Rank deficiency in sparse random GF[2] matrices

Let M be a random m×n matrix with binary entries and i.i.d. rows. The weight (i.e., number of ones) of a row has a specified probability distribution, with the row chosen uniformly at random given its weight. Let N(n,m) denote the number of left null vectors in {0,1}m for M (including the zero vector), where addition is mod 2. We take n,m→∞, with m/n→α>0, while the weight distribution converges weakly to that of a random variable W on {3,4,5,…}. Identifying M with a hypergraph on n vertices, we define the 2-core of M as the terminal state of an iterative algorithm that deletes every row incident to a column of degree 1. We identify two thresholds α∗ and α−, and describe them analytically in terms of the distribution of W. Threshold α∗ marks the infimum of values of α at which n−1logE[N(n,m)] converges to a positive limit, while α− marks the infimum of values of α at which there is a 2-core of non-negligible size compared to n having more rows than non-empty columns. We have 1/2≤α∗≤α−≤1, and typically these inequalities are strict; for example when W=3 almost surely, α∗≈0.8895 and α−≈0.9179. The threshold of values of α for which N(n,m)≥2 in probability lies in [α∗,α−] and is conjectured to equal α−. The random row-weight setting gives rise to interesting new phenomena not present in the case of non-random W that has been the focus of previous work

Forensic bitemark identification: weak foundations, exaggerated claims

Abstract Several forensic sciences, especially of the pattern-matching kind, are increasingly seen to lack the scientific foundation needed to justify continuing admission as trial evidence. Indeed, several have been abolished in the recent past. A likely next candidate for elimination is bitemark identification. A number of DNA exonerations have occurred in recent years for individuals convicted based on erroneous bitemark identifications. Intense scientific and legal scrutiny has resulted. An important National Academies review found little scientific support for the field. The Texas Forensic Science Commission recently recommended a moratorium on the admission of bitemark expert testimony. The California Supreme Court has a case before it that could start a national dismantling of forensic odontology. This article describes the (legal) basis for the rise of bitemark identification and the (scientific) basis for its impending fall. The article explains the general logic of forensic identification, the claims of bitemark identification, and reviews relevant empirical research on bitemark identification—highlighting both the lack of research and the lack of support provided by what research does exist. The rise and possible fall of bitemark identification evidence has broader implications—highlighting the weak scientific culture of forensic science and the law's difficulty in evaluating and responding to unreliable and unscientific evidence