Extreme Functions with an Arbitrary Number of Slopes

The infinite relaxations in integer programming were introduced by Gomory and Johnson to provide a general framework for the theory of cutting planes: the so-called valid functions, and in particular the minimal and extreme functions, can be seen as automatic rules for the generation of cuts. However, while many extreme functions are piecewise linear and therefore easy to describe, the set of extreme functions turns out to have a very complicated mathematical structure, as several extreme functions are known that exhibit a somewhat pathological behavior. In this paper we show that if some smoothness assumption is imposed on an extreme function $$\pi $$, then $$\pi $$ is necessarily piecewise linear. More precisely, we show that if a continuous extreme function for the Gomory–Johnson one-dimensional infinite group relaxation is a piecewise $${\mathcal {C}}^2$$ function, then it is a piecewise linear function.

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Gunluk (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous.

We study the norm of point evaluation at the origin in the Paley–Wiener space PWp for 0 < p < ∞, i.e., we search for the smallest positive constant C, called Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document}, such that the inequality |f(0)|p≤C∥f∥pp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vert f(0)\\vert ^{p}\\leq C \\Vert f\\Vert _{p}^{p}$$\\end{document} holds for every f in PWp. We present evidence and prove several results supporting the following monotonicity conjecture: The function p↦Cp/p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\mapsto {\\mathscr{C}}_{p}/p$$\\end{document} is strictly decreasing on the half-line (0, ∞). Our main result implies that Cp<p/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p} < p/2$$\\end{document} for 2 < p < ∞, and we verify numerically that Cp>p/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p} > p/2$$\\end{document} for 1 ≤ p < 2. We also estimate the asymptotic behavior of Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} as p → ∞ and as p → 0+. Our approach is based on expressing Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} as the solution of an extremal problem. Extremal functions exist for all 0 < p < ∞; they are real entire functions with only real zeros, and the extremal functions are known to be unique for 1 ≤ p < ∞. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau–Pollak–Slepian operator of time-frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range 1 < p < ∞, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} and the extremal functions.

This is a survey on the infinite group problem, an infinite-dimensional relaxation of integer linear optimization problems introduced by Ralph Gomory and Ellis Johnson in their groundbreaking papers titled "Some continuous functions related to corner polyhedra I, II" [Math. Programming 3 (1972), 23-85, 359-389]. The survey presents the infinite group problem in the modern context of cut generating functions. It focuses on the recent developments, such as algorithms for testing extremality and breakthroughs for the k-row problem for general k >= 1 that extend previous work on the single-row and two-row problems. The survey also includes some previously unpublished results; among other things, it unveils piecewise linear extreme functions with more than four different slopes. An interactive companion program, implemented in the open-source computer algebra package Sage, provides an updated compendium of known extreme functions.

We consider Gomory and Johnson’s infinite group model with a single row. Valid inequalities for this model are expressed by valid functions and it has been recently shown that any valid function is dominated by some nonnegative valid function, modulo the affine hull of the model. Within the set of nonnegative valid functions, extreme functions are the ones that cannot be expressed as convex combinations of two distinct valid functions. In this paper we construct an extreme function $$\pi :\mathbb {R}\rightarrow [0,1]$$ whose graph is dense in $$\mathbb {R}\times [0,1]$$. Therefore $$\pi $$ is discontinuous everywhere.

For positive integers d and k we define m(d, k) to be the maximum positive integer m such that the Cayley digraph of the finite cyclic group of order m, generated by a set of k positive integers has diameter less than or equal to d. In this paper we establish a new lower bound for m(2, k). We also use this lower bound to obtain a lower bound for m(d, k) for any given positive integer d and sufficiently large k.

Coset-minimal groups

We consider extremal correlation functions, involving arbitrary number of BPS (chiral or twisted chiral) operators and exactly one anti-BPS operator in 2D mathcal{N} = (2, 2) theories. These correlators define the structure constants in the rings generated by the BPS operators with their operator product expansions. We present a way of computing these correlators from the sphere partition function of a deformed theory using localization. Relating flat space and sphere correlators is nontrivial due to operator mixing on the sphere induced by conformal anomaly. We discuss the supergravitational source of this complication and a resolution thereof. Finally, we demonstrate the process for the Quintic GLSM and the Landau-Ginzburg minimal models.

