Backward Shift Research Articles

KCC Analysis of a One-Dimensional System During Catastrophic Shift of the Hill Function: Douglas Tensor in the Nonequilibrium Region

This paper considers the stability of a one-dimensional system during a catastrophic shift described by the Hill function. Because the shifting process goes through a nonequilibrium region, we applied the theory of Kosambi, Cartan, and Chern (KCC) to analyze the stability of this region based on the differential geometrical invariants of the system. Our results show that the Douglas tensor, one of the invariants in the KCC theory, affects the robustness of the trajectory during a catastrophic shift. In this analysis, the forward and backward shifts can have different Jacobi stability structures in the nonequilibrium region. Moreover, the bifurcation curve of the catastrophic shift can be interpreted geometrically, as the solution curve where the nonlinear connection and the deviation curvature become zero. KCC analysis also shows that even if the catastrophic pattern itself is similar, the stability structure in the nonequilibrium region is different in some cases, from the viewpoint of the Douglas tensor.

Toeplitz kernels and the backward shift

In this paper we study the kernels of Toeplitz operators on both the scalar and the vector-valued Hardy space for 1<p<∞. We show existence of a minimal kernel for any element of the vector-valued Hardy space and we determine a symbol for the corresponding Toeplitz operator. In the scalar case we give an explicit description of a maximal function for a given Toeplitz kernel which has been decomposed in to a certain form. In the vectorial case we show not all Toeplitz kernels have a maximal function and in the case of p=2 we find the exact conditions for when a Toeplitz kernel has a maximal function. For both the scalar and vector-valued Hardy space we study the minimal Toeplitz kernel containing multiple elements of the Hardy space, which in turn allows us to deduce an equivalent condition for a function in the Smirnov class to be cyclic for the backward shift.

Representing Kernels of Perturbations of Toeplitz Operators by Backward Shift-Invariant Subspaces

It is well known that the kernel of a Toeplitz operator is nearly invariant under the backward shift $S^*$. This paper shows that kernels of finite-rank perturbations of Toeplitz operators are nearly $S^*$-invariant with finite defect. This enables us to apply a recent theorem by Chalendar--Gallardo--Partington to represent the kernel in terms of backward shift-invariant subspaces, which we identify in several important cases.

Properties of Shape-Invariant Tridiagonal Hamiltonians

As is known, a nonnegative-definite Hamiltonian H that has a tridiagonal matrix representation in a basis set allows defining forward (and backward) shift operators that can be used to determine the matrix representation of the supersymmetric partner Hamiltonian H(+) in the same basis. We show that if the Hamiltonian is also shape-invariant, then the matrix elements of the Hamiltonian are related such that the energy spectrum is known in terms of these elements. It is also possible to determine the matrix elements of the hierarchy of supersymmetric partner Hamiltonians. Moreover, we derive the coherent states associated with this type of Hamiltonian and illustrate our results with examples from well-studied shape-invariant Hamiltonians that also have a tridiagonal matrix representation.

Effect of decompression range on decompression limit of cervical laminoplasty.

