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ABSTRACT The Kronecker product enlightens a sophisticated and esthetically pleasing algebraic structure that is a fundamental framework for diverse applications, such as signal processing, image analysis, and quantum computing. This study, proposing an iterative methodology grounded in Wirtinger calculus and polar decomposition, delves into the nearest Kronecker product problem under orthonormal constraints. The iterative approach ensures global convergence toward a local solution from any initial value. Numerical experiments are given to show the efficiency and effectiveness of our approach. Mainly, when the target matrix in the objective function is decomposable, the recovery of the original structure is achievable. [ABSTRACT FROM AUTHOR]

This brief contribution provides an overview of the Hill–Ogden generalised strain tensors, and some considerations on their representation in generalised (curvilinear) coordinates, in a fully covariant formalism that is adaptable to a more general theory on Riemannian manifolds. These strains may be naturally defined with covariant components or naturally defined with contravariant components. Each of these two macro-families is best suited to a specific geometrical context. [ABSTRACT FROM AUTHOR]

We consider a class of eigenvector-dependent nonlinear eigenvalue problems (NEPv) without the unitary invariance property. Those NEPv commonly arise as the first-order optimality conditions of a particular type of optimization problems over the Stiefel manifold, and previously, special cases have been studied in the literature. Two necessary conditions, a definiteness condition and a rank-preserving condition, on an eigenbasis matrix of the NEPv that is a global optimizer of the associated optimization problem are revealed, where the definiteness condition has been known for the special cases previously investigated. We show that, locally close to the eigenbasis matrix satisfying both necessary conditions, the NEPv can be reformulated as a unitarily invariant NEPv, the so-called aligned NEPv , through a basis alignment operation — in other words, the NEPv is locally unitarily invariantizable. Numerically, the NEPv is naturally solved by a self-consistent field (SCF)-type iteration. By exploiting the differentiability of the coefficient matrix of the aligned NEPv, we establish a closed-form local convergence rate for the SCF-type iteration and analyze its level-shifted variant. Numerical experiments confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we give new characterizations of self-adjoint and normal operators on a Hilbert space H . Among other results, we show that if H is a finite-dimensional Hilbert space and T ∈ B (H) , then T is self-adjoint if and only if there exists p > 0 such that | T | p ≤ | Re (T) | p . If in addition, T and Re T are invertible, then T is self-adjoint if and only if log | T | ≤ log | Re (T) | . Considering the polar decomposition T = U | T | of T ∈ B (H) , we show that T is self-adjoint if and only if T is p-hyponormal (log-hyponormal) and U is self-adjoint. Also, if T = U | T | ∈ B (H) is a log-hyponormal operator and the spectrum of U is contained within the set of vertices of a regular polygon, then T is necessarily normal. [ABSTRACT FROM AUTHOR]

A result of Nayak asserts that lim m → ∞ | A m | 1 / m exists for each n × n complex matrix A , where | A | = (A ⁎ A) 1 / 2 , and the limit is given in terms of the spectral decomposition. We extend the result of Nayak, namely, we prove that the limit of lim m → ∞ | B A m C | 1 / m exists for any n × n complex matrices A , B , and C , where B and C are nonsingular; the limit is obtained and is independent of B. We then provide generalization in the context of real semisimple Lie groups. [ABSTRACT FROM AUTHOR]

In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for 1 < p ≤ q < ∞ 1

We propose a novel technique for multi-image encryption and hiding schemes under an optical asymmetric framework using structured fingerprint phase masks (SFPMs) in the gyrator transform (GT) domain and three-step phase-shifting digital holography (PSDH). A SFPM contains unique features of fingerprint and structured phases of the optical vortex beam, which provides enhanced security in the cryptosystem. To encrypt multiple images, GT-based phase truncation and phase reservation techniques have been used in the first level of security, whereas three-step PSDH has been used to obtain the final cipher text. The cipher text is embedded in the host image to perform the watermarking process. In this process, the host is further decomposed into three parts in which anyone from the last two parts can be used for watermark embedding, and the first part is stored as the key. The use of polar decomposition in the watermarking process provides an additional layer of security. Numerical simulations and experimental results are presented to support the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics. [ABSTRACT FROM AUTHOR]

