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Mathematics concept
mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher
Rectifiable_set
Topics referred to by the same term
edges, and cutting off its vertices at those points Rectifiable curve, in mathematics Rectifiable set, in mathematics GHK flux equation#Rectification, in
Rectification
by replacing differentiability requirements with those provided by rectifiable sets, while maintaining the general algebraic structure usually seen in
Varifold
Measure-theoretic generalization of the concept of a tangent space
tangent spaces is very closely related to that of rectifiable sets. Loosely speaking, rectifiable sets are precisely those for which approximate tangent
Approximate_tangent_space
Distance along a curve
polygonal chains. The curves for which this limit exists are called rectifiable curves, and the process of determining their arc length in this way is
Arc_length
Set of points on a line segment with certain topological properties
Mattila, Pertti (25 February 1999). Geometry of Sets and Measures in Euclidean Space: Fractals and rectifiability. Cambridge studies in advanced mathematics
Cantor_set
French mathematician
Uniform rectifiability and quasiminimizing sets of arbitrary codimension, Memoirs AMS 2000 with Stephen Semmes: Singular integrals and rectifiable sets in
Guy_David_(mathematician)
Topological space that locally resembles Euclidean space
surfaces. A rectifiable set generalizes the idea of a piecewise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in
Manifold
Study of geometric properties of sets through measure theory
theory: Hausdorff measure and Hausdorff dimension Rectifiable sets (or Radon measures), which are sets with the least possible regularity required to admit
Geometric_measure_theory
Manifold upon which it is possible to perform calculus
attempts. A rectifiable set generalizes the idea of a piece-wise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in
Differentiable_manifold
To find the minimal surface with a given boundary
problem in arbitrary dimension and codimension for any collection of rectifiable sets satisfying a combination of general homological, cohomological or homotopical
Plateau's_problem
In Euclidean space, a measure of that set's "size"
_{\mathbb {R} ^{n}\setminus K}|\nabla u|^{2}\mathrm {d} x.} If S is a rectifiable hypersurface completely enclosing K, then the harmonic capacity can be
Capacity_of_a_set
Theorem in calculus relating line and double integrals
{\displaystyle \Gamma } be a rectifiable curve in the plane and let Δ Γ ( h ) {\displaystyle \Delta _{\Gamma }(h)} be the set of points in the plane whose
Green's_theorem
American mathematician (born 1989)
generalization to rectifiable curves of the carpenter's rule problem for polygons. In the project, he showed that every rectifiable Jordan curve in the
John_Pardon
Mathematical theorem
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
Rademacher's_theorem
optimization. In its simplest and original form, it asks which plane sets are subsets of rectifiable curves of finite length. Whereas the original traveling salesman
Analyst's traveling salesman theorem
Analyst's_traveling_salesman_theorem
Computationally feasible measure in real-valued number of dimensions
the same value to the set A as well as its closure. If A is a closed m-rectifiable set in Rn, given as the image of a bounded set from Rm under a Lipschitz
Minkowski_content
Chinese mathematician (born 1958)
"singular set". Using techniques developed by Fanghua Lin in the study of harmonic maps, Tian showed that the singular set is a rectifiable set. In the
Tian_Gang
American mathematician
247–286. David, Guy; Semmes, Stephen: Analysis of and on uniformly rectifiable sets. Mathematical Surveys and Monographs, 38. American Mathematical Society
Stephen_Semmes
m-rectifiable set. Density of a set at Encyclopedia of Mathematics Rectifiable set at Encyclopedia of Mathematics Pertti Mattila, Geometry of sets and
Hausdorff_density
Hungarian mathematician
