Decoupling approximation of P- and S-wave phase velocities in orthorhombic media

Characterizing the kinematics of seismic waves in elastic orthorhombic media involves nine independent parameters. All wave modes, P-, S1-, and S2-waves, are intrinsically coupled. Since the P-wave propagation in orthorhombic media is weakly dependent on the three S-wave velocity parameters, they are set to zero under the acoustic assumption. The number of parameters required for the corresponding acoustic wave equation is thus reduced from nine to six, which is very practical for the inversion algorithm. However, the acoustic wavefields generated by the finite-difference scheme suffer from two types of S-wave artifacts, which may result in noticeable numerical dispersion and even instability issues. Avoiding such artifacts requires a class of spectral methods based on the low-rank decomposition. To implement a six-parameter pure P-wave approximation in orthorhombic media, we develop a novel phase velocity approximation approach from the perspective of decoupling P- and S-waves. In the exact P-wave phase velocity expression, we find that the two algebraic expressions related to the S1- and S2-wave phase velocities play a negligible role. After replacing these two algebraic expressions with the designed constant and variable respectively, the exact P-wave phase velocity expression is greatly simplified and naturally decoupled from the characteristic equation. Similarly, the number of required parameters is reduced from nine to six. We also derive an approximate S-wave phase velocity equation, which supports the coupled S1- and S2-waves and involves nine independent parameters. Error analyses based on several orthorhombic models confirm the reasonable and stable accuracy performance of the proposed phase velocity approximation. We further derive the approximate dispersion relations for the P-wave and the S-wave system in orthorhombic media. Numerical experiments demonstrate that the corresponding P- and S-wavefields are free of artifacts and exhibit good accuracy and stability.

Algebraic apparatus of a Sequences of the Derivation

Предлагаемая работа нацелена на искоренение недостатков Последовательностей Вывода ({\bf ПВ}), присущих программам вычислительных машин. Постановка задачи и предшествующие исследования {\bf ПВ} составляют содержание работ [10-12]. На завершающем этапе статьи {\bf ПВ} заменяются Функционально-Эквивалентными ({\bf ФЭ}) им алгебраическими выражениями, на входе в которые имеют место те же Функциональные Зависимости ({\bf ФЗ}), что и в исходных {\bf ПВ}. Результат работы, соответствующий её целям, содержится в Теореме 3. The offered work is aimed at eradication of shortcomings of the Sequences of the Derivation ({\bf SD}) inherent in programs of computers. The establish of the problem and the previous researches {\bf SD} make the content of works [10-12]. At the final stage of this article {\bf SD} are replaced with algebraic expressions wich is Functionally Equivalent ({\bf FE}) to them on which entrance the same Functional Dependences ({\bf FD}), as in initial {\bf SD}. The result of work answering her purpose contains in the Theorem 3.

An Alternative Approach to Geometric Harmonization of the Great Mosque in Córdoba

This research constitutes an alternative to proportional composition studies of the original Great Mosque and its four extensions in Córdoba, based on diagonals of a square and rectangles in ratio 1:√2 and 1:√3 (Fernández-Puertas, 2000, 2008). The method for this alternative research consists of graphic analyses by iteration of hypothetical products of the golden section in AutoCAD 2D software conducted on architectural drawings of the original Great Mosque and its four extensions, in reconstruction, according to measurements from the relevant literature. The alternative method insists on geometric harmonization derived from only one starting length in all drawings of same scale. It resulted in the production of a single harmonization pattern based on the golden section, with an additional sequence of a√2/Φn, successively developed for the original monument and its four extensions. It also includes otherwise excluded basic composition elements (minarets) and reveals otherwise hidden proportional qualities. The alternative approach enabled a deduction of algebraic expressions having only one variable for all drawings of the same scale. Their arithmetic values and deviations from real dimensions are calculated. Geometric harmonization by golden section with another starting length is applied to the drawing of the elevation at a different scale.

Hyperbolically symmetric sources in Palatini f(R) gravity

A thorough examination of static hyperbolically symmetric matter configuration in the context of Palatini f(R) gravitational theory has been carried out in this manuscript. Following the work of Herrera et al. (Phys. Rev. D 103: 024037, 2021) we worked out the modified gravitational equations and matching conditions using the Palatini technique of variation in Einstein–Hilbert action. It is found from the evaluations that the energy density along with the contribution of dark source terms is inevitably negative which is quite useful in explaining several quantum field effects, because negative energies are closely linked with the quantum field theory. Such negative energies may also assist in time-travel to the past and formation of artificial wormholes. Furthermore, we evaluated the algebraic expressions for the mass of interior hyperbolical geometry and total energy budget, i.e., the Tolman mass of the considered source. Also, the structure scalars are evaluated to analyze the properties of matter configuration. Few analytical techniques are also presented by considering several cases to exhibit the exact analytical static solutions of the modified gravitational equations.

