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Shirali Kadyrov

Publications and source records attributed to Shirali Kadyrov.

At least 19 recordsLinked to original sources
arXiv2026

Effective Intrinsic Ergodicity for renewal-type potentials on S-gap shifts

We establish effective intrinsic ergodicity for renewal-type potentials on one-sided \(S\)-gap shifts. Inducing on the one-symbol cylinder \([1]\) reduces the system to a full shift over the alphabet \(S\), where the induced potential becomes a one-symbol potential and the equilibrium measure is Bernoulli. The associated renewal equation has a unique solution \(P\), and under the condition \(P>ϕ(0^\infty)\) (automatic when \(S\) is infinite), we show that \(P\) is the topological pressure and that the potential admits a unique equilibrium state \(μ_ϕ\). Our main result is an effective intrinsic ergodicity estimate: invariant measures whose free energy is within \(Δ\) of the pressure are \(O(\sqrtΔ)\)-close to \(μ_ϕ\) when tested against Hölder observables. As an application, every finite-word cylinder of positive \(μ_ϕ\)-measure yields a uniform pressure gap for the set of orbits avoiding that cylinder, leading in the entropy case to strict entropy and Hausdorff-dimension gaps.

math.DS↗
arXiv2026

Limit-Cycle Replication via Chebyshev Pullbacks and a Quadratic Ceiling for Separable Schemes

Let \(H(n)\) denote the Hilbert number, i.e.\ the maximal number of limit cycles of planar polynomial vector fields of degree \(\le n\). A classical lower-bound mechanism for \(H(n)\) is \emph{replication}: one pulls back a vector field by a polynomial map and lifts each existing limit cycle to several disjoint copies while controlling the resulting degree. In this paper we give a fully self-contained replication theorem based on the separable Chebyshev covering \[ Φ(u,v)=(T_m(u),T_m(v)). \] Using the \(m\) monotone full branches of \(T_m\) on \((-1,1)\), we prove that every degree-\(\le n\) polynomial vector field with \(k\) limit cycles gives rise to a degree-\(\le nm+m-1\) polynomial vector field with at least \(m^2k\) limit cycles. Consequently, \[ H(nm+m-1)\ge m^2H(n)\qquad (m\ge 2). \] We then extend the construction to general separable pullbacks \((u,v)\mapsto (p(u),p(v))\), show that Chebyshev attains the maximal possible branch count among degree-\(m\) separable pullbacks, and prove a quadratic ceiling for replication-only schemes: if one iterates separable pullbacks and no additional limit cycles are created beyond those forced by lifting, then the number of resulting limit cycles is at most quadratic in the final degree. This shows that superquadratic lower bounds, such as the known \(n^2\log n\)-type bounds, necessarily require mechanisms beyond pure separable replication. Finally, combining our replication theorem with the strongest currently published seed bounds, we obtain new explicit lower estimates in several degrees, including \begin{gather*} H(14)\ge 252,\qquad H(29)\ge 1080,\\ H(31)\ge 1380,\qquad H(39)\ge 2012. \end{gather*}

math.DS↗
arXiv2025

Fractional Heat Kernel for Semi-Supervised Graph Learning with Small Training Sample Size

In this work, we introduce novel algorithms for label propagation and self-training using fractional heat kernel dynamics with a source term. We motivate the methodology through the classical correspondence of information theory with the physics of parabolic evolution equations. We integrate the fractional heat kernel into Graph Neural Network architectures such as Graph Convolutional Networks and Graph Attention, enhancing their expressiveness through adaptive, multi-hop diffusion. By applying Chebyshev polynomial approximations, large graphs become computationally feasible. Motivating variational formulations demonstrate that by extending the classical diffusion model to fractional powers of the Laplacian, nonlocal interactions deliver more globally diffusing labels. The particular balance between supervision of known labels and diffusion across the graph is particularly advantageous in the case where only a small number of labeled training examples are present. We demonstrate the effectiveness of this approach on standard datasets.

cs.LG↗
arXiv2025

A Simple and Reproducible Hybrid Solver for a Truck-Drone VRP with Recharge

We study last-mile delivery with one truck and one drone under explicit battery management: the drone flies at twice the truck speed; each sortie must satisfy an endurance budget; after every delivery the drone recharges on the truck before the next launch. We introduce a hybrid reinforcement learning (RL) solver that couples an ALNS-based truck tour (with 2/3-opt and Or-opt) with a small pointer/attention policy that schedules drone sorties. The policy decodes launch-serve-rendezvous triplets with hard feasibility masks for endurance and post-delivery recharge; a fast, exact timeline simulator enforces launch/recovery handling and computes the true makespan used by masked greedy/beam decoding. On Euclidean instances with $N{=}50$, $E{=}0.7$, and $R{=}0.1$, the method achieves an average makespan of \textbf{5.203}$\pm$0.093, versus \textbf{5.349}$\pm$0.038 for ALNS and \textbf{5.208}$\pm$0.124 for NN -- i.e., \textbf{2.73\%} better than ALNS on average and within \textbf{0.10\%} of NN. Per-seed, the RL scheduler never underperforms ALNS on the same instance and ties or beats NN on two of three seeds. A decomposition of the makespan shows the expected truck-wait trade-off across heuristics; the learned scheduler balances both to minimize the total completion time. We provide a config-first implementation with plotting and significance-test utilities to support replication.

cs.LG↗
arXiv2025

Development and optimization of physics-informed neural networks for solving partial differential equations