BackgroundsCervical posterior decompression surgery is used to relieve ventral compression indirectly by incorporating a backward shift of the spinal cord, and this indirect decompression is bound to be limited. This study aimed to determine the decompression limit of posterior surgery and the effect of the decompression range.MethodsWe retrospectively reviewed the data of 129 patients who underwent cervical open-door laminoplasty through 2008 to 2012 and were grouped as follows: C4–C7 (n = 11), C3–C6 (n = 61), C3–C7 (n = 32), and C2–C7 (n = 25). According to the relative location of spinal levels within a decompression range, the type of decompression at a given level was categorized as external decompression (ED; achieved at the levels located immediately external to the decompression range margin), internal decompression (ID; achieved at the levels located immediately internal to the decompression range margin), and central decompression (CD; achieved at the levels located in the center, far from the decompression range margin). The vertebral–cord distance (VCD) was used to evaluate the decompression limit. The C2–C7 angle and VCD on post-operative magnetic resonance images were analyzed and compared between groups. The relationship between VCD and decompression type was analyzed. Moreover, the relationship between the magnitude of the ventral compressive factor and the probability of post-operative residual compression at each level for different decompression ranges was studied.ResultsThere was no significant kyphosis in cervical curvature (> −5°), and there was no significant difference among the groups (F = 2.091, P = 0.105). The VCD of a specific level depended on the decompression type of the level and followed this pattern: ED < ID < CD (P < 0.05). The decompression type of a level was sometimes affected by the decompression range. For a given magnitude of the ventral compressive factor, the probability of residual compression was lower for the group with the larger VCD at this level.ConclusionsOur study suggests that the decompression range affected the decompression limit by changing the decompression type of a particular level. For a given cervical spinal level, the decompression limit significantly varied with decompression type as follows: ED < ID < CD. CD provided maximal decompression limit for a given level. A reasonable range of decompression could be determined based on the relationship between the magnitude of the ventral compressive factor and the decompression limits achieved by different decompression ranges.

Backward shift invariant subspaces in reproducing kernel Hilbert spaces

In this note, we describe the backward shift invariant subspaces for an abstract class of reproducing kernel Hilbert spaces. Our main result is inspired by a result of Sarason concerning de Branges-Rovnyak spaces (the non-extreme case). Furthermore, we give new applications in the context of the range space of co-analytic Toeplitz operators and sub-Bergman spaces.

Algebras of frequently hypercyclic vectors

AbstractWe show that the multiples of the backward shift operator on the spaces , , or c0, when endowed with coordinatewise multiplication, do not possess frequently hypercyclic algebras. More generally, we characterize the existence of algebras of ‐hypercyclic vectors for these operators. We also show that the differentiation operator on the space of entire functions, when endowed with the Hadamard product, does not possess frequently hypercyclic algebras. On the other hand, we show that for any frequently hypercyclic operator T on any Banach space, is algebrable for a suitable product, and in some cases it is even strongly algebrable.

A Beurling theorem for almost-invariant subspaces of the shift operator

A complete characterization of nearly-invariant subspaces of finite defect for the backward shift operator acting on the Hardy space is provided in the spirit of Hitt and Sarason's theorems. As a corollary we describe the almost-invariant subspaces for the shift and its adjoint.

Algebrability of the set of hypercyclic vectors for backward shift operators

We study the existence of algebras of hypercyclic vectors for weighted backward shifts on Fréchet sequence spaces that are algebras when endowed with coordinatewise multiplication or with the Cauchy product. As a particular case, we obtain that the sets of hypercyclic vectors for Rolewicz's and MacLane's operators are algebrable.

On the direct sum of two bounded linear operators and subspace-hypercyclicity

AbstractIn this paper, we show that if the direct sum of two opera-tors is subspace-hypercyclic (satisfies subspace hypercyclic crite-rion), then both operators are subspace-hypercyclic (satisfy sub-space hypercyclic criterion). Moreover, if an operator T satisfiessubspace-hypercyclic criterion, then so T⊕T does. Also, we obtainthat under certain conditions, if T ⊕T is hypercyclic then T satis-fies subspace-hypercyclic criterion and, the subspace-hypercyclicoperators satisfy subspace-hypercyclic criterion which gives the“subspace-hypercyclic” analogue of Theorem 2.3. (in Hereditar-ily hypercyclic operators. J. Funct. Anal., 167:94–112, 1999 by P.B´es and A. Peris). 1 Introduction A bounded linear operator T on a separable Hilbert space H is hyper-cyclic if there is a vector x ∈ H such that Orb(T,x) = {T n x : n ≥ 0}is dense in H, such a vector x is called hypercyclic for T. The firstexample of a hypercyclic operator on a Banach space was constructedby Rolewicz in 1969 [11]. He showed that if B is the backward shift onl

