Orthogonality Theorem Research Articles

An Example on Computing the Irreducible Representation of Finite Metacyclic Groups by Using Great Orthogonality Theorem Method

Representation theory is a study of real realizations of the axiomatic systems of abstract algebra. For any group, the number of possible representative sets of matrices is infinite, but they can all be reduced to a single fundamental set, called the irreducible representations of the group. This paper focuses on an example of finite metacyclic groups of class two of order 16. The irreducible representation of that group is found by using Great Orthogonality Theorem Method.

On Some Basic Theorems of Continuous Module Homomorphisms between Random Normed Modules

We first prove the resonance theorem, closed graph theorem, inverse operator theorem, and open mapping theorem for module homomorphisms between random normed modules by simultaneously considering the two kinds of topologies—the -topology and the locally -convex topology for random normed modules. Then, for the future development of the theory of module homomorphisms on complete random inner product modules, we give a proof with better readability of the known orthogonal decomposition theorem and Riesz representation theorem in complete random inner product modules under two kinds of topologies. Finally, to connect module homomorphism between random normed modules with linear operators between ordinary normed spaces, we give a proof with better readability of the known result connecting random conjugate spaces with classical conjugate spaces, namely, , where and are a pair of Hölder conjugate numbers with a random normed module, the random conjugate space of the corresponding (resp., ) space derived from (resp., ), and the ordinary conjugate space of

Mobility and kinematic simulations of cyclically symmetric deployable truss structures

Based on the mechanism of four-fold rigid origami, this study proposes a type of deployable truss structures that consist of repetitive basic parts and retain full cyclic symmetry in the folding/deployment process. On the basis of the irreducible representations and the great orthogonality theorem, symmetry-adapted analysis using group theory is described to identify the symmetry of mobility and kinematic behavior. Equivalent three-dimensional pin-jointed frameworks are employed for the symmetric structures. To verify that the structures can be foldable while retaining their full symmetries, numerical simulations on a series of structures with different symmetries and geometries are carried out. An artificial damping is introduced to stabilize the nonlinear folding behavior with singularity. Symmetry-adapted mobility analysis reveals that the structures of this type can be continuously folded with one degree-of-freedoms. Numerical simulations using the nonlinear iterative method accurately predict the folding behavior, as the results agree very well with the theoretic value.

Control theorem of a universal uniform-rotating magnetic vector for capsule robot in curved environment

For realizing non-contact steering swimming of a capsule robot in curved environment filled with viscous liquid, based on spatial orthogonal superposition theorem of alternating magnetic vectors, an innovative physical method is proposed, which employs three-axis orthogonal square Helmholtz coils fed with three phase sine currents to create a universal uniform magnetic spin vector as energy source. According to the antiphase sine current superposition theorem generalized in this paper, an effective control method for successively adjusting the orientation and the rotating direction of the universal magnetic spin vector is proposed. For validating its feasibility and controllability, three-axis Helmholtz coils, power source and an innovative capsule robot prototype were manufactured, experiments were conducted in both spiral pipe and animal intestine. It was demonstrated that the orientation and the rotational direction of the universal uniform-magnetic spin vector can be adjusted successively through digital control and steering swimming of the capsule robot in spiral intestine can be achieved successfully. The breakthrough of the universal rotating uniform-magnetic vector will push forward the development of modern physics and biomedical engineering

State estimation for discrete-time markov jump linear systems based on orthogonal projective theorem

In this paper, state estimation problem for discrete-time Markov jump linear systems is considered. Based on orthogonal projective theorem, a novel suboptimal algorithm for state estimate of discrete-time Markov jump linear systems in the sense of minimum mean square error estimate is proposed. The proposed suboptimal algorithm is recursive and finite-dimensionally computable. Computer simulations are carried out to evaluate the performance of the proposed suboptimal algorithm.

Criteria for the single-valued metric generalized inverses of multi-valued linear operators in banach spaces

Let X, Y be Banach spaces and M be a linear subspace in X × Y = {{x, y}|x ∈ X, y ∈ Y}. We may view M as a multi-valued linear operator from X to Y by taking M(x) = {y|{x, y} ∈ M}. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M. The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.

