arXiv subjects

Michael Ruzhansky

Publications and source records attributed to Michael Ruzhansky.

At least 37 records · Page 2Linked to original sources

arXiv2025

Nonlinear Dirac equations on noncompact quantum graphs with potentials: Multiplicity and Concentration

In this paper, we study the existence and multiplicity of solutions to the following class of nonlinear Dirac equations (NLDE) on noncompact quantum graphs: \[ -i\,\varepsilon c\,σ_1\,\partial_x u + m c^2 σ_3 u + V(x)\,u = f(|u|)\,u, \quad x\in \mathcal{G}, \tag{P} \] where $V:\mathcal{G}\to\mathbb{R}$ and $f:\mathbb{R}\to\mathbb{R}$ are continuous, $\varepsilon>0$ is a semiclassical parameter, $m>0$ denotes the mass, and $c>0$ the speed of light. Here $σ_1,σ_3$ are Pauli matrices, and $\mathcal{G}$ is a noncompact quantum graph. We prove that when $\varepsilon$ is sufficiently small, the number of solutions to $(P)$ is at least the number of global minima of $V$. Moreover, these solutions exhibit semiclassical concentration: as $\varepsilon\to0$, their concentration points approach the set of global minima of $V$.

math.AP↗

arXiv2025

Numerical Approaches for Identifying the Time-Dependent Potential Coefficient in the Diffusion Equation

We address the inverse problem of identifying a time-dependent potential coefficient in a one-dimensional diffusion equation subject to Dirichlet boundary conditions and a nonlocal integral overdetermination constraint reflecting spatially averaged measurements. After establishing well-posedness for the forward problem and deriving an a priori estimate that ensures uniqueness and continuous dependence on the data, we prove existence and uniqueness for the inverse problem. To compute numerically the unknown coefficient, we propose and compare three numerical methods: an integration-based scheme, a Newton-Raphson iterative solver, and a physics-informed neural network (PINN). Numerical experiments on both exact and noisy data demonstrate the accuracy, robustness, and efficiency of each approach.

math.NA↗

arXiv2025

Fujita exponent for heat equation with Hörmander vector fields

In this paper, we show global existence and non-existence results for the heat equation with some of the squares of smooth vector fields on $\Rn$ satisfying Hörmander's rank condition with a non-linearity of the form $f(u)$, where $f$ is a suitable function and $u$ is the solution. In particular, when $f(u)=u^p$, we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type $φ(t)f(u)$.

math.AP↗

arXiv2025

Nonrelativistic limit of bound-state solutions for nonlinear Dirac equation on noncompact quantum graphs

In this paper, we investigate the nonrelativistic limit and qualitative properties of bound-state solutions for the nonlinear Dirac equation (NLDE) defined on noncompact quantum graphs: \[ -i c \frac{d}{d x} σ_1 ψ+m c^2 σ_3 ψ-ωψ=g(|ψ|) ψ, \quad \text { in } \mathcal{G} \] where $ g : \mathbb{R}\rightarrow\mathbb{R} $ is a continuous nonlinear function, $ c>0 $ represents the speed of light, $ m>0 $ is the particle's mass, $ ω\in\mathbb{R} $ is related to the frequency, $ σ_1 $ and $ σ_3 $ denote the Pauli matrices, and $\mathcal{G}$ is a noncompact quantum graph. We establish the existence of bound-state solutions to the NLDE on $\mathcal{G}$, and prove that these solutions converge toward the corresponding bound-state solutions of a nonlinear Schrödinger equation (NLS) in the nonrelativistic limit (i.e., as the speed of light $ c \to \infty $) for particles of small mass. Furthermore, we prove uniform boundedness and exponential decay properties of the NLDE solutions, uniformly in $ c $, thereby offering insight into their asymptotic behavior.

math.AP↗

arXiv2025

Semi-sparsity Generalization for Variational Mesh Denoising

In this paper, we propose a new variational framework for 3D surface denoising over triangulated meshes, which is inspired by the success of semi-sparse regularization in image processing. Differing from the uniformly sampled image data, mesh surfaces are typically represented by irregular, non-uniform structures, which thus complicate the direct application of the standard formulation and pose challenges in both model design and numerical implementation. To bridge this gap, we first introduce the discrete approximations of higher-order differential operators over triangle meshes and then develop a semi-sparsity regularized minimization model for mesh denoising. This new model is efficiently solved by using a multi-block alternating direction method of multipliers (ADMM) and achieves high-quality simultaneous fitting performance -- preserving sharp features while promoting piecewise-polynomial smoothing surfaces. To verify its effectiveness, we also present a series of experimental results on both synthetic and real scanning data, showcasing the competitive and superior results compared to state-of-the-art methods, both visually and quantitatively.

