Nonlinear Spectral Theories and Solvability of Nonlinear Hammerstein Equations

AMS Subj. Classification: 47J10, 47H30, 47H10

A Fredholm-type theory for third-kind linear integral equations

The third-kind linear integral equation Image where g(t) vanishes at a finite number of points in (a, b), is considered. In general, the Fredholm Alternative theory [[5.]] does not hold good for this type of integral equation. However, imposing certain conditions on g(t) and K(t, t′), the above integral equation was shown [[1.], 49–57] to obey a Fredholm-type theory, except for a certain class of kernels for which the question was left open. In this note a theory is presented for the equation under consideration with some additional assumptions on such kernels.

An Abstract Second Kind Fredholm Integral Equation with Degenerated Kernel

Mathematics Subject Classification: 44A40, 45B05

An integral equation method for a boundary value problem arising in unsteady water wave problems

In this paper we consider the 2D Dirichlet boundary value problem for Laplace’s equation in a non-locally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragmen-Lindelof arguments. The main result is to propose a boundary integral equation formulation, to prove equivalence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalisation of the Fredholm alternative in Arens et al [J. Int. Equ. Appl. 15 (2003) pp. 1-35]. This then leads to an existence proof for the boundary value problem. Keywords. Boundary integral equation method, Water waves, Laplace’s

On the solvability of second kind integral equations on the real line

This paper considers general second kind integral equations of the form(in operator form φ − kφ = ψ), where the functions k and ψ are assumed known, with ψ ∈ Y, the space of bounded continuous functions on R, and k such that the mapping s → k(s, · ), from R to L1(R), is bounded and continuous. The function φ ∈ Y is the solution to be determined. Conditions on a set W ⊂ BC(R, L1(R)) are obtained such that a generalised Fredholm alternative holds: If W satisfies these conditions and I − k is injective for all k ∈ W then I − k is also surjective for all k ∈ W and, moreover, the inverse operators (I − k) − 1 on Y are uniformly bounded for k ∈ W. The approximation of the kernel in the integral equation by a sequence (kn) converging in a weak sense to k is also considered and results on stability and convergence are obtained. These general theorems are used to establish results for two special classes of kernels: k(s, t) = κ(s − t)z(t) and k(s, t) = κ(s − t)λ(s − t, t), where κ ∈ L1(R), z ∈ L∞(R), and λ ∈ BC((R\{0}) × R). Kernels of both classes arise in problems of time harmonic wave scattering by unbounded surfaces. The general integral equation results are here applied to prove the existence of a solution for a boundary integral equation formulation of scattering by an infinite rough surface and to consider the stability and convergence of approximation of the rough surface problem by a sequence of diffraction grating problems of increasingly large period.

The impedance boundary value problem for the Helmholtz equation in a half-plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].

Some uniform stability and convergence results for integral equations on the real line and projection methods for their solution

The paper considers second kind equations of the form (abbreviated x=y + K2x) in which and the factor z is bounded but otherwise arbitrary so that equations of Wiener-Hopf type are included as a special case. Conditions on a set are obtained such that a generalized Fredholm alternative is valid: if W satisfies these conditions and I − Kz, is injective for each z ε W then I − Kz is invertible for each z ε W and the operators (I − Kz)−1 are uniformly bounded. As a special case some classical results relating to Wiener-Hopf operators are reproduced. A finite section version of the above equation (with the range of integration reduced to [−a, a]) is considered, as are projection and iterated projection methods for its solution. The operators (where denotes the finite section version of Kz) are shown uniformly bounded (in z and a) for all a sufficiently large. Uniform stability and convergence results, for the projection and iterated projection methods, are obtained. The argument generalizes an idea in collectively compact operator theory. Some new results in this theory are obtained and applied to the analysis of projection methods for the above equation when z is compactly supported and k(s − t) replaced by the general kernel k(s,t). A boundary integral equation of the above type, which models outdoor sound propagation over inhomogeneous level terrain, illustrates the application of the theoretical results developed.

Il problema di Dirichlet per equazioni lineari ellittiche in forma di divergenza

The purpose of this dissertation is to prove that the Dirichlet problem in a bounded domain is uniquely solvable for elliptic equations in divergence form. The proof can be achieved by Hilbert space methods based on generalized or weak solutions. Existence and uniqueness of a generalized solution for the Dirichlet problem follow from the Fredholm alternative and weak maximum principle.

