Showing papers on "Symmetry (geometry) published in 1992"

Recognition and Structure from one 2D Model View: Observations on Prototypes, Object Classes and Symmetries

Abstract: In this note we discuss how recognition can be achieved from a single 2D model view exploiting prior knowledge of an object''s structure (e.g. symmetry). We prove that for any bilaterally symmetric 3D object one non- accidental 2D model view is sufficient for recognition. Symmetries of higher order allow the recovery of structure from one 2D view. Linear transformations can be learned exactly from a small set of examples in the case of "linear object classes" and used to produce new views of an object from a single view.

The perils of Cc : comparing the frequencies of falsely assigned space groups with their general population

Abstract: Compilation of 221 space group corrections from a false low symmetry (FS) to a higher true symmetry (TS) shows that higher symmetry is often overlooked in only a few space-group types.

Addendum to ``Integrability of Open Spin Chains with Quantum Algebra Symmetry''

Abstract: We show that the quantum-algebra-invariant open spin chains associated with the affine Lie algebras $A^{(1)}_n$ for $n>1$ are integrable. The argument, which applies to a large class of other quantum-algebra-invariant chains, does not require that the corresponding $R$ matrix have crossing symmetry.

On the Busemann‐Petty problem about convex, centrally symmetric bodies in

Abstract: Let A and B be two compact, convex sets in ℝ n , each symmetric with respect to the origin 0. L is any ( n - l)-dimensional subspace. In 1956 H. Busemann and C. M. Petty (see [6]) raised the question: Does vol ( A ⌒ L ) B ⌒ L ) for every L imply vol ( A ) B )? The answer in case n = 2 is affirmative in a trivial way. Also in 1953 H. Busemann (see [4]) proved that if A is any ellipsoid the answer is affirmative. In fact, as he observed in [5], the answer is still affirmative if A is an ellipsoid with 0 as center of symmetry and B is any compact set containing 0.

Orientational Effects and Component Processes in Symmetry Detection

Abstract: In previous research on symmetry detection, factors contributing to orientational effects (axis and virtual lines connecting symmetrically positioned dots) and component processes (axis selection and pointwise evaluation) have always been confounded. The reason is the restriction to bilateral symmetry (BS), with pointwise correspondences being orthogonal to the axis of symmetry. In our experiments, subjects had to discriminate random dot patterns from symmetries defined by combining 12 axis orientations (every 15°) with seven reflection angles (0°, yielding BS, and three clockwise and counterclockwise 15° steps, yielding skewed symmetry, SS). In Experiment 1, with completely randomized trial order, a significant interaction between axis and skewing angle was obtained, indicating that classically observed orientational effects are restricted to BS and that the orientation of the pointwise correspondences is important. These basic findings were replicated in three subsequent experiments, which differed in t...

Intensity and Edge-Based Symmetry Detection Applied to Car-Following

Abstract: We present two methods for detecting symmetry in images, one based directly on the intensity values and another one based on a discrete representation of local orientation. A symmetry finder has been developed which uses the intensity-based method to search an image for compact regions which display some degree of mirror symmetry due to intensity similarities across a straight axis. In a different approach, we look at symmetry as a bilateral relationship between local orientations. A symmetryenhancing edge detector is presented which indicates edges dependent on the orientations at two different image positions. SEED, as we call it, is a detector element implemented by a feedforward network that holds the symmetry conditions. We use SEED to find the contours of symmetric objects of which we know the axis of symmetry from the intensity-based symmetry finder. The methods presented have been applied to the problem of visually guided car-following. Real-time experiments with a system for automatic headway control on motorways have been successful.

Quadratic algebras and dynamics in curved spaces. II. The Kepler problem

Abstract: The symmetry aspects of the Kepler problem in a space of constant negative curvature are considered. It is shown that the algebra of the hidden symmetry reduces to the quadratic Racah algebraQR(3), and this makes it possible to express the coefficients of the overlapping of the wave functions in the spherical and parabolic coordinates in terms of Wilson-Racah polynomials. It is shown that the dynamical symmetry algebra that generates the spectrum is the quadratic Jacobi algebraQJ(3). Its ladder operators permit explicit construction of wave functions in the coordinate representation with the ground state as the starting point.