We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles. [ABSTRACT FROM AUTHOR]

Interaction of polarized light with healthy and abnormal regions of tissue reveals structural information associated with its pathological condition. Even a slight variation in structural alignment can induce a change in polarization property, which can play a crucial role in the early detection of abnormal tissue morphology. We propose a transmission-based Stokes-Mueller microscope for quantitative analysis of the microstructural properties of the tissue specimen. The Stokes-Mueller based polarization microscopy provides significant structural information of tissue through various polarization parameters such as degree of polarization (DOP), degree of linear polarization (DOLP), and degree of circular polarization (DOCP), anisotropy (r) and Mueller decomposition parameters such as diattenuation, retardance and depolarization. Further, by applying a suitable image processing technique such as Machine learning (ML) output images were analysed effectively. The support vector machine image classification model achieved 95.78% validation accuracy and 94.81% testing accuracy with polarization parameter dataset. The study's findings demonstrate the potential of Stokes-Mueller polarimetry in tissue characterization and diagnosis, providing a valuable tool for biomedical applications. [ABSTRACT FROM AUTHOR]

Significance: Quantitative optical polarimetry has received considerable recent attention owing to its potential for being an efficient diagnosis and characterizing tool with potential applications in biomedical research and various other disciplines. In this regard, it is crucial to validate various Mueller matrix (MM) decomposition methods, which are utilized to extract and quantify the intrinsic individual polarization anisotropy properties of various complex optical media. Aim: To quantitatively compare the performance of both polar and differential MM decomposition methods for probing the structural and morphological changes in complex optical media through analyzing their intrinsic individual polarization parameters, which are extracted using the respective decomposition algorithms. We also intend to utilize the decomposition-derived anisotropy parameters to distinguish among the cervical tissues with different grades of cervical intraepithelial neoplasia (CIN) and to characterize the healing efficiency of an organic crystal. Approach: Polarization MM of the cervical tissues with different grades of CIN and the different stages of the self-healing crystal are recorded with a home-built MM imaging setup in the transmission detection geometry with a spatial resolution of ≈400 nm. The measured MMs are then processed with both the polar and differential MM decomposition methods to extract the individual polarization parameters of the respective samples. The derived polarization parameters are further analyzed to validate and compare the performance of both the MM decomposition methods for probing and characterizing the structural changes in the respective investigated optical media through their decomposition-derived intrinsic individual polarization properties. Results: Pronounced differences in the decomposed-derived polarization anisotropy parameters are observed for cervical tissue sections with different grades of CIN. While a significant increase in the depolarization parameter ðΔÞ is obtained with the increment of CIN stages for both the polar [Δ = 0.32 for CIN grade one (CIN-I) and Δ = 0.53 for CIN grade two (CIN-II))] and differential (Δ = 0.35 for CIN-I and Δ = 0.56 for CIN-II) decomposition methods, a trend reversal is seen for the linear diattenuation parameter ðdLÞ, indicating the structural distortion in the cervical morphology due to the CIN disease. More importantly, with the differential decomposition algorithm, the magnitude of the derived dL parameter decreases from 0.26 to 0.19 with the progression of CIN, which was not being probed by the polar decomposition method. Conclusion: Our results demonstrate that the differential decomposition of MM holds certain advantages over the polar decomposition method to characterize and probe the structural changes in the cervical tissues with different grades of CIN. Although the quantified individual polarization parameters obtained through both the MM decomposition methods can be used as useful metrics to characterize various optical media, in case of complex turbid media such as biological tissues, incorporation of the differential decomposition technique may yield more efficient information. Also, the study highlights the utilization of MM polarimetry with an appropriate decomposition technique as an efficient diagnostic and characterizing tool in the realm of biomedical clinical research, and various other disciplines. [ABSTRACT FROM AUTHOR]

