p-adic numbers. p-adic dynamics an application of p-adic analysis
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Branch of algebraic geometry
varieties. p-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields
Arithmetic_geometry
specifically in p-adic analysis, the Artin–Hasse exponential, introduced by Emil Artin and Helmut Hasse in 1928, is the power series given by E p ( x ) = exp
Artin–Hasse_exponential
Mathematics of real numbers and real functions
York: Wiley, ISBN 978-0-471-31716-6. Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58
Real_analysis
Highest power of p dividing a given number
the p-adic valuation or p-adic order of an integer n is the exponent of the highest power of the prime number p that divides n. It is denoted ν p ( n
P-adic_valuation
Branch of mathematics
monogenic or Clifford analytic functions. p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting
Mathematical_analysis
American mathematician
1998) was an American mathematician, known for his application of p-adic analysis to local zeta functions, and in particular for a proof of the first
Bernard_Dwork
Number
2013. Retrieved 4 April 2018. Foerster 1980, p. 3. Foerster 1980, p. 21. Cheng 2017, p. 47. Foerster 1980, p. 136. Herman, Edwin; Strang, Gilbert; et al
0
Israeli mathematician and professor
– Israel Institute of Technology. He is known for his work in p-adic analysis, p-adic quantum mechanics, and non-additive geometry, including the field
Shai_Haran
mathematical analysis (3., [Nachdr.] ed.). New York: McGraw-Hill. pp. 47, 52–54. ISBN 978-0-07-054235-8. Koblitz, Neal. (1984). P-adic Numbers, p-adic Analysis, and
Complete_field
The zeros of a linear recurrence relation mostly form a regularly repeating pattern
with values in any field of characteristic zero. Its known proofs use p-adic analysis and are non-constructive. Let K {\displaystyle K} be a field of characteristic
Skolem–Mahler–Lech_theorem
Sum of an (infinite) geometric progression
11996214. ISSN 0025-570X. Robert, Alain M. (2000). A Course in p {\displaystyle p} -adic Analysis. Graduate Texts in Mathematics. Vol. 198. New York, USA: Springer-Verlag
Geometric_series
Algorithm for finding zeros of functions
used cubic approximations. In p-adic analysis, the standard method to show a polynomial equation in one variable has a p-adic root is Hensel's lemma, which
Newton's_method
German mathematician (1903–1988)
fields of transcendental number theory, diophantine approximation, p-adic analysis, and the geometry of numbers. Mahler was a student at the universities
Kurt_Mahler
Theory in number theory
alternative proofs of partial cases of the Grothendieck conjecture without using p-adic Hodge theory. Combinatorial anabelian geometry helps to study various aspects
Anabelian_geometry
Function which measures the "size" of elements in a field or integral domain
cases. Koblitz, Neal (1984). P-adic numbers, p-adic analysis, and zeta-functions (2nd ed.). New York: Springer-Verlag. p. 1. ISBN 978-0-387-96017-3. Retrieved
Absolute_value_(algebra)
Test for the divergence of an infinite series
test is often checked first due to its ease of use. In the case of p-adic analysis the term test is a necessary and sufficient condition for convergence
Nth-term_test
American mathematician (1933–2013)
director of undergraduate studies for 30 years. His research areas were p-adic analysis and representation theory. He created several programs to improve the
Paul_Sally
Algebraic structure with addition, multiplication, and division
algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly used and studied in mathematics, particularly in number
Field_(mathematics)
Particular kind of algebraic structure
also be defined over fields of p {\displaystyle p} -adic numbers. This is part of p {\displaystyle p} -adic analysis. The prototypical example of a Banach
Banach_algebra
All numbers between two given numbers
numerical analysis, including adaptive mesh refinement, multigrid methods and wavelet analysis. Another way to represent such a structure is p-adic analysis (for
Interval_(mathematics)
Function in algebra
1967, p. 2. Emil Artin Geometric Algebra, pages 47 to 49, via Internet Archive Robert, Alain M. (2000), A Course in p-adic Analysis, Springer, p. 129,
Valuation_(algebra)
Mathematical property of algebraic structures
Verslag Afd. Natuurk. (52): 74–84. MR 0015678. Neal Koblitz, "p-adic Numbers, p-adic Analysis, and Zeta-Functions", Springer-Verlag,1977. Shell, Niel, Topological
