Rigorous Theorems Research Articles

Run-to-run product quality control of batch processes

Batch processes have been increasingly used in the production of low volume and high value added products. Consequently, optimization control in batch processes is crucial in order to derive the maximum benefit. In this paper, a run-to-run product quality control based on iterative learning optimization control is developed. Moreover, a rigorous theorem is proposed and proven in this paper, which states that the tracking error under the optimal iterative learning control (ILC) law can converge to zero. In this paper, a typical nonlinear batch continuous stirred tank reactor (CSTR) is considered, and the results show that the performance of trajectory tracking is gradually improved by the ILC.

Decomposing Neural Synchrony: Toward an Explanation for Near-Zero Phase-Lag in Cortical Oscillatory Networks

BackgroundSynchronized oscillation in cortical networks has been suggested as a mechanism for diverse functions ranging from perceptual binding to memory formation to sensorimotor integration. Concomitant with synchronization is the occurrence of near-zero phase-lag often observed between network components. Recent theories have considered the importance of this phenomenon in establishing an effective communication framework among neuronal ensembles.Methodology/Principal FindingsTwo factors, among possibly others, can be hypothesized to contribute to the near-zero phase-lag relationship: (1) positively correlated common input with no significant relative time delay and (2) bidirectional interaction. Thus far, no empirical test of these hypotheses has been possible for lack of means to tease apart the specific causes underlying the observed synchrony. In this work simulation examples were first used to illustrate the ideas. A quantitative method that decomposes the statistical interdependence between two cortical areas into a feed-forward, a feed-back and a common-input component was then introduced and applied to test the hypotheses on multichannel local field potential recordings from two behaving monkeys.Conclusion/SignificanceThe near-zero phase-lag phenomenon is important in the study of large-scale oscillatory networks. A rigorous mathematical theorem is used for the first time to empirically examine the factors that contribute to this phenomenon. Given the critical role that oscillatory activity is likely to play in the regulation of biological processes at all levels, the significance of the proposed method may extend beyond systems neuroscience, the level at which the present analysis is conceived and performed.

Analyzing safety properties of hybrid processing systems: A case study on an industrial evaporator

While current approaches for the safety verification (understood here as the verification that the trajectories of the system remain in a prescribed set in the state space) of hybrid systems yield rigorous proofs for system safety, their applicability is restricted to relatively small systems. In this paper, the safety properties of a logic-controlled industrial processing system with hybrid dynamics are investigated using two optimization-based approaches. In the first approach, the hybrid system is regarded as a black box (i.e. only the input–output behavior is considered), and optimization-based techniques are used to compute worst-case scenarios. The second approach employs a semi-analytical technique by combining rigorous theorem proving that is applied to the dynamic equations of the process with the computation of worst-case scenarios using global non-linear optimization techniques. It is shown that both approaches are able to determine safety-critical evolutions for a rigorous hybrid model of the industrial-scale evaporation system.

Existence Theorems For Tendon-Reinforced Thin Wrinkled Membranes Subjected to a Hydrostatic Pressure Load

In this paper, we establish rigorous existence theorems for a mathematical model of a tendon-reinforced thin wrinkled membrane that is subjected to a shape dependent hydrostatic pressure load. We are motivated by the problem of determining the equilibrium shape of a strained high altitude large scientific balloon. This problem has a number of unique features. The balloon is very thin (20-40 μm), especially when compared with its diameter (over 100 meters). The balloon is unable to support compressive stresses and, instead, wrinkles or forms folds of excess material. Our approach can be adapted to a wide variety of inflatable structures, but we will focus on two types of high altitude balloons, a zero-pressure natural shape balloon and a super-pressure pumpkin-shaped balloon. We outline the shape finding process for these two classes of balloon designs, formulate the problem of a strained balloon in an appropriate Sobolev space setting, establish rigorous existence theorems using direct methods in the calculus of variations, and present numerical studies to complement our theoretical results.

Optimal Iterative Learning Control for Batch Processes Based on Linear Time-varying Perturbation Model

A batch-to-batch optimal iterative learning control (ILC) strategy for the tracking control of product quality in batch processes is presented. The linear time-varying perturbation (LTVP) model is built for product quality around the nominal trajectories. To address problems of model-plant mismatches, model prediction errors in the previous batch run are added to the model predictions for the current batch run. Then tracking error transition models can be built, and the ILC law with direct error feedback is explicitly obtained. A rigorous theorem is proposed, to prove the convergence of tracking error under ILC. The proposed methodology is illustrated on a typical batch reactor and the results show that the performance of trajectory tracking is gradually improved by the ILC.

