Low-dimensional Groups Research Articles

Cohomology-developed matrices: constructing families of weighing matrices and automorphism actions

The aim of this work is to construct families of weighing matrices via their automorphism group action. The matrices can be reconstructed from the 0, 1, 2-cohomology groups of the underlying automorphism group. We use this mechanism to (re)construct the matrices out of abstract group datum. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley conference, the projective space, the Grassmannian, and the flag variety weighing matrices. We develop a general theory relying on low-dimensional group cohomology for constructing automorphism group actions and in turn obtain structured matrices that we call cohomology-developed matrices. This ‘cohomology development’ generalizes the cocyclic and group developments. The algebraic structure of modules of cohomology-developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of quasiproducts, which is a generalization of the Kronecker product.

High-Dimensional Granger Causality Tests with an Application to VIX and News

Abstract We study Granger causality testing for high-dimensional time series using regularized regressions. To perform proper inference, we rely on heteroskedasticity and autocorrelation consistent (HAC) estimation of the asymptotic variance and develop the inferential theory in the high-dimensional setting. To recognize the time-series data structures, we focus on the sparse-group LASSO (sg-LASSO) estimator, which includes the LASSO and the group LASSO as special cases. We establish the debiased central limit theorem for low-dimensional groups of regression coefficients and study the HAC estimator of the long-run variance based on the sg-LASSO residuals. This leads to valid time-series inference for individual regression coefficients as well as groups, including Granger causality tests. The treatment relies on a new Fuk–Nagaev inequality for a class of τ-mixing processes with heavier than Gaussian tails, which is of independent interest. In an empirical application, we study the Granger causal relationship between the VIX and financial news.

Stability of Clusters in the Second-Order Kuramoto Model on Random Graphs

The Kuramoto model of coupled phase oscillators with inertia on Erdos-Renyi graphs is analyzed in this work. For a system with intrinsic frequencies sampled from a bimodal distribution we identify a variety of two cluster patterns and study their stability. To this end, we decompose the description of the cluster dynamics into two systems: one governing the (macro) dynamics of the centers of mass of the two clusters and the second governing the (micro) dynamics of individual oscillators inside each cluster. The former is a low-dimensional ODE whereas the latter is a system of two coupled Vlasov PDEs. Stability of the cluster dynamics depends on the stability of the low-dimensional group motion and on coherence of the oscillators in each group. We show that the loss of coherence in one of the clusters leads to the loss of stability of a two-cluster state and to formation of chimera states. The analysis of this paper can be generalized to cover states with more than two clusters and to coupled systems on W-random graphs. Our results apply to a model of a power grid with fluctuating sources.

Large-scale benchmark study of survival prediction methods using multi-omics data.

Multi-omics data, that is, datasets containing different types of high-dimensional molecular variables, are increasingly often generated for the investigation of various diseases. Nevertheless, questions remain regarding the usefulness of multi-omics data for the prediction of disease outcomes such as survival time. It is also unclear which methods are most appropriate to derive such prediction models. We aim to give some answers to these questions through a large-scale benchmark study using real data. Different prediction methods from machine learning and statistics were applied on 18 multi-omics cancer datasets (35 to 1000 observations, up to 100 000 variables) from the database ‘The Cancer Genome Atlas’ (TCGA). The considered outcome was the (censored) survival time. Eleven methods based on boosting, penalized regression and random forest were compared, comprising both methods that do and that do not take the group structure of the omics variables into account. The Kaplan–Meier estimate and a Cox model using only clinical variables were used as reference methods. The methods were compared using several repetitions of 5-fold cross-validation. Uno’s C-index and the integrated Brier score served as performance metrics. The results indicate that methods taking into account the multi-omics structure have a slightly better prediction performance. Taking this structure into account can protect the predictive information in low-dimensional groups—especially clinical variables—from not being exploited during prediction. Moreover, only the block forest method outperformed the Cox model on average, and only slightly. This indicates, as a by-product of our study, that in the considered TCGA studies the utility of multi-omics data for prediction purposes was limited. Contact: moritz.herrmann@stat.uni-muenchen.de, +49 89 2180 3198 Supplementary information: Supplementary data are available at Briefings in Bioinformatics online. All analyses are reproducible using R code freely available on Github.

Inference for High-Dimensional Regressions With Heteroskedasticity and Auto-correlation

We study Granger causality testing for high-dimensional time series using regularized regressions. To perform proper inference, we rely on heteroskedasticity and autocorrelation consistent (HAC) estimation of the asymptotic variance and develop the inferential theory in the high-dimensional setting. To recognize the time series data structures we focus on the sparse-group LASSO estimator, which includes the LASSO and the group LASSO as special cases. We establish the debiased central limit theorem for low dimensional groups of regression coefficients and study the HAC estimator of the long-run variance based on the sparse-group LASSO residuals. This leads to valid time series inference for individual regression coefficients as well as groups, including Granger causality tests. The treatment relies on a new Fuk-Nagaev inequality for a class of $\tau$-mixing processes with heavier than Gaussian tails, which is of independent interest. In an empirical application, we study the Granger causal relationship between the VIX and financial news.

Tiling sets and spectral sets over finite fields

We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to 2 it has recently been proven that the Fuglede conjecture holds (see [6]). In this paper we study this question in higher dimensions over prime fields and provide some results and counterexamples. In particular we prove the existence of spectral sets which do not tile in Zp5 for all odd primes p and Zp4 for all odd primes p such that p≡3 mod 4. Although counterexamples in low dimensional groups over cyclic rings Zn were previously known they were usually for non-prime n or a small, sporadic set of primes p rather than general constructions. This paper is a result of a Research Experience for Undergraduates program ran at the University of Rochester during the summer of 2015 by A. Iosevich, J. Pakianathan and G. Petridis.

