Initial value methods in the theory of Fredholm integral equations

This paper deals with the problem of establishing the equivalence of a family of integral equations of Fredholm type with kernels that depend on a parameter and a related Cauchy system of integrodifferential equations. We also show how the Cauchy problem can be given an abstract formulation as an initial value problem in a complex Banach space.This research was supported by the University of Nevada at Las Vegas, Research Grant No. 4503.     

Bi-orthogonal Polynomials on the Unit Circle, Regular Semi-Classical Weights and Integrable Systems

The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of bi-orthogonal polynomials and associated functions. In particular, for the case of regular semi-classical weights on the unit circle

Branch-and-price for <Emphasis Type="Italic">p</Emphasis>-cluster editing

Given an input graph, the p-cluster editing problem consists of minimizing the number of editions, i.e., additions and/or deletions of edges, so as to create p vertex-disjoint cliques (clusters). In order to solve this \({\mathscr {NP}}\)-hard problem, we propose a branch-and-price algorithm over a set partitioning based formulation with exponential number of variables. We show that this formulation theoretically dominates the best known formulation for the problem. Moreover, we compare the performance of three mathematical formulations for the pricing subproblem, which is strongly \({\mathscr {NP}}\)-hard. A heuristic algorithm is also proposed to speedup the column generation procedure. We report improved bounds for benchmark instances available in the literature.     

A global convergent derivative-free method for solving a system of non-linear equations

Finding all zeros of a system of \(m \in \mathbb {N}\) real non-linear equations in \(n \in \mathbb {N}\) variables often arises in engineering problems. Using Newtons’ iterative method is one way to solve the problem; however, the convergence order is at most two, it depends on the starting point, there must be as many equations as variables and the function F, which defines the system of nonlinear equations F(x)=0 must be at least continuously differentiable. In other words, finding all zeros under weaker conditions is in general an impossible task. In this paper, we present a global convergent derivative-free method that is capable to calculate all zeros using an appropriate Schauder base. The component functions of F are only assumed to be Lipschitz-continuous. Therefore, our method outperforms the classical counterparts.     

First boundary-value problem for quasilinear (A, 281-01281-01281-01) -elliptic equations

The formulation of the first boundary-value problem is given for a large class of equations in divergent form, admitting a fixed degeneration on any subset of the domain of variation of the independent variables. One finds the conditions for the existence of generalized solutions of this problem and one establishes some results regarding the uniqueness and the continuous dependence of the solutions on the given data.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 84, pp. 45–88, 1979.     

Well-posedness of the water-waves equations

We prove that the water-waves equations (i.e., the inviscid Euler equations with free surface) are well-posed locally in time in Sobolev spaces for a fluid layer of finite depth, either in dimension or under a stability condition on the linearized equations. This condition appears naturally as the Lévy condition one has to impose on these nonstricly hyperbolic equations to insure well-posedness; it coincides with the generalized Taylor criterion exhibited in earlier works. Similarly to what happens in infinite depth, we show that this condition always holds for flat bottoms. For uneven bottoms, we prove that it is satisfied provided that a smallness condition on the second fundamental form of the bottom surface evaluated on the initial velocity field is satisfied.

We work here with a formulation of the water-waves equations in terms of the velocity potential at the free surface and of the elevation of the free surface, and in Eulerian variables. This formulation involves a Dirichlet-Neumann operator which we study in detail: sharp tame estimates, symbol, commutators and shape derivatives. This allows us to give a tame estimate on the linearized water-waves equations and to conclude with a Nash-Moser iterative scheme.

     

A nonlinear programming approach to a very large hydroelectric system optimization

A large scale hydroelectric system optimization is considered and solved by using a non-linear programming method. The largest numerical case involves approximately 6 000 variables, 4 000 linear equations, 11 000 linear and nonlinear inequality constraints and a nonlinear objective function. The solution method is based on

1. partial elimination of independent variables by solving linear equations,
2. essentially unconstrained optimization of a compound function that consists of the objective function, nonlinear inequality constraints and part of the linear inequality constraints. The compound function is obtained via penalty formulation.

The algorithm takes full advantage of the problem's structure and provides useful solutions for real life problems that, in general, are defined over empty feasible regions.     

