Boundary properties of fractional objects: Flexibility of linear equations and rigidity of minimal graphs

AbstractThe main goal of this article is to understand the trace properties of nonlocal minimal graphs in {\mathbb{R}^{3}}, i.e. nonlocal minimal surfaces with a graphical structure.We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum.This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can be prescribed arbitrarily, up to a small error.We remark that, at a formal level, the linearization of the trace of a nonlocal minimal graph is given by the fractional normal derivative of a fractional Laplace problem, therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type present very specific properties which are strikingly different from those of other problems of fractional type which are apparently similar, but diverse in structure, and the nonlinear case given by the nonlocal minimal graphs turns out to be significantly more rigid than its linear counterpart.

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On Ramsey-Minimal Infinite Graphs

The Electronic Journal of Combinatorics ◽

10.37236/10046 ◽

2021 ◽

Vol 28 (1) ◽

Author(s):

Jordan Barrett ◽

Valentino Vito

Keyword(s):

Ramsey Theory ◽

Infinite Graphs ◽

Minimal Graph ◽

Finite Graphs ◽

Minimal Graphs ◽

Finite Case ◽

Ramsey Minimal Graph ◽

General Conditions

For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study graphs $F$ such that $F \to (G,H)$, i.e. every red-blue coloring of the edges of $F$ produces either a red $G$ or a blue $H$. We generalize this study to infinite graphs $G$, $H$; in particular, we want to determine if there is a minimal such $F$. This problem has strong connections to the study of self-embeddable graphs: infinite graphs which properly contain a copy of themselves. We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph. We use these to prove, for example, that if $G=S_\infty$ is an infinite star and $H=nK_2$, $n \geqslant 1$ is a matching, then the pair $(S_\infty,nK_2)$ admits no Ramsey-minimal graphs.

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MINIMAL GRAPHS WITH DISCONTINUOUS BOUNDARY VALUES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788708000335 ◽

2009 ◽

Vol 86 (1) ◽

pp. 75-95 ◽

Cited By ~ 3

Author(s):

ROBERT HUFF ◽

JOHN MCCUAN

Keyword(s):

Boundary Component ◽

Global Solutions ◽

Jump Discontinuity ◽

Boundary Values ◽

Minimal Graph ◽

Closed Curves ◽

Minimal Graphs ◽

Minimal Surface Equation ◽

Discontinuous Boundary ◽

Annular Domains

AbstractWe construct global solutions of the minimal surface equation over certain smooth annular domains and over the domain exterior to certain smooth simple closed curves. Each resulting minimal graph has an isolated jump discontinuity on the inner boundary component which, at least in some cases, is shown to have nonvanishing curvature.

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Estimation and Marginalization Using the Kikuchi Approximation Methods

Neural Computation ◽

10.1162/0899766054026693 ◽

2005 ◽

Vol 17 (8) ◽

pp. 1836-1873 ◽

Cited By ~ 23

Author(s):

Payam Pakzad ◽

Venkat Anantharam

Keyword(s):

Approximation Method ◽

Message Passing ◽

Belief Propagation ◽

Iterative Algorithms ◽

Product Distribution ◽

Stationary Points ◽

Minimal Graph ◽

Minimal Graphs ◽

Minimum Number ◽

Free Case

In this letter, we examine a general method of approximation, known as the Kikuchi approximation method, for finding the marginals of a product distribution, as well as the corresponding partition function. The Kikuchi approximation method defines a certain constrained optimization problem, called the Kikuchi problem, and treats its stationary points as approximations to the desired marginals. We show how to associate a graph to any Kikuchi problem and describe a class of local message-passing algorithms along the edges of any such graph, which attempt to find the solutions to the problem. Implementation of these algorithms on graphs with fewer edges requires fewer operations in each iteration. We therefore characterize minimal graphs for a Kikuchi problem, which are those with the minimum number of edges. We show with empirical results that these simpler algorithms often offer significant savings in computational complexity, without suffering a loss in the convergence rate. We give conditions for the convexity of a given Kikuchi problem and the exactness of the approximations in terms of the loops of the minimal graph. More precisely, we show that if the minimal graph is cycle free, then the Kikuchi approximation method is exact, and the converse is also true generically. Together with the fact that in the cycle-free case, the iterative algorithms are equivalent to the well-known belief propagation algorithm, our results imply that, generically, the Kikuchi approximation method can be exact if and only if traditional junction tree methods could also solve the problem exactly.

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Convex Combinations of Minimal Graphs

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2012/724268 ◽

2012 ◽

Vol 2012 ◽

pp. 1-9

Author(s):

Michael Dorff ◽

Ryan Viertel ◽

Magdalena Wołoszkiewicz

Keyword(s):

Convex Combination ◽

Sufficient Conditions ◽

Gauss Map ◽

Minimal Graph ◽

Periodic Surface ◽

Minimal Graphs

Given a collection of minimal graphs,M1,M2,…,Mn, with isothermal parametrizations in terms of the Gauss map and height differential, we give sufficient conditions onM1,M2,…,Mnso that a convex combination of them will be a minimal graph. We will then provide two examples, taking a convex combination of Scherk's doubly periodic surface with the catenoid and Enneper's surface, respectively.

