Abstract: In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, ≥ e ℋ for some e > 0 in a Hilbert space 210B to an abstract bucklingpr oblem operator.

Abstract: The contribution of this paper is three-fold: first, we propose a novel scheme for generalized minor subspace extraction by extending an idea of dimension reduction technique. The key of this scheme is the reduction of the problem for extracting the ith (i ? 2) minor generalized eigenvector of the original matrix pencil to that for extracting the first minor generalized eigenvector of a matrix pencil of lower dimensionality. The proposed scheme can employ any algorithm capable of estimating the first minor generalized eigenvector. Second, we propose a pair of such iterative algorithms and analyze their convergence properties in the general case where the generalized eigenvalues are not necessarily distinct. Third, by using these algorithms inductively, we present adaptive implementations of the proposed scheme for estimating an orthonormal basis of the generalized minor subspace. Numerical examples show that the proposed adaptive subspace extraction algorithms have better numerical stability than conventional algorithms.

Abstract: We extend the notion of narrow operators to nonlinear maps on vector lattices. The main objects are orthogonally additive operators and, in particular, abstract Uryson operators. Most of the results extend known theorems obtained by O. Maslyuchenko, V. Mykhaylyuk and the second named author published in Positivity 13 (2009), pp. 459--495, for linear operators. For instance, we prove that every orthogonally additive laterally-to-norm continuous C-compact operator from an atomless Dedekind complete vector lattice to a Banach space is narrow. Another result asserts that the set U_{on}^{lc}(E,F) of all order narrow laterally continuous abstract Uryson operators is a band in the vector lattice of all laterally continuous abstract Uryson operators from an atomless vector lattice E with the principal projection property to a Dedekind complete vector lattice F. The band generated by the disjointness preserving laterally continuous abstract Uryson operators is the orthogonal complement to U_{n}^{lc}(E,F).

Abstract: We consider second-order differential-difference equations in bounded domains in the case where several degenerate difference operators enter the equation. The degeneration leads to the fact that the multiplicity of the zero eigenvalue for the corresponding differential-difference operator becomes infinite. Regularity of generalized solutions for such equations is known to fail in the interior of the domain. However, we prove that the projections of solutions onto the orthogonal complement to the kernel of the “leading” difference operator remain regular in certain subdomains which form a decomposition of the original domain.

Abstract: It is shown that for any full column rank matrix X 0 with more rows than columns there is a neighborhood [Formula: see text] of X 0 and a continuous function f on [Formula: see text] such that f(X) is an orthogonal complement of X for all X in [Formula: see text]. This is used to derive a distribution free goodness of fit test for covariance structure analysis. This test was proposed some time ago and is extensively used. Unfortunately, there is an error in the proof that the proposed test statistic has an asymptotic χ (2) distribution. This is a potentially serious problem, without a proof the test statistic may not, in fact, be asymptoticly χ (2). The proof, however, is easily fixed using a continuous orthogonal complement function. Similar problems arise in other applications where orthogonal complements are used. These can also be resolved by using continuous orthogonal complement functions.

Abstract: We propose a scheme for cloning an arbitrary unknown two-qubit state and its orthogonal complement state with the assistance from the state preparer. Our scheme includes two stages. The first stage requires a quantum teleportation process, in which an arbitrary unknown two-qubit state can be deterministically teleported from the sender to the receiver with χ-type entangled states as the quantum channel. In the second stage, with the assistance of the state preparer, either a perfect copy or an orthogonal complement state of an arbitrary unknown two-qubit state can be obtained with a certain probability.

Abstract: After fixing a nondegenerate bilinear form on a vector space V, we define a ℤ2-action on the manifold of flags F in V by taking a flag to its orthogonal complement. When V is of dimension 3 we check that the Crepant Resolution Conjecture of J. Bryan and T. Graber holds: the genus zero (orbifold) Gromov–Witten potential function of [F/ℤ2] agrees (up to unstable terms) with the genus zero Gromov–Witten potential function of a crepant resolution Y of the quotient scheme F/ℤ2, after setting a quantum parameter to −1, making a linear change of variables, and analytically continuing coefficients. We explicitly compute several invariants for the orbifold and the resolution, then argue that these determine the others via basic properties of Gromov–Witten invariants.

