Abstract:
The purpose of this study was to assess usefulness of local binary patterns (LBP) and related texture features, namely completed local binary patterns (CLBP) and local ternary patterns (LTP), for the classification of emphysema subtypes on low-dose CT images. Fifty patients (34 men and 16 women; age, 67.5 ± 10.1 years) who underwent low-dose CT (60 mAs) were included. They were comprised of 17 never smokers, 13 smokers without COPD, and 20 smokers with COPD. By consensus reading of low-dose CT images from these patients, two radiologists selected 3681 nonoverlapping regions of interest (ROIs) and annotated them as one of the following three classes: normal tissue, centrilobular emphysema, and paraseptal emphysema. From these ROIs, histogram of CT densities, LBP, CLBP, and LTP were calculated, and the 3 types of texture histograms were concatenated with the CT density histogram. These 3 types of histograms (referred to as combined LBP, combined CLBP, and combined LTP) were used to classify ROI using linear support vector machine. For each type of the combined histogram, the accuracy of classification was determined by patient-based 10-fold cross validation. The best accuracy of combined LBP, combined CLBP, and combined LTP were 81.36%, 82.99%, and 83.29%, respectively. Compared to the classification accuracies obtained with combined LBP, those with combined LTP or combined CLBP were consistently improved. In conclusion, the results of this study suggest that, on low-dose CT, LTP and CLBP were more useful for the classification of emphysema subtypes than LBP.

Abstract:
The purpose of this study was to develop methodology to segment tumors on ^{18}F-fluorodeoxyg- lucose (FDG) positron emission tomography (PET) images. Sixty-four metastatic bone tumors were included. Graph cut was used for tumor segmentation, with segmentation energy divided into unary and pairwise terms. Locally connected conditional random fields (LCRF) were proposed for the pairwise term. In LCRF, three-dimensional cubic window with length L was set for each voxel, and voxels within the window were considered for the pairwise term. Three other types of segmentation were applied: region-growing based on 35%, 40%, and 45% of the tumor maximum standardized uptake value (RG35, RG40, and RG45, respectively), SLIC superpixels (SS), and region-based active contour models (AC). To validate the tumor segmentation accuracy, dice similarity coefficients (DSC) were calculated between the result of each technique and manual segmentation. Differences in DSC were tested using the Wilcoxon signed-rank test. Mean DSCs for LCRF at L = 3, 5, 7, and 9 were 0.784, 0.801, 0.809, and 0.812, respectively. Mean DSCs for the other techniques were: RG35, 0.633; RG40, 0.675; RG45, 0.689; SS, 0.709; and AC, 0.758. The DSC differences between LCRF and other techniques were statistically significant (p < 0.05). Tumor segmentation was reliably performed with LCRF.

Abstract:
In this paper, we reconsider a formula of Mellin. We present a formula which relates the sum of two positive real numbers $m, n$ to their product $mn$. We apply this formula to derivation of a relationship involving the Hurwitz zeta-function. Then we define a series function (stemming from the proved relationship) and discuss an analogy in regard to the Lindel"{o}f hypothesis. Finally, a proof of the Lindel"{o}f hypothesis of the Riemann zeta-function is deduced from this analogy.

Abstract:
This paper has been withdrawn by the author. It has been shown by G.A. Edgar that curves created by the presented method are not continuous at \theta = 0. See http://www.math.ohio-state.edu/~edgar/equichord/ .

Abstract:
Let $\varphi(x, u, s)$ denote the Lerch zeta-function defined for $\text{Re}(s) > 1$ and $x, u \in (0, 1)$ by the series $\varphi(x, u, s) := \sum_{n \geq 0}\exp[2 \pi i n x](n + u)^{-s}$, and let $\varphi^{*}(x, u, s) := \varphi(x, u, s) - u^{-s}$. Furthermore, let $[a, b]$ denote the straight line in the complex plane from the point $a$ to $b$ and $\gamma$ be any real number in $(0, \pi)$. In this paper, we investigate integrals of the form $\int_{[e^{i \gamma}, 0] + [0, 1]} \varphi^{*}(x, u, 1/2 + \beta - it) u^{-1 + \epsilon} du$, where $\beta \in (0, 1/2)$, $\epsilon > 0$, and $a_{1}, x \in (0, 1)$.

Abstract:
In the paper, we first prove a sufficient condition for the Riemann hypothesis which involves the order of magnitude of the partial sum of the Liouville function. Then we show a formula which is curiously related to the proved sufficient condition.

Abstract:
Let $\lfloor t \rfloor$ denote the greatest positive integer less than or equal to a given positive real number $t$ and $\vartheta(t)$ the Chebyshev $\vartheta$-function. In this paper, we prove a certain asymptotic relationship involving $\vartheta(t) - \lfloor t \rfloor $ and $t^{1/2}$.

Abstract:
Let $p_{k}$ denote the $k$-th prime and $d(p_{k}) = p_{k} - p_{k - 1}$, the difference between consecutive primes. We denote by $N_{\epsilon}(x)$ the number of primes $\leq x$ which satisfy the inequality $d(p_{k}) \leq (\log p_{k})^{2 + \epsilon}$, where $\epsilon > 0$ is arbitrary and fixed, and by $\pi(x)$ the number of primes less than or equal to $x$. In this paper, we first prove a theorem that $\lim_{x \to \infty} N_{\epsilon}(x)/\pi(x) = 1$. A corollary to the proof of the theorem concerning gaps between consecutive squarefree numbers is stated.