Semi-hyperbolic mappings in Banach spaces

The definition of semi-hyperbolic dynamical systems generated by Lipschitz continuous and not necessarily invertible mappings in Banach spaces is presented in this thesis. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional. Bi-shadowing is a combination of the concepts of shadowing and inverse shadowing and is usually used to compare pseudo-trajectories calculated by a computer with the true trajectories. In this thesis, the concept of bi-shadowing in a Banach space is defined and proved for semi-hyperbolic dynamical systems generated by Lipschitz mappings. As an application to the concept of bishadowing, linear delay differential equations are shown to be bi-shadowing with respect to pseudo-trajectories generated by nonlinear small perturbations of the linear delay equation. This shows robustness of solutions of the linear delay equation with respect to small nonlinear perturbations. Complicated dynamical behaviour is often a consequence of the expansivity of a dynamical system. Semi-hyperbolic dynamical systems generated by Lipschitz mappings on a Banach space are shown to be exponentially expansive, and explicit rates of expansion are determined. The result is applied to a nonsmooth noninvertible system generated by delay differential equation. It is shown that semi-hyperbolic mappings are locally φ-contracting, where -0 is the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it is φ-contracting and has no spectral values on the unit circle. The definition of φ-bi-shadowing is given and it is shown that semi-hyperbolic mappings in Banach spaces are φ-bi-shadowing with respect to locally condensing continuous comparison mappings. The result is applied to linear delay differential equations of neutral type with nonsmooth perturbations. Finally, it is shown that a small delay perturbation of an ordinary differential equation with a homoclinic trajectory is ‘chaotic’.

Grain size effect on the warm deformation behaviour of a Ti-IF steel

The effect of grain size on the warm deformation behaviour of a titanium stabilized interstitial free (IF) steel was investigated using hot torsion. The initial work hardening regime is followed by the development of a broad stress peak after which work softening occurs. The hypothetical saturation stress (Estrin–Mecking model) and the stress at final strain were relatively insensitive to grain size. However, the strain to the peak stress was strongly dependent on the grain size at low values of the Zener–Hollomon parameter. A simple phenomenological approach, using a combined Estrin–Mecking model and an Avrami type equation, was used to model the flow curves. The hypothetical saturation stress, the stress at final strain and the strain to peak stress were modelled using three different hyperbolic sine laws. A comparison with independent data from the literature shows that the apparent activation energy of deformation determined in this work (Q=372 kJ/mol) can be used to rationalize the steady-state stress in compression data found in the literature.

Knowledge extraction from a mixed transfer function artificial neural network

One of the main problems with Artificial Neural Networks (ANNs) is that their results are not intuitively clear. For example, commonly used hidden neurons with sigmoid activation function can approximate any continuous function, including linear functions, but the coefficients (weights) of this approximation are rather meaningless. To address this problem, current paper presents a novel kind of a neural network that uses transfer functions of various complexities in contrast to mono-transfer functions used in sigmoid and hyperbolic tangent networks. The presence of transfer functions of various complexities in a Mixed Transfer Functions Artificial Neural Network (MTFANN) allow easy conversion of the full model into user-friendly equation format (similar to that of linear regression) without any pruning or simplification of the model. At the same time, MTFANN maintains similar generalization ability to mono-transfer function networks in a global optimization context. The performance and knowledge extraction of MTFANN were evaluated on a realistic simulation of the Puma 560 robot arm and compared to sigmoid, hyperbolic tangent, linear and sinusoidal networks.

Convergence analysis of multi-stage PIC detectors

This paper analyses the convergence behaviour of the parallel interference cancellation (PIC) detector in code division multiple access (CDMA) systems. Using the results from previous stability analysis of an iterated-map neural network, the paper derives a general condition from which the sufficient condition for convergence of the PIC detector with tentative decision functions that are monotonically increasing at a sublinear rate can be calculated. As examples, the paper derives the sufficient conditions for convergence of the PIC detector with the clip decision and the hyperbolic tangent decision functions. The paper also examines the convergence behaviour of the PIC detector with hyperbolic tangent decision function via computer simulation and compares it with the analytical results.

