Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

We present a mathematical model of perceptual completion and formation of subjective surfaces, which is at the same time inspired by the architecture of the visual cortex, and is the lifting in the 3-dimensional rototranslation group of the phenomenological variational models based on elastica functional. The initial image is lifted by the simple cells to a surface in the rototranslation group and the completion process is modeled via a diffusion driven motion by curvature. The convergence of the motion to a minimal surface is proved. Results are presented both for modal and amodal completion in classic Kanizsa images.

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A Cortical Based Model of Perceptual Completion in the Roto-Translation Space

In this window, all groups are assumed finite. Here we collect a number of results that play a significant role in the book (further material of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).

Finite Group Theory

https://www.intlpress.com/site/pub/files/_fulltext/journals/jsg/2005/0003/0004/JSG-2005-0003-0004-a009.pdf

Complex and Kähler structures on Compact Solvmanifolds

We derive a unifying framework for the a posteriori error analysis of control constrained linear-quadratic optimal control problems. We consider finite element discretizations with discretized and nondiscretized control. A fundamental error equivalence drastically simplifies the a posteriori error analysis for optimal control problems. It basically remains to apply error estimators for the linear state and adjoint problem. We give several examples, including stabilized discretizations, and investigate the quality of the estimators and the performance of the adaptive iteration by selected numerical experiments.

A Posteriori Error Analysis of Optimal Control Problems with Control Constraints

/pdf/geometries-on-surfaces-lie-transformation-groups-5c574quvfa.pdf

Geometries on Surfaces: Lie Transformation Groups

The projective, Mobius, Laguerre, and Minkowski planes over the real numbers are just a few examples of a host of fundamental classical topological geometries on surfaces. This book summarizes all known major results and open problems related to these classical point-line geometries and their close (nonclassical) relatives. Topics covered include: classical geometries; methods for constructing nonclassical geometries; classifications and characterizations of geometries. This work is related to many other fields including interpolation theory, convexity, the theory of pseudoline arrangements, topology, the theory of Lie groups, and many more. The authors detail these connections, some of which are well-known, but many much less so. Acting both as a reference for experts and as an accessible introduction for graduate students, this book will interest anyone wishing to know more about point-line geometries and the way they interact.

Geometries on Surfaces

Publisher Summary
This chapter focuses on topological circle geometries. The geometry of all circles on the real 2-sphere in 3-dimensional Euclidean space (i.e. the intersection of the sphere with Euclidean planes) has since long been investigated under various aspects. Another model of the same geometry is obtained by stereographic projection from one point of the sphere onto a plane not passing through the point of projection. This leads to the geometry of Euclidean lines extended by an infinite point and Euclidean circles in the real plane. Similarly, one considers extensions of the geometry of Euclidean lines and Euclidean parabolae with given direction of axis or Euclidean hyperbolae with given direction of asymptotes. An algebraic description of these geometries leads to chain geometries. It discusses the planar situation and takes a more geometric point of view. An axiomatization is given and thus a generalization of these geometries which leads to so-called circle planes. Since affine and projective planes occur as derived incidence structures, the structure theory and classification of topological locally compact connected finite-dimensional affine and projective planes plays a crucial role.

Chapter 24 - Topological Circle Geometries

We give some easy to use criteria for deciding whether a Mobius plane, Laguerre plane or Minkowski plane, given in some normal form, is 2-dimensional. As an application of our results we prove that a Laguerre plane or Minkowski plane with a given topology on the point set is a (topological) 2-dimensional plane if and only if each derived affine plane at points of at least one parallel class is 2-dimensional with respect to the induced topology. To appear in Beitreige fur Algebra and Geometrie. CRITERIA FOR TWO-DIMENSIONAL CIRCLE PLANES B. POLSTER AND G. F. STEINKE Department of Mathematics University of Canterbury Christchurch, New Zealand ABSTRACT. We give some easy to use criteria for deciding whether a Mobius plane, Laguerre We give some easy to use criteria for deciding whether a Mobius plane, Laguerre plane or Minkowski plane, given in some normal form, is 2-dimensional. As an application of our results we prove that a Laguerre plane or Minkowski plane with a given topology on the point set is a (topological) 2-dimensional plane if and only if each derived affine plane at points of at least one parallel class is 2-dimensional with respect to the induced topology.

Criteria for two-dimensional circle planes

We describe two methods to combine sets of lines of different 2-dimensional projective planes into line sets of new 2-dimensional projective planes. Using these methods we describe several ways in which sets of circles of different 2-dimensional circle planes can be combined into circle sets of new 2-dimensional circle planes.

Günter F. Steinke

Papers

Geometries on Surfaces: Lie Transformation Groups

Geometries on Surfaces

Chapter 24 - Topological Circle Geometries

Criteria for two-dimensional circle planes

Cut and paste in $2$-dimensional projective planes and circle planes