arXiv subjects

Fedor Bakharev

Publications and source records attributed to Fedor Bakharev.

7 recordsLinked to original sources

arXiv2026

Monotonicity of discrete spectra of Dirichlet Laplacian in 3-dimensional layers

We investigate monotonicity properties of eigenvalues of the Dirichlet Laplacian in polyhedral layers of fixed width. We establish that eigenvalues below the essential spectrum threshold monotonically depend on geometric parameters defining the polyhedral layer, generalizing previous results known for planar V-shaped waveguides and conical layers. Moreover, we demonstrate non-monotone spectral behavior arising from asymmetric geometric perturbations, providing an explicit example where unfolding the polyhedral layer unexpectedly leads to the emergence of discrete eigenvalues. The limiting behavior of eigenvalues as the geometric parameters approach critical configurations is also rigorously analyzed.

math.SP↗

arXiv2025

Fractional Laplacian in bended strip

The spectral properties of the restricted fractional Laplacian with Dirichlet boundary conditions in a smoothly bent waveguide is investigated. The existence of eigenvalues below the threshold of the continuous spectrum is proved, generalizing classical results known for the local Laplace operator. Our approach utilizes the Caffarelli--Silvestre extension, addressing the specific geometric difficulties arising from the operator non-locality. The sufficient conditions on the curvature magnitude and distribution to ensure the existence of these trapped modes is established.

math.SP↗

arXiv2024

Velocity of viscous fingers in miscible displacement: Intermediate concentration

We investigate one-phase flow in porous medium corresponding to a miscible displacement process in which the viscosity of the injected fluid is smaller than the viscosity in the reservoir fluid, which frequently leads to the formation of a mixing zone characterized by thin fingers. The mixing zone grows in time due to the difference in speed between its leading and trailing edges. The transverse flow equilibrium (TFE) model provides estimates of these speeds. We propose an enhancement for the TFE estimates, and provide its theoretical justification. It is based on the assumption that an intermediate concentration exists near the tip of the finger, which allows to reduce the integration interval in the speed estimate. Numerical simulations were conducted that corroborate the new estimates within the computational fluid dynamics model. The refined estimates offer greater accuracy than those provided by the original TFE model.

physics.flu-dyn↗

arXiv2024

Fractional Laplacian in V-shaped waveguide

The spectral properties of the restricted fractional Dirichlet Laplacian in ${\sf V}$-shaped waveguides are studied. The continuous spectrum for such domains with cylindrical outlets is known to occupy the ray $[Λ_\dagger, +\infty)$ with the threshold corresponding to the smallest eigenvalue of the cross-sectional problems. In this work the presence of a discrete spectrum at any junction angle is established along with the monotonic dependence of the discrete spectrum on the angle.

math.SP↗

arXiv2023

Optimizing substructure search: a novel approach for efficient querying in large chemical databases

Substructure search in chemical compound databases is a fundamental task in cheminformatics with critical implications for fields such as drug discovery, materials science, and toxicology. However, the increasing size and complexity of chemical databases have rendered traditional search algorithms ineffective, exacerbating the need for scalable solutions. We introduce a novel approach to enhance the efficiency of substructure search, moving beyond the traditional full-enumeration methods. Our strategy employs a unique index structure: a tree that segments the molecular data set into clusters based on the presence or absence of certain features. This innovative indexing mechanism is inspired by the binary Ball-Tree concept and demonstrates superior performance over exhaustive search methods, leading to significant acceleration in the initial filtering process. Comparative analysis with the existing Bingo algorithm reveals the efficiency and versatility of our approach. Although the current implementation does not affect the verification stage, it has the potential to reduce false positive rates. Our method offers a promising avenue for future research, meeting the growing demand for fast and accurate substructure search in increasingly large chemical databases.

cs.DB↗

arXiv2023

Spectra of Dirichlet Laplacian in 3-dimensional polyhedral layers

The structure of the spectrum of the three-dimensional Dirichlet Laplacian in the 3D polyhedral layer of fixed width is studied. It appears that the essential spectrum is defined by the smallest dihedral angle that forms the boundary of the layer while the discrete spectrum is always finite. An example of a layer with the empty discrete spectrum is constructed. The spectrum is proved to be nonempty in regular polyhedral layer.

math.SP↗

arXiv2013

Spectral gaps for the linear surface wave model in periodic channels

We consider the linear water-wave problem in a periodic channel which consists of infinitely many identical containers connected with apertures of width $ε$. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the continuous spectrum. We show that the for small apertures there exists a large number of gaps and also find asymptotic formulas for the position of the gaps as $ε\to 0$: the endpoints are determined within corrections of order $ε^{3/2}$. The width of the first bands is shown to be $O(ε)$. Finally, we give a sufficient condition which guarantees that the spectral bands do not degenerate into eigenvalues of infinite multiplicity.

math.SP↗