Connections and Divisions in On Certainty

The central claim of this paper is that On Certainty (1969)\ud is not to be seen as a "work� of Wittgenstein, but as an\ud assemblage from the enormous collection of material that\ud constitutes the Nachlass (cf. Stern 1996; Ackermann 1998;\ud Hintikka 1991). To substantiate this assertion, I will\ud critically compare the publication to its sources, namely\ud manuscripts 172, 174, 175, 176 and 177 of the Nachlass. I\ud will trace and value the decisions G.E.M. Anscombe and\ud G.H. von Wright have made in gathering this collection.\ud Their claim is twofold: first, Wittgenstein "marked it [On\ud Certainty] off in his notebooks as a separate topic�, and\ud second "it constitutes a single sustained treatment of the\ud topic� (Wittgenstein 1969, Preface). I will show that the first\ud claim is incorrect: not only are Wittgenstein"s "marks�\ud ambiguous, but the editors applied their own demarcations\ud in these notebooks as well. Their second claim is in need\ud of amplification. Wittgenstein"s concern with epistemological\ud issues is not limited to these five manuscripts: other\ud notebooks of the same period also contain entries that\ud strongly relate to topics discussed in On Certainty. I will\ud discuss examples from the undated notebooks MSS 169,\ud 170 and 171, published in Last Writings on the Philosophy\ud of Psychology, Vol. II: The "Inner� and the "Outer� (1992).\ud These show that it is difficult to make strict divisions in the\ud Nachlass in general, and in the late writings of the years\ud 1949-1951 in particular

Deep Limits of Residual Neural Networks

Neural networks have been very successful in many applications; we often, however, lack a theoretical understanding of what the neural networks are actually learning. This problem emerges when trying to generalise to new data sets. The contribution of this paper is to show that, for the residual neural network model, the deep layer limit coincides with a parameter estimation problem for a nonlinear ordinary differential equation. In particular, whilst it is known that the residual neural network model is a discretisation of an ordinary differential equation, we show convergence in a variational sense. This implies that optimal parameters converge in the deep layer limit. This is a stronger statement than saying for a fixed parameter the residual neural network model converges (the latter does not in general imply the former). Our variational analysis provides a discrete-to-continuum $\Gamma$-convergence result for the objective function of the residual neural network training step to a variational problem constrained by a system of ordinary differential equations; this rigorously connects the discrete setting to a continuum problem

Copolymer-homopolymer blends: global energy minimisation and global energy bounds

We study a variational model for a diblock-copolymer/homopolymer blend. The energy functional is a sharp-interface limit of a generalisation of the Ohta-Kawasaki energy. In one dimension, on the real line and on the torus, we prove existence of minimisers of this functional and we describe in complete detail the structure and energy of stationary points. Furthermore we characterise the conditions under which the minimisers may be non-unique. In higher dimensions we construct lower and upper bounds on the energy of minimisers, and explicitly compute the energy of spherically symmetric configurations.Comment: 31 pages, 6 Postscript figures; to be published in Calc. Var. Partial Differential Equations. Version history: Changes in v2 w.r.t v1 only concern metadata. V3 contains some minor revisions and additions w.r.t. v2. V4 corrects a confusing typo in one of the formulas of the appendix. V5 is the definitive version that will appear in prin

Stability of monolayers and bilayers in a copolymer-homopolymer blend model

We study the stability of layered structures in a variational model for diblock copolymer-homopolymer blends. The main step consists of calculating the first and second derivative of a sharp-interface Ohta-Kawasaki energy for straight mono- and bilayers. By developing the interface perturbations in a Fourier series we fully characterise the stability of the structures in terms of the energy parameters. In the course of our computations we also give the Green's function for the Laplacian on a periodic strip and explain the heuristic method by which we found it.Comment: 40 pages, 34 Postscript figures; second version has some minor corrections; to appear in "Interfaces and Free Boundaries

Anisotropic Total Variation Regularized L^1-Approximation and Denoising/Deblurring of 2D Bar Codes

We consider variations of the Rudin-Osher-Fatemi functional which are particularly well-suited to denoising and deblurring of 2D bar codes. These functionals consist of an anisotropic total variation favoring rectangles and a fidelity term which measure the L^1 distance to the signal, both with and without the presence of a deconvolution operator. Based upon the existence of a certain associated vector field, we find necessary and sufficient conditions for a function to be a minimizer. We apply these results to 2D bar codes to find explicit regimes ---in terms of the fidelity parameter and smallest length scale of the bar codes--- for which a perfect bar code is recoverable via minimization of the functionals. Via a discretization reformulated as a linear program, we perform numerical experiments for all functionals demonstrating their denoising and deblurring capabilities.Comment: 34 pages, 6 figures (with a total of 30 subfigures); errors corrected in Version 3, see Errata 1.1, 4.4, and 6.6 (v3 numbering) for more informatio

National Innovation System: an account of Thailand

The study of National Innovation Systems (NIS) is quite recent and Lundvall suggested a first integrated framework in 1988. Most of the previous studies on NIS examine the roles, interactions of various institutions such as government, universities, private firms, financial intermediaries to study the NIS of a nation. Whilst discussing the attributes of NIS and analysing the Thai innovation system the systematic problems of the Thai NIS are explored. A framework is presented which mainly focuses on building the capabilities of Thailand to increase its economic performance