Washington. The title of her talk is The Kakeya needle problem for rectifiable sets. She is included in a deck of playing cards featuring notable women
Marianna_Csörnyei
Region with boundary of finite measure
\partial ^{*}E} . Finally, ∂ ∗ E {\displaystyle \partial ^{*}E} is (n-1)-rectifiable and the restriction of (n-1)-dimensional Hausdorff measure H n − 1 {\displaystyle
Caccioppoli_set
Shape that blocks all lines of sight
line segments and rectifiable curves. Most research on this problem assumes that the given set K {\displaystyle K} is a convex set. When it is not convex
Opaque_set
that the set E is flat on most small scales. In particular, if the power in the integral is larger, our set is smoother than just being rectifiable Let p
Menger_curvature
Physics theorem of interacting particles
"Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets", Transactions of the American Mathematical Society, 360 (3): 1559–1580
Poppy-seed_bagel_theorem
Mathematical curve whose shape is a fractal
takes the form of a fractal. In general, fractal curves are nowhere rectifiable — that is, they do not have finite length — and every subarc longer than
Fractal_curve
manifolds. Tangent measures (introduced by David Preiss in his study of rectifiable sets) are a useful tool in geometric measure theory. For example, they are
Tangent_measure
Extended measure of size in mathematics
commonly accepted term to describe a set whose Jordan content is defined. Munkres (1991) suggests the term "rectifiable" as a generalization of the use of
Peano–Jordan_measure
Romanian-American mathematician
Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets (with Steve Hofmann, Marius Mitrea, and Andrew J. Morris, Memoirs of
Dorina_Mitrea
Concept in complex analysis
of sets and measures in Euclidean spaces. Cambridge University Press. ISBN 0-521-65595-1. Pajot, Hervé (2002). Analytic Capacity, Rectifiability, Menger
Analytic_capacity
American mathematician
Particular themes included surface area, rectifiability of sets, and the extent to which one could substitute rectifiability for smoothness in the classical analysis
Herbert_Federer
Lecture series established by the Association for Women in Mathematics
2021 (see above) 2022 Marianna Csörnyei The Kakeya needle problem for rectifiable sets 2023 Laura DeMarco Rigidity and uniformity in algebraic dynamics 2024
Noether_Lecture
Theorem in complex analysis
on U ¯ {\textstyle {\overline {U}}} and γ {\displaystyle \gamma } a rectifiable simple loop in U ¯ {\displaystyle \textstyle {\overline {U}}} . The
Cauchy's_integral_theorem
Theorem about mass-minimizing surfaces
minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two", Bulletin of the American Mathematical
Almgren_regularity_theorem
Counterintuitive observation
approximated by small straight segments with a definite limit is termed a rectifiable curve. Benoit Mandelbrot devised an alternative measure of length for
Coastline_paradox
Mathematical space with a notion of distance
essential use of the metric. For example, a curve in a metric space is rectifiable (has finite length) if and only if it has a Lipschitz reparametrization
Metric_space
integer rectifiable current is defined as a countable sum of currents formed in this respect. An integral current is an integer rectifiable current whose
Flat_convergence
Mathematical idealization of the trace left by a moving point
{\displaystyle t_{0}<t_{1}<\ldots <t_{n}} of [ a , b ] {\displaystyle [a,b]} . A rectifiable curve is a curve with finite length. A curve γ : [ a , b ] → X {\displaystyle
Curve
inside a simple polygon or a rectifiable simple closed curve. Let P {\displaystyle P} be a simple polygon or a rectifiable simple closed curve, and let
Relative_convex_hull
Generalization of volume to non-integer number of dimensions
\mathbb {R} ^{n}} is said to be m {\displaystyle m} -rectifiable if it is the image of a bounded set in R m {\displaystyle \mathbb {R} ^{m}} under a Lipschitz
Hausdorff_measure
On smallest surface enclosing two volumes
until 2002. The proof combines multiple ingredients. Compactness of rectifiable currents (a generalized definition of surfaces) shows that a solution
