When are Multiples of Polygonal Numbers again Polygonal Numbers?

Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation P=mP' is satisfied by infinitely many pairs of triangular numbers P, P'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit in quadratic number fields, finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP', P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P, P', P'' of polygonal numbers, and give examples of such solutions.Comment: 17 pages, 1 figure, 2 tables. New version added a table of solutions to the second proble

Queering Veganism: A Biographical, Visual and Autoethnographic Study of Animal Advocacy.

I am vegan. This means I eschew animal products (such as meat, dairy and eggs) for ethical reasons. Academic interest in animal advocacy is expanding, as evidenced in the emerging field of Critical Animal Studies (Taylor and Twine, 2014). However, concurrent with a ‘criminalization’ of legitimate protest since 9/11 (Gilmore, 2013), empirical research suggests a tendency for mainstream media sources to ridicule, misrepresent and discredit vegans (Cole and Morgan, 2011). I examine the events and experiences that have been significant in shaping the biographies of vegan animal advocates. I use biographical interviews with twelve (12) vegans alongside visual methods, and autoethnography. Participants created comics -the narrative juxtaposition of words and images- about their lives, and I created an ‘autoethnographic’ comic about my biography as a vegan researcher, thus examining animal advocacy from a reflexive, situated vegan perspective. I found that vegan identity is often subject to normalizing processes (Foucault, 1977), and is necessarily fluid across social situations (as evidenced in descriptions of ‘coming out’ vegan). Vegan identity is performed and achieved in various embodied ways. These processes intersect with other social structures such as gender and sexuality. Access to cultural narratives about veganism is also significant in the experience of participants. The project contributes to the diverse fields of Biographical Research and Critical Animal Studies, adding rich biographical and visual data to existing empirical evidence around animal advocacy. It sets a precedent for the potential use of comics in research, particularly in connection with queer methodological approaches that challenge existing representational forms and focus on fluidity. It also offers novel applications for autoethnographic and visual biographical approaches

Real-Time Control of a Virtual Hand Using Surface Electromyography

Most multi-articulate prostheses allow the user to control the prosthesis through a range of pre-determined grip patterns with fixed force outputs. Although these pre-determined movements can make the prosthesis more reliable, user commands are limited to these grips and cannot be controlled naturally in real-time. Using surface electromyography (sEMG) and a modified Kalman Filter, upper limb amputees can intuitively control arm prostheses with independent, proportional control. We created an inexpensive sleeve of 32, dry sEMG electrodes (plus reference and ground) and built a graphical user interface in Matlab to train and control an 8 degree-of-freedom virtual arm (MuJoCo, Roboti). First, the user trains the Kalman filter by mimicking a predetermined set of movements while recording muscle signals. The mean-absolute-value of all possible differential pairs (528 features) was calculated over a 300-ms window and aligned with the movement data. Gram Schmidt forward selection identified the 48 most unique and useful features which were used to train a steady-state Kalman filter and control the virtual arm in real-time. We incorporated thresholds and a latching filter to reduce noise in the system. This system will enable our lab to study proportional control algorithms and low-cost, non-invasive sensory feedback in a virtual environment

Impedance Tube Alternative via the Transfer Function Method

Typical industry-ready impedance tube systems for measuring sound absorption cost between $20,000 and$60,000. The scope of this project was to explore an industrial impedance tube and its functions as well as iteratively reduce the cost of each component of the system to its lowest possible conclusion without severely sacrificing quality of data acquisition. Replacement of the hardware and software with “off-the-shelf” alternatives and custom-made components can reduce the overall cost by as much as two orders of magnitude in order to be used in the education sector, mainly high schools. We find that the designed tube and driver replacements resulted in less than optimal performance as compared to an industrial system. We also explore alternatives to the proprietary software used for our application and improvements to the originally designed system

Tuning transcriptional regulation through signaling: A predictive theory of allosteric induction

