Zolotarev's proof of Gauss reciprocity and Jacobi symbols

2000 Mathematics Subject Classification: Primary 11A15.

Reciprocity : where art meets the community : action research in response to artistic encounters and relationships

This practice-led research project examines some of the factors and issues facing artists working in the public domain who wish to engage with the community as audience. Using the methodology of action research, the three major creative projects in this study use art as a socio-political tool with the aim of providing an effective vehicle for broadening awareness, understanding forms of social protest and increasing tolerance for diversity. The three projects: Floodline November 7, 2004, Look in, Look out, and The Urban Terrorist Project, dealt with issues of marginalisation of communities, audiences and graffiti artists respectively. The artist/researcher is outlined as both creator and collaborator in the work. Processes included ephemeral elements, such as temporary installation and performance, as well as interactive elements that encouraged direct audience involvement as part of the work. In addition to the roles of creator and collaborator, both of which included audience as well as artist, the presence of an outside entity was evident. Whether local, legal authorities or prevailing attitudes, outside entities had an unavoidable impact on the processes and outcomes of the work. Each project elicited a range of responses from their respective audiences; however, the overarching concept of reciprocity was seen to be the crucial factor in conception, artistic methods and outcomes.

Application of a new discreet form of Gauss’ theorem for measuring volume

Volume measurements are useful in many branches of science and medicine. They are usually accomplished by acquiring a sequence of cross sectional images through the object using an appropriate scanning modality, for example x-ray computed tomography (CT), magnetic resonance (MR) or ultrasound (US). In the cases of CT and MR, a dividing cubes algorithm can be used to describe the surface as a triangle mesh. However, such algorithms are not suitable for US data, especially when the image sequence is multiplanar (as it usually is). This problem may be overcome by manually tracing regions of interest (ROIs) on the registered multiplanar images and connecting the points into a triangular mesh. In this paper we describe and evaluate a new discreet form of Gauss’ theorem which enables the calculation of the volume of any enclosed surface described by a triangular mesh. The volume is calculated by summing the vector product of the centroid, area and normal of each surface triangle. The algorithm was tested on computer-generated objects, US-scanned balloons, livers and kidneys and CT-scanned clay rocks. The results, expressed as the mean percentage difference ± one standard deviation were 1.2 ± 2.3, 5.5 ± 4.7, 3.0 ± 3.2 and −1.2 ± 3.2% for balloons, livers, kidneys and rocks respectively. The results compare favourably with other volume estimation methods such as planimetry and tetrahedral decomposition.

Tragedy and assimilation : reciprocity between surface and subject

Here we unveil a tragic triptych of three Australian women painfully painted onto the walls of interior surfaces. The woman at the centre of the triptych is Florence Broadhurst whose tragic death still remains a mystery. To the right is Australian skin illustrator Emma Hack who recreates Broadhurst’s wallpapers, mimicking their colourful patterns onto live models. Hack perfectly assimilates the models’ body into the wallpaper, camouflaging bodies except for small hints at something more in the foreground. In the process of Hack’s images, the models become statues, standing painfully still holding their breath for minutes at a time. The third woman, to the left of the triptych, is the fictional character Candy from the 2006 Australian film Candy. Candy’s traumatic struggle with addiction ends with her conveying her pain in a poem she writes on the walls of her home; culminating her tragic story into a disturbed domestic wall surface. This research tries to understand this relationship with the surface through tragedy as a reciprocal agreement between surface and subject and not a permanent transference between one state and another. What the surface provides in times of personal struggle and turmoil is a method for us to come to terms with out material existence.

Gender and other determinants of trust and reciprocity in an experimental labour market amongst Chinese students

Due to economic and demographic changes highly educated women play an important role on the Chinese labour market. Gender has been shown to be an important characteristic that influences behaviour in economic experiments, as have, to a lesser degree, academic major, age and income. We provide a study looking at trust and reciprocity and their determinants in a labour market laboratory experiment. Our experimental data is based on two games, the Gift Exchange Game (GEG) and a variant of this game (the Wage Promising Game, WPG) where the employer's wage offer is non-binding and the employer can choose the wage freely after observing the workers effort. We and that women are less trusting and reciprocal than men in the GEG while this cannot be found in the WPG. Letting participants play the GEG and the WPG, allows us to disentangle reciprocal and risk attitudes. While in the employer role, it seems to be that risk attitude is the main factor, this is not confirmed analysing decisions in the worker role.

Beyond ethnography: Engagement and reciprocity as foundations for design research out here

This paper explores an emerging paradigm for HCI design research based primarily upon engagement, reciprocity and doing. Much HCI research begins with an investigatory and analytic ethnographic approach before translating to design. Design may come much later in the process and may never benefit the community that is researched. However in many settings it is difficult for researchers to access the privileged ethnographer position of observer and investigator. Moreover rapid ethnographic research often does not seem the best or most appropriate course of action. We draw upon a project working with a remote Australian Aboriginal community to illustrate an alternative approach in Indigenous research, where the notion of reciprocity is first and foremost. We argue that this can lead to sustainable designs, valid research and profound innovation. This paper received the ACM CHI Best Paper Award, which is awarded to the top 1% of papers submitted to the ACM CHI conference.

