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In mathematics, a Weierstrass ring, named by Nagata after Karl Weierstrass, is a commutative local ring that is Henselian, pseudo-geometric, and such
Weierstrass_ring
Weierstrass. Bolzano–Weierstrass theorem Casorati–Weierstrass theorem Weierstrass method Enneper–Weierstrass parameterization Lindemann–Weierstrass theorem
List of things named after Karl Weierstrass
List_of_things_named_after_Karl_Weierstrass
Local theory of several complex variables
the idea of factorization in some ring R as u·w, where u is a unit and w is some sort of distinguished Weierstrass polynomial. Carl Siegel has disputed
Weierstrass preparation theorem
Weierstrass_preparation_theorem
Mathematical theorem in the study of analysis
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly
Stone–Weierstrass_theorem
Infinite sum that is considered independently from any notion of convergence
as a product topology. The ring of formal power series with coefficients in a complete local ring satisfies the Weierstrass preparation theorem. Formal
Formal_power_series
ai−1) for all i. Weierstrass ring A Weierstrass ring is local ring that is Henselian, pseudo-geometric, and such that any quotient ring by a prime ideal
Glossary of commutative algebra
Glossary_of_commutative_algebra
Extension of the factorial function
every complex number z. The definition for the gamma function due to Weierstrass is also valid for all complex numbers z except non-positive integers:
Gamma_function
Algebraic curve in mathematics
field K, x and y the Weierstrass coordinates. Then there are only finitely many points of E(K) whose x-coordinate is in the ring of integers OK. The properties
Elliptic_curve
Type of mathematical curve
the left or to the right is needed for having a true Weierstrass form. Singular cubics in Weierstrass form Isolated point y2 = x3 − x2 semicubical parabola
Plane_cubic_curve
Doughnut-shaped surface of revolution
coplanar with the circle. The main types of tori include ring tori, horn tori, and spindle tori. A ring torus is sometimes colloquially referred to as a doughnut
Torus
Major unsolved problem in transcendental number theory
for this more general result was given by Carl Weierstrass in 1885. This so-called Lindemann–Weierstrass theorem implies the transcendence of the numbers
Schanuel's_conjecture
Formal power series with coefficients tending to 0
(reduced) Banach algebra called an affinoid algebra. Some key results are: (Weierstrass division) Let g∈Tn be a ξn-distinguished series of order s; i.e., g=∑ν=0∞gνξnν
Restricted_power_series
Mathematical constant
of 10: log210=1log102≈3.321928095 (OEIS: A020862). By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and
Natural_logarithm_of_2
Branch of mathematics
assumption he called Dirichlet's principle, which in 1870 was questioned by Weierstrass. Much later, in 1900, Hilbert justified Riemann's approach by developing
Abstract_algebra
_{j=0}^{d-1}h_{j}(x')x_{n}^{j}} . While the ring of analytic functions and the ring of formal power series both satisfy the Weierstrass division property, the same is
Quasi-analytic_function
Root-finding algorithm for polynomials
In numerical analysis, the Weierstrass method or Durand–Kerner method, discovered by Karl Weierstrass in 1891 and rediscovered independently by Durand
Durand–Kerner_method
Array of numbers
1, Ch. III, p. 96. Knobloch (1994). Hawkins (1975). Kronecker 1897 Weierstrass 1915, pp. 271–286 & Miller (1930). Bôcher (2004). Hawkins (1972). van
Matrix_(mathematics)
In algebra, integer associated to a module
and denominator. Hilbert–Poincaré series Weil divisor Chow ring Intersection theory Weierstrass factorization theorem Serre's multiplicity conjectures Hilbert
Length_of_a_module
Theory of a class of elliptic curves
{\displaystyle Y\to \pm iY,\quad X\to -X} in line with the action of i on the Weierstrass elliptic functions. More generally, consider the lattice Λ, an additive
Complex_multiplication
Type of mathematical space
when it was rediscovered by Karl Weierstrass. In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for
