Search

We investigate the structure of normal subgroups of the unit group of a quaternion algebra over a pythagorean field. [ABSTRACT FROM AUTHOR]

The orientation distribution function (ODF) in classical texture analysis is defined on the rotation group SO(3). For polycrystalline aggregates with crystallite symmetry defined by a crystallographic point group G cr which is not a subgroup of SO(3), the improper group G cr is routinely replaced by its proper peer (i.e., a subgroup of SO(3)) in the same Laue class. In this note we examine how the texture coefficients obtained from such a practice are related to their counterparts that pertain to the corresponding ODF defined on the orthogonal group O(3) as it should. [ABSTRACT FROM AUTHOR]

In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group O 3 on the space R [ x , y , z ] 2 d of ternary forms of even degree 2d. The construction relies on two key ingredients: on the one hand, the Slice Lemma allows us to reduce the problem to determining the invariants for the action on a subspace of the finite subgroup B 3 of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed B 3 -equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the B 3 -invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the O 3 -invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed B 3 -invariants to determine the O 3 -orbit locus and provide an algorithm for the inverse problem of finding an element in R [ x , y , z ] 2 d with prescribed values for its invariants. These computational issues are relevant in brain imaging. [ABSTRACT FROM AUTHOR]

For variational problems with O (N) -symmetry, the existence of several geometrically distinct solutions has been shown by use of group theoretic approach in the previous papers. It was done by a crafty choice of a family H i ⊂ O (N) subgroups such that the fixed point subspaces E H i ⊂ E of the action in a corresponding functional space are linearly independent, next restricting the problem to each E H i and using the Palais symmetry principle. In this work, we give a thorough explanation of this approach showing a correspondence between the equivalence classes of such subgroups, partial orthogonal flags in ℝ N , and unordered partitions of the number N. By showing that spaces of functions invariant with respect to different classes of groups are linearly independent, we prove that the amount of series of geometrically distinct solutions obtained in this way grows exponentially in N , in contrast to logarithmic, and linear growths of earlier papers. [ABSTRACT FROM AUTHOR]

The generating graph of a finite group is a structure which can be used to encode certain information about the group. It was introduced by Liebeck and Shalev and has been further investigated by Lucchini, Maróti, Roney-Dougal, and others. We investigate maximal cocliques (totally disconnected induced subgraphs of the generating graph) in for q a prime power and provide a classification of the "large" cocliques when q is prime. We then provide an interesting geometric example which contradicts this result when q is not prime and illustrate why the methods used for the prime case do not immediately extend to the prime-power case with the same result. [ABSTRACT FROM AUTHOR]

We examine a pair of dynamical systems on the plane induced by a pair of spanning trees in the Cayley graph of the Super-Apollonian group of Graham, Lagarias, Mallows, Wilks, and Yan. The dynamical systems compute Gaussian rational approximations to complex numbers and are “reflective” versions of the complex continued fractions of A. L. Schmidt. They also describe a reduction algorithm for Lorentz quadruples, in analogy to work of Romik on Pythagorean triples. For these dynamical systems, we produce an invertible extension and an invariant measure, which we conjecture is ergodic. We consider some statistics of the related continued fraction expansions, and we also examine the restriction of these systems to the real line, which gives a reflective version of the usual continued fraction algorithm. Finally, we briefly consider an alternate setup corresponding to a tree of Lorentz quadruples ordered by arithmetic complexity. [ABSTRACT FROM AUTHOR]

Let Γn be an n × n Haar-invariant orthogonal matrix. Let Zn be the p × q upper-left submatrix of Γn, where p = pn and q = qn are two positive integers. Let Gn be a p × q matrix whose pq entries are independent standard normals. In this paper we consider the distance between √ nZn and Gn in terms of the total variation distance, the Kullback-Leibler distance, the Hellinger distance, and the Euclidean distance. We prove that each of the first three distances goes to zero as long as pq/n goes to zero, and not so if (p, q) sits on the curve pq = σn, where σ is a constant. However, it is different for the Euclidean distance, which goes to zero provided pq²/n goes to zero, and not so if (p, q) sits on the curve pq² = σn. A previous work by Jiang (2006) shows that the total variation distance goes to zero if both p/ √ n and q/ √ n go to zero, and it is not true provided p = c √ n and q = d √ n with c and d being constants. One of the above results confirms a conjecture that the total variation distance goes to zero as long as pq/n → 0 and the distance does not go to zero if pq = σn for some constant σ. [ABSTRACT FROM AUTHOR]

