A note on a forcing related to the S-space problem in the extension with a coherent Suslin tree

2015 ◽  
Vol 61 (3) ◽  
pp. 169-178
Author(s):  
Teruyuki Yorioka
Keyword(s):  
Space Problem ◽  
Suslin Tree

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

A Hilbert Space Problem Book. By Paul R. Haimos

1984 ◽  
Vol 91 (9) ◽  
pp. 592-594
Author(s):  
Peter Fillmore ◽  
Nigel Higson
Keyword(s):  
Hilbert Space ◽  
Space Problem ◽  
Problem Book

A Hilbert Space Problem Book

1989 ◽  
Vol 73 (465) ◽  
pp. 259
Author(s):  
Philip Maher ◽  
P. R. Halmos
Keyword(s):  
Hilbert Space ◽  
Space Problem ◽  
Problem Book

Structured null space problem

1998 ◽  
Author(s):  
Franklin T. Luk ◽  
Kai-Bor Yu
Keyword(s):  
Null Space ◽  
Space Problem

On Σ1 well-orderings of the universe

1976 ◽  
Vol 41 (1) ◽  
pp. 167-170
Author(s):  
Leo Harrington ◽  
Thomas Jech
Keyword(s):  
Simple Formula ◽  
Generic Extension ◽  
Joint Paper ◽  
Suslin Tree ◽  
The Universe

The constructible universe L of Gödel [2] has a natural well-ordering <L given by the order of construction; a closer look reveals that this well-ordering is definable by a Σ1 formula. Cohen's method of forcing provides several examples of models of ZF + V ≠ L which have a definable well-ordering but none is definable by a relatively simple formula.Recently, Mansfield [7] has shown that if a set of reals (or hereditarily countable sets) has a Σ1, well-ordering then each of its elements is constructible. A question has thus arisen whether one can find a model of ZF + V ≠ L that has a Σ1 well-ordering of the universe. We answer this question in the affirmative.The main result of this paper isTheorem. There is a model of ZF + V ≠ L which has a Σ1 well-ordering.The model is a generic extension of L by adjoining a branch through a Suslin tree with certain properties. The branch is a nonconstructible subset of ℵ1. Note that by Mansfield's theorem, the model must not have nonconstructible subsets of ω.Our results can be generalized in several directions. We note that in particular, we can get a model with a Σ1 well-ordering that is not L[X] for any set X. As one might expect from a joint paper by a recursion theorist and a set theorist, the proof consists of a construction and a computation.

scholarly journals A Three-Section-Settlement Calculation Method for Composite Foundation Reinforced by Geogrid-Encased Stone Columns

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Binhui Ma ◽  
Zhiyong Hu ◽  
Zhuo Li ◽  
Kai Cai ◽  
Minghua Zhao ◽  
...  
Keyword(s):  
Calculation Method ◽  
Element Model ◽  
Summation Method ◽  
Composite Foundation ◽  
Stone Columns ◽  
Engineering Practice ◽  
Space Problem ◽  
Relative Slip ◽  
Settlement Calculation ◽  
Slip Displacement

The analysis of the bearing characteristics and deformation mechanism of composite foundation reinforced with geogrid-encased stone columns is presented in order to obtain its settlement calculation method. The settlement of composite foundation is divided into three sections which are the reinforced section, unreinforced section, and underlying stratum. Based on Hooke’s law of space problem and the thoughts of the layer-wise summation method, the relative slip displacement between pile and soil of reinforced section without plastic zone is analyzed. The settlement of reinforced section is calculated by the layered iteration method based on the pile element model. The compatibility of vertical and radial deformations of unreinforced section is analyzed based on the pile-soil element model. The settlement of underlying stratum is still calculated by the layer-wise summation method. Finally, two engineering examples are analyzed and the results show that the settlement calculated by the presented method is close to the measured one. The method overcomes the defect that the calculated results by the other existing methods are more dangerous and it is more feasible and can be applied in engineering practice.

A Generalized Model for Adhesive Contact Between a Rigid Cylinder and a Transversely Isotropic Substrate

2012 ◽  
Vol 80 (1) ◽  
Author(s):  
Haimin Yao
Keyword(s):  
Adhesion Force ◽  
Transversely Isotropic ◽  
Generalized Solution ◽  
Adhesive Contact ◽  
Equilibrium Conditions ◽  
Space Problem ◽  
Rigid Cylinder ◽  
Adhesion Behavior ◽  
Griffith’S Criterion ◽  
Resistant Moment

In this paper, a solution to the quasi-static adhesive contact problem between a rigid cylinder and a transversely isotropic substrate is extended to the most general case by taking adhesion hysteresis into account. An analytical solution to the contact stress is obtained by solving the integral equations established on the basis of the Green's function for the two-dimensional transversely isotropic half-space problem. By using equilibrium conditions and Griffith's criterion, the adhesion force and resistant moment to rolling are determined as functions of contact geometries and material properties of the contacting solids. Detailed discussions on the adhesion force and resistant moment are presented for some specific cases, revealing adhesion behaviors that have not been predicted by previous models. As the most generalized solution to the discussed problem, our results would have extensive applications in predicting the adhesion behavior between solids undergoing sophisticated mechanical loadings.

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