Allosteric regulation is found across all domains of life, yet we still lack simple, predictive theories that directly link the experimentally tunable parameters of a system to its input-output response. To that end, we present a general theory of allosteric transcriptional regulation using the Monod-Wyman-Changeux model. We rigorously test this model using the ubiquitous simple repression motif in bacteria by first predicting the behavior of strains that span a large range of repressor copy numbers and DNA binding strengths and then constructing and measuring their response. Our model not only accurately captures the induction profiles of these strains but also enables us to derive analytic expressions for key properties such as the dynamic range and $[EC_{50}]$. Finally, we derive an expression for the free energy of allosteric repressors which enables us to collapse our experimental data onto a single master curve that captures the diverse phenomenology of the induction profiles.Comment: Substantial revisions for resubmission (3 new figures, significantly elaborated discussion); added Professor Mitchell Lewis as another author for his continuing contributions to the projec

#Notallcops: Exploring ‘Rotten Apple’ Narratives In Media Reporting Of Lush’s 2018 ‘Spycops’ Undercover Policing Campaign

This article offers a commentary on the media framing of high-street ‘ethical cosmetics’ firm Lush’s 2018 ‘paid to lie’ campaign. The viral nature of Lush’s intervention into the undercover policing of activism in the UK highlights the significance of media reporting in the construction of narratives surrounding policing and activism. Based on a qualitative content analysis of articles published online in the immediate aftermath of the campaign launch, this article argues that the intensely polarised debate following Lush’s ‘paid to lie’ campaign was representative of a wider discursive framing battle that persists. Within this battle, the state and police establishment promote ‘rotten apple’ explanations of the undercover policing scandal, which seek to individualise blame, and shirk institutional accountability (Punch, 2003). This is significant as identifying systemic dimensions to the spycops scandal is a key focus for activists involved in the on-going Undercover Policing Inquiry (UCPI) (Schlembach, 2016)

Comics and Visual Biography: Sequential Art in Social Research

This article describes the use of comics in a mixed methods biographical research project aimed at understanding the experiences of vegans (i.e. people who eschew animal products such as meat, dairy and eggs). It begins with a discussion of Comics Studies as a growing interdisciplinary field of academic inquiry, and attempts to trace a connection between this and Visual Sociology more broadly. It then provides examples of the way in which comics were used in the project and the rationale underpinning this. Participants were asked to create comics about their lives, which aimed to supplement biographical interviews that had already taken place, eliciting rich ‘visual biographical’ data that an interview would not produce. Comics were also used as a mode of representation, whereby a ‘visual autoethnography’ was produced, outlining the author’s reflexive autobiographical relationship with veganism, and telling the story of the research. This project presented challenges, specifically around participation, ethics and anonymity, and data analysis. Despite this, the comics produced provided an unusual and valuable insight into the lives and experiences of vegans. To conclude, this article argues that the visual biographical data yielded through the use of the Comics medium represents a valuable tool in visual sociology

The Energetics of Molecular Adaptation in Transcriptional Regulation

Mutation is a critical mechanism by which evolution explores the functional landscape of proteins. Despite our ability to experimentally inflict mutations at will, it remains difficult to link sequence-level perturbations to systems-level responses. Here, we present a framework centered on measuring changes in the free energy of the system to link individual mutations in an allosteric transcriptional repressor to the parameters which govern its response. We find the energetic effects of the mutations can be categorized into several classes which have characteristic curves as a function of the inducer concentration. We experimentally test these diagnostic predictions using the well-characterized LacI repressor of Escherichia coli, probing several mutations in the DNA binding and inducer binding domains. We find that the change in gene expression due to a point mutation can be captured by modifying only a subset of the model parameters that describe the respective domain of the wild-type protein. These parameters appear to be insulated, with mutations in the DNA binding domain altering only the DNA affinity and those in the inducer binding domain altering only the allosteric parameters. Changing these subsets of parameters tunes the free energy of the system in a way that is concordant with theoretical expectations. Finally, we show that the induction profiles and resulting free energies associated with pairwise double mutants can be predicted with quantitative accuracy given knowledge of the single mutants, providing an avenue for identifying and quantifying epistatic interactions.Comment: 11 pages, 6 figures, supplemental info. available via http://rpgroup.caltech.edu/mwc_mutant