An alternative approach to study nonlinear inviscid flow over arbitrary bottom topography

This paper deals with a new approach to study the nonlinear inviscid flow over arbitrary bottom topography. The problem is formulated as a nonlinear boundary value problem which is reduced to a Dirichlet problem using certain transformations. The Dirichlet problem is solved by applying Plemelj-Sokhotski formulae and it is noticed that the solution of the Dirichlet problem depends on the solution of a coupled Fredholm integral equation of the second kind. These integral equations are solved numerically by using a modified method. The free-surface profile which is unknown at the outset is determined. Different kinds of bottom topographies are considered here to study the influence of bottom topography on the free-surface profile. The effects of the Froude number and the arbitrary bottom topography on the free-surface profile are demonstrated in graphical forms for the subcritical flow. Further, the nonlinear results are validated with the results available in the literature and compared with the results obtained by using linear theory. (C) 2015 Elsevier Inc. All rights reserved.

Lead-germanate glasses and fibers: A practical alternative to tellurite for nonlinear fiber applications

We report on the fabrication of novel lead-germanate glasses and fibers. We have characterized these glasses in terms of their thermal properties, Raman spectra and refractive indices (both linear and nonlinear) and present them as viable alternatives to tellurite glasses for applications requiring highly nonlinear optical fibers. © 2013 Optical Society of America.

Nonlinear limit of alternative method to 2 × 2 MIMO for LTE RoF system

Owing to the limited cell size of eNodeB (eNB), the relay node has emerged as an attractive solution for the long-term evolution (LTE) system. The nonlinear limit of the alternative method to multipleinput and multiple-output (MIMO) based on frequency division multiplexing (FDM) for orthogonal FDM (OFDM) is analysed over varying transmission spans. In this reported work, it is shown that the degradation pattern over the linear, intermixing and nonlinear propagation regions is consistent for the 2 and the 2.6 GHz bands. The proposed bands experienced a linear increase in the error vector magnitude (EVM) for both the linear and the nonlinear regions proportional to the increasing transmission spans. In addition, an optical launch power between -2 and 2 dBm achieved a significantly lower EVM than the LTE limit of 8% for the 10-60 km spans. © The Institution of Engineering and Technology 2014.

Fiber Bragg grating strain modulation based on nonlinear string transverse force amplifier

The only effective method of Fiber Bragg Grating (FBG) strain modulation has been by changing the distance between its two fixed ends. We demonstrate an alternative being more sensitive to force based on the nonlinear amplification relationship between a transverse force applied to a stretched string and its induced axial force. It may improve the sensitivity and size of an FBG force sensor, reduce the number of FBGs needed for multi-axial force monitoring, and control the resonant frequency of an FBG accelerometer.

Variations in acculturation and Australian physical education teacher education students' receptiveness to an alternative pedagogical approach to games teaching

Background The development of intelligent, thinking performers as a central theme in Physical Education curriculum documents worldwide has highlighted the need for an evolution of teaching styles from the dominant reproductive approach. This has prompted an Australian university to change the content and delivery of a games unit within their Physical Education Teacher Education (PETE) course and adopt a productive student centred approach that is compatible with current curriculum directives. The significance of prospective physical educators’ biographies on their receptiveness to this pedagogical innovation was studied to help recognise and understand potential differences and subsequently guide programme development to help improve the impact of teacher education. Purpose To investigate whether past school and sporting experiences are powerful influences on Australian PETE recruits’ initial perspectives about effective physical education teaching practice and their receptiveness to an alternative pedagogical approach. Participants and Setting 49 first year pre-service PETE students (53% male; 47% female; mean age 18.88 ± 1.57 years) undertaking a compulsory unit on games teaching at an Australian university volunteered to take part in the study and were grouped according to their highest level of representation in games, either school/club (n=13), regional (n=20), or state/national (n=16). Students experienced the constraints-led approach as learners and teachers during an 8-week games unit informed by nonlinear pedagogy and underpinned by motor learning theory. Data collection and Analysis Prior to the commencement of the unit participants completed part A of a two part mixed response questionnaire aimed at gathering data about their physical education and sporting background. The data were summarised using descriptive statistics. Pre and post intervention, participants completed part B responding, via Likert Scale with their opinion of the importance of each sub-component of the traditional reproductive style for an effective games teaching session. This resulted in a traditional reproductive games teaching belief score. For each sub-component, participants were invited to respond in more detail to justify their opinions. A one-way between groups analysis of variance (ANOVA), Tukey’s HSD Post Hoc Test and a two - tailed, paired samples t test were used to analyse the quantitative data. Content analysis was used to analyse the qualitative data. Findings The traditional, reproductive approach was the most frequently reported teaching approach used by the physical education teachers and sports coaches of participants in all groups. Prior to the commencement of the alternate games unit, participants in each representative level group held very strong custodial traditional reproductive games teaching beliefs. After experiencing the alternative games unit there were statistically significant differences in the traditional reproductive games teaching belief mean scores for each group, This combined with participants’ qualitative responses indicated a receptiveness to the alternative pedagogy. Conclusions The results of this present study show that, contrary to previous research undertaken in North America, in Australia, it is possible for PETE educators to change beliefs in order to overcome the constraint of acculturation and provide PETE students with the knowledge, understanding and belief in an alternate approach to teaching games in physical education compatible with curriculum documents.