A novel suboptimal algorithm for state estimation of Markov jump linear systems

This paper is concerned with state estimation problem for Markov jump linear systems where the disturbances involved in the systems equations and measurement equations are assumed to be Gaussian noise sequences. Based on two properties of conditional expectation, orthogonal projective theorem is applied to the state estimation problem of the considered systems so that a novel suboptimal algorithm is obtained. The novelty of the algorithm lies in using orthogonal projective theorem instead of Kalman filters to estimate the state. A numerical comparison of the algorithm with the interacting multiple model algorithm is given to illustrate the effectiveness of the proposed algorithm.

Descriptor recursive estimation for multiple sensors with different delay rates

In this article, the problem of recursive estimation is studied for a class of descriptor systems measured by multiple sensors with different delay rates. The delay phenomenon occurs in a random way and the delay rate for each sensor is characterised by an individual binary switching sequence obeying a conditional probability distribution. The original descriptor system is transformed into two non-descriptor stochastic subsystems with autocorrelated and cross-correlated noises by using the singular-value decomposition and the state augmentation. Applying an innovation analysis approach and the orthogonal projection theorem, recursive estimators including filter, predictor and smoother are first derived for each subsystem and the process noise. Then, based on the newly obtained recursive state estimators and process noise estimators, the recursive filter, predictor and smoother are obtained for the original descriptor system measured by multiple sensors with different delay rates. A numerical simulation example is exploited to show the effectiveness of the proposed approaches.

Local–global principles for embedding of fields with involution into simple algebras with involution

In this paper we prove local–global principles for the existence of an embedding (E, σ) ↪ (A, τ) of a given global field E endowed with an involutive automorphism σ into a simple algebra A given with an involution τ in all situations except where A is a matrix algebra of even degree over a quaternion division algebra and τ is orthogonal (Theorem A of the introduction). Rather surprisingly, in the latter case we have a result which in some sense is opposite to the local–global principle, viz. algebras with involution locally isomorphic to (A, τ) are distinguished by their maximal subfields invariant under the involution (Theorem B of the introduction). These results can be used in the study of classical groups over global fields. In particular, we use Theorem B to complete the analysis of weakly commensurable Zariski-dense S -arithmetic groups in all absolutely simple algebraic groups of type different from D_4 which was initiated in our paper [23]. More precisely, we prove that in a group of type D_n , n even > 4 , two weakly commensurable Zariski-dense S -arithmetic subgroups are actually commensurable. As indicated in [23], this fact leads to results about length-commensurable and isospectral compact arithmetic hyperbolic manifolds of dimension 4n + 7 , with n ≥ 1 . The appendix contains a Galois-cohomological interpretation of our embedding theorems.

Thermodynamic coarsening of dislocation mechanics and the size-dependent continuum crystal plasticity

Starting from the standard coarsening of dislocation kinematics, we derive the size-dependent continuum crystal plasticity by systematic thermodynamic coarsening of dislocation mechanics. First, we observe that the energies computed from different kinematic descriptions are different. Then, we consider systems without boundary dissipation (relaxation) and derive the continuum approximation for the free energy of elastic–plastic crystals. The key elements are: the two-dimensional nature of dislocation pile-ups at interfaces, the localized nature of the coarsening error in energy, and, the orthogonal decomposition theorem for compatible and incompatible elastic strain fields. Once the energy landscape is defined, the boundary dissipation is estimated from the height of energy barriers. The characteristic lengths are the average slip plane spacing for each slip system. They may evolve through the double-cross slip mechanism. The theory features the slip-dependent interface free energy and interface dissipation for penetrable interfaces. The main constitutive parameters are derived from elasticity. The exception is the dependence of interface energy on slip plane orientation, which is determined from numerical results. The theory requires no higher order boundary conditions. The only novel boundary conditions are kinematic, involving slip relaxation on the two sides of an interface.

Chebyshev type lattice path weight polynomials by a constant term method

We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary ‘decorated’ weights as well as an arbitrary ‘background’ weight. Our CT theorem, like Viennot's lattice path theorem from which it is derived primarily by a change of variable lemma, is expressed in terms of orthogonal polynomials which in our applications of interest often turn out to be non-classical. Hence, we also present an efficient method for finding explicit closed-form polynomial expressions for these non-classical orthogonal polynomials. Our method for finding the closed-form polynomial expressions relies on simple combinatorial manipulations of Viennot's diagrammatic representation for orthogonal polynomials. In the course of the paper we also provide a new proof of Viennot's original orthogonal polynomial lattice path theorem. The new proof is of interest because it uses diagonalization of the transfer matrix, but gets around difficulties that have arisen in past attempts to use this approach. In particular we show how to sum over a set of implicitly defined zeros of a given orthogonal polynomial, either by using properties of residues or by using partial fractions. We conclude by applying the method to two lattice path problems important in the study of polymer physics as the models of steric stabilization and sensitized flocculation.

