A matrix solution to any polygon equation

Ion conducting polymer-silica hybrid ionogels obtained via non-aqueous sol-gel route

A comparative systematic study addresses the specific features of polymerization of aniline adsorbed on the Nylon matrices in the solutions containing monomer and in the monomer-free solutions. The characteristics of the formed composite materials are investigated. In the presence of aniline in the solution, a uniform and conducting composite material is obtained and its electrical conductivity reversibly depends on pH of the medium and is several orders of magnitude higher than the electrical conductivity of the composite material prepared by the polymerization of aniline in the Nylon matrix in the monomer-free solution.

Origin mechanism of pitting corrosion induced by cerium inclusions

Research was undertaken to determine the effects of particle size and matrix solution composition on matrix interferences for slurry nebulisation analysis of clay minerals by inductively coupled plasma atomic emission spectrometry. Four size fractions of Grundite illite and Georgia kaolinite 3 were used for these analyses. The spectrometer was calibrated with multi-element solution standards. Elemental recoveries were determined by comparing the results obtained using slurry nebulisation with results for elemental determination from hydrofluoric acid digests of the samples. A differential effect of particle size on elemental recoveries was observed for samples prepared in HNO3 and NaCl matrix solutions. For the Grundite illite, Si and Al recoveries significantly decreased with increases in particle size, Mg and Fe recoveries decreased to a lesser extent and Ca recoveries were independent of particle size. These differences were attributed to the displacement of various amounts of Fe, Mg and Ca from clay particles into the matrix solutions by matrix solution cations and the more efficient transport of the matrix solutions than the particles to the plasma. In addition, Al recoveries were from 2 to 21% lower than Si recoveries for samples prepared in the HNO3 matrix solutions. Aluminium recoveries significantly increased with increases in observation height, which indicates that Al recoveries were limited by the incomplete dissociation -excitation of clay particles within the plasma. Furthermore, Al recoveries were significantly greater for samples prepared in 0.1 M NaCl than for samples prepared in 1 M HNO3. This effect is tentatively attributed to the enhanced collisional dissociation of clay particles. Matrix interferences may introduce substantial bias in results of slurry nebulisation analyses. This bias, however, can be minimised by restricting slurry nebulisation analyses to samples with particle sizes less than 2 µm and by using 0.1 M NaCl matrix solutions.

Previous studies have shown that matrix-assisted laser desorption/ionization-imaging mass spectrometry (MALDI-IMS) is useful for studying the distribution of various small metabolites, particularly lipids. However, in this technique, selective ionization of the target molecules is imperative, particularly when analyzing small molecules. Since the sample clean-up procedures available for the MALDI-IMS of small metabolites are limited, the tissue sample will contain numerous molecular species other than the target molecules. These molecules will compete for ionization resulting in severe ion suppression. Hence, it is necessary to develop and optimize a sample preparation protocol for the target molecules. In this study, through model experiments using reference compounds, we optimized the composition of the matrix solution used for positively charged lipids in terms of the concentration of the organic solvent and presence/absence of alkali metal salts. We demonstrated that a high concentration of organic solvent in the matrix solution favors the preferential detection of lipids over peptides. The presence of alkali metal salts in the matrix solution was favorable for the detection of polar lipids, while a salt-free matrix solution was suitable for the detection of nonpolar lipids. Furthermore, potassium salts added to the matrix solution caused merging of various lipid adducts (adducts with proton, sodium, and potassium) into one single potassiated species. Using the optimized protocols, we selectively analyzed phosphatidylcholine (PC) and triacylglycerol (TG) with different fatty acid compositions in a rat kidney section.

