Isoperimetric Problem and Meta-fibonacci Sequences

Let G = (V,E) be a simple, finite, undirected graph. For S ⊆ V, let $\delta(S,G) = \{ (u,v) \in E : u \in S \mbox { and } v \in V-S \}$ and $\phi(S,G) = \{ v \in V -S: \exists u \in S$ , such that (u,v) ∈ E} be the edge and vertex boundary of S, respectively. Given an integer i, 1 ≤ i ≤ ∣ V ∣, the edge and vertex isoperimetric value at i is defined as b e (i,G) =  min S ⊆ V; |S| = i |δ(S,G)| and b v (i,G) =  min S ⊆ V; |S| = i |φ(S,G)|, respectively. The edge (vertex) isoperimetric problem is to determine the value of b e (i, G) (b v (i, G)) for each i, 1 ≤ i ≤ |V|. If we have the further restriction that the set S should induce a connected subgraph of G, then the corresponding variation of the isoperimetric problem is known as the connected isoperimetric problem. The connected edge (vertex) isoperimetric values are defined in a corresponding way. It turns out that the connected edge isoperimetric and the connected vertex isoperimetric values are equal at each i, 1 ≤ i ≤ |V|, if G is a tree. Therefore we use the notation b c (i, T) to denote the connected edge (vertex) isoperimetric value of T at i. Hofstadter had introduced the interesting concept of meta-fibonacci sequences in his famous book “Gödel, Escher, Bach. An Eternal Golden Braid”. The sequence he introduced is known as the Hofstadter sequences and most of the problems he raised regarding this sequence is still open. Since then mathematicians studied many other closely related meta-fibonacci sequences such as Tanny sequences, Conway sequences, Conolly sequences etc. Let T 2 be an infinite complete binary tree. In this paper we related the connected isoperimetric problem on T 2 with the Tanny sequences which is defined by the recurrence relation a(i) = a(i − 1 − a(i − 1)) + a(i − 2 − a(i − 2)), a(0) = a(1) = a(2) = 1. In particular, we show that b c (i, T 2) = i + 2 − 2a(i), for each i ≥ 1. We also propose efficient polynomial time algorithms to find vertex isoperimetric values at i of bounded pathwidth and bounded treewidth graphs.

A FEM-Based Forward Solver for Studying the Forward Problem of Electrical Impedance Tomography (EIT) with A Practical Biological Phantom

A Finite Element Method based forward solver is developed for solving the forward problem of a 2D-Electrical Impedance Tomography. The Method of Weighted Residual technique with a Galerkin approach is used for the FEM formulation of EIT forward problem. The algorithm is written in MatLAB7.0 and the forward problem is studied with a practical biological phantom developed. EIT governing equation is numerically solved to calculate the surface potentials at the phantom boundary for a uniform conductivity. An EIT-phantom is developed with an array of 16 electrodes placed on the inner surface of the phantom tank filled with KCl solution. A sinusoidal current is injected through the current electrodes and the differential potentials across the voltage electrodes are measured. Measured data is compared with the differential potential calculated for known current and solution conductivity. Comparing measured voltage with the calculated data it is attempted to find the sources of errors to improve data quality for better image reconstruction.

Problem-based learning in an e-learning environment: A case study at Griffith University School of Medicine

Increasing numbers of medical schools in Australia and overseas have moved away from didactic teaching methodologies and embraced problem-based learning (PBL) to improve clinical reasoning skills and communication skills as well as to encourage self-directed lifelong learning. In January 2005, the first cohort of students entered the new MBBS program at the Griffith University School of Medicine, Gold Coast, to embark upon an exciting, fully integrated curriculum using PBL, combining electronic delivery, communication and evaluation systems incorporating cognitive principles that underpin the PBL process. This chapter examines the educational philosophies and design of the e-learning environment underpinning the processes developed to deliver, monitor and evaluate the curriculum. Key initiatives taken to promote student engagement and innovative and distinctive approaches to student learning at Griffith promoted within the conceptual model for the curriculum are (a) Student engagement, (b) Pastoral care, (c) Staff engagement, (d) Monitoring and (e) Curriculum/Program Review. © 2007 Springer-Verlag Berlin Heidelberg.

