arXiv subjects

Enhao Liu

Publications and source records attributed to Enhao Liu.

6 recordsLinked to original sources

arXiv2026

Active phase-space topology unifies depletion and alignment in bacterial flows

Transport at small scales is classically understood within an equilibrium framework, where dispersion theory successfully describes shear-enhanced diffusion for passive particles in the continuum limit. However, as most bacteria can move on their own, their motility in flows, inherently out of thermal equilibrium, fundamentally challenges this framework. A minimal, predictive unified theory of bacterial transport in low-Reynolds-number flows remains lacking. Here, from first principles, we develop an analytical hydrodynamic model that enforces consistent no-flux boundary conditions and uses the method of images to characterize the flow-wall coupling. The model quantitatively reproduces measured bacterial distributions and reveals a hydrodynamic locking mechanism accompanied by mean-drift invariance -- an active counterpart to Taylor dispersion. We clarify that shear-induced depletion and alignment are dual manifestations of a single active phase-space topology, ruling out explanations based solely on the local shear magnitude. The theory is validated against microfluidic experiments spanning multiple bacterial species and shear geometries, from one-dimensional to fully three-dimensional flows. Our findings establish a unified phase-space framework for bacterial hydrodynamics, advancing the fundamental understanding of active matter.

physics.flu-dyn↗

arXiv2024

Bipath Persistence as Zigzag Persistence

Persistence modules that decompose into interval modules are important in topological data analysis because we can interpret such intervals as the lifetime of topological features in the data. We can classify the settings in which persistence modules always decompose into intervals, by a recent result of Aoki, Escolar and Tada: these are standard single-parameter persistence, zigzag persistence, and bipath persistence. No other setting offers such guarantees. We show that a bipath persistence module can be decomposed via a closely related infinite zigzag persistence module, understood as a covering. This allows us to translate techniques of zigzag persistence, like recent advancements in its efficient computation by Dey and Hou, to bipath persistence. In addition, and again by the relation with the infinite zigzag, we can define an interleaving and bottleneck distance on bipath persistence. In turn, the algebraic stability of zigzag persistence implies the algebraic stability of bipath persistence.

math.AT↗

arXiv2024

Interval Multiplicities of Persistence Modules

For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ of $\mathbf{P}$, we give a formula for the multiplicity $d_M(V_I)$ of the interval module $V_I$ in the indecomposable decomposition of $M$ in terms of the ranks of matrices consisting of structure linear maps of $M$. This generalizes the corresponding formula for 1-dimensional persistence modules. As applications, the formula enables us to compute the maximal interval-decomposable direct summand of $M$, to decide whether $M$ is interval-decomposable, and to detect properties determined by prescribed interval summands without decomposing $M$. We also give criteria, in terms of top and socle supports along minimal projective resolutions and injective coresolutions of $M$, restricting the intervals that can occur as direct summands of $M$ and thereby reduce the number of intervals to be computed in practice. Moreover, the formula tells us which morphisms of $\mathbf{P}$ are essential to compute $d_M(V_I)$. This leads to the notion of an order-preserving map $\zeta \colon Z \to \mathbf{P}$ essentially covering $I$, for which the multiplicity is preserved under the induced restriction functor $R \colon \operatorname{mod} \mathbf{P} \to \operatorname{mod} Z$. When $Z$ is of Dynkin type $\mathbb{A}$, also known as a zigzag poset, this allows the multiplicity to be computed more efficiently from the filtration level of topological spaces, without computing all structure linear maps of $M$. Finally, we give a formula for $d_M(V_I)$ in terms of a projective (or injective) (co)presentation of $M$. In the 2D-grid case, this is more practical since such resolutions can be computed from the filtration level of topological spaces.

math.RT↗

arXiv2024

Curse of Dimensionality on Persistence Diagrams

The stability of persistent homology has led to wide applications of the persistence diagram as a trusted topological descriptor in the presence of noise. However, with the increasing demand for high-dimension and low-sample-size data processing in modern science, it is questionable whether persistence diagrams retain their reliability in the presence of high-dimensional noise. This work aims to study the reliability of persistence diagrams in the high-dimension low-sample-size data setting. By analyzing the asymptotic behavior of persistence diagrams for high-dimensional random data, we show that persistence diagrams are no longer reliable descriptors of low-sample-size data under high-dimensional noise perturbations. We refer to this loss of reliability of persistence diagrams in such data settings as the curse of dimensionality on persistence diagrams. Next, we investigate the possibility of using normalized principal component analysis as a method for reducing the dimensionality of the high-dimensional observed data to resolve the curse of dimensionality. We show that this method can mitigate the curse of dimensionality on persistence diagrams. Our results shed some new light on the challenges of processing high-dimension low-sample-size data by persistence diagrams and provide a starting point for future research in this area.

math.ST↗

arXiv2024

Interval Replacements of Persistence Modules

We define two notions. The first one is a $rank\ compression\ system$ $\xi$ for a finite poset $\mathbf{P}$ that assigns each interval subposet $I$ to an order-preserving map $\xi_I \colon I^{\xi} \to \mathbf{P}$ satisfying some conditions, where $I^{\xi}$ is a connected finite poset. An example is given by the $total$ compression system that assigns each $I$ to the inclusion of $I$ into $\mathbf{P}$. The second one is an $I$-$rank$ of a persistence module $M$ under $\xi$, the family of which is called the $interval\ rank\ invariant$ of $M$ under $\xi$. A compression system $\xi$ makes it possible to define the $interval\ replacement$ (also called the interval-decomposable approximation) not only for 2D persistence modules but also for any persistence modules over any finite poset. We will show that the forming of the interval replacement preserves the interval rank invariant, which is a stronger property than the preservation of the usual rank invariant. Moreover, to know what is preserved by the replacement explicitly, we will give a formula of the $I$-rank of $M$ under $\xi$ in terms of the structure linear maps of $M$ for any compression system $\xi$. The formula leads us to a concept of essential cover, which gives us a sufficient condition for the $I$-rank of $M$ under $\xi$ to coincide with that under another compression system $\zeta$. This is applied to the case where $\xi = \mathrm{tot}$, the value of $I$-rank under which is equal to the generalized rank invariant introduced by Kim--M\'emoli, to give an alternative proof of the Dey--Kim--M\'emoli theorem computing the generalized rank invariant by using a zigzag path.

math.RT↗

arXiv2021

A Bayesian spatio-temporal nowcasting model for public health decision-making and surveillance

As COVID-19 spread through the United States in 2020, states began to set up alert systems to inform policy decisions and serve as risk communication tools for the general public. Many of these systems, like in Ohio, included indicators based on an assessment of trends in reported cases. However, when cases are indexed by date of disease onset, reporting delays complicate the interpretation of trends. Despite a foundation of statistical literature to address this problem, these methods have not been widely applied in practice. In this paper, we develop a Bayesian spatio-temporal nowcasting model for assessing trends in county-level COVID-19 cases in Ohio. We compare the performance of our model to the current approach used in Ohio and the approach that was recommended by the Centers for Disease Control and Prevention. We demonstrate gains in performance while still retaining interpretability using our model. In addition, we are able to fully account for uncertainty in both the time series of cases and in the reporting process. While we cannot eliminate all of the uncertainty in public health surveillance and subsequent decision-making, we must use approaches that embrace these challenges and deliver more accurate and honest assessments to policymakers.

stat.AP↗