Minimal Subspace Research Articles

Decoupling of Measurable Signals via Self-Bounded Controlled Invariant Subspaces: Minimal Unassignable Dynamics of Feedforward Units for Prestabilized Systems

A dynamic feedforward scheme allows measurable signal decoupling to be solved independently of other problems simultaneously present in the design of a control system like, e.g., stabilization, robustness, insensitivity to disturbances. The synthesis procedure, based on the properties of self-bounded controlled invariant subspaces, ensures the minimal complexity of the feedforward unit, in terms of minimal unassignable dynamics and minimal dynamic order, in the case of left-invertible systems and, on some specific conditions, also in the case of non-left-invertible systems. The output dynamic feedback set up to guarantee stability (or, more generally, robustness or insensitivity properties) does not affect the complexity of the feedforward unit, since the peculiar layout where the feedback unit receives an additional input from the precompensator preserves, in the extended system, exactly the same set of unassignable internal eigenvalues of the minimal self-bounded controlled invariant subspace as that defined for the original system. The overall control structure turns out to be a two-degree-of-freedom controller completely devised in the geometric context

Minimal subspaces and isomorphically homogeneous sequences in a Banach space

If a Banach space is saturated with subspaces with a Schauder basis, which embed into the linear span of any subsequence of their basis, then it contains a minimal subspace. It follows that any Banach space is either ergodic or contains a minimal subspace.

Quantum channel capacities: Multiparty communication

We analyze different aspects of multiparty communication over quantum memoryless channels and generalize some of key results known from bipartite channels to that of multiparty scenario. In particular, we introduce multiparty versions of minimal subspace transmission fidelity and entanglement transmission fidelity. We also provide alternative, local, versions of fidelities and show their equivalence to the global ones in context of capacity regions defined. The equivalence of two different capacity notions with respect to two types of the fidelities is proven. In analogy to bipartite case it is shown, via sufficiency of isometric encoding theorem, that additional classical forward side channel does not increase capacity region of any quantum channel with $k$ senders and $m$ receivers which represents a compact unit of general quantum networks theory. The result proves that recently provided capacity region of multiple access channel ([M. Horodecki et al, Nature {\bf 436} 673 (2005)], [J.Yard et al, quant-ph/0501045]) is optimal also in the scenario of additional support of forward classical communication.

Kinematic sets for real-time robust articulated object tracking

In this article, a new approach is given for real-time visual tracking of a class of articulated non-rigid objects in 3D. The main contribution of this paper consists in symmetrically modeling the motion and velocity of an articulated object via a novel kinematic set approach. This is likened to a Lagrange–d'Alembert formulation in classical physics. The advantages of this new model over pre-existing methods include improved precision, robustness and efficiency, leading to real-time performance. Furthermore, a general class of mechanical joints can be considered and the method can track objects where previous approaches have failed due to a lack of visual information. In summary, a joint configuration is modeled by using Pfaffian velocity constraints. The configuration and location of a joint is then used to build a general Jacobian Matrix, which relates individual rigid body velocities (twists) to an underlying minimal subspace. A closed loop control law is then derived in order to minimize a set of distance errors in the image and estimate the system parameters. The tracking is locally based upon efficient distance criterion. Experimental results show prismatic, rotational and helical type links and eight general parameters. A statistical M-estimation technique is applied to improve robustness. A monocular camera system was used as a real-time sensor to verify the theory.

Quantum State Reconstruction of Many Body System Based on Complete Set of Quantum Correlations

We propose and study a universal approach for the reconstruction of quantum states of many body systems from symmetry analysis. The concept of minimal complete set of quantum correlation functions (MCSQCF) is introduced to describe the state reconstruction. As an experimentally feasible physical object, the MCSQCF is mathematically defined through the minimal complete subspace of observables determined by the symmetry of quantum states under consideration. An example with broken symmetry is analyzed in detail to illustrate the idea.

Self-Bounded Controlled Invariant Subspaces in Model Following by Output Feedback: Minimal-Order Solution for Nonminimum-Phase Systems

Self-bounded controlled invariant subspaces are shown to play a crucial role in the synthesis of minimal-order dynamic regulators achieving model following by output feedback with stability. The approach, completely embedded in the geometric context, provides insight into the internal eigenstructure of the minimal self-bounded invariant subspace, thus leading to an effective treatment of nonminimum-phase systems.

SIGNAL DECOUPLING WITH PREVIEW: PERFECT SOLUTION FOR NONMINIMUM-PHASE SYSTEMS IN THE GEOMETRIC APPROACH CONTEXT

The problem of making the output insensitive to an exogenous input signal possibly known with preview is tackled in the geometric approach context. The definition of minimal preview for decoupling is introduced. Necessary and sufficient constructive conditions for decoupling with minimal preview are proved by means of simple geometric arguments. The structural and the stabilizability conditions are considered separately. The minimal complexity of the solution is guaranteed by using the minimal self-bounded controlled invariant subspace. In the presence of unstable dynamics of that subspace, a steering along zeros technique completely devised in the state-space allows the solution with internal stability to be nonetheless achieved. Implementation is obtained by resorting to finite impulse response systems.

