Projective equivalence of homogenous binary polynomials

In this paper a class of Abelian groups which includes the torsion totally projective groups, S-groups, and balanced projectives is studied. It is shown that this class of groups has a complete set of invariants.

Local invariants for a class of mixed states

We study the equivalence of bipartite mixed states under local unitary transformations. A complete set of invariants under local unitary transformations is presented for a class of mixed states in CM ⊗ CN composite systems. It is shown that two density matrices in this class are locally equivalent if and only if all these invariants have equal values for these density matrices.

Feedback invariants for linear dynamical systems over a principal ideal domain

In this work, the stability of the Rossby–Haurwitz (RH) waves from the subspace \(\mathbf {H}_{1}\oplus \mathbf {H}_{n}\) is considered (\(n\ge 2\)) where \(\mathbf {H}_{k}\) is the subspace of the homogeneous spherical polynomials of degree k. A conservation law for arbitrary perturbations of the RH wave is derived, and all perturbations are divided into three invariant sets \(\mathbf {M}_{-}^{n}\), \(\mathbf {M}_{0}^{n}\) and \(\mathbf {M} _{+}^{n}\) in which the mean spectral number \(\chi (\psi ^{\prime })\) of any perturbation \(\psi ^{\prime }\) is less than, equal to or greater than \( n(n+1) \), respectively. In turn, the set \(\mathbf {M}_{0}^{n}\) is divided into the invariant subsets \(\mathbf {H}_{n}\) and \(\mathbf {M}_{0}^{n}{\setminus } \mathbf {H}_{n}\). Quotient spaces and norms of the perturbations are introduced, a hyperbolic law for the perturbations belonging to the sets \(\mathbf {M}_{-}^{n}\) and \(\mathbf {M}_{+}^{n}\) is derived, and a geometric interpretation of variations in the kinetic energy of perturbations is given. It is proved that any non-zonal RH wave from \(\mathbf {H}_{1}\oplus \mathbf {H} _{n}\) (\(n\ge 2\)) is Liapunov unstable in the invariant set \(\mathbf {M} _{-}^{n}\). Also, it is shown that a stationary RH wave from \(\mathbf {H} _{1}\oplus \mathbf {H}_{n}\) may be exponentially unstable only in the invariant set \(\mathbf {M}_{0}^{n}{\setminus } \mathbf {H}_{n}\), while any perturbation of the invariant set \(\mathbf {H}_{n}\) conserves its form with time and hence is neutral. Since a Legendre polynomial flow \(aP_{n}(\mu )\) and zonal RH wave \(-\,\omega \mu +aP_{n}(\mu )\) are particular cases of the RH waves of \(\mathbf {H} _{1}\oplus \mathbf {H}_{n}\), the major part of the stability results obtained here is also true for them.

We construct an algebraic-combinatorial model of the SOS compartment of the EGFR biochemical network. A Petri net is used to construct an initial representation of the biochemical decision making network, which in turn defines a hyperdigraph. We observe that the linear algebraic structure of each hyperdigraph admits a canonical set of algebraic-combinatorial invariants that correspond to the information flow conservation laws governing a molecular kinetic reaction network. The linear algebraic structure of the hyperdigraph and its sets of invariants can be generalized to define a discrete algebraic-geometric structure, which is referred to as an oriented matroid. Oriented matroids define a polyhedral optimization geometry that is used to determine optimal subpaths that span the nullspace of a set of kinetic chemical reaction equations. Sets of constrained submodular path optimizations on the hyperdigraph are objectively obtained as a spanning tree of minimum cycle paths. This complete set of subcircuits is used to identify the network pinch points and invariant flow subpaths. We demonstrate that this family of minimal circuits also characteristically identifies additional significant biochemical reaction pattern features. We use the SOS Compartment A of the EGFR biochemical pathway to develop and demonstrate the application of our algebraic-combinatorial mathematical modeling methodology.