Beyond Gauss: Image-set matching on the Riemannian manifold of PDFs

State-of-the-art image-set matching techniques typically implicitly model each image-set with a Gaussian distribution. Here, we propose to go beyond these representations and model image-sets as probability distribution functions (PDFs) using kernel density estimators. To compare and match image-sets, we exploit Csiszar´ f-divergences, which bear strong connections to the geodesic distance defined on the space of PDFs, i.e., the statistical manifold. Furthermore, we introduce valid positive definite kernels on the statistical manifold, which let us make use of more powerful classification schemes to match image-sets. Finally, we introduce a supervised dimensionality reduction technique that learns a latent space where f-divergences reflect the class labels of the data. Our experiments on diverse problems, such as video-based face recognition and dynamic texture classification, evidence the benefits of our approach over the state-of-the-art image-set matching methods.

Reciprocity relation in scattering by media subjected to an external magnetic field

Krishnan's reciprocity theorem in colloid optics, ρ{variant}u=1+l/ρ{variant}h/1+1/ρ{variant}v is generalised for the case when the scattering medium is subjected to an external orienting field. It is shown theoretically that a general relation of the type IBA=I′AB results in this case, where IBA is the intensity of the component of the scattered light having its electric vector inclined at an angle B to the vertical with the incident light polarised at an angle A to the vertical, the external field direction being parallel to the incident beam. I′AB is the corresponding intensity with the magnetic field parallel of the scattered ray. Experimental verification of the above generalisation is also given.

Gauss law commutator in the chirally gauged Wess-Zumino-Witten model

We evaluate the commutator of the Gauss law constraints starting from the chirally gauged Wess-Zumino-Witten action. The calculations are done at tree level, i.e. by evaluating corresponding Poisson brackets. The results are compared with commutators obtained by others directly from the gauged fermionic theory, and with Faddeev's results based on cohomology.

FAST AND ACCURATE BILATERAL FILTERING USING GAUSS-POLYNOMIAL DECOMPOSITION

The bilateral filter is a versatile non-linear filter that has found diverse applications in image processing, computer vision, computer graphics, and computational photography. A common form of the filter is the Gaussian bilateral filter in which both the spatial and range kernels are Gaussian. A direct implementation of this filter requires O(sigma(2)) operations per pixel, where sigma is the standard deviation of the spatial Gaussian. In this paper, we propose an accurate approximation algorithm that can cut down the computational complexity to O(1) per pixel for any arbitrary sigma (constant-time implementation). This is based on the observation that the range kernel operates via the translations of a fixed Gaussian over the range space, and that these translated Gaussians can be accurately approximated using the so-called Gauss-polynomials. The overall algorithm emerging from this approximation involves a series of spatial Gaussian filtering, which can be efficiently implemented (in parallel) using separability and recursion. We present some preliminary results to demonstrate that the proposed algorithm compares favorably with some of the existing fast algorithms in terms of speed and accuracy.

Essays on cooperation and reciprocity

This dissertation comprises three essays that use theory-based experiments to gain understanding of how cooperation and efficiency is affected by certain variables and institutions in different types of strategic interactions prevalent in our society.

Chapter 2 analyzes indefinite horizon two-person dynamic favor exchange games with private information in the laboratory. Using a novel experimental design to implement a dynamic game with a stochastic jump signal process, this study provides insights into a relation where cooperation is without immediate reciprocity. The primary finding is that favor provision under these conditions is considerably less than under the most efficient equilibrium. Also, individuals do not engage in exact score-keeping of net favors, rather, the time since the last favor was provided affects decisions to stop or restart providing favors.

Evidence from experiments in Cournot duopolies is presented in Chapter 3 where players indulge in a form of pre-play communication, termed as revision phase, before playing the one-shot game. During this revision phase individuals announce their tentative quantities, which are publicly observed, and revisions are costless. The payoffs are determined only by the quantities selected at the end under real time revision, whereas in a Poisson revision game, opportunities to revise arrive according to a synchronous Poisson process and the tentative quantity corresponding to the last revision opportunity is implemented. Contrasting results emerge. While real time revision of quantities results in choices that are more competitive than the static Cournot-Nash, significantly lower quantities are implemented in the Poisson revision games. This shows that partial cooperation can be sustained even when individuals interact only once.

Chapter 4 investigates the effect of varying the message space in a public good game with pre-play communication where player endowments are private information. We find that neither binary communication nor a larger finite numerical message space results in any efficiency gain relative to the situation without any form of communication. Payoffs and public good provision are higher only when participants are provided with a discussion period through unrestricted text chat.