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Non-linear generalization of a Hilbert space
In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently
Hadamard_space
used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds (simply connected complete Riemannian manifolds
Busemann_function
Type of metric space in mathematics
{CAT} (0)} spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard. Originally, Aleksandrov called these spaces “ R k {\displaystyle
CAT(k)_space
boundary of an Hadamard space (also called a complete CAT(0) space). It is named after Jacques Tits. Let (X, d) be an Hadamard space. Two geodesic rays
Tits_metric
On the structure of complete Riemannian manifolds of non-positive sectional curvature
locally convex metric spaces. The Cartan–Hadamard theorem in conventional Riemannian geometry asserts that the universal covering space of a connected complete
Cartan–Hadamard_theorem
Type of vector space in math
on "Hilbert Space". Mathematics portal Fundamental theorem of Hilbert spaces – On surjectivity of linear map to anti-dual Hadamard space – Non-linear
Hilbert_space
sectional curvature. By Cartan–Hadamard theorem all Cartan–Hadamard manifolds are diffeomorphic to the Euclidean space R n . {\displaystyle \mathbb {R}
Hadamard_manifold
function Hadamard code Hadamard's dynamical system Hadamard manifold Hadamard matrix Hadamard space Hadamard Transform and Hadamard gate Hadamard variance
List of things named after Jacques Hadamard
List_of_things_named_after_Jacques_Hadamard
Relation between distances of four points
spaces to arbitrary metric spaces. The spaces where it remains valid are called the Ptolemaic spaces; they include the inner product spaces, Hadamard
Ptolemy's_inequality
Error-correcting code
Mars back to Earth from the NASA space probe Mariner 9. Because of its unique mathematical properties, the Hadamard code is not only used by engineers
Hadamard_code
Technique in quantum computation
{\displaystyle U} is a unitary gate acting on the space of | ψ ⟩ {\displaystyle |\psi \rangle } . The Hadamard test produces a random variable whose image is
Hadamard_test
Cartan connection Cartan-Hadamard space is a complete, simply-connected, non-positively curved Riemannian manifold. Cartan–Hadamard theorem is the statement
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
tree-graded metric spaces all of whose pieces are isometric to euclidean spaces. Hruska, G. Christopher; Kleiner, Bruce (2005-08-08). "Hadamard spaces with isolated
Tree-graded_space
Theorem
In mathematics, Hadamard's inequality (also known as Hadamard's theorem on determinants) is a result first published by Jacques Hadamard in 1893. It is
Hadamard's_inequality
Topics referred to by the same term
following topics in mathematics: Hadamard space, a geodesically complete metric space of non-positive curvature Cartan–Hadamard theorem, a result on the topology
Hadamard_(disambiguation)
isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan–Hadamard manifolds. The conjecture, which is named
Cartan–Hadamard_conjecture
Concept in topology
implies that G/K is a Hadamard space, i.e. a complete metric space satisfying a weakened form of the parallelogram rule in a Euclidean space. Uniqueness can
Maximal_compact_subgroup
In mathematics, the Hadamard derivative is a concept of directional derivative for maps between Banach spaces. It is particularly suited for applications
Hadamard_derivative
Manifold that "locally looks like" Euclidean space
more general setting of discrete cocompact groups of isometries of Hadamard spaces. This provides a far-reaching generalisation of Bieberbach's theorem
Flat_manifold
Basic circuit in quantum computing
sometimes included in instruction sets. The Hadamard or Walsh-Hadamard gate, named after Jacques Hadamard (French: [adamaʁ]) and Joseph L. Walsh, acts
Quantum_logic_gate
Theorem in complex analysis
In complex analysis, a branch of mathematics, the Hadamard three-line theorem is a result about the behaviour of holomorphic functions defined in regions
Hadamard_three-lines_theorem
French mathematician (1929–2007)
occupation) Georges Bruhat, and brother of physicist Yvonne Choquet-Bruhat. Hadamard space O'Connor, John J.; Robertson, Edmund F., "François Bruhat", MacTutor
François_Bruhat
Area of mathematics
nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional
Functional_analysis
Concept in mathematics
to the hyperbolic plane). The hyperbolic plane (and more generally any Hadamard manifolds of sectional curvature ≤ − 1 {\displaystyle \leq -1} ) is 2 {\displaystyle
Hyperbolic_metric_space
Space in mathematics and theoretical physics
Hadamard: "the shortest path between two truths in the real domain passes through the complex domain." Therefore when studying four-dimensional space
Twistor_space
French mathematician (1878–1973)
mathematics by Jacques Hadamard. Hadamard recognised the potential of young Maurice and decided to tutor him on an individual basis. After Hadamard moved to the
René_Maurice_Fréchet
Property of differential equations describing physical phenomena
initial or boundary values. These criteria were first introduced by Jacques Hadamard in 1902. Examples of archetypal well-posed problems include the Dirichlet
Well-posed_problem
Smallest convex set containing a given set
intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the
Convex_hull
Concept in the solution of linear partial differential equations
operators based on power series developments was discovered by Jacques Hadamard. It can be applied to the Laplace operator, the wave equation and the heat
Parametrix
Topological space with only one nontrivial homotopy group
fundamental group of M. This is a consequence of the Cartan–Hadamard theorem. An infinite lens space L ( ∞ , q ) {\displaystyle L(\infty ,q)} given by the quotient
Eilenberg–MacLane_space
Geometric inequality applicable to any closed curve
{R} ^{n})} . Hadamard manifolds are complete simply connected manifolds with nonpositive curvature. Thus they generalize the Euclidean space R n {\displaystyle
Isoperimetric_inequality
Generalized manifold
has length at least 6. This condition, well known from the theory of Hadamard spaces, depends only on the underlying complex of groups. When the universal
Orbifold
Theorem
In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem, in which we can express a smooth
Hadamard's_lemma
Concept in mathematics
Walsh functions, the Walsh system, the Walsh series, and the fast Walsh–Hadamard transform are all named after the American mathematician Joseph L. Walsh
Walsh_function
Cartan–Hadamard theorem, which has been generalized to geodesic metric spaces by Mikhail Gromov and Hans Werner Ballmann. This class of aspherical spaces subsumes
Aspherical_space
Quantum variations of random walks
operator, the operator itself is called the "Hadamard coin" and the resulting quantum walk is called the "Hadamard walk". If the walker is initialized at the
Quantum_walk
Structure in combinatorial mathematics
constructed using the field with 11 elements, and is the Hadamard 2-design associated to the size 12 Hadamard matrix; see Paley construction I. Algebraically this
Block_design
Conversion of a matrix or a tensor to a vector
an algebra homomorphism from the space of n × n matrices with the Hadamard (entrywise) product to Cn2 with its Hadamard product: vec ( A ∘ B ) = vec
Vectorization_(mathematics)
Branch of mathematical analysis
the derivative instead of the integral. The Hadamard fractional integral was introduced by Jacques Hadamard and is given by the following formula, D a
Fractional_calculus
Concept in quantum information theory
mutually unbiased complex Hadamard matrices. An example of a one parameter family of Hadamard matrices in a 4-dimensional Hilbert space is H 4 ( ϕ ) = 1 2 [
Mutually_unbiased_bases
List of quantum computing algorithms
of the discrete Fourier transform, used in several quantum algorithms Hadamard transform Transform used in many quantum circuits and query algorithms
List_of_quantum_algorithms
Conjecture on zeros of the zeta function
to prove all of them but one [the Riemann Hypothesis itself]. — Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition Riemann's
Riemann_hypothesis
Mathematical operation in linear algebra
considered as vectors, or, equivalently the sum of the entries of the Hadamard product Hadamard product of two matrices of the same size, resulting in a matrix
Matrix_multiplication
Quantum field theory on a lattice
models of quantum field theory. This involves studying field theory on a space or spacetime that has been discretised onto a lattice. Although most lattice
