The Tchebyshev Transforms of the First and Second Kind

An in-depth study of the Tchebyshev transforms of the first and second kind of a poset is taken. The Tchebyshev transform of the first kind is shown to preserve desirable combinatorial properties, including EL-shellability and nonnegativity of the cd-index. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg, and Readdy omega map of oriented matroids. The Tchebyshev transform of the second kind U is a Hopf algebra endomorphism on the space of quasisymmetric functions which, when restricted to Eulerian posets, coincides with Stembridge’s peak enumerator. The complete spectrum of U is determined, generalizing the work of Billera, Hsiao, and van Willigenburg. The type B quasisymmetric function of a poset is introduced and, like Ehrenborg’s classical quasisymmetric function of a poset, it is a comodule morphism with respect to the quasisymmetric functions QSym. Finally, similarities among the omega map, Ehrenborg’s r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps which occur naturally in the setting of combinatorial Hopf algebras.     

Colored Posets and Colored Quasisymmetric Functions

The colored quasisymmetric functions, like the classic quasisymmetric functions, are known to form a Hopf algebra with a natural peak subalgebra. We show how these algebras arise as the image of the algebra of colored posets. To effect this approach, we introduce colored analogs of P-partitions and enriched P-partitions. We also frame our results in terms of Aguiar, Bergeron, and Sottile’s theory of combinatorial Hopf algebras and its colored analog.     

Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups

Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a q-analog of Euler?s exponential generating function formula for the Eulerian numbers (Shareshian and Wachs, 2010 [17]). They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the wreath product of the cyclic group with the symmetric group, also known as the group of colored permutations. We use this group to introduce colored Eulerian quasisymmetric functions, which are a generalization of Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, which reduces to a formula of Shareshian and Wachs for the Eulerian quasisymmetric functions. We show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics, which generalize the Shareshian–Wachs q-analog of Euler?s formula, formulas of Foata and Han, and a formula of Chow and Gessel.     

Row-Strict Quasisymmetric Schur Functions

Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions, called the quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions, called the row-strict quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as quasisymmetic Schur functions are generated through fillings of composition diagrams. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships.     

Eulerian quasisymmetric functions

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising q-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This q-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain q-analogs, (q,p)-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Björner and Welker.     

Notes on L-/M-convex functions and the separation theorems

The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions and for M-convex/concave functions as generalizations of Frank’s discrete separation theorem for submodular/supermodular set functions and Edmonds’ matroid intersection theorem. This paper shows the equivalence between Murota’s L-convex functions and Favati and Tardella’s submodular integrally convex functions, and also gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis. Received: November 27, 1997 / Accepted: December 16, 1999?Published online May 12, 2000     

Green’s functions for stationary problems with nonlocal boundary conditions

In this paper, we investigate Green’s functions for various stationary problems with nonlocal boundary conditions. We express the Green’s function per Green’s function for a problem with classical boundary conditions. This property is illustrated by various examples. Properties of Green’s functions with nonlocal boundary conditions are compared with those for classical problems. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-73/09.     

Quasisymmetric Schur functions

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the t parameter from Hall-Littlewood theory.     

A matroid-friendly basis for the quasisymmetric functions

A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the quasisymmetric functions associated to matroids by the Hopf algebra morphism F of Billera, Jia, and Reiner [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646]. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F distinguishes isomorphism classes of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer to the Hilbert basis question raised in [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646] is given.     

A generalization of Siegel’s method

We extend C.L. Siegel’s method of proving the Dedekind-eta function transformation by integrating some selected functions over a positively oriented polygon, generalizing Siegel’s integration over a parallelogram. As consequences, we obtain a generalization of the Dedekind-eta function transformation and generalizations of other transformation formulas.     

Experiments with an interactive paired comparison simplex method for molp problems

In this paper, an interactive paired comparison simplex based method formultiple objective linear programming (MOLP) problems is developed and compared to other interactive MOLP methods. The decision maker (DM)’s utility function is assumed to be unknown, but is an additive function of his known linearized objective functions. A test for ‘utility efficiency’ for MOLP problems is developed to reduce the number of efficient extreme points generated and the number of questions posed to the DM. The notion of ‘strength of preference ’ is developed for the assessment of the DM’s unknown utility function where he can express his preference for a pair of extreme points as ‘strong ’, ‘weak ’, or ‘almost indifferent ’. The problem of ‘inconsistency of the DM’ is formalized and its resolution is discussed. An example of the method and detailed computational results comparing it with other interactive MOLP methods are presented. Several performance measures for comparative evaluations of interactive multiple objective programming methods are also discussed. All rights reserved. This study, or parts thereof, may not be reproduced in any form without written permission of the authors.     

