Neighbors of the Origin for Four by Three Matrices

Scarf has defined a neighborhood system for families of integer programs where the right-hand side is allowed to vary. This system depends on a matrix A of constraint and objective function coefficients of the integer programs. This paper characterizes the set of neighbors of the origin when A is four by three; showing that it may be described as the set of integer vectors in a union of two-dimensional polyhedra, where the number of polyhedra is quadratic in the bit size of A

The Frobenius Problem and Maximal Lattice Free Bodies

Let p = ( p 1 ,…, p n ) be a vector of positive integers whose greatest common divisor is unity. The Frobenius problem is to find the largest integer f * which cannot be written as a non-negative integral combination of the p i . In this note we relate the Frobenius problem to the topic of maximal lattice free bodies and describe an algorithm for n = 3

Shortest Integer Vectors

Let A be a fixed integer matrix of size m by n and consider all b for which the body is full dimensional. We examine the set of shortest non-zero integral vectors with respect to the family of norms. We show that the number of such shortest vectors is polynomial in the bit size of A , for fixed n . We also show the existence, for any n , of a family of matrices M for which the number of shortest vectors has as a lower bound a polynomial in the bit size of M of the same degree at the polynomial bound

The Topological Structure of Maximal Lattice Free Convex Bodies: The General Case

Given a generic m x n matrix A , the simplicial complex K ( A ) is defined to be the collection of simplices representing maximal lattice point free convex bodies of the form { x : Ax \u3c b }. The main result of this paper is that the topological space associated with K ( A ) is homeomorphic with R m -1

Neighbors of the Origin for Four by Three Matrices

Scarf has defined a neighborhood system for families of integer programs where the right-hand side is allowed to vary. This system depends on a matrix A of constraint and objective function coefficients of the integer programs. This paper characterizes the set of neighbors of the origin when A is four by three; showing that it may be described as the set of integer vectors in a union of two-dimensional polyhedra, where the number of polyhedra is quadratic in the bit size of A.Integer programming, neighborhood systems

An Implementation of the Generalized Basis Reduction Algorithm for Integer Programming

In recent years many advances have been made in solution techniques for specially structured 0–1 integer programming problems. In contrast, very little progress has been made on solving general (mixed integer) problems. This, of course, is not true when viewed from the theoretical side: Lenstra (1981) made a major breakthrough, obtaining a polynomial-time algorithm when the number of integer variables is fixed. We discuss a practical implementation of a Lenstra-like algorithm, based on the generalized basis reduction method of Lovasz and Scarf (1988). This method allows us to avoid the ellipsoidal approximations required in Lenstra’s algorithm. We report on the solution of a number of small (but difficult) examples, up to 100 integer variables. Our computer code uses the linear programming optimizer CPlex as a subroutine to solve the linear programming problems that arise

Experimental Evidence of Dioxole Unimolecular Decay Pathway for Isoprene-Derived Criegee Intermediates

Ozonolysis of isoprene, one of the most abundant volatile organic compounds emitted into the Earth’s atmosphere, generates two four-carbon unsaturated Criegee intermediates, methyl vinyl ketone oxide (MVK-oxide) and methacrolein oxide (MACR-oxide). The extended conjugation between the vinyl substituent and carbonyl oxide groups of these Criegee intermediates facilitates rapid electrocyclic ring closures that form five-membered cyclic peroxides, known as dioxoles. This study reports the first experimental evidence of this novel decay pathway, which is predicted to be the dominant atmospheric sink for specific conformational forms of MVK-oxide (anti) and MACR-oxide (syn) with the vinyl substituent adjacent to the terminal O atom. The resulting dioxoles are predicted to undergo rapid unimolecular decay to oxygenated hydrocarbon radical products, including acetyl, vinoxy, formyl, and 2-methylvinoxy radicals. In the presence of O₂, these radicals rapidly react to form peroxy radicals (ROO), which quickly decay via carbon-centered radical intermediates (QOOH) to stable carbonyl products that were identified in this work. The carbonyl products were detected under thermal conditions (298 K, 10 Torr He) using multiplexed photoionization mass spectrometry (MPIMS). The main products (and associated relative abundances) originating from unimolecular decay of anti-MVK-oxide and subsequent reaction with O₂ are formaldehyde (88 ± 5%), ketene (9 ± 1%), and glyoxal (3 ± 1%). Those identified from the unimolecular decay of syn-MACR-oxide and subsequent reaction with O₂ are acetaldehyde (37 ± 7%), vinyl alcohol (9 ± 1%), methylketene (2 ± 1%), and acrolein (52 ± 5%). In addition to the stable carbonyl products, the secondary peroxy chemistry also generates OH or HO₂ radical coproducts