OUTPUT FEEDBACK MODEL MATCHING THROUGH SELF-BOUNDED CONTROLLED INVARIANT SUBSPACES

Model matching by output feedback is completely treated in the geometric approach framework. Self-bounded controlled invariant subspaces are shown to play a crucial role in the synthesis of minimal-order dynamic regulators achieving model matching by output feedback with stability. The approach provides insight into the internal eigenstructure of the minimal self-bounded controlled invariant subspace, thus paving the way to an effective treatment of nonminimum-phase systems.

Ergodic Banach spaces

We show that any Banach space contains a continuum of non-isomorphic subspaces or a minimal subspace. We define an ergodic Banach space X as a space such that E 0 Borel reduces to isomorphism on the set of subspaces of X, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis which is complementably universal for the family of its block-subspaces. We also use our methods to get uniformity results. We show that an unconditional basis of a Banach space, of which every block-subspace is complemented, must be asymptotically c 0 or ℓ p , and we deduce some new characterisations of the classical spaces c 0 and ℓ p .

Incomparable, non-isomorphic and minimal Banach spaces

A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if $E_0$ does not reduce to isomorphism of th

Lattice-subspaces and positive bases in function spaces

Let x1,...,xn be linearly independent, positive elements of the space RΩ of the real valued functions defined on a set Ω and let X be the vector subspace of RΩ generated by the functions xi. We study the problem: Does a finite-dimensional minimal lattice-subspace (or equivalently a finite-dimensional minimal subspace with a positive basis) of RΩ which contains X exist? To this end we define the function β(t)=\(\frac{1}{{z\left( t \right)}}\)(x1(t),x2(t),...,xn(t)), where z(t)=x1(t)+x2(t)+...+xn(t), which we call basic function and takes values in the simplex Δn of \(\mathbb{R}_ + ^n \). We prove that the answer to the problem is positive if and only if the convex hull K of the closure of the range of β is a polytope. Also we prove that X is a lattice-subspace (or equivalently X has positive basis) if and only if, K is an (n-1)-simplex. In both cases, using the vertices of K, we determine a positive basis of the minimal lattice-subspace. In the sequel, we study the case where Ω is a convex set and x1,x2,...,xn are linear functions. This includes the case where xi are positive elements of a Banach lattice, or more general the case where xi are positive elements of an ordered space Y. Based on the linearity of the functions xi we prove some criteria by means of which we study if K is a polytope or not and also we determine the vertices of K. Finally note that finite dimensional lattice-subspaces and therefore also positive bases have applications in economics.

Linear Optimization in C (Ω) and Portfolio Insurance

Suppose that X is a subspace of C ( z ) generated by n linearly independent positive elements of C ( z ). In this article we study the problem of minimization of a positive linear functional p of X in X , under a finite number of linear inequalities. This problem does not have always a solution and if a solution exists we cannot determine it. In this article we show that if X is contained in a finite dimensional minimal lattice-subspace Y of C ( z ) (or equivalently, if X is contained in a finite dimensional minimal subspace Y of C ( z ) with a positive basis) and m = dim Y , then the minimization problem has a solution and we determine the solutions by solving an equivalent linear programming problem in . Finally note that this minimization problem has an important application in the portfolio insurance which was the motivation for the preparation of this article.

Subsymmetric sequences and minimal spaces

We show that every Banach space saturated with subsymmetric basic sequences contains a minimal subspace.

Representations of Complete Uniform Spaces via Uniform Domains

In this paper, we show that complete uniform spaces can be represented domain-theoretically. We introduce the notion of a uniform domain, which is an ω-algebraic domain with some uniform structure on the set K(D) of finite elements of D. It is proved that when (X,μ) is a complete uniform space of countable weight, there is a uniform domain D such that X is the retract of the set L(D) of limit elements of D. On the other hand, in every uniform domain D, there exists a minimal subspace M(D) of L(D) on which K(D) induces a uniformity structure. Thus, a uniform domain can be considered as a set with a particular kind of base of a uniformity. Since every infinite increasing sequences in K(D) identifies one element of M(D), through a labelling of edges of K(D), we obtain an admissible representation of a uniform space in a uniform domain. We also show that such a representation is a proper representation.

Necessary Conditions for a Rigorous Minimal Diabatic Potential Matrix

In this article, one of the more important dilemmas in molecular physics is considered: given a matrix of the nonadiabatic coupling terms of any desired dimension, what is the minimal sub-Hilbert space for which diabatization is still valid. This problem was addressed by one of us before (Baer, M. Chem. Phys. Lett. 2000, 329, 450), but it was recently established that the suggested criteria therein lead to subspaces that are too large to be of any use. In this article, we discuss the conditions that have to be satisfied to reach the minimal subspace. We have found that these conditions are related to the spatial distribution of the various nonadiabatic coupling terms. Thus, if nonadiabatic coupling terms for the relevant states overlap only slightly in configuration space, the required size of the subspace for diabatization can be reduced significantly. As an example, we consider the C2H molecule.

