Functional Analytic Approach Research Articles

A Quality Function Deployment Approach Through Management and Technology Development

Corrugated paper industry is an indispensable industry and it has been developed for more than fifty years. Hence, it is important for a firm to find its core competence, study effective business strategies, and form a long-term competitive advantage. However, few researchers focus on the harmonizing between the operational problems and solutions from customers’ and suppliers’ angles. The purposes of this research are to analyze frequently encountered problems and to find appropriate solutions to upgrade production efficiency and business performance for the corrugated paper industry in Taiwan. This study mixes Quality Function Deployment and Analytic Hierarchy Process approach to find some operational problems and provide the solutions based on experts' experience. The findings are projected to deliver some significant contributions; academically, by professionally conducting further theoretical validation and by adopting suitable strategies to fit the needs of customers.

Maternal blood pressure trajectories and associations with gestational age at birth: a functional data analytic approach.

Research has revealed group-level differences in maternal blood pressure trajectories across pregnancy. These trajectories are typically constructed using clinical blood pressure data and multivariate statistical methods that are prone to bias and ignore the functional, dynamic process underlying a single blood pressure observation. The aim of this study was to use functional data analysis to explore blood pressure variation across pregnancy, and multivariate methods to examine whether trajectories are related to gestational age at birth. Clinical blood pressure observations were available from 370 women who participated in a longitudinal pregnancy cohort study conducted in Montreal, Quebec, Canada. Functional data analysis was used to smooth blood pressure data and then to conduct a functional principal component analysis to examine predominant modes of variation. Three eigenfunctions explained greater than 95% of the total variance in blood pressure. The first accounted for approximately 80% of the variance and was characterized by a prolonged-decrease trajectory in blood pressure; the second explained 10% of the variance and captured a late-increase trajectory; and the third accounted for approximately 7% of the variance and captured a mid-decrease trajectory. The prolonged-decrease trajectory of blood pressure was associated with older, and late-increase with younger gestational age at birth. Functional data analysis is a useful method to model repeated maternal blood pressure observations and many other time-related cardiovascular processes. Results add to previous research investigating blood pressure trajectories across pregnancy through identification of additional, potentially clinically important modes of variation that are associated with gestational age at birth.

The Dynamic Relationship Between Alpha and Beta Power and Next-Day Suicidal Ideation in Individuals With Treatment-Resistant Depression

BackgroundNocturnal wakefulness has emerged as a potential predictor of short-term suicide risk. This analysis used dynamic temporal patterns in alpha and beta power and global sleep metrics to explore the possible link between next-day suicidal ideation (NDSI) and wakefulness measures in unmedicated participants with treatment-resistant depression. MethodsThirty-three medication-free participants with treatment-resistant depression completed overnight polysomnography. Alpha and beta spectral power as functions over time were used to represent arousal-related components of the dynamic sleep process. A functional data analytic approach (multilevel functional principal component analysis [MFPCA]) was used to preserve the oscillatory nature of the data; MFPCA PC scores were then associated with NDSI. Associations between NDSI and polysomnography-defined wakefulness after sleep onset, sleep efficiency, and total sleep time were also evaluated. ResultsNDSI had the strongest relationship with the second beta PC score (slope = 0.09 [90% credible interval, 0.03 to 0.14]), which represented an oscillating pattern that reflected disturbed sleep. The first PCs from both alpha and beta MFPCAs represented the overall magnitude of power and were most closely associated with traditional polysomnography metrics but were not related to NDSI. Results were equivocal for wakefulness after sleep onset with NDSI and did not support a relationship between NDSI and either sleep efficiency or total sleep time, highlighting the value of information contained in oscillating electroencephalogram patterns for identifying physiological links between nocturnal wakefulness and NDSI. ConclusionsThis study leveraged the dynamic nature of wakefulness-related electroencephalogram frequencies and provides a potential electrophysiological link between suicidal ideation and wakefulness during sleep in individuals with treatment-resistant depression.

A Quality Function Deployment Approach Through Management and Technology Development

Corrugated paper industry is an indispensable industry and it has been developed for more than fifty years. Hence, it is important for a firm to find its core competence, study effective business strategies, and form a long-term competitive advantage. However, few researchers focus on the harmonizing between the operational problems and solutions from customers’ and suppliers’ angles. The purposes of this research are to analyze frequently encountered problems and to find appropriate solutions to upgrade production efficiency and business performance for the corrugated paper industry in Taiwan. This study mixes Quality Function Deployment and Analytic Hierarchy Process approach to find some operational problems and provide the solutions based on experts' experience. The findings are projected to deliver some significant contributions; academically, by professionally conducting further theoretical validation and by adopting suitable strategies to fit the needs of customers.

