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One of several theorems linking the sizes of different ideal class groups
theory, a reflection theorem or Spiegelungssatz (German for reflection theorem – see Spiegel and Satz) is one of a collection of theorems linking the
Reflection_theorem
Kind of proposition in mathematics
forms of the reflection principle depending on exactly what is meant by "resemble". Weak forms of the reflection principle are theorems of Zermelo–Fraenkel
Reflection_principle
On distances between points on a circle
Applications of the three-gap theorem include the study of plant growth and musical tuning systems, and the theory of light reflection within a mirrored square
Three-gap_theorem
Relation between sides of a right triangle
In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle
Pythagorean_theorem
equations) Reflection theorem (algebraic number theory) Ribet's theorem (elliptic curves) Robin's theorem (number theory) Rosser's theorem (number theory)
List_of_theorems
Discrete group type in group theory
Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a
Reflection_group
generated by transpositions (ij), which act by reflections on V. On the other hand, by the main theorem of symmetric functions, the algebra of invariants
Chevalley–Shephard–Todd theorem
Chevalley–Shephard–Todd_theorem
Scripps National Spelling Bee winner
mathematics from Princeton University, with a dissertation titled "Reflection theorems for number rings". He held a two-year post-doctoral position at the
Evan_O'Dorney
Continued fraction closely related to the Rogers–Ramanujan identities
And therefore following result appears: In the next step we use the reflection theorem for the continued fraction R again: R [ exp ( − π ) ] ⊕ R [ exp
Rogers–Ramanujan continued fraction
Rogers–Ramanujan_continued_fraction
Theorem in computability theory
recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first
Kleene's_recursion_theorem
Limitative results in mathematical logic
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Proof all ranked voting rules have spoilers
Arrow's impossibility theorem is a key result in social choice theory, proved by American economist Kenneth Arrow. It shows that no procedure for group
Arrow's_impossibility_theorem
Mathematical rule for inverting probabilities
Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes (/beɪz/), gives a mathematical rule for inverting conditional probabilities
Bayes'_theorem
Euclidean Wightman distributions
taking a reflection and complex conjugating all the fields, then the previous quantity has to be nonnegative. The Osterwalder–Schrader theorem states that
Schwinger_function
Theorems that help decompose a finite group based on prime factors of its order
specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow
Sylow_theorems
Geometric theorem regarding circles and triangles
and the Reflection Triangle" (PDF). Forum Geometricorum. 3: 105–111. John Rigby (1997). "Brief notes on some forgotten geometrical theorems". Mathematics
Kosnita's_theorem
Mathematic theorem
composition of at most n reflections. Indefinite orthogonal group Coordinate rotations and reflections Householder reflections Chasles' theorem Gallier, Jean H
Cartan–Dieudonné_theorem
Topics referred to by the same term
function f and a constant a Reflection theorem, one of a collection of theorems about the sizes of class groups Schwarz reflection principle, a way to extend
Reflection principle (disambiguation)
Reflection_principle_(disambiguation)
Theorem in mathematical logic
logic, the Paris–Harrington theorem states that a certain claim in Ramsey theory, namely the strengthened finite Ramsey theorem, which is expressible in
Paris–Harrington_theorem
naming an excess of things and concepts after Euler, some discoveries and theorems are attributed to the first person to have proved them after Euler. Euler's
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
German mathematician (1810–1893)
conjecture Kummer's transformation of series Ideal number Regular prime Reflection theorem Principalization McElroy, Tucker (2005). A to Z of Mathematicians
Ernst_Kummer
Theorem about consecutive perfect powers
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1842
Catalan's_conjecture
Theorem in Euclidean geometry
theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone. This theorem refers
Mohr–Mascheroni_theorem
Distribution result for probability mathematics
distribution as the reflection of the subsequent path about the value a. More formally, the reflection principle refers to a theorem concerning the distribution
Reflection principle (Wiener process)
Reflection_principle_(Wiener_process)
Proof assistant
of the four color theorem, which was completed in 2002. Their work led to the development of the SSReflect ("Small Scale Reflection") package, which was
Rocq
Mathematics principle in complex analysis
In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic
Schwarz_reflection_principle
Invariance under simultaneous charge conjugation, parity transformation and time reversal
explicit proofs, so this theorem is sometimes known as the Lüders–Pauli theorem. At about the same time, and independently, this theorem was also proved by
