arXiv subjects

Yavdat Sh. Il'yasov

Publications and source records attributed to Yavdat Sh. Il'yasov.

3 recordsLinked to original sources

arXiv2026

Cone Rayleigh Levels: Finite Perturbations and Certified Control

We study the reuse of positive trial profiles under finite perturbations of nonsymmetric matrix pencils B-λG. The lower and upper cone Rayleigh levels need not coincide and are defined without requiring positive eigenvectors. In the positive orthant with positive diagonal $G$, we derive computable perturbation bounds and determine the exact worst-case trial gap over prescribed independent entrywise perturbation classes. This yields the largest uniform radius meeting a given trial-gap tolerance for fixed profiles, certifying trial-value accuracy without recomputation. Experiments on a nonnegative operator and five signed matrices compare sufficient and optimal radii, directional thresholds, and cold- and warm-start recomputation. Several signed cases exhibit severely limited uniform profile reuse despite the optimality of the radius. Exact rational checks verify the reported bounds and worst-case constructions for stored numerical inputs. A finite-budget linear program and differentiable constraints illustrate applications to verified control and learning.

cs.LG↗

arXiv2026

On the asymptotics of ground states for a boundary value problem for the equation $-\varepsilon Δ_p u = a|u|^{q-2}u - b|u|^{γ-2}u$

We study a singularly perturbed Dirichlet problem for the $p$-Laplacian with competing superlinear terms, \[ -\varepsilon Δ_p u = a(x)|u|^{q-2}u - b(x)|u|^{γ-2}u, \qquad u|_{\partialΩ}=0, \] where $1 0$, and $\varepsilon>0$ is small. By means of the nonlinear Rayleigh quotient method, we introduce two critical parameter values, $\varepsilon^*$ and $\varepsilon_e^*$, related respectively to the Nehari manifold and to the zero energy level. We prove the nonexistence of nontrivial weak solutions for $\varepsilon>\varepsilon^*$, and the existence of at least two positive weak solutions for $0<\varepsilon<\varepsilon_e^*$; one of them is a ground state. The main result describes the asymptotic behaviour of ground states as $\varepsilon\to 0^+$. If, in addition, $a\geqσ_a>0$, then every family of positive ground states $u_\varepsilon$ converges in measure in $Ω$ to the explicit profile \[ \bar u_0(x) = \left(\frac{a(x)}{b(x)}\right)^{1/(γ-q)}. \] Moreover, \[ u_\varepsilon\to\bar u_0 \quad\text{strongly in }L^r(Ω), \qquad 1\le r<γ, \] and \[ u_\varepsilon\rightharpoonup\bar u_0 \quad\text{weakly in }L^r(Ω), \qquad 1<r\leγ. \]

math.AP↗

arXiv2019

Separating of critical points on the Nehari manifold via the nonlinear generalized Rayleigh quotients

In this paper, we deal with equations of variational form which Nahari manifolds can contain more than two different types of critical points. We introduce a method of separating critical points on the Nahari manifold, based on the use of nonlinear generalized Rayleigh quotients. The method is illustrated by establishing the existence of positive solutions, ground states and multiplicity results for a two-parameter nonlinear elliptic boundary problem with polynomial nonlinearities.

math.AP↗