arXiv subjects

Changjian Liu

Publications and source records attributed to Changjian Liu.

15 recordsLinked to original sources

arXiv2026

Multi-channel Uplift Policy Learning

E-commerce platforms must allocate fixed marketing budgets across multiple channels to maximize business utility. However, standard predict-then-optimize (PTO) paradigms fail in this compositional space due to observational confounding and severe extrapolation. We formulate this challenge as a simplex-constrained uplift decision problem and propose ReAlloc, a fast-slow causal framework. Specifically, an agile Orthogonal Teacher extracts unbiased local gradients from short-term logs, while an Explanation-Guided Student distills them into a structured marginal field over long-term horizons. This design enables support-aware, conservative decisions that capture cross-channel substitutions. Extensive simulations and large-scale online A/B tests on Taobao platform demonstrate that ReAlloc achieves simultaneous lifts in both pay order and income.

cs.LG↗

arXiv2026

OpFlow: Learning Opportunity-Conditioned Choice Potentials for Robust OD Flow Prediction

Origin-destination (OD) flow prediction is central to urban analytics, yet deep models trained on raw counts remain vulnerable to distribution shift. The core problem is that raw count supervision cannot distinguish transferable choice mechanisms from environment-specific shortcuts. Raw OD count mixes two objects: how much demand an origin produces and how that demand is allocated across destinations. We argue that the transferable object is the exposure-to-choice law that maps spatial conditions to relative destination preferences. We propose OpFlow, a mechanism-constrained framework that learns row-centered choice potentials and reconstructs flows by combining the induced allocation with a separately calibrated origin scale. Under distribution shift, spatial exposures and the induced allocations are allowed to vary; what transfers is the conditional map from exposure states to relative choice potentials. Theoretically, we characterize the identifiable row-centered potential and show that classical spatial interaction laws are restricted log-potential cases. Controlled synthetic shifts and a real-world experiment show OpFlow improves robustness under environment shifts.

cs.LG↗

arXiv2025

The discontinuous planar piecewise linear system with two nodes has at most two limit cycles

This paper investigates the multiplicity and the number of limit cycles for planar piecewise linear system divided into two regions by a straight line and each linear subsystem has a node. Through constructing Poincare half maps and a successor function, and analyzing the properties of the successor function, we can derive that this system has at most two limit cycles, counting the multiplicities of limit cycles.

math.DS↗

arXiv2024

Sufficient conditions for the n-dimensional real Jacobian conjecture

The real Jacobian conjecture was posed by Randall in 1983. This conjecture asserts that if $F=\left(f_1,\ldots ,f_n\right):\mathbb{R}^n\rightarrow\mathbb{R}^n$ is a polynomial map such that $\det DF\left(\mathbf{x}\right)\neq0$ for all $\mathbf{x}\in\mathbb{R}^n$, then $F$ is injective. This investigation mainly consists of two parts. Firstly, we use the qualitative theory of dynamical systems to give an alternate proof of the polynomial version of the $n$-dimensional Hadamard's theorem. Secondly, we present some algebraic sufficient conditions for the $n$-dimensional real Jacobian conjecture. Our results not only extend the main result of [J. Differential Equations {\bf 260} (2016), 5250-5258] to quasi-homogeneous type, but also generalize it from $\mathbb{R}^2$ to $\mathbb{R}^n$. As a coproduct of our proof process, we solve an open problem formulated by Braun, Giné and Llibre in [J. Differential Equations {\bf 260} (2016), 5250-5258].

math.DS↗

arXiv2024

Stability for a family of planar systems with nilpotent critical points

Consider a family of planar polynomial systems $\dot x = y^{2l-1} - x^{2k+1}, \dot y =-x +m y^{2s+1},$ where $l,k,s\in\mathbb{N^*},$ $2\le l \le 2s$ and $m\in\mathbb{R}.$ We study the center-focus problem on its origin which is a monodromic nilpotent critical point. By directly calculating the generalized Lyapunov constants, we find that the origin is always a focus and we complete the classification of its stability. This includes the most difficult case: $s=kl$ and $m=(2k+1)!!/(2kl+1)!_{(2l)}.$ In this case, we prove that the origin is always unstable. Our result extends and completes a previous one.