Compact_space
American mathematician (born 1949)
Pennsylvania under the supervision of Dock-Sang Rim. Her dissertation was Weierstrass Points and Monomial Curves. The title of her 2014 Falconer lecture was
Marie_A._Vitulli
Power series with negative powers
named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass had previously described it in a paper written in 1841 but not published
Laurent_series
Banach fixed-point theorem Banach–Tarski paradox Basel problem Bolzano–Weierstrass theorem Brouwer fixed-point theorem Buckingham π theorem (proof in progress)
List_of_mathematical_proofs
Algebraic variety in a projective space
variety of dimension 1, i.e., from an elliptic curve. In fact, the Weierstrass's elliptic function ℘ {\displaystyle \wp } attached to L satisfies a certain
Projective_variety
American mathematician
notes of Weierstrass's 1879 course, and Bolza's teaching. Bolza went on to supervise Bliss's Ph.D. thesis, The Geodesic Lines on the Anchor Ring, completed
Gilbert_Ames_Bliss
valuation Discrete valuation ring I-adic topology Weierstrass preparation theorem Noetherian ring Hilbert's basis theorem Artinian ring Ascending chain condition
List of commutative algebra topics
List_of_commutative_algebra_topics
Distance from zero to a number
notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude
Absolute_value
Russian mathematician (1850–1891)
with Weierstrass for three more years. In 1874 she presented three papers—on partial differential equations, on the dynamics of Saturn's rings, and on
Sofya_Kovalevskaya
Type of mathematical expression
differentiable function locally looks like a polynomial function, and the Stone–Weierstrass theorem, which states that every continuous function defined on a compact
Polynomial
Result in field theory about zeros of formal power series
corollary of the Weierstrass preparation theorem is that f ( x ) {\displaystyle f(x)} has exactly N {\displaystyle N} zeros in the valuation ring of the algebraic
Strassmann's_theorem
Polynomial remainder theorem See also Theory of equations below. Polynomial ring Greatest common divisior of two polynomials Symmetric function Homogeneous
List_of_polynomial_topics
coordinate ring of X. Algebraic geometry occupied a central place in the mathematics of the last century. The deepest results of Abel, Riemann, Weierstrass, many
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Number with a real and an imaginary part
is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass. Later classical writers on the general theory include Richard Dedekind
Complex_number
Van Vleck's theorem (mathematical analysis) Weierstrass–Casorati theorem (complex analysis) Weierstrass factorization theorem (complex analysis) Appell–Humbert
List_of_theorems
Fiber bundle whose fibers are projective spaces
{P} ^{1}}(6)\oplus {\mathcal {O}}_{\mathbb {P} ^{1}})} defined by the Weierstrass equation y 2 z + a 1 x y z + a 3 y z 2 = x 3 + a 2 x 2 z + a 4 x z 2
Projective_bundle
Theorem about smooth complex functions
mathematics, the Malgrange preparation theorem is an analogue of the Weierstrass preparation theorem for smooth functions. It was conjectured by René
Malgrange_preparation_theorem
Algebraic structure in linear algebra
1] can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem. In contrast, the space of all continuous functions
Vector_space
Equivalence class of objects sharing local properties at a point in a topological space
under consideration. The Weierstrass preparation theorem implies that rings of germs of holomorphic functions are Noetherian rings. It can also be shown
Germ_(mathematics)
Counterintuitive mathematical object
Weierstrass function, a function that is continuous everywhere but differentiable nowhere. The sum of a differentiable function and the Weierstrass function
Pathological_(mathematics)
Particular kind of algebraic structure
closed. But the image of the Gelfand representation is dense by the Stone–Weierstrass theorem. Conway 1990, Example VII.1.8. Conway 1990, Example VII.1.9.
