Analytic Functions and Integrable Hierarchies–Characterization of Tau Functions

We prove the dispersionless Hirota equations for the dispersionless Toda, dispersionless coupled modified KP and dispersionless KP hierarchies using an idea from classical complex analysis. We also prove that the Hirota equations characterize the tau functions for each of these hierarchies. As a result, we establish the links between the hierarchies.     

On a Class of Optimal Control Problems with State Jumps

In this paper, we consider a class of optimal control problems in which the dynamical system involves a finite number of switching times together with a state jump at each of these switching times. The locations of these switching times and a parameter vector representing the state jumps are taken as decision variables. We show that this class of optimal control problems is equivalent to a special class of optimal parameter selection problems. Gradient formulas for the cost functional and the constraint functional are derived. On this basis, a computational algorithm is proposed. For illustration, a numerical example is included.     

Some Remarks on the Minty Vector Variational Inequality

In this paper, we establish some relations between a Minty vector variational inequality and a vector optimization problem under pseudoconvexity or pseudomonotonicity, respectively. Our results generalize those of Ref. 1.     

Persistence in discrete optimization under data uncertainty

An important question in discrete optimization under uncertainty is to understand the persistency of a decision variable, i.e., the probability that it is part of an optimal solution. For instance, in project management, when the task activity times are random, the challenge is to determine a set of critical activities that will potentially lie on the longest path. In the spanning tree and shortest path network problems, when the arc lengths are random, the challenge is to pre-process the network and determine a smaller set of arcs that will most probably be a part of the optimal solution under different realizations of the arc lengths. Building on a characterization of moment cones for single variate problems, and its associated semidefinite constraint representation, we develop a limited marginal moment model to compute the persistency of a decision variable. Under this model, we show that finding the persistency is tractable for zero-one optimization problems with a polynomial sized representation of the convex hull of the feasible region. Through extensive experiments, we show that the persistency computed under the limited marginal moment model is often close to the simulated persistency value under various distributions that satisfy the prescribed marginal moments and are generated independently.     

The 2-hamiltonian cubes of graphs

This paper deals with the problem of characterizing the pairs of vertices x,y in a connected graph G such that G³ - {x,y} is hamiltonian, where G³ is the cube of G. It is known that the cube G³ is 2-hamiltonian if G is 2-connected. In this paper, we first prove the stronger result that G³ - {x,y} is hamiltonian if either x or y is not a cut-vertex of G, and then proceed to characterize those cut-vertices x and y of G such that G³ -{x,y} is hamiltonian. As a simple consequence of these, we obtain Schaar's characterization of a connected graph G such that G³ is 2-hamiltonian.     

On a Conjecture on Directed Cycles in a Directed Bipartite Graph

Let D = (V ₁, V ₂; A) be a directed bipartite graph with |V ₁| = |V ₂| = n ≥ 2. Suppose that d _D (x) + d _D (y) ≥ 3n for all x ∈ V ₁ and y ∈ V ₂. Then, with one exception, D contains two vertex-disjoint directed cycles of lengths 2n ₁ and 2n ₂, respectively, for any positive integer partition n = n ₁ + n ₂. This proves a conjecture proposed in [9].     

Approximate Augmented Lagrangian Functions and Nonlinear Semidefinite Programs

In this paper, an approximate augmented Lagrangian function for nonlinear semidefinite programs is introduced. Some basic properties of the approximate augmented Lagrange function such as monotonicity and convexity are discussed. Necessary and sufficient conditions for approximate strong duality results are derived. Conditions for an approximate exact penalty representation in the framework of augmented Lagrangian are given. Under certain conditions, it is shown that any limit point of a sequence of stationary points of approximate augmented Lagrangian problems is a KKT point of the original semidefinite program and that a sequence of optimal solutions to augmented Lagrangian problems converges to a solution of the original semidefinite program.     

Crystal Structure of the Sought-After Tetrameric [(PDBP)AgCl]4 Cluster (PDBP=5-Phenyldibenzophosphole)

The sought-after member of the [(PDBP) _n AgX] _m (n, m=1,4; 2,2; 3,1; PDBP=5-Phenyldibenzophosphole, X=halides) series, the tetrameric [(PDBP)AgCl]₄ cluster has been prepared and structurally characterized. The [P₄Ag₄Cl₄] cluster core of [(PDBP)AgCl]₄ bears striking similarity to that of [(Ph₃P)AgCl]₄.     

Catalytic asymmetric allylation of aldehydes via a chiral indium(III) complex

A chiral indium complex has been discovered to effect high enantioselectivities in the addition of allyltributyl stannanes to aldehydes. The allylation of a variety of aromatic, alpha,beta-unsaturated and aliphatic aldehydes resulted in good yields and high enantioselectivities (90-96% ee).