Nowadays, many image watermarking algorithms based on matrix decomposition have been proposed. In this paper, the properties of polar decomposition in the field of image watermarking is found, that is, embedding the watermark information into diagonal elements of the semi-positive definite matrix can improve the invisibility of the watermark. However, the diagonal elements have poor robustness. Based on this, lifting wavelet transform, which is the second generation wavelet transform, and Hadamard transform are used to improve the robustness of watermarking. In addition, image geometric correction algorithms are also combined to resist geometric attacks. The robustness and comparison experiments show that the proposed algorithm is superior to some state-of-the-art algorithms in some attacks. [ABSTRACT FROM AUTHOR]

We classify Chabauty limits of groups fixed by various (abstract) involutions over SL (2 , F) , where F is a finite field-extension of Q p , with p ≠ 2 . To do so, we first classify abstract involutions over SL (2 , F) with F a quadratic extension of Q p , and prove p-adic polar decompositions with respect to various subgroups of p-adic SL 2 . Then we classify Chabauty limits of: SL (2 , F) ⊂ SL (2 , E) where E is a quadratic extension of F, of SL (2 , R) ⊂ SL (2 , C) , and of H θ ⊂ SL (2 , F) , where H θ is the fixed point group of an F-involution θ over SL (2 , F) . [ABSTRACT FROM AUTHOR]

The analogue of polar coordinates in the Euclidean space, a polar decomposition in a metric space, if well-defined, can be very useful in dealing with integrals with respect to a sufficiently regular measure. In this note we handle the technical details associated with such polar decompositions. Supplementary Information: The online version contains the Armenian language version of the article available at https://doi.org/10.1007/s10958-023-06674-w. [ABSTRACT FROM AUTHOR]

Using the polar decomposition of a bounded linear operator A defined on a complex Hilbert space, we obtain several numerical radius inequalities of the operator A , which generalize and improve the earlier related ones. Among other bounds, we show that if w (A) is the numerical radius of A , then w (A) ≤ 1 2 ‖ A ‖ 1 / 2 ‖ | A | t + | A ⁎ | 1 − t ‖ , for all t ∈ [ 0 , 1 ]. Also, we obtain some upper bounds for the numerical radius involving the spectral radius and the Aluthge transform of operators. It is shown that w (A) ≤ ‖ A ‖ 1 / 2 (1 2 ‖ | A | + | A ⁎ | 2 ‖ + 1 2 ‖ A ˜ ‖) 1 / 2 , where A ˜ = | A | 1 / 2 U | A | 1 / 2 is the Aluthge transform of A and A = U | A | is the polar decomposition of A. Other related results are also provided. [ABSTRACT FROM AUTHOR]

Geometric algebras of dimension n<6$$ n<6 $$ are becoming increasingly popular for the modeling of 3D and 3 + 1D geometry. With this increased popularity comes the need for efficient algorithms for common operations such as normalization, square roots, and exponential and logarithmic maps. The current work presents a signature‐agnostic analysis of these common operations in all geometric algebras of dimension n<6$$ n<6 $$ and gives efficient numerical implementations in the most popular algebras ℝ4,ℝ3,1,ℝ3,0,1$$ {\mathbb{R}}_4,{\mathbb{R}}_{3,1},{\mathbb{R}}_{3,0,1} $$, and ℝ4,1$$ {\mathbb{R}}_{4,1} $$, in the hopes of lowering the threshold for adoption of geometric algebra solutions by code maintainers. [ABSTRACT FROM AUTHOR]

In the setting of adjointable operators on Hilbert $ C^* $ C ∗ -modules, this paper deals with the polar decomposition of the product of three operators. The relationship between the polar decompositions associated with three operators is clarified. Based on this relationship, a formula for the polar decomposition of a multiplicative perturbation of an operator is provided. In addition, a full characterization is made on the polar decomposition of the product of three operators. Some applications of this characterization are also dealt with. [ABSTRACT FROM AUTHOR]