Archimedean_property
Product of numbers from 1 to n
{\displaystyle p} -adic valuation of a factorial". A Course in p {\displaystyle p} -adic Analysis. Graduate Texts in Mathematics. Vol. 198. New York: Springer-Verlag
Factorial
Study of discrete mathematical structures
objects include transcendental numbers, diophantine approximation, p-adic analysis and function fields. Algebraic structures occur as both discrete examples
Discrete_mathematics
American mathematician and cryptographer
of Waterloo people Gross–Koblitz formula — (1984) [1977]. p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics. Vol. 58
Neal_Koblitz
forms, computational complexity theory, algebraic combinatorics and p-adic analysis. Sergei Evdokimov was born in Leningrad (now Saint Petersburg, Russia)
Sergei_Evdokimov
Russian mathematician (1923–2012)
physics, quantum field theory, numerical analysis, generalized functions, several complex variables, p-adic analysis, multidimensional Tauberian theorems
Vasily_Vladimirov
fields were originally introduced in p-adic analysis since the fields Q p {\displaystyle \mathbb {Q} _{p}} of p-adic numbers are locally compact topological
Locally_compact_field
Complement of an open subset
for many examples, including the Cantor set and spaces arising in p-adic analysis. In algebraic number theory, topological groups over non-Archimedean
Closed_set
Undergraduate math course at Harvard University
weeks of point-set topology and special topics (for instance, in 1994, p-adic analysis was taught by Wilfried Schmid), students would take a quiz. As of 2012
Math_55
Academic fields of study or professions
Non-standard analysis Ordinary differential equations p-adic analysis Partial differential equations Real analysis Calculus (outline) Probability theory Ergodic
Outline of academic disciplines
Outline_of_academic_disciplines
Result in number theory showing congruences involving Bernoulli numbers
to define the p-adic zeta function. The simplest form of Kummer's congruence states that B h h ≡ B k k ( mod p ) whenever h ≡ k ( mod p − 1 ) {\displaystyle
Kummer's_congruence
Mahler's compactness theorem (geometry of numbers) Mahler's theorem (p-adic analysis) Maier's theorem (analytic number theory) Mann's theorem (number theory)
List_of_theorems
Canadian-American mathematician (born 1974)
representation theory of quadratic forms, to interpolation problems and p-adic analysis, to the study of ideal class groups of algebraic number fields, and
Manjul_Bhargava
Special character in number theory
ISBN 978-0-387-49922-2, MR 2312337 Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58
Teichmüller_character
French mathematician (1924–1987)
mathematician who worked on the theory of Lie groups in the context of p-adic analysis. Born in Paris, Lazard studied at the University of Paris–Sorbonne
Michel_Lazard
Finite extension of the rationals
tools such as intermediate value theorem at the archimedean places and p-adic analysis at the nonarchimedean places) can be used. This implication does not
Algebraic_number_field
Number whose square ends in the same digits
base, i)) Arithmetic dynamics Kaprekar number p-adic number p-adic analysis Zero-divisor See Gérard Michon's article at "spherical number"
Automorphic_number
Fraction with denominator a power of two
Fractional and integral parts of p {\displaystyle p} -adic numbers", A Course in p {\displaystyle p} -adic Analysis, Graduate Texts in Mathematics, vol
Dyadic_rational
Type of topological space in mathematics
p-adic numbers is locally compact, because it is homeomorphic to the Cantor set minus one point. Thus locally compact spaces are as useful in p-adic analysis
Locally_compact_space
Type of metric space
Similar ideas can be found in domain theory. p-adic analysis makes heavy use of the ultrametric nature of the p-adic metric. In condensed matter physics, the
Ultrametric_space
Mathematical integration method
of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions. Let : f : Z p → C p {\displaystyle f:\mathbb {Z} _{p}\to
Volkenborn_integral
On all absolute values of rational numbers
\mathbb {Q} } is equivalent to either the usual real absolute value or a p-adic absolute value. An absolute value on the rational numbers is a function
Ostrowski's_theorem
Series of mathematics textbooks
