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Journal articles on the topic 'Mathlie for mathematica'

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Published: 4 June 2021

Last updated: 19 February 2022

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1

Welchman-Tischler, Rosamond. "Making Mathematical Connections." Arithmetic Teacher 39, no. 9 (May 1992): 12–17. http://dx.doi.org/10.5951/at.39.9.0012.

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Mathematically literate students should view mathematics as a way of looking at their environment that aids understanding and adds insight This attitude toward mathe matics can be fostered in the daily routines of the classroom. Mathematical experiences need not be restricted to the “math period” but can be incorporated throughout the school day. The importance of making mathematical connections, both within mathematics and between mathematics and other curriculum areas, is emphasized by the inclusion of “mathematical connection” as one of the curriculum standards for school mathematics (NCTM 1989). This article how how a simple manipulative device useful for taking attendance can be used to exercise mathematical thinking processes in a variety of contexts at different grade levels throughout elementary school.

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2

Semenikhina, Olena, and Volodymyr Proshkin. "APPLICATION OF COMPUTER MATHEMATICAL TOOLS IN PROFESSIONAL PREPARATION OF FUTURE TEACHERS OF MATHEMATICS." OPEN EDUCATIONAL E-ENVIRONMENT OF MODERN UNIVERSITY, no. 4 (2018): 60–73. http://dx.doi.org/10.28925/2414-0325.2018.4.6073.

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The article deals with the problem of improving the quality of the future teacher of mathematics by means of information technologies. There are two classes of software tools of mathematical direction such as 1) systems of computer mathematics which use traditional notation and methods of writing formulas (Maple, MatLab, Maxima, etc.), these systems are particularly effective in solving various applications, primarily mathematical modeling tasks in science and technology; 2) programs of dynamic mathematics which provide the possibility of dynamic changes of the original mathematical design, the study of the set of its numerical characteristics or their relations in dynamics (GeoGebra, Cabri, Mathkit, and the like). It is established that under the programs of dynamic mathematics it is necessary to understand the means of computer visualization of mathematical knowledge, which provide dynamic operation of various mathematical objects and the possibility of operative obtaining information about their properties. The term "computer mathematical tool" is specified as a virtual mechanism or algorithm of a computer program, or the program itself, which is used to create or study mathematical objects or their components through the numerical and geometric characteristics of the objects themselves. The methodical features of the use of computer mathematical tools have been distinguished: the possibility of dynamic changes in the drawing and observation of the trajectories of individual points opens the way for organizing the experimental and creative activity of all participants in the educational process; the ability to pre-install the tools or to expand the toolkit that simplifies the process of building or limiting tools; the computer frees the teacher from a debilitating check of the construction, and so on. The results of the experiment related to the study of the influence of the programs of dynamic mathematics on the level of academic students’ achievements have been presented. The positive influence of the use of dynamic mathematics programs and the corresponding mathematical tools on the level of mathematical preparation of schoolchildren has been proved. The necessity of preparing the future teacher of mathematics for using this tool in the process of professional activity has been indicated. The requirements of the standard of pedagogical education, the needs of school maths in the main school and the distribution of computer tools within the framework of training a modern mathematics teacher have been specified.

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3

IMAFUJI, NORIKO, and MITSUYUKI OCHIAI. "COMPUTER AIDED KNOT THEORY USING MATHEMATICA AND MATHLINK." Journal of Knot Theory and Its Ramifications 11, no. 06 (September 2002): 945–54. http://dx.doi.org/10.1142/s0218216502002062.

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We introduce a computer tool called Knot2000(K2K) which was developed for the purpose of support for the research of knot theory. K2K is a package on Mathematica in which consists of 19 functions and it has already been opened to the public with other external programs and data files. In this paper, we will describe focusing on the usages of each functions and some examples of effective ways to use K2K, and show its availability.

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4

Parmar, Rakesh K., Gradimir V. Milovanović, and Tibor K. Pogány. "Multi-parameter Mathieu, and alternating Mathieu series." Applied Mathematics and Computation 400 (July 2021): 126099. http://dx.doi.org/10.1016/j.amc.2021.126099.

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5

Janhunen, Pekka, Douglas Stein, and Robert S. Wolff. "MathHDF: MathLink-Based Distributed Visualization between Mathematica and HDF Files." Computers in Physics 7, no. 3 (1993): 290. http://dx.doi.org/10.1063/1.4823178.

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6

Ryabov, Yu A. "Numeric-analytical construction of Mathieu functions." Mathematica Bohemica 124, no. 1 (1999): 15–28. http://dx.doi.org/10.21136/mb.1999.125982.

