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Graphical authentication holds some potential as an alternative to the ubiquitous password. Graphical authentication mechanisms typically present users with one or more challenge sets composed of a number of images: one target image surrounded by distractor images. Unfortunately, this means it tends to be more time-consuming than password entry and to alleviate this, we need to streamline the process as much as possible to maximize efficiency. The distractors must be chosen with care so as to ensure that users do not become confused by similarities with the target image. It is especially challenging to achieve this filtering with minimalist image types, such as hand-drawn doodles.

This paper explores the issues related to filtering the distractor images used in graphical authentication mechanisms using minimalist images. We present an algorithm for automatically classifying minimalist images in terms of visual similarity. The principles outlined here can also be used to assess the similarity of other minimalist image types such as signatures and handwritten numerals.

The twin group $T_n$ is a right angled Coxeter group generated by $n-1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this paper, we study some properties of twin groups whose analogues are well known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups $T_n$ have $R_{\infty }$-property and are not co-Hopfian for $n\ge 3$.

We construct an Alexander-type invariant for oriented doodles from a deformation of the Tits representation of the twin group and from the Chebyshev polynomials of the second kind. Like the Alexander polynomial, our invariant vanishes on unlinked doodles with more than one component. We also include values of our invariant on several doodles.

In this paper, we introduce two new polynomial invariants $Q_D(t)$ and $A_D(t)$ for one-component virtual doodles. We will also show that these polynomial invariants are not invariants of flat virtual knots.

We show that a 1-parameter family of real analytic map germs $f_t:({ℝ}^2,0)\to ({ℝ}^3,0)$ with isolated instability is topologically trivial if it is excellent and the family of double point curves $D(f_t)$ in $({ℝ}^2,0)$ is topologically trivial. In particular, we deduce that $f_t$ is topologically trivial when the Milnor number $\mu (D(f_t))$ is constant.