Multiplier Function Research Articles

Termination proofs for term rewriting systems by lexicographic path orderings imply multiply recursive derivation lengths

It is shown that a termination proof for a term rewriting system using a lexicographic path ordering yields a multiply recursive bound on the length of derivations, measured in the depth of the starting term. This result is essentially optimal since for every multiply recursive function ƒ a rewrite system (which reduces under the lexicographic path ordering) can be found such that its derivation length cannot be bounded by ƒ.

ON ENTROPY-BASED METHODS FOR NONLINEAR PROGRAMMING PROBLEMS

Abstract This paper presents further theoretical analysis and developments of entropy-based methods for mathematical programming. A new generalized multiplier function is given which approximates uniformly the maximum function, and from which the aggregate function is derived in a different way. Some new methods for minimax and constrained nonlinear programming problems are proposed. The convergence properties for these methods are discussed.

Resolving power of a discrete electrode detector array

The resolving power of a new high resolution one-dimensional ion detector array has been studied as a function of the microchannel plate electron multiplier (MCP) voltage and the detector pulse height discrimination level. Experimental data have been analysed to give a simple representation of both the resolving power and the region of stable operation of the array. The conditions necessary for high resolution and stable operation are explained and the implications of the results for future design of high resolution arrays are outlined.

Some Geometric Properties of Spaces Associated with Multiple Stable Integrals

We investigate properties of vector lattices of multiply integrable functions with respect to a symmetric stable process.

Some geometric properties of spaces associated with multiple stable integrals

We investigate properties of vector lattices of multiply integrable functions with respect to a symmetric stable process.

Differential dynamic programming applied to continuous optimal control problems with state variable inequality constraints

The purpose of this article is to show that the differential dynamic programming (DDP) algorithm may be readily adapted to cater for state inequality constrained continuous optimal control problems. In particular, a new approach using a multiplier penalty function scheme incorporated with the DDP algorithm is shown to be effective. The DDP algorithm, implemented in conjunction with a multiplier penalty function scheme, is compared to an established DDP algorithm variant and the gradient-restoration method.

A switched gate synaptic weighting circuit for optically coupled neural networks

This paper presents results obtained with the use of a switched gate circuit realization of the multiplier function for an optically coupled synaptic weighting element. The results include both simulation and experimental measurements. The optically coupled synapse circuit employs a photosensitive element to input weighting coefficients to the neural network.

Optimal control of the feed voltage of a dipole antenna

An iterative algorithm is described for the construction of the input voltage of a dipole antenna for a specified electric field response at a given far-zone point. A functional is optimized subject to the state equation, which is an ordinary differential equation. The state equation used is Pocklington's equation where the control is the input voltage waveform and the state is the current distribution on the antenna. The Lagrange multiplier function, which is introduced by augmenting the functional, satisfies the adjoint Pocklington's equation. In the descent algorithm these two equations are solved at each step. All calculations are carried out in the time domain, so there is no need for additional specification related to frequency domain. However, the procedure implicitly determines the frequency bandwidth of the signals used. >

MODELING THE INCREMENTAL COST AND WATER CONVERSION FUNCTIONS FOR HYDRO-THERMAL COORDINATION STUDIES

ABSTRACT The solution of an optimization problem using variational means relies on Lagrange multiplier functions, called the adjoint variables to augment the physical constraints to the cost function. The multipliers usually have an economic interpretation, and in the optimal economic operation of an electric power system, one set of multipliers signifies the incremental cost of power delivered at a bus, and the other set relates to the water worth at a given reservoir. The solution of the optimization problem involves iterative techniques to obtain the optimal strategy in terms of power generations and water releases, as well as the adjoint variables. Good initial estimates are of paramount importance to a successful implementation of any iteration scheme specially those that are Newton based. There is an observed pattern of dependence of the adjoint variables and the system state as described by power demands and water availability. Finding such relationships is useful for generation planning activities...

Application of the calculus of variations to financing alternatives

The problem considered is to choose from a finite set of inter-related investment and financing alternatives and also levels of consumption/work over time in order to maximise a utility functional. Each investment and financing option is characterised by its cash flow over time. An inter-temporal budget requirement operates continuously. Application of Calculus of Variations leads to consideration of the Euler-Lagrange equations combined with Kulm-Tucker conditions. It is shown that the solution (also when there are logical dependencies present) requires the maximisation of a Generalised Net Present Wealth measure in which the discount factor is formed from an integral of the Lagrangean multiplier function.

Analogue computation using VLSI neural network devices

Results are presented demonstrating analogue neural network computation using VLSI pulse-stream devices. Synapse weight storage is dynamic, and the multiply and add functions are performed by a switched capacitor circuit. Used on a mobile robot localisation task, the neural VLSI devices achieve results within 1.2% of those of a SUN workstation.

Stability of efficiency evaluations in data envelopment analysis

Efficiency evaluations in data envelopment analysis are shown to be stable for arbitrary perturbations in the convex hulls of input and output data. Also, the corresponding restricted Lagrange multiplier functions are shown to be continuous. The results are proved using point-to-set mappings and a particular region of stability from input optimization.

Why Does the Brain Have So Many Visual Areas?

