Skew-symmetric Research Articles

Socle degrees for local cohomology modules of thickenings of maximal minors and sub-maximal Pfaffians

Let S S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I I be the determinantal ideal of maximal minors or P f Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S S -module structures of E x t S j ( S / I t , S ) Ext^j_S(S/I^t, S) and E x t S j ( S / P f t , S ) Ext^j_S(S/Pf^t, S) , from which we get the degrees of generators of these E x t Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H m j ( S / I t ) H^j_\mathfrak {m}(S/I^t) .

Emergent Hypotrochoidal and Epitrochoidal Formations in a Swarm of Agents

This paper proposes a decentralised controller for a swarm of agents that generates hypotrochoidal and epitrochoidal curves in 3D space. The dynamics of the agents is modelled as a double integrator where the control input, i.e. the acceleration profile, contains two terms that account for the interactions between the agents in terms of position and velocity, respectively. The term related to the interaction between velocities is weighted by a skew-symmetric matrix responsible for coupling the velocities displacements. Such a matrix defines an axis of rotation centred on the centroid of the swarm, around which the swarm organises itself to generate the desired trajectories, which can be selected by properly setting two free parameters of the model. In particular, the motion of each agent is analysed by separating its dynamics along the rotation axis from that in the perpendicular plane. It is shown that the agents achieve consensus or periodic motion along the axis of rotation while performing the desired evolutions in the perpendicular plane. Numerical simulations are presented to illustrate the proposed framework.

Teaching rotational physics with bivectors

Angular momentum is traditionally taught as a (pseudo)vector quantity, tied closely to the cross product. This approach is familiar to experts but challenging for students, and full of subtleties. Here, we present an alternative pedagogical approach: angular momentum is described using bivectors which can be visualized as “tiles” with area and orientation, and whose components form an antisymmetric matrix. Although bivectors have historically been studied in specialized contexts like spacetime classification or geometric algebra, they are no more complicated to understand than cross products. The bivector language provides a more fundamental definition for rotational physics and opens the door to understanding rotations in relativity and in extra dimensions.

Eigenproblem Basics and Algorithms

Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.

Symplectic partially hyperbolic automorphisms of 6-torus

We study topological properties of automorphisms of a 6-dimensional torus T6 generated by integer matrices with simple eigenvalues being symplectic with respect to either the standard symplectic structure in R6 or a nonstandard symplectic structure given by an integer skew-symmetric non-degenerate matrix. Such a symplectic matrix generates a partially hyperbolic automorphism of the torus, if its eigenvalues lie both outside and on the unit circle. We study transitive and decomposable cases possible here and present a classification in both cases.

A between-cluster approach for clustering skew-symmetric data

In order to investigate exchanges between objects, a clustering model for skew-symmetric data is proposed, which relies on the between-cluster effects of the skew-symmetries that represent the imbalances of the observed exchanges between pairs of objects. The aim is to detect clusters of objects that share the same behaviour of exchange so that origin and destination clusters are identified. The proposed model is based on the decomposition of the skew-symmetric matrix pertaining to the imbalances between clusters into a sum of a number of off-diagonal block matrices. Each matrix can be approximated by a skew-symmetric matrix by using a truncated Singular Value Decomposition (SVD) which exploits the properties of the skew-symmetric matrices. The model is fitted in a least-squares framework and an efficient Alternating Least Squares algorithm is provided. Finally, in order to show the potentiality of the model and the features of the resulting clusters, an extensive simulation study and an illustrative application to real data are presented.

Rees algebra of maximal order Pfaffians and its diagonal subalgebras

Given a skew-symmetric matrix X, the Pfaffian of X is defined as the square root of the determinant of X. In this article, we give the explicit defining equations of the Rees algebra of a Pfaffian ideal I generated by the maximal order Pfaffians of a generic skew-symmetric matrix. We further prove that all diagonal subalgebras of the corresponding Rees algebra of I are Koszul. We also look at Rees algebras of Pfaffian ideals of linear type associated with certain sparse skew-symmetric matrices. In particular, we consider the tridiagonal matrices and identify the corresponding Pfaffian ideals to be of Gröbner linear type and as the vertex cover ideals of unmixed bipartite graphs. As an application of our results, we conclude that all their ordinary and symbolic powers have linear quotients.

Cooperation of robot manipulators with motion constraint by real-time RNN-based finite-time fault-tolerant control

The nonlinear cooperative dynamics of m robots with motion constraint and faulty input saturation is addressed. To the best of our knowledge, the motion constraint for the cooperative control of multiple robots is first discussed as one of the designed concepts such that the proposed cooperative control system is more practical and its performance is also improved. Together with a finite-time concept, a real-time recurrent-neural-network-based finite-time fault-tolerant cooperation constraint control (RT-RNN-FTFTCCC) is designed to tackle the aggregately dynamic uncertainties brought about by motion constraint error, cooperation uncertain dynamics, disturbance torque, faulty input saturation, unsatisfying skew-symmetric matrix condition, and others. Moreover, the stabilized learning law includes two terms to cover all the range of filtering cooperation errors such that learning weights converge even in uncertain circumstances. Finally, the stability analysis of the overall cooperation control system is achieved by Lyapunov stability theory. In comparison to adaptive compensation control, recurrent-neural-network-based control without the motion constraint compensation, and different stabilized learning laws, the cooperative control of bi-robotic arms of 3-link in-plane with new constraint validates the improvement and robustness of the proposed control.

