Structure Of Matrices Research Articles

Joint Gramian inversion of gravity, magnetic and gravity gradient tensor data for the modelling of Mesoproterozoic iron oxide deposits, southeast Missouri, USA

SUMMARY Within the iron metallogenic province of southeast Missouri, USA, there are several mines that contain not only economic iron resources, magnetite and/or hematite, but also contain rare earth elements, copper and gold. An area including three major deposits, Pea Ridge, Bourbon and Kratz Spring, was selected for detailed modelling for the upper crustal magnetic susceptibility and density structures. For the study area, ground gravity and high-resolution airborne magnetic and gravity gradiometry data sets are available. An efficient and novel joint inversion algorithm for the simultaneous inversion of these multiple data sets is presented. The Gramian coupling constraint is used to correlate the reconstructed density and magnetic susceptibility models. The implementation relies on the structures of the sensitivity matrices and an efficient minimization algorithm to achieve significant reductions in the memory requirements and computational costs. Consequently, it is feasible to use a laptop computer for the inversion of multiple data sets, each containing thousands of data points, for the recovery of models on the study area, each including approximately one million model parameters. This is the first time that these multiple data sets have been simultaneously inverted for this area. The L1-norm stabilizer is used to provide compact and focused images of the ore deposits. For contrast, independent inversions of each data set are also discussed. In general, our results provide new insights about the concealed ore deposits in the Mesoproterozoic basement rocks of southeast Missouri. Both short- and long-wavelength anomalies exist in the recovered models; these provide a high-resolution image of the subsurface. The geometry and physical properties of the known deposits are determined very well. Additionally, some unknown concealed deposits are revealed; these could be economically valuable and should be considered in future geophysical and geological investigations.

Quark mass hierarchies and CP violation in A4 × A4 × A4 modular symmetric flavor models

We study A4× A4× A4 modular symmetric flavor models to realize quark mass hierarchies and mixing angles without fine-tuning. Mass matrices are written in terms of modular forms. At modular fixed points τ = i∞ and ω, A4 is broken to Z3 residual symmetry. When the modulus τ is deviated from the fixed points, modular forms show hierarchies depending on their residual charges. Thus, we obtain hierarchical structures in mass matrices. Since we begin with A4× A4× A4, the residual symmetry is Z3× Z3× Z3 which can generate sufficient hierarchies to realize quark mass ratios and absolute values of the CKM matrix |VCKM| without fine-tuning. Furthermore, CP violation is studied. We present necessary conditions for CP violation caused by the value of τ. We also show possibilities to realize observed values of the Jarlskog invariant JCP, quark mass ratios and CKM matrix |VCKM| simultaneously, if (10) adjustments in coefficients of Yukawa couplings are allowed or moduli values are non-universal.

Structure and Approximation Properties of Laplacian-Like Matrices

Many of today’s problems require techniques that involve the solution of arbitrarily large systems Atextbf{x}=textbf{b}. A popular numerical approach is the so-called Greedy Rank-One Update Algorithm, based on a particular tensor decomposition. The numerical experiments support the fact that this algorithm converges especially fast when the matrix of the linear system is Laplacian-Like. These matrices that follow the tensor structure of the Laplacian operator are formed by sums of Kronecker product of matrices following a particular pattern. Moreover, this set of matrices is not only a linear subspace it is a Lie sub-algebra of a matrix Lie Algebra. In this paper, we characterize and give the main properties of this particular class of matrices. Moreover, the above results allow us to propose an algorithm to explicitly compute the orthogonal projection onto this subspace of a given square matrix A in mathbb {R}^{Ntimes N}.