Tensor interpolation is an essential step for tensor data analysis in various fields of application and scientific disciplines. In the present work, novel interpolation schemes for general, that is, symmetric or non‐symmetric, invertible square tensors are proposed. Critically, the proposed schemes rely on a combined polar and spectral decomposition of the tensor data T=RQTΛQ$$ \boldsymbol{T}=\boldsymbol{R}{\boldsymbol{Q}}^T\kern0.00em \boldsymbol{\Lambda} \boldsymbol{Q} $$, followed by an individual interpolation of the two rotation tensors R$$ \boldsymbol{R} $$ and Q$$ \boldsymbol{Q} $$ and the positive definite diagonal eigenvalue tensor Λ$$ \boldsymbol{\Lambda} $$ resulting from this decomposition. Two different schemes are considered for a consistent rotation interpolation within the special orthogonal group 핊핆(3), either based on relative rotation vectors or quaternions. For eigenvalue interpolation, three different schemes, either based on the logarithmic weighted average, moving least squares or logarithmic moving least squares, are considered. It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, that is, R$$ \boldsymbol{R} $$ and Q$$ \boldsymbol{Q} $$ remain orthogonal tensors and Λ$$ \boldsymbol{\Lambda} $$ remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity). Based on selected numerical examples considering the interpolation of either symmetric or non‐symmetric tensors, the proposed schemes are compared to existing approaches such as Euclidean, Log‐Euclidean, Cholesky and Log‐Cholesky interpolation. In contrast to these existing methods, the proposed interpolation schemes result in smooth and monotonic evolutions of tensor invariants such as determinant, trace, fractional anisotropy (FA), and Hilbert's anisotropy (HA). Moreover, a consistent spatial convergence behavior is confirmed for first‐ and second‐order realizations of the proposed schemes. The present work is mainly motivated by the frequently occurring necessity for remeshing or mesh adaptivity when applying the finite element method to complex problems of nonlinear continuum mechanics with inelastic constitutive behavior, which requires the consistent interpolation of tensor‐valued history data for the transfer between different meshes. However, the proposed schemes are very general in nature and suitable for the interpolation of general invertible second‐order square tensors independent of the specific application. [ABSTRACT FROM AUTHOR]

In this paper, we generalize and refine several operator inequalities involving the joint numerical radius and the joint operator norm of spherical Aluthge transform to generalized spherical Aluthge transforms. Moreover, we investigate the link between nontrivial joint invariant subspaces of the generalized spherical Aluthge transform and the original commuting d -tuples of bounded operators. [ABSTRACT FROM AUTHOR]

In this article we present the polar decomposition of a bounded linear operator defined on an infinite-dimensional complex Hilbert space with respect to a conjugation. As a result, the polar decomposition for C-symmetric, C-skew-symmetric, and C-normal operators are obtained. As an another consequence the Godič –Lucenko theorem is obtained. We give two applications of the polar decomposition theorem. First, we obtain the polar decomposition of bounded antilinear operators and discuss various cases. Second, we derive an inequality for the bounded linear operator based on the Hölder-McCarthy's inequality and derive an inequality for C-normal operators. [ABSTRACT FROM AUTHOR]

Let \mathcal {B}(H) be the algebra of all bounded operators on a Hilbert space H. Let T=V|T| be the polar decomposition of an operator T\in \mathcal {B}(H). The mean transform of T is defined by M(T)=\frac {T+|T|V}{2}. In this paper, we discuss several properties related to the spectrum, the kernel, the image, and the polar decomposition of mean transform. Moreover, we investigate the image and preimage by the mean transform of some class of operators such as positive, normal, unitary, hyponormal, and co-hyponormal operators. [ABSTRACT FROM AUTHOR]

We study Hilbert C ∗ -modules over a C ∗ -algebra A for which the Banach A-dual module carries a natural structure of Hilbert A-module. In this direction we prove that if A is monotone complete, M and N are Hilbert A-modules, M is self-dual, and both T : M → N and its Banach A-dual T ′ : N ′ → M ′ have trivial kernels and cokernels then M ≅ N ′ . With the help of this result, for a monotone complete C ∗ -algebra A, we prove that the index of any A-Fredholm operator can be calculated as the difference of its kernel and cokernel as in the Hilbert space case. [ABSTRACT FROM AUTHOR]

