arXiv subjects

Anh Do

Publications and source records attributed to Anh Do.

5 recordsLinked to original sources

arXiv2026

Optimal AB-program for Poincaré--Sobolev and logarithmic Sobolev inequalities on hyperbolic spaces

In this paper, we prove that we can subtract the full spectral gap $\left(\frac{N-1}{p}\right)^{p}$ of the $p$-Laplacian and keep the sharp Sobolev constant $S(N,p)$ of $\mathbb{R}^N$ at the same time in a large family of $p$. For $p=2$, this was proved by Benguria, Frank and Loss in dimension three, and it was shown by Hebey to be false in dimensions $N\geq4$. For $p\neq2$, it was proved by Nguyen for $N\geq4$ and $\frac{2N}{N-1}\leq p<N$. By employing, among other tools, a suitable conjugation that maps the radial $p$-Dirichlet integral on $\mathbb{H}^N$ onto its Euclidean counterpart on $\mathbb{R}^N$ while preserving the $L^p$ norm, together with a one-dimensional tangent line inequality, we establish such sharp Poincaré--Sobolev inequalities for $N=3$ and $2\leq p<3$, for $N=4$ and $\frac{1+\sqrt{13}}{2}\leq p<4$, and for $N\geq5$ and $\min\{3,\frac{1+\sqrt{4N-3}}{2}\}\leq p<N$. In particular, in all cases we also obtain an explicit remainder term. Moreover, since the change of function keeps the $L^p$ norm, we prove the sharp logarithmic Sobolev inequalities with the full spectral gap and the sharp Euclidean constant for $N=3$ and $2\leq p\leq3$, and for $N\geq4$ on an explicit interval of $p$. For $p\geq3$, we also use the change of variable $t=\sinhρ$ and the Hardy inequality, which give a second remainder term. At the endpoint $p=N$, this remainder gives a Hardy--Trudinger--Moser inequality on $\mathbb{H}^N$ under a weaker constraint than the one of Liang, Lu, Wang and Yang, in every dimension $N\geq3$. Finally, we show that for $1<p<\frac{1+\sqrt{4N-3}}{2}$, which includes all $1<p<2$, the full spectral gap can still be subtracted with some positive constant in front of the Sobolev term, and that this constant is smaller than $S(N,p)^{p}$ when $p<\min\{2,\sqrt N\}$. Together with our other results, this shows that the full spectral gap can be subtracted for every $1<p<N$.

math.DG↗

arXiv2026

Optimal stability hierarchies of the Heisenberg Uncertainty Principle for solenoidal fields and of the second order Caffarelli--Kohn--Nirenberg inequalities

In 2018, V. Maz'ya proposed 75 open problems in analysis and PDEs. One of them is about the best constant for the Heisenberg Uncertainty Principle for solenoidal vector fields motivated by questions in hydrodynamics. This was answered by Cazacu, Flynn and Lam in dimension two and subsequently by Hamamoto in all dimensions $N\ge3$ by establishing sharp inequalities with explicit constants. Nevertheless, the much harder stability problem remains open in all dimensions. The first main result in this paper is to establish a sharp stability hierarchy by proving that its deficit controls the distance to the set of extremals, with the sharp stability constant $\frac12(N-\sqrt{N^2-4N+12})$ for $N\ge4$ and $1$ for $N=3$, together with chains of remainder terms measuring the distance to explicit larger families of poloidal and toroidal fields. The second main result is the second order $L^2$-Caffarelli--Kohn--Nirenberg inequality with the weights $|x|^{-2a}$ and $|x|^{-2b}$ on the line $1+a+b=0$, for which we obtain the stability constant $\frac12\min\{4(1+a),\sqrt{(N+2a)^2+4N-4}-(N+2a)\}$. This CKN inequality is sharp whenever the second number is the smaller one. We develop a novel method: the fourth order one-dimensional problems attached to the spherical modes are transformed, by a Fourier--Hankel transform of real order, into first order inequalities whose sharp stability is that of weighted Gaussian Poincaré inequalities. The inverse transform produces the extremals in terms of Kummer's function. As a byproduct of our new approach, we also obtain substantially simpler proofs, with sharp remainder terms and equality cases, of Hamamoto's one-dimensional inequality and of his sharp uncertainty principle for solenoidal fields.

math.AP↗

arXiv2026

A Fourier approach to the sharp stability of Heisenberg Uncertainty Principles of higher and fractional orders: a new perspective

We show that, on the Fourier side, the sharp second order Heisenberg Uncertainty Principle (HUP) and its sharp stability for functions are nothing but the first order $L^{2}$-Caffarelli--Kohn--Nirenberg inequality and its sharp stability, applied to their Fourier transforms. This gives a proof in a few lines of the sharp stability estimate that we established recently by a much longer argument. The same observation then produces two one-parameter families of sharp fractional Heisenberg Uncertainty Principles, in which the gradient is replaced by a fractional power of the Laplacian: for each of them we compute the sharp constants, characterize all the optimizers, and establish the sharp stability estimates. The stability constant of the first family is equal to $1$ for every nonnegative order and every dimension. Negative orders, where the fractional Laplacian becomes a Riesz potential, are treated as well. Finally, we show that the deficit carries much more information than the distance to the optimizers alone: it controls an explicit chain of successive remainder terms whose constants are the successive spectral gaps of an explicit operator. For the classical Heisenberg Uncertainty Principle this gives three remainder terms with optimal explicit constants, and for the second order HUP a chain of four remainder terms measured against explicit confluent hypergeometric profiles and coupled through one single set of parameters. As applications, we also establish the chain of stability of the HUP for curl-free vector fields.

math.AP↗

arXiv2024

Detecting Omissions in Geographic Maps through Computer Vision

This paper explores the application of computer vision technologies to the analysis of maps, an area with substantial historical, cultural, and political significance. Our focus is on developing and evaluating a method for automatically identifying maps that depict specific regions and feature landmarks with designated names, a task that involves complex challenges due to the diverse styles and methods used in map creation. We address three main subtasks: differentiating maps from non-maps, verifying the accuracy of the region depicted, and confirming the presence or absence of particular landmark names through advanced text recognition techniques. Our approach utilizes a Convolutional Neural Network and transfer learning to differentiate maps from non-maps, verify the accuracy of depicted regions, and confirm landmark names through advanced text recognition. We also introduce the VinMap dataset, containing annotated map images of Vietnam, to train and test our method. Experiments on this dataset demonstrate that our technique achieves F1-score of 85.51% for identifying maps excluding specific territorial landmarks. This result suggests practical utility and indicates areas for future improvement.

cs.CV↗

arXiv2022

Improving Generative Flow Networks with Path Regularization

Generative Flow Networks (GFlowNets) are recently proposed models for learning stochastic policies that generate compositional objects by sequences of actions with the probability proportional to a given reward function. The central problem of GFlowNets is to improve their exploration and generalization. In this work, we propose a novel path regularization method based on optimal transport theory that places prior constraints on the underlying structure of the GFlowNets. The prior is designed to help the GFlowNets better discover the latent structure of the target distribution or enhance its ability to explore the environment in the context of active learning. The path regularization controls the flow in GFlowNets to generate more diverse and novel candidates via maximizing the optimal transport distances between two forward policies or to improve the generalization via minimizing the optimal transport distances. In addition, we derive an efficient implementation of the regularization by finding its closed form solutions in specific cases and a meaningful upper bound that can be used as an approximation to minimize the regularization term. We empirically demonstrate the advantage of our path regularization on a wide range of tasks, including synthetic hypergrid environment modeling, discrete probabilistic modeling, and biological sequence design.

cs.LG↗