Narrow Results

This paper addresses the dilation problem on (dual) frames for Krein spaces. We characterize Riesz bases for Krein spaces and equivalence ($(J_1,J_2)$-unitary equivalence) between frames for Krein spaces; prove that every frame (dual frame pair) for a Krein space can be dilated to a Riesz basis (dual Riesz basis pair) for a larger Krein space, and that the corresponding $J$-orthogonal complementary frame ($J$-joint complementary frame) is unique up to equivalence ($(J_1,J_2)$-joint equivalence). Also we illustrate that two equivalent Parseval frames for Krein spaces need not be $(J_1,J_2)$-unitarily equivalent and that not every Parseval frame can be dilated to a $J$-orthonormal basis for a larger Krein space, and derive a result on matrices of finite size as application.

This paper addresses the weaving theory of operator-valued frames (OPV-frames). We give a rigorous proof of the equivalence between “weakly woven” and “woven” of OPV-frames; estimate the optimal universal OPV-frame bounds of all weavings; and prove that OPV-frame and its dual OPV-frame are woven. Also, using examples, we show that “woven property” does not have transmissibility, and that a collection of pairwise weaving frames need not be woven. Finally, we give a sufficient condition for a collection of adjacent weaving OPV-frames to be woven.

We show how to map gravitational theories formulated in the Jordan frame to the Einstein frame at the quantum field theoretical level considering quantum fields in curved space–time. As an example, we consider gravitational theories in the Jordan frame of the type F(ϕ, R) = f(ϕ)R-V(ϕ) and perform the map to the Einstein frame. Our results can easily be extended to any gravitational theory. We consider the Higgs inflation model as an application of our results.

Motivated by frame-vector for a unitary system, we study a class of cyclic operators on a separable Hilbert space which is called frame-cyclic operators. The orbit of such an operator on some vector, namely frame-cyclic vector, is a frame. Some properties of these operators on finite- and infinite-dimensional Hilbert spaces and their relations with cyclic and hypercyclic operators are established. A lower and upper bound for the norm of a self-adjoint frame-cyclic operator is obtained. Also, construction of the set of frame-cyclic vectors is considered. Finally, we deal with Kato’s approximation of frame-cyclic operators and discuss their frame-cyclic properties.

In order to achieve optimally sparse approximations of signals exhibiting anisotropic singularities, the shearlet systems that are systems of functions generated by one generator with dilation, shear transformation and translation operators applied to it were introduced. In this paper, we will construct the shearlet systems that are not only Parseval frames for $L^2({ℝ}^2)$ but they are also obtained from an $AB$-MRA associated with wavelet multiresolution, and by using this approach, we obtain the corresponding filters for these systems. For this purpose, the tensor product of the corresponding wavelet scaling function and a compact support bump function is used to construct the scaling function associated with the shearlet.

In this research, a seismic retrofitting method for chevron-braced frames (CBFs) is proposed. The key idea here is to prevent the buckling of the chevron braces via a conventional construction technique that involves a hysteretic energy-dissipating element installed between the braces and the connected beam. The energy-dissipating element is designed to yield prior to buckling of the braces, thereby preventing the lateral stiffness and strength degradation of the CBF caused by buckling, while effectively dissipating the earthquake input energy. Nonlinear static pushover, time history and damage analyses of the CBF and retrofitted CBF (RCBF) are conducted to assess the performance of the RCBF compared with that of the CBF. The results of the analyses reveal that the proposed retrofitting method can efficiently alleviate the detrimental effects of earthquakes on the CBF. The RCBF has a more stable lateral force–deformation behavior with enhanced energy dissipation capability than the CBF. For small-to-moderate intensity ground motions, the maximum interstory drift of the RCBF is close to that of the CBF. But, for high intensity ground motions, it is considerably smaller than that of the CBF. Compared with the CBF under medium-to-large intensity ground motions, the RCBF experiences significantly less damage due to prevention of buckling of the braces.

This paper presents the results of dynamic responses and fire resistance of concrete-filled steel tubular (CFST) frame structures in fire conditions by using the nonlinear finite element method. Both strength and stability criteria are considered in the collapse analysis. The frame structures are constructed with circular CFST columns and steel beams of I-sections. In order to validate the finite element solutions, the numerical results are compared with those from a fire resistance test on CFST columns. The finite element model is then adopted to simulate the behavior of frame structures in fire. The structural responses of the frames, including the critical temperature and fire-resisting limit time, are obtained for the ISO-834 standard fire. Parametric studies are carried out to show their influence on the load capacity of the frame structures in fire. Suggestions and recommendations are presented for possible adoption in future construction and design of similar structures.

Low-cost robotic welding and wide availability of high strength steel plates of grades over 500MPa make the use of tapered members an economical alternative to conventional prismatic members for modern steel structures, as experienced by the authors in some practical projects in Hong Kong and Macau. This paper proposes a new and efficient numerical method for modal and elastic time-history analysis of the frames with tapered sections. A series of non-prismatic elements is derived on the basis of analytical expressions, and the exact consistent mass and tangent stiffness matrices are formulated. Five common types of tapered sections for practical applications, namely the circular solid, circular hollow, rectangular solid, rectangular hollow and doubly symmetric-I sections, are studied. Contrary to the conventional method using the approximate assumptions for the section properties along the member length, this research analytically expresses the flexural rigidity and cross-sectional area for the stiffness and mass matrices of an element. Further, the techniques for obtaining the dynamic performances, such as natural vibrations and time-history responses, of non-prismatic members are investigated. Finally, three examples are conducted for validating and verifying the accuracy of the proposed formulations. The present work can be used in the dynamic response analysis of frame structures with tapered sections in seismic zones.

