scholarly journals On Pleijel’s Nodal Domain Theorem for Quantum Graphs

2021 ◽  
Author(s):  
Matthias Hofmann ◽  
James B. Kennedy ◽  
Delio Mugnolo ◽  
Marvin Plümer
Keyword(s):  
Finite Subset ◽  
Broad Class ◽  
Quantum Graphs ◽  
Nodal Domain ◽  
Natural Conditions ◽  
Metric Graphs ◽  
Nodal Domains ◽  
Accumulation Points ◽  
Special Cases ◽  
Set Of Points

AbstractWe establish metric graph counterparts of Pleijel’s theorem on the asymptotics of the number of nodal domains $$\nu _n$$ ν n of the nth eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with $$L^1$$ L 1 -potentials and a variety of vertex conditions as well as the p-Laplacian with natural vertex conditions, and without any assumptions on the lengths of the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. Among other things, these results characterise the accumulation points of the sequence $$(\frac{\nu _n}{n})_{n\in \mathbb {N}}$$ ( ν n n ) n ∈ N , which are shown always to form a finite subset of (0, 1]. This extends the previously known result that $$\nu _n\sim n$$ ν n ∼ n generically, for certain realisations of the Laplacian, in several directions. In particular, in the special cases of the Laplacian with natural conditions, we show that for graphs any graph with pairwise commensurable edge lengths and at least one cycle, one can find eigenfunctions thereon for which $${\nu _n}\not \sim {n}$$ ν n ≁ n ; but in this case even the set of points of accumulation may depend on the choice of eigenbasis.

scholarly journals On fully supported eigenfunctions of quantum graphs

2021 ◽  
Vol 111 (6) ◽  
Author(s):  
Marvin Plümer ◽  
Matthias Täufer
Keyword(s):  
Analogous Result ◽  
Metric Graphs ◽  
Nodal Domains ◽  
One Dimensional ◽  
Orthonormal Sequence ◽  
Multiplicity One ◽  
Higher Dimensional ◽  
Special Cases ◽  
Complicated Topology ◽  
The One

AbstractWe prove that every metric graph which is a tree has an orthonormal sequence of generic Laplace-eigenfunctions, that are eigenfunctions corresponding to eigenvalues of multiplicity one and which have full support. This implies that the number of nodal domains $$\nu _n$$ ν n of the n-th eigenfunction of the Laplacian with standard conditions satisfies $$\nu _n/n \rightarrow 1$$ ν n / n → 1 along a subsequence and has previously only been known in special cases such as mutually rationally dependent or rationally independent side lengths. It shows in particular that the Pleijel nodal domain asymptotics from two- or higher dimensional domains cannot occur on these graphs: Despite their more complicated topology, they still behave as in the one-dimensional case. We prove an analogous result on general metric graphs under the condition that they have at least one Dirichlet vertex. Furthermore, we generalize our results to Delta vertex conditions and to edgewise constant potentials. The main technical contribution is a new expression for a secular function in which modifications to the graph, to vertex conditions, and to the potential are particularly easy to understand.

scholarly journals Structure and Uncertainty in Discrete Choice Models

2003 ◽  
Vol 11 (4) ◽  
pp. 316-344 ◽  
Author(s):  
Curtis S. Signorino
Keyword(s):  
Discrete Choice ◽  
Private Information ◽  
Statistical Models ◽  
Broad Class ◽  
Choice Models ◽  
Discrete Choice Models ◽  
Social Scientists ◽  
Sources Of Uncertainty ◽  
Special Cases

Social scientists are often confronted with theories in which one or more actors make choices over a discrete set of options. In this article, I generalize a broad class of statistical discrete choice models, with both well-known and new nonstrategic and strategic special cases. I demonstrate how to derive statistical models from theoretical discrete choice models and, in doing so, I address the statistical implications of three sources of uncertainty: agent error, private information about payoffs, and regressor error. For strategic and some nonstrategic choice models, the three types of uncertainty produce different statistical models. In these cases, misspecifying the type of uncertainty leads to biased and inconsistent estimates, and to incorrect inferences based on estimated probabilities.

scholarly journals Is there contextuality in behavioural and social systems?