Direct kinetic measurements and theoretical predictions of an isoprene-derived Criegee intermediate

Isoprene has the highest emission into Earth’s atmosphere of any nonmethane hydrocarbon. Atmospheric processing of alkenes, including isoprene, via ozonolysis leads to the formation of zwitterionic reactive intermediates, known as Criegee intermediates (CIs). Direct studies have revealed that reactions involving simple CIs can significantly impact the tropospheric oxidizing capacity, enhance particulate formation, and degrade local air quality. Methyl vinyl ketone oxide (MVK-oxide) is a four-carbon, asymmetric, resonance-stabilized CI, produced with 21 to 23% yield from isoprene ozonolysis, yet its reactivity has not been directly studied. We present direct kinetic measurements of MVK-oxide reactions with key atmospheric species using absorption spectroscopy. Direct UV-Vis absorption spectra from two independent flow cell experiments overlap with the molecular beam UV-Vis-depletion spectra reported recently [M. F. Vansco, B. Marchetti, M. I. Lester, J. Chem. Phys. 149, 44309 (2018)] but suggest different conformer distributions under jet-cooled and thermal conditions. Comparison of the experimental lifetime herein with theory indicates only the syn-conformers are observed; anti-conformers are calculated to be removed much more rapidly via unimolecular decay. We observe experimentally and predict theoretically fast reaction of syn-MVK-oxide with SO₂ and formic acid, similar to smaller alkyl-substituted CIs, and by contrast, slow removal in the presence of water. We determine products through complementary multiplexed photoionization mass spectrometry, observing SO₃ and identifying organic hydroperoxide formation from reaction with SO₂ and formic acid, respectively. The tropospheric implications of these reactions are evaluated using a global chemistry and transport model

Direct kinetic measurements and theoretical predictions of an isoprene-derived Criegee intermediate

Isoprene has the highest emission into Earth’s atmosphere of any nonmethane hydrocarbon. Atmospheric processing of alkenes, including isoprene, via ozonolysis leads to the formation of zwitterionic reactive intermediates, known as Criegee intermediates (CIs). Direct studies have revealed that reactions involving simple CIs can significantly impact the tropospheric oxidizing capacity, enhance particulate formation, and degrade local air quality. Methyl vinyl ketone oxide (MVK-oxide) is a four-carbon, asymmetric, resonance-stabilized CI, produced with 21 to 23% yield from isoprene ozonolysis, yet its reactivity has not been directly studied. We present direct kinetic measurements of MVK-oxide reactions with key atmospheric species using absorption spectroscopy. Direct UV-Vis absorption spectra from two independent flow cell experiments overlap with the molecular beam UV-Vis-depletion spectra reported recently [M. F. Vansco, B. Marchetti, M. I. Lester, J. Chem. Phys. 149, 44309 (2018)] but suggest different conformer distributions under jet-cooled and thermal conditions. Comparison of the experimental lifetime herein with theory indicates only the syn-conformers are observed; anti-conformers are calculated to be removed much more rapidly via unimolecular decay. We observe experimentally and predict theoretically fast reaction of syn-MVK-oxide with SO₂ and formic acid, similar to smaller alkyl-substituted CIs, and by contrast, slow removal in the presence of water. We determine products through complementary multiplexed photoionization mass spectrometry, observing SO₃ and identifying organic hydroperoxide formation from reaction with SO₂ and formic acid, respectively. The tropospheric implications of these reactions are evaluated using a global chemistry and transport model