Narrow Results

In this paper, we present a method for calculation of spin groups elements for known pseudo-orthogonal group elements with respect to the corresponding two-sheeted coverings. We present our results using the Clifford algebra formalism in the case of arbitrary dimension and signature, and then explicitly using matrices, quaternions, and split-quaternions in the cases of all possible signatures $(p,q)$ of space up to dimension $n=p+q=3$. The different formalisms are convenient for different possible applications in physics, engineering, and computer science.

We study the integrals of type , depending on a matrix a ∈ M_{p × q}(ℕ), whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary expansion" formula from the case a ∈ M_{2 × q}(2ℕ) to the general case a ∈ M_{p × q}(ℕ), (2) the construction of the "best algebraic normalization" of I(a), in the case a ∈ M_{2 × q}(ℕ), (3) an explicit formula for I(a), for diagonal matrices a ∈ M_{3 × 3}(ℕ), (4) a modeling result in the case a ∈ M_{1 × 2}(ℕ), in relation with the Euler–Rodrigues formula. Most proofs use various combinatorial techniques.

Human action recognition is very important and significant research work in numerous fields of science, for example, human–computer interaction, computer vision and crime analysis. In recent years, relative geometry features have been widely applied to the description of relative relation of body motion. It brings many benefits to action recognition such as clear description, abundant features etc. But the obvious disadvantage is that the extracted features severely rely on the local coordinate system. It is difficult to find a bijection between relative geometry and skeleton motion. To overcome this problem, many previous methods use relative rotation and translation between all skeleton pairs to increase robustness. In this paper we present a new motion representation method. It establishes a motion model based on the relative geometry with the aid of special orthogonal group SO(3). At the same time, we proved that this motion representation method can establish a bijection between relative geometry and motion of skeleton pairs. After the motion representation method in this paper is used, the computation cost of action recognition reduces from the two-way relative motion (motion from A to B and B to A) to one-way relative motion (motion from A to B or B to A) between any skeleton pair, namely, permutation problem $P_n^2$ is simplified into combinatorics problem $C_n^2$. Finally, the experimental results of the three motion datasets are all superior to present skeleton-based action recognition methods.

The Wishart distribution on a homogeneous cone is a generalization of the Riesz distribution on a symmetric cone which corresponds to a given graph. The paper extends to this distribution, the famous Olkin and Rubin characterization of the ordinary Wishart distribution on symmetric matrices.

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.

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 $H_i\subset O(N)$ subgroups such that the fixed point subspaces $E^{H_i}\subset E$ 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.

We study the injectivity of the map (of pointed sets) $U{m}_{r+1}(R)/E_{r+1}(R)\to S{O}_{2(r+1)}(R)/E{O}_{2(r+1)}(R)$, which we had constructed in [12]. We obtain a necessary and sufficient condition for the kernel to be non-trivial under some additional assumptions on $r$.

We answer the question of splitting of the normalizer of a maximal torus in orthogonal groups of type $B_n$ and $D_n$ over finite fields and algebraically closed fields of positive characteristic.

In this paper, we consider the integral orthogonal group with respect to the quadratic form of signature $(2,3)$ given by $\left(\right.\begin{array}{cc}0\hfill & 1\hfill \\ 1\hfill & 0\hfill \end{array}\left)\right.\perp \left(\right.\begin{array}{cc}0\hfill & 1\hfill \\ 1\hfill & 0\hfill \end{array}\left)\right.\perp (-2N)$ for square-free $N\in \mathbb{N}$. The associated Hecke algebra is commutative and also the tensor product of its primary components, which turn out to be polynomial rings over $\mathbb{Z}$ 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 ${\mathrm{\text{Sp}}}_2(\mathbb{Q})$ fails to be commutative if $N>1$.

We show that the additive Borcherds lifts of vector-valued non-holomorphic Eisenstein series are orthogonal non-holomorphic Eisenstein series for $O(2,l)$. Using this we give another proof that they have a meromorphic continuation, calculate their Fourier expansion and show that they have a functional equation analogous to the classical case. Moreover, we will investigate the image of Borcherds lift and give a sufficient condition for surjectivity.

We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups $\mathrm{\text{O(N)}}$, $\mathrm{\text{U(N)}}$ and $\mathrm{\text{Sp(2N)}}$. For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal and symplectic cases. Asymptotic expansions in strong and weak non-unitarity regimes are obtained. Using the connection to matrix hypergeometric functions, we establish limit theorems for the log-modulus of the characteristic polynomial evaluated on the unit circle.