Banach_algebra
Analytic function on the upper half-plane with a certain behavior under the modular group
see e.g. "DLMF: §23.15 Definitions ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions". dlmf.nist.gov. Retrieved 2023-07-07
Modular_form
Type of image blur produced by a Gaussian function
image with a Gaussian function. This is also known as a two-dimensional Weierstrass transform. By contrast, convolving by a circle (i.e., a circular box
Gaussian_blur
Infinitely detailed mathematical structure
century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the word fractal in the 20th century with a
Fractal
Type of complex number
1. The numbers π and e are not algebraic numbers (see the Lindemann–Weierstrass theorem). If a polynomial with rational coefficients is multiplied through
Algebraic_number
Projective analogue of the spectrum of a ring
algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties
Proj_construction
Fractal named after mathematician Benoit Mandelbrot
Orbit trap Pickover stalk Plotting algorithms for the Mandelbrot set Weierstrass–Mandelbrot function "Mandelbrot set". Lexico UK English Dictionary. Oxford
Mandelbrot_set
Symbol representing a mathematical object
nowhere differentiable continuous function. To solve this problem, Karl Weierstrass introduced a new formalism consisting of replacing the intuitive notion
Variable_(mathematics)
Relation between algebraic varieties and polynomial ideals
Nullstellensatz holds for Tate algebras. Krivine–Stengle Positivstellensatz Weierstrass Nullstellensatz Tao, Terence (2007-11-26). "Hilbert's nullstellensatz"
Hilbert's_Nullstellensatz
Element of a nonstandard model of the reals, which can be infinite or infinitesimal
the Bolzano-Weierstrass theorem, the property (1) of ultrafilters turns out to be crucial. The finite elements F of ∗ℝ form a local ring, and in fact
Hyperreal_number
Mathematical function
define other modular forms. In particular the modular discriminant of the Weierstrass elliptic function with ω2=τω1 can be defined as Δ(τ)=(2πω1)12η(τ)24 and
Dedekind_eta_function
Instantaneous rate of change (mathematics)
a monotone or a Lipschitz function), this is true. However, in 1872, Weierstrass found the first example of a function that is continuous everywhere but
Derivative
Modular function in mathematics
This lattice corresponds to the elliptic curve y2=4x3−g2(τ)x−g3(τ) (see Weierstrass elliptic functions). Note that j is defined everywhere in H as the modular
J-invariant
in 1872 by Karl Weierstrass, and in fact examples had been found earlier of functions that were nowhere differentiable (see Weierstrass function). According
List_of_conjectures
Oscillatory error in Fourier series
convergent Fourier coefficients would be uniformly convergent by the Weierstrass M-test and would thus be unable to exhibit the above oscillatory behavior
Gibbs_phenomenon
Simple curve of Euclidean geometry
task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (π) is a transcendental number, rather
Circle
Ring homomorphism from the cobordism ring of manifolds to another ring
In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding
Genus of a multiplicative sequence
Genus_of_a_multiplicative_sequence
of things named after Stanislaw Ulam List of things named after Karl Weierstrass List of things named after André Weil List of things named after Hermann
Lists_of_mathematics_topics
Used to count, measure, and label
with the work of Augustin-Louis Cauchy, Charles Méray (1869), Karl Weierstrass (1872), Eduard Heine (1872), Georg Cantor (1883), and Richard Dedekind
Number
Mathematics independent of applications
professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent. After Weierstrass, by the end of 19th century
Pure_mathematics
Topological structure in number theory
This ring is a 2-dimensional complete Noetherian regular local ring, and in particular a unique factorization domain. It follows from the Weierstrass preparation
Iwasawa_algebra
Branch of functional analysis
from polynomial to continuous functional calculus by using the Stone–Weierstrass theorem. The crucial fact here is that, for a bounded self adjoint operator
Borel_functional_calculus
23 mathematical problems stated in 1900
