arXiv subjects

Sung Min Lee

Publications and source records attributed to Sung Min Lee.

13 recordsLinked to original sources

arXiv2026

Average twin prime conjecture for elliptic curves in arithmetic progressions

In 1988, Koblitz conjectured an asymptotic formula for the number of primes $p \le x$ for which the reduction of an elliptic curve over $\mathbb{Q}$ has prime order. Building on the work of Balog, Cojocaru, and David, who proved this conjecture on average in 2011, we extend the result to primes $p$ lying in arithmetic progressions. Our asymptotic formula holds uniformly for moduli up to a fixed power of $\log x$. We also show that the resulting average constant matches the theoretical constant predicted by Lee, Mayle, and Wang in 2025 using Galois representations.

math.NT↗

arXiv2026

Divisibility Biases in the Orders of Elliptic Curve Reductions

Let $E$ be an elliptic curve over the rationals. In 2004, Cojocaru proved, using the Chebotarev density theorem, that the set of primes $p \leq x$ for which $m$ divides $\#E_p(\mathbb{F}_p)$ has a natural density. In 2009, Banks and Shparlinski proved an averaged version of this result over families of elliptic curves. In this article, we give a more explicit analysis of these densities. In particular, we show that, for Serre curves, the density of primes $p$ for which $m \mid \#E_p(\mathbb{F}_p)$ is approximately $1/φ(m)$, and is always greater than $1/m$ for every $m \geq 2$. Thus, the orders $\#E_p(\mathbb{F}_p)$ exhibit a bias toward divisibility by $m$. Finally, based on Jones' method, we prove that the average of the individual $m$-divisibility densities coincides with the average density proposed by Banks and Shparlinski.

math.NT↗

arXiv2026

On the Density of Coprime Reductions of Elliptic Curves

Given non-CM elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$, we study the natural density of primes $p$ of good reduction for which the orders of the groups $E_1(\mathbb{F}_p)$ and $E_2(\mathbb{F}_p)$ are coprime. This problem may be viewed as an elliptic curve analogue of the classical question concerning the density of coprime integer pairs. Motivated by Zywina's refinement of the Koblitz conjecture, we formulate a conjecture for the density of such primes. We prove that the series defining this constant converges and admits an almost Euler product expansion. In the case of Serre pairs, we give a closed formula for the constant and use it to prove a moments result describing the distribution of these constants as $(E_1, E_2)$ varies.

math.NT↗

arXiv2025

On the average congruence class bias for cyclicity and divisibility of the groups of $\mathbb{F}_p$-points of elliptic curves

In 2009, W. D. Banks and I. E. Shparlinski studied the average densities of primes $p \leq x$ for which the reductions of elliptic curves of small height modulo $p$ satisfy certain arithmetic properties, namely cyclicity and divisibility of the number of points by a fixed integer $m$. In this paper, we refine their results, restricting the primes $p$ under consideration to lie in an arithmetic progression $k \bmod{n}$. Furthermore, for a fixed modulus $n$, we investigate statistical biases among the different congruence classes $k \bmod{n}$ of primes satisfying the aforementioned properties.

math.NT↗

arXiv2024

Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves

The cyclicity and Koblitz conjectures ask about the distribution of primes of cyclic and prime-order reduction, respectively, for elliptic curves over $\mathbb{Q}$. In 1976, Serre gave a conditional proof of the cyclicity conjecture, but the Koblitz conjecture (refined by Zywina in 2011) remains open. The conjectures are now known unconditionally "on average" due to work of Banks--Shparlinski and Balog--Cojocaru--David. Recently, there has been a growing interest in the cyclicity conjecture for primes in arithmetic progressions (AP), with relevant work by Akbal--Güloğlu and Wong. In this paper, we adapt Zywina's method to formulate the Koblitz conjecture for primes in AP and refine a theorem of Jones to establish results on the moments of the constants in both the cyclicity and Koblitz conjectures for AP. In doing so, we uncover a somewhat counterintuitive phenomenon: On average, these two constants are oppositely biased over congruence classes. Finally, in an accompanying repository, we give Magma code for computing the constants discussed in this paper.

math.NT↗

arXiv2023

On the acyclicity of reductions of elliptic curves modulo primes in arithmetic progressions