2016 ◽  
Vol 374 (2058) ◽  
pp. 20150099 ◽  
Author(s):  
E. N. Dzhafarov ◽  
Ru Zhang ◽  
Janne Kujala
Keyword(s):  
Quantum Physics ◽  
Broad Class ◽  
Social Systems ◽  
Sufficient Conditions ◽  
Type Systems ◽  
Binary Outcomes ◽  
Special Cases ◽  
Einstein Podolsky Rosen ◽  
Cyclic Systems ◽  
Necessary And Sufficient

Most behavioural and social experiments aimed at revealing contextuality are confined to cyclic systems with binary outcomes. In quantum physics, this broad class of systems includes as special cases Klyachko–Can–Binicioglu–Shumovsky-type, Einstein–Podolsky–Rosen–Bell-type and Suppes–Zanotti–Leggett–Garg-type systems. The theory of contextuality known as contextuality-by-default allows one to define and measure contextuality in all such systems, even if there are context-dependent errors in measurements, or if something in the contexts directly interacts with the measurements. This makes the theory especially suitable for behavioural and social systems, where direct interactions of ‘everything with everything’ are ubiquitous. For cyclic systems with binary outcomes, the theory provides necessary and sufficient conditions for non-contextuality, and these conditions are known to be breached in certain quantum systems. We review several behavioural and social datasets (from polls of public opinion to visual illusions to conjoint choices to word combinations to psychophysical matching), and none of these data provides any evidence for contextuality. Our working hypothesis is that this may be a broadly applicable rule: behavioural and social systems are non-contextual, i.e. all ‘contextual effects’ in them result from the ubiquitous dependence of response distributions on the elements of contexts other than the ones to which the response is presumably or normatively directed.

Variance Regularization in Sequential Bayesian Optimization

2020 ◽  
Vol 45 (3) ◽  
pp. 966-992
Author(s):  
Michael Jong Kim
Keyword(s):  
Error Bounds ◽  
Broad Class ◽  
Planning Horizon ◽  
Bayesian Optimization ◽  
Model Parameters ◽  
Regularization Term ◽  
Problem Class ◽  
Special Cases ◽  
The Impact ◽  
Bayesian Dynamic Programming

Sequential Bayesian optimization constitutes an important and broad class of problems where model parameters are not known a priori but need to be learned over time using Bayesian updating. It is known that the solution to these problems can in principle be obtained by solving the Bayesian dynamic programming (BDP) equation. Although the BDP equation can be solved in certain special cases (for example, when posteriors have low-dimensional representations), solving this equation in general is computationally intractable and remains an open problem. A second unresolved issue with the BDP equation lies in its (rather generic) interpretation. Beyond the standard narrative of balancing immediate versus future costs—an interpretation common to all dynamic programs with or without learning—the BDP equation does not provide much insight into the underlying mechanism by which sequential Bayesian optimization trades off between learning (exploration) and optimization (exploitation), the distinguishing feature of this problem class. The goal of this paper is to develop good approximations (with error bounds) to the BDP equation that help address the issues of computation and interpretation. To this end, we show how the BDP equation can be represented as a tractable single-stage optimization problem that trades off between a myopic term and a “variance regularization” term that measures the total solution variability over the remaining planning horizon. Intuitively, the myopic term can be regarded as a pure exploitation objective that ignores the impact of future learning, whereas the variance regularization term captures a pure exploration objective that only puts value on solutions that resolve statistical uncertainty. We develop quantitative error bounds for this representation and prove that the error tends to zero like o(n-1) almost surely in the number of stages n, which as a corollary, establishes strong consistency of the approximate solution.

scholarly journals Tracking the critical points of curves evolving under planar curvature flows