Completing Hermann Schubert's Work on the Enumerative Geometry of Cuspidal Cubics in ℙ3

Abstract

We outline the results of our revisiting Hermann Schubert's work on the enumerative geometry of cuspidal cubics in ?³(Sec. 23 of his Kalkül der abzählenden Geometrie, Teubner, [1879] Schubert, H. 1879. Kalkül der abzählenden Geometrie Teubner. Rep. in 1979 by Springer-Verlag [Google Scholar]. Rep. in 1979 by Springer-Verlag). There are three main aspects that we would like to point to. First, we describe the spaces parameterizing cuspidal cubics in ?³, as well as several different degenerations, using modern algebraic geometry language and techniques. Then we get formulas, by means of today's intersection theory, for the relevant relations among conditions and degenerations, and for allthe intersection numbers in which Schubert was in principle interested. And finally there is the computational aspect, which has been an adventure on its own: the computations have been performed by means of the mathematical computation system OmegaMath, together with the WITmodule. They are discussed briefly in the final Section, with references to detailed information, and here we would just like to say that one of our motivations has been to test that system in what has turned out to be an interesting computational project. Our final table for the cuspidal cubics, which has the 19778 nonzero numbers involving the nine first-order conditions considered by Schubert, fully confirms the fraction of numbers computed by Schubert, as listed on pages 140–143 of the Kalkül. (For those interested in getting our table, please see the indications in the last section of the full paper.) For an assessment of whether or not the numbers computed by Schubert are fully representative of the problems involved in computing all of them, see the Remark at the end of Sec. 3.     

Schubert polynomials and Arakelov theory of orthogonal flag varieties

We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of the orthogonal flag variety ${\mathfrak X={\rm SO}_N/B}$ . We use these polynomials to describe the arithmetic Schubert calculus on ${\mathfrak X}$ . Moreover, we give a method to compute the natural arithmetic Chern numbers on ${\mathfrak X}$ , and show that they are all rational numbers.     

Probabilistic Formulation of the Emergency Service Location Problem

The problem of locating emergency service facilities is studied under the assumption that the locations of incidents (accidents, fires, or customers) are random variables. The probability distribution for rectilinear travel time between a new facility location and the random location of the incident P _i is developed for the case of P _i being uniformly distributed over a rectangular region. The location problem is considered in a discrete space. A deterministic formulation is obtained and recognized to be a set cover problem. Probabilistic variation of the central facility location problem is also presented.An example and some computational experience are provided to emphasize the impact of the probabilistic formulation on the location decision.     

Formulas for Lagranigian and orthogonal degeneracy loci; \widetilde Q-polynomial approach

The main goal of the paper is to give explicit formulas for the fundamental classes of Schubert subschemes in Lagrangian and orthogonal Grassmannians of maximal isotropic subbundles as well as some globalizations of them. The used geometric tools overlap appropriate desingularizations of such Schubert subschemes and Gysin maps for such Grassmannian bundles. The main algebraic tools are provided by the families of and -polynomials introduced and investigated in the present paper. The key technical result of the paper is the computation of the class of the (relative) diagonal in isotropic Grassmannian bundles based on the orthogonality property of and polynomials. Some relationships with quaternionic Schubert varieties and Schubert polynomials for classical groups are also discussed.     

The municipal financial planning model: A simultaneous regression equations and goal programming approach

Most of today's city managers are concerned about municipal financial problems. In trying to resolve these problems, scientific planning tools are needed to examine the optimality of resource allocation. For the municipal financial policy planners, the following two points are important.

• 1.(1) Statistical aspects. Since there are many economic variables in municipal financial problems, it is necessary to clarify the relationship among these variables and to infer the parameters through a statistical approach.
• 2.(2) Mathematical aspects. Policy-planners must specify the optimal value of these variables so as to attain the multiple goals of a local government.

Econometric models, especially the simultaneous equations approach, are appropriate for statistical analysis; whereas a goal-programming formulation may be used for mathematical aspects of the problem.In this paper, we propose and show that these two models can be combined. We call this the GPE model.The GPE model is applied to Urawa City. The Urawa model is composed of 6 structural equations 9 variables and 4 scenarios from the standpoint of future insight of Urawa City.From the Urawa Case, we conclude that the GPE model may be a practical tool for the municipal financial planning of other local governments.     

Monodromy and K-theory of Schubert curves via generalized jeu de taquin

We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves \(S(\lambda _\bullet )\), which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In Levinson (One-dimensional Schubert problems with respect to osculating flags, 2016, doi: 10.4153/CJM-2015-061-1), it was shown that the real geometry of these curves is described by the orbits of a map \(\omega \) on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of \({\mathbb {RP}}^1\), with \(\omega \) as the monodromy operator. We provide a fast, local algorithm for computing \(\omega \) without rectifying the skew tableau and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong’s genomic tableaux (Pechenik and Yong in Genomic tableaux, 2016. arXiv:1603.08490), which enumerate the K-theoretic Littlewood–Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of \(S(\lambda _\bullet )\).     

