Concepts Of Functional Analysis Research Articles

A Concise Tutorial on Functional Analysis for Applications to Signal Processing

Functional analysis is a well-developed field in the discipline of Mathematics, which provides unifying frameworks for solving many problems in applied sciences and engineering. In particular, several important topics (e.g., spectrum estimation, linear prediction, and wavelet analysis) in signal processing had been initiated and developed through collaborative efforts of engineers and mathematicians who used results from Hilbert spaces, Hardy spaces, weak topology, and other topics of functional analysis to establish essential analytical structures for many subfields in signal processing. This paper presents a concise tutorial for understanding the theoretical concepts of the essential elements in functional analysis, which form a mathematical framework and backbone for central topics in signal processing, specifically statistical and adaptive signal processing. The applications of these concepts for formulating and analyzing signal processing problems may often be difficult for researchers in applied sciences and engineering, who are not adequately familiar with the terminology and concepts of functional analysis. Moreover, these concepts are not often explained in sufficient details in the signal processing literature; on the other hand, they are well-studied in textbooks on functional analysis, yet without emphasizing the perspectives of signal processing applications. Therefore, the process of assimilating the ensemble of pertinent information on functional analysis and explaining their relevance to signal processing applications should have significant importance and utility to the professional communities of applied sciences and engineering. The information, presented in this paper, is intended to provide an adequate mathematical background with a unifying concept for apparently diverse topics in signal processing. The main objectives of this paper from the above perspectives are summarized below: (1) Assimilation of the essential information from different sources of functional analysis literature, which are relevant to developing the theory and applications of signal processing. (2) Description of the underlying concepts in a way that is accessible to non-specialists in functional analysis (e.g., those with bachelor-level or first-year graduate-level training in signal processing and mathematics). (3) Signal-processing-based interpretation of functional-analytic concepts and their concise presentation in a tutorial format.

Reproducing Kernel Hilbert spaces for wave optics: tutorial.

An introduction to the Hilbert spaces that are endowed with a reproducing kernel is presented on using the mathematical tools of Fourier optics and coherence theory. After giving the basic definition of such spaces, some examples are worked out to show how the inner product can take different forms depending on the particular function space one works with. The basic rule to build a reproducing kernel Hilbert space (RKHS) is then presented together with the basic properties of those spaces. Eigenfunctions and eigenvalues of the reproducing kernel are then illustrated and lead to the important integral representation of the reproducing kernel. The latter is used to present pseudomodal expansions and generalized forms of sampling. The concluding section offers some thoughts on the applications of RKHSs in wave optics. An appendix presents an introduction to treatments using more advanced concepts of functional analysis.

A Systematic Review of Psychometric Transactional Analysis Instruments

Since the founding of transactional analysis (TA) by Eric Berne in the 1950s, many psychometric instruments have been developed to operationalize TA concepts. Several studies have provided general reviews of these, but the studies are not systematic and do not assess the psychometric quality of the instruments. We conducted a systematic literature review of all psychometric instruments operationalizing TA concepts by searching in Pubmed, Medline, PsycInfo, APA articles, Web-of-Knowledge, and scholar.google.com. The study quality was evaluated with the consensus-based standards for the selection of health measurement instruments (COSMIN). Of the 12,287 initial search hits, we selected 56 instruments in 263 studies. Among those, the Schema Mode Inventory, the Tokyo Egogram, the Adjective Check List, the ANINT-A36, and the Life Position Scale were found to have fair to good overall COSMIN quality. Except for the Life Position Scale, they operationalize ego states (Parent, Adult, Child) and the functional analysis concepts of Critical Parent, Nurturing Parent, and Adapted/Free Child. The Life Position Scale operationalized the positive and negative positions toward self and others. Other questionnaires had poor to fair COSMIN quality, usually because of a lack of studies. Meta-analyses indicated that psychopathology and physical symptoms are associated with strong Critical Parent, strong Adapted Child, weak Integrated Adult, negative coping styles, and negative positions toward oneself and others. Many instruments need further validation and translation into more languages. The Schema Mode Inventory, the Tokyo Egogram, the Adjective Check List, and the Life Position Scale are the best validated questionnaires. These could be used by researchers to test treatment effectiveness and by psychotherapists to examine the problems and etiology of clients.