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and, for a prime $p$ of good reduction for $E$ let $\tilde{E}_p$ denote the reduction of $E$ modulo $p$. Inspired by an elliptic curve analogue of Artin's primitive root conjecture posed by S. Lang and H. Trotter in 1977, J-P. Serre adapted methods of C. Hooley to prove a GRH-conditional asymptotic formula for the number of primes $p \leq x$ for which the group $\tilde{E}_p(\mathbb{F}_p)$ is cyclic. More recently, Akbal and Gülo$\breve{\text{g}}$lu considered the question of cyclicity of $\tilde{E}_p(\mathbb{F}_p)$ under the additional restriction that $p$ lie in an arithmetic progression. In this note, we study the issue of which arithmetic progressions $a \bmod n$ have the property that, for all but finitely many primes $p \equiv a \bmod n$, the group $\tilde{E}_p(\mathbb{F}_p)$ is not cyclic, answering a question of Akbal and Gülo$\breve{\text{g}}$lu on this issue.

math.NT↗

arXiv2020

Deep Learning-Based Solvability of Underdetermined Inverse Problems in Medical Imaging

Recently, with the significant developments in deep learning techniques, solving underdetermined inverse problems has become one of the major concerns in the medical imaging domain. Typical examples include undersampled magnetic resonance imaging, interior tomography, and sparse-view computed tomography, where deep learning techniques have achieved excellent performances. Although deep learning methods appear to overcome the limitations of existing mathematical methods when handling various underdetermined problems, there is a lack of rigorous mathematical foundations that would allow us to elucidate the reasons for the remarkable performance of deep learning methods. This study focuses on learning the causal relationship regarding the structure of the training data suitable for deep learning, to solve highly underdetermined inverse problems. We observe that a majority of the problems of solving underdetermined linear systems in medical imaging are highly non-linear. Furthermore, we analyze if a desired reconstruction map can be learnable from the training data and underdetermined system.

eess.IV↗

arXiv2019

Deep learning for undersampled MRI reconstruction

This paper presents a deep learning method for faster magnetic resonance imaging (MRI) by reducing k-space data with sub-Nyquist sampling strategies and provides a rationale for why the proposed approach works well. Uniform subsampling is used in the time-consuming phase-encoding direction to capture high-resolution image information, while permitting the image-folding problem dictated by the Poisson summation formula. To deal with the localization uncertainty due to image folding, very few low-frequency k-space data are added. Training the deep learning net involves input and output images that are pairs of Fourier transforms of the subsampled and fully sampled k-space data. Numerous experiments show the remarkable performance of the proposed method; only 29% of k-space data can generate images of high quality as effectively as standard MRI reconstruction with fully sampled data.

stat.ML↗

arXiv2019

Improving learnability of neural networks: adding supplementary axes to disentangle data representation

Over-parameterized deep neural networks have proven to be able to learn an arbitrary dataset with 100$\%$ training accuracy. Because of a risk of overfitting and computational cost issues, we cannot afford to increase the number of network nodes if we want achieve better training results for medical images. Previous deep learning research shows that the training ability of a neural network improves dramatically (for the same epoch of training) when a few nodes with supplementary information are added to the network. These few informative nodes allow the network to learn features that are otherwise difficult to learn by generating a disentangled data representation. This paper analyzes how concatenation of additional information as supplementary axes affects the training of the neural networks. This analysis was conducted for a simple multilayer perceptron (MLP) classification model with a rectified linear unit (ReLU) on two-dimensional training data. We compared the networks with and without concatenation of supplementary information to support our analysis. The model with concatenation showed more robust and accurate training results compared to the model without concatenation. We also confirmed that our findings are valid for deeper convolutional neural networks (CNN) using ultrasound images and for a conditional generative adversarial network (cGAN) using the MNIST data.

cs.LG↗

arXiv2018

Automatic evaluation of fetal head biometry from ultrasound images using machine learning

Ultrasound-based fetal biometric measurements, such as head circumference (HC) and biparietal diameter (BPD), are commonly used to evaluate the gestational age and diagnose fetal central nervous system (CNS) pathology. Since manual measurements are operator-dependent and time-consuming, there have been numerous researches on automated methods. However, existing automated methods still are not satisfactory in terms of accuracy and reliability, owing to difficulties in dealing with various artifacts in ultrasound images. This paper focuses on fetal head biometry and develops a deep-learning-based method for estimating HC and BPD with a high degree of accuracy and reliability. The proposed method effectively identifies the head boundary by differentiating tissue image patterns with respect to the ultrasound propagation direction. The proposed method was trained with 102 labeled data set and tested to 70 ultrasound images. We achieved a success rate of 92.31% for HC and BPD estimations, and an accuracy of 87.14 % for the plane acceptance check.