Lattice_field_theory
\epsilon } -biased sample space via the connection mentioned above. Concatenating Algebraic geometric codes with the Hadamard code gives an ϵ {\displaystyle
Small-bias_sample_space
Theorem in topology
Jacques Hadamard, and Émile Borel. The ensuing discussions convinced Brouwer of the importance of a better understanding of Euclidean spaces, and were
Brouwer_fixed-point_theorem
polynomial-time decoding complexity. Concatenated codes became widely used in space communications in the 1970s. The field of channel coding is concerned with
Concatenated error correction code
Concatenated_error_correction_code
One of several theorems in different areas of mathematics
different). Mohammad Ghomi generalized Schur's theorem to curves in Cartan-Hadamard manifolds. In linear algebra, Schur’s theorem is referred to as either
Schur's_theorem
completeness Cauchy condensation test Cauchy's convergence test Cauchy–Hadamard theorem Cauchy product Cauchy's radical test Cauchy ratio test Cauchy sequence
List of things named after Augustin-Louis Cauchy
List_of_things_named_after_Augustin-Louis_Cauchy
Matrix equal to its conjugate-transpose
\rangle } denotes the standard inner product operation in complex coordinate space, a Hermitian form defined by ⟨ v , w ⟩ = v H w {\displaystyle \langle
Hermitian_matrix
Euclidean space admits at least two umbilical points. Cartan–Hadamard conjecture: can the classical isoperimetric inequality for subsets of Euclidean space be
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Length in a vector space
\|_{p}^{p-1}}}\right)^{\top }.} where ∘ {\displaystyle \circ } denotes Hadamard product and | ⋅ | {\displaystyle |\cdot |} is used for absolute value of
Norm_(mathematics)
Branch of differential geometry
with sectional curvature K ≥ C, diameter ≤ D and volume ≥ V. The Cartan–Hadamard theorem states that a complete simply connected Riemannian manifold M with
Riemannian_geometry
Type of group used in topology and geometric group theory
universal cover, which is a Cartan-Hadamard manifold. More generally, fundamental groups of compact, locally CAT(0) metric spaces are CAT(0) groups, as a consequence
CAT(0)_group
Class of error-correcting code
Hadamard code is a [ 2 r , r , 2 r − 1 ] 2 {\displaystyle [2^{r},r,2^{r-1}]_{2}} linear code and is capable of correcting many errors. Hadamard code
Linear_code
Radius of the event horizon of a Schwarzschild black hole
Einstein's field equations that corresponds to the radius of a sphere in flat space that has the same surface area as that of the event horizon of a Schwarzschild
Schwarzschild_radius
for the Hadamard product of power series.[citation needed] ∂ 1. Boundary of a topological subspace: If S is a subspace of a topological space, then its
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Index of articles associated with the same name
in an algebra over a field. A Lie bracket for vectors in a Lie algebra. Hadamard product – entrywise or elementwise product of tuples of scalar coordinates
Vector_multiplication
Linear algebra matrix
whose rows are pairwise orthogonal is called a circulant Hadamard matrix; the circulant Hadamard matrix conjecture asserts that no such matrix exists for
Circulant_matrix
Formula in matrix theory
In matrix theory, the Hadamard variation formula is a set of differential equations for how the eigenvalues of a time-varying Hermitian matrix with distinct
Hadamard_variation_formula
Degree of differentiability of a function or map
short descriptions of redirect targets Hadamard's lemma – TheoremPages displaying short descriptions with no spaces Non-analytic smooth function – Mathematical
Smoothness
Markov chain in which all states can be absorbing
N_{2}:=N(2N_{\operatorname {dg} }-I_{t})-N_{\operatorname {sq} },} where Nsq is the Hadamard product of N with itself (i.e. each entry of N is squared). The variance
Absorbing_Markov_chain
First spacecraft to enter orbit around Mars (1971–1972)
considerations had to be put into choosing an FEC, and it was decided to use a Hadamard code for Mariner 9. Each image pixel was represented as a six-bit binary
Mariner_9
Property of objects which are scaled or mirrored versions of each other
