Explicit Parametric Form Research Articles

On the Structure of the Bursting Region for a Neuronal Model

We examine the question of whether the bursting region of a neuronal model is simply or multiply connected in the parameter space. Taking into account that the presence of slow oscillations of intracellular calcium concentration is a necessary condition for a bursting mode, we analyze the subysystem responsible for the slow component of bursting. Explicit parametric form of the Hopf bifurcation is given. It enables us to reduce the problem to determining the number of the bifurcation curves by finding the number of discontinuities of the expressions giving the bifurcation parameter values. The results obtained suggest that the bursting region is simply connected over the physiologically possible parameter range.

Modifications of the EM algorithm for survival influenced by an unobserved stochastic process

Let Y=( Y t ) t ≥0) be an unobserved random process which influences the distribution of a random variable T which can be interpreted as the time to failure. When a conditional hazard rate corresponding to T is a quadratic function of covariates, Y, the marginal survival function may be represented by the first two moments of the conditional distribution of Y among survivors. Such a representation may not have an explicit parametric form. This makes it difficult to use standard maximum likelihood procedures to estimate parameters - especially for censored survival data. In this paper a generalization of the EM algorithm for survival problems with unobserved, stochastically changing covariates is suggested. It is shown that, for a general model of the stochastic failure model, the smoothing estimates of the first two moments of Y are of a specific form which facilitates the EM type calculations. Properties of the algorithm are discussed.

Equilibrium polymerization of chains and rings: A bicritical phenomenon.

The equilibrium polymerization of chains and rings together is a ($n=1$) critical phenomenon with aspects of bicriticality, governed by exponents not found in any symmetric $\mathrm{O}(n)$ model. The bicritical nature of the critical point is the result of a competition between a transition to form long-chain polymers and a transition to form an infinite-rings condensate. As a result the fraction of monomers incorporated in chains and in rings varies with temperature according to a different power law than that for the total fraction of polymerized material and is governed by an exponent that depends on the crossover exponent $\ensuremath{\varphi}$ for quadratic anisotropy for the $\mathrm{O}(n)$ vector model of magnetism in the limit $n\ensuremath{\rightarrow}1$. Moreover, the fraction of monomers in polymeric rings is found to decrease with temperature with infinite slope just beyond the transition in the polymerized phase. Similarly, the geometry of large polymers is described by a power law $R\ensuremath{\sim}{N}^{\frac{\ensuremath{\nu}}{\ensuremath{\varphi}}}$, where $R$ is the distance spanned by a polymer segment of $N$ monomers and where the exponent is not the de Gennes result $\ensuremath{\nu}(n=0)$ for chains alone, but rather also depends on the crossover exponent $\ensuremath{\varphi}$ for $n=1$. We study the equilibrium polymerization of chains and rings together using a variety of techniques, including simple equilibrium theory, a lattice model, field-theoretic correspondence to magnetism and a direct renormalization-group calculation on a polymer model. An explicit parametric form for the equation of state is presented to lowest order in $\ensuremath{\epsilon}$. Results are obtained for the fraction of polymeric material in chains and rings and for various correlation functions of interest in considering the geometry and distribution of chains and rings.

Dead-beat control of linear discrete-time systems

The paper considers the problem of determining the various classes of state-feed back dead-beat controllers for linear multivariable discrete-time systems. Unlike earlier work, attention is not restricted to the special case of minimum-time dead-beat controllers. A general dead-beat control problem is posed and solved. The solution is based principally on an approach of closed-loop eigenstructure assignment. The minimum of free parameters in each class are specified and used in an explicit parametric form for the feedback gain matrix. The concept of the greatest minimum of free parameters is introduced. Finally, the method is illustrated by a numerical example

Multivariable closed-loop deadbeat control: a polynomial-matrix approach

The closed-loop deadbeat servo problem (CDSP), considered in this paper, consists of the synthesis of a linear, output feedback controller such that the control signal and tracking error both vanish, after a finite period of time, for every reference sequence from a prespecified class and for every initial state of a plant and the controller. The closed-loop structure is determined by studying necessary and sufficient conditions for deadbeat tracking performance. A new theorem asserts that if an open-loop deadbeat control strategy exists for every initial state of the plant and every reference function from a given class, then CDSP is solvable and all desired control laws are found in an explicit parametric form by solving simple, unilateral, linear equations in polynomial matrices. On the basis of this theorem a design algorithm is developed. Asymptotic stability of the closed-loop system exhibiting deadbeat properties is demonstrated. A numerical example is given to illustrate the usefulness and computational efficiency of the new design algorithm presented.

On eigenstructure assignment in linear multivariable systems

The eigenvalue-assignment approach of Brogan [1], [2] is generalized and extended to the assignment of the entire closed-loop eigenstructure of linear multivariable systems. The set of assignable eigenvectors and generalized eigenvectors emerges naturally in the solution and is given, moreover, in an explicit parametric form. Two numerical examples are worked out to demonstrate the application of the procedure.