Capturing Goodwillie's Derivative

Recent work of Biedermann and Roendigs has translated Goodwillie's calculus of functors into the language of model categories. Their work focuses on symmetric multilinear functors and the derivative appears only briefly. In this paper we focus on understanding the derivative as a right Quillen functor to a new model category. This is directly analogous to the behaviour of Weiss's derivative in orthogonal calculus. The immediate advantage of this new category is that we obtain a streamlined and more informative proof that the n-homogeneous functors are classified by spectra with an action of the symmetric group on n objects. In a later paper we will use this new model category to give a formal comparison between the orthogonal calculus and Goodwillie's calculus of functors.

A comprehensive comparison between generalized incidence calculus and Dempster-Shafer theory of evidence.

Dealing with uncertainty problems in intelligent systems has attracted a lot of attention in the AI community. Quite a few techniques have been proposed. Among them, the Dempster-Shafer theory of evidence (DS theory) has been widely appreciated. In DS theory, Dempster's combination rule plays a major role. However, it has been pointed out that the application domains of the rule are rather limited and the application of the theory sometimes gives unexpected results. We have previously explored the problem with Dempster's combination rule and proposed an alternative combination mechanism in generalized incidence calculus. In this paper we give a comprehensive comparison between generalized incidence calculus and the Dempster-Shafer theory of evidence. We first prove that these two theories have the same ability in representing evidence and combining DS-independent evidence. We then show that the new approach can deal with some dependent situations while Dempster's combination rule cannot. Various examples in the paper show the ways of using generalized incidence calculus in expert systems.

Boundary conditions in the simplest model of linear and second harmonic magneto-optical effects

This paper is concerned with linear and nonlinear magneto- optical effects in multilayered magnetic systems when treated by the simplest phenomenological model that allows their response to be represented in terms of electric polarization, The problem is addressed by formulating a set of boundary conditions at infinitely thin interfaces, taking into account the existence of surface polarizations. Essential details are given that describe how the formalism of distributions (generalized functions) allows these conditions to be derived directly from the differential form of Maxwell's equations. Using the same formalism we show the origin of alternative boundary conditions that exist in the literature. The boundary value problem for the wave equation is formulated, with an emphasis on the analysis of second harmonic magneto-optical effects in ferromagnetically ordered multilayers. An associated problem of conventions in setting up relationships between the nonlinear surface polarization and the fundamental electric field at the interfaces separating anisotropic layers through surface susceptibility tensors is discussed. A problem of self- consistency of the model is highlighted, relating to the existence of resealing procedures connecting the different conventions. The linear approximation with respect to magnetization is pursued, allowing rotational anisotropy of magneto-optical effects to be easily analyzed owing to the invariance of the corresponding polar and axial tensors under ordinary point groups. Required representations of the tensors are given for the groups infinitym, 4mm, mm2, and 3m, With regard to centrosymmetric multilayers, nonlinear volume polarization is also considered. A concise expression is given for its magnetic part, governed by an axial fifth-rank susceptibility tensor being invariant under the Curie group infinityinfinitym.

Experimental characterization of the nonlinear optical and magneto-optical properties of interfaces

An effective ellipsometric technique to determine parameters that characterize second-harmonic optical and magneto-optical effects in centrosymmetric media within the electric-dipole approximation is proposed and outlined in detail. The parameters, which are ratios of components of the nonlinear-surface-susceptibility tensors, are obtained from experimental data related to the state of polarization of the second-harmonic-generated radiation as a function of the angle between the plane of incidence and the polarization plane of the incident, linearly polarized, fundamental radiation. Experimental details of the technique are described. A corresponding theoretical model is given as an example for a single isotropic surface assuming polycrystalline samples. The surfaces of air-Au and air-Ni (in magnetized and demagnetized states) have been investigated ex situ in ambient air, and the results are presented. A nonlinear, least-squares-minimization fitting procedure between experimental data and theoretical formulas has been shown to yield realistic, unambiguous results for the ratios corresponding to each of the above materials. Independent methods for verifying the validity of the fitting parameters are also presented. The influence of temporal variations at the surfaces on the state of polarization (due to adsorption, contamination, or oxidation) is also illustrated for the demagnetized air-Ni surface. (C) 2005 Optical Society of America