Cowen-Douglas function and its application on chaos

In this paper, on $${\mathbb {D}}$$ we define Cowen–Douglas function introduced by the Cowen–Douglas operator $$M_\phi ^*$$ on Hardy space $${\mathcal {H}}^2({\mathbb {D}})$$ , moreover, we give a necessary and sufficient condition to determine when $$\phi$$ is a Cowen–Douglas function, where $$\phi \in {\mathcal {H}}^\infty ({\mathbb {D}})$$ and $$M_{\phi }$$ is the associated multiplication operator on $${\mathcal {H}}^{2}({\mathbb {D}})$$ . Then, we give some applications of Cowen–Douglas function on chaos, such as its application on the inverse problem of chaos for $$\phi (T)$$ , where $$\phi$$ is a Cowen–Douglas function and T is the backward shift operator on the Hilbert space $${\mathcal {L}}^2({\mathbb {N}})$$ .

Reiterative $$m_{n}$$-Distributional Chaos of Type s in Fréchet Spaces

The main aim of this paper is to consider various notions of (dense) $$m_{n}$$ -distributional chaos of type s and (dense) reiterative $$m_{n}$$ -distributional chaos of type s for general sequences of linear not necessarily continuous operators in Frechet spaces. Here, $$(m_{n})$$ is an increasing sequence in $$[1,\infty )$$ satisfying $$\liminf _{n\rightarrow \infty }\frac{m_{n}}{n}>0$$ and s could be $$0,1,2,2+,2\frac{1}{2},3,1+,2-,2_{Bd},2_{Bd}+$$ . We investigate $$m_{n}$$ -distributionally chaotic properties and reiteratively $$m_{n}$$ -distributionally chaotic properties of some special classes of operators like weighted forward shift operators and weighted backward shift operators in Frechet sequence spaces, considering also continuous analogues of introduced notions and some applications to abstract partial differential equations.

N-hypercontractivity and similarity of Cowen-Douglas operators

When the backward shift operator on a weighted space Hw2={f=∑j=0∞ajzj:∑j=0∞|aj|2wj<∞} is an n-hypercontraction, we prove that the weights must satisfy the inequalitywj+1wj≤1+jn+j. As an application of this result, it is shown that such an operator cannot be subnormal. We also give an example to illustrate the important role that the n-hypercontractivity assumption plays in determining the similarity of Cowen-Douglas operators in terms of the curvatures of their eigenvector bundles.

The Forward and Backward Shift on the Lipschitz Space of a Tree

We initiate the study of the forward and backward shifts on the Lipschitz space of an undirected tree, $$\mathcal {L}$$, and on the little Lipschitz space of an undirected tree, $$\mathcal {L}_0$$. We determine that the forward shift is bounded both on $$\mathcal {L}$$ and on $$\mathcal {L}_0$$ and, when the tree is leafless, it is an isometry; we also calculate its spectrum. For the backward shift, we determine when it is bounded on $$\mathcal {L}$$ and on $$\mathcal {L}_0$$, we find the norm when the tree is homogeneous, we calculate the spectrum for the case when the tree is homogeneous, and we determine, for a general tree, when it is hypercyclic.

Invariance of distributional chaos for backward shifts

A sufficient and necessary condition ensuring that the backward shift operator on the K\"{o}the sequence space admits an invariant distributionally $\varepsilon$-scrambled set for some $\varepsilon>0$ is obtained, improving the main results in [F. Mart\'{\i}nez-Gim\'{e}nez, P. Oprocha, A. Peris, J. Math. Anal. Appl., {\bf 351} (2009), 607--615].

Li-Yorke chaotic eigen set of the backward shift operator on ℓ2(𝕅)

Abstract In this paper, we prove our main result that the Li-Yorke chaotic eigen set of a positive integer multiple of the backward shift operator on ℓ2 (𝕅) is a disk in the complex plane 𝔺 and the union of such Li-Yorke chaotic eigen set’s is the whole complex plane 𝔺.