Conversion of the hot torsion test results into flow curve with multiple regimes of hardening

The method of Fields and Backofen has been commonly used to reduce the data obtained by hot torsion test into flow curves. The method, however, is most suitable for materials with monotonic strain hardening behaviour. Other methods such as Stüwe’s method, tubular specimens, differential testing and the inverse method, each suffer from similar drawbacks. It is shown in the current work that for materials with multiple regimes of hardening any method based on an assumption of constant hardening indices introduces some errors into the flow curve obtained from the hot torsion test. Therefore such methods do not enable accurate prediction of onset of recrystallisation where slow softening occurs. A new method to convert results from the hot torsion test into flow curves by taking into account the variation of constitutive parameters during deformation is presented. The method represents the torque twist data by a parametric linear least square model in which Euler and hyperbolic coefficients are used as the parameters. A closed form relationship obtained from the mathematical representation of the data is employed next for flow stress determination. Two different solution strategies, the method of normal equations and singular value decomposition, were used for parametric modelling of the data with hyperbolic basis functions. The performance of both methods is compared. Experimental data obtained by FHTTM, a flexible hot torsion test machine developed at IROST, for a C–Mn austenitic steel was used to demonstrate the method. The results were compared with those obtained using constant strain and strain rate hardening characteristics.

Application of real and functional analysis to solve boundary value problems

This thesis is about using appropriate tools in functional analysis arid classical analysis to tackle the problem of existence and uniqueness of nonlinear partial differential equations. There being no unified strategy to deal with these equations, one approaches each equation with an appropriate method, depending on the characteristics of the equation. The correct setting of the problem in appropriate function spaces is the first important part on the road to the solution. Here, we choose the setting of Sobolev spaces. The second essential part is to choose the correct tool for each equation. In the first part of this thesis (Chapters 3 and 4) we consider a variety of nonlinear hyperbolic partial differential equations with mixed boundary and initial conditions. The methods of compactness and monotonicity are used to prove existence and uniqueness of the solution (Chapter 3). Finding a priori estimates is the main task in this analysis. For some types of nonlinearity, these estimates cannot be easily obtained, arid so these two methods cannot be applied directly. In this case, we first linearise the equation, using linear recurrence (Chapter 4). In the second part of the thesis (Chapter 5), by using an appropriate tool in functional analysis (the Sobolev Imbedding Theorem), we are able to improve previous results on a posteriori error estimates for the finite element method of lines applied to nonlinear parabolic equations. These estimates are crucial in the design of adaptive algorithms for the method, and previous analysis relies on, what we show to be, unnecessary assumptions which limit the application of the algorithms. Our analysis does not require these assumptions. In the last part of the thesis (Chapter 6), staying with the theme of choosing the most suitable tools, we show that using classical analysis in a proper way is in some cases sufficient to obtain considerable results. We study in this chapter nonexistence of positive solutions to Laplace's equation with nonlinear Neumann boundary condition. This problem arises when one wants to study the blow-up at finite time of the solution of the corresponding parabolic problem, which models the heating of a substance by radiation. We generalise known results which were obtained by using more abstract methods.

Natural convection inside dome shaped enclosures

In the present paper the analysis of heat transfer and free convective motion have been carried out numerically for dome shaped enclosures. The solution method is based on the finite element technique with the frontal solver and is used to examine the flow parameters and the heat transfer characteristics inside dome shaped enclosures of various offsets. In formulating the solution a general conic equation is considered to represent the dome of circular, elliptical, parabolic and hyperbolic shapes. The numerical results indicate that the circular and elliptical shapes of dome give higher heat transfer rate and offset of the dome effects convective heat transfer quite significantly. However, beyond 0.3 top dome offset, the change in overall heat transfer rate is not significant. In addition, the convective phenomenon influenced by a dome shaped cover results in establishing a secondary core region even at a moderate Rayleigh number when compared with an equivalent rectangular enclosure. A good comparison between the present numerical predictions and the previous published data is achieved.