For the variation of the polarized Mueller matrix of oil smoke particles under different relative humidity levels, the polarized single scattering characteristics of oil smoke particles are studied by using the Mie scattering theory, and the multiple scattering simulation is implemented with the Monte Carlo method. Variation in relative humidity is achieved by changing in mixing of the oil smoke and the water fog particles during the same dry particle filling time. Using the 36 sets of polarized Mueller matrices method, the Mueller matrix patterns of oil smoke were calculated for four conditions of 0%, 10%, 40%, and 95% relative humidity, respectively. We can verify the simulation’s correctness from the simulation and the experimental results. Specifically, as the relative humidity increases, the size of the Mueller matrix pattern increases, and the patterns of m22, m33, and m44 related to the depolarization characteristic change significantly. Furthermore, the scattering depolarization coefficients of the Mueller matrix polar decomposition increase with the increasing relative humidity, with the ability of depolarization being continuously enhanced. This method determines differences in relative humidity using intuitive measurements of stabilized scattering patterns, which can present a theoretical basis for the impact of environmental variation on polarization detection.

We present a generalization of the pseudoinverse operation to pairs of matrices, as opposed to single matrices alone. We note the fact that the Singular Value Decomposition can be used to compute the ordinary Moore-Penrose pseudoinverse. We present an analogue of the Singular Value Decomposition for pairs of matrices, which we show is inadequate for our purposes. We then present a more sophisticated analogue of the SVD which includes features of the Jordan Normal Form, which we show is adequate for our purposes. This analogue of the SVD, which we call the Jordan SVD, was already presented in a previous paper by us called 'Matrix decompositions over the double numbers'. We adopt the idea presented in that same paper that a pair of matrices is actually a single matrix over the double-complex number system. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel dual-user image authentication algorithm based on the double fractional Mellin transform. We perform a security analysis of the nonlinear cryptosystem based on the fractional Mellin transform and show its vulnerability. In the proposed algorithm, polar decomposition and sparse multiplexing are additionally applied to generate a ciphertext. During the encryption process, polar decomposition generates two private keys that can be utilized on the dual-user authentication platform. The proposed scheme has a large key space and is robust against several attacks such as contamination attacks (noise and occlusion), brute force attacks, plain-text attacks, and special iterative attacks. In addition, we carry out a comparison with a similar existing scheme for the proposed algorithm. Simulated results indicate that the proposed authentication algorithm is feasible and robust. [ABSTRACT FROM AUTHOR]

We devise a spectral divide-and-conquer scheme for matrices that are self-adjoint with respect to a given indefinite scalar product (i.e., pseudosymmetic matrices). The pseudosymmetric structure of the matrix is preserved in the spectral division such that the method can be applied recursively to achieve full diagonalization. The method is well suited for structured matrices that come up in computational quantum physics and chemistry. In this application context, additional definiteness properties guarantee a convergence of the matrix sign function iteration within two steps when Zolotarev functions are used. The steps are easily parallelizable. Rirthermore, it is shown that the matrix decouples into symmetric definite eigenvalue problems after just one step of spectral division. [ABSTRACT FROM AUTHOR]

Let C be a conjugation on a complex separable Hilbert space H. A bounded linear operator T is said to be C-normal if C T ∗ T C = T T ∗ . In this paper, first, we give a representation of C-normal operators on finite dimensional Hilbert space and later extend it to compact C-normal operators on infinite-dimensional separable Hilbert spaces. In the end, we discuss the eigenvalue problem for C-normal operators and show that every compact C-normal operator has a solution for the eigenvalue problem. [ABSTRACT FROM AUTHOR]