CPT_symmetry
Theorem in differential topology
The hairy ball theorem of algebraic topology (formally, the Sphere Vector Field Theory, sometimes called the hedgehog theorem) states that there is no
Hairy_ball_theorem
Well-quasi-ordering of finite trees
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under
Kruskal's_tree_theorem
Mapping from a Euclidean space to itself
Cartan–Dieudonné theorem. Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes
Reflection_(mathematics)
Every rigid motion is a screw displacement
In kinematics, Chasles' theorem, or Mozzi–Chasles' theorem, says that the most general rigid body displacement can be produced by a screw displacement
Chasles'_theorem_(kinematics)
Special mathematical function
]^{4}\}} Lemniscatic example for the fifth power theorem: A next example for the fifth power theorem: If two positive numbers a {\displaystyle a} and
Nome_(mathematics)
Shape with four equal sides and angles
number of equal-area triangles, a result of Monsky's theorem. Cross's theorem or Vecten's theorem states that, for a triangle formed by the sides of three
Square
Principle in compass and straightedge constructions
In geometry, the compass equivalence theorem is an important statement in compass and straightedge constructions. The tool advocated by Plato in these
Compass_equivalence_theorem
Theorem in plane geometry
In geometry, Hjelmslev's theorem, named after Johannes Hjelmslev, is the statement that if points P, Q, R... on a line are isometrically mapped to points
Hjelmslev's_theorem
Group of geometric symmetries with at least one fixed point
Dihedral groups Dn of n-fold rotation and reflection groups Applying the crystallographic restriction theorem restricts n to values 1, 2, 3, 4, and 6 for
Point_group
Relationship between derivatives and integrals
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at every
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Equality of areas of a sliced disk
"Reflection groups and the pizza theorem", Algebra i Analiz (in Russian), 33 (6): 1–8 Brailov, Yury (2022), "Reflection groups and the pizza theorem"
Pizza_theorem
Mathematical transformation that preserves distances
transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid
Rigid_transformation
Theorem in optics that explains light propagation in a medium
well as refraction, reflection, and diffraction). It is named after Paul Peter Ewald and Carl Wilhelm Oseen, who proved the theorem in crystalline and
Ewald–Oseen extinction theorem
Ewald–Oseen_extinction_theorem
Interactive theorem prover software
computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs
Proof_assistant
Isometry group of Euclidean space
The Euclidean group E(n) comprises all translations, rotations, and reflections of E n {\displaystyle \mathbb {E} ^{n}} ; and arbitrary finite combinations
Euclidean_group
Geometrical theorem relating the lengths of two segments that divide a triangle
In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that
Angle_bisector_theorem
Simple curve of Euclidean geometry
equation, known as the equation of the circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the
Circle
Theorem concerning spontaneous symmetry breaking
In theoretical physics, the Vafa–Witten theorem, named after Cumrun Vafa and Edward Witten, is a theorem that shows that vector-like global symmetries
Vafa–Witten_theorem
Theorem on the orders of subgroups
In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is
Lagrange's theorem (group theory)
Lagrange's_theorem_(group_theory)
Formula for number of orbits of a group action
sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma, or the orbit-counting theorem, is a result in group theory that is often
Burnside's_lemma
Geometric symmetry operation
transforms as direct sums of rotations and reflections, which follows from the spectral theorem, for instance. "Reflections in Lines". new.math.uiuc.edu. Retrieved
Point_reflection
Overview of and topical guide to geometry
progression Geometric shape Pi Angular velocity Linear velocity De Moivre's theorem Similar triangles Unit circle Point Line and Ray Plane Bearing Angle Degree
Outline_of_geometry
Election result probability theorem
is popularly known as André's reflection method, although André did not use any reflections. Bertrand's ballot theorem is related to the cycle lemma.
Bertrand's_ballot_theorem
pseudoreflection generalizes the concepts of reflection and complex reflection and is simply called reflection by some mathematicians. It plays an important
Pseudoreflection
Generalizations of the Riemann zeta function
{\displaystyle \Lambda } , the result follows. For k = 3 {\displaystyle k=3} , the theorem says ∑ σ ∈ Σ 3 S ( i σ ( 1 ) , i σ ( 2 ) , i σ ( 3 ) ) = ζ ( i 1 ) ζ (
Multiple_zeta_function
This is a list of misnamed theorems in mathematics. It includes theorems (and lemmas, corollaries, conjectures, laws, and perhaps even the odd object)
List_of_misnamed_theorems
English mathematician (1907–1969)
College, Cambridge Known for Davenport–Erdős theorem Davenport–Schinzel sequences Davenport–Schmidt theorem Hasse–Davenport relations Children James H.
