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Zhichun Zhai

Publications and source records attributed to Zhichun Zhai.

At least 19 recordsLinked to original sources
arXiv2024

Learning Image Derived PDE-Phenotypes from fMRI Data

Partial Differential Equations (PDEs) model various physical phenomena, such as electromagnetic fields and fluid mechanics. Methods like Sparse Identification of Nonlinear Dynamics (SINDy) and PDE-Net 2.0 have been developed to identify and model PDEs based on data using sparse optimization and deep neural networks, respectively. While PDE models are less commonly applied to fMRI data, they hold the potential for uncovering hidden connections and essential components in brain activity. Using the ADHD200 dataset, we applied Canonical Independent Component Analysis (CanICA) and Uniform Manifold Approximation (UMAP) for dimensionality reduction of fMRI data. We then used Sparse Ridge Regression to identify PDEs from the reduced data, achieving high accuracy in classifying attention deficit hyperactivity disorder (ADHD). The study demonstrates a novel approach to extracting meaningful features from fMRI data for neurological disorder analysis to understand the role of oxygen transport (delivery $\&$ consumption) in the brain during neural activity relevant for studying intracranial pathologies.

q-bio.NC↗
arXiv2023

Wiener type regularity for non-linear integro-differential equations

The primary purpose of this paper is to study the Wiener-type regularity criteria for non-linear equations driven by integro-differential operators, whose model is the fractional $p-$Laplace equation. In doing so, with the help of tools from potential analysis, such as fractional relative Sobolev capacities, Wiener type integrals, Wolff potentials, $(α,p)-$barriers, and $(α,p)-$balayages, we first prove the characterizations of the fractional thinness and the Perron boundary regularity. Then, we establish a Wiener test and a generalized fractional Wiener criterion. Furthermore, we also prove the continuity of the fractional superharmonic function, the fractional resolutivity, a connection between $(α,p)-$potentials and $(α,p)-$Perron solutions, and the existence of a capacitary function for an arbitrary condenser.

math.AP↗
arXiv2022

Robust Optimal Designs when Missing Data Happen at Random

In this article, we investigate the robust optimal design problem for the prediction of response when the fitted regression models are only approximately specified, and observations might be missing completely at random. The intuitive idea is as follows: We assume that data are missing at random, and the complete case analysis is applied. To account for the occurrence of missing data, the design criterion we choose is the mean, for the missing indicator, of the averaged (over the design space) mean squared errors of the predictions. To describe the uncertainty in the specification of the real underlying model, we impose a neighborhood structure on the deterministic part of the regression response and maximize, analytically, the \textbf{M}ean of the averaged \textbf{M}ean squared \textbf{P}rediction \textbf{E}rrors (MMPE), over the entire neighborhood. The maximized MMPE is the ``worst'' loss in the neighborhood of the fitted regression model. Minimizing the maximum MMPE over the class of designs, we obtain robust ``minimax'' designs. The robust designs constructed afford protection from increases in prediction errors resulting from model misspecifications.

stat.ME↗
arXiv2022

Strengthened Fractional Sobolev Type Inequalities in Besov Spaces

The purpose of this article is twofold. The first is to strengthen fractional Sobolev type inequalities in Besov spaces via the classical Lorentz space. In doing so, we show that the Sobolev inequality in Besov spaces is equivalent to the fractional Hardy inequality and the iso-capacitary type inequality. Secondly, we will strengthen fractional Sobolev type inequalities in Besov spaces via capacitary Lorentz spaces associated with Besov capacities. For this purpose, we first study the embedding of the associated capacitary Lorentz space to the classical Lorentz space. Then, the embedding of the Besov space to the capacitary Lorentz space is established. Meanwhile, we show that these embeddings are closely related to the iso-capacitary type inequalities in terms of a new-introduced fractional $(β, p, q)$-perimeter. Moreover, characterizations of more general Sobolev type inequalities in Besov spaces have also been established.

math.AP↗
arXiv2022

Fractional Besov Trace/Extension Type Inequalities via the Caffarelli-Silvestre extension