Secrets of Triangles. Prometheus Books. p. 22. Jacobs 1974, pp. 384–393. Hadamard, Jacques (2008). Lessons in Geometry, Vol. I: Plane Geometry. American
Similarity_(geometry)
Branch of mathematics
function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906
Topology
Mathematical operation on vectors in 3D space
Press. p. 94. ISBN 0-521-00551-5. Shuangzhe Liu; Gõtz Trenkler (2008). "Hadamard, Khatri-Rao, Kronecker and other matrix products". Int J Information and
Cross_product
Theorem in quantum information science
factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem as
No-cloning_theorem
Property of a mathematical matrix
the Hadamard product is, M ∘ N ≥ 0 {\displaystyle M\circ N\geq 0} (this result is often called the Schur product theorem). Regarding the Hadamard product
Definite_matrix
System of rules to convert information into another form or representation
Reed–Muller, Walsh–Hadamard, Bose–Chaudhuri–Hochquenghem, Turbo, Golay, algebraic geometry codes, low-density parity-check codes, and space–time codes. Error
Code
Vector operation
which takes a pair of matrices as input and produces a block matrix The Hadamard product, which is the element-wise product Standard matrix multiplication
Outer_product
Theorem on operator interpolation
spaces", Mat. Sb., New Series (in Russian), 66 (108): 473–482. Thorin, G. O. (1948), "Convexity theorems generalizing those of M. Riesz and Hadamard with
Riesz–Thorin_theorem
Mathematics of smooth surfaces
embedded in E3, the Gauss map provides an explicit diffeomorphism. As Hadamard observed, in this case the surface is convex; this criterion for convexity
Differential geometry of surfaces
Differential_geometry_of_surfaces
Cartan–Brauer–Hua theorem Cartan–Dieudonné theorem Cartan–Hadamard manifold Cartan–Hadamard theorem Cartan–Iwahori decomposition[citation needed] Cartan-Iwasawa-Malcev
List of things named after Élie Cartan
List_of_things_named_after_Élie_Cartan
Generalization of the concept of directional derivative
_{\Omega }\int _{0}^{1}F'(u+s\tau \psi )\,\psi \,ds\,dx.\end{aligned}}} Hadamard derivative Derivative (generalizations) – Fundamental construction of differential
Gateaux_derivative
Field of mathematics and science based on non-linear systems and initial conditions
mentioned by hadamard 1898 which derives a hill differential equation with negative curvature, and therefore a hyperbolic geometry in phase space, i.e. divergent
Chaos_theory
Method of analysis applied to problems wave propagation
constructing the surface tangent to the secondary wavelets. In 1900, Jacques Hadamard observed that Huygens's principle was broken when the number of spatial
Huygens–Fresnel_principle
Triangle with at least two sides congruent
(1922). Montroll (2009). Hadamard (2008), p. 23. Guinand (1984). Harris & Stöcker (1998), p. 78. Salvadori & Wright (1998). Hadamard (2008), Exercise 5, p
Isosceles_triangle
Mathematical result
\end{array}}\right]} , where ∘ {\displaystyle \circ } is the element-wise (Hadamard) product. Such computations have been used to efficiently compute polynomial
Johnson–Lindenstrauss_lemma
French mathematician, physicist and engineer (1854–1912)
by the fact that he worked so often by visual representation. Jacques Hadamard wrote that Poincaré's research demonstrated marvelous clarity and Poincaré
Henri_Poincaré
Type of mathematical array
2, 2) if and only if there exists a Hadamard matrix of order 4λ. To proceed in one direction, let H be a Hadamard matrix of order 4m in standardized form
Orthogonal_array
Optimization algorithm
attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties
Gradient_descent
Form of matter
it, making it indefinitely stable at STP. Witold Rybczynski and Jacques Hadamard developed an equation to calculate the velocity of bubbles that rise in
Foam
Characterization of how many integers are prime
at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by
Prime_number_theorem
On distances from opposite corners to a point inside a rectangle
Tournament solutions Archived 2018-12-22 at the Wayback Machine, Problem 28. Hadamard, Jacques (2008), Lessons in Geometry: Plane geometry, American Mathematical
British_flag_theorem
Description in Riemannian geometry
Cartan–Hadamard theorem: if M is a complete manifold with non-positive sectional curvature, then its universal cover is diffeomorphic to a Euclidean space.
