A Synthesis of 2-(6′-Carboxy-2′-cis-Hexenyl)-4-Hydroxycyclopent-2-EN-1-ONE,a Prostagiandin Intermediate1

Several recent publications² report the synthesis of prostaglandin E₁ (PGE₁, 2) and some of its derivatives by conjugate addition reactions to ether-ester forms of hydroxycyclopentenone acid 1.³ The simplicity of this approach makes its application to the preparation of prostaglandins of the PG₂- and PG₃ - series (cis-Δ⁵)⁴ an attractive alternative to the existing elegant methods.⁵ We now wish to report a five-step synthesis of the requisite hydroxycyclopentenone precursor 3 from the readily available⁶ lactone 4.     

Action Quantization,Energy Quantization,and Time Parametrization

The additional information within a Hamilton–Jacobi representation of quantum mechanics is extra, in general, to the Schrödinger representation. This additional information specifies the microstate of \(\psi \) that is incorporated into the quantum reduced action, W. Non-physical solutions of the quantum stationary Hamilton–Jacobi equation for energies that are not Hamiltonian eigenvalues are examined to establish Lipschitz continuity of the quantum reduced action and conjugate momentum. Milne quantization renders the eigenvalue J. Eigenvalues J and E mutually imply each other. Jacobi’s theorem generates a microstate-dependent time parametrization \(t-\tau =\partial _E W\) even where energy, E, and action variable, J, are quantized eigenvalues. Substantiating examples are examined in a Hamilton–Jacobi representation including the linear harmonic oscillator numerically and the square well in closed form. Two byproducts are developed. First, the monotonic behavior of W is shown to ease numerical and analytic computations. Second, a Hamilton–Jacobi representation, quantum trajectories, is shown to develop the standard energy quantization formulas of wave mechanics.     

Using a Facility Location Algorithm to Determine Optimum Cast Bloom Lengths

The dimensions of a bloom, which is a rectangular piece of steel, are critical for efficiently and effectively rolling the bloom into a finished structural shape (I-beam) for sale to the customer. To achieve maximum productivity and yield, the bloom size (thickness, width and length) to be rolled on a finishing mill into a structural shape must be determined by steel deformation experts. Suppose, for a particular finishing mill, these ‘rolled' blooms are all produced from a ‘cast' bloom of the same cross-section but with many different lengths. It is necessary to consolidate the many ‘cast' bloom length-metallurgical grade combinations to a number that can be managed by the casting operation and bloom stockyard without significantly impacting productivity and yield. An uncapacitated facility location problem formulation and algorithm were used to solve this problem. The way in which this approach was used to solve a real-world application is discussed.