A study of fractional order Ambartsumian equation involving exponential decay kernel

<abstract> Recently, non-singular fractional operators have a significant role in the modeling of real-world problems. Specifically, the Caputo-Fabrizio operators are used to study better dynamics of memory processes. In this paper, under the non-singular fractional operator with exponential decay kernel, we analyze the Ambartsumian equation qualitatively and computationally. We deduce the result of the existence of at least one solution to the proposed equation through Krasnoselskii's fixed point theorem. Also, we utilize the Banach fixed point theorem to derive the result concerned with unique solution. We use the concept of functional analysis to show that the proposed equation is Ulam-Hyers and Ulam-Hyers-Rassias stable. We use an efficient analytical approach to compute a semi-analytical solution to the proposed problem. The convergence of the series solution to an exact solution is proved through non-linear analysis. Lastly, we present the solution for different fractional orders. </abstract>

Prognostics and Health Management in the Oil &amp; Gas Industry - A Step Change

Traditionally the life cycle phases of a typical project in the Oil &amp; Gas industry can be broken down into: Feasibility &amp; Concept Design; Front End Engineering Design (FEED); Detail Design; Manufacture and Construction; Installation and Commissioning; Operation. A range of Process Hazard Analysis (PHA) tools are used throughout these phases in hazard identification and analysis, to understand the effects to the system as a result of the hazards and to eliminate/reduce/mitigate the identified hazards. These PHA tools are typically carried out independent from one another albeit from the same base documentation (e.g. P&amp;ID’s) and they are subsequently documented independently e.g. Hazardous &amp; Operability (HAZOP) study, Risk Analysis, Layer of Protection Analysis (LOPA), Failure Mode, Effects and Criticality Analysis (FMECA). Prognostics and Health Management (PHM) is a life cycle concept introducing an integrated approach to the health management of a system through the design and operation cycles. The design cycle can be broken down into: System Design; Identification and Analysis of Risk; Failure Modelling and how it affects the System in terms of Reliability/Availability/Maintainability. The operation cycle can be broken down into: Real-time Monitoring; Diagnostics; Failure Prediction; Determine Maintenance Strategies. Parallels can be drawn between life cycle phases in both the traditional approach and a PHM approach during project development. Therefore, if projects can be developed within a PHM environment, this may lead to greater integration between the stakeholders, with the potential for a technically superior product developed with cost and efficiency savings. The aim of this paper is to review the concepts of Functional Analysis within a PHM environment, to assess its suitability and applicability during the engineering design and operation of an Oil-Injected Rotary Screw Compressor. PHA tools, sensor set design and the organisation interfaces that exist between the stakeholders represented throughout the life cycle of a project shall be reviewed. The outcomes from the Functional Analysis modelling application, Maintenance Aware Design environment (MADe™) in terms of sensor set design and PHA tools shall be compared with the existing processes used in the traditional approach throughout the life cycle of a project. The paper shall also review the organisational interfaces that exist between the stakeholders that are represented throughout the life cycle of the project and how these relationships can deliver a safe, functional system that maximises reliability, availability and maintainability of the asset whilst facilitating fault detection/isolation and potential fault prediction. A functional model of the compressor package was created using the MADe™ functional modelling application. Functional analyses of key components were performed and sensors set options were reviewed for applicability and suitability. Key components were identified for functional failure analysis either from Offshore Reliability Data Handbook (OREDA, 2015), documentation supplied by the Original Equipment Manufacturer (OEM) or from a FMECA report derived from the functional model’s PHA suite of tools. The key roles and responsibilities of each of the stakeholders were analysed to explore shared responsibility, business outcomes and engineering/technical deliverables. The results demonstrate that functional analysis software applications can facilitate sensor set design to detect and isolate faults associated with the system’s components and that those are comparable – with some manual adjustment – with the traditional approaches used in sensor set design. The results also show that a functional analysis approach is a viable tool in the PHA process i.e. it can generate FMECA reports and can be used to complement HAZOP studies. Finally using a PHM functional modelling application can benefit the main stakeholders in terms of demonstrating reliability, availability and maintainability of equipment whilst realising cost savings and improved efficiency.