Assignment methods for incidence calculus

Incidence calculus is a mechanism for probabilistic reasoning in which sets of possible worlds, called incidences, are associated with axioms, and probabilities are then associated with these sets. Inference rules are used to deduce bounds on the incidence of formulae which are not axioms, and bounds for the probability of such a formula can then be obtained. In practice an assignment of probabilities directly to axioms may be given, and it is then necessary to find an assignment of incidence which will reproduce these probabilities. We show that this task of assigning incidences can be viewed as a tree searching problem, and two techniques for performing this research are discussed. One of these is a new proposal involving a depth first search, while the other incorporates a random element. A Prolog implementation of these methods has been developed. The two approaches are compared for efficiency and the significance of their results are discussed. Finally we discuss a new proposal for applying techniques from linear programming to incidence calculus.

Constructing probabilistic ATMS using extended incidence calculus

This paper discusses the relations between extended incidence calculus and assumption-based truth maintenance systems (ATMSs). We first prove that managing labels for statements (nodes) in an ATMS is equivalent to producing incidence sets of these statements in extended incidence calculus. We then demonstrate that the justification set for a node is functionally equivalent to the implication relation set for the same node in extended incidence calculus. As a consequence, extended incidence calculus can provide justifications for an ATMS, because implication relation sets are discovered by the system automatically. We also show that extended incidence calculus provides a theoretical basis for constructing a probabilistic ATMS by associating proper probability distributions on assumptions. In this way, we can not only produce labels for all nodes in the system, but also calculate the probability of any of such nodes in it. The nogood environments can also be obtained automatically. Therefore, extended incidence calculus and the ATMS are equivalent in carrying out inferences at both the symbolic level and the numerical level. This extends a result due to Laskey and Lehner.

Two-stage mixed discrete-continuous identification of radial basis function (RBF) neural models for nonlinear systems

The identification of nonlinear dynamic systems using radial basis function (RBF) neural models is studied in this paper. Given a model selection criterion, the main objective is to effectively and efficiently build a parsimonious compact neural model that generalizes well over unseen data. This is achieved by simultaneous model structure selection and optimization of the parameters over the continuous parameter space. It is a mixed-integer hard problem, and a unified analytic framework is proposed to enable an effective and efficient two-stage mixed discrete-continuous; identification procedure. This novel framework combines the advantages of an iterative discrete two-stage subset selection technique for model structure determination and the calculus-based continuous optimization of the model parameters. Computational complexity analysis and simulation studies confirm the efficacy of the proposed algorithm.

Spin-Based Diagnostic of Nanostructure in Copper Phthalocyanine-C-60 Solar Cell Blends

Nanostructure and molecular orientation play a crucial role in determining the functionality of organic thin films. In practical devices, such as organic solar cells consisting of donor-acceptor mixtures, crystallinity is poor and these qualities cannot be readily determined by conventional diffraction techniques, while common microscopy only reveals surface morphology. Using a simple nondestructive technique, namely, continuous-wave electron paramagnetic resonance spectroscopy, which exploits the well-understood angular dependence of the g-factor and hyperfine tensors, we show that in the solar cell blend of C-60 and copper phthalocyanine (CuPc)-for which X-ray diffraction gives no information-the CuPc, and by implication the C-60, molecules form nanoclusters, with the planes of the CuPc molecules oriented perpendicular to the film surface. This information demonstrates that the current nanostructure in CuPc:C-60 solar cells is far from optimal and suggests that their efficiency could be considerably increased by alternative film growth algorithms.

Model categories for orthogonal calculus

We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. Thus we develop model structures for the category of n-polynomial and n-homogeneous functors, along with Quillen pairs relating them. We then classify n-homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an O(n)-action. This improves upon the classification theorem of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra.