Schauder estimates for sub-elliptic equations

We establish Holder estimates of second derivatives for a class of sub-elliptic partial differential operators in \({\mathbb{R}^{N}}\) of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the Xj’s are smooth vector fields in \({\mathbb{R}^{N}}\), and aij is a uniformly elliptic matrix. It is assumed that the Xj’s satisfy homogeneity conditions with respect to a group of dilations δr which yield the existence of a composition law \({\circ}\) in \({\mathbb{R}^{N}}\) making the triplet \({\mathbb G=(\mathbb{R}^{N},\circ,\delta_{r})}\) an homogeneous Lie group on which the Xj’s are left translation invariant. The Holder norms are defined in terms of this composition law. The main tools used are the Taylor formula for smooth functions on \({\mathbb{G}}\), some properties of the corresponding Taylor polynomials, and an orthogonality theorem that extends to homogeneous Lie groups a classical theorem of Calderon and Zygmund in the Euclidean setting.

The cardinal orthogonal scaling function and sampling theorem in the wavelet subspaces

In this paper, we derive that there is a relation between the lowpass filter coefficient and wavelet’s samples in its integer points when a scaling function is a cardinal orthogonal scaling function. And we give some examples in which the lowpass filter coefficients are constructed from the wavelets. Then, we discuss the symmetry property of cardinal orthogonal scaling function, and give some useful characterizations. Therefore, we construct a family of cardinal orthogonal scaling functions with exponential decay and higher approximation order. Furthermore, some examples with exponential decay and higher approximation order are given, and they will testify our results. In the end, we deduce that there no exists a cardinal orthogonal scaling function with exponential decay, high approximation order and symmetry property in this family of cardinal orthogonal scaling functions constructed in our paper.

Irreducible Representations of Groups of Order 8

Irreducible representations of a group provide the ways for labelling orbitals, determining molecular orbitals formation and determining vibrational motions for a molecule. A set of irreducible representations represents the ways a particular bond, atom or sets of atoms may respond to a given set of symmetry operations. In this paper, the irreducible representations of all groups of order 8, namely $D_4$, $Q$, $C_8$, $C_2 \times C_4$ and $C_2 \times C_2 \times C_2$ are obtained using Burnside method and Great Orthogonality Theorem method. Then, comparisons of the two methods are made. Keywords: Irreducible representation; group; Burnside method; Great Orthogonality Theorem method.

Finite-energy solutions of mixed 3d div-curl systems

This paper describes the existence and representation of certain finite energy ( L 2 (L^2 -)solutions of weighted div-curl systems on bounded 3D regions with C 2 C^2 -boundaries and mixed boundary data. Necessary compatibility conditions on the data for the existence of solutions are described. Subject to natural integrability assumptions on the data, it is then shown that there exist L 2 L^2 -solutions whenever these compatibility conditions hold. The existence results are proved by using a weighted orthogonal decomposition theorem for L 2 L^2 -vector fields in terms of scalar and vector potentials. This representation theorem generalizes the classical Hodge-Weyl decomposition. With this special choice of the potentials, the mixed div-curl problem decouples into separate problems for the scalar and vector potentials. Variational principles for the solutions of these problems are described. Existence theorems, and some estimates, for the solutions of these variational principles are obtained. The unique solution of the mixed system that is orthogonal to the null space of the problem is found and the space of all solutions is described. The second part of the paper treats issues concerning the non-uniqueness of solutions of this problem. Under additional assumptions, this space is shown to be finite dimensional and a lower bound on the dimension is described. Criteria that prescribe the harmonic component of the solution are investigated. Extra conditions that determine a well-posed problem for this system on a simply connected region are given. A number of conjectures regarding the results for bounded regions with handles are stated.