ABC on IBC

This is a tutorial introducing basic concepts of Information-Based Complexity (IBC) with the emphasis on tractability of multivariate problems. IBC deals with the complexity of computing approximate solutions of continuous problems. No prior knowledge of IBC is required from the reader, however, some knowledge of basic concepts of functional analysis and measure theory will be very helpful.This text is based on tutorials delivered by the author during the workshop “IBC-Paris 2017: Information-Based Complexity, High-Dimensional Problems” in Paris, March 2017, and the mini-semester at the Erwin Schrödinger Institute “Information-Based Complexity and Tractability of Multivariate Problems” in Vienna, September–October 2017.

Using Functional Analysis as a Framework to Guide Individualized Treatment for Negative Symptoms.

Although numerous interventions are available for negative symptoms, outcomes have been unsatisfactory with pharmacological and psychological interventions producing changes of only limited clinical significance. Here, we argue that because negative symptoms occur as a complex syndrome caused and maintained by numerous factors that vary between individuals they are unlikely to be treated effectively by the present “one size fits all” approaches. Instead, a well-founded selection of those interventions relevant to each individual is needed to optimize both the efficiency and the efficacy of existing approaches. The concept of functional analysis (FA) can be used to structure existing knowledge so that it can guide individualized treatment planning. FA is based on stimulus—response learning mechanisms taking into account the characteristics of the organism that contribute to the responses, their consequences and the contingency with which consequences are tied to the response. FA can thus be flexibly applied to the level of individual patients to understand the factors causing and maintaining negative symptoms and derive suitable interventions. In this article we will briefly introduce the concept of FA and demonstrate—exemplarily—how known psychological and biological correlates of negative symptoms can be incorporated into its framework. We then outline the framework's implications for individual assessment and treatment. Following the logic of FA, we argue that a detailed assessment is needed to identify the key factors causing or maintaining negative symptoms for each individual patient. Interventions can then be selected according to their likelihood of changing these key factors and need to take interactions between different factors into account. Supplementary case vignettes exemplify the usefulness of functional analysis for individual treatment planning. Finally, we discuss and point to avenues for future research guided by this model.

Spatial and temporal variabilities of rainfall data using functional data analysis

The main concern of this study is to build a functional data object from discrete rainfall observations by looking at how rainfall fluctuates, both spatially and temporally, in the form of smoothing curves. The functional data methods employed in this study are able to extract additional information contained in the function and its derivatives which may not be normally available from traditional statistical methods. Functional concepts such as functional descriptive statistics and functional analysis of variance were applied to describe the spatial and temporal rainfall variations at the stations and at any time throughout the year. This study involves 32 rainfall stations in Peninsular Malaysia and rainfall records for 32 years. Eleven basis functions were used to describe the unimodal rainfall pattern for stations in the East Peninsula, while five and seven bases were required to describe the rainfall pattern for stations in the northwest, west, southwest, and central regions of the peninsula. Based on the location and scale curves, the highest mean and the highest variability of rainfall were observed during the northeast monsoon flow. On the other hand, the concept of functional analysis of variance allows the detailed information in determining when, in a time series, differences may exist in rainfall profiles between two or more regions. In general, the findings suggested that the rainfall profiles of the regions are very dependent on their geographical and spatial locations, as well as the monsoon effect, which reflects the time of year.

Linear bounded phase coordinate control problems in 2-Banach spaces

In this paper, we have used some fundamental concepts of functional analysis to tackle a class of optimization problems. A class of optimal control for linear bounded phase coordinate control problems in 2-Banach spaces has been defined and solved. Also, we define the concepts of regularity and normality and using this we prove the existence and uniqueness of solutions. Finally, we give the complete characterization of the solution in terms of a hyper-plane.