We consider quantum mechanics written in hydrodynamic formulation for the case of relativistic spinor fields to study their velocity: within such a hydrodynamic formulation it is possible to see that the velocity as is usually defined can not actually represent the tangent vector to the trajectories of particles. We propose an alternative definition for this tangent vector and hence for the trajectories of particles, which we believe to be new and the only one possible. We discuss how these results are a necessary step to take in order to face further problems, like the definition of trajectories for multi-particle systems or ensembles, as they happen to be useful in many applications and interpretations of quantum mechanics. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new partial order called CS partial order, which is based on the core partial order and star partial order. We give some characteristics of CS partial order by using polar decomposition. Then we apply the polar-like decomposition to introduce another new partial order called WC (weak CS) partial order, which is an extension of the CS partial order. We illustrate the relationships of the two partial orders with some well-known partial orders, such as minus, L¨owner, GL, CL partial order. At the end of the paper, the application of CS and WC partial orders is briefly discussed from the perspective of quantum physics. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new multiuser nonlinear optical cryptosystem using fractional-order vortex speckle (FOVS) patterns as security keys. In conventional optical cryptosystems, mostly random phase masks are used as the security keys which are prone to various attacks such as brute force attack. In the current study, the FOVSs are generated optically by the scattering of the fractional-order vortex beam, known for azimuthal phase and helical wavefronts, through a ground glass diffuser. FOVSs have a remarkable property that makes them almost impossible to replicate. In the input plane, the amplitude image is first phase encoded and then modulated with the FOVS phase mask to obtain the complex image. This complex image is further processed to obtain the encrypted image using the proposed method. Two private security keys are obtained through polar decomposition which enables the multi-user capability in the cryptosystem. The robustness of the proposed method is tested against existing attacks such as the contamination attack and known-plaintext attack. Numerical simulations confirm the validity and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we establish necessary and sufficient conditions for the existence of line segments (or flats) in the sphere of the nuclear norm via the notion of simultaneous polarization and a refined expression for the subdifferential of the nuclear norm. This is then leveraged to provide (point-based) necessary and sufficient conditions for uniqueness of solutions for minimizing the nuclear norm over an affine subspace. We further establish an alternative set of sufficient conditions for uniqueness, based on the interplay of the subdifferential of the nuclear norm and the range of the problem-defining linear operator. Finally, we show how to transfer the uniqueness results for the original problem to a whole class of nuclear norm-regularized minimization problems with a strictly convex fidelity term. [ABSTRACT FROM AUTHOR]

As an important deterministic error of the inertial measurement unit (IMU), the installation error has a serious impact on the navigation accuracy of the strapdown inertial navigation system (SINS). The impact becomes more severe in a highly dynamic application environment. This paper proposes a new IMU calibration model based on polar decomposition. Using the new model, the installation error is decomposed into a nonorthogonal error and a misalignment error. The compensation of the IMU calibration model is decomposed into two steps. First, the nonorthogonal error is compensated, and then the misalignment error is compensated. Based on the proposed IMU calibration model, we used a three-axis turntable to calibrate three sets of strapdown inertial navigation systems (SINS). The experimental results show that the misalignment errors are larger than the nonorthogonal errors. Based on the experimental results, this paper proposes a new method to simplify the installation error. This simplified method defines the installation error matrix as an antisymmetric matrix composed of three misalignment errors. The navigation errors caused by the proposed simplified calibration model are compared with the navigation errors caused by the traditional simplified calibration model. The 48-h navigation experiment results show that the proposed simplified calibration model is superior to the traditional simplified calibration model in attitude accuracy, velocity accuracy, and position accuracy. [ABSTRACT FROM AUTHOR]

