Extension Of Matrix Research Articles

Preparation of Gradient HEA-DAC/HPA Hydrogels by Limited Domain Swelling Method.

Hydrogels are widely used in biological dressing, tissue scaffolding, drug delivery, sensors, and other promising applications owing to their water-rich soft structures, biocompatibility, and adjustable mechanical properties. However, most of the conventional hydrogels are isotropic. The anisotropic structures existed widely in the organizational structure of plants and animals, which played a crucial role in biological systems. In this work, a method of limited domain swelling to prepare anisotropic hydrogels is proposed. Through spatially controlled swelling, the extension direction of hydrogels can be limited by a tailored mold, further achieving anisotropic hydrogels with concentration gradients. The external solution serves as a swelling solution to promote swelling and extension of the hydrogel matrix in a mold which can control the extension direction. Due to the diversity of external solutions, the method can be applied to prepare a variety of stimulus-responsive polymers. The limited domain swelling method is promising for the construction of anisotropic hydrogels with different structures and properties.

On the neighbourhood degree sum-based Laplacian energy of graphs

A useful extension of the Laplacian matrix is proposed here and the corresponding modification of the Laplacian energy (LE) is presented. The neighbourhood degree sum-based Laplacian energy (LNE) is produced by means of the eigenvalues of the newly introduced neighbourhood degree sum-based Laplacian matrix (LN). We investigate the mathematical properties of LNE by comparing it with the Laplacian energy. The role of LNE in structure–property modelling of molecules is demonstrated by statistical approach. The formulation of LNE is not ad hoc; rather, its chemical significance exerts that it outperforms LE in modelling physiochemical behaviours of molecules. Mathematical properties of LN are also revealed by finding crucial bounds of its eigenvalues with characterizing extremal graphs.

Matriglycan maintains t-tubule structural integrity in cardiac muscle

Maintaining the structure of cardiac membranes and membrane organelles is essential for heart function. A critical cardiac membrane organelle is the transverse tubule system (called the t-tubule system) which is an invagination of the surface membrane. A unique structural characteristic of the cardiac muscle t-tubule system is the extension of the extracellular matrix (ECM) from the surface membrane into the t-tubule lumen. However, the importance of the ECM extending into the cardiac t-tubule lumen is not well understood. Dystroglycan (DG) is an ECM receptor in the surface membrane of many cells, and it is also expressed in t-tubules in cardiac muscle. Extensive posttranslational processing and O-glycosylation are required for DG to bind ECM proteins and the binding is mediated by a glycan structure known as matriglycan. Genetic disruption resulting in defective O-glycosylation of DG results in muscular dystrophy with cardiorespiratory pathophysiology. Here, we show that DG is essential for maintaining cardiac t-tubule structural integrity. Mice with defects in O-glycosylation of DG developed normal t-tubules but were susceptible to stress-induced t-tubule loss or severing that contributed to cardiac dysfunction and disease progression. Finally, we observed similar stress-induced cardiac t-tubule disruption in a cohort of mice that solely lacked matriglycan. Collectively, our data indicate that DG in t-tubules anchors the luminal ECM to the t-tubule membrane via the polysaccharide matriglycan, which is critical to transmitting structural strength of the ECM to the t-tubules and provides resistance to mechanical stress, ultimately preventing disruptions in cardiac t-tubule integrity.

New results over Zappa-Szep products via a recent semigroup

In [17], the authors established a new semigroup N as an extension of both Rees matrix and completely zero-simple semigroups. In this paper, by taking into account the Zappa-Sz\'{e}p product obtained by special subsemigroups of N, we will expose some new distinguishing theoretical results on this product.

Recursive posterior Cramér–Rao lower bound on Lie groups

The recursive posterior Cramér–Rao Lower Bound (CRLB) introduced by Tichavsky et al. (1998) is an efficient tool to estimate the minimum variance of an estimator. Lately, filters and observers on matrix Lie groups have raised interest in control and nonlinear estimation. However, the recursive CRLB applies to variables belonging to the Euclidean space, and a recursive formulation on matrix Lie groups was not introduced yet. Hence, this paper derives a recursive formula of the Fisher information matrix on Lie groups, which reveals to be a natural extension of the information matrix on the Euclidean space. This generic result is applied to nonlinear Gaussian systems on Lie groups and tested on a navigation problem. The proposed recursive CRLB is consistent with state-of-the-art filters and shows a representative behavior with respect to the estimation errors. This paper provides a simple method to recursively compute the minimal variance of an estimator on matrix Lie groups. It enables to assess the behavior of an estimator, which is fundamental to implement robust algorithms.

