A sum sequence modulo n is a sequence S = ( s 1 , s 2 , … , s d ) of elements in Z / n Z such that every x ∈ Z / n Z can be represented as s i + s j , i < j , in the same number λ of ways. For example, ( 0 , 1 , 2 , 4 ) is a sum sequence modulo 6 with λ = 1 . We examine polynomials associated with sum sequences using tools from number theory, combinatorics and Galois theory. In particular, we give a complete characterization of sum sequences and their associated polynomials. We also describe some variations on these ideas and mention several possible generalizations to arbitrary finite groups.

The papers reprinted in this chapter are concerned with enumeration within a geometric lattice. They are closely related to papers on the Tutte decomposition reprinted in the next chapter.

A theorem is established that provides necessary and sufficient conditions in order that a locally finite bipartite graph have a subgraph whose valences lie in prescribed intervals. This theorem is applied to the study of flows in locally finite directed graphs. In particular, generalizations of the max-flow min-cut theorem and of the circulation theorem are obtained.

The main theorem of this memorandum gives necessary and sufficient conditions for an infinite family of sets with only finitely many infinite members to have a transversal. We also show that the existence of a transversal of an infinite family of sets with only countably many infinite members is equivalent to the existence of a function from the subsets of the index set of the family to the cardinal numbers having certain properties.

Previous article Next article Approximating One Convex Function by AnotherJon Folkman and Norman ShapiroJon Folkman and Norman Shapirohttps://doi.org/10.1137/0116080PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. J. Clasen, The numerical solution of the chemical equilibrium problem, RM- 4345, The RAND Corporation, Santa Monica, California, 1965 Google Scholar[2] George B. Dantzig, , Jon Folkman and , Norman Shapiro, On the continuity of the minimum sets of a continuous function, J. Math. Anal. Appl., 17 (1967), 519–548 10.1016/0022-247X(67)90139-4 MR0207426 (34:7241) 0153.49201 CrossrefISIGoogle Scholar[3] E. A. Guggenheim, Thermodynamics: an Advanced Treatment for Chemists and Physicists, North-Holland, Amsterdam, 1967, and John Wiley, New York CrossrefGoogle Scholar[4] N. Z. Shapiro and , L. S. Shapley, Mass action laws and the Gibbs free energy function, J. Soc. Indust. Appl. Math., 13 (1965), 353–375 10.1137/0113020 MR0180121 (31:4356) LinkISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Molecular Simulation of Chemical Reaction Equilibrium by Computationally Efficient Free Energy Minimization23 August 2018 | ACS Central Science, Vol. 4, No. 9 Cross Ref Computing complex chemical equilibria by generalized linear programmingMathematical and Computer Modelling, Vol. 10, No. 7 Cross Ref Computational Aspects of Chemical Equilibrium in Complex Systems Cross Ref A Generalized Technique for Eliminating Species in Complex Chemical Equilibrium CalculationsN. Z. Shapiro12 July 2006 | SIAM Journal on Applied Mathematics, Vol. 17, No. 5AbstractPDF (954 KB) Volume 16, Issue 5| 1968SIAM Journal on Applied Mathematics History Submitted:06 February 1968Published online:28 July 2006 InformationCopyright © 1968 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0116080Article page range:pp. 993-997ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

: This memorandum obtains necessary and sufficient conditions so that the solution of a constrained minimization problem will vary continuously when the constraints and objective function are varied. It also obtains special results in the case that the constraints are linear inequalities.