lecture—which, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, is its
Hilbert's_problems
Algebra associated to any vector space
confusion. This axiomatization of areas is due to Leopold Kronecker and Karl Weierstrass; see Bourbaki (1989b, Historical Note). For a modern treatment, see Mac
Exterior_algebra
Integral transform useful in probability theory, physics, and engineering
among the first to study the Laplace transform rigorously in the Karl Weierstrass school of analysis, and apply it to the study of differential equations
Laplace_transform
the lengths of its sides. Nowhere differentiable function called also Weierstrass function: continuous everywhere but not differentiable even at a single
List_of_types_of_functions
Day of the year
English-Australian politician, 5th Premier of Tasmania (died 1884) 1815 – Karl Weierstrass, German mathematician and academic (died 1897) 1825 – Charles Lavigerie
October_31
Concepts from linear algebra
the corresponding result for skew-symmetric matrices. Finally, Karl Weierstrass clarified an important aspect in the stability theory started by Laplace
Eigenvalues_and_eigenvectors
Modular unit in mathematics
has class number one. Let a be an ideal of R with generator α. For a Weierstrass model of E, define Θ a ( P ) = α − 12 Δ E N a − 1 ∏ a P = 0 , P ≠ 0 (
Elliptic_unit
Field of knowledge
computer networks. In the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly focused their research on internal
Mathematics
Weixiao (2018). "Hausdorff dimension of the graphs of the classical Weierstrass functions". Mathematische Zeitschrift. 289 (1–2): 223–266. arXiv:1505
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
German polymath (1646–1716)
infinitesimals in mathematics was frowned upon by followers of Karl Weierstrass, but survived in science and engineering, and even in rigorous mathematics
Gottfried_Wilhelm_Leibniz
Algorithm for integer factorization
with a ≠ 0. Calculate b=yP2−xP3−axP. The elliptic curve E is then in Weierstrass form given by y2=x3+ax+b and by using projective coordinates the elliptic
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
Post-quantum cryptographic algorithm
and is not patented. The j-invariant of an elliptic curve given by the Weierstrass equation y2=x3+ax+b is given by the formula: j(E)=17284a34a3+27b2. Isomorphic
Supersingular isogeny key exchange
Supersingular_isogeny_key_exchange
Mathematical behavior near singularities
(of a punctured disk) König, Wolfgang; Sprekels, Jürgen (2015). Karl Weierstraß (1815–1897): Aspekte seines Lebens und Werkes – Aspects of his Life and
Monodromy
Projective variety that is also an algebraic group
important contributors to the theory of abelian functions were Riemann, Weierstrass, Frobenius, Poincaré, and Picard. The subject was very popular at the
Abelian_variety
Blancmange curve Triflake[citation needed] Vicsek fractal von Koch curve Weierstrass function Z-order curve von Koch curve with random interval von Koch curve
List_of_mathematical_shapes
Decomposition of periodic functions
L2([−π,π]). The density of their span is a consequence of the Stone–Weierstrass theorem, but follows also from the properties of classical kernels like
Fourier_series
Complex-differentiable (mathematical) function
of view, the set of holomorphic functions on an open set is a commutative ring and a complex vector space. Additionally, the set of holomorphic functions
Holomorphic_function
Algorithm in the theory of elliptic curves
2007-12-20 Laska, Michael (1982), "An Algorithm for Finding a Minimal Weierstrass Equation for an Elliptic Curve", Mathematics of Computation, 38 (157):
Tate's_algorithm
discouraged, and the preferred representation is U+00C5 'capital letter A with ring above', which has the same glyph. IJ and ij: The use of U+0132
List of XML and HTML character entity references
List_of_XML_and_HTML_character_entity_references
curve, cubic curve Elliptic function, Jacobi's elliptic functions, Weierstrass's elliptic functions Elliptic integral Complex multiplication Weil pairing
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Infinite sum
can be integrated term by term. Tests for uniform convergence include Weierstrass' M-test, Abel's uniform convergence test, Dini's test, and the Cauchy