Importance. The current stage of technological development of society is characterized by the intensive integration of artificial intelligence (AI) technologies into professional spheres. AI-based technical solutions make it possible to automate some routine processes and free up time for humans to solve other more important and complex issues. Gradually, the interaction of specialistswith AI tools to solve professional problems is becoming a daily practice. In this regard, the training of qualified personnel at the university for the realities of today is impossible without integrating professionally oriented AI tools into the student learning process. Law is one of the activity fields in which modern AI technologies are able to take on many professional tasks. At the same time, the systematic integration of AI-based technical solutions into the university’s law student training process is impossible without a comprehensive study of the entire range of AI tools and their professionally oriented potential. The purpose of the work is to develop a matrix of AI-based technical solutions used in the professional training of future lawyers.Materials and Methods. The study is conducted on the expert assessment method basis. This allowed the authors to: a) identify a list of professional tasks solved by lawyers in the field of professional activity; b) based on the identified tasks, develop a matrix of AI-based technical solutions used in the professional training of future lawyers. The materials are scientific papers on pedagogy, methods of teaching foreign languages and specialized disciplines, published in scientific journals indexed in the Ministry of National Security (Scopus and Web of Science), as well as those included in the list of the Higher Attestation Commission of the Russian Federation (K1, K2), the Federal State Educational Standard for Higher Education in the field of Law. The AI tools widely used among current lawyers, which they use in their professional activities to solve professional problems, are used as practical materials.Results and Discussion. A matrix of AI-based technical solutions used in the professional training of future lawyers has been developed. The matrix is presented according to twelve professional tasks that lawyers solve in the course of their professional activities. The main and most accessible AI-based technical solutions for teachers of specialized disciplines that can help lawyers solve professional problems are the following: Legal AI tools, Legal Document Generator and DocZilla AI are used to draw up contracts (lease, sale, employment agreements, etc.), DocZilla AI and Genie AI – for the analysis and comparison of document editions, Mistral AI and LexisNexis – for checking documents for errors and contradictions, ROSS Intelligence and WestLaw – to search for relevant court decisions and analyze use cases, TrademarkVision and PatentPal – to search for similar trademarks, Perplexity AI – to analyze license agreements, Legalese Decoder, ChatGPT, YandexGPT, GigaChat and DeepSeek – to simplify legal terms for clients (colleagues, students), Canva and MidJourney – to visualize processes (for example, judicial meetings), LegalAI and Jasper AI – for legal advice, Perplexity AI, ChatGPT, YandexGPT, GigaChat and DeepSeek – for mathematical calculations (taxes, insurance payments, etc.), MidJourney – to create images of suspects, Legalese Decoder and Mistral AI are used for conducting examinations (handwriting, ballistic, etc.).Conclusion. The research novelty is the development of a matrix of AI-based technical solutions used in the professional training of future lawyers. The perspective of the conducted research lies in the development of step-by-step methods of teaching aspects of specialized disciplines based on the students’practice with specific technical solutions based on AI.

In spite of the potential advantages of the 'matrix solution' to DGLAP, an instability in the set of linear decomposition equations obtained from it, when decomposing the structure function data, hinders its application. Here, we pre-constrain the solutions of the decomposition equations, based on the principle of positivity of the parton distributions and well-known facts such as the increasing nature of gluons with decreasing values of Bjorken x. Thus, reasonable solutions for evolved parton distributions may be extracted within this framework.

Formulation of Matrix Solutions for Use in Matrix-Assisted Laser Desorption/Ionization of Biomolecules

Summary Amounts of inorganic phosphate (P) sorbed by two unfertilized soils, during times less than required to reach equilibrium, were affected by the ionic strength and cation species of the matrix solution. For non‐equilibrium conditions the amounts of P sorbed increased with increasing ionic strength and were greater with Ca 2+ than Na + . For higher P additions, resulting in equilibrium solution P concentrations greater than 30 to 40μrnole 1 −1 , the effects of the matrix solution on P sorption were maintained at equilibrium, whereas at lower P additions the dependence of sorption on matrix solution composition was eliminated at equilibrium. Equilibrium sorption isotherms for each soil and matrix solution were described by three Langmuir equations, which corresponded to distinct concentration ranges or regions (I, II, and III) on the overall isotherm. The free energies of sorption (Δ G ) for each region, were essentially independent of the soil matrix solution. The sorption maxima for regions I and II of the isotherm for a particular soil were also virtually independent of the matrix solution used. The sorption maximum for region III, however, was markedly dependent on the matrix solution, implying a potential‐determining (p.d.) sorption mechanism.

This paper introduces a new algorithm for solving the matrix Riccati equation. Differential equations for the eigenvalues and eigenvectors of the solution matrix are developed in which their derivatives are expressed in terms of the eigenvalues and eigenvectors themselves and not as functions of the solution matrix. The solution of these equations yields, then, the time behavior of the eigenvalues and eigenvectors of the solution matrix. A reconstruction of the matrix itself at any desired time is immediately obtained through a trivial similarity transformation. This algorithm serves two purposes. First, being a square-root solution, it entails all the advantages of square root algorithms such as nonnegativedefiniteness and accuracy. Secondly, it furnishes the eigenvalues and eigenvectors of the solution matrix continuously without resorting to the complicated route of solving the equation directly and then decomposing the solution matrix into its eigenvalues and eigenvectors. The accuracy and the stability of the new algorithm are demonstrated through an example. It is shown that the algorithm works in a case where the ordinary algorithm fails and the closed-form solution cannot be computed as a result of numerical difficulties.