Deformation measurement is crucial for understanding the mechanical responses of rock masses; however, full-scale tests on rock masses are difficult and costly. With the development of 3D printing technology, small-scale tests provide a promising solution. Nevertheless, traditional DIC codes usually fail due to the existence of discontinuities such as cracks and shear bonds, and the correlation also rapidly drops when large deformation occurs, which is an obstacle for further application of the DIC in small-scale tests on rock. In this work, a discrete digital image correlation (DDIC), based on the principle of DIC and the discrete finite element method (DFEM), is introduced to address discontinuities and large deformations. The triangular mesh with shrunken internal nodes was selected to address the discontinuities. A scheme of updating the reference image based on the preset threshold was employed to solve the decorrelation caused by large deformation. The rigid rotation of the strain field was removed by introducing polar decomposition in DDIC. The effectiveness and reliability of the proposed DIC were verified through several small-scale tests including dislocation, separation, and large deformation. The results showed that the updated triangular mesh can adequately trace and adapt to the discontinuous deformation and large deformation of specimen, while correlation coefficients can still fall within a relatively high confidence interval. Overall, DDIC with small-scale tests might provide a promising solution to the difficulty of preparing large specimens in rock mechanics studies and provide measurement data for the verification and calibration of discontinuous numerical methods. Highlights: A novel digital image correlation method is proposed to address the discontinuous deformation. Combined with an automatic updating scheme, the subset distortion caused by large deformation is solved. The objective strain fields can be obtained by introducing polar decomposition. The method provides measurement data for the verification and calibration of numerical tools. [ABSTRACT FROM AUTHOR]

In this paper, we present a refined convergence analysis for a simple yet powerful method for computing a symmetric low-rank orthogonal approximation of a symmetric tensor proposed in the literature. The significance is that the assumption guaranteeing the global convergence is vastly relaxed to only on an input parameter of this algorithm. [ABSTRACT FROM AUTHOR]

This work studies solution methods for approximating a given tensor by a sum of R rank-1 tensors with one or more of the latent factors being orthonormal. Such a problem arises from applications such as image processing, joint singular value decomposition, and independent component analysis. Most existing algorithms are of the iterative type, while algorithms of the approximation type are limited. By exploring the multilinearity and orthogonality of the problem, we introduce an approximation algorithm in this work. Depending on the computation of several key subproblems, the proposed approximation algorithm can be either deterministic or randomized. The approximation lower bound is established, both in the deterministic and the expected senses. The approximation ratio depends on the size of the tensor, the number of rank-1 terms, and is independent of the problem data. When reduced to the rank-1 approximation case, the approximation bound coincides with those in the literature. Moreover, the presented results fill a gap left in Yang (SIAM J Matrix Anal Appl 41:1797–1825, 2020), where the approximation bound of that approximation algorithm was established when there is only one orthonormal factor. Numerical studies show the usefulness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We introduce the notion of a tractable pair of operators as well as that of the generalized parallel sum in the setting of adjointable operators on Hilbert C ∗ -modules. Some significant results about the parallel sum known for matrices and Hilbert space operators are extended to the case of the generalized parallel sum. In particular, a factorization theorem on the parallel sum is proved, and a common upper bound of two positive operators is constructed in the Hilbert C ∗ -module case. The harmonic mean for positive operators on Hilbert C ∗ -modules is also dealt with. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new multiuser nonlinear optical cryptosystem using fractional-order vortex speckle (FOVS) patterns as security keys. In conventional optical cryptosystems, mostly random phase masks are used as the security keys which are prone to various attacks such as brute force attack. In the current study, the FOVSs are generated optically by the scattering of the fractional-order vortex beam, known for azimuthal phase and helical wavefronts, through a ground glass diffuser. FOVSs have a remarkable property that makes them almost impossible to replicate. In the input plane, the amplitude image is first phase encoded and then modulated with the FOVS phase mask to obtain the complex image. This complex image is further processed to obtain the encrypted image using the proposed method. Two private security keys are obtained through polar decomposition which enables the multi-user capability in the cryptosystem. The robustness of the proposed method is tested against existing attacks such as the contamination attack and known-plaintext attack. Numerical simulations confirm the validity and feasibility of the proposed method.