Mathematics background of engineering students in Northern Ireland and Finland

The Organisation for Economic Co-operation and Development investigated numeracy proficiency among adults of working age in 23 countries across the world. Finland had the highest mean numeracy proficiency for people in the 16 – 24 age group while Northern Ireland’s score was below the mean for all the countries. An international collaboration has been undertaken to investigate the prevalence of mathematics within the secondary education systems in Northern Ireland and Finland, to highlight particular issues associated with transition into university and consider whether aspects of the Finnish experience are applicable elsewhere. In both Northern Ireland and Finland, at age 16, about half of school students continue into upper secondary level following their compulsory education. The upper secondary curriculum in Northern Ireland involves a focus on three subjects while Finnish students study a very wide range of subjects with about two-thirds of the courses being compulsory. The number of compulsory courses in maths is proportionally large; this means that all upper secondary pupils in Finland (about 55% of the population) follow a curriculum which has a formal maths content of 8%, at the very minimum. In contrast, recent data have indicated that only about 13% of Northern Ireland school leavers studied mathematics in upper secondary school. The compulsory courses of the advanced maths syllabus in Finland are largely composed of pure maths with a small amount of statistics but no mechanics. They lack some topics (for example, in advanced calculus and numerical methods for integration) which are core in Northern Ireland. This is not surprising given the much broader curriculum within upper secondary education in Finland. In both countries, there is a wide variation in the mathematical skills of school leavers. However, given the prevalence of maths within upper secondary education in Finland, it is to be expected that young adults in that country demonstrate high numeracy proficiency.

Dynamic identification of building structures: new strategies:Meccanica dei materiali e delle strutture

In this paper the evolution of a time domain dynamic identification technique based on a statistical moment approach is presented. This technique can be used in the case of structures under base random excitations in the linear state and in the non linear one. By applying Itoˆ stochastic calculus, special algebraic equations can be obtained depending on the statistical moments of the response of the system to be identified. Such equations can be used for the dynamic identification of the mechanical parameters and of the input. The above equations, differently from many techniques in the literature, show the possibility of obtaining the identification of the dissipation characteristics independently from the input. Through the paper the first formulation of this technique, applicable to non linear systems, based on the use of a restricted class of the potential models, is presented. Further a second formulation of the technique in object, applicable to each kind of linear systems and based on the use of a class of linear models, characterized by a mass proportional damping matrix, is described.

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L-2(G)) of bounded linear operators on L-2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E subset of G is a set of multiplicity if and only if the set E* = {(s,t) is an element of G x G : ts(-1) is an element of E} is a set of operator multiplicity. Analogous results are established for M-1-sets and M-0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function psi : G -> C defines a closable multiplier on the reduced C*-algebra G(r)*(G) of G if and only if Schur multiplication by the function N(psi): G x G -> C, given by N(psi)(s, t) = psi(ts(-1)), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L-2(G). Similar results are obtained for multipliers on VN(C).

Belief change with noisy sensing in the situation calculus

Situation calculus has been applied widely in arti?cial intelligence to model and reason about actions and changes in dynamic systems. Since actions carried out by agents will cause constant changes of the agents’ beliefs, how to manage
these changes is a very important issue. Shapiro et al. [22] is one of the studies that considered this issue. However, in this framework, the problem of noisy sensing, which often presents in real-world applications, is not considered. As a
consequence, noisy sensing actions in this framework will lead to an agent facing inconsistent situation and subsequently the agent cannot proceed further. In this paper, we investigate how noisy sensing actions can be handled in iterated
belief change within the situation calculus formalism. We extend the framework proposed in [22] with the capability of managing noisy sensings. We demonstrate that an agent can still detect the actual situation when the ratio of noisy sensing actions vs. accurate sensing actions is limited. We prove that our framework subsumes the iterated belief change strategy in [22] when all sensing actions are accurate. Furthermore, we prove that our framework can adequately handle belief introspection, mistaken beliefs, belief revision and belief update even with noisy sensing, as done in [22] with accurate sensing actions only.