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7

Winsberg, Eric. "Wittgenstein, Finitism, and the Foundations of Mathematics. Mathieu Marion." Philosophy of Science 67, no. 3 (September 2000): 533–36. http://dx.doi.org/10.1086/392796.

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8

Floyd, Juliet. "Mathieu Marion. Wittgenstein, Finitism, and the Foundations of Mathematics." Philosophia Mathematica 10, no. 1 (February 1, 2002): 67–88. http://dx.doi.org/10.1093/philmat/10.1.67.

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9

Pitcher, Neil. "Educational software in mathematics: developing and using a mathwise module." International Journal of Mathematical Education in Science and Technology 29, no. 5 (September 1998): 709–20. http://dx.doi.org/10.1080/0020739980290507.

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10

Gnanachandra, P., and A. M. Kumar. "TOPOLOGICAL GROUPS: VIRTUE OF PRE-OPEN SETS." Advances in Mathematics: Scientific Journal 10, no. 1 (January 20, 2021): 385–89. http://dx.doi.org/10.37418/amsj.10.1.38.

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In this paper, we introduce notions of $\mathit{p}$-topological group and $\mathit{p}$-irresolute topological group which are generalizations of the notion topological group. We discuss the properties of $\mathit{p}$-topological group with illustrated examples. Also, we prove that translation and inversion in $\mathit{p}$-topological group are $\mathit{p}$-homeomorphism.

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11

Sauer, Tilman. "Hilberts Ruf nach Bern." Gesnerus 57, no. 3-4 (November 27, 2000): 182–205. http://dx.doi.org/10.1163/22977953-0570304004.

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Attempts to find a successor for the late Berne professor of mathematics Johann Heinrich Graf in winter and spring 1919 proved fruitless when, unexpectedly, the renowned Gottingen mathematician David Hilbert intimated that he himself would, under certain conditions, accept a call to Berne. Subsequent negotiations between Hilbert and the Bernese and Prussian authorities mainly concerned the question of salary, with offers of unusually high compensation both in Berne and in Gottingen. Although Hilbert eventually declined for financial reasons, it is argued that his consi-derations may well have been rather complex and reflect both his own situation in Gottingen and the general state of affairs of university mathe-matics in Germany and Switzerland after the First World War.

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12

Pogány, Tibor K., H. M. Srivastava, and Živorad Tomovski. "Some families of Mathieu a-series and alternating Mathieu a-series." Applied Mathematics and Computation 173, no. 1 (February 2006): 69–108. http://dx.doi.org/10.1016/j.amc.2005.02.044.

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13

Baumann, G. "MathLie a program of doing symmetry analysis." Mathematics and Computers in Simulation 48, no. 2 (December 1998): 205–23. http://dx.doi.org/10.1016/s0378-4754(98)00143-8.

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14

Shikhmurzaev, Yulii D., and Grigori M. Sisoev. "Spiralling liquid jets: verifiable mathematical framework, trajectories and peristaltic waves." Journal of Fluid Mechanics 819 (April 24, 2017): 352–400. http://dx.doi.org/10.1017/jfm.2017.169.

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The dynamics of a jet of an inviscid incompressible liquid spiralling out under the action of centrifugal forces is considered with both gravity and the surface tension taken into account. This problem is of direct relevance to a number of industrial applications, ranging from the spinning disc atomization process to nanofibre formation. The mathematical description of the flow by necessity requires the use of a local curvilinear non-orthogonal coordinate system centred around the jet’s baseline, and we present the general formulation of the problem without assuming that the jet is slender. To circumvent the inconvenience inherent in the non-orthogonality of the local coordinate system, the orthonormal Frenet basis is used in parallel with the local non-orthogonal basis, and the equation of motion, with the velocity considered with respect to the local coordinate system, is projected onto the Frenet basis. The variation of the latter along the baseline is then described by the Frenet equations which naturally brings the baseline’s curvature and torsion into the equations of motion. This technique allows one to handle different line-based non-orthogonal curvilinear coordinate systems in a straightforward and mathematically transparent way. An analysis of the slender-jet approximation that follows the general formulation shows how a set of ordinary differential equations describing the jet’s trajectory can be derived in two cases: $\mathit{We}=O(1)$ and $\unicode[STIX]{x1D716}\mathit{We}=O(1)$ as $\unicode[STIX]{x1D716}\rightarrow 0$, where $\unicode[STIX]{x1D716}$ is the ratio of characteristic length scales across and along the jet and $\mathit{We}$ is the Weber number. A one-dimensional model for the propagation of nonlinear peristaltic disturbances along the jet is derived in each of these cases. A critical review of the work published on this topic is presented showing where errors typically occur and how to identify and avoid them.