As an important deterministic error of the inertial measurement unit (IMU), the installation error has a serious impact on the navigation accuracy of the strapdown inertial navigation system (SINS). The impact becomes more severe in a highly dynamic application environment. This paper proposes a new IMU calibration model based on polar decomposition. Using the new model, the installation error is decomposed into a nonorthogonal error and a misalignment error. The compensation of the IMU calibration model is decomposed into two steps. First, the nonorthogonal error is compensated, and then the misalignment error is compensated. Based on the proposed IMU calibration model, we used a three-axis turntable to calibrate three sets of strapdown inertial navigation systems (SINS). The experimental results show that the misalignment errors are larger than the nonorthogonal errors. Based on the experimental results, this paper proposes a new method to simplify the installation error. This simplified method defines the installation error matrix as an antisymmetric matrix composed of three misalignment errors. The navigation errors caused by the proposed simplified calibration model are compared with the navigation errors caused by the traditional simplified calibration model. The 48-h navigation experiment results show that the proposed simplified calibration model is superior to the traditional simplified calibration model in attitude accuracy, velocity accuracy, and position accuracy.

The symmetric eigenvalue decomposition and the singular value decomposition (SVD) are fundamental matrix decompositions with many applications. Conventional algorithms for computing these decompositions are suboptimal in view of recent trends in computer architectures which require minimizing communication together with arithmetic costs. Spectral divide-and-conquer algorithms, which recursively decouple the problem into two smaller subproblems, can achieve both requirements. Such algorithms can be constructed with the polar decomposition playing two key roles: it forms a bridge between the symmetric eigendecomposition and the SVD, and its connection to the matrix sign function naturally leads to spectral-decoupling. For computing the polar decomposition, the scaled Newton and QDWH iterations are two of the most popular algorithms, as they are backward stable and converge in at most nine and six iterations, respectively. Following this framework, we develop a higher-order variant of the QDWH iteration for the polar decomposition. The key idea of this algorithm comes from approximation theory: we use the best rational approximant for the scalar sign function due to Zolotarev in 1877. The algorithm exploits the extraordinary property enjoyed by the sign function that a high-degree Zolotarev function (best rational approximant) can be obtained by appropriately composing low-degree Zolotarev functions. This lets the algorithm converge in just two iterations in double-precision arithmetic, with the whopping rate of convergence seventeen. The resulting algorithms for the symmetric eigendecompositions and the SVD have higher arithmetic costs than the QDWH-based algorithms, but are better suited for parallel computing and exhibit excellent numerical backward stability.

Some characterizations of products of two projections on a Hilbert space are generalized to the case of products of a finite number of projections on a Hilbert C ∗ -module. An example is constructed to show that in the Hilbert C ∗ -module case, X and its subset X ⊥ can be different, where X denotes the set of all products of two projections on a Hilbert C ∗ -module, and X ⊥ consists of those elements in X that have the polar decomposition. Some new phenomena are revealed when an operator is taken in X instead of X ⊥ . Some refinements are made on norms of differences of two projections. [ABSTRACT FROM AUTHOR]

The Aluthge transformation is generalized in the viewpoint of the axiom of operator means by using double operator integrals. It includes the mean transformation which is defined by S. H. Lee, W. Y. Lee and J. Yoon. Next we shall give some properties of it. Especially, we shall show that the n -th iteration of mean transformation of an invertible matrix converges to a normal matrix. Inclusion relations among numerical ranges of induced Aluthge transformations by some operator means are considered. [ABSTRACT FROM AUTHOR]

Rickart C ∗ -algebras are unital and satisfy polar decomposition. We proved that if a unital C ∗ -algebra A satisfies polar decomposition and admits "good" faithful tracial states then A is a Rickart C ∗ -algebra. Via polar decomposition we characterized tracial states among all states on a Rickart C ∗ -algebra. We presented the triangle inequality for Hermitian elements and traces on Rickart C ∗ -algebra. For a block projection operator and a trace on a Rickart C ∗ -algebra we proved a new inequality. As a corollary, we obtain a sharp estimate for a trace of the commutator of any Hermitian element and a projection. Also we give a characterization of traces in a wide class of weights on a von Neumann algebra. [ABSTRACT FROM AUTHOR]