Diagonalization of the cross-product matrix

The paper considers diagonalization of the cross-product matrices, i.e., skew-symmetric matrices of order three. A procedure to determine a nonsingular matrix, which yields the diagonalization is indicated. Furthermore, a method to derive the inverse of a diagonalizing matrix is proposed by means of a formula for the Moore–Penrose inverse of any matrix, which is columnwise partitioned into two matrices having disjoint ranges. This rather nonstandard method to obtain the inverse of a nonsingular matrix is appealing, as it can be applied to any diagonalizing matrix, and not only of those originating from diagonalization of the cross-product matrices. The paper provides also comments and examples demonstrating applicability of the diagonalization procedure to calculate roots of a cross-product matrix.

A Model to Study Time Lagged Interactions, Source Connectivity and Source Activities Using Multi-channel EEG.

A computational model to examine time lagged interactions; identify number of interacting pairs of neuronal sources; and determine source activities from multi-channel EEG measurements is described. It is based on the imaginary part of the cross spectrum between the EEG channels. The imaginary part of the cross spectrum between the EEG channels provides the most suitable property that reflects the presence of interacting sources. The model assumes that not all sources are activated simultaneously and that there is a time lag amongst some of them. A new analytical expression derived for the imaginary part of cross spectrum between channels shows that it is different from the zero lag case. A method is then proposed to identify time lag interactions, by studying its variation as a function of frequency. Assuming pair wise interaction between sources, the model shows that simultaneous diagonalization at different frequencies of symmetric matrices formed by multiplying the anti-symmetric matrix of the imaginary part of cross spectrum with its transpose will provide information on the number of interacting source pairs as a function of frequency. The matrix that simultaneously diagonalizes all the symmetric matrices is identified as the mixing matrix. This can be used to obtain the source activities.

Determinantal properties of Boolean graphs using recursive approach

The aim of this paper is to study the determinant and inverse of the adjacency matrices of weighted and directed versions of Boolean graphs. Our approach is recursive. We describe the adjacency matrix of a weighted Boolean graph in terms of the adjacency matrix of a smaller-sized weighted Boolean graph. This allows us to compute the determinant and inverse of the adjacency matrix of a weighted Boolean graph recursively. In particular, we show that the determinant of a directed Boolean graph is 1. Further, using a classical theorem of Cayley which expresses the determinant of any skew-symmetric matrix as a square of its Pfaffian, we show that for any directed Boolean graph, the characteristic polynomial has all its even degree coefficients strictly positive with the odd ones being zero.

On the consistency of the matrix equation X T A X = B when B is skew-symmetric: improving the previous characterization

We provide a necessary and sufficient condition for the matrix equation to be consistent, when A is an arbitrary complex square matrix and B is skew-symmetric. This problem is equivalent to find the largest dimension of a subspace in which the bilinear form A is symplectic. The necessity is valid for any A and B as above, whereas the sufficiency is proved to be valid for any skew-symmetric matrix B and for all complex square matrices A whose canonical form for congruence (CFC) does not contain blocks . The provided condition improves the one in [Borobia A, Canogar R, De Terán F. Lin Multilin Algebra, 2022. DOI:10.1080/03081087.2022.2093825], because it includes the case where CFC(A) includes symmetric blocks, and it is given in terms of the size of A and the rank of its symmetric and skew-symmetric parts. More precisely, if A is , we prove that the equation is consistent if and only if , where is the dimension of .

Ornstein-Uhlenbeck operators and holomorphic representations for a class of bounded symmetric domains

Let G be the group of holomorphic automorphisms of a bounded symmetric domain D. We discuss the correspondence between unitarizing measures for scalar-valued holomorphic discrete series representations of G and Ornstein-Uhlenbeck operators on D. We complement the results obtained for the domains in [1] and [6] by showing that for Cartan domain II of n×n symmetric complex matrices and domain III of skew-symmetric complex matrices Z such that ZZ⁎<I, the vector fields in the infinitesimal holomorphic representation are the entries of Hua matrices. Similar results stay valid for some Kähler submanifolds M of D, assuming that the entries of the matrices in M depend linearly on fixed independent entries and M is isomorphic to a product of Cartan domains.