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15

Vladimirov, V. A. "Magnetohydrodynamic drift equations: from Langmuir circulations to magnetohydrodynamic dynamo?" Journal of Fluid Mechanics 698 (March 23, 2012): 51–61. http://dx.doi.org/10.1017/jfm.2012.40.

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AbstractWe derive the closed system of averaged magnetohydrodynamic (MHD) equations for general oscillating flows. The used small parameter of our asymptotic theory is the dimensionless inverse frequency, and the leading term for a velocity field is chosen to be purely oscillating. The employed mathematical approach combines the two-timing method and the notion of a distinguished limit. The properties of commutators are used to simplify calculations. The derived averaged equations are similar to the original MHD equations, but surprisingly (instead of the commonly expected Reynolds stresses) a drift velocity plays a part of an additional advection velocity. In the special case of a vanishing magnetic field $\mathbi{h}\equiv 0$, the averaged equations produce the Craik–Leibovich equations for Langmuir circulations (which can be called ‘vortex dynamo’). We suggest that, since the mathematical structure of the full averaged equations for $\mathbi{h}\not = 0$ is similar to those for $\mathbi{h}\equiv 0$, these full equations could lead to a possible mechanism of MHD dynamo, such as the generation of the magnetic field of the Earth.

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16

Zhang, Xiao Tao, Ji Yuan Yang, and Jin Xuan Xie. "A Study on Elliptical Gas Flow in Double Porosity Gas Reservoirs." Applied Mechanics and Materials 275-277 (January 2013): 576–83. http://dx.doi.org/10.4028/www.scientific.net/amm.275-277.576.

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Elliptical flow exists in the near vertical fracture area, or in anisotropic reservoirs, and the classical radial flow models cannot provide complete analysis for elliptical flow. This paper presents a mathematic model for gas elliptical flow in double porosity gas reservoirs. The differential equation of gas flow is written in Mathieu equation, so that the solution can be expressed also by Mathieu functions. The numerical calculation of relevant Mathieu functions , and its derivative is done to obtain the dimensionless pseudo pressure drop in Laplace space. Property sensitivities of double porosity systems including combination of , parameter related to anisotropy, storativity ratio , interporosity flow coefficient are studied by using the Laplace numerical inversion. The new solution not only includes factors considered for classic solutions in previous papers, but also incorporates the effect of reservoir anisotropy. It is kind of modification and supplementation to classic solutions.

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17

Gerhold, Stefan, and Živorad Tomovski. "Asymptotic expansion of Mathieu power series and trigonometric Mathieu series." Journal of Mathematical Analysis and Applications 479, no. 2 (November 2019): 1882–92. http://dx.doi.org/10.1016/j.jmaa.2019.07.029.

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18

Joseph, M. A. "Book Review. Wittgenstein, Finitism, and the Foundations of Mathematics Mathieu Marion." Mind 110, no. 438 (April 1, 2001): 501–4. http://dx.doi.org/10.1093/mind/110.438.501.

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19

Tomovski, Zivorad. "New integral and series representations of the generalized Mathieu series." Applicable Analysis and Discrete Mathematics 2, no. 2 (2008): 205–12. http://dx.doi.org/10.2298/aadm0802205t.

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By using some recently investigated fourier sine integral representations for the Mathieu type series (see [4]), new integral and series representations are derived here for certain general families of Mathieu type series.

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20

DANILENKO, ALEXANDRE I., and MARIUSZ LEMAŃCZYK. "K-property for Maharam extensions of non-singular Bernoulli and Markov shifts." Ergodic Theory and Dynamical Systems 39, no. 12 (April 5, 2018): 3292–321. http://dx.doi.org/10.1017/etds.2018.14.

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It is shown that each conservative non-singular Bernoulli shift is either of type $\mathit{II}_{1}$ or $\mathit{III}_{1}$. Moreover, in the latter case the corresponding Maharam extension of the shift is a $K$-automorphism. This extends earlier results obtained by Kosloff for equilibrial shifts. Non-equilibrial shifts of type $\mathit{III}_{1}$ are constructed. We further generalize (partly) the main results to non-singular Markov shifts.

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21

Gannon, Terry. "Much ado about Mathieu." Advances in Mathematics 301 (October 2016): 322–58. http://dx.doi.org/10.1016/j.aim.2016.06.014.

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22

Jiang, Xiaoyan, and Jianguo Sun. "Local geometric properties of the lightlike Killing magnetic curves in de Sitter 3-space." AIMS Mathematics 6, no. 11 (2021): 12543–59. http://dx.doi.org/10.3934/math.2021723.