Extension of adjacency matrix in QSPR analysis

The utilization of extended adjacency matrices to investigate various combinatorial properties of graph is one of the most recent research directions in the spectral graph theory. This work proposes a useful modification to the classical adjacency matrix, as well as associated spectral radius and energy. When compared to existing descriptors, the novel energy and Estrada index are established to be significant molecular descriptors in structure–property modelling and isomer discrimination by means of quantitative structure property relationship (QSPR) analysis. Following that, mathematical attributes of the energy are demonstrated by estimating its tight bounds with identifying extremal graphs.

The Linear Algebra of (r, β)-Stirling Matrices

This paper establishes the linear algebra of the $(r, \beta)$-Stirling matrix. Along the way, this paper derives various identities, such as its factorization and relationship to the Pascal matrix and the Stirling matrix of the second kind. Additionally, this paper develops a natural extension of the Vandermonde matrix, which can be used to study and evaluate successive power sums of arithmetic progressions.

Block-MUSIC in blade tip timing: Performance study of block snapshot matrix

Periodic nonuniform sampling can be viewed as a combination of uniform sampling and non-uniform sampling. Blade tip timing (BTT) technique conforms to this sampling pattern, where non-uniformity and uniformity are caused by non-uniform probe layout and rotational motion, respectively. By introducing block structure into multiple signal classification (MUSIC) for the extension of snapshot matrix, Block-MUSIC has been proved to be anti-aliasing and can filter out synchronous frequency in BTT application. However, the theoretical justification of the block structure is lacking. In this paper, a performance study of Block-MUSIC is conducted from three aspects: identifiability, stability, and resolution. Additionally, a pseudo 2-dimensional signal model is proposed to represent the periodic nonuniform sampled signal under the assumption of equally spaced probes. To support the performance study, simulations and experiments are carried out to demonstrate how the size of the snapshot matrix, the noise level, and the amplitude of sinusoidal signals affect algorithm performance. Simulation codes are available at https://github.com/superjdg/block-music4btt.

On a Response Matrix Solver for Slab-Geometry Neutral Particle Transport Problems in the Discrete Ordinates and Energy Multigroup Formulations Considering Non-Uniform Interior Sources

We present in this work an extension of the Response Matrix (RM) method for the numerical solution of slab-geometry neutral particle transport equation in the discrete ordinates (S) and energy multigroup formulations considering non-uniform sources. By using the term non-uniform we mean that the particle source is not spatially uniform inside the regions that compose the domain. The extended RM method, differently from the conventional RM method, is based on the solution of “source-independent” auxiliary problems (Green’s functions). The solution of these auxiliary problems is used in conjunction with the given non-uniform source to generate the sweeping matrices for the extended RM method. Numerical results with respect to both uniform and non-uniform source problems are given to illustrate the efficiency of the offered extended RM method.

Dependency Matrix-Based Fast Fault Diagnosis Using Search Algorithm

The dependency matrix is widely used in fault diagnosis as a valid diagnostic model. However, the conventional dependency matrix-based diagnosis method pays more attention to the extension and improvement of the dependency matrix, and the timeliness is rarely discussed. To solve the problem and improve the diagnosis efficiency, fast diagnosis method using two search algorithms is proposed. The hash function is introduced, and the dependency matrix is transformed into the key-value pairs. The binary search algorithm is brought in to locate the desired failure mode by splitting the dependency matrix. Thus, the fault diagnosis is converted into a search process, which can improve the diagnosis efficiency by making full use of the advantages of the search algorithms. Finally, a numerical simulation is utilized to verify the proposed method. The proposed methods are applied to matrices of different sizes. The results show that the hash-based diagnosis method has significantly better efficiency, making it more suitable for onboard diagnosis. On the other hand, the binary search-based method is advantageous from a memory perspective only when the matrix size is small.

