On the notions of facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem

We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We then separate the three notions using discontinuous examples.Comment: 18 pages, 2 figure

Software for cut-generating functions in the Gomory--Johnson model and beyond

We present software for investigations with cut generating functions in the Gomory-Johnson model and extensions, implemented in the computer algebra system SageMath.Comment: 8 pages, 3 figures; to appear in Proc. International Congress on Mathematical Software 201

The Dynamics of Metropolitan Housing Prices

This article is the winner of the Innovative Thinking ‘‘Thinking Out of the Box’’ manuscript prize (sponsored by the Homer Hoyt Advanced Studies Institute) presented at the 2001 American Real Estate Society Annual Meeting. This study examines the dynamics of real housing price appreciation in 130 metropolitan areas across the United States. The study finds that real housing price appreciation is strongly influenced by the growth of population and real changes in income, construction costs and interest rates. The study also finds that stock market appreciation imparts a strong current and lagged wealth effect on housing prices. Housing appreciation rates also are found to vary across areas because of location-specific fixed-effects; these fixed effects represent the residuals of housing price appreciation attributable to location. The magnitudes of the fixed-effects in particular cities are positively correlated with restrictive growth management policies and limitations on land availability.

The structure of the infinite models in integer programming

The infinite models in integer programming can be described as the convex hull of some points or as the intersection of halfspaces derived from valid functions. In this paper we study the relationships between these two descriptions. Our results have implications for corner polyhedra. One consequence is that nonnegative, continuous valid functions suffice to describe corner polyhedra (with or without rational data)

Reverse Chv\'atal-Gomory rank

We introduce the reverse Chv\'atal-Gomory rank r*(P) of an integral polyhedron P, defined as the supremum of the Chv\'atal-Gomory ranks of all rational polyhedra whose integer hull is P. A well-known example in dimension two shows that there exist integral polytopes P with r*(P) equal to infinity. We provide a geometric characterization of polyhedra with this property in general dimension, and investigate upper bounds on r*(P) when this value is finite.Comment: 21 pages, 4 figure

Approximation of corner polyhedra with families of intersection cuts

We study the problem of approximating the corner polyhedron using intersection cuts derived from families of lattice-free sets in $\mathbb{R}^n$. In particular, we look at the problem of characterizing families that approximate the corner polyhedron up to a constant factor, which depends only on $n$ and not the data or dimension of the corner polyhedron. The literature already contains several results in this direction. In this paper, we use the maximum number of facets of lattice-free sets in a family as a measure of its complexity and precisely characterize the level of complexity of a family required for constant factor approximations. As one of the main results, we show that, for each natural number $n$, a corner polyhedron with $n$ basic integer variables and an arbitrary number of continuous non-basic variables is approximated up to a constant factor by intersection cuts from lattice-free sets with at most $i$ facets if $i> 2^{n-1}$ and that no such approximation is possible if $i \leq 2^{n-1}$. When the approximation factor is allowed to depend on the denominator of the fractional vertex of the linear relaxation of the corner polyhedron, we show that the threshold is $i > n$ versus $i \leq n$. The tools introduced for proving such results are of independent interest for studying intersection cuts

Subtropical Real Root Finding

We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients. Then we employ linear programming to heuristically find roots. There is a specialized variant for roots with exclusively positive coordinates, which is of considerable interest for applications in chemistry and systems biology. An implementation of our method combining the computer algebra system Reduce with the linear programming solver Gurobi has been successfully applied to input data originating from established mathematical models used in these areas. We have solved several hundred problems with up to more than 800000 monomials in up to 10 variables with degrees up to 12. Our method has failed due to its incompleteness in less than 8 percent of the cases

Model mass spectrometric study of competitive interactions of antimicrobial bisquaternary ammonium drugs and aspirin with membrane phospholipids

