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Junxian Li

Publications and source records attributed to Junxian Li.

52 records · Page 3Linked to original sources

Uncertainty-aware Traffic Prediction under Missing Data

Traffic prediction is a crucial topic because of its broad scope of applications in the transportation domain. Recently, various studies have achieved promising results. However, most studies assume the prediction locations have complete or at least partial historical records and cannot be extended to non-historical recorded locations. In real-life scenarios, the deployment of sensors could be limited due to budget limitations and installation availability, which makes most current models not applicable. Though few pieces of literature tried to impute traffic states at the missing locations, these methods need the data simultaneously observed at the locations with sensors, making them not applicable to prediction tasks. Another drawback is the lack of measurement of uncertainty in prediction, making prior works unsuitable for risk-sensitive tasks or involving decision-making. To fill the gap, inspired by the previous inductive graph neural network, this work proposed an uncertainty-aware framework with the ability to 1) extend prediction to missing locations with no historical records and significantly extend spatial coverage of prediction locations while reducing deployment of sensors and 2) generate probabilistic prediction with uncertainty quantification to help the management of risk and decision making in the down-stream tasks. Through extensive experiments on real-life datasets, the result shows our method achieved promising results on prediction tasks, and the uncertainty quantification gives consistent results which highly correlated with the locations with and without historical data. We also show that our model could help support sensor deployment tasks in the transportation field to achieve higher accuracy with a limited sensor deployment budget.

cs.LG↗

Single-image based deep learning for precise atomic defects identification

Defect engineering has been profoundly employed to confer desirable functionality to materials that pristine lattices inherently lack. Although single atomic-resolution scanning transmission electron microscopy (STEM) images are widely accessible for defect engineering, harnessing atomic-scale images containing various defects through traditional image analysis methods is hindered by random noise and human bias. Yet the rise of deep learning (DL) offering an alternative approach, its widespread application is primarily restricted by the need for large amounts of training data with labeled ground truth. In this study, we propose a two-stage method to address the problems of high annotation cost and image noise in the detection of atomic defects in monolayer 2D materials. In the first stage, to tackle the issue of data scarcity, we employ a two-state transformation network based on U-GAT-IT for adding realistic noise to simulated images with pre-located ground truth labels, thereby infinitely expanding the training dataset. In the second stage, atomic defects in monolayer 2D materials are effectively detected with high accuracy using U-Net models trained with the data generated in the first stage, avoiding random noise and human bias issues. In both stages, we utilize segmented unit-cell-level images to simplify the model's task and enhance its accuracy. Our results demonstrate that not only sulfur vacancies, we are also able to visualize oxygen dopants in monolayer MoS2, which are usually overwhelmed by random background noise. As the training was based on a few segmented unit-cell-level realistic images, this method can be readily extended to other 2D materials. Therefore, our results outline novel ways to train the model with minimized datasets, offering great opportunities to fully exploit the power of machine learning (ML) applicable to a broad materials science community.

cond-mat.mtrl-sci↗

Simultaneous extreme values of zeta and L-functions

We show that distinct primitive L-functions can achieve extreme values simultaneously on the critical line. Our proof uses a modification of the resonance method and can be applied to establish simultaneous extreme central values of L-functions in families.

math.NT↗

Reinforcement Learning Approaches for Traffic Signal Control under Missing Data

The emergence of reinforcement learning (RL) methods in traffic signal control tasks has achieved better performance than conventional rule-based approaches. Most RL approaches require the observation of the environment for the agent to decide which action is optimal for a long-term reward. However, in real-world urban scenarios, missing observation of traffic states may frequently occur due to the lack of sensors, which makes existing RL methods inapplicable on road networks with missing observation. In this work, we aim to control the traffic signals in a real-world setting, where some of the intersections in the road network are not installed with sensors and thus with no direct observations around them. To the best of our knowledge, we are the first to use RL methods to tackle the traffic signal control problem in this real-world setting. Specifically, we propose two solutions: the first one imputes the traffic states to enable adaptive control, and the second one imputes both states and rewards to enable adaptive control and the training of RL agents. Through extensive experiments on both synthetic and real-world road network traffic, we reveal that our method outperforms conventional approaches and performs consistently with different missing rates. We also provide further investigations on how missing data influences the performance of our model.

cs.LG↗

Additive problems with almost prime squares

We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes p-1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.

math.NT↗

Correlations of values of random diagonal forms

We study the value distribution of diagonal forms in k variables and degree d with random real coefficients and positive integer variables, normalized so that mean spacing is one. We show that the l-correlation of almost all such forms is Poissonian when k is large enough depending on l and d.

math.NT↗

Joint value distribution of $L$-functions on the critical line

In this paper, we discuss the joint value distribution of $L$-functions in a suitable class. We obtain joint large deviations results in the central limit theorem for these $L$-functions and some mean value theorems, which give evidence that different $L$-functions are "statistically independent".

math.NT↗

Uniform Titchmarsh divisor problems

Asymptotic formulae for Titchmarsh-type divisor sums are obtained with strong error terms that are uniform in the shift parameter. This applies to more general arithmetic functions such as sums of two squares, improving the error term in the representation of the number as a sum of a prime and two squares, and to Fourier coefficients of cusp forms, generalizing a result of Pitt.

math.NT↗

Lower bounds for discrete negative moments of the Riemann zeta function

We prove lower bounds for the discrete negative $2k$th moment of the derivative of the Riemann zeta function for all fractional $k\geqslant 0$. The bounds are in line with a conjecture of Gonek and Hejhal. Along the way, we prove a general formula for the discrete twisted second moment of the Riemann zeta function. This agrees with a conjecture of Conrey and Snaith.