Double_bubble_theorem
American mathematician
controlled singular sets. This work was extended with Wenshuai Jiang in order to prove sharp rectifiability of the singular sets. During this time Naber
Aaron_Naber
Geometric shape formed from squares
3390/a15050164. Golomb, Polyominoes, chapter 8 Reid, Michael. "References for Rectifiable Polyominoes". Archived from the original on 2004-01-16. Retrieved 2007-05-11
Polyomino
Shape subdivided into copies of itself
laying equilateral triangles and squares edge-to-edge. If a polyomino is rectifiable, that is, able to tile a rectangle, then it will also be a rep-tile,
Rep-tile
Result in integral geometry
"random" line intersects it. Suppose γ {\displaystyle \gamma } is a rectifiable plane curve. Given an oriented line ℓ, let n γ {\displaystyle n_{\gamma
Crofton_formula
Concept in geometry
if it is agreed that the circumference of the circle is measured as a rectifiable curve by means of the integral C = 2 ∫ − R R R d x R 2 − x 2 = 2 R ∫
Area_of_a_circle
Number, approximately 3.14
analysis is contour integration of a function over a positively oriented (rectifiable) Jordan curve γ. A form of Cauchy's integral formula states that if a
Pi
Real-valued number of spatial dimensions
snowflake. It has a topological dimension of 1, but it is by no means rectifiable: the length of the curve between any two points on the Koch snowflake
Fractal_dimension
Finnish mathematician (born 1948)
problem by Xavier Tolsa. His book Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability is now a widely cited and a standard textbook
Pertti_Mattila
Type of plane curve
without any requirement of symmetry. Every bounded convex curve is a rectifiable curve, meaning that it has a well-defined finite arc length, and can
Convex_curve
fractal Mandelbrot set Menger sponge Minkowski–Bouligand dimension Multifractal analysis Olbers' paradox Perlin noise Power law Rectifiable curve Scale-free
Index of fractal-related articles
Index_of_fractal-related_articles
Types of mappings in mathematics
_{x_{0}}^{x_{1}}f(x)\;\mathrm {d} x} L p {\displaystyle L^{p}} norm of a function on a set E {\displaystyle E} f ↦ ( ∫ E | f | p d x ) 1 / p {\displaystyle f\mapsto
Functional_(mathematics)
Spanish mathematician
on the so-called David-Semmes problem involving Riesz transforms and rectifiability. In 2002 he was awarded the Salem Prize. In 2006 in Madrid he was an
Xavier_Tolsa
Theorem in analysis
analysis Mattila, Pertti (1999). Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. ISBN 978-0-521-65595-8. Croft, Hallard (1982)
Lebesgue's_density_theorem
Concept of complex analysis
{\displaystyle U_{0}} . Letting γ {\displaystyle \gamma } be a closed rectifiable curve in U 0 {\displaystyle U_{0}} , and denoting the residue of f
Residue_theorem
the set of positive density of the mass measure of T. As a consequence of deep theorems of Ambrosio–Kirchheim, X is then a countably Hm rectifiable metric
Intrinsic_flat_distance
Provides integral formulas for all derivatives of a holomorphic function
The circle γ {\displaystyle \gamma } can be replaced by any closed rectifiable curve in U {\displaystyle U} that has winding number one about a {\displaystyle
Cauchy's_integral_formula
Geometric inequality applicable to any closed curve
published a short proof using the Fourier series that applies to arbitrary rectifiable curves (not assumed to be smooth). An elegant direct proof based on comparison
Isoperimetric_inequality
Mathematical inequality in Sobolev space theory
path connected. Indeed, between any pair of points there must exist a rectifiable path with length comparable to the distance of the points. Much deeper
Poincaré_inequality
NP-hard problem in combinatorial optimization
what conditions may a subset E of Euclidean space be contained in a rectifiable curve (that is, when is there a curve with finite length that visits
Travelling_salesman_problem
Continuous function that is not absolutely continuous
function is non-decreasing, and so in particular its graph defines a rectifiable curve. Scheeffer (1884) showed that the arc length of its graph is 2