Pengarna eller livet? : En kvalitativ och longitudinell studie om långtidsarbetslösa unga i ett aktörsperspektiv

Your money or your life? A qualitative follow-up study of the young unemployed from an actor perspective is a qualitative and longitudinal study following 36 unemployed young people in Helsinki over a span of ten years. The purpose of the study is to shed light on how a few young people view employment/unemployment and their lives and future, how they as unemployed perceive their encounters with society, and how society supports them. Four so-called key informants were followed at a finer level of empirical detail. They were chosen for the thematic interviews because of their different personalities, starting points and preferences. Although some differences were expected, what the results show is quite striking. The individual stories raise a number of questions about differences between young people, about society s view of the young unemployed, and about the principles behind the so-called activation policy and how society s support is distributed. The key informants descriptions underline that the group young unemployed does not consist of individuals who are alike but that life is complex, that paid work and unemployment can be perceived very differently, and that background and unofficial support can have consequences for self-perception and for ways of looking at the future, vocational choices, paid work and activation policy. Margaret S. Archer s theory of Morphogenesis and Barbara Cruikshank s theory of constructing democracies compose the study s theoretical framework. The key informants stories give a picture of a formal support system that, even though it puts part of the responsibility for unemployment on the individuals themselves, in the name of fairness and equality, treats them in an impersonal way, not giving their personal situation and wishes much weight. As a consequence, those who share the dominant values of society do well, while others who do not are faced with difficulties. The bigger the gap between society s and the individual s values, the bigger the risk to be met by little understanding and by penalties. And vice versa: Those who initially have the right values and know how to deal with authorities get heard and their opinions get accepted. The informants ask for a more personal encounter, which could improve both the atmosphere and the clients experiences of being heard. Still the risk of having a more individualistic system should be addressed, as a new system might generate new winners, but just as well give new losers. Finally, we have to ask if the so-called activation policy is looking for answers primarily to a macro-level problem on the micro-level. If it does not produce more jobs, its support for the unemployed will be insignificant. It is not enough to think about what to do at the grassroots level to make the system more functional and support job-seeking. If the current rate of unemployment endures, the quality of life of the unemployed should be addressed. A first step could be taken by placing less guilt on the unemployed. Instead of talking about activating the unemployed, discussion should be targeted at removing structural impediments to employment. If we want to have less polarisation between the those with paid work and those without, who often struggle with low incomes, we need to include the macro-level in the discussion. What does high unemployment mean in a work-based society, where the individual s self-perception and important social forms of support are linked to labour income? And what can be done at the macro-level to change this undesirable condition at the micro-level? Keywords: Unemployment, Youth, Public interventions, Activation policy, Individual actors, Qualitative, Longitudinal, Holistic, Helsinki, Finland

Effect of boundary condition description on convergence of solution in a boundary-value problem

It is well known that the numerical accuracy of a series solution to a boundary-value problem by the direct method depends on the technique of approximate satisfaction of the boundary conditions and on the stage of truncation of the series. On the other hand, it does not appear to be generally recognized that, when the boundary conditions can be described in alternative equivalent forms, the convergence of the solution is significantly affected by the actual form in which they are stated. The importance of the last aspect is studied for three different techniques of computing the deflections of simply supported regular polygonal plates under uniform pressure. It is also shown that it is sometimes possible to modify the technique of analysis to make the accuracy independent of the description of the boundary conditions.

The Problem of Implementation and its Relation to the Philosophy of Cognitive Science

According to certain arguments, computation is observer-relative either in the sense that many physical systems implement many computations (Hilary Putnam), or in the sense that almost all physical systems implement all computations (John Searle). If sound, these arguments have a potentially devastating consequence for the computational theory of mind: if arbitrary physical systems can be seen to implement arbitrary computations, the notion of computation seems to lose all explanatory power as far as brains and minds are concerned. David Chalmers and B. Jack Copeland have attempted to counter these relativist arguments by placing certain constraints on the definition of implementation. In this thesis, I examine their proposals and find both wanting in some respects. During the course of this examination, I give a formal definition of the class of combinatorial-state automata , upon which Chalmers s account of implementation is based. I show that this definition implies two theorems (one an observation due to Curtis Brown) concerning the computational power of combinatorial-state automata, theorems which speak against founding the theory of implementation upon this formalism. Toward the end of the thesis, I sketch a definition of the implementation of Turing machines in dynamical systems, and offer this as an alternative to Chalmers s and Copeland s accounts of implementation. I demonstrate that the definition does not imply Searle s claim for the universal implementation of computations. However, the definition may support claims that are weaker than Searle s, yet still troubling to the computationalist. There remains a kernel of relativity in implementation at any rate, since the interpretation of physical systems seems itself to be an observer-relative matter, to some degree at least. This observation helps clarify the role the notion of computation can play in cognitive science. Specifically, I will argue that the notion should be conceived as an instrumental rather than as a fundamental or foundational one.