Skew quasisymmetric Schur functions and noncommutative Schur functions

Recently a new basis for the Hopf algebra of quasisymmetric functions QSym, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset LC that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions NSym. This basis of NSym is dual to the basis of quasisymmetric Schur functions and its elements are the pre-image of the Schur functions under the forgetful map χ:NSym→Sym. We prove that the multiplicative structure constants of the noncommutative Schur functions, equivalently the coefficients of the skew quasisymmetric Schur functions when expanded in the quasisymmetric Schur basis, are nonnegative integers, satisfying a Littlewood–Richardson rule analogue that reduces to the classical Littlewood–Richardson rule under χ.As an application we show that the morphism of algebras from the algebra of Poirier–Reutenauer to Sym factors through NSym. We also extend the definition of Schur functions in noncommuting variables of Rosas–Sagan in the algebra NCSym to define quasisymmetric Schur functions in the algebra NCQSym. We prove these latter functions refine the former and their properties, and project onto quasisymmetric Schur functions under the forgetful map. Lastly, we show that by suitably labeling LC, skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets.     

Sensitivity analysis incorporating fuzzy evaluation for scaling constants of multiattribute utility functions

A multiattribute utility function can be represented by a function of single-attribute utility functions if the decision maker’s preference satisfies additive independence or mutually utility independence. Additive independence is a preference condition stronger than mutually utility independence, and the multiattribute utility function is in the additive form if the former condition is satisfied, otherwise it is in the multiplicative form. In this paper, we propose a method for sensitivity analysis of multiattribute utility functions in multiplicative form, taking into account the imprecision of the decision maker’s judgment in the procedures for determining scaling constants (attribute weights).     

A proximal cutting plane method using Chebychev center for nonsmooth convex optimization

An algorithm is developed for minimizing nonsmooth convex functions. This algorithm extends Elzinga–Moore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones, thus removing compactness assumption. Our method is to Elzinga–Moore’s algorithm what a proximal bundle method is to Kelley’s algorithm. Instead of lower approximations used in proximal bundle methods, the present approach is based on some objects regularizing translated functions of the objective function. We propose some variants and using some academic test problems, we conduct a numerical comparative study with Elzinga–Moore algorithm and two other well-known nonsmooth methods.      

Multiplicity Free Schur, Skew Schur, and Quasisymmetric Schur Functions

In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F-multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F-multiplicity free quasisymmetric Schur functions with one or two terms in the expansion, or one or two parts in the indexing composition. This identifies composition shapes such that all standard composition tableaux of that shape have distinct descent sets. We conclude by providing such a classification for quasisymmetric Schur function families, giving a classification of Schur functions that are in some sense almost F-multiplicity free.     

The two-time Green’s function and the diagram technique

Based on constructing the equations of motion for the two-time Green’s functions, we discuss calculating the dynamical spin susceptibility and correlation functions in the Heisenberg model. Using a Mori-type projection, we derive an exact Dyson equation with the self-energy operator in the form of a multiparticle Green’s function. Calculating the self-energy operator in the mode-coupling approximation in the ferromagnetic phase, we reproduce the results of the temperature diagram technique, including the correct formula for low-temperature magnetization. We also consider calculating the spin fluctuation spectrum in the paramagnetic phase in the framework of the method of equations of motion for the relaxation function.     

Moduli of quadrilaterals and extremal quasiconformal extensions of quasisymmetric functions

We establish a relationship between Strebel boundary dilatation of a quasisymmetric function of the unit circle and indicated by the change in the module of the quadrilaterals with vertices on the circle. By using general theory of universal Teichmüller space, we show that there are many quasisymmetric functions of the circle have the property that the smallest dilatation for a quasiconformal extension of a quasisymmetric function of the unit circle is larger than indicated by the change in the module of quadrilaterals with vertices on the circle. Received: December 8, 1996     

The computational complexity of bilevel assignment problems

In bilevel optimization problems there are two decision makers, the leader and the follower, who act in a hierarchy. Each decision maker has his own objective function, but there are common constraints. This paper deals with bilevel assignment problems where each decision maker controls a subset of edges and each edge has a leader’s and a follower’s weight. The edges selected by the leader and by the follower need to form a perfect matching. The task is to determine which edges the leader should choose such that his objective value which depends on the follower’s optimal reaction is maximized. We consider sum- and bottleneck objective functions for the leader and follower. Moreover, if not all optimal reactions of the follower lead to the same leader’s objective value, then the follower either chooses an optimal reaction which is best (optimistic rule) or worst (pessimistic rule) for the leader. We show that all the variants arising if the leader’s and follower’s objective functions are sum or bottleneck functions are NP-hard if the pessimistic rule is applied. In case of the optimistic rule the problem is shown to be NP-hard if at least one of the decision makers has a sum objective function.