Fixed-point Model Research Articles

Tarskian and Kripkean truth

A theory of the transfinite Tarskian hierarchy of languages is outlined and compared to a notion of partial truth by Kripke. It is shown that the hierarchy can be embedded into Kripke’s minimal fixed point model. From this results on the expressive power of both approaches are obtained.

Bottom-up evaluation of datalog with negation

Declarative semantics gives the meaning of a logic program in terms of properties, while the procedural semantics gives the meaning in terms of the execution or evaluation of the program. From the database point of view, the procedural semantics of the program is equally important. This paper focuses on the study of the bottom-up evaluation of the WFM semantics ofdatalog − programs. To compute the WFM, first, the stability transformation is revisited, and a new operatorO P and its fixpoint are defined. Based on this, a fixpoint semantics, called oscillating fixpoint model semantics, is defined. Then, it is shown that for anydatalog − program the oscillating fixpoint model is identical to its WFM. So, the oscillating fixpoint model can be viewed as an alternative (constructive) definition of WFM. The underlying operation (or transformation) for reaching the oscillating fixpoint provides a potential of bottom-up evaluation. For the sake of computational feasibility, the strongly range-restricted program is considered, and an algorithm used to compute the oscillating fixpoint is described.

A renormalization group scaling analysis for compressible two-phase flow

Computational solutions to the Rayleigh–Taylor fluid mixing problem, as modeled by the two-fluid two-dimensional Euler equations, are presented. Data from these solutions are analyzed from the point of view of Reynolds averaged equations, using scaling laws derived from a renormalization group analysis. The computations, carried out with the front tracking method on an Intel iPSC/860, are highly resolved and statistical convergence of ensemble averages is achieved. The computations are consistent with the experimentally observed growth rates for nearly incompressible flows. The dynamics of the interior portion of the mixing zone is simplified by the use of scaling variables. The size of the mixing zone suggests fixed-point behavior. The profile of statistical quantities within the mixing zone exhibit self-similarity under fixed-point scaling to a limited degree. The effect of compressibility is also examined. It is found that, for even moderate compressibility, the growth rates fail to satisfy universal scaling, and moreover, increase significantly with increasing compressibility. The growth rates predicted from a renormalization group fixed-point model are in a reasonable agreement with the results of the exact numerical simulations, even for flows outside of the incompressible limit.

Loss circuit switched communication network-performance analysis and dynamic routing

We investigate a loss circuit switched communication network with state-dependent dynamic routing strategy, wherein the state of the network at the time of call arrival determines whether or not the call is accepted and, if accepted, its route. We develop an approximate approach to the network performance analysis. The approach enhances the Fixed Point Model by treating multiple solutions of the Fixed Point Equations. Assuming that the multiple solutions correspond to the long-living network modes, we develop the aggregated Markov chain that describes the network transitions among these modes. We also propose and discuss a new state-dependent dynamic routing strategy which we call Least-Expected-Blocking strategy (LEB). LEB accepts an incoming call only if this results in a decrease in the expected blocking probability and it chooses a route that yields the maximum decrease. The new strategy outperforms the previously known strategies by the criterion of network steady blocking probability.

State-dependent routing on symmetric loss networks with trunk reservations. I

The aggregated-least-busy-alternative (ALBA), a distributed, state-dependent, dynamic routing strategy for circuit-switched loss networks is discussed. The networks considered are symmetric and fully connected. The offered calls form Poisson streams, and routes have at most two links. In ALBA(K), the states of each link are lumped into K (K>or=2) aggregates, and the route of each call is determined by local information on the aggregate states of the links of the alternate routes at the time of the call's arrival. The last aggregate is always the set of states reserved for direct traffic. A fixed-point model for ALBA(K) for general K is presented. The particular case of ALBA in which there is no aggregation is least busy alternative (LBA); ALBA(2) represents the other extreme of aggregation. Simulation and analytic results for LBA are compared. An asymptotic scaling based on the fixed-point models is also discussed. It is shown that there is a dichotomy in network behavior: if the offered traffic is below a threshold, then the network loss probability decreases exponentially with increasing network size, and above the threshold, performance is poor. >

A note on the existence and characteristics of fair deposit insurance premia

This paper treats the fair deposit insurance premium as a fixed point of the value of insurance per dollar of deposits. Using the standard model of the value of deposit insurance and treating the premium as an up-front cost to a bank it is shown that the fixed-point premium exists and is unique under fairly general conditions. It is shown that ignoring the premium as an up-front cost may lead to underestimation of the fair premium. In addition, the fixed-point model suggests that premium rates should vary with the ratio of deposits to total liabilities.

State-dependent routing on symmetric loss networks with trunk reservations. II: Asymptotics, optimal design

We investigate a distributed, state-dependent, dynamic routing strategy for circuit-switched loss networks which we have called Aggregated-Least-Busy-Alternative (ALBA). The networks considered are symmetric and fully connected, the offered calls form Poisson streams and routes have at most two links. In ALBA(K), the states of each link are lumped intoK (K ≥ 2) aggregates and the route of each call is determined by local information on the aggregate-states of the links of the alternate routes. The last aggregate is always the set of states reserved for direct traffic and defined by the trunk reservation parameter. The particular case of ALBA in which there is no aggregation is Least-Busy-Alternative (LBA); ALBA(2) represents the other extreme of aggregation. We consider two separate asymptotic scalings based on Fixed Point Models for ALBA(K) which were obtained and investigated in an earlier paper. In the first, it is assumed that the number of network nodes, the offered traffic and trunk group size are all large; their ratios have been chosen to reflect practical interest. The results show that there exists a threshold which delineates fundamentally different behavior: for offered traffic below the threshold, the network loss probability decreases exponentially with increasing network size, while above the threshold the decrease is only polynomial. In the related second asymptotic scaling, the asymptotically optimum trunk reservation parameter is obtained as the solution of a simple equation. Such asymptotically optimal designs are compared to the outputs of exhaustive numerical searches for some realistically sized networks and found to perform very well.