cs.CG↗

arXiv2025

Functional calculus for Safarov pseudo-differential operators

Given a smooth, closed Riemannian manifold $(M,g)$ equipped with a linear connection $\nabla$ (not necessarily metric), we develop the holomorphic functional calculus for operators belonging to the global pseudo-differential classes $Ψ_{ρ, δ}^m\left(Ω^κ, \nabla, τ\right)$ introduced by Safarov. As a consequence of our main result, we establish a Szegö type-theorem, derive asymptotic expansion of the heat kernel trace, and calculate some associated spectral $ζ$-functions.

math.AP↗

arXiv2025

A Closed-form Approximation for Impulse Response of Fractionally Damped Oscillators

We consider a fractionally damped oscillator, where the damping term is expressed by the Caputo fractional derivative of order $β\in (0,1).$ The impulse response of this oscillator can be expressed in terms of the bivariate Mittag-Leffler function consisting of a double infinite series. Although this series is uniformly convergent, its numerical implementation suffers from computational instabilities. In this contribution, we propose an approximate closed-form solution that avoids these numerical pitfalls while maintaining a reasonable accuracy. The resulting approximation is computationally efficient and robust, making it suitable for practical engineering applications.

math.CA↗

arXiv2025

Cylindrical extensions of critical Sobolev type inequalities and identities

In this paper, we investigate cylindrical extensions of critical Sobolev type (improved Hardy) inequalities and identities in the style of Badiale-Tarantello [BT02], which in a special case give a critical Hardy inequality and its stability results. We also obtain higher-order identities, which interestingly include well-known numbers like double factorial, Oblong numbers, and Stirling numbers of the second kind. All functional identities are obtained in $L^{p}$ for $p\in (1,\infty)$ without the real-valued function assumption, which gives a simple and direct understanding of the corresponding inequalities as well as the nonexistence of nontrivial extremizers. As applications, we obtain Caffarelli-Kohn-Nirenberg type inequalities with logarithmic weights, which in a particular case give the critical case of the Heisenberg-Pauli-Weyl type uncertainty principle. We also discuss these results in the setting of Folland and Stein's homogeneous Lie groups. A special focus is devoted to stratified Lie groups, where Sobolev type inequalities become intricately intertwined with the properties of sub-Laplacians and more general subelliptic partial differential equations. The obtained results are already new even in the classical Euclidean setting with respect to the range of parameters and the arbitrariness of the choice of any homogeneous quasi-norm. Most inequalities are obtained with sharp constants.

math.AP↗

arXiv2025

Schatten-von Neumann classes of tensors of invariant operators

In this work we study Schatten-von Neumann classes of tensor products of invariant operators on Hilbert spaces. In the first part we first deduce some spectral properties for tensors of anharmonic oscillators thanks to the knowledge on corresponding Schatten-von Neumann properties. In the second part we specialised on tensors of invariant operators. In the special case where a suitable Fourier analysis associated to a fixed partition of a Hilbert space into finite dimensional subspaces is available we also give the corresponding formulae in terms of symbols. We also give a sufficient condition for Dixmier traceability for a class of finite tensors of pseudo-differential operators on the flat torus.

math.FA↗

arXiv2025

Very weak solutions of the heat equation with anisotropically singular time-dependent diffusivity

We investigate the heat equation with a time-dependent, anisotropic, and potentially singular diffusivity tensor. Since weak (in the Sobolev sense) or distributional solutions may not exist in this setting, we employ the framework of very weak solutions to establish the existence and uniqueness of solutions to the heat equation with singular, anisotropic, time-dependent diffusivity.

math.AP↗

arXiv2025

Shifted Pitt and uncertainty inequalities on Riemannian symmetric spaces of noncompact type

Our primary objective is to study Pitt-type inequalities on Riemannian symmetric spaces $\mathbb{X}$ of noncompact type, as well as within the framework of Jacobi analysis. Inspired by the spectral gap of the Laplacian on $\mathbb{X}$, we introduce the notion of a \textit{shifted} Pitt's inequality as a natural and intrinsic analogue tailored to symmetric spaces, capturing key aspects of the underlying non-Euclidean geometry. In the rank one case (in particular, for hyperbolic spaces), we show that the sufficient condition for the \textit{shifted} Pitt's inequality matches the necessary condition in the range $p \leq q \leq p'$, yielding a sharp characterization of admissible polynomial weights with non-negative exponents. In the Jacobi setting, we modify the transform so that the associated measure exhibits polynomial volume growth. This modification enables us to fully characterize the class of polynomial weights with non-negative exponents for which Pitt-type inequalities hold for the modified Jacobi transforms. As applications of the \textit{shifted} Pitt's inequalities, we derive $L^2$-type Heisenberg-Pauli-Weyl uncertainty inequalities and further establish generalized $L^p$ versions. Moreover, the geometric structure of symmetric spaces allows us to formulate a broader version of the uncertainty inequalities previously obtained by Ciatti-Cowling-Ricci in the setting of stratified Lie groups.