Richard H. Crowell, Ralph H. Fox (1977, ISBN 978-0-387-90272-2) p-adic Numbers, p-adic Analysis, and Zeta-Functions, Neal Koblitz (1984, 2nd ed., ISBN 978-0-387-96017-3)
Graduate_Texts_in_Mathematics
Mathematics of varieties with integer coordinates
geometry Arakelov geometry Hindry & Silverman 2000, p. vii, Preface. Hindry & Silverman 2000, p. viii, Preface. "Mordell : Review: Serge Lang, Diophantine
Diophantine_geometry
German mathematician
internazionale per la ricerca matematica; Congress on "p-adic Analysis" (1990). P-adic analysis : proceedings of the international conference held in Trento
Siegfried_Bosch
Infinite series that diverges
p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics, vol. 58. Springer-Verlag. pp. chapter I, exercise 16, p. 20. ISBN 0-387-96017-1
1_+_2_+_4_+_8_+_⋯
Fundamental construction of differential calculus
arbitrary algebraic varieties, instead of just smooth manifolds. In p-adic analysis, the usual definition of derivative is not quite strong enough, and
Generalizations of the derivative
Generalizations_of_the_derivative
Operation in differential geometry
analytic functions between real or complex domains, to p-adic analysis, and to other areas of analysis. Let C ∞ ( R n , R m ) {\displaystyle C^{\infty }({\mathbb
Jet_(mathematics)
Russian–American mathematician
Chicago Press, 1992. Russian edition, Faktorial Press, Moscow, 2002. p-adic Analysis Compared with Real, Student Mathematical Library, vol. 37, American
Svetlana_Katok
International specialist organization
Annual Gödel Lecture 1993 1993 Angus Macintyre, Logic of Real and p-adic Analysis: Achievements and Challenges The Third Annual Gödel Lecture 1992 1992
Association for Symbolic Logic
Association_for_Symbolic_Logic
Award in mathematical logic
Shoenfield, The Priority Method. 1993 Angus Macintyre, Logic of Real and p-adic Analysis: Achievements and Challenges. 1994 Donald A. Martin, L(R): A Survey
Gödel_Lecture
consequences. Dwork's method Bernard Dwork used distinctive methods of p-adic analysis, p-adic algebraic differential equations, Koszul complexes and other techniques
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
American mathematician
theory, with specific interests in p-adic analysis and arithmetic geometry. In particular, he developed a theory of p-adic integration analogous to the classical
Robert_F._Coleman
for indigenous students Yvette Amice (1936–1993), French expert on p-adic analysis who became president of the French mathematical society Divsha Amirà
List_of_women_in_mathematics
Type of zeta function
1017/is010004028jkt103. Sources François Bruhat (1963). Lectures on some aspects of p-adic analysis. Tata Institute of Fundamental Research. Serre, Jean-Pierre (1969–1970)
Arithmetic_zeta_function
space Metric topology Manhattan distance Ultrametric space P-adic numbers, p-adic analysis Open ball Bounded subset Pointwise convergence Metrization
List of general topology topics
List_of_general_topology_topics
Analogue of a complex analytic space over a nonarchimedean field
on uniformizing p-adic elliptic curves with bad reduction using the multiplicative group. In contrast to the classical theory of p-adic analytic manifolds
Rigid_analytic_space
Belgian mathematician (1936–2008)
July 2008) was a Belgian mathematician known as a pioneer of p-adic functional analysis, and particularly for her work on locally convex topological vector
Nicole_De_Grande-De_Kimpe
Swiss mathematician
of p-adic analysis of one variable (except the rationality of the zeta function of an algebraic variety over a finite field and the theory of p-adic differential
Alain_M._Robert
Infinite sum that is considered independently from any notion of convergence
seen as the (x)-adic completion of the polynomial ring R [ x ] , {\displaystyle R[x],} in the same way as the p-adic integers are the p-adic completion of
Formal_power_series
American mathematician
research interests include algebraic number theory, Diophantine analysis, p-adic analysis, geometry of numbers, and the theory of continued fractions. He
Edward_Burger
Chebotarev's density theorem Totally real field Local field p-adic number p-adic analysis Adele ring Idele group Idele class group Adelic algebraic group
List of algebraic number theory topics
List_of_algebraic_number_theory_topics
Alternative decimal expansion of 1
p {\displaystyle p} -adic numbers are an alternative number system of interest in number theory. Like the real numbers, the p {\displaystyle p} -adic
0.999...