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Eszter Fehér ◽  
Gábor Domokos ◽  
Bernd Krauskopf
Keyword(s):  
Distance Function ◽  
Critical Points ◽  
Broad Class ◽  
Computational Cost ◽  
Radial Distance ◽  
Test Case ◽  
Case Examples ◽  
Planar Curve ◽  
Special Cases ◽  
Two Parameters

<p style='text-indent:20px;'>We are concerned with the evolution of planar, star-like curves and associated shapes under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flow. This family of flows has two parameters that control one constant and one curvature-dependent component for the velocity in the direction of the normal to the curve. The Andrews-Bloore flow includes as special cases the well known Eikonal, curve-shortening and affine shortening flows, and for positive parameter values its evolution shrinks the area enclosed by the curve to zero in finite time. A question of key interest has been how various shape descriptors of the evolving shape behave as this limit is approached. Star-like curves (which include convex curves) can be represented by a periodic scalar polar distance function <inline-formula><tex-math id="M1">\begin{document}$ r(\varphi) $\end{document}</tex-math></inline-formula> measured from a reference point, which may or may not be fixed. An important question is how the numbers and the trajectories of critical points of the distance function <inline-formula><tex-math id="M2">\begin{document}$ r(\varphi) $\end{document}</tex-math></inline-formula> and of the curvature <inline-formula><tex-math id="M3">\begin{document}$ \kappa(\varphi) $\end{document}</tex-math></inline-formula> (characterized by <inline-formula><tex-math id="M4">\begin{document}$ dr/d\varphi = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ d\kappa /d\varphi = 0 $\end{document}</tex-math></inline-formula>, respectively) evolve under the Andrews-Bloore flows for different choices of the parameters.</p><p style='text-indent:20px;'>We present a numerical method that is specifically designed to meet the challenge of computing accurate trajectories of the critical points of an evolving curve up to the vicinity of a limiting shape. Each curve is represented by a piecewise polynomial periodic radial distance function, as determined by a chosen mesh; different types of meshes and mesh adaptation can be chosen to ensure a good balance between accuracy and computational cost. As we demonstrate with test-case examples and two longer case studies, our method allows one to perform numerical investigations into subtle questions of planar curve evolution. More specifically — in the spirit of experimental mathematics — we provide illustrations of some known results, numerical evidence for two stated conjectures, as well as new insights and observations regarding the limits of shapes and their critical points.</p>

scholarly journals Torsion points and height jumping in higher-dimensional families of abelian varieties

2019 ◽  
Vol 15 (09) ◽  
pp. 1801-1826 ◽  
Author(s):  
David Holmes
Keyword(s):  
Infinite Order ◽  
Abelian Varieties ◽  
Uniform Boundedness ◽  
Torsion Points ◽  
Small Height ◽  
Higher Dimensional ◽  
Special Cases ◽  
Set Of Points

In 1983, Silverman and Tate showed that the set of points in a 1-dimensional family of abelian varieties where a section of infinite order has “small height” is finite. We conjecture a generalization to higher-dimensional families, where we replace “finite” by “not Zariski dense.” We show that this conjecture would imply the uniform boundedness conjecture for torsion points on abelian varieties. We then prove a few special cases of this new conjecture.

scholarly journals Instability of Cylindrical Shells Subjected to Axisymmetric Moving Loads

1966 ◽  
Vol 33 (2) ◽  
pp. 289-296 ◽  
Author(s):  
G. A. Hegemier
Keyword(s):  
Cylindrical Shell ◽  
Constant Velocity ◽  
Broad Class ◽  
Infinite Length ◽  
Functional Difference ◽  
Moving Loads ◽  
Loading Function ◽  
Double Laplace Transform ◽  
Special Cases ◽  
The Stability

Using a Donnell-type nonlinear theory and the stability in the small concept of Poincare´, the instability of an infinite-length cylindrical shell subjected to a broad class of axisymmetric loads moving with constant velocity is studied. Special cases of the general loading function include the moving-ring, step, and decayed-step loads. The analysis is carried out with a double Laplace transform, functional-difference technique. Numerical results are presented for the case of the moving-ring load.