On nonlocal minimal graphs

We develop a functional analytic approach for the study of nonlocal minimal graphs. Through this, we establish existence and uniqueness results, a priori estimates, comparison principles, rearrangement inequalities, and the equivalence of several notions of minimizers and solutions.

Optimal boundary control of Saint-Venant equations with arbitrary friction and space-varying slope

Abstract This paper is concerned with the optimal boundary control for the one-dimensional Saint-Venant equations with arbitrary friction and space-varying slope. By the Dubovitskii and Milyutin functional analytical approach, the Pontryagin maximum principles of the optimal control systems equipped with two boundary control variables are investigated and the first-order necessary optimality conditions are presented in both the fixed and the free final horizon cases, respectively. Finally, a remark on numerical solution is made for illustrating how to apply the obtained results to the investigational optimal boundary control problem.

Pontryagin's maximum principle for optimal control of an extrusion process

In this article, two optimal control problems of an extrusion process are considered in the isothermal case. The controlled system consists of a hyperbolic partial differential equation coupled with a nonlinear ordinary differential equation. Equipped with two control variables, the system describes the evolution of a moving interface between a fully filled zone and a partially filled zone. The Dubovitskii and Milyutin functional analytical approach is adopted in the investigation of the Pontryagin maximum principle for the optimal control problem. In both fixed and free final horizon cases, the corresponding necessary optimality conditions are, respectively, established. A remark is then made for illustrating the applicability of the obtained results.

Thermodynamic Formalism for Random Non-uniformly Expanding Maps

We develop a quenched thermodynamic formalism for a wide class of random maps with non-uniform expansion, where no Markov structure, no uniformly bounded degree or the existence of some expanding dynamics is required. We prove that every measurable and fibered $$C^1$$ -potential at high temperature admits a unique equilibrium state which satisfies a weak Gibbs property, and has exponential decay of correlations. The arguments combine a functional analytic approach for the decay of correlations (using Birkhoff cone methods) and Carathéodory-type structures to describe the relative pressure of not necessary compact invariant sets in random dynamical systems. We establish also a variational principle for the relative pressure of random dynamical systems.

Analytic methods for reachability problems

We consider a function-analytic approach to study synchronizing automata, primitive and ergodic matrix families. This gives a new way to establish some criteria for primitivity and for ergodicity of families of nonnegative matrices. We introduce a concept of canonical partition and use it to construct a polynomial-time algorithm for finding a positive matrix product and an ergodic matrix product whenever they exist. This also provides a generalization of some results of the Perron-Frobenius theory from one nonnegative matrix to families of matrices. Then we define the h-synchronizing automata and prove that the existence of a reset tuple is polynomially decidable. The question whether the functional-analytic approach can be extended to the h-primitivity is addressed and several open problems are formulated.

Optimal control of vibrations of two nonlinear Gao beams connected with a joint

This paper is concerned with optimal control of vibrations of two uniform elastic or viscoelastic nonlinear Gao beams that are connected with a joint. The dynamic contact is modelled with the Signorini non-penetration or unilateral conditions in which the stops are assumed to be perfectly rigid. By the Dubovitskii and Milyutin functional analytical approach, we derive the Pontryagin maximum principle for the optimal control problem with multiple equality and multiple inequality constraints in fixed final horizon case. A remark is made on how to use the obtained results in the case of a quadratic cost functional.

Blood Pressure Trajectories Across Pregnancy and Associations with Gestational Age at Birth: A Functional Data Analytic Approach

Research has revealed important, group-level differences in maternal blood pressure trajectories across pregnancy. These trajectories are typically constructed using multivariate statistical methods and while studies have produced useful information, these methods ignore several sources of statistical bias and the functional, dynamic processes that give rise to a single blood pressure observation.

A Functional Analytic Approach to Infinite Dimensional Stochastic Linear Systems

In this paper, we study infinite dimensional stochastic systems having both unbounded control and observation operators. First of all, using a semigroup approach, we give another take of the well-p...

Modeling internal waves in a two-layer fluid flow

This study presents mathematical model of the internal waves and examines wave propagation in a two-layer fluid flow. Elements of the functional-analytical approach are used to develop the model. A flat unsteady motion of a two-layer liquid under a cover over a flat bottom is considered. The fluid is assumed to be ideal and incompressible. Internal waves are caused by external pressure application to the interface between the layers, oscillation of the flat bottom and disturbances in the flow. The power of the wave source is characterized by dimensionless parameter ε. The problem is formulated, and its solution is based on asymptotic analysis for 0<ε<1.

