arXiv subjects

Eli Appleboim

Publications and source records attributed to Eli Appleboim.

6 recordsLinked to original sources

arXiv2024

Quasi Normality and PL Approximation of Least Area Surfaces in 3-Manifolds

This} paper presents relations between least area and normal surfaces, embedded in either a Euclidean or hyperbolic $3$-manifold. A relaxed version of normal surfaces, termed quasi-normal, is introduced, and it is shown that under appropriate conditions, every embedded least area surface is quasi-normal with respect to a fine enough fat triangulation of the $3$-manifold. In addition, it is shown that the intersections of a least area surface with the tetrahedra of such fine enough triangulation, even when not as simple as in the case of normal surfaces, are also well behaved. Finally, it is shown that a least area surface, when considered as a quasi normal surface, gives rise to a sequence of piecewise flat surfaces termed as flat-associated surfaces, and this sequence converges to the given least area surface and approximates its area.

math.GT↗

arXiv2024

Functional Dimensionality of Koopman Eigenfunction Space

This work presents the general form solution of Koopman Partial Differential Equation and shows that its functional dimensionality is finite. The dimensionality is as the dimensionality of the dynamics. Thus, the representation of nonlinear dynamics as a linear one with a finite set of Koopman eigenfunctions without error is possible. This formulation justifies the flowbox statement and provides a simple numerical method to find such representation.

math.AP↗

arXiv2023

A Minimal Set of Koopman Eigenfunctions -- Analysis and Numerics

Research on Koopman operator theory has focused on three key areas for several decades: the mathematical structure of the Koopman eigenfunction space, the basis of this space, and the ability to represent nonlinear dynamics as linear. This study provides a thorough and comprehensive framework for these topics, including theoretical, analytical, and numerical approaches. A novel mathematical structure is introduced, which outlines permissible actions on the infinite set of Koopman Eigenfunction, under which this set is closed. Notions of generating and independent sets of Koopman eigenfunctions are defined. In addition, notions of a minimal generating set, and a maximal independent set are defined and are shown to be equivalent. This structure defines conditions for independence within the set of Koopman eigenfunctions. This independent set can be interpreted as a new coordinate system in which the dynamical system is linear. The theory also highlights the equivalence of a minimal set, flowbox representation, and conservation laws. Finally, the presented theory is supported by numerical experiments.

math.DS↗

arXiv2010

Geometric approach to sampling and communication

Relationships that exist between the classical, Shannon-type, and geometric-based approaches to sampling are investigated. Some aspects of coding and communication through a Gaussian channel are considered. In particular, a constructive method to determine the quantizing dimension in Zador's theorem is provided. A geometric version of Shannon's Second Theorem is introduced. Applications to Pulse Code Modulation and Vector Quantization of Images are addressed.

cs.IT↗

arXiv2003

Digital Version of Green`s Theorem and its Application to The Coverage Problem in Formal Verification

We present a novel scheme to the coverage problem, introducing a quantitative way to estimate the interaction between a block and its enviroment.This is achieved by setting a discrete version of Green`s theorem, specially adapted for Model Checking based verification of integrated circuits.This method is best suited for the coverage problem since it enables one to quantify the incompleteness or, on the other hand, the redundancy of a set of rules, describing the model under verification.Moreover this can be done continuously throughout the verification process, thus enabling the user to pinpoint the stages at which incompleteness/redundancy occurs. Although the method is presented locally on a small hardware example, we additionally show its possibility to provide precise coverage estimation also for large scale systems. We compare this method to others by checking it on the same test-cases.

cs.SC↗

arXiv1999

Finite Type Invariants of Links with Fixed Linking Matrix

In this paper we introduce two theories of finite type invariants for framed links with fixed linking matrix. We show that these thepries are related to the theory of Vassiliev invariants of framed links. We also study the corresponding spaces of ``chord diagrams''.

math.GT↗