arXiv subjects
Publications and source records attributed to Yu-Chen Sun.
In this paper we give a short interval version of the Balog-Ruzsa theorem concerning bounds for the $L_1$ norm of the exponential sum over $r$-free numbers. As an application, we give a lower bound for the $L_1$ norm of the exponential sum defined with the Möbius function. Namely we show that $$\int_{\mathbb T} \left|\sum_{|n-N|<H} μ(n)e(n α)\right| d α\gg H^{\frac{1}{6}}$$ when $H \gg N^{\frac{9}{17} + \varepsilon}$.
The COVID-19 pandemic is one of the most challenging healthcare crises during the 21st century. As the virus continues to spread on a global scale, the majority of efforts have been on the development of vaccines and the mass immunization of the public. While the daily case numbers were following a decreasing trend, the emergent of new virus mutations and variants still pose a significant threat. As economies start recovering and societies start opening up with people going back into office buildings, schools, and malls, we still need to have the ability to detect and minimize the spread of COVID-19. Individuals with COVID-19 may show multiple symptoms such as cough, fever, and shortness of breath. Many of the existing detection techniques focus on symptoms having the same equal importance. However, it has been shown that some symptoms are more prevalent than others. In this paper, we present a multimodal method to predict COVID-19 by incorporating existing deep learning classifiers using convolutional neural networks and our novel probability-based weighting function that considers the prevalence of each symptom. The experiments were performed on an existing dataset with respect to the three considered modes of coughs, fever, and shortness of breath. The results show considerable improvements in the detection of COVID-19 using our weighting function when compared to an equal weighting function.
Let ${\mathcal C}_n$ be the set of all permutation cycles of length $n$ over $\{1,2,\ldots,n\}$. Let $${\mathfrak f}_n(q):=\sum_{σ\in{\mathcal C}_{n+1}}q^{{\mathrm maj}\,σ} $$ be a $q$-analogue of the factorial $n!$, where ${\mathrm maj}$ denotes the major index. We prove a $q$-analogue of Wilson's congruence $$ {\mathfrak f}_{n-1}(q)\equivμ(n)\pmod{Φ_n(q)}, $$ where $μ$ denotes the Möbius function and $Φ_n(q)$ is the $n$-th cyclotomic polynomial.
Lagrange's four squares theorem is a classical theorem in number theory. Recently, Z.-W. Sun found that it can be further refined in various ways. In this paper we study some conjectures of Sun and obtain various refinements of Lagrange's theorem. We show that any nonnegative integer can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+z+w$ (or $x+y+z+2w$, or $x+2y+3z+w$) a square (or a cube). Also, every $n=0,1,2,\ldots$ can be represented by $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+3z$ (or $x+2y+3z$) a square (or a cube), and each $n=0,1,2,\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $(10w+5x)^2+(12y+36z)^2$ (or $x^2y^2+9y^2z^2+9z^2x^2$) a square. We also provide an advance on the 1-3-5 conjecture of Sun. Our main results are proved by a new approach involving Euler's four-square identity
For a fixed positive integer $m$ and any partition $m = m_1 + m_2 + \cdots + m_e$ , there exists a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{k}},$$ with the property that partial sums of the series $\{\frac{1}{n_i}\}_{i=1}^{k}$ can only represent the integers with the form $\sum_{i\in I}m_i$, where $I\subset\{1,...,e\}$.
Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$ for some small constant $c_m>0$. Furthermore, we also obtain a related result concerning the least primes in arithmetic progressions.
We prove that the primes of the form $x^2+y^2+1$ contain arbitrarily long non-trivial arithmetic progressions.