Problem of transport in billiards with infinite horizon

We consider particles transport in the Sinai billiard with infinite horizon. The simulation shows that the transport is superdiffusive in both continuous and discrete time. Also, it is shown that the moments do not converge to the Gaussian moments even in the logarithmically renormalized time scale, at least for a fairly long computational time. These results are discussed with respect to the existent rigorous theorems. Similar results are obtained for the stadium billiard.

Contractile stress generation by actomyosin gels

The tension generated by randomly distributed myosin minifilaments in an actin gel is evaluated using a rigorous theorem relating the surface forces acting on the gel to the forces exerted by the myosins. The maximum tension generated per myosin depends strongly on the lengths of the myosin minifilaments and the actin filaments. The result is used to place an upper bound on the tension that can be generated during cytokinesis. It is found that actomyosin contraction by itself generates too little force for ring contraction during cytokinesis unless the actin filaments are tightly crosslinked into inextensible units much longer than a single actin filament.

Heisenberg’s uncertainty relation may be violated in a single measurement

The well-known Heisenberg’s uncertainty relation is an inequality between uncertainties of canonically conjugate observables in a given state. In this interpretation, the Heisenberg’s uncertainty relation is a rigorous mathematical theorem and is, therefore, always valid. However, the same inequality is often applied in the situation of measurement, where it is illustrated in a quite dierent way. The uncertainty relation is then an inequality connecting the precision (resolution) of the measurement of one observable and the uncertainty of the conjugate observable in the state arising after the measurement. It turns out that in such an interpretation the Heisenberg’s inequality may be violated for some measurement readouts that emerge with small but finite probabilities. Making use of the uncertainties averaged in a special way over all possible measurement readouts, one may formulate an inequality of the type of Heisenberg’s inequality but valid for any measurement.

On Hilfer's objection to the fractional time diffusion equation

Hilfer [Physica A 329 (2003) 35] claims to give an example of a continuous time random walk (CTRW) model with long-tailed waiting time probability density that approaches a Gaussian behavior in the continuum limit. Rigorous limit theorems, derived previously, show however that in the limit of long-time such a CTRW converges to a non-Gaussian behavior. We discuss two types of continuum limits for the CTRW model: the fractional continuum limit and the one introduced by Hilfer. We show that the fractional limit yields the correct long-time behavior of the CTRW, while Hilfer's continuum limit does not. We discuss a general approach to find a continuum limit of the CTRW process.

Phenomenological analysis connecting proton–proton and antiproton–proton elastic scattering

Based on the behavior of the elastic scattering data, we introduce an almost model-independent parametrization for the imaginary part of the scattering amplitude, with the energy and momentum transfer dependences inferred on empirical basis and selected by rigorous theorems and bounds from axiomatic quantum field theory. The corresponding real part is analytically evaluated by means of dispersion relations, allowing connections between particle-particle and particle-antiparticle scattering. Simultaneous fits to proton-proton and antiproton-proton experimental data in the forward direction and also including data beyond the forward direction, lead to a predictive formalism in both energy and momentum transfer. We compare our extrapolations with predictions from some popular models and discuss the applicability of the results in the normalization of elastic rates that can be extracted from present and future accelerator experiments (Tevatron, RHIC and LHC).

Random Cluster Models on the Triangular Lattice

We study percolation and the random cluster model on the triangular lattice with 3-body interactions. Starting with percolation, we generalize the star–triangle transformation: We introduce a new parameter (the 3-body term) and identify configurations on the triangles solely by their connectivity. In this new setup, necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non-percolation. Next we apply this set of ideas to the q > 1 random cluster model: We derive duality relations for the suitable random cluster measures, prove necessary and sufficient conditions for them to have positive correlations, and finally prove some rigorous theorems concerning phase transitions.

Analysis of open system Carnot cycle and state functions

A first principle analysis of an open system thermodynamical Carnot cycle is provided and the results are compared to those proposed by Gibbs for open systems. The Kelvin–Clausius statement concerning heat transfer for reversible cycles is taken as an axiom, from which several rigorous theorems are proven. An equation is derived that resembles a Gibbs–Duhem relation equating convected entropies, from which two distinguishable forms of entropy are proven to exist for such systems; this questions prevailing developments which presume a singular or characteristic entropic form that couple all work and heat flows, such as in Onsager first-order thermodynamics. In particular, a closed path undergone by the system does not return the environment to the initial state for one of these entropic forms. The Biot assertion that the entropy contribution due to diathermal heat transfer does not form a state function is therefore contradicted and a local entropy is shown to exist. Several other new composite state functions for work and heat flow are shown to exist for open systems. From Gibbs’ results, it is suggested that the ensuing chemical potentials used routinely may possibly ignore heat effects. The functions developed here are suitable for application since the functions are proven to exist, rather than presumed to exist.