Sectional_curvature
Symmetric arrangement of finite sets
is built around balanced incomplete block designs (BIBDs), Hadamard matrices and Hadamard designs, symmetric BIBDs, Latin squares, resolvable BIBDs, difference
Combinatorial_design
Method of mass spectrometry
in TOF mass spectrometers and in ion mobility spectrometers, as well as Hadamard transform TOF mass spectrometers. The Bradbury–Nielsen shutter is ideal
Time-of-flight mass spectrometry
Time-of-flight_mass_spectrometry
Russian-French mathematician
topological restrictions (such as the Cheeger–Gromoll soul theorem or Cartan–Hadamard theorem) on geodesically complete Riemannian manifolds of positive or negative
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Mathematical operation on matrices
{B} )+(\mathbf {A} \mathbf {C} )\otimes [\mathbf {B} ,\mathbf {D} ]} . Hadamard product (element-wise multiplication): The mixed-product property also
Kronecker_product
Matrix whose only nonzero elements are on its main diagonal
{T}}\right)\circ \mathbf {I} ,} where ∘ {\displaystyle \circ } represents the Hadamard product, and 1 is a constant vector with elements 1. The inverse matrix-to-vector
Diagonal_matrix
{z}}_{1}\circ {\vec {z}}_{2}} , where ∘ {\displaystyle \circ } denotes the Hadamard product of the same-length vectors. These properties guarantee the approximate
HEAAN
Surjective bounded operator on a Hilbert space preserving the inner product
{\displaystyle {\frac {1}{\sqrt {n}}}} times a Hadamard matrix. In general, any operator in a Hilbert space that acts by permuting an orthonormal basis is
Unitary_operator
Mathematical form
have a tensor product. Other kinds of products in linear algebra include: Hadamard product Kronecker product The product of tensors: Wedge product or exterior
Product_(mathematics)
Algorithm to approximate functions
Chebyshev approximation. Mathematics portal Hadamard's lemma – TheoremPages displaying short descriptions with no spaces Laurent series – Power series with negative
Remez_algorithm
Process in quantum computing
similarly constructed and is equivalent to the bit-flip code up to transversal Hadamard gates. Consider the situation in which we want to transmit the state of
Quantum_error_correction
Process of calculating the causal factors that produced a set of observations
Of the three conditions for a well-posed problem suggested by Jacques Hadamard (existence, uniqueness, and stability of the solution or solutions) the
Inverse_problem
Scheme for controlling errors in data over noisy communication channels
is of practical interest Goppa code, used in the McEliece cryptosystem Hadamard code Hagelbarger code Hamming code Latin square based code for non-white
Error_correction_code
conjecture Borel–Carathéodory theorem Corona theorem Hadamard three-circle theorem Hardy space Hardy's theorem Maximum modulus principle Nevanlinna theory
List of complex analysis topics
List_of_complex_analysis_topics
Quantum states of two qubits
the simplest takes a computational basis as the input, and contains a Hadamard gate and a CNOT gate (see picture). As an example, the pictured quantum
Bell_state
Spectral Geometry Phenomenon
inequality as a complete simply connected space form of constant sectional curvature. In particular, if the Cartan–Hadamard conjecture holds, then the Faber–Krahn
Rayleigh–Faber–Krahn inequality
Rayleigh–Faber–Krahn_inequality
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