Norman Shapiro

ON THE CONTINUITY OF THE MINIMUM SET OF A CONTINUOUS FUNCTION

Abstract : This memorandum obtains necessary and sufficient conditions so that the solution of a constrained minimization problem will vary continuously when the constraints and objective function are varied. It also obtains special results in the case that the constraints are linear inequalities.

Values of Large Games, I: A Limit Theorem

This paper gives an analysis of what happens in a weighted majority game when a block of votes is broken up and distributed among a large number of players. It is shown that the value of the game to the other players converges to a limit as the size of the largest fragment tends to zero. An explicit expression is given for the limit.

Implicit function theorems for mathematical programming and for systems of inequalities

Implicit function formulas for differentiating the solutions of mathematical programming problems satisfying the conditions of the Kuhn—Tucker theorem are motivated and rigorously demonstrated. The special case of a convex objective function with linear constraints is also treated with emphasis on computational details. An example, an application to chemical equililibrium problems, is given.Implicit function formulas for differentiating the unique solution of a system of simultaneous inequalities are also derived.

Privacy and security in databank systems: measures of effectiveness, costs, and protector-intruder interactions

The nearly seven years of concern with data privacy and security in computerized information systems have produced a variety of hardware and software techniques for protecting sensitive information against unauthorized access or modification. However, systematic procedures for cost-effective implementation of these safeguards are still lacking.

Some security principles and their application to computer security

objects. One may hide money, or encrypt data. The protection mechanisms themselves may be concealed or kept secret. The object here is to keep information needed by a violator from him, so that he will not know all he needs to know to mount an attack. In summary, we have introduced the notions of a barrier, detection, concealment, and a guard to achieve security, and mentioned the ancillary notions of identification, apprehension and counterforce. There are a wide variety of techniques and mechanisms which embody some or all of these principles. Furthermore, in many systems a degree of security can be achieved by using several mechanisms, which provides much greater security than that provided by the individual mechanisms used alone. One of the best examples is that of a safe in an area that is under surveillance. The combination of surveillance and a safe provides a much greater degree of security than either alone would provide. The opportunity for detection is greatly enhanced by the presence of the safe since the time it takes to break into the safe increases the opportunity for detection. On the other hand, the fact that the safe is, or may be, under surveillance limits the attacks that can be mounted on it since the violator no longer has undisturbed access to it. People are often an integral part of the protection mechanisms in a security system. There is a class of vulnerabilities associated with people; they are subject to physical attack by the violator, and they can be subverted or deceived. In addition, a person who is involved in security can himself become a violator of the system. Special precautions are often taken to insure that the people in the system will function correctly. A particularly good example of this is the design of systems so that for critical elements two people must be involved in any attempt to defeat the system. Some vaults, for instance, require both a key and a combination, both of which are not possessed by the same individual. It is intended that a conspiracy be required to defeat the system, on the grounds that a conspiracy is much less likely than an attempted violation on the part of a single individual. It is important to consider security from the point of view of the potential violator. He may seek to obtain information of value to him or to modify information that somebody else will use because there is some expected value to him as a consequence of the modificaton. He may be dissuaded from doing so because he estimates that the costs are unacceptable. The first cost is the direct cost in time, effort, and money of carrying out his plans. Both strong protection mechanisms and concealment mechanisms, such as cryptography, may impose unacceptable costs in one or more of these measures. In addition, detection and apprehension may have costs associated with them that are uncertain to the violator but whose deterrence value may be substantial. The violator may be deterred by the social stigma associated with detection or by the penalties which may follow as a consequence of detection. Detection may occur while the violation of security is in progress or afterwards. If detection does occur, it can cause the violator to fail to achieve his objective even if he is not identified. Because the penalties we mentioned above occur only if the violator is correctly identified,

Chemical Equilibrium Problems with Unbounded Constraint Sets

Abstract : An investigation of the use of mathematical models to explore the chemical aspects of physiological systems; this deals with the theoretical and computational aspects of understanding the chemistry of human physiological function. The question of existence of solutions to problems having unbounded constraint sets is investigated by relating their existence (or nonexistence) to a property of a solution to an auxiliary chemical equilibrium problem with a bounded constraint set. An example system is selected consisting of gases in contact with an aqueous buffer solution at a uniform total hydrostatic pressure and temperature. The numerical problem of determining the amount of CO2 to be added to achieve a specified partial pressure of CO2 in the gas phase, and its effects on the composition of the total system, is solved by using a procedure suggested by the concept of unbounded constraint sets. Findings may apply to design of artificial life-support systems needed in extraterrestrial environments related to Air Force missions.

A 'propagative' approach to sensitivity analysis

It is shown that the computational cost of traditional approaches to sensitivity analysis is logically unnecessary and can be largely avoided by propagating and combining sensitivities during a computation, rather than recomputing them. This propagative approach to sensitivity analysis is described and the algorithm implemented to explore its potential is presented. Initial results indicate that this approach has tremendous potential, reducing a combinatorial process to a linear one. In addition, it is noted that the approach has implications beyond sensitivity analysis: it suggests a novel computational paradigm in which functions replace themselves by approximations when they are first called and these approximations are used for the remainder of a computation, e.g. to improve performance. Sensitivity analysis is simply one instance of this approach, using linear approximations based on partial derivatives; however, the approach and the computational environment implemented allow arbitrary approximations to be used.<<ETX>>

A METHOD FOR SOLVING CERTAIN STIFF DIFFERENTIAL EQUATIONS

Certain differential equations that arise when solving chemical kinetics problems which have widely differing time constants are analyzed by a method that implicitly separates the fast reacting components from the remaining components of the system. A method for determining equilibrium of the fast reacting components is provided, and these values provide the initial conditions for an associated system of differential equations that yields a first and second order solution of the original system. Computational experiments have shown that a suitably chosen method for solving the modified system may be superior to a direct solution with a stiff differential equation solver.

ON MEMBRANE EQUILIBRIA

Abstract : The study mathematically investigates equilibrium properties of certain two-phase chemical systems. The different types of equilibrium compositions and the conditions that govern them are determined. The effects of linear variation in the chemical inputs are studied; e.g., the effects on a multisubstance fluid, separated by a semipermeable membrane, to which another fluid is slowly added. Among other results, it is shown that the relative sizes of the two phases will always change monotonically, but that the absolute sizes may fluctuate.

Approximating One Convex Function by Another

Abstract : The report describes a simple iterative procedure for approximating one convex function, relative to a given constraint set, by another convex function, having the same constraint set, plus an appropriate linear function. This procedure is particularly useful when efficient digital computer programs are already available for minimizing functions that differ from some other convex function by a linear function. A theorem is presented that gives sufficient conditions for such a procedure to succeed. (Author)