Eigenvectors of the inertia tensor and perceiving the orientation of a hand-held object by dynamic touch.

Abstract: Subjects wielded an object, hidden from view, and reported the orientation in which the object was positioned in the hand. The object consisted of a stem with two branches forming a V attached perpendicularly to the stem’s distal end. The branches were differentially weighted so that the same spatial orientation of the object was associated with different orientations of its principal (symmetry) axes or eigenvectors. Perceived orientation was found to be dependent on the eigenvectors of the object’s inertia tensor, computed about the point of rotation in the wrist, rather than on its spatial orientation. The results underscore the significance of the inertia tensor to understanding the perception of spatial properties by dynamic touch.

The Quantum Double in Integrable Quantum Field Theory

Abstract: Various aspects of recent works on affine quantum group symmetry of integrable 2d quantum field theory are reviewed and further clarified. A geometrical meaning is given to the quantum double, and other properties of quantum groups. Multiplicative presentations of the Yangian double are analyzed.

An algorithm for determining the size of symmetry groups

Abstract: To determine the symmetry group of pointor Lie-symmetries of a differential equation is of great theoretical and practical importance, in particular for determining closed form solutions. There does not seem to exist an algorithm that finds this group in general. However, it is always possible to determine thesize of the symmetry group. In this article an algorithm is described that determines for any system of algebraic partial differential equations the number of parameters if the symmetry group is finite, and the number of unspecified functions and its arguments if it is infinite. To this end the so calleddetermining system is transformed into aninvolutive system by means of a critical-pair/completion algorithm similar like it is applied for computing Grobner bases in polynomial ideal theory. The foundation for obtaining this form is the theory of Riquier and Janet for partial differential equations. The algorithmInvolution System has been implemented in several computer algebra systems as part of the packageSPDE. Various results that have been obtained by applying it are presented as well. If symmetry analysis is considered as part of the more general process of obtaining the best possible information on the solutions of a differential equation, the algorithm described in this article removes the heuristics which is usually involved in making the transition from analytical to numerical methods.

Quantum affine symmetry in vertex models

Abstract: We study the higher spin anologs of the six vertex model on the basis of its symmetry under the quantum affine algebra $U_q(\slth)$. Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin $1/2$, and that the $n$-particle space has an RSOS-type structure rather than a simple tensor product of the $1$-particle space. This agrees with the picture proposed earlier by Reshetikhin.

Molecular replacement real‐space averaging

Abstract: Structure determination of macromolecules often depends on phase improvement and phase extension by use of real-space averaging of electron density related by noncrystallographic symmetry. Although techniques for such procedures have been described previously [Bricogne (1976). Acta Cryst. A32, 832-847; Johnson (1978). Acta Cryst. B34, 576-577], modern computer architecture and experience with these methods have suggested changes and improvements. Two unit cells are considered: (1) the p-cell corresponding to the actual crystal structure(s) being determined (there would be more than one of these if the molecule crystallizes in more than one crystal form) and (2) the h-cell corresponding to the molecule in a standard orientation with respect to which the molecular symmetry axes are defined. Averaging can proceed entirely within the p-cell, referring to the h-cell only in as far as knowledge of the molecular symmetry is required. It is also possible to place the averaged molecule back into the h-cell, where it can be used to redefine the molecular envelope or for displaying a suitably chosen asymmetric unit of the molecule. Techniques are discussed for automatically selecting a molecular envelope which is consistent with packing considerations within the p-cell and which retains the symmetry of the molecular point group. The electron density map to be averaged is divided into bricks for storage in virtual memory. Roughly as many bricks as there are noncrystallographic asymmetric units per crystallographic asymmetric unit need to be retained in memory at one time. This procedure minimizes paging problems and avoids double sorting. Use of eight-point interpolation permits storing the map at grid points separated by no more than 1/2.5 of the resolution limit to obtain rapid convergence.