Natural convection in domed porous enclosures : non-darcian flow

The natural-convection flow and associated heat transfer in a fluid-saturated porous medium have been investigated using the generalized porous medium approach for a dome-shaped enclosure. Many new features have been predicted with the connective heat transfer and the shape of the top dome cover. The solutions are obtained for a wide range of Darcy and Rayleigh numbers for different offsets and eccentricities of the top dome covers. The detailed parametric study reveals that there is a significant change in heat transfer rate when the offset is between 0.2 and 0.4. Different shapes of conic section, such as circular, elliptical, parabolic, and hyperbolic are used for the top dome cover, and their effects on natural convection and heat transfer rates are studied.

A non-darcian numerical modeling in domed enclosures filled with heat-generating porous media

Numerical study of the natural-convection flow and heat transfer in a dome-shaped, heat-generating, porous enclosure is considered. The general conic equation for the top dome is used to consider various conical top sections such as circular, elliptical, parabolic, and hyperbolic. The individual effect of fluid Rayleigh, Darcy, and heat-generating parameters on flow patterns and heat transfer rates are analyzed and presented. The predicted results show that the heat-generating parameter has the most significant contribution toward the growth of bicellular core flow. Moreover, there is significant change in temperature distribution in comparison to rectangular enclosures, due to the existence of the domed-shape top adiabatic cover. The results also show that, regardless of Darcy and Rayleigh values, a flat adiabatic top cover tends to yield the highest value of Nusselt number, followed by circular, elliptical, parabolic, and hyperbolic top covers, respectively.

Numerical investigation of natural convection inside complex enclosures

In this article, the analyses of heat transfer and free convective motion have been carried out numerically for various structures. The solution is based on a finite element method with the frontal solver to examine the flow parameters and heat transfer characteristics. Several dome configurations--such as flat, inclined, and dome shapes--are considered for the top of the enclosure. A general conic equation is considered to represent the dome as circular, elliptical, parabolic, or hyperbolic shape. The findings from this study indicate that the convective phenomenon is greatly influenced by the shape of the top cover dome and tends to form a secondary core even at a moderate Rayleigh number when compared with an equivalent rectangular enclosure. In addition, the circular and elliptical shapes of the dome give higher heat transfer rate. The effect of various "offset" of the dome and inclined roof on convective heat transfer is also found to be quite significant. However, beyond 0.3 of offset of the top cover for the dome and inclined roof, the change in overall heat transfer rate is minimal. The heat transfer coefficients of dome shaped and inclined roof enclosures are given and discussed.

The untameable logic of sacrifice

Paolo Diego Bubbio's Sacrifice in the Post-Kantian Tradition offers a valuable and insightful discussion of the place of sacrifice plays in nineteenth century European philosophy, setting the stage for its emergence as a central theme in subsequent continental thought. Bubbio offers a strong case for the claim that the foundational move of the post-Kantian tradition is a fundamentally kenotic one. Bubbio is also critical of certain excesses in the way sacrifice is discussed in more recent work. However, the case of Kierkegaard in particular suggests kenosis is not so easily kept within the comfortable boundaries Bubbio prescribes for it: its excesses may be an integral part, rather than a hyperbolic distortion, of the logic of sacrifice.

Quantum emission from defects in single-crystalline hexagonal boron nitride

Bulk hexagonal boron nitride (hBN) is a highly nonlinear natural hyperbolic material that attracts major attention in modern nanophotonics applications. However, studies of its optical properties in the visible part of the spectrum and quantum emitters hosted by bulk hBN have not been reported to date. In this work, we study the emission properties of hBN crystals in the red spectral range using sub-band-gap optical excitation. Quantum emission from defects is observed at room temperature and characterized in detail. Our results advance the use of hBN in quantum nanophotonics technologies and enhance our fundamental understanding of its optical properties.