Sampling Random Spanning Trees Faster than Matrix Multiplication

We present an algorithm that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in $\tilde{O}(n^{4/3}m^{1/2}+n^{2})$ time (The $\tilde{O}(\cdot)$ notation hides $\operatorname{polylog}(n)$ factors). The tree is sampled from a distribution where the probability of each tree is proportional to the product of its edge weights. This improves upon the previous best algorithm due to Colbourn et al. that runs in matrix multiplication time, $O(n^\omega)$. For the special case of unweighted graphs, this improves upon the best previously known running time of $\tilde{O}(\min\{n^{\omega},m\sqrt{n},m^{4/3}\})$ for $m \gg n^{5/3}$ (Colbourn et al. '96, Kelner-Madry '09, Madry et al. '15). The effective resistance metric is essential to our algorithm, as in the work of Madry et al., but we eschew determinant-based and random walk-based techniques used by previous algorithms. Instead, our algorithm is based on Gaussian elimination, and the fact that effective resistance is preserved in the graph resulting from eliminating a subset of vertices (called a Schur complement). As part of our algorithm, we show how to compute $\epsilon$-approximate effective resistances for a set $S$ of vertex pairs via approximate Schur complements in $\tilde{O}(m+(n + |S|)\epsilon^{-2})$ time, without using the Johnson-Lindenstrauss lemma which requires $\tilde{O}( \min\{(m + |S|)\epsilon^{-2}, m+n\epsilon^{-4} +|S|\epsilon^{-2}\})$ time. We combine this approximation procedure with an error correction procedure for handing edges where our estimate isn't sufficiently accurate

EVASION OF MORTGAGE MORATORIA BY PROSECUTION OF PERSONAL REMEDIES

For reasons political, social and economic which have never been fully analyzed, the moratory legislation of the last five years has shown special favor to the debtor whose obligation is secured by mortgage of land, and this legislation is faintly echoed in moratory decisions which have no direct statutory foundation. Granted that contracts have sometimes received similar treatment, that banks and insurance companies have also enjoyed indulgence, and that the new chapters of the Bankruptcy Act extend asylum to all and sundry, yet the mortgage is so far favored that the word moratorium brings to mind this case before all others

Note and Comment

Sales - Liability for the Presence of Mice and Other Uncommon Things in Food - A group of recent decisions presents a somewhat farcical conformity with Montesquiet\u27. thesis that law may vary with time and geography. It strikingly illustrates, also, the importance of the particular theory of liability upon which a suit is predicated. The unusual similarity in detail of the operative facts of these cases lends peculiar emphasis to the difference in the judgments rendered

Chen-Ruan cohomology of ADE singularities

We study Ruan's \textit{cohomological crepant resolution conjecture} for orbifolds with transversal ADE singularities. In the $A_n$-case we compute both the Chen-Ruan cohomology ring $H^*_{\rm CR}([Y])$ and the quantum corrected cohomology ring $H^*(Z)(q_1,...,q_n)$. The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between $H^*_{\rm CR}([Y])$ and $H^*(Z)(-1)$ in the $A_1$-case, verifying Ruan's conjecture. In the $A_n$-case, the family $H^*(Z)(q_1,...,q_n)$ is not defined for $q_1=...=q_n=-1$. This implies that the conjecture should be slightly modified. We propose a new conjecture in the $A_n$-case which we prove in the $A_2$-case by constructing an explicit isomorphism.Comment: This is a short version of my Ph.D. Thesis math.AG/0510528. Version 2: chapters 2,3,4 and 5 has been rewritten using the language of groupoids; a link with the classical McKay correpondence is given. International Journal of Mathematics (to appear

Evaluation of a Low-Cost, PC-Based Driving Simulator to Assess Persons with Cognitive Impairments Due to Brain Injury