math.NT↗

The Surprising Accuracy of Benford's Law in Mathematics

Benford's law is an empirical ``law'' governing the frequency of leading digits in numerical data sets. Surprisingly, for mathematical sequences the predictions derived from it can be uncannily accurate. For example, among the first billion powers of $2$, exactly $301029995$ begin with digit 1, while the Benford prediction for this count is $10^9\log_{10}2=301029995.66\dots$. Similar ``perfect hits'' can be observed in other instances, such as the digit $1$ and $2$ counts for the first billion powers of $3$. We prove results that explain many, but not all, of these surprising accuracies, and we relate the observed behavior to classical results in Diophantine approximation as well as recent deep conjectures in this area.

math.PR↗

Almost Beatty Partitions

Given $0<α<1$, the Beatty sequence of density $α$ is the sequence $B_α=(\lfloor n/α\rfloor)_{n\in\mathbb{N}}$. Beatty's theorem states that if $α,β$ are irrational numbers with $α+β=1$, then the Beatty sequences $B_α$ and $B_β$ partition the positive integers, that is, each positive integer belongs to exactly one of these two sequences. On the other hand, Uspensky showed that this result breaks down completely for partitions into three (or more) sequences: There does not exist a single triple $(α,β,γ)$ such that the Beatty sequences $B_α,B_β,B_γ$ partition the positive integers. In this paper we consider the question of how close we can come to a three-part Beatty partition by considering "almost" Beatty sequences, that is, sequences that represent small perturbations of an "exact" Beatty sequence. We first characterize all cases in which there exists a partition into two exact Beatty sequences and one almost Beatty sequence with given densities, and we determine the approximation error involved. We then give two general constructions that yield partitions into one exact Beatty sequence and two almost Beatty sequences with prescribed densities, and we determine the approximation error in these constructions. Finally, we show that in many situations these constructions are best-possible in the sense that they yield the closest approximation to a three-part Beatty partition.

math.NT↗

Leading Digits of Mersenne Numbers

It has long been known that sequences such as the powers of $2$ and the factorials satisfy Benford's Law; that is, leading digits in these sequences occur with frequencies given by $P(d)=\log_{10}(1+1/d)$, $d=1,2,\dots,9$. In this paper, we consider the leading digits of the Mersenne numbers $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime. In light of known irregularities in the distribution of primes, one might expect that the leading digit sequence of $\{M_n\}$ has \emph{worse} distribution properties than "smooth" sequences with similar rates of growth, such as $\{2^{n\log n}\}$. Surprisingly, the opposite seems to be the true; indeed, we present data, based on the first billion terms of the sequence $\{M_n\}$, showing that leading digits of Mersenne numbers behave in many respects \emph{more regularly} than those in the above smooth sequences. We state several conjectures to this effect, and we provide an heuristic explanation for the observed phenomena based on classic models for the distribution of primes.

math.NT↗

A local Benford Law for a class of arithmetic sequences

It is well-known that sequences such as the Fibonacci numbers and the factorials satisfy Benford's Law, that is, leading digits in these sequences occur with frequencies given by $P(d)=\log_{10}(1+1/d)$, $d=1,2,\dots,9$. In this paper, we investigate leading digit distributions of arithmetic sequences from a local point of view. We call a sequence locally Benford distributed of order $k$ if, roughly speaking, $k$-tuples of consecutive leading digits behave like $k$ independent Benford-distributed digits. This notion refines that of a Benford distributed sequence, and it provides a way to quantify the extent to which the Benford distribution persists at the local level. Surprisingly, most sequences known to satisfy Benford's Law have rather poor local distribution properties. In our main result we establish, for a large class of arithmetic sequences, a "best-possible" local Benford Law, that is, we determine the maximal value $k$ such that the sequence is locally Benford distributed of order $k$. The result applies, in particular, to sequences of the form $\{a^n\}$, $\{a^{n^d}\}$, and $\{n^β a^{n^α}\}$, as well as the sequence of factorials $\{n!\}$ and similar iterated product sequences.

math.NT↗

On distinct consecutive $r$-differences

Suppose $A\subset \mathbb{R}$ of size $k$ has distinct consecutive $r$--differences, that is for $1 \leq i \leq k -r$, the $r$--tuples $$(a_{i+1} - a_i , \ldots , a_{i+r} - a_{i + r -1})$$ are distinct. Then for any finite $B \subset \mathbb{R}$, one has $$|A+B| \gg_r |A||B|^{1/(r+1)}.$$ Utilizing de Bruijn sequences, we show this inequality is sharp up to the constant. Moreover, for the sequence $\{nα\}$, a sharp upper bound for the size of the distinct consecutive $r$--differences is obtained, which generalizes Steinhaus' three gap theorem. A dual problem on the consecutive $r$--differences of the returning times for some $ϕ\in \mathbb{R}$ defined by $\{T : \{Tθ\}<ϕ\}$ is also considered, which generalizes a result of Slater.

math.NT↗

A lower bound for the least prime in an arithmetic progression

Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek lower bounds for $P(k)$. In particular, we show that for almost every $k$ one has $P(k) \gg ϕ(k) \log k \log_2 k \log_4 k / \log_3 k,$ answering a question of Ford, Green, Konyangin, Maynard, and Tao. We rely on their recent work on large gaps between primes. Our main new idea is to use sieve weights to capture not only primes, but also small multiples of primes. We also give a heuristic which suggests that $\liminf_{k} \frac{P(k)}{ ϕ(k) \log^2 k} = 1.$

math.NT↗