Fixed point model for adaptive token passing bus protocol

The performance of an adaptive token passing bus protocol is analysed by modelling the behaviour of each node as an M/GI/1 queue and deriving a set of fixed point equations. The technique is shown to be applicable to the analysis of a number of variants of the token passing bus protocol which arise in communication networks and computer integrated manufacturing systems. The validity of the independence assumption underlying this approach is investigated by first examining the autocorrelation function of the simulated time series of token passing cycles. This indicates that cycles are not highly correlated except at high traffic loads. Mean message transmission times predicted by the models are compared with the corresponding results of simulation experiments and the performance of the adaptive protocol is compared quantitatively with that of the standard protocol.

Validation of the chaotic mixing renormalization group fixed point

The statistical behavior for chaotic mixing of Rayleigh-Taylor unstable interfaces is characterized by the merging of small bubbles (the portions of a light fluid adjacent to a heavy fluid) into large bubbles. The length scales change dynamically as a consequence of bubble merger. We study such statistical behavior quantitatively using a renormalization group fixed point model. Our study shows good agreement among the results of the renormalization group model, experimental data and direct numerical simulation of the two-fluid Euler equations.

Fixed point models of loss networks

AbstractIn this paper we review a simple class of fixed point models for loss networks. We illustrate how these models can readily deal with heterogeneous call types and with simple dynamic routing strategies, and we outline some of the recent mathematical advances in the study of such models. We describe how fixed point models lead to a natural and tractable definition of the implied cost of carrying a call, and how this concept is related to issues of routing and capacity expansion in loss networks.

A Fixed Point Characterization for Bias of Autoregressive Estimators

Least squares estimators of the coefficients of an autoregression of known, finite order are biased to order $1/T$, where $T$ is the sample length, unless the observed time series is generated by a unique model for that order. The coefficients of this special model are the point of a linear mapping defined by the bias of the least squares estimator. Separate results are given for with known mean and unknown mean. The fixed point models for different orders of autoregression are least squares approximations to an infinite-order autoregression which is unique but for arbitrary scaling. Explicit expressions are given for the coefficients of the point at each order. The autocorrelation function and spectral density of the underlying infinite-order process are also presented. Numerical calculations suggest similar properties hold for Yule-Walker estimators. Implications for bootstrapping autoregressive are discussed.

A MODEL FOR ELECTROSTATIC INTERACTIONS AMONG CHARGED SITES OF WATER-SOLUBLE, ORGANIC POLYIONS

A mathematical model for electrostatic interactions among the charged sites of a flexible polyion was developed from a fixed-point model of the spatial distribution of ionizable sites. Each polyion was represented by a series of segments, and each segment was assumed to contain an equal number of ionizable sites at equally spaced intervals. Derivation of the model was based on the assumption that significant interactions occur only within a segment and between adjacent segments. The flexible nature of the polyion was simulated by varying the value of the angle between adjacent segments, which was treated as a function of the average charge per segment as well as the ionic strength of the solution. A subroutine that calculated an energy of electrical interaction among the charged sites was incorporated into a “mixture model” (Sposito 1981) for the water-soluble organic compounds extracted from a sludge-soil mixture. The correction factor obtained from calculated electrostatic interactions increased the conditional stability constants of protonation for model ligands as the fraction of ionized ligands increased. The magnitude of change in the condition stability constant exhibited in a test of the model has implications with respect to the use of titration data for elucidating the chemical properties of polyions and to the development of “mixture models.”

Chaotic nature of the spin-glass phase.

The description of the spin-glass phase in terms of a $T=0$ fixed point is shown to imply a "chaotic" phase in which the relative orientations of spins with large separations $L\ensuremath{\gtrsim}{L}^{*}$ are sensitive to small changes $\ensuremath{\delta}T$ in the temperature or $\ensuremath{\delta}J$ in the bond strengths, where ${L}^{*}\ensuremath{\propto}\frac{1}{{(\ensuremath{\delta}T)}^{\frac{1}{\ensuremath{\zeta}}}}$ or $\frac{1}{{(\ensuremath{\delta}J)}^{\frac{1}{\ensuremath{\zeta}}}}$, respectively, $\ensuremath{\zeta}=\frac{{d}_{s}}{2\ensuremath{-}y}$, and $\ensuremath{-}y$ and ${d}_{S}$ are the thermal eigenvalue at the $T=0$ fixed point and the "interfacial (fractal) dimension," respectively.

Regular trees and the free iterative theory

This paper establishes the relationship between regular trees, equationally defined trees, and Elgot's iterative algebraic theories; in particular, it provides a regular tree characterization of the free iterative theory and an equational tree characterization of the free iterative theory. These tree characterizations have implications for decision problems involving flowchart schemes interpreted in iterative theories and they also help to explicate the connections between order theoretic and non-order theoretic fixed point models for iteration.

Theoretical note: A fixed-point model for short-term memory

Most current models for short-term memory assume at least two separate storage systems. A unitary model for paired associates that seems capable of accounting for at least the gross empirical phenomena is proposed. Transitions both into and out of the recall state are possible, and the fixed point of the transition matrix determines the asymptote of the short-term retention curve. Experimental evidence is presented to document the contention that a steady state is achieved by practiced subjects after a single presentation of a list of paired associates.