A trace formula for isometric pairs

It is well known that for every isometry , This fact for the shift operator is a basis for many important developments in operator theory and topology. In this paper we prove an analogous formula for a pair of isometries , namely

where is the complete anti-symmetric sum and is the Fredholm index of the pair . The major tool is what we call the fringe operator. Two examples are considered.

     

A proper generalized decomposition approach for optical flow estimation

This paper introduces the use of the proper generalized decomposition (PGD) method for the optical flow (OF) problem in a classical framework of Sobolev spaces, ie, optical flow methods including a robust energy for the data fidelity term together with a quadratic penalizer for the regularization term. A mathematical study of PGD methods is first presented for general regularization problems in the framework of (Hilbert) Sobolev spaces, and their convergence is then illustrated on OF computation. The convergence study is divided in two parts: (a) the weak convergence based on the Brézis-Lieb decomposition and (b) the strong convergence based on a growth result on the sequence of descent directions. A practical PGD-based OF implementation is then proposed and evaluated on freely available OF data sets. The proposed PGD-based OF approach outperforms the corresponding non-PGD implementation in terms of both accuracy and computation time for images containing a weak level of information, namely, low image resolution and/or low signal-to-noise ratio (SNR).     

The isometric extension of “into” mappings on unit spheres of AL-spaces

In this paper, we show that if V0 is an isometric mapping from the unit sphere of an AL-space onto the unit sphere of a Banach space E, then V0 can be extended to a linear isometry defined on the whole space.     

Invariant decomposition of functions with respect to commuting invertible transformations

Consider arbitrary elements. We characterize those functions that decompose into the sum of -periodic functions, i.e., with . We show that has such a decomposition if and only if for all partitions with consisting of commensurable elements with least common multiples one has .

Actually, we prove a more general result for periodic decompositions of functions defined on an Abelian group ; in fact, we even consider invariant decompositions of functions with respect to commuting, invertible self-mappings of some abstract set .

We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.

     

The generalized triangular decomposition

Given a complex matrix , we consider the decomposition , where is upper triangular and and have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where is an upper triangular matrix with the eigenvalues of on the diagonal. We show that any diagonal for can be achieved that satisfies Weyl's multiplicative majorization conditions:

where is the rank of , is the -th largest singular value of , and is the -th largest (in magnitude) diagonal element of . Given a vector which satisfies Weyl's conditions, we call the decomposition , where is upper triangular with prescribed diagonal , the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.

     

The single-valued extension property for sums and products of commuting operators

It is shown that the sum and the product of two commuting Banach space operators with Dunford's property (C) have the single-valued extension property.     

On the strong cell decomposition property for weakly o‐minimal structures

We consider a class of weakly o‐minimal structures admitting an o‐minimal style cell decomposition, for which one can construct certain canonical o‐minimal extension. The paper contains several fundamental facts concerning the structures in question. Among other things, it is proved that the strong cell decomposition property is preserved under elementary equivalences. We also investigate fiberwise properties (of definable sets and definable functions), definable equivalence relations, and conditions implying elimination of imaginaries.     

A space decomposition method for parabolic equations

A convergence proof is given for an abstract parabolic equation using general space decomposition techniques. The space decomposition technique may be a domain decomposition method, a multilevel method, or a multigrid method. It is shown that if the Euler or Crank–Nicolson scheme is used for the parabolic equation, then by suitably choosing the space decomposition, only O(| log τ |) steps of iteration at each time level are needed, where τ is the time-step size. Applications to overlapping domain decomposition and to a two-level method are given for a second-order parabolic equation. The analysis shows that only a one-element overlap is needed. Discussions about iterative and noniterative methods for parabolic equations are presented. A method that combines the two approaches and utilizes some of the good properties of the two approaches is tested numerically. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 27–46, 1998     

Taylor spectral invariance for crisscross commuting pairs on Banach spaces

A theorem on the commuting property of Taylor's spectrum for crisscross commuting pairs is proved in this paper.

     

A-调和方程很弱解的正则性

In this paper, the following result is given by using Hodge decomposition and weak reverse H(o)lder inequality: For every r1 with p-(2n 1100n2p(23 n/(p-1) 1)b/a)-1＜r1＜p,there exists the exponent r2 = r2(n,r1,p) ＞ p, such that for every very weak solution u ∈W1r1,loc(Ω) to A-harmonic equation, u also belongs to W1r2,loc(Ω) . In particular, u is the weak solution to A-harmonic equation in the usual sense.     

