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Hadamard transform
About: Hadamard transform is a research topic. Over the lifetime, 7262 publications have been published within this topic receiving 94328 citations.
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TL;DR: This work uses several new number theoretic results to rule out many of the known open cases of the circulant Hadamard matrix conjecture and settles the only known open case of the Barker sequence conjecture.
Abstract: We obtain several new number theoretic results which improve the field descent method. We use these results to rule out many of the known open cases of the circulant Hadamard matrix conjecture. In particular, the only known open case of the Barker sequence conjecture is settled.
43 citations
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TL;DR: In this paper, the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry is considered and an approximate diagonalization and a microlocal decomposition of the Cauche evolution using a time-dependent version of the pseu-dodifferential calculus is given.
Abstract: We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter and Kerr-Kruskal spacetimes. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseu-dodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a one-parameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.
43 citations
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TL;DR: The Douglas--Rachford algorithm and its parallel version are extended to symmetric Hadamard manifolds to solve the convex minimization problem and it is shown that proper convex lower semicontinuous functions can have expansive reflections.
Abstract: We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation--like regularizing term. To solve the convex minimization problem, we extend the Douglas--Rachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the Douglas--Rachford algorithm is reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for Krasnoselski--Mann iterations imply the convergence of the Douglas--Rachford algorithm. Unfortunately, these general results do not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions---namely, several distance-like functions and an indicator function ...
43 citations
07 Mar 2012
43 citations
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TL;DR: In this paper, a new identity for differentiable functions is derived and a consequence of the identity is that the author establishes some new general inequalities containing all of the Hermite-Hadamard and Simpson-like types for functions whose derivatives in absolute value at certain power are harmonically convex.
Abstract: A new identity for differentiable functions is derived. A consequence of the identity is that the author establishes some new general inequalities containing all of the Hermite-Hadamard and Simpson-like types for functions whose derivatives in absolute value at certain power are harmonically convex. Some applications to special means of real numbers are also given.
43 citations