Concept image and concept definition in mathematics with particular reference to limits and continuity

The concept image consists of all the cognitive structure in the individual's mind that is associated with a given concept. This may not be globally coherent and may have aspects which are quite different from the formal concept definition. The development of limits and continuity, as taught in secondary school and university, are considered. Various investigations are reported which demonstrate individual concept images differing from the formal theory and containing factors which cause cognitive conflict

Exactly Marginal Deformations of N=4 SYM and of its Supersymmetric Orbifold Descendants

In this paper we study exactly marginal deformations of field theories living on D3-branes at low energies. These theories include N=4 supersymmetric Yang-Mills theory and theories obtained from it via the orbifolding procedure. We restrict ourselves only to orbifolds and deformations which leave some supersymmetry unbroken. A number of new families of N=1 superconformal field theories are found. We analyze the deformations perturbatively, and also by using general arguments for the dimension of the space of exactly marginal deformations. We find some cases where the space of perturbative exactly marginal deformations is smaller than the prediction of the general analysis at least up to three-loop order), and other cases where the perturbative result (at low orders) has a non-generic form.Comment: 25 pages, 1 figure. v2: added preprint number, references adde

Planar quark scattering at strong coupling and universality

We discuss scattering of fundamental matter in the planar and strong coupling limit via the AdS/CFT correspondence, generalizing the recently proposed calculation for adjoint matter due to Alday and Maldacena [arXiv:0705.0303]. Color decomposition of quark amplitudes is a key property allowing to repeat the procedure in the case of fundamental matter and to derive the relation of these strong coupling amplitudes to minimal area problems. We present the results for two different D3-D7 systems, one is only conformal in the planar limit and the other is exactly conformal. Our results suggest a universal behavior of scattering amplitudes at strong coupling and planar limit (both for gluons and quarks).Comment: 13 pages, 4 figures, JHEP format. v2: added references and minor corrections. v3: following arXiv:0710.0393 we change our claim about a minimal surface solution without spike singularities. We make the appropriate corrections where necessar

Exceptional Indices

Recently a prescription to compute the superconformal index for all theories of class S was proposed. In this paper we discuss some of the physical information which can be extracted from this index. We derive a simple criterion for the given theory of class S to have a decoupled free component and for it to have enhanced flavor symmetry. Furthermore, we establish a criterion for the "good", the "bad", and the "ugly" trichotomy of the theories. After interpreting the prescription to compute the index with non-maximal flavor symmetry as a residue calculus we address the computation of the index of the bad theories. In particular we suggest explicit expressions for the superconformal index of higher rank theories with E_n flavor symmetry, i.e. for the Hilbert series of the multi-instanton moduli space of E_n.Comment: 33 pages, 11 figures, v2: minor correction

Better the donor you know?:A qualitative study of renal patients' views on ‘altruistic’ live-donor kidney transplantation

AbstractBackgroundIn the UK there is a short-fall between individuals requiring a renal transplant and kidneys available for transplantation. Non-directed ‘altruistic’ living kidney donation has emerged as a strategy for bridging this gap between supply and demand, with the number increasing each year.ObjectiveThis study aimed to explore the views of potential recipients towards non-directed ‘altruistic’ live-donor kidney transplantation.MethodsSemi-structured interviews with 32 UK deceased-donor kidney transplant recipients were performed. Interviews explored willingness to consider directed and non-directed live-donor kidney transplants (LDKTs). Interviews were recorded, transcribed verbatim and transcripts were analysed using the constant comparison method described in Grounded Theory.ResultsFor those not willing to accept a non-directed ‘altruistic’ LDKT, the following themes were identified: i) Prioritising other recipients above self; ii) Fear of acquiring an unknown donor's characteristics, and iii) Concern for the donor – unnecessary risk. For those willing to accept a non-directed ‘altruistic’ LDKT the following themes were identified: iv) Prioritising known above unknown persons, v) Belief that they are as deserving as other potential recipients, and vi) Advantages of a LDKT.ConclusionsDrawing on ‘gift exchange theory’, this study contributes to our understanding of the experience of the intended recipient of a gift. The anonymity of the donor-recipient appears to be seen as a benefit of non-directed ‘altruistic’ live-donor transplants, freeing recipients from the obligations of the gift. However, those who feel unworthy of the ‘gifted transplant’ are concerned about the donor and by the lack of opportunity for direct reciprocity. Highlighting the ‘reciprocal benefits’ reported by donors may allow individuals whose preference is a live-donor transplant to accept one if offered. These insights provide the transplant community with targets for intervention, through which the concerns of potential recipients might be addressed

A k-shell decomposition method for weighted networks

We present a generalized method for calculating the k-shell structure of weighted networks. The method takes into account both the weight and the degree of a network, in such a way that in the absence of weights we resume the shell structure obtained by the classic k-shell decomposition. In the presence of weights, we show that the method is able to partition the network in a more refined way, without the need of any arbitrary threshold on the weight values. Furthermore, by simulating spreading processes using the susceptible-infectious-recovered model in four different weighted real-world networks, we show that the weighted k-shell decomposition method ranks the nodes more accurately, by placing nodes with higher spreading potential into shells closer to the core. In addition, we demonstrate our new method on a real economic network and show that the core calculated using the weighted k-shell method is more meaningful from an economic perspective when compared with the unweighted one.Comment: 17 pages, 6 figure

On fluctuations of closed string tachyon solitons

We discuss fluctuations on solitons in the dilaton/graviton/tachyon system using the low energy effective field theory approach. It is shown that closed string solitons are free of tachyons in this approximation, regardless of the exact shape of the tachyon potential.Comment: 13 pages, 1 figure, uses JHEP3.cl

From Matrices to Strings and Back

We discuss an explicit construction of a string dual for the Gaussian matrix model. Starting from the matrix model and employing Strebel differential techniques we deduce hints about the structure of the dual string. Next, following these hints a worldheet theory is constructed. The correlators in this string theory are assumed to localize on a finite set of points in the moduli space of Riemann surfaces. To each such point one associates a Feynman diagram contributing to the correlator in the dual matrix model, and thus recasts the worldsheet expression as a sum over Feynman diagrams.Comment: 27 pages, 3 figure

Extremal Correlators and Hurwitz Numbers in Symmetric Product Orbifolds

We study correlation functions of single-cycle chiral operators in the symmetric product orbifold of N supersymmetric four-tori. Correlators of twist operators are evaluated on covering surfaces, generally of different genera, where fields are single-valued. We compute some simple four-point functions and study how the sum over inequivalent branched covering maps splits under OPEs. We then discuss extremal n-point correlators, i.e. correlators of n-1 chiral and one anti-chiral operators. They obey simple recursion relations involving numbers obtained from counting branched covering maps with particular properties. In most cases we are able to solve explicitly the recursion relations. Remarkably, extremal correlators turn out to be equal to Hurwitz numbers.Comment: 36 pages, 3 figures, v2: minor improvement

A Spin Chain for the Symmetric Product CFT_2

We consider "gauge invariant" operators in Sym^N T^4, the symmetric product orbifold of N copies of the 2d supersymmetric sigma model with T^4 target. We discuss a spin chain representation for single-cycle operators and study their two point functions at large N. We perform systematic calculations at the orbifold point ("tree level"), where non-trivial mixing is already present, and some sample calculations to first order in the blow-up mode of the orbifold ("one loop").Comment: 52 pages, 10 figure