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.     

Generalized Kato decomposition, single-valued extension property and approximate point spectrum

In this paper, we define the generalized Kato spectrum of an operator, and obtain that the generalized Kato spectrum differs from the semi-regular spectrum on at most countably many points. We study the localized version of the single-valued extension property at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point λ₀∈C in the case that λ₀I−T admits a generalized Kato decomposition. From this characterization we shall deduce several results on cluster points of some distinguished parts of the spectrum.     

Explicitly given pairs of dual frames with compactly supported generators and applications to irregular B-splines

We consider systems of functions appearing by letting a class of modulations act on a countable collection of functions. These systems correspond to shift-invariant systems, considered on the Fourier side. We provide sufficient conditions for the system to be a frame, as well as an explicit construction of a class of frames and associated duals. We use the result to construct frames based on B-splines with knot sequences satisfying a natural condition, as well as explicitly given duals.     

Hyponormality and subnormality for powers of commuting pairs of subnormal operators

Let H₀ (respectively H_∞) denote the class of commuting pairs of subnormal operators on Hilbert space (respectively subnormal pairs), and for an integer k?1 let Hk denote the class of k-hyponormal pairs in H₀. We study the hyponormality and subnormality of powers of pairs in Hk. We first show that if (T₁,T₂)∈H₁, the pair may fail to be in H₁. Conversely, we find a pair (T₁,T₂)∈H₀ such that but (T₁,T₂)∉H₁. Next, we show that there exists a pair (T₁,T₂)∈H₁ such that is subnormal (for all m,n?1), but (T₁,T₂) is not in H_∞; this further stretches the gap between the classes H₁ and H_∞. Finally, we prove that there exists a large class of 2-variable weighted shifts (T₁,T₂) (namely those pairs in H₀ whose cores are of tensor form (cf. Definition 3.4)), for which the subnormality of and does imply the subnormality of (T₁,T₂).     

A note on the local theta correspondence for unitary similitude dual pairs

Following Roberts? work in the case of orthogonal-symplectic similitude dual pairs, we study the local theta correspondence for unitary similitude dual pairs over a p-adic field.     

Computing a logarithm of a unitary matrix with general spectrum

We analyze an algorithm for computing a skew‐Hermitian logarithm of a unitary matrix and also skew‐Hermitian approximate logarithms for nearly unitary matrices. This algorithm is very easy to implement using standard software, and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near ? 1, lead to very non‐Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force skew‐Hermitian output creates accuracy issues, which are avoided by the considered algorithm. A modification is introduced to deal properly with the J‐skew‐symmetric unitary matrices. Applications to numerical studies of topological insulators in two symmetry classes are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Wavelet bases for a set of commuting unitary operators

Let (U=U ₁, ...,U _d ) be an orderedd-tuple of distinct, pairwise commuting, unitary operators on a complex Hilbert space , and letX:={x ₁, ...,x _r } such that is a Riesz basis of the closed linear spanV ₀ of . Suppose there is unitary operatorD on such thatV ₀ D V ₀ =:V ₁ andU ⁿ D=DU ^An for alln ^d , whereA is ad ×d matrix with integer entries and := det(A) 0. Then there is a subset inV ₁, withr( – 1) vectors, such that is a Riesz basis ofW ₀, the orthogonal complement ofV ₀ inV ₁. The resulting multiscale and decomposition relations can be expressed in a Fourier representation by one single equation, in terms of which the duality principle follows easily. These results are a consequence of an extension, to a set of commuting unitary operators, of Robertson's Theorems on wandering subspace for a single unitary operator [24]. Conditions are given in order that is a Riesz basis ofW ₀. They are used in the construction of a class of linear spline wavelets on a four-direction mesh.     

A delay decomposition approach to stability analysis of neutral systems with time-varying delay

This paper presents novel stability criteria for neutral systems with time-varying delay. By developing a delayed decomposition approach, information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities (LMIs). Then, based on the Lyapunov method, delay-dependent stability criteria are devised by taking the relationship between terms in the Leibniz-Newton formula into account. Criteria are derived in terms of LMIs, which can be easily solved by using various convex optimization algorithms. Three illustrative numerical examples are given to show less conservatism of our obtained results and the effectiveness of the proposed method.