math.FA↗

arXiv2025

Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces

In this work, we study the Sobolev inequality on noncommutative Euclidean spaces. As a simple consequence, we obtain the Gagliardo-Nirenberg type inequality and as its application we show global well-posedness of nonlinear PDEs in the noncommutative Euclidean space. Moreover, we show that the logarithmic Sobolev inequality is equivalent to the Nash inequality for possibly different constants in this noncommutative setting by completing the list in noncommutative Varopoulos's theorem in [37]. Finally, we present a direct application of the Nash inequality to compute the time decay for solutions of the heat equation in the noncommutative setting.

math.AP↗

arXiv2025

Oscillating singular integral operators on graded Lie groups revisited

In this work, we extend the Euclidean theory of oscillating singular integrals due to Fefferman and Stein in \cite{Fefferman1970,FeffermanStein1972} to arbitrary graded Lie groups. Our approach reveals the strong compatibility between the geometric measure theory of a graded Lie group and the Fourier analysis associated with Rockland operators. Our criteria are presented in terms of the oscillating Fefferman condition of the kernel of the operator and its group Fourier transform. One of the novelties of this work is that we use the infinitesimal representation of a Rockland operator to measure the decay of the Fourier transform of the kernel.

math.FA↗

arXiv2025

PointExplainer: Towards Transparent Parkinson's Disease Diagnosis

Deep neural networks have shown potential in analyzing digitized hand-drawn signals for early diagnosis of Parkinson's disease. However, the lack of clear interpretability in existing diagnostic methods presents a challenge to clinical trust. In this paper, we propose PointExplainer, an explainable diagnostic strategy to identify hand-drawn regions that drive model diagnosis. Specifically, PointExplainer assigns discrete attribution values to hand-drawn segments, explicitly quantifying their relative contributions to the model's decision. Its key components include: (i) a diagnosis module, which encodes hand-drawn signals into 3D point clouds to represent hand-drawn trajectories, and (ii) an explanation module, which trains an interpretable surrogate model to approximate the local behavior of the black-box diagnostic model. We also introduce consistency measures to further address the issue of faithfulness in explanations. Extensive experiments on two benchmark datasets and a newly constructed dataset show that PointExplainer can provide intuitive explanations with no diagnostic performance degradation. The source code is available at https://github.com/chaoxuewang/PointExplainer.

cs.CV↗

arXiv2025

Brezis-Nirenberg type problems associated with nonlinear superposition operators of mixed fractional order

This paper aims to study the Brezis-Nirenberg type problem driven by the nonlinear superposition of operators of the form $$A_{μ, p}u:=\int_{[0,1]}(-Δ)_{p}^{s} u\,\, d μ(s),$$ where $μ$ denotes the signed measure over $[0, 1]$. We consider nonlinear nonlocal equations associated with $A_{μ, p}$, involving critical nonlinearity and lower-order perturbation. Using variational techniques, we establish existence results for the critical problem by employing weak lower semicontinuity arguments under general assumptions on the perturbation term. We discuss the multiplicity results when the perturbation term vanishes at the origin. Additionally, when the lower-order term is a pure power function, we examine the Brezis-Nirenberg-type problem using the mountain pass technique. Furthermore, we address the existence of solutions to subcritical problems associated with $A_{μ, p}.$ Our findings are novel, even in the case of the sum of two distinct fractional $p$-Laplacians or a combination of a fractional $p$-Laplacian with a classical $p$-Laplacian. More generally, our framework is sufficiently broad to accommodate finite sums of different fractional $p$-Laplacians as well as cases involving fractional Laplacians with ``wrong" signs. A key contribution of this study is the development of a unified approach that systematically addresses these problems by incorporating a broad class of operators and lower-order perturbation terms within a common theoretical framework. The results remain new even in the case of linear superposition of fractional operators of different orders.

math.AP↗

arXiv2025

Hardy-Hilbert type inequalities on homogeneous groups-An introduction and generalization to the kernel case

There is a lot of information available concerning Hardy-Hilbert type inequalities in one or more dimensions. In this paper we introduce the development of such inequalities on homogeneous groups. Moreover, we point out a unification of several of the Hardy-Hilbert type inequalities in the classical case to a general kernel case. Finally, we generalize these results to the homogeneous group case.

math.FA↗

arXiv2025

Singular Klein-Gordon equation on a bounded domain

In this paper, we consider the wave equation for the Laplace operator with potential, initial data, and nonhomogeneous Dirichlet boundary condition. We establish a weak solution by using traces and extension domains. We also establish the existence, uniqueness and consistency of the very weak solution for the wave equation with singularities in the potential, initial data, source term, boundary and boundary condition.

math.AP↗

arXiv2025

Zero modes and Dirac-(logarithmic) Sobolev-type inequalities

We study the decay rate of the zero modes of the Dirac operator with a matrix-valued potential that is considered here without any regularity assumptions, compared to the existing literature. For the Dirac operator and for Clifford-valued functions we prove the $L^p$-$L^2$ Dirac Sobolev inequality with explicit constant, as well as the $L^p$-$L^q$ Dirac-Sobolev inequalities. We prove its logarithmic counterpart for $q=2$, extending it to its Gaussian version of Gross, as well as show Nash and Poincaré inequalities in this setting, with explicit values for constants.

math.AP↗