Optimal attitude control of a rigid body

Being mainly interested in the control of satellites, we investigate the problem of maneuvering a rigid body from a given initial attitude to a desired final attitude at a specified end time in such a way that a cost functional measuring the overall angular velocity is minimized.This problem is solved by applying a recent technique of Jurdjevic in geometric control theory. Essentially, this technique is just the classical calculus of variations approach to optimal control problems without control constraints, but formulated for control problems on arbitrary manifolds and presented in coordinate-free language. We model the state evolution as a differential equation on the nonlinear state spaceG=SO(3), thereby completely circumventing the inevitable difficulties (singularities and ambiguities) associated with the use of parameters such as Euler angles or quaternions. The angular velocities _k about the body's principal axes are used as (unbounded) control variables. Applying Pontryagin's Maximum Principle, we lift any optimal trajectorytg^*(t) to a trajectory onT ^*G which is then revealed as an integral curve of a certain time-invariant Hamiltonian vector field. Next, the calculus of Poisson brackets is applied to derive a system of differential equations for the optimal angular velocitiest _k ^* (t); once these are known the controlling torques which need to be applied are determined by Euler's equations.In special cases an analytical solution in closed form can be obtained. In general, the unknown initial values _k ^* (t₀) can be found by a shooting procedure which is numerically much less delicate than the straightforward transformation of the optimization problem into a two-point boundary-value problem. In fact, our approach completely avoids the explicit introduction of costate (or adjoint) variables and yields a differential equation for the control variables rather than one for the adjoint variables. This has the consequence that only variables with a clear physical significance (namely angular velocities) are involved for which gooda priori estimates of the initial values are available.     

Constraint Control of Nonholonomic Mechanical Systems

We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control. We focus on Suslov’s problem, which is defined as the motion of a rigid body with a vanishing projection of the body frame angular velocity on a given direction \(\varvec{\xi }\). We derive the optimal control formulation, first for an arbitrary group, and then in the classical realization of Suslov’s problem for the rotation group \(\textit{SO}(3)\). We show that it is possible to control the system using the constraint \(\varvec{\xi }(t)\) and demonstrate numerical examples in which the system tracks quite complex trajectories such as a spiral.     

Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras

The aim of this article is to attach to the set of L-S paths of type in a canonical way a basis of the corresponding representation . This basis has some nice algebraic-geometric properties. For example, it is compatible with restrictions to Schubert varieties and has the ``standard monomial property'. As a consequence we get new simple proofs of the normality of Schubert varieties, the surjectivity of the multiplication map or the restriction map for sections of a line bundle on Schubert varieties. Other applications to the defining ideal of Schubert varieties and associated Groebner basis will be discussed in a forthcoming paper.

     

Erratum: A tight formulation for uncapacitated lot-sizing with stock upper bounds

For an n-period uncapacitated lot-sizing problem with stock upper bounds, stock fixed costs, stock overload and backlogging, we present a tight extended shortest path formulation of the convex hull of solutions with O\((n^2)\) variables and constraints, also giving an O\((n^2)\) algorithm for the problem. This corrects and extends a formulation in Section 4.4 of our article “Lot-sizing with production and delivery time windows”, Mathematical Programming A, 107:471–489, 2006, for the problem with just stock upper bounds.     

Grassmann bundles

Summary In the classical theory of the Grassmann Variety there are three principal results. The Basis Theorem asserts that the Chow ring has a selfdual linear basis of classes. Determinantal Formulawhich expresses any basic class as a determinant in the special classes. Finally the ring structure is elucidated by Pieri's Formulawhich expresses the intersection of a basic class and a special class in terms of the basic classes. Here we show how all these results can be established also for the Chow ring of a Grassmann bundle. There are however some differences. In the classical case the basic classes are Schubert classes: this is impossible in the general case as there need not be enough Schubert classes to provide a basis and in the general case there is a pair of dual bases which both reduce to the Schubert basis in the classical case. In addition to these generalizations of the classical results we also enlarge on the theory of Schubert classes developed in the important paper of Kempf and Laksov [4].Following them we shall henceforth use the phrase « determinantal formula » to mean their formula for Schubert classes and our generalization of it to « improper » Schubert classes.     

Belousov equations on ternary quasigroups

Summary The study of Belousov equations in binary quasigroups was initiated by V. D. Belousov. Krape and Taylor showed that every finite set of Belousov equations was equivalent to a single Belousov equation which was in some sense no longer than any single member of the set. This led to the concept of an irreducible Belousov equation, that is one which is not equivalent to an equation with fewer variables. Krape and Taylor determined the structure of the irreducible equations by establishing a correspondence between them and specific polynomials overZ ₂.In this paper it is shown that the structure of the ternary equations is richer than the binary counterpart, although the main result is similar to the binary case in as far as a system of ternary Belousov equations is equivalent to a single Belousov equation which is no longer than any member of the system or the system is equivalent to a pair of equations each with three variables.     

A Pieri-type formula for the -theory of a flag manifold

We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special Schubert classes are indexed by a cycle which has either the form or the form , and are pulled back from a Grassmannian projection. Our formulas are in terms of certain labeled chains in the -Bruhat order on the symmetric group and are combinatorial in that they involve no cancellations. We also show that the multiplicities in the Pieri formula are naturally certain binomial coefficients.