Frame Operator Research Articles

New Properties of Frames

Let H be a finite-dimentional complex Hilbert space and l2(H) is the space of square summable sequences in H. We will give a new characterization of a frame for H, we give our definition of a frame for the Hilbert space l2(H), we also define and give the properties of the frame operator. We equally show that our definition is equivalent to the definition of a frame for the Hilbert space H. Finally, we give a way to construct frames for l2(Hn) from frames for l2(Hp) such that p<n via fusion frame theory.

Data reduction for low energy nuclear physics experiments using data frames

Low energy nuclear physics experiments are transitioning towards fully digital data acquisition systems. Realizing the gains in flexibility afforded by these systems relies on equally flexible data reduction techniques. In this paper, methods utilizing data frames and in-memory techniques to work with data, including data from self-triggering, digital data acquisition systems, are discussed within the context of a Python package, sauce. It is shown that data frame operations can encompass common analysis needs and allow interactive data analysis. Two event building techniques, dubbed referenced and referenceless event building, are shown to provide a means to transform raw list mode data into correlated multi-detector events. These techniques are demonstrated in the analysis of two example data sets.

Codebase release 2.0 for sauce

Low energy nuclear physics experiments are transitioning towards fully digital data acquisition systems. Realizing the gains in flexibility afforded by these systems relies on equally flexible data reduction techniques. In this paper, methods utilizing data frames and in-memory techniques to work with data, including data from self-triggering, digital data acquisition systems, are discussed within the context of a Python package, sauce. It is shown that data frame operations can encompass common analysis needs and allow interactive data analysis. Two event building techniques, dubbed referenced and referenceless event building, are shown to provide a means to transform raw list mode data into correlated multi-detector events. These techniques are demonstrated in the analysis of two example data sets.

A Study on Urbanization Promotion Area Incorporation by Population Frame and Urban Facility Location Plan Operation

A Study on Urbanization Promotion Area Incorporation by Population Frame and Urban Facility Location Plan Operation

Optimization of Rafter and Frame structures

Objective. Trusses and frames are widely used in the design and construction of buildings and structures, since they have high load-bearing capacity and rigidity. At the same time, they have a significant mass. In order to reduce the mass and cost of rafter and frame structures, this article develops optimal design and calculation schemes using steels of different strengths. Method. The study is based on the theory and methods of design optimization. By studying the features of the frame and truss operation, the most loaded elements of the structures are determined. In accordance with this, steels of different strengths are selected, which ensures a decrease in the mass of the structure and its cost. Comparing various design and calculation schemes, the optimal distribution of steels of different strengths along the span of the structures is proposed. Result. The developed design and calculation schemes make it possible to reduce the mass and cost of frames and trusses by optimally distributing steels of different grades. Conclusion. The proposed solutions can find application in the practice of designing and constructing metal structures. Structural schemes of trusses and frames have been developed, where steels of different strengths are used, which in terms of weight and cost compare favorably with corresponding structures made of steels of the same strength (one grade).

An unbounded operator theory approach to lower frame and Riesz-Fischer sequences

Frames and orthonormal bases are important concepts in functional analysis and linear algebra. They are naturally linked to bounded operators. To describe unbounded operators those sequences might not be well suited. This has already been noted by von Neumann in the 1920ies. But modern frame theory also investigates other sequences, including those that are not naturally linked to bounded operators. The focus of this manuscript will be two such kind of sequences: lower frame and Riesz-Fischer sequences. We will discuss the inter-relation of those sequences. We will fill a hole existing in the literature regarding the classification of these sequences by their synthesis operator. We will use the idea of generalized frame operator and Gram matrix and extend it. We will use that to show properties for canonical duals for lower frame sequences, such as a minimality condition regarding its coefficients. We will also show that other results that are known for frames can be generalized to lower frame sequences. Finally, we show that the converse of a well-known result is true, i.e. that minimal lower frame sequences are equivalent to complete Riesz-Fischer sequences, without any further assumptions.To be able to tackle these tasks, we had to revisit the concept of invertibility (in particular for non-closed operators). In addition, we are able to define a particular adjoint, which is uniquely defined for any operator.