scholarly journals Quantum graphs and dimensional crossover: the honeycomb

2019 ◽  
Vol 10 (1) ◽  
pp. 109-122 ◽  
Author(s):  
Riccardo Adami ◽  
Simone Dovetta ◽  
Alice Ruighi
Keyword(s):  
Ground States ◽  
Dimensional Crossover ◽  
Quantum Graphs ◽  
Graph Structure ◽  
Two Dimensional ◽  
Grid Graph ◽  
Metric Graphs ◽  
Square Grid ◽  
One Dimensional ◽  
Threshold Phenomena

Abstract We summarize features and results on the problem of the existence of Ground States for the Nonlinear Schrödinger Equation on doubly-periodic metric graphs. We extend the results known for the two–dimensional square grid graph to the honeycomb, made of infinitely-many identical hexagons. Specifically, we show how the coexistence between one–dimensional and two–dimensional scales in the graph structure leads to the emergence of threshold phenomena known as dimensional crossover.

scholarly journals Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Bruno Mera ◽  
Anwei Zhang ◽  
Nathan Goldman
Keyword(s):  
Quantum Physics ◽  
Berry Phase ◽  
Broad Class ◽  
Topological States Of Matter ◽  
Quantum Geometry ◽  
Special Cases ◽  
Chern Numbers ◽  
Spatial Dimensions ◽  
Topological States ◽  
Engineered Systems

Quantum geometry has emerged as a central and ubiquitous concept in quantum sciences, with direct consequences on quantum metrology and many-body quantum physics. In this context, two fundamental geometric quantities are known to play complementary roles:~the Fubini-Study metric, which introduces a notion of distance between quantum states defined over a parameter space, and the Berry curvature associated with Berry-phase effects and topological band structures. In fact, recent studies have revealed direct relations between these two important quantities, suggesting that topological properties can, in special cases, be deduced from the quantum metric. In this work, we establish general and exact relations between the quantum metric and the topological invariants of generic Dirac Hamiltonians. In particular, we demonstrate that topological indices (Chern numbers or winding numbers) are bounded by the quantum volume determined by the quantum metric. Our theoretical framework, which builds on the Clifford algebra of Dirac matrices, is applicable to topological insulators and semimetals of arbitrary spatial dimensions, with or without chiral symmetry. This work clarifies the role of the Fubini-Study metric in topological states of matter, suggesting unexplored topological responses and metrological applications in a broad class of quantum-engineered systems.

scholarly journals Influence of Natural Conditions on the Imaging of the Bottom of the Gdańsk Bay by Means of the Side Scan Sonar

2018 ◽  
Vol 25 (s1) ◽  
pp. 104-110
Author(s):  
Dominika Witos-Okrasińska ◽  
Grażyna Grelowska ◽  
Eugeniusz Kozaczka
Keyword(s):  
Systematic Errors ◽  
Random Errors ◽  
Sediment Characteristics ◽  
Natural Conditions ◽  
Correct Operation ◽  
Side Scan Sonar ◽  
Special Cases ◽  
Sea Grass ◽  
Maritime Infrastructure ◽  
Support Devices

Abstract The interest in underwater resources is the reason for the development of modern hydroacoustic systems, including side sonars, which find numerous applications such as: research of seabed morphology and sediment characteristics, preparation of sea sediment maps, and even in special cases of biocenoses such as sea grass meadows, detection of specific targets at the bottom such as shipwrecks, mines, identification of suitable sites for maritime infrastructure. Such applications require precise information about the position of the objects to be observed. Errors affecting the depiction of the bottom using hydroacoustic systems can be divided into errors associated with improper operation of measuring and support devices, systematic errors and random errors. Systematic errors result from the changing conditions prevailing in the analyzed environment affecting the measurement system. The errors affecting the correct operation of hydroacoustic systems can include: changing angle of inclination of the beam caused by the vessel’s movement on the wave or refraction connected to changes in the sound speed as the depth function.

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