math.DS↗

arXiv2024

The stability of smooth solitary waves for the $b$-family of Camassa-Holm equations

The $b$-family of Camassa-Holm ($b$-CH) equation is a one-parameter family of PDEs, which includes the completely integrable Camassa-Holm and Degasperis-Procesi equations but possesses different Hamiltonian structures. Motivated by this, we study the existence and the orbital stability of the smooth solitary wave solutions with nonzero constant background to the $b$-CH equation for the special case $b=1$, whose the Hamiltonian structure is different from that of $b\neq1$. We establish a connection between the stability criterion for the solitary waves and the monotonicity of a singular integral along the corresponding homoclinic orbits of the spatial ODEs. We verify the latter analytically using the framework for the monotonicity of period function of planar Hamiltonian systems, which shows that the smooth solitary waves are orbitally stable. In addition, we find that the existence and orbital stability results for $0 1$, particularly the stability criteria are the same. Finally, combining with the results for the case $b>1$, we conclude that the solitary waves to the $b$-CH equation is structurally stable under the variation of %with respect to the parameter $b>0$.

math.AP↗

arXiv2023

On the non-existence of isochronous centers in planar discontinuous differential systems

The determination of whether a center is isochronous or not is a well-known problem in the qualitative theory of planar systems. In this paper, we explore planar piecewise discontinuous differential systems characterized by a straight switching line $y=0$ and $x=0$, respectively. Our investigation reveals that such systems do not possess an isochronous center at the origin.

math.DS↗

arXiv2023

Maximum number of limit cycles for Abel equation having coefficients with linear trigonometric functions

This paper devotes to the study of the classical Abel equation $\frac{dx}{dt}=g(t)x^{3}+f(t)x^{2}$, where $g(t)$ and $f(t)$ are trigonometric polynomials of degree $m\geq1$. We are interested in the problem that whether there is a uniform upper bound for the number of limit cycles of the equation with respect to $m$, which is known as the famous Smale-Pugh problem. In this work we generalize an idea from the recent paper (Yu, Chen and Liu, arXiv:$2304.13528$, $2023$) and give a new criterion to estimate the maximum multiplicity of limit cycles of the above Abel equations. By virtue of this criterion and the previous results given by {Á}lvarez et al. and Bravo et al., we completely solve the simplest case of the Smale-Pugh problem, i.e., the case when $g(t)$ and $f(t)$ are linear trigonometric, and obtain that the maximum number of limit cycles, is three.

math.CA↗

arXiv2023

The number of limit cycles of Josephson equation

In this paper, the existence and number of non-contractible limit cycles of the Josephson equation $β\frac{d^{2}Φ}{dt^{2}}+(1+γ\cos Φ)\frac{dΦ}{dt}+\sin Φ=α$ are studied, where $ϕ\in \mathbb S^{1}$ and $(α,β,γ)\in \mathbb R^{3}$. Concretely, by using some appropriate transformations, we prove that such type of limit cycles are changed to limit cycles of some Abel equation. By developing the methods on limit cycles of Abel equation, we prove that there are at most two non-contractible limit cycles, and the upper bound is sharp. At last, combining with the results of the paper (Chen and Tang, J. Differential Equations, 2020), we show that the sum of the number of contractible and non-contractible limit cycles of the Josephson equation is also at most two, and give the possible configurations of limit cycles when two limit cycles appear.

math.CA↗

arXiv2023

Centers and invariant straight lines of planar real polynomial vector fields and its configurations

In the paper, we first give the least upper bound formula on the number of centers of planar real polynomial Hamiltonian vector fields. This formula reveals that the greater the number of invariant straight lines of the vector field and the less the number of its centers. Then we obtain some rules on the configurations of centers of planar real polynomial Hamiltonian Kolmogorov vector fields when the number of centers is exactly the least upper bound. As an application of these results, we give an affirmative answer to a conjecture on the topological classification of configurations for the cubic Hamiltonian Kolmogorov vector fields with four centers. Moreover, we discuss the relationship between the number of centers of planar real polynomial vector fields and the existence of limit cycles, and prove that cubic real polynomial Kolmogorov vector fields have no limit cycles if the number of its centers reaches the maximum. More precisely, it is shown that the cubic real polynomial Kolmogorov vector field must have an elementary first integral in $\mathbb{R}^2\setminus\{xy=0\}$ if it has four centers, and the number of configurations of its centers is one more than that of the cubic polynomial Hamiltonian Kolmogorov vector fields.