Erasure Recovery Matrices for Data Erasures and Rearrangements

When studying signal reconstruction, the frames are often selected in advance as encoding tools. However, in practical applications, this encoding frame may be subject to attacks by intermediaries and generate errors. To solve this problem, in this paper, the erasure recovery matrices for data erasures and rearrangements are analyzed. Unlike the previous research, first of all, we introduce a kind of frame and its erasure recovery matrix M so that MI,Λ=Im×m, where Im×m is a unit matrix. In this case, we do not need to invert the matrix of the frame operator and the erasure recovery matrix, and this greatly simplifies reconstruction problems and calculations. Then three different construction algorithms of the above erasure recovery matrix M and the frame are proposed, and each of them has advantages. Furthermore, some restrictions on M so that the constructed frame and erasure recovery matrix M can recover coefficients from rearrangements are imposed. We prove that in some cases, the above M and frame can recover coefficients stably from m rearrangements.

Generalized fusion frame in quaternionic Hilbert spaces

The notion of a generalized fusion frame in quaternionic Hilbert space is introduced. A characterization of generalized fusion frame in quaternionic Hilbert space with the help of frame operator is being discussed. Finally, g-fusion frame in quaternionic Hilbert space using invertible bounded right Q-linear operator on quaternionic Hilbert space is constructed.

Some properties of weaving $ K $-frames in $ n $-Hilbert space

<p>$ K $-frames are more generalized than ordinary frames, particularly in terms of their weaving properties. The study of weaving $ K $-frames in Hilbert space has already been explored. Given the significance of $ n $-Hilbert spaces in functional analysis, it is essential to study weaving $ K $-frames in $ n $-Hilbert spaces. In this paper, we introduced the notion of weaving $ K $-frames in $ n $-Hilbert spaces and obtained some new properties for these frames using operator theory methods. First, the concept of weaving $ K $-frames in $ n $-Hilbert spaces is developed, and examples are given. By virtue of auxiliary operators, such as the preframe operator, analysis operator, and frame operator, some new properties and characterizations of these frames are presented, and several new methods for their construction are given. Stability and perturbation results are discussed and new inequalities are established as applications.</p>

Operation al-Aqsa Flood: A Rupture in the History of the Palestinian Resistance and Its Implications

This paper argues that the al-Aqsa Flood operation launched by the Izz ad-Din al-Qassam Brigades –the military wing of Hamas– on the morning of October 7 has led to a psychological and epistemological rupture in the Palestinian-Israeli conflict and Middle East politics as the operation marked a paradigm shift in the philosophy of the resistance. The infiltration of al-Qassam into the occupied territories by land, sea, and air was a clear sign of a change in the strategy of the resistance to continue the active struggle against the aggressive expansionism and aggression of the Zionist Israeli government. In this context, the study tries to frame Operation al-Aqsa Flood and analyze the dimensions of the rupture. Besides this, it will also outline the possible impact of the operation on global and regional politics, considering that the ongoing process will change the political balance in the Middle East.

An Introduction to the Construction of Subfusion Frames

Fusion frames and subfusion frames are generalizations of frames in the Hilbert spaces. In this paper, we study subfusion frames and the relations between the fusion frames and subfusion frame operators. Also, we introduce new construction of subfusion frames. In particular, we study atomic resolution of the identity on the Hilbert spaces and derive new results.

Controlled Integral Frames for Hilbert C*-Modules

The notion of controlled frames for Hilbert spaces were introduced by Balazs, Antoine and Grybos to improve the numerical efficiency of iterative algorithms for inverting the frame operator. Controlled frame theory has a great revolution in recent years. This theory have been extended from Hilbert spaces to Hilbert C∗-modules. In this paper we introduce and study the extension of this notion to integral frame for Hilbert C∗-modules. Also we give some characterizations between integral frame in Hilbert C∗-modules.

Quantitative bounds for unconditional pairs of frames

We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [22]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that for all C>0 and N∈N the following is true: Let (xj)j=1N and (fj)j=1N be sequences in a finite dimensional Hilbert space which satisfy ‖xj‖=‖fj‖ for all 1≤j≤N and‖∑j=1Nεj〈x,fj〉xj‖≤C‖x‖,for all x∈ℓ2M and |εj|=1. If the frame operator for (fj)j=1N has eigenvalues λ1≥...≥λM and β>0 is such that λ1≤βM−1∑j=1Mλj then (fj)j=1N has Bessel bound 27β2C. The same holds for (xj)j=1N.