Abstract: Fusion frames are a generalized form of frames in Hilbert spaces. In the present paper we introduce Bessel subfusion sequences and subfusion frames and we investigate the relationship between their operation. Also, the definition of the orthogonal complement of subfusion frames and the definition of the completion of Bessel fusion sequences are provided and several results related with these notions are shown.

Abstract: A two-level method in space and time for the time-dependent Navier-Stokes equations is considered in this article. The approximate solution uM∈HM is decomposed into the large eddy component v∈Hm(m < M) and the small eddy component w∈H. We obtain the large eddy component v by solving a standard Galerkin equation in a coarse-level subspace Hm with a time step length k, whereas the small eddy component w is derived by solving a linear equation in an orthogonal complement subspace H with a time step length pk, where p is a positive integer. The analysis shows that our two-level scheme has long-time stability and can reach the same accuracy as the standard Galerkin method in fine-level subspace HM for an appropriate configuration of p and m. Moreover, some numerical examples are provided to complement our theoretical analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Abstract: Let T denote an operator on a Hilbert space H and let {Λj} be a g-frame for the orthogonal complement of the kernel NT . We construct a sequence of operators {φn} of the form φn(. )= n=1 g n (.)Λj which converges to the psuedoinverse T † of T in the strong operator topology as n →∞ . The operators {φn} can be found using finite-dimensional methods. We also prove an adaptive iterative version of the result. Mathematics Subject Classification: Primary 42C99; Secondary 46B99, 46C99

Abstract: Kinematic modeling of a tree-type robotic system is presented in this chapter. In order to obtain kinematic constraints, a tree-type topology is first divided into a set of modules. The kinematic constraints are then obtained between these modules by introducing the concepts of module-twist, module-joint-rate, etc. This helps in obtaining the generic form of the Decoupled Natural Orthogonal Complement (DeNOC) matrices for a tree-type system with the help of module-to-module velocity transformations. Using the present derivation, link-to-link velocity transformation (Saha 1999a, b) turns out to be a special case of the module-to-module velocity transformation (Shah et al. 2012a) presented in this chapter.

Abstract: Contour regression, a method for estimating the central subspace in regression, is based on estimating contour directions of small variation in the response. These directions span the orthogonal complement of the central subspace and can be extracted according to two measures of variation in the response: simple and general contour regression (SCR and GCR). When the elliptically contoured distribution and mild assumptions hold, the contour regression approach in comparison with existing sufficient dimension reduction methods suggests exhaustiveness of the central space, keeping n -consistency. In addition, the contour-based approach proves robust to violations of departures from ellipticity. In this paper, two kernel simple and general contour regressions (KSCR and KGCR) are proposed and compared with SCR and GCR.

Abstract: Manufacturing science focuses on understanding problems from the perspective of the stakeholders involved and then applying manufacturing science as needed. We investigate semi-orthogonal frame wavelets and Parseval frame wavelets in with a dilation factor. We show that every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces. This result is obtained through considering the basicquestion as to when the orthogonal complement of an afffine subspace in another one is still affine subspace.The definition of multiple pseudofames for subspaces with integer translation is proposed. The notion of a generalized multiresolution structure of is also introduced. The construction of a generalized multireso-lution structure of Paley-Wiener subspaces of is investigated.

Abstract: Readers likely recall the following elementary geometric statement: if the sides of an angle α in a Euclidean plane are orthogonal to the sides of an angle β, then α=β or α+β=180°; in other words, |cos α|=|cos β|. A multidimensional generalization of this theorem relates to an interaction of skew symmetry and orthogonality features, i.e., properties of determinants and other multilinear functions, and Euclidean spaces; we presume readers are familiar with these features within the scope of a common university course, and on this basis we develop the tools necessary to prove this generalization. These tools have multiple applications. For examples, they allow us to extend a definition of the angle between straight lines or between hyperplanes to k-dimensional planes of ℝ n for every 0