Brain injury due to accident or stroke frequently results in cognitive impairment, reducing an individual’s ability to judge driving situations accurately. Rehabilitation professionals typically use a combination of clinical and on-road tests to determine whether an individual is safe to drive. Weighing the safety of the community, the candidate, and the driving evaluator, these on-road tests are often conducted under road, traffic and weather conditions less demanding than those that a driver might face in the “real world,” and thus may offer less than complete information regarding the candidate’s responses to such real-world driving challenges. Indeed, individuals with mild cognitive deficits may perform adequately under such testing conditions but unsafely when driving challenges increase. Complicating this situation further, those with mild to moderate acquired cognitive impairments may be largely unaware of their own limitations, and thus more intolerant of perceived delays or challenges to their desire to drive again. Although continuing advances have improved performance and fidelity while significantly reducing costs, most interactive driving simulators remain too expensive for widespread clinical application. In a project funded by the National Center for Medical Rehabilitation Research, National Institutes of Health, we sought to determine, on a pilot basis, whether a low-cost, PC-based driving simulator could provide clinicians with information useful to their efforts to assess the safe ability to drive of individuals with cognitive impairments. We developed two comprehensive simulator-based driving scenarios, one quite basic and one more challenging, and pilot-tested them on ten subjects – five with moderate cognitive impairments, and five age and sex matched-controls without impairment. The “simple” scenario was developed to match the essential demands of the first half of an existing on-road driving evaluation; the “complex” scenario was based on the second half of the on-road evaluation into which more demanding, but still common, driving challenges were integrated. Road types, lane widths, pavement markings, traffic signals, horizontal and vertical curvature, and the proximal built environment were all created in simulation to provide a convincing generic representation of the on-road test. Challenges incorporated into the “complex” phase of the scenario, which were absent from the “simple” phase, included traffic events such as: cross-traffic failing to stop at a STOP sign; pedestrians crossing the driver’s path; vehicles suddenly pulling out in front of the subject from the road shoulder; opposing thru traffic appearing suddenly from behind slower moving vehicles as the subject attempted to turn left; slower moving lead vehicles causing passing decisions; traffic streams forcing gap acceptance decisions; etc. Results from the simulator were compared to results from the on-road evaluation. In addition, data gathered from subject exit interviews was used to judge simulator verisimilitude and efficacy in changing self-awareness of deficit. Because the cognitive impairments associated with brain injury often reduce the individual’s awareness of his or her own limitations, we looked at evidence that performance on the simulator could contribute to an individual’s own understanding of his or her driving strengths and weaknesses. The results of the pilot study will lead to an enhancement of simulator capabilities, and to a comprehensive clinical trial at multiple sites. This paper will present the findings of this pilot investigation and an overview of the expanded clinical study

Note and Comment

The Basis of Relief from Penalties and Forfeitures - The equitable principle of relief from penalties and forfeitures is so far elementary as almost to defy analysis. Many, perhaps most, of the judicial explanations of the principle have based it upon interpretation or construction, appealing to the doctrine that equity regards intent rather than form. Yet a logical application of this doctrine would lead to results very different from those which have actually been arrived at in the decisions. Thus, a stipulation in a mortgage that the mortgagor waives his equity of redemption can hardly be interpreted as meaning that he does not waive it, yet all such stipulations are ignored and redemption granted, nevertheless. Again, a penalty for breach of contract cannot be saved by the most solemn declaration that it is intended as liquidated damages. It must be conceded that many cases have actually been, decided on the interpretation theory, producing such enormities as Iowa Land Co. v. Mickel, 41 Ia. 4o2 (sale of land, $15a out of$6oo paid, $4ooo in improvements, second instalment of$150 one day late due to a misunderstanding; held forfeited), and Doctornan v. Schroeder, 114 AUt. 8io (N. J., 1921; sale of land, $IOOO paid, last instalment of$iooo thirty minutes late; held forfeited). But such cases represent the minority view, and one applicable only to instalment sales, no court pursuing this course in mortgage or liquidated damage cases

Calorimetry of Bose-Einstein condensates

We outline a practical scheme for measuring the thermodynamic properties of a Bose-Einstein condensate as a function of internal energy. We propose using Bragg scattering and controlled trap manipulations to impart a precise amount of energy to a near zero temperature condensate. After thermalisation the temperature can be measured using standard techniques to determine the state equation $T(U,N,\omega)$. Our analysis accounts for interaction effects and the excitation of constants of motion which restrict the energy available for thermalisation.Comment: 6 pages, 1 figure. Updated to published versio