arXiv subjects
Publications and source records attributed to Bogdan Georgiev.
Recent progress in recommender system research has shown the importance of including temporal representations to improve interpretability and performance. Here, we incorporate temporal representations in continuous time via recurrent point process for a dynamical model of reviews. Our goal is to characterize how changes in perception, user interest and seasonal effects affect review text.
Service system dynamics occur at the interplay between customer behaviour and a service provider's response. This kind of dynamics can effectively be modeled within the framework of queuing theory where customers' arrivals are described by point process models. However, these approaches are limited by parametric assumptions as to, for example, inter-event time distributions. In this paper, we address these limitations and propose a novel, deep neural network solution to the queuing problem. Our solution combines a recurrent neural network that models the arrival process with a recurrent generative adversarial network which models the service time distribution. We evaluate our methodology on various empirical datasets ranging from internet services (Blockchain, GitHub, Stackoverflow) to mobility service systems (New York taxi cab).
We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any nodal domain $Ω_λ$ almost fully contains a ball of radius $\sim \frac{1}{\sqrtλ}$. This also gives a slight refinement of a result by Mangoubi, concerning the inradius of nodal domains (\cite{Man2}). On the other hand, we also prove that no nodal domain can be contained in a reasonably thin tubular neighbourhood of unions of finitely many surfaces inside $M$.
We consider a Laplace eigenfunction $φ_λ$ on a smooth closed Riemannian manifold, that is, satisfying $-Δφ_λ= λφ_λ$. We introduce several observations about the geometry of its vanishing (nodal) set and corresponding nodal domains. First, we give asymptotic upper and lower bounds on the volume of a tubular neighbourhood around the nodal set of $φ_λ$. This extends previous work of Jakobson and Mangoubi in case $ (M, g) $ is real-analytic. A significant ingredient in our discussion are some recent techniques due to Logunov (cf. \cite{L1}). Second, we exhibit some remarks related to the asymptotic geometry of nodal domains. In particular, we observe an analogue of a result of Cheng in higher dimensions regarding the interior opening angle of a nodal domain at a singular point. Further, for nodal domains $Ω_λ$ on which $φ_λ$ satisfies exponentially small $L^\infty$ bounds, we give some quantitative estimates for radii of inscribed balls.
Let $Ω\subset \mathbb{R}^n$ be a bounded domain satisfying a Hayman-type asymmetry condition, and let $ D $ be an arbitrary bounded domain referred to as "obstacle". We are interested in the behaviour of the first Dirichlet eigenvalue $ λ_1(Ω\setminus (x+D)) $. First, we prove an upper bound on $ λ_1(Ω\setminus (x+D)) $ in terms of the distance of the set $ x+D $ to the set of maximum points $ x_0 $ of the first Dirichlet ground state $ ϕ_{λ_1} > 0 $ of $ Ω$. In short, a direct corollary is that if \begin{equation} μ_Ω:= \max_{x}λ_1(Ω\setminus (x+D)) \end{equation} is large enough in terms of $ λ_1(Ω) $, then all maximizer sets $ x+D $ of $ μ_Ω$ are close to each maximum point $ x_0 $ of $ ϕ_{λ_1} $. Second, we discuss the distribution of $ ϕ_{λ_1(Ω)} $ and the possibility to inscribe wavelength balls at a given point in $ Ω$. Finally, we specify our observations to convex obstacles $ D $ and show that if $ μ_Ω$ is sufficiently large with respect to $ λ_1(Ω) $, then all maximizers $ x+D $ of $ μ_Ω$ contain all maximum points $ x_0 $ of $ ϕ_{λ_1(Ω)} $.
We study solutions of uniformly elliptic PDE with Lipschitz leading coefficients and bounded lower order coefficients. We extend previous results of A. Logunov concerning nodal sets of harmonic functions and, in particular, prove polynomial upper bounds on interior nodal sets of Steklov eigenfunctions in terms of the corresponding eigenvalue $ λ$.
We consider the Schrödinger operator $$-\frac{d^2}{d x^2} + V \qquad \mbox{on an interval}~~[a,b]~\mbox{with Dirichlet boundary conditions},$$ where $V$ is bounded from below and prove a lower bound on the first eigenvalue $λ_1$ in terms of sublevel estimates: if $ w_V(y) = |I_y|,\text{ where } I_y := \left\{ x \in [a,b]: V(x) \leq y \right\},$ then $$ λ_1 \geq \frac{1}{250} \min_{y > \min V}{\left(\frac{1}{w_V(y)^2} + y\right)}.$$ The result is sharp up to a universal constant if $\left\{ x \in [a,b]: V(x) \leq y \right\}$ is an interval for the value of $y$ solving the minimization problem. An immediate application is as follows: let $Ω\subset \mathbb{R}^2$ be a convex domain with inradius $ρ$ and diameter $D$ and let $u:Ω\rightarrow \mathbb{R}$ be the first eigenfunction of the Laplacian $-Δ$ on $Ω$ with Dirichlet boundary conditions on $\partial Ω$. We prove $$ \| u \|_{L^{\infty}} \lesssim \frac{1}{ρ^{}} \left( \fracρ{D} \right)^{1/6} \|u\|_{L^2},$$ which answers a question of van den Berg in the special case of two dimensions.
We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions $ ϕ_λ$ on a closed Riemannian manifold $ (M,g) $. First, in the real-analytic case we present an improvement of the currently best known bounds, due to Mangoubi (\cite{Man1}). Furthermore, using recent results of Hezari (\cite{Hezari}, \cite{Hezari2}) we obtain $ \log $-type improvements in the case of negative curvature and improved bounds for $ (M,g) $ possessing an ergodic geodesic flow. Second, we discuss the relation between the distribution of the $ L^2 $ norm of an eigenfunction $ ϕ_λ$ and the inner radius of the corresponding nodal domains. In the spirit of the works of Colding-Minicozzi and Jakobson-Mangoubi, we consider a covering of good cubes and show that, if a nodal domain is sufficiently well covered by good cubes, then its inner radius is large.