Characterization of physical properties, volatile compounds and aroma profiles of different salted egg yolk lipids

Salted egg yolks (SEY) have a desirable and unique flavor with multiple underlying applications in food processing, and their abundant lipids contribute to a creamy and pleasant aroma. However, it is important to maintain the stability of the SEY flavor, which depends to a large extent on the egg species and the processing method. This study aimed to extract different SEY lipids with conventional solvents, analyze the fatty acid composition, and screen the volatile compounds to elucidate the flavor differences between salted hen eggs and duck eggs. Compared to ethanol extraction, acetone-extracted lipids had lower acid value and viscosity, and almost had no phospholipid content. Fatty acid analysis revealed that the highest content of fatty acid in SEY lipids was oleic acid, followed by palmitic acid and linoleic acid, while there were significant variations of different SEY lipids in the fatty acid profiles. The volatile compounds were identified by headspace solid-phase microextraction-gas chromatography-mass spectrometry (HS-SPME-GC–MS), and the overall odor was detected by the electronic nose (E-nose). A total of 27 volatile compounds were analyzed in SEY lipids and divided into 8 chemical classes. The aldehydes, furans and pyrazines were decreased, and the hydrocarbons were increased compared with untreated SEY. The combination of the physical properties and flavor evaluation of SEY lipids could provide a theoretical basis for the extension of the characteristic flavor matrix in SEY.

The strong truncated Hamburger moment problem with and without gaps

The strong truncated Hamburger moment problem (STHMP) of degree (−2k1,2k2) asks to find necessary and sufficient conditions for the existence of a positive Borel measure, supported on R, such that βi=∫xidμ(−2k1≤i≤2k2). The first solution of the STHMP, covering also its matrix generalization, was established by Simonov [60], who used the operator approach and described all solutions in terms of self-adjoint extensions of a certain symmetric operator. Using the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we give an alternative solution of the STHMP and describe concretely all minimal solutions, i.e., solutions having the smallest support. Then, using the equivalence with the STHMP of degree (−2k,2k), we obtain the solution of the 2–dimensional truncated moment problem (TMP) of degree 2k with variety xy=1, first solved by Curto and Fialkow [22]. Our addition to their result is the fact previously known only for k=2, that the existence of a measure is equivalent to the existence of a flat extension of the moment matrix. Further on, we solve the STHMP of degree (−2k1,2k2) with one missing moment in the sequence, i.e., β−2k1+1 or β2k2−1, which also gives the solution of the TMP with variety x2y=1 as a special case, first studied by Fialkow in [33].

On a Solution of the Multidimensional Truncated Matrix-Valued Moment Problem

We will consider the multidimensional truncated p times p Hermitian matrix-valued moment problem. We will prove a characterisation of truncated p times p Hermitian matrix-valued multisequence with a minimal positive semidefinite matrix-valued representing measure via the existence of a flat extension, i.e., a rank preserving extension of a multivariate Hankel matrix (built from the given truncated matrix-valued multisequence). Moreover, the support of the representing measure can be computed via the intersecting zeros of the determinants of matrix-valued polynomials which describe the flat extension. We will also use a matricial generalisation of Tchakaloff’s theorem due to the first author together with the above result to prove a characterisation of truncated matrix-valued multisequences which have a representing measure. When p = 1, our result recovers the celebrated flat extension theorem of Curto and Fialkow. The bivariate quadratic matrix-valued problem and the bivariate cubic matrix-valued problem are explored in detail.

Lyapunov functionals for linear neutral delay systems in the Hale's form

We construct Lyapunov functionals for linear neutral delay systems in the Hale's form. Our construction is made for no differentiable initial functions, as required for such a class of systems, but not thus considered in the existing works. The functionals depend on a special matrix-valued function which is the natural extension of the Lyapunov matrix for retarded delay systems. The constructed functionals allow deriving a converse result for exponential stability of the systems and exponential estimates for the solutions.

The potential of using SWCNTs, MWCNTs and CNFs capable of increasing the composite material dimensional and technological stability as modifiers of a polymer matrix

In this study, the effect of carbon nanofibers (CNN), single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs) on the warpage expected value and warpage dispersion of a plate made of a fibrous composite material are investigated. Laminates with fiber disorientation were considered as asymmetric and the model proposed by Dano and Hyer was used to evaluate the warpage. The results obtained confirm that the addition of carbon nanoparticles as a modifier to the polymer matrix of a fibrous composite material can increase the dimensional stability (the mathematical expectation of the standard deviation) and the technological stability of the reinforcement scheme (the variance of the standard deviation of the composite plate). Modeling of the warpage of plate, taking into account the possible disorientation of the fiber, showed a decrease in the warpage dispersion by 12.6 and 26.6 % with the modification of the SWNTs and MWCNTs, respectively. The coefficient of thermal extension (CTE) of a nanostructured polymer matrix with various fillers were experimentally determined. It was found that carbon nanomodifiers are more effective as compensators for the thermal expansion of the polymer matrix in composite laminates reinforced with carbon fibers than the polymer matrix without macrofibers. The addition of 0.05 % SWCNTs, 1 % MWCNTs to the epoxy resin reduces the CTE by 9.7 and 15.4 %, respectively. At the same time, the addition of a similar amount of nanoparticles to the epoxy matrix of the fiber composite reduces the CTE in the transverse direction by 15.56 and 35.8 %, respectively. On the basis of the obtained results, the dependences of the transverse CTE of the polymer composite material, the mathematical expectation of the standard deviation, and the variance of the standard deviation of the composite plate form accuracy on the concentration of the modifier were constructed. According to the obtained data, it can be concluded that in order to reduce the mathematical expectation and the variance of the warping of the composite material, there is an effective concentration, the increase of which is impractical, despite the further decrease in the transversal CTE.

