A Generalization of Martin's Axiom

We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $\omega_1$ that don't seem to have been considered in the literature before.Comment: 36 page

Vickrey Auctions for Irregular Distributions

The classic result of Bulow and Klemperer \cite{BK96} says that in a single-item auction recruiting one more bidder and running the Vickrey auction achieves a higher revenue than the optimal auction's revenue on the original set of bidders, when values are drawn i.i.d. from a regular distribution. We give a version of Bulow and Klemperer's result in settings where bidders' values are drawn from non-i.i.d. irregular distributions. We do this by modeling irregular distributions as some convex combination of regular distributions. The regular distributions that constitute the irregular distribution correspond to different population groups in the bidder population. Drawing a bidder from this collection of population groups is equivalent to drawing from some convex combination of these regular distributions. We show that recruiting one extra bidder from each underlying population group and running the Vickrey auction gives at least half of the optimal auction's revenue on the original set of bidders

A universal characterization of higher algebraic K-theory

In this paper we establish a universal characterization of higher algebraic K-theory in the setting of small stable infinity categories. Specifically, we prove that connective algebraic K-theory is the universal additive invariant, i.e., the universal functor with values in spectra which inverts Morita equivalences, preserves filtered colimits, and satisfies Waldhausen's additivity theorem. Similarly, we prove that non-connective algebraic K-theory is the universal localizing invariant, i.e., the universal functor that moreover satisfies the "Thomason-Trobaugh-Neeman" localization theorem. To prove these results, we construct and study two stable infinity categories of "noncommutative motives"; one associated to additivity and another to localization. In these stable infinity categories, Waldhausen's S. construction corresponds to the suspension functor and connective and non-connective algebraic K-theory spectra become corepresentable by the noncommutative motive of the sphere spectrum. In particular, the algebraic K-theory of every scheme, stack, and ring spectrum can be recovered from these categories of noncommutative motives. In order to work with these categories of noncommutative motives, we establish comparison theorems between the category of spectral categories localized at the Morita equivalences and the category of small idempotent-complete stable infinity categories. We also explain in detail the comparison between the infinity categorical version of Waldhausen K-theory and the classical definition. As an application of our theory, we obtain a complete classification of the natural transformations from higher algebraic K-theory to topological Hochschild homology (THH) and topological cyclic homology (TC). Notably, we obtain an elegant conceptual description of the cyclotomic trace map.Comment: Various revisions and correction

Simple, optimal and efficient auctions

Proceedings of the 7th International Workshop, WINE 2011, Singapore, December 11-14, 2011.We study the extent to which simple auctions can simultaneously achieve good revenue and efficiency guarantees in single-item settings. Motivated by the optimality of the second price auction with monopoly reserves when the bidders’ values are drawn i.i.d. from regular distributions [12], and its approximate optimality when they are drawn from independent regular distributions [11], we focus our attention to the second price auction with general (not necessarily monopoly) reserve prices, arguably one of the simplest and most intuitive auction formats. As our main result, we show that for a carefully chosen set of reserve prices this auction guarantees at least 20% of both the optimal welfare and the optimal revenue, when the bidders’ values are distributed according to independent, not necessarily identical, regular distributions. We also prove a similar guarantee, when the values are drawn i.i.d. from a—possibly irregular—distribution.National Science Foundation (U.S.) (award CCF-0953960)National Science Foundation (U.S.) (CCF-1101491

The relationship of femoral neck shaft angle and adiposity to greater trochanteric pain syndrome in women. A case control morphology and anthropometric study