A novel constructive procedure to low-order Fornasini–Marchesini model realization

In this paper, a novel constructive procedure is proposed for the realization of a low-order Fornasini–Marchesini state-space model for a given multidimensional system. In particular, essential properties for the constructions of a backward shift space and a resolvent invariant space associated with the Gleason’s problem from a given transfer function matrix are investigated and some new sufficient conditions for such constructions are developed. Then, based on these conditions, a systematic constructive procedure is proposed for the multidimensional Fornasini–Marchesini model realization. It turns out that the new constructive procedure can generate Fornasini–Marchesini models with much lower orders and even the minimal realizations for a much larger class of multidimensional systems than the existing methods. Nontrivial examples are also provided to illustrate the basic idea as well as the effectiveness of the proposed procedure.

Kernels of Toeplitz Operators and Rational Interpolation

The kernel of a Toeplitz operator on the Hardy class H2 in the unit disk is a nearly invariantsubspace of the backward shift operator, and, by D. Hitt’s result, it has the form g · Kω where ω is an inner function, Kω = H2 ⊝ ωH2, and g is an isometric multiplier on Kω. We describe the functions ω and g for the kernel of the Toeplitz operator with symbol .$$ \overline{\theta}\varDelta $$ where θ is an inner function and Δ is a finite Blaschke product.

Structure and Dynamics of a Site-Specific Labeled Fc Fragment with Altered Effector Functions.

Antibody-drug conjugates (ADCs) are a class of biotherapeutic drugs designed as targeted therapies for the treatment of cancer. Among the challenges in generating an effective ADC is the choice of an effective conjugation site on the IgG. One common method to prepare site-specific ADCs is to engineer solvent-accessible cysteine residues into antibodies. Here, we used X-ray diffraction and hydrogen-deuterium exchange mass spectroscopy to analyze the structure and dynamics of such a construct where a cysteine has been inserted after Ser 239 (Fc-239i) in the antibody heavy chain sequence. The crystal structure of this Fc-C239i variant at 0.23 nm resolution shows that the inserted cysteine structurally replaces Ser 239 and that this causes a domino-like backward shift of the local polypeptide, pushing Pro 238 out into the hinge. Proline is unable to substitute conformationally for the wild-type glycine at this position, providing a structural reason for the previously observed abolition of both FcγR binding and antibody-dependent cellular cytotoxicity. Energy estimates for the both the FcγR interface (7 kcal/mol) and for the differential conformation of proline (20 kcal/mol) are consistent with the observed disruption of FcγR binding, providing a quantifiable case where strain at a single residue appears to disrupt a key biological function. Conversely, the structure of Fc-C239i is relatively unchanged at the intersection of the CH2 and CH3 domains; the site known to be involved in binding of the neonatal Fc receptor (FcRn), and an alignment of the Fc-C239i structure with an Fc structure in a ternary Fc:FcRn:HSA (human serum albumin) complex implies that these favorable contacts would be maintained. Hydrogen deuterium exchange mass spectroscopy (HDX-MS) data further suggest a significant increase in conformational mobility for the Fc-C239i protein relative to Fc that is evident even far from the insertion site but still largely confined to the CH2 domain. Together, the findings provide a detailed structural and dynamic basis for previously observed changes in ADC functional binding to FcγR, which may guide further development of ADC designs.

Independence of Vector-Valued Functions Associated with Kernels of Block Hankel Operators

For a block Hankel operator $$H_\Phi $$ with a matrix function symbol $$\Phi \in L^2_{M_{n\times m}}$$ , it is well-known that $$\ker H_\Phi = \Theta H^2_{\mathbb {C}^r}$$ for a natural number r and an $$m\times r$$ matrix inner function $$\Theta $$ if $$\ker H_\Phi $$ is nonempty. It will be shown that the size of the matrix inner function $$\Theta $$ associated with $$\ker H_\Phi $$ is closely related with a certain degree of independence of the column vectors of $$\Phi $$ , which will be defined in this paper. As an important application, the shape of shift invariant, or, backward shift invariant subspaces of $$H^2_{\mathbb {C}^n}$$ generated by finite elements will be studied.