physics.med-ph↗

arXiv2018

CT sinogram-consistency learning for metal-induced beam hardening correction

This paper proposes a sinogram consistency learning method to deal with beam-hardening related artifacts in polychromatic computerized tomography (CT). The presence of highly attenuating materials in the scan field causes an inconsistent sinogram, that does not match the range space of the Radon transform. When the mismatched data are entered into the range space during CT reconstruction, streaking and shading artifacts are generated owing to the inherent nature of the inverse Radon transform. The proposed learning method aims to repair inconsistent sinograms by removing the primary metal-induced beam-hardening factors along the metal trace in the sinogram. Taking account of the fundamental difficulty in obtaining sufficient training data in a medical environment, the learning method is designed to use simulated training data and a patient-type specific learning model is used to simplify the learning process. The feasibility of the proposed method is investigated using a dataset, consisting of real CT scan of pelvises containing hip prostheses. The anatomical areas in training and test data are different, in order to demonstrate that the proposed method extracts the beam hardening features, selectively. The results show that our method successfully corrects sinogram inconsistency by extracting beam-hardening sources by means of deep learning. This paper proposed a deep learning method of sinogram correction for beam hardening reduction in CT for the first time. Conventional methods for beam hardening reduction are based on regularizations, and have the fundamental drawback of being not easily able to use manifold CT images, while a deep learning approach has the potential to do so.

physics.med-ph↗

arXiv2017

Automatic Estimation of Fetal Abdominal Circumference from Ultrasound Images

Ultrasound diagnosis is routinely used in obstetrics and gynecology for fetal biometry, and owing to its time-consuming process, there has been a great demand for automatic estimation. However, the automated analysis of ultrasound images is complicated because they are patient-specific, operator-dependent, and machine-specific. Among various types of fetal biometry, the accurate estimation of abdominal circumference (AC) is especially difficult to perform automatically because the abdomen has low contrast against surroundings, non-uniform contrast, and irregular shape compared to other parameters.We propose a method for the automatic estimation of the fetal AC from 2D ultrasound data through a specially designed convolutional neural network (CNN), which takes account of doctors' decision process, anatomical structure, and the characteristics of the ultrasound image. The proposed method uses CNN to classify ultrasound images (stomach bubble, amniotic fluid, and umbilical vein) and Hough transformation for measuring AC. We test the proposed method using clinical ultrasound data acquired from 56 pregnant women. Experimental results show that, with relatively small training samples, the proposed CNN provides sufficient classification results for AC estimation through the Hough transformation. The proposed method automatically estimates AC from ultrasound images. The method is quantitatively evaluated, and shows stable performance in most cases and even for ultrasound images deteriorated by shadowing artifacts. As a result of experiments for our acceptance check, the accuracies are 0.809 and 0.771 with the expert 1 and expert 2, respectively, while the accuracy between the two experts is 0.905. However, for cases of oversized fetus, when the amniotic fluid is not observed or the abdominal area is distorted, it could not correctly estimate AC.

cs.CV↗

arXiv2017

Machine-learning-based nonlinear decomposition of CT images for metal artifact reduction

Computed tomography (CT) images containing metallic objects commonly show severe streaking and shadow artifacts. Metal artifacts are caused by nonlinear beam-hardening effects combined with other factors such as scatter and Poisson noise. In this paper, we propose an implant-specific method that extracts beam-hardening artifacts from CT images without affecting the background image. We found that in cases where metal is inserted in the water (or tissue), the generated beam-hardening artifacts can be approximately extracted by subtracting artifacts generated exclusively by metals. We used a deep learning technique to train nonlinear representations of beam-hardening artifacts arising from metals, which appear as shadows and streaking artifacts. The proposed network is not designed to identify ground-truth CT images (i.e., the CT image before its corruption by metal artifacts). Consequently, these images are not required for training. The proposed method was tested on a dataset consisting of real CT scans of pelvises containing simulated hip prostheses. The results demonstrate that the proposed deep learning method successfully extracts both shadowing and streaking artifacts.

physics.med-ph↗