alpha-Synuclein (alpha-syn) phosphorylation at serine 129 is characteristic of Parkinson disease (PD) and related alpha-synulceinopathies. However, whether phosphorylation promotes or inhibits alpha-syn aggregation and neurotoxicity in vivo remains unknown. This understanding is critical for elucidating the role of alpha-syn in the pathogenesis of PD and for development of therapeutic strategies for PD. To better understand the structural and molecular consequences of Ser-129 phosphorylation, we compared the biochemical, structural, and membrane binding properties of wild type alpha-syn to those of the phosphorylation mimics (S129E, S129D) as well as of in vitro phosphorylated alpha-syn using a battery of biophysical techniques. Our results demonstrate that phosphorylation at Ser-129 increases the conformational flexibility of alpha-syn and inhibits its fibrillogenesis in vitro but does not perturb its membrane-bound conformation. In addition, we show that the phosphorylation mimics (S129E/D) do not reproduce the effect of phosphorylation on the structural and aggregation properties of alpha-syn in vitro. Our findings have significant implications for current strategies to elucidate the role of phosphorylation in modulating protein structure and function in health and disease and provide novel insight into the underlying mechanisms that govern alpha-syn aggregation and toxicity in PD and related alpha-synulceinopathies.

Uniform matrix deposition on tissue samples for matrix-assisted laser desorption/ionization (MALDI) is key for reproducible analyte ion signals. Current methods often result in nonhomogenous matrix deposition, and take time and effort to produce acceptable ion signals. Here we describe a fully-automated method for matrix deposition using an enclosed spray chamber and spray nozzle for matrix solution delivery. A commercial air-atomizing spray nozzle was modified and combined with solenoid controlled valves and a Programmable Logic Controller (PLC) to control and deliver the matrix solution. A spray chamber was employed to contain the nozzle, sample, and atomized matrix solution stream, and to prevent any interference from outside conditions as well as allow complete control of the sample environment. A gravity cup was filled with MALDI matrix solutions, including DHB in chloroform/methanol (50:50) at concentrations up to 60mg/mL. Various samples (including rat brain tissue sections) were prepared using two deposition methods (spray chamber, inkjet). A linear ion trap equipped with an intermediate-pressure MALDI source was used for analyses. Optical microscopic examination showed a uniform coating of matrix crystals across the sample. Overall, the mass spectral images gathered from tissues coated using the spray chamber system were of better quality and more reproducible than from tissue specimens prepared by the inkjet deposition method.

We improve the construction of exact matrix solutions for nonlinear wave equations by using unitary anti-Hermitian and anticommuting matrices. We prove the theorem that constructs the matrix functions un satisfying the nonlinear wave equation for a set of special potentials. In this case, the graph of complex solution u1 has a soliton-like form with a finite number of coils. Exponential representation of matrix solutions un is associated with continuous rotations that can be used for describing intrinsic rotations and state changes of elementary particles. We also prove the theorem on the decomposition of continuous rotation (described by solution u2) onto three simultaneous rotations about coordinate vectors. Each of the three constructed matrix solutions u3 is also decomposed into the triplet of elementary matrix solutions.

This paper shows how to write node or mesh analysis linear circuit equations by inspection of the circuit schematic diagram and obtains two different matrix solutions of these equations. The linear circuit can have resistances or impedances, controlled sources, ideal operational amplifiers, or mutually coupled coils. The first matrix solution finds the node-voltage or mesh-current vector in terms of matrix operations with the inspection matrices. Also, this method gives a matrix solution for any arbitrary output vector in terms of the node-voltage or mesh-current solution vector, the independent-source vector, and the inspection matrices. The second matrix solution method finds the solution for a vector consisting of all node voltages or mesh currents, dependent sources, controlling variables, and any output variable(s) using a single matrix equation. Matrix methods of circuit analysis are now appropriate for student use because of the existence of calculators capable of solving large matrices and the availability of inexpensive math programs for personal computers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We are extending the formation of matrix solutions un for linear and nonlinear wave equations by construction of unitary anti-Hermitian-anti-commuting matrices up to the eighth order. We use Clifford algebras C(0,n) with periodicity in modulo 8 to construct coupled matrix solutions. We also propose to use the matrix solutions for describing the intrinsic rotations of particles.