A new proof of the colored branch Theorem

The colored branch theorem is a result from graph theory that has been described first by Minty. It states the existence of certain meshes and cuts in a graph, whose edges are colored red, green and blue, respectively. The theorem, sometimes also referred to as the lemma of the colored arcs, can be utilized to give short and elegant proofs of many other theorems in graph and circuit theory and has therefore turned out to be of vital importance. We present a new set theoretic proof of the colored branch theorem, that reveals its relationship to the orthogonality theorem, another well-known fundamental result about meshes and cuts in a graph.

Influence of a mesoscopic bath on quantum coherence

For a quantum double well system interacting with a mesoscopic bath, it is shown that a ``bath'' consisting of a single particle is sufficient to substantially reduce tunneling between the two wells. This is demonstrated by considering an ammonia molecule in the center of a ring; in addition to halving the maser line frequency, there is an increase in intensity by four orders of magnitude. The tunneling varies nonmonotonically with the number $N$ of electrons in the ring, reflecting the changing electronic correlations. Although the tunneling is reduced for small $N$, it turns around and grows to its free value for large $N$. This is shown to not violate Anderson's orthogonality theorem. Experimental implementations are discussed.

Response of a tethered aerostat to simulated turbulence

Aerostats are lighter-than-air vehicles tethered to the ground by a cable and used for broadcasting, communications, surveillance, and drug interdiction. The dynamic response of tethered aerostats subject to extreme atmospheric turbulence often dictates survivability. This paper develops a theoretical model that predicts the planar response of a tethered aerostat subject to atmospheric turbulence and simulates the response to 1000 simulated hurricane scale turbulent time histories. The aerostat dynamic model assumes the aerostat hull to be a rigid body with non-linear fluid loading, instantaneous weathervaning for planar response, and a continuous tether. Galerkin’s method discretizes the coupled aerostat and tether partial differential equations to produce a non-linear initial value problem that is integrated numerically given initial conditions and wind inputs. The proper orthogonal decomposition theorem generates, based on Hurricane Georges wind data, turbulent time histories that possess the sequential behavior of actual turbulence, are spectrally accurate, and have non-Gaussian density functions. The generated turbulent time histories are simulated to predict the aerostat response to severe turbulence. The resulting probability distributions for the aerostat position, pitch angle, and confluence point tension predict the aerostat behavior in high gust environments. The dynamic results can be up to twice as large as a static analysis indicating the importance of dynamics in aerostat modeling. The results uncover a worst case wind input consisting of a two-pulse vertical gust.

Metric generalized inverse for linear manifolds and extremal solutions of linear inclusion in Banach spaces

Let X, Y be Banach spaces and M a linear manifold in X×Y={{x,y}∣x∈X, y∈Y} . The central problem which motivates many of the concepts and results of this paper is the problem of characterization and construction of all extremal solutions of a linear inclusion y∈ M( x). First of all, concept of metric operator parts and metric generalized inverses for linear manifolds are introduced and investigated, and then, characterizations of the set of all extremal or least extremal solutions in terms of metric operator parts and metric generalized inverses of linear manifolds are given by the methods of geometry of Banach spaces. The principal tool in this paper is the generalized orthogonal decomposition theorem in Banach spaces.

New variants of the criss-cross method for linearly constrained convex quadratic programming

In this paper, S. Zhang's [Eur. J. Oper. Res. 116 (1999) 607] new and more flexible criss-cross type algorithms (with LIFO and most-often-selected-variable pivot rules) are generalized for linearly constrained convex primal–dual quadratic programming problems. These criss-cross type algorithms are different from the one described in Klafszky and Terlaky [Math. Oper. und Stat. Ser. Optim. 24 (1992) 127]. Even though the finiteness proof of these new criss-cross type algorithms is similar to the original one for the algorithm of Klafszky and Terlaky (in the sense that both these proofs are based on the orthogonality theorem), more cases have to be considered due to the flexibility of pivot (LIFO/most-often-selected-variable) rules, which requires a deeper and more careful analysis. When the primal–dual problem is a linear programming problem (no quadratic terms in the objective function), the structure of the corresponding linear complementarity problem is simpler (i.e. the matrix of the problem is skew-symmetric). For such problem pairs, our proof of finiteness simplifies to the proof of Illés and Mészáros' [Yugoslav J. Oper. Res. 11 (2001) 17] and provides a new finiteness proof for S. Zhang's criss-cross type algorithms.