Implementation of Basic Operations for Sparse Matrices when Solving a Generalized Eigenvalue Problem in the ACELAN-COMPOS Complex

Introduction. The widespread use of piezoelectric materials in various industries stimulates the study of their physical characteristics and determines the urgency of such research. In this case, modal analysis makes it possible to determine the operating frequency and the coefficient of electromechanical coupling of piezoelectric elements of various devices. These indicators are of serious theoretical and applied interest. The study was aimed at the development of numerical methods for solving the problem of determining resonance frequencies in a system of elastic bodies. To achieve this goal, we needed new approaches to the discretization of the problem based on the finite element method and the execution of the software implementation of the selected method in C# on the .net platform. Current solutions were created in the context of the ACELAN-COMPOS class library. The known methods of solving the generalized eigenvalue problem based on matrix inversion are not applicable to large-dimensional matrices. To overcome this limitation, the presented scientific work implemented the logic of constructing mass matrices and created software interfaces for exchanging data on eigenvalue problems with pre- and postprocessing modules.Materials and Methods. A platform was used to implement numerical methods .net and the C# programming language. Validation of the research results was carried out through comparing the values found with solutions obtained in well-known SAE packages (computer-aided engineering). The created routines were evaluated in terms of performance and applicability for large-scale tasks. Numerical experiments were carried out to validate new algorithms in small-dimensional problems that were solved by known methods in MATLAB. Next, the approach was tested on tasks with a large number of unknowns and taking into account the parallelization of individual operations. To avoid finding the inverse matrix, a modified Lanczos method was programmatically implemented. We examined the formats for storing matrices in RAM: triplets, CSR, СSC, Skyline. To solve a system of linear algebraic equations (SLAE), an iterative symmetric LQ method adapted to these storage formats was used.Results. New calculation modules integrated into the class library of the ACELAN-COMPOS complex were developed. Calculations were carried out to determine the applicability of various formats for storing sparse matrices in RAM and various methods for implementing operations with sparse matrices. The structure of stiffness matrices constructed for the same task, but with different renumbering of nodes of a finite element grid, was graphically visualized. In relation to the problem of the theory of electroelasticity, data on the time required to perform basic operations with stiffness matrices in various storage formats were summarized and presented in the form of a table. It has been established that the renumbering of grid nodes gives a significant increase in performance even without changing the internal structure of the matrix in memory. Taking into account the objectives of the study, the advantages and weaknesses of known matrix storage formats were named. Thus, CSR was optimal when multiplying a matrix by a vector, SKS was optimal when inverting a matrix. In problems with the number of unknowns of the order of 103, iterative methods for solving a generalized eigenvalue problem won in speed. The performance of the software implementation of the Lanczos method was evaluated. The contribution of all operations to the total solution time was measured. It has been found that the operation of solving SLAE takes up to 95% of the total time of the algorithm. When solving the SLAE by symmetric LQ method, the greatest computational costs were needed to multiply the matrix by a vector. To increase the performance of the algorithm, parallelization with shared memory was resorted to. When using eight threads, the performance gain increased by 40–50%.Discussion and Conclusion. The software modules obtained as part of the scientific work were implemented in the ACELAN-COMPOS package. Their performance for model problems with quasi-regular finite element grids was estimated. Taking into account the features of the structures of the stiffness and mass matrices obtained through solving the generalized eigenvalue problem for an electroelastic body, the preferred methods for their processing were determined.

Vison transformer adapter-based hyperbolic embeddings for multi-lesion segmentation in diabetic retinopathy

Diabetic Retinopathy (DR) is a major cause of blindness worldwide. Early detection and treatment are crucial to prevent vision loss, making accurate and timely diagnosis critical. Deep learning technology has shown promise in the automated diagnosis of DR, and in particular, multi-lesion segmentation tasks. In this paper, we propose a novel Transformer-based model for DR segmentation that incorporates hyperbolic embeddings and a spatial prior module. The proposed model is primarily built on a traditional Vision Transformer encoder and further enhanced by incorporating a spatial prior module for image convolution and feature continuity, followed by feature interaction processing using the spatial feature injector and extractor. Hyperbolic embeddings are used to classify feature matrices from the model at the pixel level. We evaluated the proposed model’s performance on the publicly available datasets and compared it with other widely used DR segmentation models. The results show that our model outperforms these widely used DR segmentation models. The incorporation of hyperbolic embeddings and a spatial prior module into the Vision Transformer-based model significantly improves the accuracy of DR segmentation. The hyperbolic embeddings enable us to better capture the underlying geometric structure of the feature matrices, which is important for accurate segmentation. The spatial prior module improves the continuity of the features and helps to better distinguish between lesions and normal tissues. Overall, our proposed model has potential for clinical use in automated DR diagnosis, improving accuracy and speed of diagnosis. Our study shows that the integration of hyperbolic embeddings and a spatial prior module with a Vision Transformer-based model improves the performance of DR segmentation models. Future research can explore the application of our model to other medical imaging tasks, as well as further optimization and validation in real-world clinical settings.