Yang-Mills fields which are not self-dual

Abstract: The purpose of this paper is to prove the existence of a new family of non-self-dual finite-energy solutions to the Yang-Mills equations on Euclidean four-space, withSU(2) as a gauge group. The approach is that of “equivariant geometry:” attention is restricted to a special class of fields, those that satisfy a certain kind of rotational symmetry, for which it is proved that (1) a solution to the Yang-Mills equations exists among them; and (2) no solution to the self-duality equations exists among them. The first assertion is proved by an application of the direct method of the calculus of variations (existence and regularity of minimizers), and the second assertion by studying the symmetry properties of the linearized self-duality equations. The same technique yields a new family of non-self-dual solutions on the complex projective plane.

Exploiting symmetry in boundary element methods

Abstract: Linear operator equations $\mathcal {L}f = g$ are considered in the context of boundary element methods, where the operator $\mathcal {L}$ is equivariant, i.e., commutes with the actions of a given finite symmetry group. By introducing a generalization of Reynolds projectors, a decomposition of the identity operator is constructed, which in turn allows the decomposition of the problem $\mathcal {L}f = g$ into a finite number of symmetric subproblems. The data function g does not need to possess any symmetry properties. It is shown that analogous reductions are possible for discretizations. An explicit construction of the corresponding reduced system matrices is given. This effects a considerable reduction in the computational complexity. For example, in the case of the isometry group of the 3-cube, the computational complexity of a direct linear equation solver for full matrices is reduced by 99.65 percent. Specific decompositions of the identity are given for most of the significant finite isometry group...

Symmetry classes of tensors associated with certain groups

Abstract: We discuss the existence of an orthogonal basis consisting of decomposable vectors for some symmetry classes of tensors associated with certain subgroups of the full symmetric group The dimensions of these symmetry classes of tensors are also given

Imaging system for charged particles

Abstract: The invention relates to an imaging system for charged particles having a correction unit for correcting an objective lens. The correction unit essentially includes a beam deflector and a mirror which reflects the incoming particle beam. A first symmetry plane of the deflector is imaged in the mirror. The mirror images this first symmetry plane at an imaging scale of 1:1 in a second symmetry plane of the deflector. At the same time, the symmetry planes are either intermediate image planes or diffraction planes. With the high symmetry of the imaging system, the condition is achieved that the aberrations of the second order caused by a one-time passthrough through the deflector are cancelled after the second passthrough. The mirror can be so adjusted that its negative chromatic aberration compensates for the chromatic aberration of the objective lens and the other lenses. The spherical aberration can be compensated independently thereof with the aid of a hexapole which is centered in a diffraction plane.

Dilaton-Axion Symmetry

Abstract: The heterotic string compactified on a six-torus is described by a low-energy effective action consisting of N=4 supergravity coupled to N=4 super Yang-Mills, a theory that was studied in detail many years ago. By explicitly carrying out the dimensional reduction of the massless fields, we obtain the bosonic sector of this theory. In the Abelian case the action is written with manifest global $O(6,6+n)$ symmetry. A duality transformation that replaces the antisymmetric tensor field by an axion brings it to a form in which the axion and dilaton parametrize an $SL(2,R)/SO(2)$ coset, and the equations of motion have $SL(2,R)$ symmetry. This symmetry, which combines Peccei--Quinn translations with Montonen--Olive duality transformations, has been exploited in several recent papers to construct black hole solutions carrying both electric and magnetic charge. Our purpose is to explore whether, as various authors have conjectured, an $SL(2,Z)$ subgroup could be an exact symmetry of the full quantum string theory. If true, this would be of fundamental importance, since this group transforms the dilaton nonlinearly and can relate weak and strong coupling.