Harold_Davenport
Characterization by prime factors of sums of two squares
In number theory, the sum of two squares theorem relates the prime decomposition of any integer n > 1 to whether it can be written as a sum of two squares
Sum_of_two_squares_theorem
Pictorial representation of the behavior of subatomic particles
x = e i k x {\displaystyle A_{kx}=e^{ikx}\,} and the Fourier inversion theorem tells you the inverse: A k x − 1 = e − i k x {\displaystyle A_{kx}^{-1}=e^{-ikx}\
Feynman_diagram
Principle relating to fluid dynamics
that Bernoulli's theorem is responsible... Unfortunately, the 'dynamic lift' involved...is not properly explained by Bernoulli's theorem. a. Babinsky, Holger
Bernoulli's_principle
Mathematical transform that expresses a function of time as a function of frequency
sufficient regularity and decay properties is given by the Fourier inversion theorem, i.e., Inverse transform The functions f {\displaystyle f} and f ^ {\displaystyle
Fourier_transform
Theorem in geometrical optics
through an arbitrary amount of reflections and refractions, then let it emerge in some other homogenous medium. The theorem states that the resulting pencil
Malus–Dupin_theorem
Theorem concerning ratios of line segments
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry
Intercept_theorem
Group of symmetries of a regular polygon
Algebraically, this is an instance of the conjugate Sylow theorem (for n odd): for n odd, each reflection, together with the identity, form a subgroup of order
Dihedral_group
Interpretation of probability
sequential use of Bayes' theorem: as more data become available, calculate the posterior distribution using Bayes' theorem; subsequently, the posterior
Bayesian_probability
Symmetry-based invariance to continuous group action
viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to another
Continuous_symmetry
Theorem in electrical engineering
between two cylinders, the transmission and reflection of light at the boundary between two media. The theorem was originally misunderstood (notably by Joule)
Maximum power transfer theorem
Maximum_power_transfer_theorem
Counting real roots of a polynomial in an interval
In mathematics, Budan's theorem is a theorem for bounding the number of real roots of a polynomial in an interval, and computing the parity of this number
Budan's_theorem
Line which touches a circle at exactly one point
circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs
Tangent_lines_to_circles
Relates the length of a median of a triangle to the lengths of its sides
Theorem via Ptolemy's Theorem". Mathematics Magazine. doi:10.1080/0025570X.2024.2385255. Rose, Mike (2007). "27. Reflections on Apollonius' Theorem"
Apollonius's_theorem
Topics referred to by the same term
telnet client .crt, X.509 Certificate filename extension Chinese remainder theorem, in number theory Crater (constellation), in astronomy Canal & River Trust
CRT
Movement with a fixed point is rotation
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the body remains
Euler's_rotation_theorem
Set with associative invertible operation
order of the reflection elements f v {\displaystyle f_{\mathrm {v} }} etc. is 2. Both orders divide 8, as predicted by Lagrange's theorem. The groups F
Group_(mathematics)
Shape made from cubes joined together
depending on whether chiral pairs of polycubes (those equivalent by mirror reflection, but not by using only translations and rotations) are counted as one
Polycube
{(\cos \theta )}^{2}.} This can be viewed as a version of the Pythagorean theorem, and follows from the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1}
List of trigonometric identities
List_of_trigonometric_identities
Examining and comparative mode of thinking
Reflection means a form of thoughtful and comparative thinking. Different types of reflection can be distinguished. On the one hand, there is self-reflection
Reflection_(philosophy)
On when a function on convex body K does not decrease if K is translated inwards
In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmetric, unimodal, non-negative
Anderson's_theorem
Concept in mathematics
vector space is a complex reflection group if and only if its ring of invariants is a polynomial ring (Chevalley–Shephard–Todd theorem). For ℓ {\displaystyle
Complex_reflection_group
Unique knot with a crossing number of four
braid (namely, the closure of the 3-string braid σ1σ2−1σ1σ2−1), and a theorem of John Stallings shows that any closed homogeneous braid is fibered. (2)
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Distance-preserving mathematical transformation
motion (translation or rotation), or a composition of a rigid motion and a reflection. Isometries are often used in constructions where one space is embedded
Isometry
Family of polynomials