Infinite series that diverges
summation methods to the series, as well as the limit of the series using the 2-adic metric. Gottfried Leibniz considered the divergent alternating series 1 −
1_−_2_+_4_−_8_+_⋯
Topological space that is maximally disconnected
homeomorphic to the set of p-adic integers. Another example, playing a key role in algebraic number theory, is the field Qp of p-adic numbers. A topological
Totally_disconnected_space
Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special polynomials. It is the p-adic counterpart to the Stone-Weierstrass
Mahler's_theorem
Field of mathematics
also called p-adic or nonarchimedean dynamics, is an analogue of complex dynamics in which one replaces the complex numbers C by a p-adic field such as
Arithmetic_dynamics
Mathematical space with a notion of distance
graphs may be viewed as metric spaces. In abstract algebra, the field of p-adic numbers is the completion of the field of rational numbers with respect
Metric_space
Harish-Chandra (1973), "Harmonic analysis on reductive p-adic groups", in Moore, Calvin C. (ed.), Harmonic analysis on homogeneous spaces (Proc. Sympos
Steinberg_representation
Used to count, measure, and label
algebraic structures are explicitly referred to as numbers (such as the p-adic numbers and hypercomplex numbers) while others are not, but this is more
Number
Function named after Harish Chandra
c-function for p-adic Lie groups. Macdonald (1968, 1971) and Langlands (1971) found an analogous product formula for the c-function of a p-adic Lie group.
Harish-Chandra's_c-function
Mathematical concept
analogous theory to the classical theory of complex modular forms and the p-adic theory of modular forms. Modular forms are analytic functions, so they admit
Modular_forms_modulo_p
Branch of functional analysis
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the
Operator_algebra
American mathematician
1977). Kottwitz works in the Langlands program, including harmonic analysis on p-adic Lie groups and automorphic forms and the general linear groups and
Robert_Kottwitz
Branch of algebra that studies commutative rings
integers, including the ordinary integers Z {\displaystyle \mathbb {Z} } ; and p-adic integers. Commutative algebra is the main technical tool of algebraic geometry
Commutative_algebra
Solving integer equations from all modular solutions
then this also yields a real solution and a p-adic solution, as the rationals embed in the reals and p-adics: a global solution yields local solutions at
Hasse_principle
Algebra based on a vector space with a quadratic form
(it is not a subalgebra). This Z2-grading plays an important role in the analysis and application of Clifford algebras. The automorphism α is called the
Clifford_algebra
Quotient of two integers
d p ) {\displaystyle (\mathbb {Q} ,d_{p})} is not complete, and its completion is the p-adic number field Q p . {\displaystyle \mathbb {Q} _{p}.}
Rational_number
Group that is also a differentiable manifold with group operations that are smooth
\mathbb {Q} } , one can define a p-adic Lie group over the p-adic numbers, a topological group which is also an analytic p-adic manifold, such that the group
Lie_group
Retrieved 15 August 2023. Analysis of Tobacco Market in Sri-Lanka (PDF) (in British English and Indian English). Colombo: ADIC Sri Lanka – Alcohol & Drug
List_of_cigarette_brands
Indexed set in mathematics
important special case is known as the I {\displaystyle I} -adic topology (or J {\displaystyle J} -adic, etc.): Let R {\displaystyle R} be a commutative ring
Filtration_(mathematics)
Number of arguments required by a function
many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency. In general, functions
Arity
Branch of mathematics
arithmetic dynamics studies iteration over the rational numbers or the p-adic numbers instead of the complex numbers. A simple example that shows some
Complex_dynamics
P ADIC-ANALYSIS
P ADIC-ANALYSIS
Female
English
(עֲדִי) Hebrew unisex name ADI means "my ornament" or "my witness."