Feller semigroups and degenerate elliptic operators III

AbstractThis paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups in the characteristic case via the Fichera function. Probabilistically, our result may be stated as follows: We construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the interior of the state space, without reaching the boundary. We make use of the Hille–Yosida–Ray theorem that is a Feller semigroup version of the classical Hille–Yosida theorem in terms of the positive maximum principle. Our proof is based on a method of elliptic regularizations essentially due to Oleĭnik and Radkevič. The weak convergence of approximate solutions follows from the local sequential weak compactness of Hilbert spaces and Mazur's theorem in normed linear spaces.

Form-perturbation theory for higher-order elliptic operators and systems by singular potentials.

We give a form-perturbation theory by singular potentials for scalar elliptic operators on [Formula: see text] of order 2m with Hölder continuous coefficients. The form-bounds are obtained from an L1 functional analytic approach which takes advantage of both the existence of m-gaussian kernel estimates and the holomorphy of the semigroup in [Formula: see text] We also explore the (local) Kato class potentials in terms of (local) weak compactness properties. Finally, we extend the results to elliptic systems and singular matrix potentials. This article is part of the theme issue 'Semigroup applications everywhere'.

Equilibrium validation in models for pattern formation based on Sobolev embeddings

<p style='text-indent:20px;'>In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable. On the other hand, numerical bifurcation searches are useful but not guaranteed to give an accurate structure, in that they could miss a portion of a branch or find a spurious branch where none exists. In a series of recent papers, we have aimed for a third option. Namely, we have developed a method of computer-assisted proofs to prove both existence and isolation of branches of equilibrium solutions. In the current paper, we extend these techniques to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three, by giving a detailed description of the analytical underpinnings of the method. Although the paper concentrates on applying the method to the Ohta-Kawasaki model, the functional analytic approach and techniques can be generalized to other parabolic partial differential equations.

Correlation based snapshot models of the archeomagnetic field

SUMMARY For the time stationary global geomagnetic field, a new modelling concept is presented. A Bayesian non-parametric approach provides realistic location dependent uncertainty estimates. Modelling related variabilities are dealt with systematically by making little subjective a priori assumptions. Rather than parametrizing the model by Gauss coefficients, a functional analytic approach is applied. The geomagnetic potential is assumed a Gaussian process to describe a distribution over functions. A priori correlations are given by an explicit kernel function with non-informative dipole contribution. A refined modelling strategy is proposed that accommodates non-linearities of archeomagnetic observables: First, a rough field estimate is obtained considering only sites that provide full field vector records. Subsequently, this estimate supports the linearization that incorporates the remaining incomplete records. The comparison of results for the archeomagnetic field over the past 1000 yr is in general agreement with previous models while improved model uncertainty estimates are provided.

Arbitrage-free modeling under Knightian uncertainty

We study the Fundamental Theorem of Asset Pricing for a general financial market under Knightian Uncertainty. We adopt a functional analytic approach which requires neither specific assumptions on the class of priors $$\mathcal {P}$$ nor on the structure of the state space. Several aspects of modeling under Knightian Uncertainty are considered and analyzed. We show the need for a suitable adaptation of the notion of No Free Lunch with Vanishing Risk and discuss its relation to the choice of an appropriate technical filtration. In an abstract setup, we show that absence of arbitrage is equivalent to the existence of approximate martingale measures sharing the same polar set of $$\mathcal {P}$$ . We then specialize our results to a discrete-time financial market in order to obtain martingale measures.

Oblique derivative problems and Feller semigroups with discontinuous coefficients

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes with an oblique derivative boundary condition for second-order, uniformly elliptic differential operators with discontinuous coefficients. More precisely, we construct Feller semigroups associated with absorption, reflection, drift and sticking (or viscosity) phenomena at the boundary. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Calderon–Zygmund theory of singular integral operators with non-smooth kernels.

Davies’ method for heat-kernel estimates: An extension to the semi-elliptic setting

We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in different directions. The “heat” kernels associated to these so-called positive-homogeneous operators are seen to arise naturally as the limits of convolution powers of complex-valued measures, just as the classical heat kernel appears in the central limit theorem. Building on the functional-analytic approach developed by E. B. Davies for higher-order uniformly elliptic operators with measurable coefficients, we formulate a general theory for (anisotropic) self-adjoint variable-coefficient operators, each comparable to a positive-homogeneous operator, and study their associated heat kernels. Specifically, under three abstract hypotheses, we show that the heat kernels satisfy off-diagonal (Gaussian-type) estimates involving the Legendre-Fenchel transform of the operator’s principle symbol. Our results extend those of E. B. Davies and G. Barbatis and partially extend results of A. F. M. ter Elst and D. Robinson.