An Interview with George B. Dantzig: The Father of Linear Programming

Albers, Donald J.; Reid, Constance

doi:10.1080/07468342.1986.11972971

Cited by 22 publications

(13 citation statements)

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“…When working in large-dimension spaces we will heed also Dantzig's remark that "one's intuition in higher dimensional space is not worth a damn!" [5]. For purposes of quantitative analysis we will rely upon Gauss' Theorema Egregium [86] to analyze the Riemannian and Kählerian geometric properties of gabion manifolds.…”

Section: The Geometric Properties Of Gabion-kähler (Gk) Manifoldsmentioning

confidence: 99%

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Practical recipes for the model order reduction, dynamical simulation and compressive sampling of large-scale open quantum systems

et al. 2009

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This article presents practical numerical recipes for simulating high-temperature and nonequilibrium quantum spin systems that are continuously measured and controlled. The notion of a "spin system" is broadly conceived, in order to encompass macroscopic test masses as the limiting case of large-j spins. The simulation technique has three stages: first the deliberate introduction of noise into the simulation, then the conversion of that noise into an informatically equivalent continuous measurement and control process, and finally, projection of the trajectory onto a Kählerian state-space manifold having reduced dimensionality and possessing a Kähler potential of multilinear (i.e., product-sum) functional form. These state-spaces can be regarded as ruled algebraic varieties upon which a projective quantum model order reduction (QMOR) is performed. The Riemannian sectional curvature of ruled Kählerian varieties is analyzed, and proved to be non-positive upon all sections that contain a rule. It is further shown that the class of ruled Kählerian state-spaces includes the Slater determinant wave-functions of quantum chemistry as a special case, and that these Slater determinant manifolds have a Fubini-Study metric that is Kähler-Einstein; hence they are solitons under Ricci flow. It is suggested that these negative sectional curvature properties geometrically account for the fidelity, efficiency, and robustness of projective trajectory simulation on ruled Kählerian state-spaces. Some implications of trajectory compression for geometric quantum mechanics are discussed. The resulting simulation formalism is used to construct a positive P -representation for the thermal density matrix and to derive a quantum limit for force noise and measurement noise in monitoring both macroscopic and microscopic test-masses; this quantum noise limit is shown to be consistent with wellestablished quantum noise limits for linear amplifiers and for monitoring linear dynamical systems. Single-spin detection by magnetic resonance force microscopy (MRFM) is then simulated, and the data statistics are shown to be those of a random telegraph signal with additive white noise, to all orders, in excellent agreement with experimental results. Then a larger-scale spin-dust model is simulated, having no spatial symmetry and no spatial ordering; the high-fidelity projection of numerically computed quantum trajectories onto low-dimensionality Kähler state-space manifolds is demonstrated. Finally, the high-fidelity reconstruction of quantum trajectories from sparse random projections is demonstrated, the onset of Donoho-Stodden breakdown at the Candès-Tao sparsity limit is observed, and methods for quantum state optimization by Dantzig selection are given.

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Section: The Geometric Properties Of Gabion-kähler (Gk) Manifoldsmentioning

confidence: 99%

“…Here we recall that dim H is the (complex) dimension of the Hilbert space within which the efflorescent GK state-space manifold of (complex) dimension dim K is embedded (see Sections 2.6.4 and 2.11, and also (5), for discussion of how to calculate dim K).…”

Section: Quantum State Reconstruction From Sparse Random Projectionsmentioning

confidence: 99%

Practical recipes for the model order reduction, dynamical simulation and compressive sampling of large-scale open quantum systems

et al. 2009

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“…Dantzig and von Neumann observed that the Min-Max theorem is implied by strong linear programming duality [10], [2]. Dantzig also provided a candidate construction for the opposite implication [10], and this was also established some decades later [1].…”

Section: Introductionmentioning

confidence: 99%

“…Samuel Karlin conjectured in 1959 that fictitious play converges at rate O(t − 1 2 ) with respect to the number of steps t. We disprove this conjecture by showing that, when the payoff matrix of the row player is the n × n identity matrix, fictitious play may converge (for some tie-breaking) at rate as slow as Ω(t − 1 n ). † Supported by a Sloan Foundation fellowship, a Microsoft Research faculty fellowship and NSF Award CCF-0953960 (CAREER) and CCF-1101491.‡ Supported by ONR grant N00014-12-1-0999.Step 1: row chooses 1 column chooses 2 row chooses 1 column chooses 1Step 2: row chooses 2 column chooses 2 row chooses 1 column chooses 2Step 3: row chooses 2 column chooses 1 row chooses 2 column chooses 2Step 4: row chooses 2 column chooses 1 row chooses 2 column chooses 2Step 5: row chooses 1 column chooses 1 row chooses 2 column chooses 1 U (5) = [2, 3] T , V (5) = [3, 2] U (5) = [2, 3] T , V (5) = [2,3]…”

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confidence: 99%