Results on the Dimension Spectra of Planar Lines

In this paper we investigate the (effective) dimension spectra of lines in the Euclidean plane. The dimension spectrum of a line L_{a,b}, sp(L), with slope a and intercept b is the set of all effective dimensions of the points (x, ax + b) on L. It has been recently shown that, for every a and b with effective dimension less than 1, the dimension spectrum of L_{a,b} contains an interval. Our first main theorem shows that this holds for every line. Moreover, when the effective dimension of a and b is at least 1, sp(L) contains a unit interval. Our second main theorem gives lower bounds on the dimension spectra of lines. In particular, we show that for every alpha in [0,1], with the exception of a set of Hausdorff dimension at most alpha, the effective dimension of (x, ax + b) is at least alpha + dim(a,b)/2. As a consequence of this theorem, using a recent characterization of Hausdorff dimension using effective dimension, we give a new proof of a result by Molter and Rela on the Hausdorff dimension of Furstenberg sets

Polynomial Space Randomness in Analysis

We study the interaction between polynomial space randomness and a fundamental result of analysis, the Lebesgue differentiation theorem. We generalize Ko\u27s framework for polynomial space computability in R^n to define weakly pspace-random points, a new variant of polynomial space randomness. We show that the Lebesgue differentiation theorem characterizes weakly pspace random points. That is, a point x is weakly pspace random if and only if the Lebesgue differentiation theorem holds for a point x for every pspace L_1-computable function

Semicomputable Points in Euclidean Spaces

We introduce the notion of a semicomputable point in R^n, defined as a point having left-c.e. projections. We study the range of such a point, which is the set of directions on which its projections are left-c.e., and is a convex cone. We provide a thorough study of these notions, proving along the way new results on the computability of convex sets. We prove realization results, by identifying computability properties of convex cones that make them ranges of semicomputable points. We give two applications of the theory. The first one provides a better understanding of the Solovay derivatives. The second one is the investigation of left-c.e. quadratic polynomials. We show that this is, in fact, a particular case of the general theory of semicomputable points

Projection Theorems Using Effective Dimension

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand\u27s projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal. We use Kolmogorov complexity to give two new results on the Hausdorff and packing dimensions of orthogonal projections onto lines. The first shows that the conclusion of Marstrand\u27s theorem holds whenever the Hausdorff and packing dimensions agree on the set E, even if E is not analytic. Our second result gives a lower bound on the packing dimension of projections of arbitrary sets. Finally, we give a new proof of Marstrand\u27s theorem using the theory of computing

Asymptotic divergences and strong dichotomy

The Schnorr-Stimm dichotomy theorem [31] concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet S. The theorem asserts that, for any such sequence S, the following two things are true. (1) If S is not normal in the sense of Borel (meaning that every two strings of equal length appear with equal asymptotic frequency in S), then there is a finite-state gambler that wins money at an infinitely-often exponential rate betting on S. (2) If S is normal, then any finite-state gambler betting on S loses money at an exponential rate betting on S. In this paper we use the Kullback-Leibler divergence to formulate the lower asymptotic divergence div(S||a) of a probability measure a on S from a sequence S over S and the upper asymptotic divergence Div(S||a) of a from S in such a way that a sequence S is a-normal (meaning that every string w has asymptotic frequency a(w) in S) if and only if Div(S||a) = 0. We also use the Kullback-Leibler divergence to quantify the total risk RiskG(w) that a finite-state gambler G takes when betting along a prefix w of S. Our main theorem is a strong dichotomy theorem that uses the above notions to quantify the exponential rates of winning and losing on the two sides of the Schnorr-Stimm dichotomy theorem (with the latter routinely extended from normality to a-normality). Modulo asymptotic caveats in the paper, our strong dichotomy theorem says that the following two things hold for prefixes w of S. (10) The infinitely-often exponential rate of winning in 1 is 2Div(S||a)|w| . (20) The exponential rate of loss in 2 is 2-RiskG(w) . We also use (10) to show that 1 - Div(S||a)/c, where c = log(1/mina¿S a(a)), is an upper bound on the finite-state a-dimension of S and prove the dual fact that 1 - div(S||a)/c is an upper bound on the finite-state strong a-dimension of S

Factors Associated with Revision Surgery after Internal Fixation of Hip Fractures

Background: Femoral neck fractures are associated with high rates of revision surgery after management with internal fixation. Using data from the Fixation using Alternative Implants for the Treatment of Hip fractures (FAITH) trial evaluating methods of internal fixation in patients with femoral neck fractures, we investigated associations between baseline and surgical factors and the need for revision surgery to promote healing, relieve pain, treat infection or improve function over 24 months postsurgery. Additionally, we investigated factors associated with (1) hardware removal and (2) implant exchange from cancellous screws (CS) or sliding hip screw (SHS) to total hip arthroplasty, hemiarthroplasty, or another internal fixation device. Methods: We identified 15 potential factors a priori that may be associated with revision surgery, 7 with hardware removal, and 14 with implant exchange. We used multivariable Cox proportional hazards analyses in our investigation. Results: Factors associated with increased risk of revision surgery included: female sex, [hazard ratio (HR) 1.79, 95% confidence interval (CI) 1.25-2.50; P = 0.001], higher body mass index (fo

Investigation of spatio‐temporal clusters of positive leptospirosis polymerase chain reaction test results in dogs in the United States, 2009 to 2016

Abstract Background Leptospirosis is a zoonotic disease of concern and an investigation of recent spatio‐temporal trends of leptospirosis in dogs in the United States is needed. Leptospira PCR testing has become increasingly used in veterinary clinical medicine and these data might provide information on recent trends of disease occurrence. Objectives To identify and describe clusters of PCR‐positive Leptospira test results in dogs in the United States. Animals Leptospira real‐time PCR test results from dogs (n = 40 118) in the United States from IDEXX Laboratories, Inc., between 2009 and 2016 were included in the analysis. Methods In this retrospective study, spatio‐temporal clusters for a real‐time PCR‐positive test were identified using the space‐time permutation scan statistic and the centroid of the zip code reported for each test. A maximum spatial window of 20% of the population at risk, and a maximum temporal window of 6 months were used. Results Seven statistically significant space‐time clusters of Leptospira real‐time PCR‐positive test results were identified across the United States: 1 each located within the states of Arizona (2016), California (2014‐2015), Florida (2010), South Carolina (2015), and 1 each located within the south‐central region (2015), midwest region (2014), and northeast region (2011). Clusters ranged from 3 to 108 dogs and were identified during all years under study, except 2009, 2012, and 2013. Conclusions and Clinical Importance The spatial and temporal components of leptospirosis in dogs in this study are similar to those in previous work. However, clusters were identified in new areas, demonstrating the complex epidemiology of this disease