Male
English
Short form of English Alexander, ALIC means "defender of mankind."
Boy/Male
African Egyptian
Righteous.
Boy/Male
Teutonic American German English Norse
Noble commander.
Boy/Male
Indian
A companion of the prophet, Also the name of the son of Hatim tiay known for his generosity, Also the son of Thabit had this name
Boy/Male
Indian
Pleasure giver, Beautiful, Adorned
Boy/Male
Arabic
Fair; judicious.
Boy/Male
Indian
A literary person, Cultured, Civilized
Boy/Male
Hebrew
Attractive; handsome; pleasure given. Adin was a biblical exile who returned to Israel from Babylon.
Boy/Male
Muslim
A companion of the prophet, Also the name of the son of Hatim tiay known for his generosity, Also the son of Thabit had this name
Boy/Male
Hebrew
Gentle; delicate.
Boy/Male
Indian
From the beginning
Male
English
Anglicized form of Hebrew Adiyn, ADIN means "dainty, delicate." In the bible, this is the name of an ancestor of a family of exiles who returned with Zerubbabel.
Boy/Male
Hebrew
noble.
Boy/Male
Indian
Pleasant
Boy/Male
Muslim
A literary person, Cultured, Civilized
Boy/Male
Indian
Judge, Honest, Upright, Justice, Sincere, Just
Male
English
Variant spelling of English Eric, ARIC means "ever-ruler."
Male
Hungarian
Hungarian form of English Philip, FÜLÖP means "lover of horses."
Boy/Male
Muslim
Pleasure giver, Beautiful, Adorned
P ADIC-ANALYSIS
P ADIC-ANALYSIS
Boy/Male
Hindu, Indian, Marathi
Mighty Superior
Boy/Male
American, British, English
Roofer
Boy/Male
Indian
Intended, Aimed at, Object, Proposed
Boy/Male
Tamil
Intelligent
Girl/Female
Tamil
Person with a good voice
Female
Japanese
(ã‚ゆã¿) Japanese name AYUMI means "pace, stroll, walk."
Boy/Male
African, Arabic, Indian, Muslim, Sanskrit, Swahili
Jewel; Gem; Pearl
Surname or Lastname
English
English : variant spelling of Machen.Spanish (MachÃn) : probably a nickname from machÃn ‘boor’, ‘lout’, often applied to a blacksmith’s apprentice.French : nickname from Old French machin ‘scheming’.
Boy/Male
Assamese, Bengali, French, German, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Son of Wind
Boy/Male
Welsh
Little.
P ADIC-ANALYSIS
P ADIC-ANALYSIS
P ADIC-ANALYSIS
P ADIC-ANALYSIS
P ADIC-ANALYSIS
a.
Related to, or derived, ammonia; -- used chiefly as a suffix; as, amic acid; phosphamic acid.
a.
Pertaining to, or derived from, the cod (Gadus); -- applied to an acid obtained from cod-liver oil, viz., gadic acid.