This work compares the advantages and limitations of the Finite Difference Method with Physics-Informed Neural Networks, showing where each can best be applied for different problem scenarios. Analysis on the L2 relative error based on one-dimensional and two-dimensional Poisson equations suggests that FDM gives far more accurate results with a relative error of 7.26 x 10-8 and 2.21 x 10-4, respectively, in comparison with PINNs, with an error of 5.63 x 10-6 and 6.01 x 10-3 accordingly. Besides forward problems, PINN is realized also for forward-inverse problems which reflect its ability to predict source term after its sufficient training. Visualization of the solution underlines different methodologies adopted by FDM and PINNs, yielding useful insights into their performance and applicability.

math.GM↗
arXiv2024

On the solutions of second order difference equations with variable coefficients

In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are elementary and based on factoring a quadratic shift operator. As an application, we obtain two new generalized continued fraction formulas for the mathematical constant $π^2$.

math.NT↗
arXiv2020

Periodic solutions and the avoidance of pull--in instability in non--autonomous micro--electro--mechanical systems

We study periodic solutions of a one-degree of freedom micro-electro-mechanical system (MEMS) with a parallel-plate capacitor under $T$--periodic electrostatic forcing. We obtain analytical results concerning the existence of $T-$ periodic solutions of the problem in the case of arbitrary nonlinear restoring force, as well as when the moving plate is attached to a spring fabricated using graphene. We then demonstrate numerically on a $T-$ periodic Poincar{é} map of the flow that these solutions are generally locally stable with large "islands" of initial conditions around them, within which the pull-in stability is completely avoided. We also demonstrate graphically on the Poincar{é} map that stable periodic solutions with higher period $nT, n>1$ also exist, for wide parameter ranges, with large "islands" of bounded motion around them, within which all initial conditions avoid the pull--in instability, thus helping us significantly increase the domain of safe operation of these MEMS models.

math.DS↗
arXiv2019

Generalized continued fraction expansions for $π$ and $e$

Recently Raayoni et al. announced various conjectures on continued fractions of fundamental constants automatically generated with machine learning techniques. In this paper we prove some of their stated conjectures for Euler number $e$ and show the equivalence of some of the listed conjectures. Moreover, we propose a simple method that can be used to generate other continued fractions using their series representations.

math.NT↗
arXiv2018

Polynomial estimates over exponential curves in $\mathbb C^2$

For any complex $α$ with non-zero imaginary part we show that Bernstein-Walsh type inequality holds on the piece of the curve $\{(e^z,e^{αz}) : z \in \mathbb C\}$. Our result extends a theorem of Coman-Poletsky \cite{CP10} where they considered real-valued $α$.

math.CV↗
arXiv2016

Effective equidistribution of periodic orbits for subshifts of finite type

We study equidistribution of certain subsets of periodic orbits for subshifts of finite type. Our results solely rely on the growth of these subsets. As a consequence, effective equidistribution results are obtained for both hyperbolic diffeomorphisms and expanding maps on compact manifolds.

math.DS↗
arXiv2014

Bernstein-Walsh inequalities in higher dimensions over exponential curves

Let ${\bf x}=(x_1,\dots,x_d) \in [-1,1]^d$ be linearly independent over $\mathbb Z$, set $K=\{(e^{z},e^{x_1 z},e^{x_2 z}\dots,e^{x_d z}): |z| \le 1\}.$ We prove sharp estimates for the growth of a polynomial of degree $n$, in terms of $$E_n({\bf x}):=\sup\{\|P\|_{Δ^{d+1}}:P \in \mathcal P_n(d+1), \|P\|_K \le 1\},$$ where $Δ^{d+1}$ is the unit polydisk. For all ${\bf x} \in [-1,1]^d$ with linearly independent entries, we have the lower estimate $$\log E_n({\bf x})\ge \frac{n^{d+1}}{(d-1)!(d+1)} \log n - O(n^{d+1});$$ for Diophantine $\bf x$, we have $$\log E_n({\bf x})\le \frac{ n^{d+1}}{(d-1)!(d+1)}\log n+O( n^{d+1}).$$ In particular, this estimate holds for almost all $\bf x$ with respect to Lebesgue measure.

math.CV↗
arXiv2014

Exceptional sets in homogeneous spaces and Hausdorff dimension

In this paper we study the dimension of a family of sets arising in open dynamics. We use exponential mixing results for diagonalizable flows in compact homogeneous spaces $X$ to show that the Hausdorff dimension of set of points that lie on trajectories missing a particular open ball of radius $r$ is at most $$\dim X + C\frac{r^{\dim X}}{\log r},$$ where $C>0$ is a constant independent of $r>0$. Meanwhile, we also describe a general method for computing the least cardinality of open covers of dynamical sets using volume estimates.

math.DS↗
arXiv2014

Escape of mass and entropy for diagonal flows in real rank one situations

Let $G$ be a connected semisimple Lie group of real rank 1 with finite center, let $Γ$ be a non-uniform lattice in $G$ and $a$ any diagonalizable element in $G$. We investigate the relation between the metric entropy of $a$ acting on the homogeneous space $Γ\backslash G$ and escape of mass. Moreover, we provide bounds on the escaping mass and, as an application, we show that the Hausdorff dimension of the set of orbits (under iteration of $a$) which miss a fixed open set is not full.

math.DS↗
arXiv2012

Amount of failure of upper-semicontinuity of entropy in noncompact rank one situations, and Hausdorff dimension

Recently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal flows on homogeneous spaces $Γ\backslash G$, where $G$ is any connected semisimple Lie group of real rank 1 with finite center and $Γ$ is any nonuniform lattice in $G$. We show that this bound is sharp and apply the methods used to establish bounds for the Hausdorff dimension of the set of points which diverge on average.

math.DS↗