Complex, Symplectic, and Contact Structures on Low-Dimensional Lie Groups

It is well known that on any Lie group, a left-invariant Riemannian structure can be defined. For other left-invariant geometric structures, for example, complex, symplectic, or contact structures, there are difficult obstructions for their existence, which have still not been overcome, although a lot of works were devoted to them. In recent years, substantial progress in this direction has been made; in particular, classification theorems for low-dimensional groups have been obtained. This paper is a brief review of left-invariant complex, symplectic, pseudo-Kahlerian, and contact structures on low-dimensional Lie groups and classification results for Lie groups of dimension 4, 5, and 6.

Möbius Transformations For Global Intrinsic Symmetry Analysis

AbstractThe goal of our work is to develop an algorithm for automatic and robust detection of global intrinsic symmetries in 3D surface meshes. Our approach is based on two core observations. First, symmetry invariant point sets can be detected robustly using critical points of the Average Geodesic Distance (AGD) function. Second, intrinsic symmetries are self‐isometries of surfaces and as such are contained in the low dimensional group of Möbius transformations. Based on these observations, we propose an algorithm that: 1) generates a set of symmetric points by detecting critical points of the AGD function, 2) enumerates small subsets of those feature points to generate candidate Möbius transformations, and 3) selects among those candidate Möbius transformations the one(s) that best map the surface onto itself. The main advantages of this algorithm stem from the stability of the AGD in predicting potential symmetric point features and the low dimensionality of the Möbius group for enumerating potential self‐mappings. During experiments with a benchmark set of meshes augmented with human‐specified symmetric correspondences, we find that the algorithm is able to find intrinsic symmetries for a wide variety of object types with moderate deviations from perfect symmetry.

An algorithm for low dimensional group homology

Given a finitely presented group $G$, Hopf's formula expresses the second integral homology of $G$ in terms of generators and relators. We give an algorithm that exploits Hopf's formula to estimate $H_2(G;k)$, with coefficients in a finite field k, and give examples using $G=SL_2$ over specific rings of integers. These examples are related to a conjecture of Quillen.

Homology with Trivial Coefficients and Universal Central Extensions of Algebras with Bracket

A 5-term exact sequence and the interpretation of low dimensional groups of homology with trivial coefficients of algebras with bracket are obtained. The endofunctor in the category of algebras with bracket which assigns to a perfect algebra with bracket its universal central extension is constructed and the characterization of the universal central extension by means of the low-dimensional homology groups is done. The conditions required to lift an automorphism or a derivation of A to A′ in a covering f:A′ ↠ A are analyzed.

Finite Dimensional Algebras with Vanishing Hochschild Cohomology

Ž . Let A be a finite dimensional algebra associative with unit over an algebraically closed field k. For a finitely generated bimodule X , the Hochschild cohomology A A iŽ . Ž . w x iŽ . groups H A, X i G 0 were introduced in 10 . We shall denote H A iŽ . iŽ . Ž . s H A, A . The vanishing of the low dimensional groups H A i F 2 has classical interpretations and plays an important role in the representaw x tion theory of algebras: Gernstenhaber 7 has shown some connections 2Ž . Ž w x. 1Ž . between H A and the deformation theory of A see also 5 ; H A is

Concept of polarization entropy in optical scattering

We consider the application of the general theory of unitary matrices to problems of wave scattering involving polarized waves. Haying outlined useful parameterizations of the low dimensional groups associated with these unitary matrices, we develop a general processing strategy, which we suggest has application in the extraction of physical information from a range of scattering matrices in optics. Examples are presented of applying the unitary matrix structure to problems of single and multiple scattering from a cloud of random particles. The techniques are best suited to characterization of depolarizing systems, where the scattered waves undergo a change of degree as well as polarization state. The degree of disorder of the system is then quantified by a scalar, the polarimetric entropy, defined from the eigenvalues of a scattering matrix that ranges from 0 for systems with zero scattering to 1 for perfect depolarizers. Further, we show that the unitary matrix parameterization can be used to extract important system information from the eigenvectors of this matrix.

Cocyclic Development of Designs

We present the basic theory of cocyclic development of designs, in which group development over a finite group G is modified by the action of a cocycle defined on G × G. Negacyclic and ω-cyclic development are both special cases of cocyclic development.Techniques of design construction using the group ring, arising from difference set methods, also apply to cocyclic designs. Important classes of Hadamard matrices and generalized weighing matrices are cocyclic.We derive a characterization of cocyclic development which allows us to generate all matrices which are cocyclic over G. Any cocyclic matrix is equivalent to one obtained by entrywise action of an asymmetric matrix and a symmetric matrix on a G-developed matrix. The symmetric matrix is a Kronecker product of back ω-cyclic matrices, and the asymmetric matrix is determined by the second integral homology group of G. We believe this link between combinatorial design theory and low-dimensional group cohomology leads to (i) a new way to generate combinatorial designs; (ii) a better understanding of the structure of some known designs; and (iii) a better understanding of known construction techniques.

Quelques calculs en cobordisme lagrangien

We consider the cobordism groups (defined by Arnold) of exact Lagrange immersions of compact manifolds in R 2n . Due to the Gromov-Lees theorem, their computation is that of the homotopy groups of Thom spectra built on the spaces U/O (unoriented case, the computation here is due to Smith and Stong) and U/SO (oriented case, we compute here the “even” part, and give informations on the “odd”part). We also compute the images of those groups in the ordinary cobordism groups, then, we study some examples: cobordism classes which can be represented by spheres, generators of the low-dimensional groups and some applications to the enumerative theory of Lagrange singularities.

Low Dimensional Group Cohomology as Monoidal Structures

Low Dimensional Group Cohomology as Monoidal Structures