Distribution in coprime residue classes of polynomially-defined multiplicative functions

An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)\in \mathbb{Z}[T]$ with $f(p)=F(p)$ for all primes $p$. We study the distribution in coprime residue classes of polynomially-defined multiplicative functions, establishing equidistribution results allowing a wide range of uniformity in the modulus $q$. For example, we show that the values $\phi(n)$, sampled over integers $n \le x$ with $\phi(n)$ coprime to $q$, are asymptotically equidistributed among the coprime classes modulo $q$, uniformly for moduli $q$ coprime to $6$ that are bounded by a fixed power of $\log{x}$.Comment: edited paragraph following Theorem 1.3, correcting a claim in the discussion of condition (i

The distribution of intermediate prime factors

Let $P^{\left(\frac 12\right)}(n)$ denote the middle prime factor of $n$ (taking into account multiplicity). More generally, one can consider, for any $\alpha \in (0,1)$, the $\alpha$-positioned prime factor of $n$, $P^{(\alpha)}(n)$. It has previously been shown that $\log \log P^{(\alpha)}(n)$ has normal order $\alpha \log \log x$, and its values follow a Gaussian distribution around this value. We extend this work by obtaining an asymptotic formula for the count of $n\leq x$ for which $P^{(\alpha)}(n)=p$, for primes $p$ in a wide range up to $x$. We give several applications of these results, including an exploration of the geometric mean of the middle prime factors, for which we find that $\frac 1x \sum_{1<n \le x} \log P^{\left(\frac 12 \right)}(n) \sim A(\log x)^{\varphi-1}$, where $\varphi$ is the golden ratio, and $A$ is an explicit constant. Along the way, we obtain an extension of Lichtman's recent work on the ``dissected'' Mertens' theorem sums $\sum_{\substack{P^+(n) \le y \\ \Omega(n)=k}} \frac{1}{n}$ for large values of $k$

Classification of bone defects using natural and synthetic X-ray images

In this thesis, we study methods to reduce the amount of data needed to create deep learning models that can detect defects in bones from X-ray images. Detecting defects in bones from X-ray images and properly annotating the images is the paramount step when it comes to corrective surgeries of bones. Annotations or labels, such as radial inclination and volar tilt are measurements that are necessary for many corrective surgeries. Generating these annotations is an arduous and manual task for medical professionals. By being able to automate the process of generating these annotations, it will be possible to reduce a significant amount of labor of these professionals. Modern deep learning models are heavily reliant upon availability of a large amount of properly labeled data for their training. In this thesis, we experimented to find methods to create appropriate synthetic data that can be combined with natural data to train deep learning models. We designed three deep learning models to generate two different forms of annotations. The first goal was to use cycle consistent generative adversarial networks to create proper synthetic images. Then we used the synthetic images to improve classifier models that can detect defects in bones. In the end, we expanded the cycle consistent generative adversarial network so that it can accommodate three input domains instead of two and called it multi-cycleGAN. We used multi-cycleGAN to segment bones from natural X-ray images. Our experiments concluded that by adding proper synthetic images with natural images, we can improve the performance of classifiers significantly and circumvent the persistent issue of unavailability of data. However, the multi-cycleGAN model did not generate a very accurate segmentation of bones. It was able to segment bones of forearm better than bones of wrist. It was able to understand the overall shape and positioning of the wrists in X-ray images but it did not produce proper segmentations of the individual fingers

Optimal Stack Layout in a Sea Container Terminal with Automated Lifting Vehicles

Container terminal performance is largely determined by its design decisions, which include the number and type of quay cranes (QCs), stack cranes (SCs), transport vehicles, vehicle travel path, and stack layout. The terminal design process is complex because it is affected by factors such as topological constraints, stochastic interactions among the quayside, vehicle transport and stackside operations. Further, the orientation of the stack layout (parallel or perpendicular to the quayside) plays an important role in the throughput time performance of the terminals. Previous studies in this area typically use deterministic optimization or probabilistic travel time models to analyze the effect of stack layout on terminal throughput times, and ignore the stochastic interactions among the resou