Uncovering block structures in large rectangular matrices

In this article, we proposed a conceptually simple, efficient, and easily implemented approach, named Block-structure Analysis on Rectangular Matrix (BARM), for learning the block structure in a large rectangular data matrix corrupted with random effects and white noise. With the possible unknown order of row or column variables, their group structures can be directly uncovered based on the singular values and singular vectors of the scaled data matrix. We also established the asymptotic properties of the proposed approach under regular conditions. Extensive experimental evaluations also demonstrate the reliability and robustness of the proposed approach.

Revealing hidden structure in time-resolved spectral matrices using multivariate analysis of the streak camera data

Two new multi-stage smoothing LRA and PARAFAC-based methods were suggested and applied to study a 3D array composed of a set of the time-resolved spectral matrices (TRSM) containing instant fluorescence spectra of the fluorescein solution in methanol measured by the streak camera with picosecond resolution. Numerical modeling showed that both methods combined with Fourier processing allowed identifying the hidden structure of the pattern obscured entirely by noise. The PARAFAC algorithm was shown to have some advantages over LRA since the averaging of matrices in the last method causes information losses.Mathematical analysis of 20 TRSM of the fluorescein solution in methanol allowed us to expand the time range of instantaneous spectra while studying fast processes.The time dependence of the statistical moments of the spectra and their second-order derivatives was associated with the molecular reorganization processes of fluorescein solvates after absorption of excitation quanta.

On Structured Random Matrices Defined by Matrix Substitutions

The structure of the random matrices introduced in this work is given by deterministic matrices—the skeletons of the random matrices—built with an algorithm of matrix substitutions with entries in a finite field of integers modulo some prime number, akin to the algorithm of one dimensional automatic sequences. A random matrix has the structure of a given skeleton if to the same number of an entry of the skeleton, in the finite field, it corresponds a random variable having, at least, as its expected value the correspondent value of the number in the finite field. Affine matrix substitutions are introduced and fixed point theorems are proven that allow the consideration of steady states of the structure which are essential for an efficient observation. For some more restricted classes of structured random matrices the parameter estimation of the entries is addressed, as well as the convergence in law and also some aspects of the spectral analysis of the random operators associated with the random matrix. Finally, aiming at possible applications, it is shown that there is a procedure to associate a canonical random surface to every random structured matrix of a certain class.

Radiative interaction of atmosphere and surface: Write-up with elements of code

In passive satellite remote sensing of the Earth, separation of the path radiance (atmosphere-only contribution) from the surface reflection remains a “significant challenge”. Recent literature names it among the gaps in radiative transfer (RT) topics that “require continued research in the near future”. The challenge comes from multiple reflections (bouncing) between the atmosphere and surface – radiative interaction. In this paper we use a known RT technique, the matrix-operator method (MOM), and a new modification of the monochromatic vector RT (vRT) code IPOL (Intensity and POLarization) to simulate the interaction of a plane-parallel atmosphere and a few widely used surface reflection models.Following the idea of the Green's function method, IPOL no longer takes the surface model parameters on input. Instead, it provides the path radiance, and the atmospheric reflection and transmission matrices as output. Despite many RT codes use the MOM formalism, this output does not seem common. The surface reflection matrix is computed externally. Therefore, this paper extends the Green's function atmospheric correction technique to the case of polarized light.Aiming clarity rather than performance, we explain in Python the structure of the surface matrices for the isotropic (Lambertian), directional unpolarized, and polarized ocean reflection models. We then combine these surface matrices and the precomputed IPOL output to get numerically accurate signal at the top of atmosphere (TOA) and test it vs. published benchmarks. Then, for each benchmark scenario we show how to get the surface from the TOA signal, i.e. perform the RT-based atmospheric correction.

