Qualitative and quantitative data envelopment analysis with interval data

In this paper, we investigate DEA with interval input-output data. First we show various extensions of efficiency and that 25 of them are essential. Second we formulate the efficiency test problems as mixed integer programming problems. We prove that 14 among 25 problems can be reduced to linear programming problems and that the other 11 efficiencies can be tested by solving a finite sequence of linear programming problems. Third, in order to obtain efficiency scores, we extend SBM model to interval input-output data. Fourth, to moderate a possible positive overassessment by DEA, we introduce the inverted DEA model with interval input-output data. Using efficiency and inefficiency scores, we propose a classification of DMUs. Finally, we apply the proposed approach to Japanese Bank Data and demonstrate its advantages.     

The value of foresight

Suppose you have one unit of stock, currently worth 1, which you must sell before time $T$. The Optional Sampling Theorem tells us that whatever stopping time we choose to sell, the expected discounted value we get when we sell will be 1. Suppose however that we are able to see $a$ units of time into the future, and base our stopping rule on that; we should be able to do better than expected value 1. But how much better can we do? And how would we exploit the additional information? The optimal solution to this problem will never be found, but in this paper we establish remarkably close bounds on the value of the problem, and we derive a fairly simple exercise rule that manages to extract most of the value of foresight.     

Exploring an unknown graph

We wish to explore all edges of an unknown directed, strongly connected graph. At each point, we have a map of all nodes and edges we have visited, we can recognize these nodes and edges if we see them again, and we know how many unexplored edges emanate from each node we have visited, but we cannot tell where each leads until we traverse it. We wish to minimize the ratio of the total number of edges traversed divided by the optimum number of traversals, had we known the graph. For Eulerian graphs, this ratio cannot be better than two, and two is achievable by a simple algorithm. In contrast, the ratio is unbounded when the deficiency of the graph (the number of edges that have to be added to make it Eulerian) is unbounded. Our main result is an algorithm that achieves a bounded ratio when the deficiency is bounded. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 265–297, 1999     

Optimal control of nonlinear evolution inclusions

In this paper, we study the optimal control of nonlinear evolution inclusions. First, we prove the existence of admissible trajectories and then we show that the set that they form is relatively sequentially compact and in certain cases sequentially compact in an appropriate function space. Then, with the help of a convexity hypothesis and using Cesari's approach, we solve a general Lagrange optimal control problem. After that, we drop the convexity hypothesis and pass to the relaxed system, for which we prove the existence of optimal controls, we show that it has a value equal to that of the original one, and also we prove that the original trajectories are dense in an appropriate topology to the relaxed ones. Finally, we present an example of a nonlinear parabolic optimal control that illustrates the applicability of our results.This research was supported by NSF Grant No. DMS-88-02688.     

Fractal properties of number systems

In this paper we study properties of the fundamental domain F of number systems in the n-dimensional real vector space. In particular we investigate the fractal structure of its boundary F. In a first step we give upper and lower bounds for its box counting dimension. Under certain circumstances these bounds are identical and we get an exact value for the box counting dimension. Under additional assumptions we prove that the Hausdorf dimension of F is equal to its box counting dimension. Moreover, we show that the Hausdorf measure is positive and fnite. This is done by applying the theory of graphdirected self similar sets due to Falconer and Bandt. Finally, we discuss the connection to canonical number systems in number felds, and give some numerical examples.     

Classification of numerical 3-semigroups by means of L-shapes

We recall L-shapes, which are minimal distance diagrams, related to weighted 2-Cayley digraphs, and we give the number and the relation between minimal distance diagrams related to the same digraph. On the other hand, we consider some classes of numerical semigroups useful in the study of curve singularity. Then, we associate L-shapes to each numerical 3-semigroup and we describe some main invariants of numerical 3-semigroups in terms of their associated L-shapes. Finally, we give a characterization of the parameters of the L-shapes associated with a numerical 3-semigroup in terms of its generators, and we use it to classify the numerical 3-semigroups of interest in curve singularity.     