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<abstract><p>In this article, we mainly discuss the local differential geometrical properties of the lightlike Killing magnetic curve $ \mathit{\boldsymbol{\gamma }}(s) $ in $ \mathbb{S}^{3}_{1} $ with a magnetic field $ \boldsymbol{ V} $. Here, a new Frenet frame $ \{\mathit{\boldsymbol{\gamma }}, \boldsymbol{ T}, \boldsymbol{ N}, \boldsymbol{ B}\} $ is established, and we obtain the local structure of $ \mathit{\boldsymbol{\gamma }}(s) $. Moreover, the singular properties of the binormal lightlike surface of the $ \mathit{\boldsymbol{\gamma }}(s) $ are given. Finally, an example is used to understand the main results of the paper.</p></abstract>

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23

Yamskulna, Gaywalee. "Mathieu–Zhao subspaces of vertex algebras." Journal of Algebra and Its Applications 18, no. 12 (November 3, 2019): 1950225. http://dx.doi.org/10.1142/s0219498819502256.

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We introduce a notion of Mathieu–Zhao subspaces of vertex algebras. Among other things, we show that for a vertex algebra [Formula: see text] and its subspace [Formula: see text] that contains [Formula: see text], [Formula: see text] is a Mathieu–Zhao subspace of [Formula: see text] if and only if the quotient space [Formula: see text] is a Mathieu–Zhao subspace of a commutative associative algebra [Formula: see text]. As a result, one can study the famous Jacobian conjecture in terms of Mathieu–Zhao subspaces of vertex algebras. In addition, for a [Formula: see text]-type vertex operator algebra [Formula: see text] that satisfies the [Formula: see text]-cofiniteness condition, we classify all Mathieu–Zhao subspaces [Formula: see text] that contain [Formula: see text].

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24

XUAN, WEI-FENG, and WEI-XUE SHI. "A NOTE ON SPACES WITH RANK 2-DIAGONAL." Bulletin of the Australian Mathematical Society 90, no. 1 (April 2, 2014): 141–43. http://dx.doi.org/10.1017/s0004972713001184.

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AbstractWe prove that if a space $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ with a rank 2-diagonal either has the countable chain condition or is star countable then the cardinality of $X$ is at most $\mathfrak{c}$.

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25

Estrada Gonzalez, Luis, and Manuel Tapia-Navarro. ""A Smack of Irrelevance" in Inconsistent Mathematics?" Australasian Journal of Logic 18, no. 5 (July 21, 2021): 503–23. http://dx.doi.org/10.26686/ajl.v18i5.6925.

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Recently, some proponents and practitioners of inconsistent mathe- matics have argued that the subject requires a conditional with ir- relevant features, i.e. where antecedent and consequent in a valid conditional do not behave as expected in relevance logics —by shar- ing propositional variables, for example. Here we argue that more fine-grained notions of content and content-sharing are needed to ex- amine the language of (inconsistent) arithmetic and set theory, and that the conditionals needed in inconsistent mathematics are not as irrelevant as it is suggested in the current literature.

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26

Curtis, R. T. "Natural constructions of the Mathieu groups." Mathematical Proceedings of the Cambridge Philosophical Society 106, no. 3 (November 1989): 423–29. http://dx.doi.org/10.1017/s0305004100068158.

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In the second half of the last century the French mathematician Emil Mathieu discovered two quintuply transitive permutation groups, now labelled M12 and M24, acting on twelve and twenty-four letters respectively. With the classification of finite simple groups complete we now know that any other quintuply transitive permutation group, on any number of letters, must contain the corresponding alternating group. Indeed, the only quadruply transitive groups, other than the alternating and symmetric groups, are the point stabilizers in M12 and M24, which are denoted by M11 and M23 respectively. To put it another way, the study of multiply (≥ 4-fold) transitive groups now means the study of the symmetric groups and the Mathieu groups. Apart from their beauty and interest in their own right the Mathieu groups are involved in many of the other sporadic simple groups: see ([2], p. 238). Thus a detailed understanding of the other exceptional groups necessitates an intimate knowledge of M12 and M24.

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GUILLERA, JESUS, and MATHEW ROGERS. "RAMANUJAN SERIES UPSIDE-DOWN." Journal of the Australian Mathematical Society 97, no. 1 (July 7, 2014): 78–106. http://dx.doi.org/10.1017/s1446788714000147.

