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In mathematics, a piecewise isometry is a dynamical system that consists of finitely many Euclidean isometries acting in different places, including rotations
Piecewise_isometry
generalizations include polygon exchanges, polyhedral exchanges and piecewise isometries. Odometer Keane, Michael (1975), "Interval exchange transformations"
Interval exchange transformation
Interval_exchange_transformation
Smooth manifold with an inner product on each tangent space
surface is called a local isometry. A property of a surface is called an intrinsic property if it is preserved by local isometries and it is called an extrinsic
Riemannian_manifold
Calculus of stochastic differential equations
Itô isometry, the use of the Doléans measure for submartingales, or the use of the Burkholder–Davis–Gundy inequalities instead of the Itô isometry. The
Itô_calculus
Multidimensional data algorithm
to have stability and performance guarantees under certain restricted isometry conditions. The incremental multi-parameter algorithm (IMP), published
Matching_pursuit
(t),\gamma (t'){\big )}={\big |}t-t'{\big |}.} Equivalently, a ray is an isometry from the "canonical ray" (the set [ 0 , ∞ ) {\displaystyle [0,\infty )}
Busemann_function
Strong form of uniform continuity
distance between all points Dini continuity Modulus of continuity Quasi-isometry Johnson–Lindenstrauss lemma – For any integer n ≥ 0 {\displaystyle n\geq
Lipschitz_continuity
Argentine physicist
segregation of granular materials, exploiting the mathematics of piecewise isometries. His research has been featured in articles and on the covers of
Julio_M._Ottino
Self-balancing binary search tree data structure
sequence of operations. The 2–3–4 tree isometry was described in 1978 by Sedgewick. With 2–3–4 trees, the isometry is resolved by a "color flip," corresponding
Red–black_tree
Type of group used in topology and geometric group theory
space The action of G {\displaystyle G} on X {\displaystyle X} is by isometries, i.e. it is a group homomorphism G ⟶ I s o m ( X ) {\displaystyle G\longrightarrow
CAT(0)_group
Mathematical theory
finitely many disjoint Jordan curves. In most applications these curves are piecewise analytic, but there is some explicit minimal regularity condition on these
Ahlfors_theory
Theorem in set-theoretic geometry
n-dimensional Euclidean space (for integral n), and G consists of all isometries of X, i.e. the transformations of X into itself that preserve the distances
Banach–Tarski_paradox
Open problem on 3x+1 and x/2 functions
_{k=0}^{\infty }\left(T^{k}(x){\bmod {2}}\right)2^{k}.} The function Q is a 2-adic isometry. Consequently, every infinite parity sequence occurs for exactly one 2-adic
Collatz_conjecture
Mathematics of smooth surfaces
Gaussian curvature is an intrinsic invariant, i.e. invariant under local isometries. This point of view was extended to higher-dimensional spaces by Riemann
Differential geometry of surfaces
Differential_geometry_of_surfaces
Semigroup in abstract algebra
that ee = e and e* = e. Every projection is a partial isometry, and for every partial isometry s, s*s and ss* are projections. If e and f are projections
Semigroup_with_involution
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
fact γ is homotopic to a piecewise smooth closed curve δ, so that ∫γ ω = ∫δ ω. On the other hand there are finitely many piecewise smooth Jordan curves δi
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
Polygon associated with a compact Riemann surface
by a diagonal piecewise geodesic segment in its interior. The final polygon has 4g equivalent vertices, with sides that are piecewise geodesic. The sides
Fundamental_polygon
Geometric surface
of a pseudospherical surface. A pseudospherical surface is a surface piecewise smoothly immersed in R 3 {\displaystyle \mathbb {R} ^{3}} with constant
Pseudosphere
: T x M → T y N {\displaystyle I:T_{x}M\rightarrow T_{y}N} be a linear isometry. This can be interpreted as isometrically mapping an infinitesimal patch
Cartan–Ambrose–Hicks_theorem
Concept in mathematics
sine and cosine functions, which are continuous, Walsh functions are piecewise constant. They take the values −1 and +1 only, on sub-intervals defined
Walsh_function
Non-orientable surface with one edge
MR 3370020. S2CID 119640200. Stillwell, John (1992). "4.6 Classification of isometries". Geometry of Surfaces. Universitext. Cham: Springer. pp. 96–98. doi:10
Möbius_strip
Branch of topology
neighbourhood that is homeomorphic to Euclidean 3-space. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions,
Low-dimensional_topology
Topics referred to by the same term
sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation Linear convergence (to converge linearly), the description
Linear_(disambiguation)
Budninskiy et al. Spectral Affine-Kernel Embeddings Sebastian Claici et al. Isometry-Aware Preconditioning for Mesh Parameterization 2016 Behrend Heeren et
Symposium on Geometry Processing
Symposium_on_Geometry_Processing
Mean position of all the points in a shape
central void. If the centroid is defined, it is a fixed point of all isometries in its symmetry group. In particular, the geometric centroid of an object
Centroid
Mathematical space
already known to be intractable. For manifolds of dimension at most 6, any piecewise linear (PL) structure can be smoothed in an essentially unique way, so
4-manifold
On tangency patterns of circles
structure of this domain is uniquely determined, up to isometry of the hyperbolic space; these isometries, when viewed in terms of their actions on the Euclidean
Circle_packing_theorem
Type of plane curve
equivalent definition of convexity for smooth curves, or more generally for piecewise smooth curves. A convex curve may be alternatively defined as a connected
Convex_curve
Intrinsic geometric structures in mathematics
invariant of the surface. Parallel transport can be extended immediately to piecewise C1 curves. When M is a surface embedded in E3, this last condition can
Riemannian connection on a surface
Riemannian_connection_on_a_surface
Locally compact topological group with an invariant averaging operation
1017/S0143385700005708 Juschenko, Kate; Monod, Nicolas (2013), "Cantor systems, piecewise translations and simple amenable groups", Annals of Mathematics, 178 (2):
Amenable_group
Generalized manifold
orbifold path is a path in the underlying space provided with an explicit piecewise lift of path segments to orbifold charts and explicit group elements identifying
Orbifold
Smallest convex set containing a given set
1986, doi:10.1007/BF01086114 Sontag, Eduardo D. (1982), "Remarks on piecewise-linear algebra", Pacific Journal of Mathematics, 98 (1): 183–201, doi:10
Convex_hull
proportional fitting Iteratively reweighted least squares Itô calculus Itô isometry Itô's lemma Jaccard index Jackknife (statistics) Jackson network Jackson's
List_of_statistics_articles
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PIECEWISE ISOMETRY
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