Geometric topology | Functional analysis | Metric spaces | 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 defined as complete CAT(0) spaces. A Hadamard space is defined to be a nonempty complete metric space such that, given any points and there exists a point such that for every point The point is then the midpoint of and In a Hilbert space, the above inequality is equality (with ), and in general an Hadamard space is said to be flat if the above inequality is equality. A flat Hadamard space is isomorphic to a closed convex subset of a Hilbert space. In particular, a normed space is an Hadamard space if and only if it is a Hilbert space. The geometry of Hadamard spaces resembles that of Hilbert spaces, making it a natural setting for the study of rigidity theorems. In a Hadamard space, any two points can be joined by a unique geodesic between them; in particular, it is contractible. Quite generally, if is a bounded subset of a metric space, then the center of the closed ball of the minimum radius containing it is called the of Every bounded subset of a Hadamard space is contained in the smallest closed ball (which is the same as the closure of its convex hull). If is the group of isometries of a Hadamard space leaving invariant then fixes the circumcenter of (Bruhat–Tits fixed point theorem). The basic result for a non-positively curved manifold is the Cartan–Hadamard theorem. The analog holds for a Hadamard space: a complete, connected metric space which is locally isometric to a Hadamard space has an Hadamard space as its universal cover. Its variant applies for non-positively curved orbifolds. (cf. Lurie.) Examples of Hadamard spaces are Hilbert spaces, the Poincaré disc, complete metric trees (for example, complete Bruhat–Tits building), with and and Hadamard manifolds, that is, complete simply-connected Riemannian manifolds of nonpositive sectional curvature. Important examples of Hadamard manifolds are simply connected nonpositively curved symmetric spaces. Applications of Hadamard spaces are not restricted to geometry. In 1998, Dmitri Burago and used CAT(0) geometry to solve a problem in dynamical billiards: in a gas of hard balls, is there a uniform bound on the number of collisions? The solution begins by constructing a configuration space for the dynamical system, obtained by joining together copies of corresponding billiard table, which turns out to be an Hadamard space. (Wikipedia).
A 10' overview of the LHC project and its research plans
From playlist The Large Hadron Collider
il Large Hadron Collider (Italiano)
Una panoramica sul progetto LHC ed i suoi campi di ricerca.
From playlist Italiano
Le "Large Hadron Collider" (français)
un apreçu du grand collisionneur de hadrons (LHC) et de son programme de recherche
From playlist Français
What is the "hadron" in the name Large Hadron Collider?
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From playlist Science Unplugged: Particle Physics
Can Scientists Work Outside The Standard Model of Physics?
Episode 3 of 4 Check us out on iTunes! http://apple.co/1TXDZAI Please Subscribe! http://bit.ly/28iQhYC There are specific things scientists are looking for when conducting the compact muon solenoid (CMS) experiment at the large hadron collider (LHC), but what about things we don't know
From playlist The Large Hadron Collider Explained
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The second run, or second season, begins at CERN's Large Hadron Collider. Can it top season one's discovery of the Higgs Boson!? See our videos from inside the LHC: http://bit.ly/LHCvideos This video features Professor Ed Copeland. See Ed's trilogy of extended interviews: http://bit.ly/C
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From playlist The Large Hadron Collider
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From playlist The Large Hadron Collider
Christian Bär: Local index theory for Lorentzian manifolds
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From playlist Mathematical Physics
Finding and Solving the Hadamard Population Conjecture
We describe our process for finding and solving the Hadamard Population conjecture. This conjecture is for all v, for all w, fht(v) dot-product fht(w) = n * population(v intersect w), where v and w are binary vectors and n is the length of all vectors. This is a submission to the #SoME2 co
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From Classical to Quantum Stochastic Process by Soham Biswas
DISCUSSION MEETING STATISTICAL PHYSICS: RECENT ADVANCES AND FUTURE DIRECTIONS (ONLINE) ORGANIZERS: Sakuntala Chatterjee (SNBNCBS, Kolkata), Kavita Jain (JNCASR, Bangalore) and Tridib Sadhu (TIFR, Mumbai) DATE: 14 February 2022 to 15 February 2022 VENUE: Online In the past few dec
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From playlist The Large Hadron Collider
Deutsch Jozsa Algorithm - Quantum Computer Programming w/ Qiskit p.3
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From playlist Quantum Computer Programming w/ Qiskit
Topics in Combinatorics lecture 5.0 --- Sets of vectors with no acute angles, and Hadamard matrices
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From playlist Topics in Combinatorics (Cambridge Part III course)
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Classical Verification of Quantum Computations - Urmila Mahadev
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From playlist Mathematics
Klaus Fredenhagen - Quantum Field Theory and Gravitation
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From playlist Trimestre: Le Monde Quantique - Colloque de clôture
Open Source Quantum Computing: Write Your Own Quantum Programs
Quantum computers are not just science fiction anymore, with many companies building increasingly more powerful quantum computers. While, concepts in quantum computing have been around for over 30 years, but it hasn't been generally accessible until recently. Despite this quantum computing
From playlist Quantum Computing
Qubits and Gates - Quantum Computer Programming w/ Qiskit p.2
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From playlist Quantum Computer Programming w/ Qiskit
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From playlist The Large Hadron Collider