Electrospinning of a Copolymer PVDF-co-HFP Solved in DMF/Acetone: Explicit Relations among Viscosity, Polymer Concentration, DMF/Acetone Ratio and Mean Nanofiber Diameter

The process of electrospinning polymer solutions depends on many entry parameters, with each having a significant impact on the overall process and where complexity prevents the expression of their interplay. However, under the assumption that most parameters are fixed, it is possible to evaluate the mutual relations between pairs or triples of the chosen parameters. In this case, the experiments were carried out with a copolymer poly(vinylidene-co-hexafluoropropylene) solved in mixed N,N’-dimethylformamide (DMF)/acetone solvent for eight polymer concentrations (8, 10, 12, 15, 18, 21, 24, and 27 wt.%) and five DMF/acetone ratios (1/0, 4/1, 2/1, 1/1, 1/2). Processing of the obtained data (viscosity, mean nanofiber diameter) aimed to determine algebraic expressions relating both to viscosity and a mean nanofiber diameter with polymer concentration, as well as DMF/acetone ratio. Moreover, a master curve relating these parameters with no fitting factors was proposed continuously covering a sufficiently broad range of concentration as well as DMF/acetone ratio. A comparison of algebraic evaluation with the experimental data seems to be very good (the mean deviation for viscosity was about 2%, while, for a mean nanofiber diameter was slightly less than 10%).

Examination of Semantic Structures Used by Teacher Candidates to Transform Algebraic Expressions into Verbal Problems

The aim of this study is to examine the semantic structures used by mathematics teacher candidates to transform algebraic expressions into verbal problems. The research is a descriptive study in the survey model, which is one of the quantitative research types. The study group of the research consists of 165 teacher candidates studying in the primary school mathematics teaching department of a state university in the south of Turkey in the 2019-2020 academic years. 73.2% of the teacher candidates in the study group are female and 26.8% are male. Criterion sampling method, one of the purposeful sampling methods, was used in the selection of teacher candidates in the study group. While the Algebraic Expression Questionnaire Form was used as the data collection tool, the evaluation rubric of verbal problems was used in the analysis of the data. As a result of the research, it has been revealed that pre-service teachers are more successful in transforming algebraic expressions into verbal problems, but they have problems in creating problems with algebraic expressions that make up systems of equations. Again in the study, it was concluded that pre-service teachers used addition and subtraction problems more than multiplication and division problems. On the other hand, when the problems in the type of addition and subtraction are examined in the study, in the type of combining and separating; It has been concluded that the category of equal groups is mostly used in the problems of multiplication and division.

Examination of Semantic Structures Used by Teacher Candidates to Transform Algebraic Expressions into Verbal Problems

The aim of this study is to examine the semantic structures used by mathematics teacher candidates to transform algebraic expressions into verbal problems. The research is a descriptive study in the survey model, which is one of the quantitative research types. The study group of the research consists of 165 teacher candidates studying in the primary school mathematics teaching department of a state university in the south of Turkey in the 2019-2020 academic years. 73.2% of the teacher candidates in the study group are female and 26.8% are male. Criterion sampling method, one of the purposeful sampling methods, was used in the selection of teacher candidates in the study group. While the Algebraic Expression Questionnaire Form was used as the data collection tool, the evaluation rubric of verbal problems was used in the analysis of the data. As a result of the research, it has been revealed that pre-service teachers are more successful in transforming algebraic expressions into verbal problems, but they have problems in creating problems with algebraic expressions that make up systems of equations. Again in the study, it was concluded that pre-service teachers used addition and subtraction problems more than multiplication and division problems. On the other hand, when the problems in the type of addition and subtraction are examined in the study, in the type of combining and separating; It has been concluded that the category of equal groups is mostly used in the problems of multiplication and division.

Mapping First to Second wave transition of covid19 Indian data via Sigmoid function and prediction of Third wave

Understanding first and second wave of covid19 Indian data along with its few selective states, we have realized a transition between two Sigmoid pattern with twice larger growth parameter and maximum values of cumulative data. As a result of those transition, time duration of second wave shrink to half of that first wave with four times larger peak values. It is really interesting that the facts can be easily understood by simple algebraic expressions of Sigmoid function. After understanding the crossing zone between first and second wave curves, a third wave Sigmoid pattern is guessed.

Congruence properties of indices of triangular numbers multiple of other triangular numbers

For any non-square integer multiplier \(k\), there is an infinity of triangular numbers multiple of other triangular numbers. We analyze the congruence properties of indices \(\xi\) of triangular numbers multiple of triangular numbers. Remainders in congruence relations \(\xi\) modulo \(k\) come always in pairs whose sum always equal \((k-1)\), always include 0 and \((k-1)\), and only 0 and \((k-1)\) if \(k\) is prime, or an odd power of a prime, or an even square plus one or an odd square minus one or minus two. If the multiplier \(k\) is twice the triangular number of \(n\), the set of remainders includes also \(n\) and \((n^{2}-1)\) and if \(k\) has integer factors, the set of remainders include multiples of a factor following certain rules. Algebraic expressions are found for remainders in function of \(k\) and its factors, with several exceptions. This approach eliminates those \(\xi\) values not providing solutions.

Archimedes of Syracuse and Sir Isaac Newton: On the Quadrature of a Parabola

Good mathematics stands the test of time. As culture changes, we often ask different questions, bringing new perspectives, but modern mathematics stands on ancient discoveries. Isaac Newton’s discovery of calculus (along with Leibniz) may seem old but is predated by Archimedes’ findings. Current mathematics students should be familiar with parabolas and simple curves; in our introductory calculus courses, we teach them to compute the areas under such curves. Our modern approach derives its roots from Newton’s work; however, we have filled in many of the gaps in the pursuit of mathematical rigor. What many students may not know is that Archimedes solved the area problem for parabolas long before the use of algebraic expressions became mainstream. Archimedes used the geometry of the ancient Greeks, which gave him a vastly different perspective. In this paper we provide both Archimedes’ and Newton’s proofs involving the quadrature of the parabola, trying to remain true to their original texts as much as feasible.