Presented herein is a matrix method for buckling analysis of general frames based on the Hencky bar-chain model comprising of rigid segments connected by hinges with elastic rotational springs. Unlike the conventional matrix method of structural analysis based on the Euler–Bernoulli beam theory, the Hencky bar-chain model (HBM) matrix method allows one to readily handle the localized changes in end restraint conditions or localized structural changes (such as local damage or local stiffening) by simply tweaking the spring stiffnesses. The developed HBM matrix method was applied to solve some illustrative example problems to demonstrate its versatility in solving the buckling problem of beams and frames with various boundary conditions and local changes. It is hoped that this easy-to-code HBM matrix method will be useful to engineers in solving frame buckling problems.

The critical pressure is determined for a trapezoidal vault with rigid members and semi-rigid joints. For maximal volume enclosed per boundary length, it is found that the critical pressure is highest when the vault symmetrical, with top three pieces 39.64% of the base length. The upper two joints should also be heavily strengthened.

Conventional beam elements ignoring distortion may overestimate the lateral resistance of frames and curved beams made of monosymmetric I-sections. This paper introduces two new distortional modes represented by mechanical couples relative to twisting and shearing of the two flanges that are opposite in directions but unequal in magnitudes. A straight beam element with nine degrees of freedom (DOFs) per node, including the conventional three translations, three rotations, warping and the new two distortions, is newly derived. This allows all the DOFs of the connected elements at a common joint to be easily transformed to the global coordinates for stiffness assembly. As a result, the warping–distortion compatibility problem that occurs in frames and curved beams is resolved. In the numerical examples, the results produced by the present beam element is demonstrated to agree excellently with the shell-element solutions for the lateral-distortional deformation of the angled frame and curved beam. It is observed that the cross-sectional distortion effect becomes extremely significant for angled frames of short unbraced length and for curved beams of high curvature.

In this paper, we have proposed and presented a method for object tracking in video that exploits new tight frame of curvelets. We constructed a method of tracking that provides a sparse expansion for typical images having smooth contours. We used curvelet coefficients to segment and tracking of object in the sequence of frames. Results obtained for tracking of moving objects in video clips demonstrated an improved performance over other recent related methods available in the literature.

Due to the significance of aquatic robotics and marine engineering, the underwater video enhancement has gained huge attention. Thus, a video enhancement method, namely Manta Ray Foraging Lion Optimization-based fusion Convolutional Neural Network (MRFLO-based fusion CNN) algorithm is developed in this research for enhancing the quality of the underwater videos. The MRFLO is developed by merging the Lion Optimization Algorithm (LOA) and Manta Ray Foraging Optimization (MRFO). The blur in the input video frame is detected and estimated through the Laplacian’s variance method. The fusion CNN classifier is used for deblurring the frame by combining both the input frame and blur matrix. The fusion CNN classifier is tuned by the developed MRFLO algorithm. The pixel of the deblurred frame is enhanced using the Type II Fuzzy system and Cuckoo Search optimization algorithm filter (T2FCS filter). The developed MRFLO-based fusion CNN algorithm uses the metrics, Underwater Image Quality Measure (UIQM), Underwater Color Image Quality Evaluation (UCIQE), Structural Similarity Index Measure (SSIM), Mean Square Error (MSE), and Peak Signal-to-Noise Ratio (PSNR) for the evaluation by varying the blur intensity. The proposed MRFLO-based fusion CNN algorithm acquired a PSNR of 38.9118, SSIM of 0.9593, MSE of 3.2214, UIQM of 3.0041 and UCIQE of 0.7881.

A necessary and sufficient condition for the perturbation of a Banach frame by a non-zero functional to be a Banach frame has been obtained. Also a sufficient condition for the perturbation of a Banach frame by a sequence in E* to be a Banach frame has been given. Finally, a necessary condition for the perturbation of a Banach frame by a finite linear combination of linearly independent functionals in E* to be a Banach frame has been given.

The notion of frame has some generalizations such as frames of subspaces, fusion frames and g-frames. In this paper, we introduce fusion frames and g-frames in Hilbert C*-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. We also generalize a perturbation result in frame theory to g-frames in Hilbert spaces. We also show that tensor product of fusion frames (g-frames) is a fusion frame (g-frame) and tensor product of resolution of identity is a resolution of identity.

This paper deals with the theory of frames for vector-valued Weyl–Heisenberg wavelets (VVWHW). We derive frame and the corresponding frame bounds for VVWHW.

This paper develops several aspects of shift-invariant spaces on locally compact abelian groups. For a second countable locally compact abelian group G we prove a useful Hilbert space isomorphism, introduce range functions and give a characterization of shift-invariant subspaces of L²(G) in terms of range functions. Utilizing these functions, we generalize characterizations of frames and Riesz bases generated by shifts of a countable set of generators from L²(ℝⁿ) to L²(G).

Frames of subspaces for Banach spaces have been introduced and studied. Examples and counter-examples to distinguish various types of frames of subspaces have been given. It has been proved that if a Banach space has a Banach frame, then it also has a frame of subspaces. Also, a necessary and sufficient condition for a sequence of projections, associated with a frame of subspaces, to be unique has been given. Finally, we consider complete frame of subspaces and prove that every weakly compactly generated Banach space has a complete frame of subspaces.

Bi-Banach frames in Banach spaces have been defined and studied. A necessary and sufficient condition under which a Banach space has a Bi-Banach frame has been given. Finally, Pseudo exact retro Banach frames have been defined and studied.

Various types of Schauder frames have been defined and studied. A necessary and sufficient condition for each type of Schauder frame is given. Finally, we give some theoretical applications of these types of Schauder frames.