Generalized orthopair fuzzy matrices

A q-rung orthopair fuzzy matrix (q-ROFM), an extension of the Pythagorean fuzzy matrix (PFM) and intuitionistic fuzzy matrix (IFM), is very helpful in representing vague information that occurs in real-world circumstances. In this paper we define some algebraic operations, such as max-min, min-max, complement, algebraic sum, algebraic product, scalar multiplication \((nA)\), and exponentiation \((A^n)\). We also investigate the algebraic properties of these operations. Furthermore, we define two operators, namely the necessity and possibility to convert q-ROFMs into an ordinary fuzzy matrix, and discuss some of their basic algebraic properties. Finally, we define a new operation(@) on q-ROFMs and discuss distributive laws in the case where the operations of \(\oplus_{q}, \otimes_{q}, \wedge_{q}\) and \(\vee_{q}\) are combined each other.

Generalized Parikh matrices of picture array

The theory of Parikh matrices and subword occurrences has led to extensive research in combinatorics on words. The extension of Parikh matrix of a word into picture array is an another interesting problem. Since the Parikh matrix does not say about the number of subword occurrences in the array. Therefore, in this paper we introduce the generalized Parikh matrix of a picture array which gives the number of subwords occurrences in the array. We also discuss some properties of generalized Parikh matrices.

A hybrid-dimensional compositional two-phase flow model in fractured porous media with phase transitions and Fickian diffusion

This paper presents an extension of Discrete Fracture Matrix (DFM) models to compositional two-phase Darcy flow accounting for phase transitions and Fickian diffusion. The hybrid-dimensional model is based on nonlinear transmission conditions at matrix fracture (mf) interfaces designed to be consistent with the physical processes. They account in particular for the saturation jump induced by the different rock types, for the Fickian diffusion in the fracture width, as well as for the thermodynamical equilibrium formulated by complementary constraints. The model is validated by numerical comparison with a reference equi-dimensional model using a TPFA approximation in space and a fully implicit Euler time integration. It is also compared with the usual approach based on an harmonic averaging of the transmissivities at mf interfaces combined with a two-point upwinding of the mobilities jumping over the mf interfaces. Our approach is shown to provide basically the same accuracy than the equi-dimensional model as opposed to the classical harmonic averaging approach which is shown to exhibit physical inconsistency. It is then applied to simulate the desaturation by suction at the interface between a fractured Callovo-Oxfordian argilite storage rock and a ventilation tunnel with data set provided by Andra.

2D Words and Generalized Parikh Matrices

The extension of Parikh vector is Parikh matrix which is a useful tool in arithmetizing words by numbers. Another interesting problem is the extension of Parikh matrix of a words to 2D words. However, Parikh matrix fails to tell about the number of subword occurrences in 2D word. Therefore, in this paper we introduce the notion of generalized Parikh matrix of 2D words which fulfill the number of subwords occurrences in an 2D words and also we obtain the properties of the generalized Parikh matrices.

On a minimal solution for the indefinite multidimensional truncated moment problem

We will consider the indefinite multidimensional truncated moment problem. Necessary and sufficient conditions for a given truncated multisequence to have a signed representing measure with minimal cardinality of its support are given by the existence of a rank preserving extension of a multivariate Hankel matrix (built from the given truncated multisequence) such that the corresponding associated polynomial ideal is real radical. This result is a special case of a more general characterisation of truncated multisequences with a minimal complex representing measure whose support is symmetric with respect to complex conjugation (which we will call quasi-complex). One motivation for our results is the fact that positive semidefinite truncated multisequences need not have a positive representing measure. Thus, our main result gives the potential for computing a signed representing measure μ with Jordan decomposition μ=μ+−μ−, where cardμ− is small. We will illustrate this point on concrete examples.