Controlling uncertain Lü system using backstepping design

In this paper, we discussed how to control Lü system with unknown parameters. Firstly we designed an observer to identify the unknown parameter of Lü system, then we used backstepping design method to control the system, and track any desired trajectory by the same way. At the same time we gave the numerical simulation for the results we had gained.     

Classification of quintic eutactic forms

From the classical Voronoi algorithm, we derive an algorithm to classify quadratic positive definite forms by their minimal vectors; we define some new invariants for a class, for which several conjectures are proposed. Applying the algorithm to dimension 5 we obtain the table of the 136 classes in this dimension, we enumerate the 118 eutactic quintic forms, and we verify the Ash formula.

     

Agent-based interpretations of classic network models

The paper is aimed at developing agent-based variants of traditional network models that make full use of concurrency. First, we review some classic models of the static structure of complex networks with the objective of developing agent-based models suited for simulating a large-scale, technology-enabled social network. Secondly, we outline the basic properties that characterize such networks. Then, we briefly discuss some classic network models and the properties of the networks they generate. Finally, we discuss how such models can be converted into agent-based models (i) to be executed more easily in heavily concurrent environments and (ii) to serve as basic blocks for more complex agent-based models. We evidence that many implicit assumptions made by traditional models regarding their execution environment are too expensive or outright impossible to maintain in concurrent environments. Consequently, we present the concurrency issues resulting from the violation of such assumptions. Then, we experimentally show that, under reasonable hypothesis, the agent-based variants maintain the main features of the classic models, notwithstanding the change of environment. Eventually, we present a meta-model that we singled out from the individual classic models and that we used to simplify the agent-oriented conversion of the traditional models. Finally, we discuss the software tools that we built to run the agent-based simulations.     

On Multiplicity-Free Skew Characters and the Schubert Calculus

In this paper we introduce a partial order on the set of skew characters of the symmetric group which we use to classify the multiplicity-free skew characters. Furthermore, we give a short and easy proof that the Schubert calculus is equivalent to that of skew characters in the following sense: If we decompose the product of two Schubert classes we get the same as if we decompose a skew character and replace the irreducible characters by Schubert classes of the ‘inverse’ partitions (Theorem 4.3).     

Rational Solutions to the KdV Equation in Terms of Particular Polynomials

Here, we construct rational solutions to the KdV equation by particular polynomials. We get the solutions in terms of determinants of the order $n$ for any positive integer $n$, and we call these solutions, solutions of the order $n$. Therefore, we obtain a very efficient method to get rational solutions to the KdV equation, and we can construct explicit solutions very easily. In the following, we present some solutions until order $10$.     

Local invertibility of adapted shifts on Wiener space,under finite energy condition

In this paper we study the connection between local invertibility and global invertibility of adapted shifts on Wiener space. First we go from the global to the local and we obtain an explicit formula for the inverse of a stopped adapted shift which is invertible. Then we take the opposite direction and we show that under finite energy condition, a shift which is locally invertible is also invertible. We work in a general framework which applies both to the classical Wiener space and to the Brownian sheet.     

Discontinuous local semiflows for Kurzweil equations leading to LaSalle's invariance principle for differential systems with impulses at variable times

In this paper, we consider an initial value problem for a class of generalized ODEs, also known as Kurzweil equations, and we prove the existence of a local semidynamical system there. Under certain perturbation conditions, we also show that this class of generalized ODEs admits a discontinuous semiflow which we shall refer to as an impulsive semidynamical system. As a consequence, we obtain LaSalle's invariance principle for such a class of generalized ODEs. Due to the importance of LaSalle's invariance principle in studying stability of differential systems, we include an application to autonomous ordinary differential systems with impulse action at variable times.     