OBJECTIVE To evaluate if pelvic or hip width predisposed women to developing greater trochanteric pain syndrome (GTPS). DESIGN Prospective case control study. PARTICIPANTS Four groups were included in the study: those gluteal tendon reconstructions (n=31, GTR), those with conservatively managed GTPS (n=29), those with hip osteoarthritis (n=20, OA) and 22 asymptomatic participants (ASC). METHODS Anterior-posterior pelvic x-rays were evaluated for femoral neck shaft angle; acetabular index, and width at the lateral acetabulum, and the superior and lateral aspects of the greater trochanter. Body mass index, and waist, hip and greater trochanter girth were measured. Data were analysed using a one-way analysis of variance (ANOVA; posthoc Scheffe analysis), then multivariate analysis. RESULTS The GTR group had a lower femoral neck shaft angle than the other groups (p=0.007). The OR (95% CI) of having a neck shaft angle of less than 134°, relative to the ASC group: GTR=3.33 (1.26 to 8.85); GTPS=1.4 (0.52 to 3.75); OA=0.85 (0.28 to 2.61). The OR of GTR relative to GTPS was 2.4 (1.01 to 5.6). No group difference was found for acetabular or greater trochanter width. Greater trochanter girth produced the only anthropometric group difference (mean (95% CI) in cm) GTR=103.8 (100.3 to 107.3), GTPS=105.9 (100.2 to 111.6), OA=100.3 (97.7 to 103.9), ASC=99.1 (94.7 to 103.5), (ANOVA: p=0.036). Multivariate analysis confirmed adiposity is associated with GTPS. CONCLUSION A lower neck shaft angle is a risk factor for, and adiposity is associated with, GTPS in women

Gorenstein homological algebra and universal coefficient theorems

We study criteria for a ring—or more generally, for a small category—to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman’s Brown–Adams representability theorem for compactly generated categories

On Albanese torsors and the elementary obstruction

We show that the elementary obstruction to the existence of 0-cycles of degree 1 on an arbitrary variety X (over an arbitrary field) can be expressed in terms of the Albanese 1-motives associated with dense open subsets of X. Arithmetic applications are given

Development and validation of a VISA tendinopathy questionnaire for greater trochanteric pain syndrome, the VISA-G

BACKGROUND Greater trochanteric pain syndrome (GTPS) is common, resulting in significant pain and disability. There is no condition specific outcome score to evaluate the degree of severity of disability associated with GTPS in patients with this condition. OBJECTIVE To develop a reliable and valid outcome measurement capable of evaluating the severity of disability associated with GTPS. METHODS A phenomenological framework using in-depth semi structured interviews of patients and medical experts, and focus groups of physiotherapists was used in the item generation. Item and format clarification was undertaken via piloting. Multivariate analysis provided the basis for item reduction. The resultant VISA-G was tested for reliability with the inter class co-efficient (ICC), internal consistency (Cronbach's Alpha), and construct validity (correlation co-efficient) on 52 naïve participants with GTPS and 31 asymptomatic participants. RESULTS The resultant outcome measurement tool is consistent in style with existing tendinopathy outcome measurement tools, namely the suite of VISA scores. The VISA-G was found to be have a test-retest reliability of ICC2,1 (95% CI) of 0.827 (0.638-0.923). Internal consistency was high with a Cronbach's Alpha of 0.809. Construct validity was demonstrated: the VISA-G measures different constructs than tools previously used in assessing GTPS, the Harris Hip Score and the Oswestry Disability Index (Spearman Rho:0.020 and 0.0205 respectively). The VISA-G did not demonstrate any floor or ceiling effect in symptomatic participants. CONCLUSION The VISA-G is a reliable and valid score for measuring the severity of disability associated GTPS.The study was funded through the Australian National University, Monash University and LaTrobe University. Prof Cook was supported by the Australian Centre for Research into Sports Injury and its Prevention, which is one of the International Research Centres for Prevention of Injury and Protection of Athlete Health supported by the International Olympic Committee (IOC). Prof Cook is a NHMRC practitioner fellow (ID 1058493)

Derived categories of cubic fourfolds

We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaffian cubics, cubics containing a plane and singular cubics, and discuss its relation to the rationality of these cubics.Comment: 18 page