Symmetry and complexity

Abstract: We examine the effect of symmetry on the complexity of Boolean functions and find a remarkably tight hierarchy. Generalizing the fact that all symmetric Boolean functions belong to (nonuniform) Z’CO, we find that the complexity of the class of Boolean functions admitting a given group of symmetries is essentially determined by a single parameter of that group. Let G be a permutation group acting on the set of n Boolean variables. Let 7(G) denote the set of Boolean functions on n variables which are invariant under G. Let ~ denote a sequence G. < Sym(Q.) of groups (/S2.[ = n). We say that the language L ~ {O, 1}* belongs to the symmetry class ~(~) if the indicator function of L n {O, l}n belongs to Y(Gn) for every n. Following Clote and Kranakis, we consider the parameter s(G), the number of orbits of G on the set {O, l}’”. We show that (a) all functions in F(G) are computable by circuits of size, polynomial in s(G) and depth, polynomial in log s(G); (b) there exist functions in 3(G) which cannot be computed by circuits of size < s(G)/(2 log s(G)). While part (b) is obtained by straightforward counting, it demonstrates that part (a) is tight. The result in particular confirms the following conjecture of Clote and Kranakis: if s(Gn ) is polynomially bounded (where G. < S.) then F(g) ~ NC (nonuniform). If in addition the groups are transitive, we prove that the left hand side is actually in TCO. oT& ~e~e=ch was pmti~y supported by NSF Gr~ts CCR 8710078 and CGR 9014562 1Department of Computer Science, University of Chicago, Chicago, Illinois 60637 2E~tv~s University, Budapest, Hungary H-lo88 31aci@cs.uchicago .edu 4beals@cs.uchicago. edu Permission to copy without fee all or part of this material is granted provided that the copies ara not made or distributed for diremt GommerGial advantaga, the ACM copyright notice and the title of the publication and its data appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or spacific permission. 24th ANNUAL ACM STOC 5/92/VICTORIA, B. C., CANADA ~ 1992 A(JM ().89791-51 2-7/92 JOO04/0438...$~ .50 The proof of the main result involves nontrivial elementary asymptotic structure theory of permutation groups; and a delicate analysis (under new circumstances) of algorithmic techniques developed largely by E. M. Luks in the context of graph isomorphism testing. In the context of isomorphism of sets under group action, uniform versions of our results are obtained.

A characterization of and new consistent tests for symmetry

Abstract: Let X and Y be independent and identically distributed random variables having a continuous distribution function. We study new consistent tests for symmetry around a known median based on the fact that the distribution of X is symmetric around 0 if, and only if, |X| and |max(X,Y)| have the same distribution.

Numbering the crystallographic variants in phase transformation

Abstract: In phase transformation one initial orientation gives rise to several final orientations called crystallographic variants. In this paper it is shown that the number of physically distinct variants depends not only on the lattice symmetry of the initial phase but also on the mutual orientation relations between the lattices. This number is lower than the order of the rotational symmetry group when commutation relations exist between some rotations of the rotational symmetry group and rotations which describe the mutual orientations between lattices. Different examples are given for illustration.

Three-Dimensional Symmetry Identification Part I : Theory

Abstract: A theoretical background necessary for symmetry identification of an arbitrary, finite, threedimensional object at any position and with arbitrary orientation'is presented. It supports construction of an algorithm for identification of a wide range of symmetry types represented by groups of proper and improper rotations and location of corresponding axes and/or planes of symmetry. The prerequisite is that the input object be represented by an octree. The proposed technique is based on examination of the octree obtained by the principal axis transform of the input octree. It is shown that the octree data structure is very convenient for symmetry evaluation, especially for objects whose symmetry types are simpler or equal in complexity with the four-fold rotational symmetry. The cases when the principal axis transform is not uniquely defined are analyzed and possible solutions are offered. Finally, extensions to multidimensional gray-scale images are briefly discussed.

An Approach to the Design of Molecular Solids. A Symmetry Analysis of the Problem

Abstract: In a molecular crystal, intermolecular interactions will correspond to specific symmetry elements. If one chooses molecules carefully, one can reliably predict specific intermolecular interactions and the corresponding symmetry operations. The symmetry operations of one and two dimensional networks of molecules can be combined to form rod and layer groups respectively. In many cases of chemical interest the sequence of moving from a molecule to a one dimensional array, then on to two and three dimensions corresponds directly to the symmetry combinations leading from the point group to rod group, to layer group and on to the space group. The structures of a number of disubstituted urea derivatives were determined and are used to illustrate these ideas of molecular design.