Spurious interface and boundary behaviour beyond physical solutions in lattice Boltzmann schemes

Composite lifting of group inequalities and an application to two-row mixing inequalities

This thesis presents recent developments concerning two open problems related to algebra and discrete geometry. In 1934, Harold S. M. Coxeter introduced Coxeter groups as abstractions of groups generated by reflections in a vector space [Cox34]. The present work lays out how a study of geometric and combinatorial properties of Coxeter groups contributed to the comprehension of the two open problems. Furthermore, this thesis presents some progress in proving the first problem, and an application of the approach introduced in this thesis for the second problem. Open Problem (Lattice for infinite Coxeter groups, Dyer [Dye11]). Is there, for each infinite Coxeter group, a complete ortholattice that contains the weak order? In Chapter 2, we study the asymptotical behaviour of roots of infinite Coxeter groups, which is part of a joint work with Christophe Hohlweg and Vivien Ripoll [HLR13]. In particular, we show that the directions of roots tend to the isotropic cone of the geometric representation of the root system. Moreover, using this framework, this thesis presents a proof that there is a complete ortholattice structure enclosing the weak order of infinite Coxeter groups of rank at most 3. Open Problem (Existence of multiassociahedra, Jonsson [Jon05]). Let k be an integer such that k &gt;= 1. Is there a polytope whose boundary complex corresponds to the simplicial complex of sets of diagonals of a convex polygon not containing k + 1 mutually crossing diagonals? In Chapter 3 we introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. This family generalizes the concept of multitriangulations of type A and B to arbitrary finite Coxeter groups. For k = 1, this simplicial complex coincides with the finite cluster complex of the given type. We study combinatorial and geometric properties of multi-cluster complexes. In particular, we show that every spherical subword complex is the link of a face of a multi-cluster complex. This work was realized jointly with Cesar Ceballos and Christian Stump [CLS13]. Finally, this approach allows us to exhibit formulas counting the number of common vertices of permutahedra and generalized associahedra for arbitrary finite Coxeter groups and Coxeter elements.

This paper aims to treat a study on the order of every element in the higher even, odd and prime order of group for composition. In fact, express order of a group and order of an element of a group in real numbers. Here we discuss the higher order of groups in different types of order, which will give us practical knowledge to see the applications of the composition. In order to find out the order of an element am∈G in which an= e= identity element, then find the least common multiple (i.e.(LCM))= λ) of m and n. The least common multiple of two numbers is the "smallest non-zero common number," which is a multiple of both the numbers. So O(am )= λ/m. Also, if G is a finite group, n is a positive integer, and a∈G then the order of the products na. When G is a finite group, every element must have finite order, but the converse is false. There are infinite groups where each element has finite order. Finally, find out the order of every element of a group in different types of the higher even, odd and prime order of group for composition.

We classify combinatorial Dyson–Schwinger equations giving a Hopf subalgebra of the Hopf algebra of Feynman graphs of the considered Quantum Field Theory. We first treat single equations with an arbitrary (eventually infinite) number of insertion operators. We distinguish two cases; in the first one, the Hopf subalgebra generated by the solution is isomorphic to the Faa di Bruno Hopf algebra or to the Hopf algebra of symmetric functions; in the second case, we obtain the dual of the enveloping algebra of a particular associative algebra (seen as a Lie algebra). We also treat systems with an arbitrary finite number of equations, with an arbitrary number of insertion operators, with at least one of degree 1 in each equation.

The classification of tensor categories of two-colored noncrossing partitions

Several authors have studied the question of when the monoid ring \(DM\) of a monoid \(M\) over a ring \(D\) is a right and/or left fir (free ideal ring), a semifir, or a \(2\)-fir (definitions recalled in §1). It is known that for \(M\) nontrivial, a necessary condition for any of these properties to hold is that \(D\) be a division ring. Under that assumption, necessary and sufficient conditions on \(M\) are known for \(DM\) to be a right or left fir, and various conditions on \(M\) have been proved necessary or sufficient for \(DM\) to be a \(2\)-fir or semifir. A sufficient condition for \(DM\) to be a semifir is that \(M\) be a direct limit of monoids which are free products of free monoids and free groups. Warren Dicks has conjectured that this is also necessary. However F. Cedó has given an example of a monoid \(M\) which is not such a direct limit, but satisfies all the known necessary conditions for \(DM\) to be a semifir. It is an open question whether for this \(M,\) the rings \(DM\) are semifirs. We note here some reformulations of the known necessary conditions for a monoid ring \(DM\) to be a \(2\)-fir or a semifir, motivate Cedó’s construction and a variant thereof, and recover Cedó’s results for both constructions. Any homomorphism from a monoid \(M\) into \(\mathbb {Z}\) induces a \(\mathbb {Z}\)-grading on \(DM,\) and we show that for the two monoids just mentioned, the rings \(DM\) are “homogeneous semifirs” with respect to all such nontrivial \(\mathbb {Z}\)-gradings; i.e., have (roughly) the property that every finitely generated homogeneous one-sided ideal is free of unique rank. If \(M\) is a monoid such that \(DM\) is an \(n\)-fir, and \(N\) a “well-behaved” submonoid of \(M,\) we prove some properties of the ring \(DN.\) Using these, we show that for \(M\) a monoid such that \(DM\) is a \(2\)-fir, mutual commutativity is an equivalence relation on nonidentity elements of \(M,\) and each equivalence class, together with the identity element, is a directed union of infinite cyclic groups or of infinite cyclic monoids. Several open questions are noted.