Maximal, Minimal, and Primary Invariant Subspaces

Let X be a complex infinite dimensional Banach space. An operator L on X is called of subcritical class, if ∑∞n=1n−3/2log+‖Ln‖<∞. Assume that T is an operator on X whose iterates have norms of polynomial growth. We prove that if T has a range of finite codimension and a left inverse of subcritical class, then every maximal invariant subspace of T has codimension one, and if T has a finite dimensional kernel and a right inverse of subcritical class, then every minimal invariant subspace of T is one dimensional. Using these results we obtain new information on the invariant subspace lattices of shifts and backward shifts on a wide class of Banach spaces of analytic functions on the unit disc. We also introduce the notion of primary invariant subspaces, and determine their structure for a large class of shifts.

Minimal realization and dynamic properties of optimal smoothers

Smoothing algorithms of various kinds have been around for several decades. However, some basic issues regarding the dynamical structure and the minimal dimension of the steady-state algorithm are still poorly understood. In this paper, we derive a realization of minimal dimension of the optimal smoother for a signal admitting a state-space description of dimension n. It is shown that the dimension of the smoothing algorithm can vary from n to 2n, depending on the zero structure of the signal model. The dynamics (pole structure) of the steady-state smoother is also characterized explicitly and is related to the zero structure of the model. We use several recent ideas from stochastic realization theory. In particular, a minimal Markovian representation of the smoother is derived, which requires solving a nonsymmetric Wiener-Hopf factorization problem. In this way, the smoother is naturally expressed as the cascade of a whitening filter and a linear filter of least possible dimension, whose state space is a minimal Markovian subspace containing the smoothed estimate x/spl I.cap/. This, among other aspects, affords a very simple calculation of the error covariance matrix of the smoother. A reduced-order two-filter implementation of the type due to Mayne (1966) and Fraser (1967) is obtained by solving a Riccati equation of reduced dimension, which is in general smaller than the dimension of the Riccati equations considered in the literature.

A minimal subspace residual method for large-scale eigenvalue problems

We present an iterative method for treating extremely large-scale eigenvalue problems. Based on an exact formula and the GMRES method, our approach generates a subspace which has the property that the residual of interior eigenpairs in the subspace is minimized. The result is that the corresponding large matrix is block-diagonalized iteratively. The accuracy of the final eigenpairs of interest is directly controlled by the accuracy of the GMRES procedure. Our method limits the number of Arnoldi iterations involved, and the dimension of the subspace, by including the residual in the subspace and minimizing it at each step of the iteration.

The restrictions of functions holomorphic in a domain to curves lying on its boundary, and discrete -spectra

We consider the operator of restriction of functions holomorphic in a ball or a polydisc to curves lying on the Shilov boundary. It turns out that any function with polynomial growth near the boundary has such a restriction if the position of the curve satisfies a certain condition: if the domain is a ball, then the curve must be transversal to the standard contact distribution on the sphere, and if the domain is a polydisc, then the curve must be monotonic increasing with respect to all coordinates in the standard coordinatization of the torus. We use assertions of this kind to obtain a simple description of discrete inclusions in spectra (of minimal invariant subspaces) for several problems of -harmonic analysis.

Stochastic Dominance, Pareto Optimality, and Equilibrium Asset Pricing

pricing. A factor subspace contains the market portfolio and is such that every marketed contingent claim is second-order stochastically dominated by a claim from the factor subspace. Conditions are given for the existence of equilibrium, and it is shown how APT and CAPM can be interpreted in the framework of the paper. If sufficiently many call options on the market portfolio are traded, then the space spanned by these options can be used as the factor subspace. We consider markets in which a collection of financial assets are traded. The collection may be finite or infinite. Markets are frictionless, and short-sales of assets are permitted. Asset payoffs are random variables which are elements of a space of contingent claims. By forming portfolios of the marketed assets, payoffs in a certain linear subspace of the space of all contingent claims can be attained. We call this subspace a space of marketed claims. We say that a closed subspace of the space of marketed claims is a factor subspace if two conditions are satisfied: first, the market portfolio payoff (i.e. the payoff of an aggregate endowment of portfolios per capita) belongs to this subspace; second, every marketed contingent claim is dominated in the sense of second-order stochastic dominance by some contingent claim in the factor subspace. We demonstrate that for each asset market economy, there is a well-defined minimal factor subspace in the sense that no proper subspace of it can be a factor subspace. If markets are complete, i.e., if a full set of Arrow securities are traded, then the price of any given asset can be expressed in terms of prices of Arrow securities or state prices. Breeden and Litzenberger (1978) have shown that state prices can be replaced by the values of options. Thus in a complete market asset prices are determined either by state prices or call options. It is established in this paper that these ideas and methods are applicable to asset pricing theories even when markets are incomplete. Making use of the argument of second-order stochastic dominance, we show that every constrained Pareto optimal portfolio allocation consists exclusively of portfolios which generate the factor subspace. The same holds for an equilibrium portfolio allocation. Thus, for an economy with a factor subspace, only a partial set of Arrow securities which is large enough to generate the factor subspace is required to price assets. This paper establishes that there exists an equilibrium in which the price of a portfolio equals the price of the dominating portfolio with the same expected return in the factor subspace. Since the dominating