Highly dispersed nanocomposites based on polymer/inorganic hybrids and nickel nanoparticles: the role of the matrix structure in the process of in situ formation

A series of polymer/inorganic hybrids based on silica sol and polyacrylamide (SiO2-g-PAAm) with different number and length of PAAm chains was obtained by radical graft polymerization of acrylamide from the surface of SiO2. The main molecular and structural parameters of the hybrids, such as the chemical composition, average radius and charge of SiO2 particles, the number of grafts per one particle and their molecular weight, the average diameter and hydrodynamic volume of hybrid particles, and the thickness of the PAA layer, were determined. For this, elemental analysis, DTGA, static light scattering, viscometry, potentiometric titration, and TEM were used. The functional properties of hybrids as hydrophilic matrices in the in situ synthesis of nickel nanoparticles by borohydride reduction of metal ions from the Ni(NO3)2·6H20 salt in an aqueous medium have been studied. Using the method of UV-Vis spectroscopy and the developed original approach, the kinetics and efficiency of the formation of NiNPs in hybrid solutions were characterized, depending on the structure and concentration of the hybrid matrices and the concentration of the metal salt. An increase in the rate of accumulation and yield of NiNPs in solutions of all hybrids was found with an increase in salt concentration in the range of 0,010-0,078 kg·m-3, as well as a predominant decrease in the reaction rate with an increase in the concentration of hybrid matrices from 0,5 to 2,0 kg·m-3. It was shown that the structure of the hybrid matrices, determined by the number and length of PAAm chains, as well as the permeability of the grafted polymer layer, was one of the key factors affecting the formation rate and yield of NiNPs. It provided greater or lesser accessibility of the active groups of the "corona" and the inorganic "core" for metal ions and reducing agent molecules. Morphological studies of purified reduction products were carried out by TEM. Based on them, the main structural elements of highly dispersed NiNPs/SiO2-g-PAAm nanocomposites were established – swollen hairy particles of hybrids with small amorphous NiNPs (1,7±0,8 nm) included in the polymer "corona".

A KKT Conditions Based Transceiver Optimization Framework for RIS-Aided Multiuser MIMO Networks

In many core problems of signal processing and wireless communications, Karush-Kuhn-Tucker (KKT) conditions based optimization plays a fundamental role. Hence we investigate the KKT conditions in the context of optimizing positive semidefinite matrix variables under nonconvex rank constraints. More explicitly, based on the properties of KKT conditions, we optimize a reconfigurable intelligent surface (RIS) aided multi-user multi-input multi-output (MU-MIMO) network. Specifically, we consider the capacity maximization and sum mean square error (MSE) minimization problems of both the RIS-aided MU-MIMO uplink (UL) and downlink (DL) under multiple weighted power constraints and rank constraints. As for the RIS-aided MU-MIMO UL, the optimal structures of the signal covariance matrices are derived based on the KKT conditions. Furthermore, an efficient procedure is designed for solving the capacity maximization and sum mean square error (MSE) minimization problems. Then the UL-DL dualities are exploited for solving the capacity maximization and MSE minimization problems of the RIS-aided MU-MIMO DL based on the results of the UL optimization. Hence in the proposed framework, the phase shifting matrix of the RIS is jointly optimized with the signal covariance matrices for both the UL and DL. Our simulation results demonstrate the performance advantages of the proposed framework.