Let $u(\cdot,\cdot)$ be the Caffarelli-Silvestre extension of $f.$ The first goal of this article is to establish the fractional trace type inequalities involving the Caffarelli-Silvestre extension $u(\cdot,\cdot)$ of $f.$ In doing so, firstly, we establish the fractional Sobolev/ logarithmic Sobolev/ Hardy trace inequalities in terms of $\nabla_{(x,t)}u(x,t).$ Then, we prove the fractional anisotropic Sobolev/ logarithmic Sobolev/ Hardy trace inequalities in terms of $ {\partial_{t} u(x,t)}$ or $(-Δ)^{-γ/2}u(x,t)$ only. Moreover, based on an estimate of the Fourier transform of the Caffarelli-Silvestre extension kernel and the sharp affine weighted $L^p$ Sobolev inequality, we prove that the $\dot{H}^{-β/2}(\mathbb{R}^n)$ norm of $f(x)$ can be controlled by the product of the weighted $L^p-$affine energy and the weighted $L^p-$norm of ${\partial_{t} u(x,t)}.$ The second goal of this article is to characterize non-negative measures $μ$ on $\mathbb{R}^{n+1}_+$ such that the embeddings $$\|u(\cdot,\cdot)\|_{L^{q_0,p_0}_μ(\mathbb{R}^{n+1})}\lesssim \|f\|_{\dotΛ^{p,q}_β(\mathbb{R}^n)}$$ hold for some $p_0$ and $q_0$ depending on $p$ and $q$ which are classified in three different cases: (1). $p=q\in (n/(n+β),1];$ (2) $(p,q)\in (1,n/β)\times (1,\infty);$ (3). $(p,q)\in (1,n/β)\times\{\infty\}.$ For case (1), the embeddings can be characterized in terms of an analytic condition of the variational capacity minimizing function, the iso-capacitary inequality of open balls, and other weak type inequalities. For cases (2) and (3), the embeddings are characterized by the iso-capacitary inequality for fractonal Besov capacity of open sets.

math.AP↗
arXiv2022

Problems related to Waring-Goldbach problem involving cubes of primes

In this note, we try to understand the recent development on the Waring-Goldbach problem involving cubes of primes. Especially, we want to determine whether integers that are either primes, squares of primes, cubes of primes, or a cube of an even number can be written as the sum of four cubes of primes. Meanwhile, we raise some problems that may deepen our understanding of the problem about the sum of four cubes of primes. Moreover, some examples suggest that almost all the cubes of integers can be written as the sum of cubes of four integers.

math.GM↗
arXiv2021

Embeddings of Function Spaces via the Caffarelli-Silvestre Extension, Capacities and Wolff potentials

Let $P_α f(x,t)$ be the Caffarelli-Silvestre extension of a smooth function $f(x): \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}_+:=\mathbb{R}^n\times (0,\infty).$ The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure $μ$ on $\mathbb{R}^{n+1}_+$ such that $f(x)\rightarrow P_α f(x,t)$ induces bounded embeddings from the Lebesgue spaces $L^p(\mathbb{R}^n)$ to the $L^q(\mathbb{R}^{n+1}_+,μ).$ On one hand, these embeddings will be characterized by using a newly introduced $L^p-$capacity associated with the Caffarelli-Silvestre extension. In doing so, the mixed norm estimates of $P_α f(x,t),$ the dual form of the $L^p-$capacity, the $L^p-$capacity of general balls, and a capacitary strong type inequality will be established, respectively. On the other hand, when $p>q>1,$ these embeddings will also be characterized in terms of the Hedberg-Wolff potential of $μ.$ Secondly, we characterize a nonnegative measure $μ$ on $\mathbb{R}^{n+1}_+$ such that $f(x)\rightarrow P_α f(x,t)$ induces bounded embeddings from the homogeneous Sobolev spaces $\dot{W}^{β,p}(\mathbb{R}^n)$ to the $L^q(\mathbb{R}^{n+1}_+,μ)$ in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.

math.AP↗
arXiv2017

Topological Data Analysis of Clostridioides difficile Infection and Fecal Microbiota Transplantation

Computational topologists recently developed a method, called persistent homology to analyze data presented in terms of similarity or dissimilarity. Indeed, persistent homology studies the evolution of topological features in terms of a single index, and is able to capture higher order features beyond the usual clustering techniques. There are three descriptive statistics of persistent homology, namely barcode, persistence diagram and more recently, persistence landscape. Persistence landscape is useful for statistical inference as it belongs to a space of $p-$integrable functions, a separable Banach space. We apply tools in both computational topology and statistics to DNA sequences taken from Clostridioides difficile infected patients treated with an experimental fecal microbiota transplantation. Our statistical and topological data analysis are able to detect interesting patterns among patients and donors. It also provides visualization of DNA sequences in the form of clusters and loops.

q-bio.QM↗
arXiv2013

On Global Regularity of 2D Generalized Magnetohydrodynamic Equations

In this article we study the global regularity of 2D generalized magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are $- ν(- \triangle)^α u$ and $- κ(-\triangle)^β b$. We show that smooth solutions are global in the following three cases: $α\geqslant 1 / 2, β\geqslant 1$; $0 \leqslant α< 1 / 2, 2 α+ β> 2$; $α\geqslant 2, β= 0$. We also show that in the inviscid case $ν= 0$, if $β> 1$, then smooth solutions are global as long as the direction of the magnetic field remains smooth enough.

math.AP↗
arXiv2013

Regularity and Capacity for the Fractional Dissipative Operator

This note is devoted to exploring some analytic-geometric properties of the regularity and capacity associated to the so-called fractional dissipative operator $\partial_t+(-Δ)^α$, naturally establishing a diagonally sharp Hausdorff dimension estimate for the blow-up set of a weak solution to the fractional dissipative equation $(\partial_t+(-Δ)^α)u(t,x)=F(t,x)$ subject to $u(0,x)=0$.