math.DS↗

arXiv2022

Orbital stability of smooth solitary waves for the $b$-family of Camassa-Holm equations

In this paper, we study the stability of smooth solitary waves for the $b$-family of Camassa-Holm equations. We verify the stability criterion analytically for the general case $b>1$ by the idea of the monotonicity of the period function for planar Hamiltonian systems and show that the smooth solitary waves are orbitally stable, which gives a positive answer to the open problem proposed by Lafortune and Pelinovsky [S. Lafortune, D. E. Pelinovsky, Stability of smooth solitary waves in the $b$-Camassa-Holm equation].

math.AP↗

arXiv2022

A proof of the uniqueness of the limit cycle of a quasi-homogeneous system

A. Gasull shared a list of 33 open problems in low dimensional dynamical systems in his work in 2021. The second part of Problem 3 is about whether the limit cycle of a quasi-homogeneous system $ \dot{x}=y,\; \dot{y}=-x^3+αx^2y+y^3 $ is unique. In this paper, we give a positive answer to this question by analysing the uniqueness of the heteroclinic separatrix at infinity.

math.DS↗

arXiv2021

On the Structure of Periodic Eigenvalues of the Vectorial $p$-Laplacian

In this paper we will solve an open problem raised by Manásevich and Mawhin twenty years ago on the structure of the periodic eigenvalues of the vectorial $p$-Laplacian. This is an Euler-Lagrangian equation on the plane or in higher dimensional Euclidean spaces. The main result obtained is that for any exponent $p$ other than $2$, the vectorial $p$-Laplacian on the plane will admit infinitely many different sequences of periodic eigenvalues with a given period. These sequences of eigenvalues are constructed using the notion of scaling momenta we will introduce. The whole proof is based on the complete integrability of the equivalent Hamiltonian system, the tricky reduction to $2$-dimensional dynamical systems, and a number-theoretical distinguishing between different sequences of eigenvalues. Some numerical simulations to the new sequences of eigenvalues and eigenfunctions will be given. Several further conjectures towards to the panorama of the spectral sets will be imposed.

math.DS↗

arXiv2019

On the maximal saddle order of p:-q resonant saddle

In this paper, we obtain some estimations of the saddle order which is the sole topological invariant of the non-integrable resonant saddles of planar polynomial vector fields of arbitrary degree $n$. Firstly, we prove that, for any given resonance $p:-q$, $(p, q)=1$, and sufficiently big integer $n$, the maximal saddle order can grow at least as rapidly as $n^2$. Secondly, we show that there exists an integer $k_0$, which grows at least as rapidly as $3n^2/2$, such that $L_{k_0}$ does not belong to the ideal generated by the first $k_0-1$ saddle values $L_1, L_2, \cdots, L_{k_0-1}$, where $L_{k}$ means the $k$-th saddle value of the given system. In particular, if $p=1$ (or $q=1$), we obtain a sharper result that $k_0$ can grow at least as rapidly as $2 n^2$.

math.CA↗

arXiv2017

Minimum Rate Sampling and Spectrum Blind Reconstruction in Random Equivalent Sampling

The random equivalent sampling (RES) is a well-known sampling technique that can be used to capture a high-speed repetitive waveform with low sampling rate. In this paper, the feasibility of spectrum-blind multiband signal reconstruction for data sampled from RES is investigated. We propose a RES sampling pattern and its corresponding mathematical model that guarantees well-conditioned reconstruction of multiband signal with unknown spectral support. We give the minimum number of RES acquisitions that hold overwhelming probability to successfully reconstruct original signal. We demonstrate that for signal with specific spectral occupation, the number of RES acquisitions and the minimum sampling rate could be approached. The signal reconstruction is studied in the framework of compressive sampling (CS) theory. The eigen-decomposition and minimum description length (MDL) criteria are adopted to adaptively estimate the dimension of signal, and the number of unknowns of reconstruction problem is reduced. Experimental results are reported to indicate that, for a spectrum-blind sparse multiband signal, the proposed reconstruction algorithm for RES is feasible and robust.

physics.ins-det↗