Telling compositeness at a distance with outer automorphisms and CP

We investigate charge-parity (CP) and non-CP outer automorphism of groups and the transformation behavior of group representations under them. We identify situations where composite and elementary states that transform in exactly the same representation of the group, transform differently under outer automorphisms. This can be instrumental in discriminating composite from elementary states solely by their quantum numbers with respect to the outer automorphism, i.e. without the need for explicit short distance scattering experiments. We discuss under what conditions such a distinction is unequivocally possible. We cleanly separate the case of symmetry constrained (representation) spaces from the case of multiple copies of identical representations in flavor space, and identify conditions under which non-trivial transformation in flavor space can be enforced for composite states. Next to composite product states, we also discuss composite states in non-product representations. Comprehensive examples are given based on the finite groups and D 8. The discussion also applies to and we scrutinize recent claims in the literature that outer automorphism with antisymmetric matrices correspond to distinct outer automorphisms. We show that outer automorphism transformations with antisymmetric matrices are related by an inner automorphism to the standard outer automorphism of . As a direct implication, no non-trivial transformation behavior can arise for composite product states under the outer automorphism of .

A note on complex conference matrices

Let Fq be the Galois field of order q=pm, p a prime number and m a positive integer. We prove in this article that for any nontrivial multiplicative character ϰ of Fq⁎ and for any b∈Fq⁎ we have∑a∈Fq⁎ϰ(a)ϰ(a+b)‾=−1. Whenever q is odd and ϰ is the Legendre symbol this formula reduces to the well-known Jacobsthal's formula. A complex conference matrix is a square matrix of order n with zero diagonal and unimodular complex numbers elsewhere such that C⁎C=(n−1)I. Paley used finite fields with odd orders q=pm, p prime and the real Legendre symbol to construct real symmetric conference matrices of orders q+1 whenever q≡1(mod4) and real skew-symmetric conference matrices of orders q+1 whenever q≡−1(mod4). In this article we extend Paley construction to the complex setting. We extend Jacobsthal's formula to all other nontrivial characters to produce a complex symmetric conference matrix of order q+1 whenever q≥4 is any prime power as well as a complex skew-symmetric conference matrix of order q+1 whenever q is any odd prime power. These matrices were constructed very recently in connection with harmonic Grassmannian codes, by use of finite fields and the character table of their additive characters. We propose here a new proof of their construction by use of the above generalized formula similarly as was done by Paley in the real case. We also classify, up to equivalence, the complex conference matrices constructed with some nontrivial characters. In particular, we prove that the complex conference matrix constructed with any nontrivial multiplicative character ϰ and that one constructed with ϰpk for any integer k=1,...m−1 are permutation equivalent. Moreover, we determine the spectrum of any complex conference matrix obtained from this construction.

Ising analogs of quantum spin chains with multispin interactions

A new family of free fermionic quantum spin chains with multispin interactions was recently introduced. Here we show that it is possible to build standard quantum Ising chains---but with inhomogeneous couplings---which have the same spectra as the novel spin chains with multispin interactions. The Ising models are obtained by associating an antisymmetric tridiagonal matrix to the polynomials that characterize the quasienergies of the system via a modified Euclidean algorithm. For the simplest nontrivial case, corresponding to the Fendley model, the phase diagram of the inhomogeneous Ising model is investigated numerically. It is characterized by gapped phases separated by critical lines with order-disorder transitions depending on the parity of the total number of energy density operators in the Hamiltonian.

Development and dynamics of a novel couple-constrained parallel wrist with three measuring force flexible fingers

AbstractA novel couple-constrained parallel wrist with three measuring force flexible fingers is designed for grabbing heavy objects and measuring grabbed forces. Its prototype is developed, its dynamics model is established, and its grabbing forces are measured. First, using the extended formulas of the skew-symmetric matrix, the kinematic formulas are derived for solving the Jacobian/Hessian matrices and the general velocity/acceleration of the moving links in the couple-constrained parallel wrist. Second, a dynamics model is established for solving the dynamic actuation forces, the couple-constrained forces, and the torque in the couple-constrained parallel wrist. Third, the theoretical solutions of the kinematics/dynamics of the couple-constrained parallel wrist are verified using a simulation mechanism. Finally, the grabbing forces of the three flexible fingers are measured and analyzed.

The Conchoidal Twisted Surfaces Constructed by Anti-Symmetric Rotation Matrix in Euclidean 3-Space

A twisted surface is a type of mathematical surface that has a nontrivial topology, meaning that it cannot be smoothly deformed into a flat surface without tearing or cutting. Twisted surfaces are often described as having a twisted or Möbius-like structure, which gives them their name. Twisted surfaces have many interesting mathematical properties and applications, and are studied in fields such as topology, geometry, and physics. In this study, a conchoidal twisted surface is formed by the synchronized anti-symmetric rotation matrix of a planar conchoidal curve in its support plane and this support plane is about an axis in Euclidean 3-space. In addition, some examples of the conchoidal twisted surface are given and the graphs of the surfaces are presented. The Gaussian and mean curvatures of this conchoidal twisted surface are calculated. Afterward, the conchoidal twisted surface formed by an involute curve and the conchoidal twisted surface formed by a Bertrand curve pair are given. Thanks to the results obtained in our study, we have added a new type of surface to the literature.

Pairs of Symmetric and Skew-Symmetric Toeplitz Matrices with Identical Squares

Certain sets consisting of pairs of symmetric and skew-symmetric Toeplitz matrices with identical squares are described.

Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming

We establish connections between the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.