The unknown in decision making: What to do about it

The unknown or “other” that affects our lives is what we usually very much want to know about to cope with uncertainty. We often suspect that it affects us with partial and indefinite evidence that it exists but we only have uncertain feelings about it. Even when we do not know what it is we would like to allow for its influence in our explaining the outcome of a decision. One way to deal with the many factors of a decision is to include the unknown as one of them and then determine its priority of influence on the outcome by comparing it with other factors. We are able to do that to the extent that we are sure of what we know and of the residual that remains outside our understanding that may also have some effect on what we do. Confidence from good understanding and past success are what we need in order to judge the potential significance of what we do not know on the outcome. We can then perform sensitivity analysis to see how much effect unknown factors can have on the stability of the choice we make. Pairwise comparisons make it possible to tackle this idea explicitly and rather simply. This note illustrates how to prioritize and test the effect of the unknown alongside the known.     

Proportionally Modular Diophantine Inequalities and Full Semigroups

A proportionally modular numerical semigroup is the set of nonnegative integer solutions to a Diophantine inequality of the type ax mod b ≤ cx. We give a new presentation for these semigroups and we relate them with a type of affine full semigroups. Next, we describe explicitly the minimal generating system for the affine full semigroups we are considering. As a consequence, we obtain generating systems for proportionally modular numerical semigroups and we exhibit several families of these semigroups in terms of their generators. Finally, we use the concept of fundamental gap to study when a proportionally modular numerical semigroup is symmetric and we propose some open problems.     

Exponentially Small Splitting for the Pendulum: A Classical Problem Revisited

In this paper, we study the classical problem of the exponentially small splitting of separatrices of the rapidly forced pendulum. Firstly, we give an asymptotic formula for the distance between the perturbed invariant manifolds in the so-called singular case and we compare it with the prediction of Melnikov theory. Secondly, we give exponentially small upper bounds in some cases in which the perturbation is bigger than in the singular case and we give some heuristic ideas how to obtain an asymptotic formula for these cases. Finally, we study how the splitting of separatrices behaves when the parameters are close to a codimension-2 bifurcation point.     

图论与矩形对角线法在钻井布局优化设计中的应用

本对钻井布局问题的研究,首先给出图论模型对问题1得出最多可利用4口旧井,井号为2、4、5、10。利用矩形对角线法对问题2得出最多可利用6口旧井,井号为1、6、7、8、9、11。同时利用矩形对角线法给出判定这些井均可利用的条件和算法。     

Forward dynamic utility functions: A new model and new results

A major obstacle in the existing models of forward dynamic utilities and investment performance evaluation is to establish the existence and uniqueness of the optimal solutions. Consequently, we present a new model of forward dynamic utilities. In doing so, we establish the existence and uniqueness of the solutions for a general (smooth) utility function, and we show that the assumptions needed for such solutions are similar to those under the backward formulation. Moreover, we provide unique viscosity solutions. We also provide discontinuous viscosity solutions. In addition, we introduce Hausdorff-continuous viscosity solutions to the portfolio model.     

Cache Oblivious Algorithms for the RMQ and the RMSQ Problems

In the Range Minimum/Maximum Query (RMQ) and Range Maximum-Sum Segment Query (RMSQ) problems, we are given an array which we can preprocess in order to answer subsequent queries. In the RMQ query, we are given a range on the array and we need to find the maximum/minimum element within that range. On the other hand, in RMSQ query, we need to return the segment within the given query range that gives the maximum sum. In this paper, we present cache oblivious optimal algorithms for both of the above problems. In particular, for both the problems, we have presented linear time data structures having optimal cache miss. The data structures can answer the corresponding queries in constant time with constant cache miss.     

A conjecture on bipartite graphical regular representations

In this paper we are concerned with the classification of the finite groups admitting a bipartite DRR and a bipartite GRR.First, we find a natural obstruction that prevents a finite group from admitting a bipartite GRR. Then we give a complete classification of the finite groups satisfying this natural obstruction and hence not admitting a bipartite GRR. Based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite GRR.Next, we prove the existence of bipartite DRRs for most of the finite groups not admitting a bipartite GRR found in this paper. Actually, we prove a much stronger result: we give an asymptotic enumeration of the bipartite DRRs over these groups. Again, based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite DRR.