Series_(mathematics)
Type of function in mathematics
series on a finite extension field K of a p-adic field converges on the ring of integers if and only if |an|K→0. The reason is the ultrametric criterion:
Analytic_function
Set without nontrivial polynomial equalities
numbers π and eπn are algebraically independent over ℚ. The Lindemann–Weierstrass theorem can often be used to prove that some sets are algebraically independent
Algebraic_independence
Georg Cantor/Charles Méray, Richard Dedekind/Joseph Bertrand and Karl Weierstrass all occurred within a few years of each other. Each has advantages and
Construction of the real numbers
Construction_of_the_real_numbers
apparatus – Ernst Heinrich Weber Weierstrass–Casorati theorem – Karl Theodor Wilhelm Weierstrass and Felice Casorati Weierstrass's elliptic functions, factorization
Scientific phenomena named after people
Scientific_phenomena_named_after_people
computer science. Augustin-Louis Cauchy, Bernhard Riemann, and Karl Weierstrass reformulated the calculus in a more rigorous fashion. Also, for the first
History_of_mathematics
field K and p a prime ideal of the ring of integers of K. We consider a minimal equation for E: a generalised Weierstrass equation whose coefficients are
Conductor of an elliptic curve
Conductor_of_an_elliptic_curve
Algebraic curve
models. One geometric characterization of hyperelliptic curves is via Weierstrass points. More detailed geometry of non-hyperelliptic curves is read from
Hyperelliptic_curve
Number representing a continuous quantity
showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), Hilbert (1893), Hurwitz, and Gordan. The concept that many points
Real_number
Space formed by the ''n''-tuples of real numbers
This sequence is bounded because ‖𝐱~k‖2=1. So because of the Bolzano–Weierstrass theorem there exists a convergent subsequence (𝐱~kj)j∈𝐍 with limit
Real_coordinate_space
Term in mathematics
greatest-lower-bound property The nested interval property The Bolzano-Weierstrass theorem The convergence of Cauchy sequences A typical real analysis university
Characterization (mathematics)
Characterization_(mathematics)
Canadian mathematician (1917–2006)
doi:10.1090/s0002-9947-1950-0032612-4. MR 0032612. —— (1950). "The Weierstrass theorem in fields with valuations". Proc. Amer. Math. Soc. 1 (3): 356–357
Irving_Kaplansky
Type of vector space in math
for continuous convex functionals, in the same way that the Bolzano–Weierstrass theorem is used for continuous functions on Rd. Among several variants
Hilbert_space
Efficient algorithm to count points on elliptic curves
field of characteristic ≠2,3 an elliptic curve can be given by a (short) Weierstrass equation y2=x3+Ax+B with A,B∈𝔽q. The set of points defined over 𝔽q
Schoof's_algorithm
Type of mathematical functions
analysis, since its characteristic phenomena weren't uncovered. The Weierstrass preparation theorem would now be classed as commutative algebra; it did
Function of several complex variables
Function_of_several_complex_variables
Uniform norm Matrix norm Spectral radius Normed division algebra Stone–Weierstrass theorem Banach algebra *-algebra B*-algebra C*-algebra Universal C*-algebra
List of functional analysis topics
List_of_functional_analysis_topics
B]/(y^{2}-x^{3}-Ax-B)} . If an elliptic curve E {\displaystyle E} is given in the Weierstrass form y 2 = x 3 + A x + B {\displaystyle y^{2}=x^{3}+Ax+B} over some field
Division_polynomials
Special functions of several complex variables
four theta functions, and could have been used by him to construct Weierstrass's elliptic functions also, since ℘(z;τ)=−(logϑ11(z;τ))″+c where the second
Theta_function
Way to divide polygon into smaller parts
in higher-dimensional mathematical analysis such as for the Bolzano–Weierstrass theorem and Heine–Borel theorem. A finite subdivision rule R {\displaystyle
Finite_subdivision_rule
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WEIERSTRASS RING
WEIERSTRASS RING
WEIERSTRASS RING
WEIERSTRASS RING
WEIERSTRASS RING
WEIERSTRASS RING
WEIERSTRASS RING
WEIERSTRASS RING
WEIERSTRASS RING
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