Mammals vary in number of visual areas from a few to 20 or more as a result of new visual areas being added to the middle levels of processing hierarchies. Having more visual areas probably increases visual abilities, perhaps in part by allowing more stimulus parameters to be considered. Proposals that each visual area computes and thereby "detects" a specific stimulus attribute have so far dealt with attributes that most mammals can detect and thus do not relate to the issue of species differences in numbers of areas. The problem of forming and maintaining complex patterns of interconnections between many different sets of distinct processing models within an area may limit multiplying functions within a field. In addition, the adding of new visual areas is a way of avoiding constraints on modifying existing visual areas that are imposed by the ongoing functional requirements. Thus, increasing the number of visual or other cortical areas is an effective and apparently common mechanism for evolving new capacities.

Proportional material damping in finite element impact analysis

A method is presented to implement proportional material damping into the combined Lagrangian multiplier and penalty function finite element technique for impact problems. The discrete impact and release conditions are modified to include extra structural damping terms. A two-dimensional analysis for the normal impact of a ring and a simulated spring is solved to illustrate the use of proportional material damping.

Nonuniqueness of families of periodic solutions in a four dimensional mapping

If we follow a family of periodic solutions along a closed path in a parameter space of two dimensions we may not return to the original solution when the parameters return to the original values. We study such nonuniqueness phenomena in simple and double period familis. Nonuniqueness appears if a closed path in the parameter space goes around a critical point. In some cases we find Riemann sheets in the same way as in multiply valued functions. In other cases the connections of various families change in a complicated way around the critical point. All these phenomena are explained analytically. At the critical point there is a ‘collision of bifurcations’. The changes of the connections of various families at such collisions of bifurcations are studied in some detail.

Riesz-Thorin theorem and l p-stability of nonlinear time-varying discrete systems

The interpolation theorem due to Riesz-Thorin is used along with Hölder's and Young's inequalities to derive some new conditions, more general than those in the literature, for the l p -stability (1 ⩽ p ⩽ ∞) of a class of nonlinear time-varying discrete systems represented by a time invariant linear discrete part G in feedback with a discrete nonlinear time-varying gain k( n) ϑ(·). These stability conditions are expressed in terms of a general multiplier (causal + anticausal) function and global upper and lower bounds on the normalized rate of growth, ( k(n + 1) k(n) ) , of the time-varying gain.

Combined Lagrangian multiplier and penalty function finite element technique for elastic impact analysis

A new finite element procedure which combines the Lagrangian multiplier method and the penalty function method for dynamic contact analysis with friction is presented in this paper. As a result of this combination, the whole contact system can be analyzed as one body in both contact and separation states. No displacement constraints are required on the striker and no extra stress due to gap elements is created. Coulomb's friction law is employed to set up sliding and sticking conditions and discrete impact and release conditions are implemented. A two-dimensional analysis for the longitudinal impact of two rods is solved as a numerical example. The average values of stress and the time durations are very close to the theoretical solution.

Converse hölder inequality and the Lp-instability of nonlinear time-varying feedback systems

WE DEAL with the Lp-instability of a nonlinear time-varying feedback system governed by the pair of equations: v(t) = x(t) - k(t)cp (y(t)),y(t) = (qJv)(t) (1) := ~ g i v ( t - ri) + g( O v ( t - r) dr i=1 for all t I> 0, where x, v, and y are respectively the input to the system, the error signal and output of the system; N is a time-invariant linear operator; qv is a first and third quadrant continuous, memoryless (monotone) nonlinearity; and k is a time-varying gain. For assumptions on the components of (1), see Section 2 below. We derive Lp-instability conditions in terms of the frequency response of ~3 and a general causal + anticausal multiplier function. The derivation is based on the converse H61der inequality and the energy balance argument as used in [1].The problem of instability of feedback systems with a single time-varying nonlinearity was initially considered by Brockett and Lee [2] who, in a Lyapunov-Chetaev setting, derived an instability version of the circle criterion [3, 4] under certain assumptions on the related linear time-invariant systemwith a constant gain in the feedback loop. It is found that this instability theorem is conservative [5, Section 5]. See [5] for references to other instability results. As far as the analysis of instability of feedback systems by functional methods is concerned,different types of results are available. Willems [6] extends the domain of operators and imbedsthe system causal operator in a noncausal operator in an attempt to prove the noncausality of the inverse of the original operator by contradiction. This technique has been explicitly used [7] but there do exist certain unresolvable difficulties [8; 9, Chapter 7]. Similar difficulties are encountered in the results of Takeda and Bergen [10, 11] and Steding and Bergen [12], who consider a subclass of inputs over which the linear time-invariant part of the system is well- behaved (and satisfies some additional conditions), and prove instability by contradiction. See [9, Chapter 7] for a complete analysis of these contributions in which, more importantly, the assumptions made on the linear time-invariant part of the system are too severe.

Necessity conditions for $L\sp p$ multipliers with power weights

It is shown that if multiplier operators are bounded on with weight ${\left | x \right |^\alpha }$ for all functions in the space ${\mathcal {S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0 and all multiplier functions in a standard Hörmander type multiplier class, then $\alpha$ must satisfy certain inequalities. This is a sequel to a previous paper in which conditions on $\alpha$ that were almost the same were shown to be sufficient for the norm inequality to hold.

Riesz-thorin and thelpstability of nonlinear time varying feedback systems

The paper is concerned with the Lpstability of a class of nonlinear time varying feedback systems, with a linear time invariant stable part g in feedback with a nonlinear time varying gain k(t)y(.). The Riesz - Thorin interpolation theorem is employed in a novel way to derive the Lp-stability conditions (for 1≤p≤∞) which are believed to be more general and less restrictive than the results of the literature. These conditions involve a (cau–sal + anti–causal) multiplier function and global conditions on the normalized rate of variation of k(t)