ON THE ORIGIN AND FUNCTIONS OF THE TERM FUNCTIONAL ANALYSIS

In this essay, we note that although Iwata, Dorsey, Slifer, Bauman, and Richman (1982) established the standard framework for conducting functional analyses of problem behavior, the term functional analysis was probably first used in behavior analysis by B. F. Skinner in 1948. We also remind readers that a functional analysis is really an experimental analysis, words that were contained in the title of Skinner's first book, The Behavior of Organisms: An Experimental Analysis (1938). We further describe how Skinner initially applied the concept of functional analysis to an understanding of verbal behavior, and we suggest that the same tactic be applied to the verbal behavior of behavior analysts, in the present case, to the term functional analysis.

Using homological methods on the base of iterated spectra in functional analysis

We introduce new concepts of functional analysis: Hausdorff spectrum and Hausdorff limit or H-limit of Hausdorff spectrum of locally convex spaces. Particular cases of regular H-limit are projective and inductive limits of separated locally convex spaces. The class of H-spaces contains Frechet spaces and is stable under forming countable inductive and projective limits, closed subspaces and quotient spaces. Moreover, for H-space an unproved variant of the closed graph theorem holds true. Homological methods are used for proving of theorems of vanishing at zero for first derivative of Hausdorff limit functor: Haus1(X)=0.

Analysis of Coordination in Multi-Agent Systems Through Partial Difference Equations

In this note, we introduce the framework of partial difference equations (PdEs) over graphs for analyzing the behavior of multi-agent systems equipped with decentralized control schemes. Both leaderless and leader-follower models are considered. PdEs mimic partial differential equations (PDEs) on graphs and can be studied by introducing concepts of functional analysis strongly inspired to the corresponding ones arising in PDEs theory. We generalize different models proposed in the literature by introducing errors in the agent dynamics and analyze agent coordination through the joint use of PdEs and automatic control tools. Moreover, for the simplest control schemes, we show that the resulting PdEs enjoy properties that are similar to those of well-known PDEs like the heat equation, thus allowing to exploit physical-based reasoning for conjecturing formation properties.

Adjusted fair scheduling and non-linear workload prediction for QoS guarantees in grid computing

In this paper, we propose an efficient non-linear task workload prediction mechanism incorporated with a fair scheduling algorithm for task allocation and resource management in Grid computing. Workload prediction is accomplished in a Grid middleware approach using a non-linear model expressed as a series of finite known functional components using concepts of functional analysis. The coefficient of functional components are obtained using a training set of appropriate samples, the pairs of which are estimated based on a runtime estimation model relied on a least squares approximation scheme. The advantages of the proposed non-linear task workload prediction scheme is that (i) it is not constrained by analysis of source code ( analytical methods), which is practically impossible to be implemented in complicated real-life applications or (ii) it does not exploit the variations of the workload statistics as the statistical approaches does. The predicted task workload is then exploited by a novel scheduling algorithm, enabling a fair Quality of Service oriented resource management so that some tasks are not favored against others. The algorithm is based on estimating the adjusted fair completion times of the tasks for task order selection and on an earliest completion time strategy for the grid resource assignment. Experimental results and comparisons with traditional scheduling approaches as implemented in the framework of European Union funded research projects GRIA and GRIDLAB grid infrastructures have revealed the outperformance of the proposed method.

Decomposition method in linear elastic problems with eigenstrain

The general theory of linearized elasticity with eigenstrain is considered with applications to continuous, discrete and discretized structures. It is shown that any eigenstrain can be uniquely decomposed into impotent and nilpotent constituents. The proven theorem on decomposition is based on the concepts of functional analysis, in particular, with respect to Hilbert functional spaces. This unique decomposition allows for the individual and independent control of stress, strain and displacement (e.g. shape control). The associated algorithm avoids the cumbersome solution of boundary-value problems with eigenstrain in connection with these control problems. Decomposition of eigenstrain opens the practically important opportunity to fully separate the control of strain and stress produced by force loading.