In this article, we consider the Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal subgroup of the orthogonal group of a non-degenerate quadratic space with a hyperbolic summand over a commutative ring, introduced by Roy. We prove a set of commutator relations among the elementary generators of the DSER elementary orthogonal group. As an application, we prove that this group is perfect and an action version of the Quillen's local-global principle for this group is proved. This affirmatively answers a question of Rao in his Ph.D. thesis. [ABSTRACT FROM AUTHOR]

A result by Hashimoto and Ueda says that the graded ring of modular forms with respect to SO(2,10) is a polynomial ring in modular forms of weights 4, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42. In this paper, we show that one may choose Eisenstein series as generators. This is done by calculating sufficiently many Fourier coefficients of the restrictions to the Hermitian half-space. Moreover, we give two constructions of the skew-symmetric modular form of weight 252. [ABSTRACT FROM AUTHOR]

Linear complementary dual (LCD) cyclic codes were referred historically to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In particular, it has been shown that binary LCD codes play an important role in implementations against side-channel attacks and fault injection attacks. In this paper, we first present a new characterization of binary LCD codes in terms of their orthogonal or symplectic basis. Using such a characterization, we solve a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes. Next, we consider the action of the orthogonal group on the set of all LCD codes, determine all possible orbits of this action, derive simple closed formulas of the size of the orbits, and present some asymptotic results on the size of the corresponding orbits. Our results show that almost all binary LCD codes are odd-like codes with odd-like duals, and about half of $q$ -ary LCD codes have orthonormal basis, where $q$ is a power of an odd prime. [ABSTRACT FROM AUTHOR]

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the integral orthogonal group with respect to the quadratic form of signature (2 , 3) given by 0 1 1 0 ⊥ 0 1 1 0 ⊥ (− 2 N) for square-free N ∈ ℕ. The associated Hecke algebra is commutative and also the tensor product of its primary components, which turn out to be polynomial rings over ℤ in two algebraically independent elements. The integral orthogonal group is isomorphic to the paramodular group of degree 2 and level N , more precisely to its maximal discrete normal extension. The results can be reformulated in the paramodular setting by virtue of an explicit isomorphism. The Hecke algebra of the non-maximal paramodular group inside Sp 2 (ℚ) fails to be commutative if N > 1. [ABSTRACT FROM AUTHOR]

We prove the existence of strongly regular graphs with parameters (216, 40, 4, 8) and (540, 187, 58, 68). We also construct a strongly regular graph with parameters (540, 224, 88, 96) that was previously unknown. Further, we construct all distance-regular graphs with at most 600 vertices, admitting a transitive action of the orthogonal group O + ( 6 , 2 ) or O − ( 6 , 2 ) . Furthermore, we show that under certain conditions an orbit matrix M of a strongly regular graph Γ can be used to define a new strongly regular graph Γ ˜ , where the vertices of the graph Γ ˜ correspond to the orbits of Γ (the rows of M ). We show that some of the obtained strongly regular graphs are related to each other in a way that one can be constructed from an orbit matrix of the other. [ABSTRACT FROM AUTHOR]

R.A. Rao–W. van der Kallen showed that the Vaserstein symbol V Γ ( S R 3 ) from the orbit space of unimodular rows of length three over the coordinate ring of the real three sphere S R 3 modulo elementary action to the elementary symplectic Witt group W E ( Γ ( S R 3 ) ) is not injective. Dhvanita R. Rao–Neena Gupta gave an uncountable family of singular real threefolds A α for which the Vaserstein symbol V A α is not injective. In this paper, we give a countable family of smooth real birationally equivalent threefolds A n for which the Vaserstein symbol V A n is not injective. [ABSTRACT FROM AUTHOR]