Non-linear principal components analysis : an alternative method for finding patterns in environmental data

The main purpose of this article is to gain an insight into the relationships between variables describing the environmental conditions of the Far Northern section of the Great Barrier Reef, Australia. Several of the variables describing these conditions had different measurement levels and often they had non-linear relationships. Using non-linear principal component analysis, it was possible to acquire an insight into these relationships. Furthermore, three geographical areas with unique environmental characteristics could be identified.

Overcoming acculturation: Physical education recruits' experiences of an alternative pedagogical approach to games teaching

Background Physical education teacher education (PETE) programmes have been identified as a critical platform to encourage the exploration of alternative teaching approaches by pre-service teachers. However, the socio-cultural constraint of acculturation or past physical education and sporting experiences results in the maintenance of the status quo of a teacher-driven, reproductive paradigm. Previous studies have reported successfully overcoming the powerful influence of acculturation, resulting in a change in PETE students’ custodial teaching beliefs and receptiveness to alternative teaching approaches. However, to date, limited information has been reported about how PETE students’ acculturation shaped their receptiveness to an alternative teaching approach. This is particularly the case for PETE recruits identified in the literature as most resistant to change. Purpose To explore the features and experiences of an alternative games teaching approach that appealed to PETE recruits’ identified as most resistant to change, requiring a specific sample of PETE recruits with strong, custodial, traditional physical education teaching beliefs, and whom are high achieving sporting products of this traditional culture. The alternative teaching approach explored in this study is the constraints-led approach (CLA), which is similar operationally to TGfU, but distinguished by a neurobiological theoretical framework (nonlinear pedagogy) that informs learning design. Participants and Setting A purposive sample of 10 Australian PETE students was recruited for the study. All participants initially had strong, custodial, traditional physical education teaching beliefs, and were successful sporting products of this teaching approach. After experiencing the CLA as learners during a games unit, participants demonstrated receptiveness to the alternative pedagogy. Data Collection and Analysis Semi-structured interviews and written reflections were sources of data collection. Each participant was interviewed separately, once prior to participation in the games unit to explore their positive physical education experiences, and then again after participation to explore the specific games unit learning experiences that influenced their receptiveness to the alternative pedagogy. Participants completed written reflections about their personal experiences after selected practical sessions. Data were qualitatively analysed using grounded theory. Findings: Thorough examination of the data resulted in establishment of two prominent themes related to the appeal of the CLA for the participants: (i) psychomotor (effective in developing skill), and (ii), inclusivity (included students of varying skill level). The efficacy of the CLA in skill development was clearly an important mediator of receptiveness for highly successful products of a traditional culture. This significant finding could be explained by three key factors: the acculturation of the participants, the motor learning theory underpinning the alternative pedagogy and the unit learning design and delivery. The inclusive nature of the CLA provided a solution to the problem of exclusion, which also made the approach attractive to participants. Conclusion PETE educators could consider these findings when introducing an alternative pedagogy aimed at challenging PETE recruits’ custodial, traditional teaching beliefs. To mediate receptiveness, it is important that the learning theory underpinning the alternative approach is operationalised in a research-informed pedagogical learning design that facilitates students’ perceptions of the effectiveness of the approach through experiencing and or observing it working.