Long-term large scale behavior and location of the attractors of the barotropic atmosphere model described by the dissipative and forced vorticity equation (VE) on a rotating sphere are studied analytically. Size of a bounded invariant set B that eventually attracts the trajectories of all the VE solutions is estimated depending on the linear drag, turbulence and spectral composition and smoothness of the forcing. If the VE forcing belongs to the set Hn of the homogeneous spherical polynomials of degree n, the solutions show quite different behavior for ideal fluid (a); nonturbulent fluid with linear drag (b), and turbulent fluid (c). For n ≫ 1, the whole space of the VE solutions is divided into sets M n + and M n − of the small and large scale fields defined by χ> n(n + 1) and χ n(n + 1) respectively (χ is the Fjörtoft average spectral number of field on a sphere), and the interface M n 0:χ = n(n + 1) that includes Hn . In cases (a) and (b), M n +, M n 0, M n − and H n are invariant sets of the VE solutions, also, in case (b), any solution of M n + or M n − tends to the intersection of M n 0 and B. In the case (c), the sets M n −, M n + and M n 0 are no longer invariant, however, Hn and the union of M n − and M n 0 remain invariant, and all stationary attractors are in the intersection of B and the set (49). In the cases (b) and (c), Hn is the basin of attraction of a steady state, limit cycle or quasi-periodic VE solution according to whether the forcing is stationary, periodic or quasi-periodic. Conditions providing for such an attractor to be global are also given. As examples, it is shown that (1) the global attractor in Hn can cyclically change its structure from a zonal to a blocking-like flow; (2) bifurcation of the global attractor from one set Hn to another can take place.

We study real nonsingular cubic hypersurfaces X ⊂ P 5 X\subset P^5 up to deformation equivalence combined with projective equivalence and prove that they are classified by the conjugacy classes of involutions induced by the complex conjugation in H 4 ( X ) H_4(X) . Moreover, we provide a graph Γ K 4 \Gamma _{K4} whose vertices represent the equivalence classes of such cubics and whose edges represent their adjacency. It turns out that the graph Γ K 4 \Gamma _{K4} essentially coincides with the graph Γ K 3 \Gamma _{K3} characterizing a certain adjacency of real nonpolarized K3-surfaces.

The contragredient equivalence for several matrices: a set of invariants

The systems considered in the paper are m-input m-output, linear, time invariant decouplable systems S = (A, B, C) which can be described by the set of differential equations $ where A∊R n×n , B∊R n×m and C∊R m×n . In § 1 the decoupling problem is stated and some results relevant to our discussion are reviewed. In § 2 we analyse the effect of an integrator decoupling feedback law (IDF) upon the controllability and the observability of the system under consideration, showing that observability is preserved under IDF if and only if $. Furthermore it is shown that if a decouplable system satisfies this condition, then it is controllable and observable. In § 3 we give our main result : for the class Ω of decouplable systems which preserve observability under IDF, the set of integers {di } ; i = 1, 2,…, m is a complete set of independent invariants under feedback and similarity transformations. Finally a canonical form (unique) is given for the elements of Ω, completely determined by the set {di } ; i = 1,…, m.

LMI RELAXATIONS FOR HOMOGENEOUS POLYNOMIAL SOLUTIONS OF PARAMETER-DEPENDENT LMIS

We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that we investigate general properties of differential invariants of partial differential operators under gauge transformations and introduce a sufficient condition for a set of invariants to be complete. We also give a some mild conditions that guarantee the existence of such a set. The proof is constructive. The method gives many examples of invariants previously known in the literature as well as many new examples including multidimensional.

Chapter 2 focuses on the guaranteed stability analysis of switched dynamical systems under arbitrary switching. As global uniform asymptotic stability is equivalent to the existence of a common Lyapunov function of the subsystems, the Lyapunov approach plays a dominant role in the stability analysis. For switched linear systems, emphasis is laid on the sets of functions that are universal in the sense that each asymptotically stable system admits a Lyapunov function from the function set. We also pay much attention to the algebraic theory of discrete-time switched linear systems, where the stability is elegantly characterized by the spectral radius of the matrix set, which generalizes the standard matrix spectral theory. To approximate the spectral radius numerically, the homogeneous polynomials are utilized to serve as common Lyapunov functions, where the sum of squares technique and the semi-definite programming are used to searching for suitable homogeneous polynomial Lyapunov functions. Finally, the more subtle issue of marginal stability is carefully examined, and its connection to the common weak Lyapunov function is established. We reveal that marginal stability admits a block triangular decomposition with clear spectral information, and this leads to an invariant set viewpoint for characterizing marginal stability and marginal instability.

Deconvolution filtering for stochastic systems via homogeneous polynomial Lyapunov functions

A new method is presented for solving equivalence problems for the extended jet of finite order of the scale-space corresponding to a 2-dimensional input image and the groups of spatially homogeneous affine and orthogonal transformations of local cartesian coordinate frames. By means of this method complete and irreducible sets of algebraic invariants are found that may describe any local and (or) multilocal affine or orthogonal invariant of scale-space. Consequently these sets may form bases for topological descriptions of scale-space.