Mig eller mej, själ eller sjel? Problem och lösningar vid transkription av svenska sångtexter

Abstract (Mig or mej, själ or sjel? Problems and solutions in the transcription of Swedish song texts): In this article I am pointing out and discussing problems and solutions concerning phonetic transcription of Swedish song texts. My material consists of 66 Swedish songs phonetically transcribed. The transcriptions were published by The Academy of Finnish Art Song in 2009. The first issue was which level of accuracy should be chosen. The transcriptions were created to be clear at a glance and suitable for the needs of interpretation of non Swedish speaking singers. The principle was to use as few signs and symbols as possible without sacrificing accuracy. Certain songs were provided with additional information whenever there was a chance of misinterpretation. The second issue was which geographic variety of the language should be visible in the transcription, Standard Swedish or Finland-Swedish? The songs in the volume are a selection of well-known works that are also of international interest. Most were composed by Jean Sibelius (1865–1957), a substantial number of whose songs were based on poems written by Finland’s national poet, Johan Ludvig Runeberg (1804–1877). Thus I chose to use the variety of Swedish language spoken in Finland, in order to reflect the cultural origin of the songs. This variety differs slightly from the variety spoken in Sweden both on prosodic and phonetic level. In singing, the note-text gives the interpretor enough information about prosody. The differences concern mostly the phonemes. A fully consequent transcript was, however, difficult to make, due to vocal requirement. So, for example, in an unstressed final syllable the vowel was often indicated as a central vowel, which in singing is given a more direct emphasis than in a literal pronunciation, even if this central vowel does not occur in spoken Finland-Swedish.

Numerical procedure for second order non-linear ordinary differential equations and application to heat transfer problem

In this paper, we have first given a numerical procedure for the solution of second order non-linear ordinary differential equations of the type y″ = f (x;y, y′) with given initial conditions. The method is based on geometrical interpretation of the equation, which suggests a simple geometrical construction of the integral curve. We then translate this geometrical method to the numerical procedure adaptable to desk calculators and digital computers. We have studied the efficacy of this method with the help of an illustrative example with known exact solution. We have also compared it with Runge-Kutta method. We have then applied this method to a physical problem, namely, the study of the temperature distribution in a semi-infinite solid homogeneous medium for temperature-dependent conductivity coefficient.

Quantum kinematics of Fermi-Dirac vacuum-plus-one-electron problem

Following Weisskopf, the kinematics of quantum mechanics is shown to lead to a modified charge distribution for a test electron embedded in the Fermi-Dirac vacuum with interesting consequences.

Roman Ingarden’s Objectivity vs. Subjectivity as a problem of Translatability

Ingarden (1962, 1964) postulates that artworks exist in an “Objective purely intentional” way. According to this view, objectivity and subjectivity are opposed forms of existence, parallel to the opposition between realism and idealism. Using arguments of cognitive science, experimental psychology, and semiotics, this lecture proposes that, particularly in the aesthetic phenomena, realism and idealism are not pure oppositions; rather they are aspects of a single process of cognition in different strata. Furthermore, the concept of realism can be conceived as an empirical extreme of idealism, and the concept of idealism can be conceived as a pre-operative extreme of realism. Both kind of systems of knowledge are mutually associated by a synecdoche, performing major tasks of mental order and categorisation. This contribution suggests that the supposed opposition between objectivity and subjectivity, raises, first of all, a problem of translatability, more than a problem of existential categories. Synecdoche seems to be a very basic transaction of the mind, establishing ontologies (in the more Ingardean way of the term). Wegrzecki (1994, 220) defines ontology as “the central domain of philosophy to which other its parts directly or indirectly refer”. Thus, ontology operates within philosophy as the synecdoche does within language, pointing the sense of the general into the particular and/or viceversa. The many affinities and similarities between different sign systems, like those found across the interrelationships of the arts, are embedded into a transversal, synecdochic intersemiosis. An important question, from this view, is whether Ingardean’s pure objectivities lie basically on the impossibility of translation, therefore being absolute self-referential constructions. In such a case, it would be impossible to translate pure intentionality into something else, like acts or products.