Structures of Weak-Mixing Matrices As a Consequence of Broken Mirror Symmetry

Structures of Weak-Mixing Matrices As a Consequence of Broken Mirror Symmetry

МЕТОД ФОРМИРОВАНИЯ СТРУКТУР ЦИФРОВЫХ ДВОЙНИКОВ ПРЕДМЕТНО-ОРИЕНТИРОВАННЫХ ОБЪЕКТОВ В ПРОСТРАНСТВЕ ОТКРЫТЫХ ИСТОЧНИКОВ НА ОСНОВЕ ФОРМАЛИЗМОВ ТЕОРИИ МНОЖЕСТВ, ГРАФОВ, ТЕОРИИ КАТЕГОРИЙ И ТЕОРИИ ПОРОЖДАЮЩИХ ЯЗЫКОВ ХОМСКОГО

For the material of this article, open sources can be presented as information posted on the Internet for multiple and unlimited use in the form of machine-readable systematized data, in formats that allow their separate automated processing. Any of the open sources is partially structured content, cha¬racterized by the fact that it consists of fuzzy overlaps and connections formalized by a number of sustainable rules. Aim. The purpose of the study is to produce a mathematical description of the rules for the decomposition of links of virtual images of content based on statistical data of comparisons in order to formulate a digital analogue of the verbal structure. The research is accompanied by the development of a modular cross-platform predictive analytics system for distributed open sources of the social digital environment based on multi-stream data processing technologies. The system is based on the creation of a prototype of a digital twin, which allows monitoring and subsequent analysis of open sources. A cross-platform digital twin can be created for various open sources. Currently, among them, social networks are a priority for the analysis of the studied data. Materials and methods. Turning to the same type of code content, it becomes necessary to formalize the rules for categorizing the semantic meanings of the structure of connections of the object of study, which can be described in the language of graph theory, have a suitable structure in Lie matrices, obey the laws of transitivity and have properties that allow you to recreate connectivity in a quasi-projection. A method is proposed that makes it possible to produce sequential calculus that does not contradict each other according to the basic rules of axiomatics. Results. The method allows for real-time analysis of the data flow of open sources, identification of digital traces of research objects, identification of the structure of their connections. Conclusion. With the help of the proposed group of algorithmized mathematical iterations, it becomes possible to create a combination of local systems with feedback subsystems of predictive analytics of open sources of Internet resources and local systems for various purposes.

OBTAINING AND RHEOLOGICAL PROPERTIES OF PECTIN ISOLATED USING MICROWAVE RADIATION

А method has been developed for extracting modified pectin from plant raw materials by using microwave radiation at the stage of washing. The rheological properties of 5.0% aqueous solutions of apple PV obtained by exposure to microwave radiation of various powers: 300, 450 and 600 W, as well as PV obtained by the traditional method at temperatures of 25, 40, 55, 70 °C were investigated. It is shown that the use of microwave radiation makes it possible to improve the method of obtaining pectins with the possibility of increasing the yield of pectin substances and regulating their structure and properties. The rheological properties of concentrated aqueous solutions of pectins obtained by exposure to microwave radiation of various powers have been studied. It is shown, that all the studied systems of PV solutions are non-Newtonian fluids with a pseudoplastic type of viscous flow anomaly, determined by the values of the microwave energy. The values of the apparent activation energies of the viscous flow of solutions, which are an indirect characteristic of the strength of the structures of the system, have been estimated. Information was obtained on the structure and changes in the structure of their concentrated solutions under the action of external forces and temperature in the course of a viscous flow. It is shown that the power values оf microwave radiation are the main factor determining the structure of molecular matrices of pectin substances and the structure of their concentrated aqueous solutions.

