scholarly journals Anchored isoperimetric profile of the infinite cluster in supercritical bond percolation is Lipschitz continuous

2020 ◽  
Vol 25 (0) ◽  
Author(s):  
Barbara Dembin
Keyword(s):  
Bond Percolation ◽  
Infinite Cluster ◽  
Lipschitz Continuous ◽  
Isoperimetric Profile

On the infinite cluster of Bernoulli bond percolation in Scherk's graph

2001 ◽  
Vol 38 (04) ◽  
pp. 828-840
Author(s):  
Dayue Chen
Keyword(s):  
Three Dimensional ◽  
The Other ◽  
Bond Percolation ◽  
Infinite Cluster ◽  
Dimensional Lattice ◽  
Supercritical Region ◽  
Other Hand

Scherk's graph is a subgraph of the three-dimensional lattice. It was shown by Markvorsen, McGuinness and Thomassen (1992) that Scherk's graph is transient. Consider the Bernoulli bond percolation in Scherk's graph. We prove that the infinite cluster is transient for p > ½ and is recurrent for p < ½. This implies the well-known result of Grimmett, Kesten and Zhang (1993) on the transience of the infinite cluster of the Bernoulli bond percolation in the three-dimensional lattice for p > ½. On the other hand, Scherk's graph exhibits a new dichotomy in the supercritical region.

Percolated Superconductivity and Electron-Electron Exchange Mechanism in Ba-La-Cu-O

1987 ◽  
Vol 01 (02) ◽  
pp. 645-648 ◽  
Author(s):  
Ruibao Tao
Keyword(s):  
Transition Temperature ◽  
Electrical Conduction ◽  
Basal Plane ◽  
Oxygen Vacancies ◽  
Electron Exchange ◽  
Exchange Mechanism ◽  
Bond Percolation ◽  
Infinite Cluster

It is found that the oxygen vacancies at the Cu-O basal plane of oxide (La1−xBax)2CuO4−y (Balacuo) will break the bonds of Cu-O-Cu to make the hopping between those coppers disappear so that the electrical conduction in the Cu-O basal plane would become a bond percolation system consisting of an infinite cluster carrying the current with a great number of finite clusters hanging around. It is favorable to create some mechanism of electron-electron exchange so that the transition temperature Tc of superconductivity could be increased significantly. The comparison with Y1Ba2Cu3O9−y is also discussed briefly.

scholarly journals On the Truncated Long-Range Percolation on Z2

2008 ◽  
Vol 45 (1) ◽  
pp. 287-291 ◽  
Author(s):  
Bernardo Nunes Borges de Lima ◽  
Artëm Sapozhnikov
Keyword(s):  
Long Range ◽  
Partial Answer ◽  
Bond Percolation ◽  
Infinite Cluster

We consider an independent long-range bond percolation on Z2. Horizontal and vertical bonds of length n are independently open with probability p_n ∈ [0, 1]. Given ∑n=1∞∏i=1n(1 − pi) < ∞, we prove that there exists an infinite cluster of open bonds of length less than or equal to N for some large but finite N. The result gives a partial answer to the truncation problem.

scholarly journals On the Truncated Long-Range Percolation on Z 2

2008 ◽  
Vol 45 (01) ◽  
pp. 287-291 ◽  
Author(s):  
Bernardo Nunes Borges de Lima ◽  
Artëm Sapozhnikov
Keyword(s):  
Long Range ◽  
Partial Answer ◽  
Bond Percolation ◽  
Infinite Cluster

We consider an independent long-range bond percolation onZ2. Horizontal and vertical bonds of lengthnare independently open with probabilityp_n ∈ [0, 1]. Given ∑n=1∞∏i=1n(1 −pi) &lt; ∞, we prove that there exists an infinite cluster of open bonds of length less than or equal toNfor some large but finiteN. The result gives a partial answer to the truncation problem.

On the infinite cluster of Bernoulli bond percolation in Scherk's graph

2001 ◽  
Vol 38 (4) ◽  
pp. 828-840
Author(s):  
Dayue Chen
Keyword(s):  
Three Dimensional ◽  
The Other ◽  
Bond Percolation ◽  
Infinite Cluster ◽  
Dimensional Lattice ◽  
Supercritical Region ◽  
Other Hand

Scherk's graph is a subgraph of the three-dimensional lattice. It was shown by Markvorsen, McGuinness and Thomassen (1992) that Scherk's graph is transient. Consider the Bernoulli bond percolation in Scherk's graph. We prove that the infinite cluster is transient forp> ½ and is recurrent forp< ½. This implies the well-known result of Grimmett, Kesten and Zhang (1993) on the transience of the infinite cluster of the Bernoulli bond percolation in the three-dimensional lattice forp> ½. On the other hand, Scherk's graph exhibits a new dichotomy in the supercritical region.

On nonhomeomorphic mappings with the inverse Poletsky inequality

2020 ◽  
Vol 17 (3) ◽  
pp. 414-436
Author(s):  
Evgeny Sevost'yanov ◽  
Serhii Skvortsov ◽  
Oleksandr Dovhopiatyi
Keyword(s):  
Boundary Behavior ◽  
Hölder Continuity ◽  
Continuous Extension ◽  
Quasiconformal Mappings ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion ◽  
Lipschitz Continuous ◽  
Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

scholarly journals Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1113
Author(s):  
Isaías Alonso-Mallo ◽  
Ana M. Portillo
Keyword(s):  
Numerical Experiments ◽  
Splitting Method ◽  
Method Of Lines ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
High Order ◽  
Time Dependent ◽  
Boundary Values ◽  
Lipschitz Continuous

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

scholarly journals No Infimum Gap and Normality in Optimal Impulsive Control Under State Constraints

2021 ◽  
Author(s):  
Giovanni Fusco ◽  
Monica Motta
Keyword(s):  
Original Problem ◽  
State Constraints ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Continuous Data ◽  
Lipschitz Continuous ◽  
Extended Sense ◽  
Unbounded Controls ◽  
Locally Lipschitz ◽  
Optimal Impulsive Control

AbstractIn this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously been explored for problems without state constraints. This paper establishes such links in the presence of state constraints and of an additional ordinary control, for locally Lipschitz continuous data.

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