I reconstruct Riemann's and Helmholtz-Lie's problems of space, from some perspectives that allow for a fruitful comparison with Weyl. In Part II. of his inaugural lecture, Riemann justifies that the infinitesimal metric is the square root of a quadratic form. Thanks to Finsler geometry, I clarify both the implicit and explicit hypotheses used for this justification. I explain that Riemann-Finsler's kind of method is also appropriate to deal with indefinite metrics. Nevertheless, Weyl shares with Helmholtz a strong commitment to the idea that the notion of group should be at the center of the foundations of geometry. Riemann missed this point, and that is why, according to Weyl, he dealt with the problem of space in a “too formal” way. As a consequence, to solve the problem of space, Weyl abandoned Riemann-Finsler's methods for group-theoretical ones. However, from a philosophical point of view, I show that Weyl and Helmholtz are in strong opposition. The meditation on Riemann's inaugural lecture, and its clear methodological separation between the infinitesimal and the finite parts of the problem of space, must have been crucial for Weyl, while searching for strong epistemological foundations for the group-theoretical methods, avoiding Helmholtz's unjustified transition from the finite to the infinitesimal. [ABSTRACT FROM AUTHOR]

Non-transitive subgroups of the orthogonal group play an important role in the non-Euclidean geometry. If G is a closed subgroup in the orthogonal group such that the orbit of a single Euclidean unit vector does not cover the (Euclidean) unit sphere centered at the origin then there always exists a non-Euclidean Minkowski functional such that the elements of G preserve the Minkowskian length of vectors. In other words the Minkowski geometry is an alternative of the Euclidean geometry for the subgroup G . It is rich of isometries if G is “close enough” to the orthogonal group or at least to one of its transitive subgroups. The measure of non-transitivity is related to the Hausdorff distances of the orbits under the elements of G to the Euclidean sphere. Its maximum/minimum belongs to the so-called lazy/busy orbits, i.e. they are the solutions of an optimization problem on the Euclidean sphere. The extremal distances allow us to characterize the reducible/irreducible subgroups. We also formulate an upper and a lower bound for the ratio of the extremal distances. As another application of the analytic tools we introduce the rank of a closed non-transitive group G . We shall see that if G is of maximal rank then it is finite or reducible. Since the reducible and the finite subgroups form two natural prototypes of non-transitive subgroups, the rank seems to be a fundamental notion in their characterization. Closed, non-transitive groups of rank n − 1 will be also characterized. Using the general results we classify all their possible types in lower dimensional cases n = 2 , 3 and 4 . Finally we present some applications of the results to the holonomy group of a metric linear connection on a connected Riemannian manifold. [ABSTRACT FROM AUTHOR]

For homogeneous higher-gradient elasticity models we discuss frame-indifference and isotropy requirements. To this end, we introduce the notions of local versus global SO(3)-invariance and identify frame-indifference (traditionally) with global left SO(3)-invariance and isotropy with global right SO(3)-invariance. For specific restricted representations, the energy may also be local left SO(3)-invariant as well as local right SO(3)-invariant. Then we turn to linear models and consider a consequence of frame-indifference together with isotropy in nonlinear elasticity and apply this joint invariance condition to some specific linear models. The interesting point is the appearance of finite rotations in transformations of a geometrically linear model. It is shown that when starting with a linear model defined already in the infinitesimal symmetric strain ε = symGrad [ u ] , the new invariance condition is equivalent to the isotropy of the linear formulation. Therefore, it may also be used in higher-gradient elasticity models for a simple check of isotropy and for extensions to anisotropy. In this respect we consider in more detail variational formulations of the linear indeterminate couple-stress model, a new variant of it with symmetric force stresses and general linear gradient elasticity. [ABSTRACT FROM AUTHOR]