Formal self duality

We study the notion of formal self duality in finite abelian groups. Formal duality in finite abelian groups has been proposed by Cohn, Kumar, Reiher and Schürmann. In this paper we give a precise definition of formally self dual sets and discuss results from the literature in this perspective. Also, we discuss the connection to formally dual codes. We prove that formally self dual sets can be reduced to primitive formally self dual sets similar to a previously known result on general formally dual sets. Furthermore, we describe several properties of formally self dual sets. Also, some new examples of formally self dual sets are presented within this paper. Lastly, we study formally self dual sets of the form $\{(x,F(x)) \ : \ x\in {\mathbb {F}}_{2^{n}}\}$ { ( x , F ( x ) ) : x ∈ F 2 n } where F is a vectorial Boolean function mapping ${\mathbb {F}}_{2^{n}}$ F 2 n to ${\mathbb {F}}_{2^{n}}$ F 2 n .

Dualities of Form: Considerations of Functional and Narrative Aspects of Design for Data Visualization

Design expertise's true value is its capacity to render complex information understandable and compelling. To make informed decisions, lay audiences depend on the clarity and resonance of data displays. But these, as created by scientists, journalists, and policy researchers, often contradict principles understood by designers to effectively and convincingly communicate data through visual means. Design education and practice insist that visual form conveys the nature of content, or data, on both objectively functional and narrative levels. This article addresses established best practices for creating clear, functional data displays, using form to clarify data's relevance and reveal its intricacy without distortion. It further advocates exploiting form's narrative potential—to qualify and contextualize information, thus enhancing how viewers internalize data, accept its veracity, and invest interest in its meaning. The author posits balancing this formal duality within data visualizations to best engage and resonate with lay audiences, and so catalyze their participation in the world around them.

Zero sign duality in visual semiotics

In this paper, the dyadic concept of zero sign in general linguistics is transformed into a triadic concept of zero sign in visual semiotics. A formal duality relation is defined between two types of visual zero sign. It is argued that the common experience of failure to perceive visual zero signs may be due to either inattention of the interpreter or camouflage of the object of the sign. We conclude that dual zero signs act as detectors in visual semiotics.

ANYON SUPERCONDUCTIVITY AND CHARGE-VORTEX DUALITY

We review recent work on a bosonic formulation of both anyon superconductivity and the fractional quantum Hall effect (FQHE). Central to this approach is the concept of charge-vortex duality in two-dimensional (2d) boson systems. A formal duality transformation which takes one from a particle to a vortex representation of a 2d boson model is described in detail. The duality transformation is employed to obtain detailed properties of both the FQHE hierarchy and a hierarchy of anyon superconducting phases.