Cantor_function
Type of mathematical functions
being continuous and separately homorphic on domain D. Each disk has a rectifiable curve γ {\displaystyle \gamma } , γ ν {\displaystyle \gamma _{\nu }}
Function of several complex variables
Function_of_several_complex_variables
Polish mathematician (1896–1972)
Habilitation at the Jagiellonian University on the basis of the thesis Rectifiable Continuums in Relation to Absolutely Continuous Functions and Mappings
Tadeusz_Ważewski
Complex-differentiable (mathematical) function
_{\gamma }f(z)\,\mathrm {d} z=0.} Here γ {\displaystyle \gamma } is a rectifiable path in a simply connected complex domain U ⊂ C {\displaystyle U\subset
Holomorphic_function
Perimeter of a circle or ellipse
a , b ] → R 2 {\displaystyle \gamma :[a,b]\to \mathbb {R} ^{2}} is a rectifiable curve, its length may be defined as L ( γ ) = sup ∑ i = 1 n | γ ( t i
Circumference
{\displaystyle \Gamma } contains some non-rectifiable curves and Γ 0 {\displaystyle \Gamma _{0}} denotes the set of rectifiable curves in Γ {\displaystyle \Gamma
Extremal_length
Distributions on spaces of differential forms
. {\displaystyle {\mathcal {E}}_{m}(M).} Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current
Current_(mathematics)
Unsolved problem about inscribing a square in a Jordan curve
parametrization of C {\displaystyle C} , as C {\displaystyle C} may not be rectifiable. Instead of a limit argument, the proof is based on relative obstruction
Inscribed_square_problem
words, the cardinality of the set of transcendentals (denoted ℶ 1 {\displaystyle \beth _{1}} ) is greater than that of the set of algebraic numbers ( ℵ 0
List_of_conjectures
or composite rectifiable contour, which encloses some region G Γ {\displaystyle G_{\Gamma }} and lies entirely within the resolvent set ρ ( A ) {\displaystyle
Riesz_projector
Twin-engined, medium-lift helicopter manufactured by Leonardo
Wolf at Eglin Air Force Base. Flight testing began in 2020. Several rectifiable deficiencies were identified, such as the positioning of the gunner in
AgustaWestland_AW139
Several sets of circles associated with Apollonius of Perga
exactly but is roughly 1.3, which is higher than that of a regular (or rectifiable) curve (d = 1) but less than that of a plane (d = 2). The Apollonian
Circles_of_Apollonius
Assignment of a vector to each point in a subset of Euclidean space
constructed analogously to the Riemann integral and it exists if the curve is rectifiable (has finite length) and the vector field is continuous. Given a vector
Vector_field
Concept in geometry/topology
is the set of finite partitions of [ 0 , 1 ] {\displaystyle [0,1]} . If the supremum is finite, we call γ {\displaystyle \gamma } a rectifiable curve.
Intrinsic_metric
Property of objects which are scaled or mirrored versions of each other
related in the same way. The relationship holds for figures that are not rectifiable as well. The ratio between the volumes of similar figures is equal to
Similarity_(geometry)
Christian denomination and charity
homosexuality is not in itself blameworthy nor is the disposition seen as rectifiable at will.... Homosexual practice however, is, in the light of Scripture
Salvation_Army
American award for mathematical analysis
cones and the singular set of minimal submanifolds. J. Diff. Geom. 38 (1993), no. 3, 585–652. Rectifiability of the singular set of energy minimizing maps
Bôcher_Memorial_Prize
Construct all metric spaces where lines resemble those on a sphere
{\displaystyle dS=d\rho \,d\varphi } . Let γ {\displaystyle \gamma } be a rectifiable curve on a plane. Then the length of γ {\displaystyle \gamma } is L =
Hilbert's_fourth_problem
Straight path on a curved surface or a Riemannian manifold
points in a length metric space are joined by a minimizing sequence of rectifiable paths, although this minimizing sequence need not converge to a geodesic
Geodesic
function of a metric space X {\textstyle X} measures "how efficiently rectifiable loops are coarsely contractible with respect to their length". For the