Generalized Nonlinear Relevance Feedback for Interactive Content-Based Retrieval and Organization

In this paper, a novel relevance feedback algorithm is proposed for improving the performance of interactive content-based retrieval systems. The algorithm recursively estimates the similarity measure, which is used for data ranking in description environments where similarity-based queries are applied, using a set of relevant/irrelevant samples feedback by the user to the system so that the adjusted response is a better approximation of the current user's information needs and preferences. In particular, using concepts of functional analysis, the similarity measure is expressed as a parametric form of known monotone increasing functional components. Then, the contribution of each functional component to the similarity measure is estimated through a recursive and efficient on-line learning algorithm so that: 1) the current user's needs and preferences, as indicated by a set of selected relevant/irrelevant samples, are satisfied as much as possible, while simultaneously 2) a minimal modification of the already estimated similarity measure is accomplished. Experimental results on a large real-life database using objective evaluation criteria, such as the precision-recall curve and the average normalized modified retrieval rank (ANMRR), indicate that the proposed scheme outperforms the compared ones. In addition, the proposed algorithm requires low computational complexity and it can be implemented in a recursive way.

Análise Funcional: Definição e Aplicação na Terapia Analítico-Comportamental

O artigo revisa alguns usos do conceito de análise funcional na análise do comportamento e na terapia comportamental, enfatizando suas relações com o modelo causal de seleção por conseqüências.Emseguida, propõe queumaaplicação clínica da análise funcional consistente com o sistema explicativo analítico-comportamental estaria apoiada em: a) selecionismo como modelo causal e funcionalismo como princípio de análise; b) externalismo como recorte de análise; c) complexidade, variabilidade, e caráter idiossincrático das relações comportamentais; d) critério pragmático na definição do nível de intervenção; e) distinção entre alcance da avaliação e alcance da intervenção.

Fundamental properties of parametric functionals in quantum chemistry

A theoretical interpretation and justification of the basic approaches proposed in parametrical methods is presented in this paper. The idea of providing a fundamental understanding of these methods can be considered as a guide for further development in this area. The application of functional analysis concepts to understanding the nature of parametric functionals in terms of the Riesz and Weiestrass's theorems is discussed. In addition, applications of virial and hypervirial theorems to constrain the type of functional and the parameters used in these methods is addressed. A justification of the use of a single determinantal wavefunction and the nature and conditions of optimized orbital basis in terms of different electronic environments involved in different interactions was also established.

A convolutional approach for two-sided solution bounding in unilateral structural elastodynamics

The paper deals with a numerical procedure, which provides upper and lower bounds for solution quantities, either global or local, for dynamic inequality problems of structural mechanics. The procedure is based on a convolutional problem formulation, a suitable time-discretization and some functional analysis concepts.

Functional analysis concepts and Hartree-Fock instability conditions II: An analysis of the eight classes of solutions

The continuation of the previous work is presented. Here the instability matrices for the eight classes of HF solutions, considering time reversal and spin symmetries, are derived and explicitly shown. These matrices are expressed in terms of LCAO coefficients. Hence they are of immediate application to the verification of instabilities in HF calculations.

Convexity in complex analysis

Convexity is the key concept of functional analysis, but apart from some notable exceptions, it has played a relatively minor role in several complex variables theory. The work of Lempert has focused attention on convex domains, and in these two lectures I will present examples involving invariant metrics in complex analysis where convexity, whether realized as in the case of bounded symmetric domains or assumed as in the case of Blp , is essential. Functional analysis brings to problems a variety of developed concepts such as complex extreme points, complex uniform convexity and an approach which is often coordinate and dimension free. I hope to illustrate these points in my lectures. Throughout this article X will denote a Banach space over the complex numbers C, BX will denote the open unit ball in X and BX its closure. We denote by ∆ the open unit disc in C. Our first example relates the maximum modulus theorem of complex analysis with the functional analytic concept of complex extreme point. A point x in X, (‖x‖ = 1), is a complex extreme point (of the unit ball) if ‖x + λy‖ ≤ 1 for all λ ∈ ∆ implies y = 0. The strong maximum modulus theorem, due to Thorp and Whitely in 1965, states the following: