scholarly journals Approximations of Metric Graphs by Thick Graphs and Their Laplacians

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 369
Author(s):  
Olaf Post
Keyword(s):  
Realistic Model ◽  
Linear Operators ◽  
Metric Graphs ◽  
Resolvent Convergence ◽  
Metric Graph ◽  
Convergence Results ◽  
Non Linear ◽  
Norm Resolvent Convergence

The main purpose of this article is two-fold: first, to justify the choice of Kirchhoff vertex conditions on a metric graph as they appear naturally as a limit of Neumann Laplacians on a family of open sets shrinking to the metric graph (“thick graphs”) in a self-contained presentation; second, to show that the metric graph example is close to a physically more realistic model where the edges have a thin, but positive thickness. The tool used is a generalization of norm resolvent convergence to the case when the underlying spaces vary. Finally, we give some hints about how to extend these convergence results to some mild non-linear operators.

scholarly journals Regular approximation of singular Sturm–Liouville problems with eigenparameter dependent boundary conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Maozhu Zhang ◽  
Kun Li ◽  
Hongxiang Song
Keyword(s):  
Boundary Conditions ◽  
Operator Theory ◽  
Resolvent Convergence ◽  
Spectral Inclusion ◽  
Regular Approximation ◽  
Special Cases ◽  
Abstract Operator ◽  
Norm Resolvent Convergence ◽  
Sturm Liouville ◽  
Spectral Exactness

AbstractIn this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.

The Beam Analysis Tool (BAT)

2011 ◽  
pp. 146-152
Author(s):  
Peter Burrage ◽  
Leslee Francis Pelton
Keyword(s):  
Chaos Theory ◽  
Teaching And Learning ◽  
Structural Engineering ◽  
Active Role ◽  
Realistic Model ◽  
Analysis Tool ◽  
Non Linear Programming ◽  
Non Linear ◽  
Linear Behaviour ◽  
Computers In Education

In Houghton’s (1989) review of educational paradigms, he highlights the gaining importance of chaos theory. Chaos theory is often characterized by the term non-linear. Chaos theory can be found in many disciplines; in structural engineering, the behaviour of a structure under earthquake loads is often seen in terms of non-linear behaviour. Another characteristic of chaos theory is unpredictability. The implications for educational theory, as Houghton suggests, is that we have a realistic model for what happens in highly interactive systems. If the process of teaching and learning is seen as a highly interactive environment, then the parallels to chaos theory can be easily seen. The nature of a lecture can change when a student asks a question. This results in a non-linear learning environment. Students affect how something is taught by their own unique ways of understanding. Houghton (1989) suggests that the use of computers in education is supported by chaos theory. He suggests that computers should play a significant and active role with learning. Chaos theory not only supports the concept of using computers in education, it suggests that with non-linear programming (e.g., hypertext), education can change from the traditional linear format to a non-linear methodology that is alive and vibrant.

scholarly journals EMBEDDING OF METRIC GRAPHS ON HYPERBOLIC SURFACES

2019 ◽  
Vol 99 (03) ◽  
pp. 508-520
Author(s):  
BIDYUT SANKI
Keyword(s):  
Euler Characteristic ◽  
Isometric Embedding ◽  
Hyperbolic Surface ◽  
Hyperbolic Surfaces ◽  
Metric Graphs ◽  
Metric Graph ◽  
Complementary Region

An embedding of a metric graph $(G,d)$ on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_{e}(G)$ of $(G,d)$ is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute $g_{e}(G)$ and show that, for every integer $g\geq g_{e}(G)$ , there is an embedding of $(G,d)$ (possibly after a rescaling of $d$ ) on a surface of genus $g$ . Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$ . The maximum essential genus $g_{e}^{\max }(G)$ of $(G,d)$ is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of $(G,d)$ , where $g_{e}(G)$ and $g_{e}^{\max }(G)$ are realised.

Complementary bounds for inner products associated with non-linear equations

1984 ◽  
Vol 96 (1-2) ◽  
pp. 135-142 ◽  
Author(s):  
R. J. Cole ◽  
J. Mika ◽  
D. C. Pack
Keyword(s):  
Real Hilbert Space ◽  
Linear Equations ◽  
Linear Operators ◽  
Inner Product ◽  
Upper And Lower Bounds ◽  
Non Linear Equation ◽  
Non Linear ◽  
Non Linear Operator ◽  
Non Linear Equations ◽  
Unknown Solution

SynopsisFunctionals are found that give upper and lower bounds to the inner product 〈g0, f〉 involving the unknown solution f of a non-linear equation T[f] = f0, with f∈H, a real Hilbert space, g0 a given function in H and f0 a given function in the range of the non-linear operator T. The method depends upon a re-ordering of terms in the expansion of T[f] about a trial function so as to transfer the non-linearity to a secondary problem that requires its own particular treatment and to enable earlier results obtained for linear operators to be used for the main part. First, bivariational bounds due to Barnsley and Robinson are re-derived. The new and more accurate bounds are given under relaxed assumptions on the operator T by introducing a third approximating function. The results are obtained from identities, thus avoiding some of the conditions imposed by the use of variational methods. The accuracy of the new method is illustrated by applying it to the problem of the heat contained in a bar.

scholarly journals A metric graph satisfying w41=1$w_4^1 = 1$ that cannot be lifted to a curve satisfying dim⁡ (W41)=1$\dim \;(W_4^1 ) = 1$

2016 ◽  
Vol 14 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Marc Coppens
Keyword(s):  
Linear Systems ◽  
Harmonic Morphism ◽  
Metric Graphs ◽  
Clifford Index ◽  
Special Linear ◽  
Metric Graph ◽  
Index 2

AbstractFor all integers g ≥ 6 we prove the existence of a metric graph G with $w_4^1 = 1$ such that G has Clifford index 2 and there is no tropical modification G′ of G such that there exists a finite harmonic morphism of degree 2 from G′ to a metric graph of genus 1. Those examples show that not all dimension theorems on the space classifying special linear systems for curves have immediate translation to the theory of divisors on metric graphs.

scholarly journals Lipschitz Regularity of Solutions to Two-phase Free Boundary Problems

2017 ◽  
Vol 2019 (7) ◽  
pp. 2204-2222 ◽  
Author(s):  
D De Silva ◽  
O Savin
Keyword(s):  
Free Boundary ◽  
Viscosity Solutions ◽  
Lipschitz Continuity ◽  
Free Boundary Problems ◽  
Linear Operators ◽  
Regularity Of Solutions ◽  
Two Phase ◽  
Boundary Problems ◽  
Lipschitz Regularity ◽  
Non Linear

AbstractWe prove Lipschitz continuity of viscosity solutions to a class of two-phase free boundary problems governed by fully non-linear operators.

scholarly journals On the constructive approximation of non-linear operators in the modelling of dynamical systems

1997 ◽  
Vol 39 (1) ◽  
pp. 1-27 ◽  
Author(s):  
A. P. Torokhti ◽  
P. G. Howlett
Keyword(s):  
Dynamical Systems ◽  
Real Number ◽  
Input Signal ◽  
Linear Operators ◽  
Non Linear ◽  
Constructive Approximation ◽  
Modelling Process ◽  
Theoretical Procedure ◽  
Small Disturbances

AbstractIn this paper we propose a systematic theoretical procedure for the constructive approximation of non-linear operators and show how this procedure can be applied to the modelling of dynamical systems. We extend previous work to show that the model is stable to small disturbances in the input signal and we pay special attention to the role of real number parameters in the modelling process. The implications of computability are also discussed. A number of specific examples are presented for the particular purpose of illustrating the theoretical procedure.

Identification of a class of non-linear operators

1978 ◽  
Vol 18 (2) ◽  
pp. 7-15
Author(s):  
M.L. Agranovskii ◽  
R.D. Baglai ◽  
K.K. Smirnov
Keyword(s):  
Linear Operators ◽  
Non Linear
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