The aim of the study is to reveal molecular mechanisms of possible activity modulation of antimicrobial bis-quaternary ammonium compounds (BQAC) and aspirin (ASP) through noncovalent competitive complexation under their combined introduction into the model systems with membrane phospholipids. Methods. Binary and triple systems containing either decamethoxinum or ethonium, or thionium and aspirin, as well as dipalmitoyl-phosphatidylcholine (DPPC) have been investigated by electrospray ionization mass spectrometry. Results. Basing on the analysis of associates recorded in the mass spectra, the types of nonocovalent complexes formed in the systems studied were determined and the supposed role of the complexation in the BQAC and ASP activity modulation was discussed. The formation of associates of BQAC dications with ASP anion is considered as one of the possible ways of deactivation of ionic forms of the medications. The formation of stable complexes of BQAC with DPPC and ASP with DPPC in binary systems as well as the complexes distribution in triple-components systems BQAC:ASP:DPPC point to the existence of competition between drugs of these two types for the binding to DPPC. Conclusions. The results obtained point to the competitive complexation in the model molecular systems containing the BQAC, aspirin and membrane phospholipids. The observed phenomenon testifies to the possibility of modulating the activity of bisquaternary antimicrobial agents and aspirin under their combined usage, due to the competition between the drugs for binding to the target membrane phospholipid molecules and also due to the formation of stable noncovalent complexes between BQAC and ASP.Мета. Вивчення молекулярних механізмів можливої модуляції активності антимікробних бісчетвертинних амонієвих сполук (БЧАС) та аспірину (АСП) внаслідок формування нековалентних комплексів під час спільного введення препаратів двох типів у модельні системи з мембранними фосфоліпідами. Методи. Дво- і трикомпонентні системи, які містять декаметоксин, етоній або тіоній, АСП і дипальмітоїлфосфатидилхолін (ДПФХ), досліджували методом мас-спектрометрії з іонізацією електроспреєм. Результати. Грунтуючись на даних аналізу асоціатів, зареєстрованих у мас-спектрах, встановлено типи нековалентних комплексів, які формуються у досліджуваних системах, та обговорено їхню можливу роль у модуляції активності БЧАС і АСП. Утворення асоціатів дикатіонів БЧАС з аніоном АСП є одним з імовірних шляхів дезактивації іонних форм препаратів. Формування стабільних комплексів БЧАС з ДПФХ та АСП з ДПФХ у двокомпонентних системах, а також розподіл комплексів у трикомпонентних системах БЧАС:АСП:ДПФХ вказують на існування конкуренції між препаратами двох типів за зв’язування з ДПФХ. Висновки. Отримані результати свідчать про конкурентне комплексоутворення у модельних молекулярних системах, що містять БЧАС, АСП і мембранні фосфоліпіди. Виявлений факт підтверджує можливість модуляції активності бісчетвертинних амонієвих протимікробних агентів і аспірину при сумісному використанні завдяки конкуренції між ліками за зв’язування з мембранними фосфоліпідами, а також внаслідок формування стабільних нековалентних комплексів між БЧАС і АСП.Цель. Изучение молекулярных механизмов возможной модуляции активности антимикробных бисчетвертичных аммониевых соединений (БЧАС) и аспирина (АСП) посредством формирования нековалентных комплексов при совместном введении препаратов двух типов в модельные системы с мембранными фосфолипидами. Методы. Двух- и трехкомпонентные системы, содержащие декаметоксин, этоний или тионий и АСП, а также дипальмитоилфосфатидилхолин (ДПФХ) исследовали методом масс-спектрометрии с ионизацией электроспреем. Результаты. На основании анализа ассоциатов, зарегистрированных в масс-спектрах, установлены типы нековалентных комплексов, образующихся в исследованных системах, а также обсуждена их предполагаемая роль в модуляции активности БЧАС и АСП. Формирование ассоциатов дикатионов БЧАС с анионом АСП является одним из возможных путей дезактивации ионных форм препаратов. Образование стабильных комплексов БЧАС с ДПФХ и АСП с ДПФХ в двухкомпонентных системах, а также распределение комплексов в трехкомпонентных системах БЧАС:АСП:ДПФХ указывают на существование конкуренции между препаратами двух типов за связывание с ДПФХ. Выводы. Полученные результаты свидетельствуют о конкурентном комплексообразовании в модельных молекулярных системах, содержащих БЧАС, АСП и мембранные фосфолипиды. Обнаруженный факт подтверждает возможность модуляции активности бисчетвертичных аммониевых противомикробных агентов и аспирина при совместном применении вследствие конкуренции между лекарствами за связывание с мембранными фосфолипидами, а также благодаря формированию нековалентных комплексов между БЧАС и АСП