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On minimal Ramsey graphs and Ramsey equivalence in multiple colours

Combinatorics Probability Computing ◽

10.1017/s0963548320000036 ◽

2020 ◽

Vol 29 (4) ◽

pp. 537-554

Author(s):

Dennis Clemens ◽

Anita Liebenau ◽

Damian Reding

Keyword(s):

Infinite Number ◽

Maximum Degree ◽

List Type ◽

Minimal Graph ◽

Induced Subgraph ◽

Minimal Graphs ◽

Large Maximum ◽

Ramsey Minimal Graph ◽

Ramsey Graphs ◽

Monochromatic Copy

AbstractFor an integer q ⩾ 2, a graph G is called q-Ramsey for a graph H if every q-colouring of the edges of G contains a monochromatic copy of H. If G is q-Ramsey for H yet no proper subgraph of G has this property, then G is called q-Ramsey-minimal for H. Generalizing a statement by Burr, Nešetřil and Rödl from 1977, we prove that, for q ⩾ 3, if G is a graph that is not q-Ramsey for some graph H, then G is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H as long as H is 3-connected or isomorphic to the triangle. For such H, the following are some consequences.For 2 ⩽ r < q, every r-Ramsey-minimal graph for H is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H.For every q ⩾ 3, there are q-Ramsey-minimal graphs for H of arbitrarily large maximum degree, genus and chromatic number.The collection $\{\mathcal M_q(H) \colon H \text{ is 3-connected or } K_3\}$ forms an antichain with respect to the subset relation, where $\mathcal M_q(H)$ denotes the set of all graphs that are q-Ramsey-minimal for H.We also address the question of which pairs of graphs satisfy $\mathcal M_q(H_1)=\mathcal M_q(H_2)$ , in which case H1 and H2 are called q-equivalent. We show that two graphs H1 and H2 are q-equivalent for even q if they are 2-equivalent, and that in general q-equivalence for some q ⩾ 3 does not necessarily imply 2-equivalence. Finally we indicate that for connected graphs this implication may hold: results by Nešetřil and Rödl and by Fox, Grinshpun, Liebenau, Person and Szabó imply that the complete graph is not 2-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.

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Metal Microstructures in Advanced CMOS Devices

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100164258 ◽

1996 ◽

Vol 54 ◽

pp. 358-359

Author(s):

L. M. Gignac ◽

K. P. Rodbell

Keyword(s):

Grain Boundaries ◽

Semiconductor Device ◽

Chemical Vapor ◽

Microstructural Characterization ◽

Metal Films ◽

Thin Metal ◽

Electrical Performance ◽

Thin Metal Films ◽

Chemical Vapor Deposited ◽

Boundary Properties

As advanced semiconductor device features shrink, grain boundaries and interfaces become increasingly more important to the properties of thin metal films. With film thicknesses decreasing to the range of 10 nm and the corresponding features also decreasing to sub-micrometer sizes, interface and grain boundary properties become dominant. In this regime the details of the surfaces and grain boundaries dictate the interactions between film layers and the subsequent electrical properties. Therefore it is necessary to accurately characterize these materials on the proper length scale in order to first understand and then to improve the device effectiveness. In this talk we will examine the importance of microstructural characterization of thin metal films used in semiconductor devices and show how microstructure can influence the electrical performance. Specifically, we will review Co and Ti silicides for silicon contact and gate conductor applications, Ti/TiN liner films used for adhesion and diffusion barriers in chemical vapor deposited (CVD) tungsten vertical wiring (vias) and Ti/AlCu/Ti-TiN films used as planar interconnect metal lines.

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Overestimations in Bounding Solutions of Perturbed Linear Equations

Linear Algebra and its Applications ◽

10.1016/s0024-3795(96)00467-3 ◽

1997 ◽

Vol 262 (1-3) ◽

pp. 55-65 ◽

Cited By ~ 2

Author(s):

J Rohn

Keyword(s):

Linear Equations

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The influence of extrinsic motivation and work experience on employee performance (Case study in the Serang Baru Sub-district, Bekasi District)

The Management Journal of Binaniaga ◽

10.33062/mjb.v5i2.387 ◽

2020 ◽

Vol 5 (02) ◽

pp. 167

Author(s):

Nur’enny Nur’enny ◽

Rahmat Hidayat

Keyword(s):

Work Experience ◽

Extrinsic Motivation ◽

Employee Performance ◽

Linear Equations ◽

Test Reliability ◽

Coefficient Of Determination ◽

District Office ◽

Normality Test ◽

Observation Methods ◽

Saturated Sample

This study aims to obtain information about extrinsic motivation and work experience and its effect on employee performance in the Serang Baru District Office. This study uses a saturated sample so that the population is the same as the sample of 80 employees, at the Serang Baru District Office. The method used is validation test, reliability test, then classical assumption test, which includes normality test and multicollinearity, as well as heteroscedasticity test, multiple linear analysis test, multiple linear equations, F test, coefficient of determination, and t test. The data of this research used observation methods and questionnaires distributed to 80 samples which were addressed to employees of the Serang Baru District Office. Based on the results of research and discussion, it can be concluded: 1) Extrinsic motivation does not affect employee performance because employees are willing to work more than expected regardless of extrinsic motivation or not. 2) Employee performance is strongly influenced by work experience. The more experience, they get while working, the more knowledge they will get. 3) Employee performance will be better with the support of experienced employees so as to increase the level of output produced. Keywords: Employee Performance, Extrinsic Motivation, Work Experience

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