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
D\subset \mathbb {R} ^{2}} is a simply connected planar domain bounded by a rectifiable curve (i.e. if H 1 ( ∂ D ) < ∞ {\displaystyle H^{1}(\partial D)<\infty
Harmonic_measure
MR 0260979 Mattila, Pertti (1995). Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability. Cambridge Studies in Advanced Mathematics
Uniformly_distributed_measure
Point of interest for complex multi-valued functions
calculated explicitly from Cauchy's integral formula. Let γ be a simple rectifiable loop in X around P. The ramification index of ƒ at P is e P = 1 2 π i
Branch_point
Hypercube partition of Euclidean space
Wavelet transform Okikiolu, Kate (1992). "Characterization of subsets of rectifiable curves in Rn". J. London Math. Soc. Series 2. 46 (2): 336–348. doi:10
Dyadic_cubes
Preiss, David (1987). "Geometry of measures in R^n: Distribution, rectifiability, and densities". The Annals of Mathematics. 125 (3): 537–643. doi:10
David_Preiss
Motion of a curve based on its curvature
more general inputs than curves, for instance by using rectifiable varifolds or the level-set method. However, these extended definitions may allow parts
Curve-shortening_flow
Area where land meets the sea or ocean
adequate to reality would be to call most arcs encountered in nature not rectifiable. Vulpiani, Angelo (2014). "Lewis Fry Richardson: scientist, visionary
Coast
Geometry problem about finding touching circles
exactly but is roughly 1.3, which is higher than that of a regular (or rectifiable) curve (d = 1) but less than that of a plane (d = 2). The Apollonian
Problem_of_Apollonius
Branch of functional analysis
open set D ⊂ C, and Γ be a rectifiable Jordan curve in D, that is, a closed curve of finite length without self-intersections. Assume that the set U of
Holomorphic functional calculus
Holomorphic_functional_calculus
American mathematician (1933–1997)
minimizing Dirichlet's integral and the regularity of area-minimizing rectifiable currents up to codimension 2, World Scientific Monograph Series in Mathematics
Frederick_J._Almgren_Jr.
Lebesgue measure, denoted e(u) dx; a jump part, supported on a rectifiable (n − 1)-dimensional set Ju of points where u has two different approximate limits
Bounded_deformation
British series of judicial decisions (2009–2014)
forfeiture of the borrower's property for what may be a trivial and rectifiable breach is penal, the true intention of the parties is that the property
Cukurova Finance International Ltd v Alfa Telecom Turkey Ltd
Cukurova_Finance_International_Ltd_v_Alfa_Telecom_Turkey_Ltd
Australian mathematician (born 1945)
zero set of solutions of general second-order elliptic partial differential equations, obtaining information on Hausdorff measure and rectifiability. By
Leon_Simon
chapter 1) Mattila, Pertti (1995). Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability. Cambridge Studies in Advanced Mathematics
Spherical_measure
Holomorphic functions in infinite dimensions
line integral of a vector-valued holomorphic function f : U → X along a rectifiable curve γ : [a, b] → U in the same way as for complex-valued holomorphic
Infinite-dimensional holomorphy
Infinite-dimensional_holomorphy
Formula E electric car race in Morocco
there." Third-place finisher Bird revealed his car was affected by a rectifiable reoccurring issue since Hong Kong and made the decision not to replace
2018_Marrakesh_ePrix
Moldovan mathematician (1942–2021)
Arhangel'skii, Alexander V.; Choban, Mitrofan M. (2010). "Remainders of rectifiable spaces". Topology and Its Applications. 157 (4): 789–799. doi:10.1016/j
Mitrofan_Cioban
American annual mathematics conference
Stein/Weinstein domains Raanan Schul (Stony Brook): Qualitative and quantitative rectifiability Ursula Hamenstädt (Bonn): A Gromov/Thurston rigidity theorem for hyperbolic
Geometry_Festival
Spanish Catholic ultraconservative politician
but to have allied with the liberals; this error, however, was still rectifiable by creating a strong, conservative alliance. Demonstrating some degree
Ramón_Nocedal
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