math.AP↗
arXiv2012

On the Euler-Poincaré equation with non-zero dispersion

We consider the Euler-Poincaré equation on $\mathbb R^d$, $d\ge 2$. For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. This settles an open problem raised by Chae and Liu \cite{Chae Liu}. Our analysis exhibits some new concentration mechanism and hidden monotonicity formula associated with the Euler-Poincaré flow. In particular we show the abundance of blowups emanating from smooth initial data with certain sign properties. No size restrictions are imposed on the data. We also showcase a class of initial data for which the corresponding solution exists globally in time.

math.AP↗
arXiv2009

Several analytic inequalities in some $Q-$spaces

In this paper, we establish separate necessary and sufficient John-Nirenberg (JN) type inequalities for functions in $Q_α^β(\mathbb{R}^{n})$ which imply Gagliardo-Nirenberg (GN) type inequalities in $Q_α(\mathbb{R}^{n}).$ Consequently, we obtain Trudinger-Moser type inequalities and Brezis-Gallouet-Wainger type inequalities in $Q_α(\mathbb{R}^{n}).$

math.AP↗
arXiv2009

Note on affine Gagliardo-Nirenberg inequalities

This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine $L^{p}-$Sobolev inequalities. The logarithmic version of affine $L^{p}-$Sobolev inequalities is verified. Moreover, An alternative proof of the affine Moser-Trudinger and Morrey-Sobolev inequalities is given. The main tools are the equimeasurability of rearrangements and the strengthened version of the classical Pólys-Szegö principle.

math.FA↗
arXiv2009

Well-posedness for fractional Navier-Stokes equations in critical spaces close to $\dot{B}^{-(2β-1)}_{\infty,\infty}(\mathbb{R}^{n})$

In this paper, we prove the well-posedness for the fractional Navier-Stokes equations in critical spaces $G^{-(2β-1)}_{n}(\mathbb{R}^{n})$ and $BMO^{-(2β-1)}(\mathbb{R}^{n}).$ Both of them are close to the largest critical space $\dot{B}^{-(2β-1)}_{\infty,\infty}(\mathbb{R}^{n}).$ In $G^{-(2β-1)}_{n}(\mathbb{R}^{n}),$ we establish the well-posedness based on a priori estimates for the fractional Navier-Stokes equations in Besov spaces. To obtain the well-posedness in $BMO^{-(2β-1)}(\mathbb{R}^{n}),$ we find a relationship between $Q_{α;\infty}^{β,-1}(\mathbb{R}^{n})$ and $BMO(\mathbb{R}^{n})$ by giving an equivalent characterization of $BMO^{-ζ}(\mathbb{R}^{n}).$

math.AP↗
arXiv2009

Strichartz type estimates for fractional heat equations

We obtain Strichartz estimates for the fractional heat equations by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-Littlewood-Sobolev inequality. We also prove an endpoint homogeneous Strichartz estimate via replacing $ L^{\infty}_{x}(\mathbb{R}^{n})$ by $BMO_{x}(\mathbb{R}^{n})$ and a parabolic homogeneous Strichartz estimate. Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue spaces with either Besov spaces or Sobolev spaces. Moreover, we establish the Strichartz estimates for the fractional heat equations with a time dependent potential of an appropriate integrability. As an application, we prove the global existence and uniqueness of regular solutions in spatial variables for the generalized Navier-Stokes system with $L^{r}(\mathbb{R}^{n})$ data.

math.AP↗
arXiv2009

Well-posedness and regularity of generalized Navier-Stokes equations in some Critical $Q-$spaces

We study the well-posedness and regularity of the generalized Navier-Stokes equations with initial data in a new critical space $Q_{α;\infty}^{β,-1}(\mathbb{R}^{n})=\nabla\cdot(Q_α^β(\mathbb{R}^{n}))^{n}, β\in({1/2},1)$ which is larger than some known critical homogeneous Besov spaces. Here $Q_α^β(\mathbb{R}^{n})$ is a space defined as the set of all measurable functions with $$\sup(l(I))^{2(α+β-1)-n}\int_{I}\int_{I}\frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2(α-β+1)}}dxdy<\infty$$ where the supremum is taken over all cubes $I$ with the edge length $l(I)$ and the edges parallel to the coordinate axes in $\mathbb{R}^{n}.$ In order to study the well-posedness and regularity, we give a Carleson measure characterization of $Q_α^β(\mathbb{R}^{n})$ by investigating a new type of tent spaces and an atomic decomposition of the predual for $Q_α^β(\mathbb{R}^{n}).$ In addition, our regularity results apply to the incompressible Navier-Stokes equations with initial data in $Q_{α;\infty}^{1,-1}(\mathbb{R}^{n}).$

math.AP↗