analog. There is an analog of the binomial theorem for q-binomial coefficients, known as the Cauchy binomial theorem: ∏ k = 0 n − 1 ( 1 + q k t ) = ∑ k = 0
Gaussian_binomial_coefficient
Identity obeyed by many special functions related to the gamma function
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit
Multiplication_theorem
1970s automated theorem prover
Logic for Computable Functions (LCF) is an interactive automated theorem prover developed at Stanford and Edinburgh by Robin Milner and collaborators
Logic for Computable Functions
Logic_for_Computable_Functions
On distance between centers of a triangle
In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by d 2 = R ( R − 2 r ) {\displaystyle
Euler's_theorem_in_geometry
Natural number
squares, 22 + 32 + 42. There are 29 pentacubes if reflections are considered distinct. The 15 and 290 theorems describes integer-quadratic matrices that describe
29_(number)
Mathematical concept
The following table gives examples of rotation and reflection matrix : Cartan–Dieudonné theorem Dihedral group Euclidean plane isometry Euclidean symmetries
Rotations and reflections in two dimensions
Rotations_and_reflections_in_two_dimensions
Generalized scaling operation in geometry
gets the identity mapping; for k = − 1 {\displaystyle k=-1} one gets the reflection at the center; for 1 / k {\displaystyle 1/k} one gets the inverse mapping
Homothety
Country in South Asia
BCE) contain the earliest extant verbal expression of the Pythagorean theorem (although very likely it had been known to the Old Babylonians.) All mathematical
India
Maximum attainable efficiency of any heat engine
Carnot's theorem, also called Carnot's rule or Carnot's law, is a principle of thermodynamics developed by Nicolas Léonard Sadi Carnot in 1824 that specifies
Carnot's theorem (thermodynamics)
Carnot's_theorem_(thermodynamics)
Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space
Continuous-time stochastic process
Continuous-time_stochastic_process
Geometric inequality applicable to any closed curve
this, in itself, does not represent a rigorous proof of the isoperimetric theorem (see external links). The solution to the isoperimetric problem is usually
Isoperimetric_inequality
Point where the incircle and nine-point circle of a triangle are tangent
theorem based on Casey's theorem on the bitangents of four circles tangent to a fifth circle was published by John Casey in 1866; Feuerbach's theorem
Feuerbach_point
Void between celestial bodies
Tadokoro, M. (1968), "A Study of the Local Group by Use of the Virial Theorem", Publications of the Astronomical Society of Japan, 20 (3): 230, Bibcode:1968PASJ
Outer_space
Geometric arrangements of points, foundational to Lie theory
If you consider the line perpendicular to any root, say β, then the reflection of R2 in that line sends any other root, say α, to another root. Moreover
Root_system
Thought experiment, to justify Bayesian probability
choice theory Mathematics of bookmaking Von Neumann-Morgenstern utility theorem Scoring rule Bovens, Luc; Hartmann, Stephan (2003). "Coherence". Bayesian
Dutch_book_arguments
Geometric shape
(equivalently, π radians, or a half-turn). It only has one line of symmetry (reflection symmetry). In non-technical usage, the term "semicircle" is sometimes
Semicircle
French officer, engineer, physicist and mathematician
theory in analytical form. His discovery of the polarization of light by reflection was published in 1809 and his theory of double refraction of light in
Étienne-Louis_Malus
Supposition or system of ideas intended to explain something
likelihood). Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory (that is, one whose theorems form a recursively enumerable
Theory
Mathematical theorem
In mathematics, Ramanujan's master theorem, named after Srinivasa Ramanujan, is a technique that provides an analytic expression for the Mellin transform
Ramanujan's_master_theorem
Geometric theorem regarding 3 circles intersecting at a point
tangent to each of the Johnson circles. The three tangent points are reflections of point H about the vertices of the Johnson triangle. The points of
Johnson_circles
Concept in statistics
uniformly distributed random phase. Where applicable, the central limit theorem dictates that at any point, the sum of these individual plane-wave contributions
Gaussian_random_field
Characterizes spherical triangles with fixed base and area
In spherical geometry, Lexell's theorem holds that every spherical triangle with the same surface area on a fixed base has its apex on a small circle
Lexell's_theorem
Subgroup of a root system's isometry group
group is this: Theorem: If Δ {\displaystyle \Delta } is base for Φ {\displaystyle \Phi } , then the Weyl group is generated by the reflections s α {\displaystyle
Weyl_group
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