Nonconvex low-rank tensor approximation with graph and consistent regularizations for multi-view subspace learning

Multi-view clustering is widely used to improve clustering performance. Recently, the subspace clustering tensor learning method based on Markov chain is a crucial branch of multi-view clustering. Tensor learning is commonly used to apply tensor low-rank approximation to represent the relationships between data samples. However, most of the current tensor learning methods have the following shortcomings: the information of the local graph is not taken into account, the relationships between different views are not shown, and the existing tensor low-rank representation takes a biased tensor rank function for estimation. Therefore, a nonconvex low-rank tensor approximation with graph and consistent regularizations (NLRTGC) model is proposed for multi-view subspace learning. NLRTGC retains the local manifold information through graph regularization, and adopts a consistent regularization between multi-views to keep the diagonal block structure of representation matrices. Furthermore, a nonnegative nonconvex low-rank tensor kernel function is used to replace the existing classical tensor nuclear norm via tensor-singular value decomposition (t-SVD), so as to reduce the deviation from rank. Then, an alternating direction method of multipliers (ADMM) which makes the objective function monotonically non-increasing is proposed to solve NLRTGC. Finally, the effectiveness and superiority of the NLRTGC are shown through abundant comparative experiments with various state-of-the-art algorithms on noisy datasets and real world datasets.

Resolution of Electronic States in Heisenberg Cluster Models within the Unitary Group Approach.

In this work ground and excited electronic states of Heisenberg cluster models, in the form of configuration interaction many-body wave functions, are characterized within the spin-adapted Graphical Unitary Group Approach framework, and relying on a novel combined unitary and symmetric group approach. Finite-size cluster models of well-defined point-group symmetry and of general local-spin are presented, including J1-J2 triangular and tetrahedral clusters, which are often used to describe magnetic interactions in biological and biomimetic polynuclear transition metal clusters with unique catalytic activity, such as nitrogen fixation and photosynthesis. We show that a unique block-diagonal structure of the underlying Hamiltonian matrix in the spin-adapted basis emerges when an optimal lattice site ordering is chosen that reflects the internal symmetries of the model investigated. The block-diagonal structure is bound to the commutation relations between cumulative spin operators and the Hamiltonian operator, that in turn depend on the geometry of the cluster investigated. The many-body basis transformation, in the form of the orbital/site reordering, exposes such commutation relations. These commutation relations represent a rigorous and formal demonstration of the block-diagonal structure in Hamiltonian matrices and the compression of the corresponding spin-adapted many-body wave functions. As a direct consequence of the block-diagonal structure of the Hamiltonian matrix, it is possible to selectively optimize electronic excited states without the overhead of calculating the lower-energy states by simply relying on the initial ansatz for the targeted wave function. Additionally, more compact many-body wave functions are obtained. In extreme cases, electronic states are precisely described by a single configuration state function, despite the curse of dimensionality of the corresponding Hilbert space. These findings are crucial in the electronic structure theory framework, for they offer a conceptual route toward wave functions of reduced multireference character, that can be optimized more easily by approximated eigensolvers and are of more facile physical interpretation. They open the way to study larger ab initio and model Hamiltonians of increasingly larger number of correlated electrons, while keeping the computational costs at their lowest. In particular, these elements will expand the potential of electronic structure methods in understanding magnetic interactions in exchange-coupled polynuclear transition metal clusters.

Breaking the Communication-Privacy-Accuracy Trilemma

Two major challenges in distributed learning and estimation are 1) preserving the privacy of the local samples; and 2) communicating them efficiently to a central server, while achieving high accuracy for the end-to-end task. While there has been significant interest in addressing each of these challenges separately in the recent literature, treatments that simultaneously address both challenges are still largely missing. In this paper, we develop novel encoding and decoding mechanisms that simultaneously achieve optimal privacy and communication efficiency in various canonical settings. In particular, we consider the problems of mean estimation and frequency estimation under <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> -local differential privacy and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula> -bit communication constraints. For mean estimation, we propose the SQKR mechanism, a scheme based on Kashin’s representation and random sampling, with order-optimal estimation error under both constraints. We further apply SQKR to distributed SGD and obtain a communication efficient and (locally) differentially private distributed SGD protocol. For frequency estimation, we present the RHR mechanism, a scheme that leverages the recursive structure of Walsh-Hadamard matrices and achieves order-optimal estimation error for all privacy levels and communication budgets. As a by-product, we also construct a distribution estimation mechanism that is rate-optimal for all privacy regimes and communication constraints, extending recent work that is limited to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$b=1$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon =O(1)$ </tex-math></inline-formula> . Our results demonstrate that intelligent encoding under joint privacy and communication constraints can yield a performance that matches the optimal accuracy achievable under either constraint alone. In other words, the optimal performance is determined by the more stringent of the two constraints, and the less stringent constraint can be satisfied for free.