Characterization of rationally sampled quaternionic dual Gabor frames

This paper addresses quaternionic dual Gabor frames under the condition that the products of time‐frequency shift parameters are rational numbers. For a general overcomplete quaternionic Gabor frame with the product of time‐frequency shift parameters not equal to , we show that its corresponding frame and translation operators do not commute, which leads to its canonical dual frame not having the Gabor structure, but it may have other dual frames with Gabor structure. We characterize when two quaternionic Gabor Bessel sequences form a pair of dual frames, and present a class of quaternionic dual Gabor frames. We also characterize quaternionic Gabor Riesz bases and prove that their canonical dual frames have Gabor structure.

Accelerated ISAR Sparse Imaging Based on Tight Frame and Projection Operator

Accelerated ISAR Sparse Imaging Based on Tight Frame and Projection Operator

Tensor product for g-fusion frame in hilbert $C^{\ast}-$modules

In this paper, we stady the tensor product of $g-$fusion frame in Hilbert $C^{\ast}-$modules and we give the frame operator for a pair of $g-$fusion bessel sequences in tensor product of Hilbert $C^{\ast}-$modules.

Space–Time Sampling for Network Observability

Designing sparse sampling strategies are crucial in having resilient estimation and control in networked systems as they make sampling less expensive and networks more resilient w.r.t where and when samples are collected. It is shown under what conditions taking coarse samples from a network will contain the same amount of information as a finer set of samples. Our goal is to estimate the initial condition of linear time-invariant networks using a set of noisy measurements. The observability condition is reformulated as the frame condition, where one can easily trace location and time stamps of each sample. We compare estimation quality of various sampling strategies using estimation measures, which depend on spectrum of the corresponding frame operators. Using properties of the minimal polynomial of the state matrix, deterministic and randomized methods are suggested to construct observability frames. Intrinsic tradeoffs assert that collecting samples from fewer subsystems dictates taking more samples (in average) per subsystem. Three scalable algorithms are developed to generate sparse space-time sampling strategies with explicit error bounds.

Double preconditioning for Gabor frame operators: Algebraic, functional analytic and numerical aspects

Double preconditioning for Gabor frame operators: Algebraic, functional analytic and numerical aspects

Hamel's field variational integrator for simulating dynamics of thin-walled geometrically exact beams with warping effects

Using discrete variational principle, a Hamel's field variational integrator (HFVI) for simulating dynamics of thin-walled geometrically exact beams with warping effects is originally proposed. Firstly, a spacetime independent interpolation way for thin-walled beam elements is established on SE(3) × R space by using Hamel's formalism. Then, a new discretization formulation of convective velocity and convective strains on Lie algebra is proposed by using frame operators, which helps to reduce the system's nonlinearity. After that, the variations of discrete convective velocity and discrete convective strain are decoupled from the cross-section configuration variables. This way can avoid the computation of nonlinear exponential map's tangent map and reduce the computation complexity significantly. According the Hamiltonian form of proposed HFVI, the warping variables and discrete convective strains can be independently solved by using discrete compatibility equations on Lie algebra. Finally, the efficiency and accuracy of HFVI is validated by six static and dynamic examples.

Nonstationary frames of translates and frames from the Weyl–Heisenberg group and the extended affine group *

In this work, we analyze Gabor frames for the Weyl–Heisenberg group and wavelet frames for the extended affine group. Firstly, we give necessary and sufficient conditions for the existence of nonstationary frames of translates. Using these conditions, we prove the existence of Gabor frames from the Weyl–Heisenberg group and wavelet frames for the extended affine group. We present a representation of functions in the closure of the linear span of a Gabor frame sequence in terms of the Fourier transform of window functions. We show that the canonical dual of frames of translates has the same structure. An approximation of the inverse of the frame operator of nonstationary frames of translates is presented. It is shown that a nonstationary frame of translates is a Riesz basis if it is linearly independent and satisfies the approximation of the inverse frame operator. Finally, we give equivalent conditions for a nonstationary sequence of translates to be linearly independent.