A computer-assisted proof on the stability of the Kolmogorov flows of incompressible viscous fluid

There exists a great number of references of bifurcations and stability for the Navier–Stokes equations. Only a few, however, provide a rigorous result which guarantees stability or instability. Our aim is to present a rigorous theorem which proves the stability of certain solutions arising in what is called the Kolmogorov problem. We accomplish this by the verified computation. The eigenvalue problem arising in the Kolmogorov problem is not self-adjoint and, accordingly, it is quite difficult to treat theoretically. Our method is a numerical approach to deal with this difficulty and numerical examples are given as a demonstration.

Scaling of the critical function for the standard map: some numerical results

The behaviour of the critical function for the breakdown of the homotopically non-trivial invariant (KAM) curves for the standard map, as the rotation number tends to a rational number, is investigated using a version of Greene's residue criterion. The results are compared with the analogous ones for the radius of convergence of the Lindstedt series, in which case rigorous theorems have been proved. The conjectured interpolation of the critical function in terms of the Bryuno function is discussed.

Past attractor in inhomogeneous cosmology

We present a general framework for analyzing spatially inhomogeneous cosmological dynamics. It employs Hubble-normalized scale-invariant variables which are defined within the orthonormal frame formalism, and leads to the formulation of Einstein's field equations with a perfect fluid matter source as an autonomous system of evolution equations and constraints. This framework incorporates spatially homogeneous dynamics in a natural way as a special case, thereby placing earlier work on spatially homogeneous cosmology in a broader context, and allows us to draw on experience gained in that field using dynamical systems methods. One of our goals is to provide a precise formulation of the approach to the spacelike initial singularity in cosmological models, described heuristically by Belinski\v{\i}, Khalatnikov and Lifshitz. Specifically, we construct an invariant set which we conjecture forms the local past attractor for the evolution equations. We anticipate that this new formulation will provide the basis for proving rigorous theorems concerning the asymptotic behavior of spatially inhomogeneous cosmological models.

Allowed Eta-Decay Modes and Chiral Symmetry

Recently, the development of chiral perturbation theory has allowed the generation of rigorous low-energy theorems for various hadronic processes based only on the chiral invariance of the underlying QCD Lagrangian. Such techniques are highly developed and well-tested in the domain of pionic and kaonic reactions. In this note we point out that with the addition of a few additional and reasonable assumptions similar predictive power is available for processes involving the eta meson

On possible surface magnetism in nanographite

Magnetism of nanographite is examined by consideration based on known rigorous theorems for the Hubbard model. For a bearded nanographite ribbon with the zigzag edge on one side and Klein's decorated edge on the other side, the surface magnetization of O(\\mu_B) per a carbon atom is exactly shown to appear for the π-electron system described by the Hubbard model. The surface magnetism appears due to existence of the edge states, which are degenerate localized states specific to these edges of nanographite. The same magnetic mechanism holds for network structures called the hypergraphite, when edges are prepared so that completely degenerate edge states appear.

Comment on “Method of handling the divergences in the radiation theory of sources that move faster than their waves” [J. Math. Phys. 40, 4331 (1999)

The “method” in question1 has repeatedly led to results which I proved false in 1996 by a short, transparent, and rigorous theorem.2 Even shorter is the remark here that inspection of the standard formula for the electromagnetic field suffices to prove the results of the “method” false.

Density-matrix functional method for electronic properties of impurities

We develop a density-matrix correlation-energy functional suitable for treating impurity sites with strong electronic correlations. The functional is based on a rigorous theorem about the form of such functionals as well as an exact inequality for the exchange-correlation energy. It is validated by comparison with exact results for small clusters, and is used to treat the electronic properties of an Anderson impurity in a random alloy.

Variational incorporation of negative-energy orbitals in relativistic electronic structure calculations

Significant effects of negative-energy orbitals in molecules with light atoms have recently been observed in calculations by perturbation theory from wave functions expressed in terms of positive-energy orbitals. We give a detailed discussion of how to incorporate, through a rigorous variational theorem, negative-energy orbitals in relativistic calculations of bound electronic states without any a priori knowledge of positive-energy orbitals, neither ad hoc potentials nor self-consistent-field (SCF) equations. The energy contribution of negative-energy orbitals, generally shown to be of positive sign, can be minimized by resorting to a minimax theorem or in other more practical ways, thus allowing large-scale configuration interaction (CI) with (+) orbitals obtained from correlated calculations, rather than from uncorrelated ones as in the conventional no-pair approach. General SCF equations are derived yielding positive-energy orbitals in the limit of the independent-particle model. The method is illustrated with relativistic CI calculations taken from the recent literature. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 461–470, 2000