The Influence of Ozone on the Structure of Biodegradable Matrices Based on Mixtures of Poly(3-Hydroxybutyrate), Polylactide, and Chitosan

The Influence of Ozone on the Structure of Biodegradable Matrices Based on Mixtures of Poly(3-Hydroxybutyrate), Polylactide, and Chitosan

Bioaccessibility and bioavailability of phytochemicals: Influencing factors, improvements, and evaluations

Many phytochemicals with beneficial physiological activities have limited bioavailability due to their poor solubility and stability characteristics. The bioactivities of phytochemicals could be impacted during liberation, digestion, absorption, distribution, metabolism, and excretion processes after their oral ingestion. And the food matrix and processing significantly affect their solubility, chemical degradation, and absorption within the gastrointestinal tract. Researches indicated that the bioavailability of phytochemicals can be enhanced by controlling the composition and structure of food matrices, as well as optimizing food processing operations. Moreover, chemical modification and encapsulation approaches were reported with ability improving the dispersibility, stability, bioavailability, and bioactivity of phytochemicals. Especially for a range of encapsulation technologies are available for this purpose, such as biopolymer nanoparticles, emulsions, liposomes, solid lipid nanoparticles, and nanostructure lipid carriers, which can be assembled from food grade carbohydrates, proteins, and/or lipids. In addition, the experimental methods commonly used to evaluate the bioaccessibility and bioavailability of phytochemicals are reviewed. Utilization of these methods allows screening of food formulations with different structures and compositions, thereby facilitating the optimization of their performance. This review aims to provide information that will guide future research and development efforts on the creation of more effective phytochemical delivery systems. • Phytochemicals experienced digestion, absorption, distribution, metabolism, etc. • Food matrix and processing influence the biological benefits of phytochemicals. • Chemical modification improves the performance of bioactive phytochemicals. • Encapsulation could enhance the bioaccessibility of phytochemicals effectively. • In vitro/vivo evaluations are used to assess the biological fate of phytochemicals.

Interrelation of Symmetry and Antisymmetry of Quasi-Orthogonal Cyclic Matrices with Prime Numbers

Quasi-orthogonal Hadamard matrices and Mersenne matrices with two and three values of the elements, used in digital data processing, are considered, as well as the basis of error-correcting codes and algorithms for transforming orthogonal images. Attention is paid to the structures of cyclic matrices with symmetries and antisymmetries. The connection between symmetry and antisymmetry of structures of cyclic Hadamard and Mersenne matrices on a orders equal to prime numbers, products of close primes, composite numbers, powers of a prime number is shown. Separately, orders equal to the degrees of the prime number 2 are distinguished, both the orders of Hadamard matrices and the basis of the composite orders of Mersenne matrices of block structures with two element values. It is shown that symmetric Hadamard matrices of cyclic and bicyclic structures, according to the extended Riser boundary, do not exist on orders above 32. Mersenne matrices of composite orders belonging to the sequence of Mersenne numbers 2k ‒ 1 nested in the sequence of orders of the main family of Mersenne matrices 4t ‒ 1 exist in a symmetric and antisymmetric form. For orders equal to the powers of a prime number, Mersenne matrices exist in the form of block-diagonal constructions with three element values. The value of prime power determines the number of blocks along the diagonal of the matrix on which the elements with the third value are located. The cyclic blocks are symmetrical and antisymmetric.