Some of Dailey’s current research interests
Helping take grapher http://granite.sru.edu/dailey09/grapher.html into
the next phase of research on navigation (gravity in graphs) http://srufaculty.sru.edu/david.dailey/gravity.html
See gravity graphs below. This year the primary emphasis should be on
visualization of "traffic" in graphs such as websites and social
networks.
Finding gravitational flavorings
of graphs that motivate simplified navigation of those graphs. Humans
use cues to navigate in our physical world. We use landmarks,
compasses, and even our sense of gravity. Consider abstract metric
spaces (like graphs). May we impose "field effects" like magnetism and
gravity into graphs so that navigation is simplified? In particular
might a single dimension of "flavoring" (gravity) of a graph , by
assigning ordinal ranks to the nodes, allow a greedy algorithm to find
a shortest path between any pair of nodes p and q in exactly d(p, q)
steps? It turns out that the N-lattice is flavorable with a single
dimension of gravity. See more at http://cs.sru.edu/~ddailey/svg/open/GravityTalk.svg
and http://srufaculty.sru.edu/david.dailey/gravity.html
Build the Parisian
street model of font-design and web layout. See paper
and software
Consider using Voronoi
tessellations, but basically, design a font-editor which also
creates rich geometries for flowed content. Think: nonrectilinear
layout models for HTML, CSS, and SVG. See also
here
for some allied issues. This year it would be nice to fix existing
issues and to add the alternative of using Voronoi diagrams instead of
line-arrangements.
Web combinatorics: Rectangular
and polygonal partitions of the plane. How many different HTML tables
are there, allowing for rotation and rescaling? How many
different ways may a rectangle be carved into N rectangles? How many
different ways may a rectangle be carved into N k-sided polygons? How
might we, algorithmically, and quickly, choose a random partition of a
rectangle into N k-sided polygons? One application: transitions (see
below). Also issues like how to best to use SVG filters and their
cousins in CSS to present things like Venn diagrams.
(See also this.)
Transitions: See these.
Develop a large collection of open source transitions (fades, wipes,
dissolves, etc.) from one image to another. Develop a simplified
grammar to describe the class of all smooth inter-image transitions
(including with memory for k previous images). Develop JavaScript which
maps those transitions to SVG, CSS3 and
<canvas>.
working on string (and graph)
similarity metrics w Dr. Whitfield and me. See any of us for a
description, but you can glimpse some of it here:
http://srufaculty.sru.edu/david.dailey/javascript/StringDistances.html
talk for
PACISE
The basic premise is this. Take a set of discrete structures (graphs,
strings, groups, discrete metric spaces). Consider the union
of all their substructures. Measure the number of occurrences of each
substructure within each structure. (For example, the number of
occurances of K1 in a graph is its number of nodes; the occurances of
K2 represents its number of lines.) Consider each substructure to be a
dimension in a vector space. These measures of structures give to each
structure a position in that vector space. The distances between any
two structures can then be calculated using, for example, a Minkowski
distance (Euclidean when r=2, Manhattan when r=1). If the Manhattan
distance is used, then the resulting metric space is discrete, hence
modelable by a graph. See embedding metric spaces in graphs, below.
fast exponentiation of integers
using Dailey numeric notation
see http://srufaculty.sru.edu/david.dailey/fibos/outline.htm
condensations of numeric fields
of irrationals http://srufaculty.sru.edu/david.dailey/javascript/exponents.html
and http://srufaculty.sru.edu/david.dailey/math.htm
psychophysical studies of
equidistant objects http://srufaculty.sru.edu/david.dailey/EquidistantStimuli.html
Each discrete metric
space (in which all distances are integers) can have its metric
properties exactly modeled by a graph, in the sense that graph
theoretic distances (shortest paths) can perfectly represent the
distances of the metric space (by introducing new "leprechaun" nodes --
sort of like inventing imaginary axes to describe certain simpler
spaces, like Euclidean n-space). One question then is "what is the
smallest graph that models a given metric space." In this 1994 paper,
encouraged by and collaborated on with Frank Harary, I proved a few
things including the very poetic sounding theorem: "the tree is the
smallest container of the metric space defined on its
leaves." “On the Graphical Containment of Discrete Metric
Spaces” David Dailey,.Discrete Mathematics, 1994, 131,
51-66. Elsevier Science.
Client-side
vectorization (using SVG filters). Maybe use <canvas> in
conjunction with SVG? How to do efficient edge detection in images in
the browser.
Organizing all the 1000
plus pages under the umbrella of http://srufaculty.sru.edu/david.dailey/svg/
The site has been growing for 15 years now (with much content predating
that and having been created over 25 years at four institutions). It is
all in need of reworking, reorganizing, styling and so forth.