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AbstractWe prove that there is a correspondence between Ramanujan-type formulas for$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1/\pi $and formulas for Dirichlet$L$-values. Our method also allows us to reduce certain values of the Epstein zeta function to rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.

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28

Lee, Ang Wee, Nayef Mohamed Ghasem, and Mohamed Azlan Hussain. "Utilization of Mathematical Software Packages in Chemical Engineering Research." ASEAN Journal of Chemical Engineering 5, no. 2 (December 1, 2005): 125. http://dx.doi.org/10.22146/ajche.50180.

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Using Fortran taken as the starting point, we are now on the sixth decade of high-level programming applications. Among the programming languages available, computer algebra systems (CAS) appear to be a good choice in chemical engineering can be applied easily. Until the emergence of CAS, the assistance from a specialized group for large-scale programming is justified. Nowadays, it is more effective for the modern chemical engineer to rely on his/her own programming ability for problem solving. In the present paper, the abilities of Polymath, Maple, Matlab, Mathcad, and Mathematica in handling differential equations are illustrated for differential-algebraic equations, large system of nonlinear differential equations, and partial differential equations. The programming of solutions with these CAS are presented, contrasted, and discussed in relation to chemical engineering problems. Keywords: Computer algebra systems (CAS),computer simulation,Mathcad, Mathematica,Mathlab and numerical methods.

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29

Parmar, Rakesh, and Tibor Pogány. "On Mathieu-type series for the unified Gaussian hypergeometric functions." Applicable Analysis and Discrete Mathematics 14, no. 1 (2020): 138–49. http://dx.doi.org/10.2298/aadm190525014p.

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The main purpose of this paper is to present closed integral form expressions for the Mathieu-type a-series and for the associated alternating versions whose terms contain a generalized p-extended Gauss' hypergeometric function. Related bounding inequalities for the p-generalized Mathieu-type series are also obtained. Finally, a set of various (known or new) special cases and consequences of the results earned are presented.

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Pogany, Tibor K., and Rakesh K. Parmar. "On p–extended Mathieu series." Sv. 22(2018)=knj. 57=knj. 534 534, no. 22 (2018): 107–17. http://dx.doi.org/10.21857/90836cwk4y.

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31

BEER, GERALD, and M. I. GARRIDO. "BORNOLOGIES AND LOCALLY LIPSCHITZ FUNCTIONS." Bulletin of the Australian Mathematical Society 90, no. 2 (May 15, 2014): 257–63. http://dx.doi.org/10.1017/s0004972714000215.

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AbstractLet$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\langle X,d \rangle $be a metric space. We characterise the family of subsets of$X$on which each locally Lipschitz function defined on$X$is bounded, as well as the family of subsets on which each member of two different subfamilies consisting of uniformly locally Lipschitz functions is bounded. It suffices in each case to consider real-valued functions.

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32

JIANG, QINHUI, and CHANGGUO SHAO. "SOLVABILITY OF FINITE GROUPS WITH FOUR CONJUGACY CLASS SIZES OF CERTAIN ELEMENTS." Bulletin of the Australian Mathematical Society 90, no. 2 (July 21, 2014): 250–56. http://dx.doi.org/10.1017/s0004972714000495.

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AbstractAssume that $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}m$ and $n$ are two positive integers which do not divide each other. If the set of conjugacy class sizes of primary and biprimary elements of a group $G$ is $\{1, m, n, mn\}$, we show that up to central factors $G$ is a $\{p,q\}$-group for two distinct primes $p$ and $q$.

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Khosravi, Behrooz, and Behnam Khosravi. "Groups with the same orders of Sylow normalizers as the Mathieu groups." International Journal of Mathematics and Mathematical Sciences 2005, no. 9 (2005): 1449–53. http://dx.doi.org/10.1155/ijmms.2005.1449.

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There exist many characterizations for the sporadic simple groups. In this paper we give two new characterizations for the Mathieu sporadic groups. LetMbe a Mathieu group and letpbe the greatest prime divisor of|M|. In this paper, we prove thatMis uniquely determined by|M|and|NM(P)|, whereP∈Sylp(M). Also we prove that ifGis a finite group, thenG≅Mif and only if for every primeq,|NM(Q)|=|NG(Q′)|, whereQ∈Sylq(M)andQ′∈Sylq(G).

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34

Yoshino, Masafumi. "Global hypoellipticity of a Mathieu operator." Proceedings of the American Mathematical Society 111, no. 3 (March 1, 1991): 717. http://dx.doi.org/10.1090/s0002-9939-1991-1042277-2.

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35

Rand, R. H., S. M. Sah, and M. K. Suchorsky. "Fractional Mathieu equation." Communications in Nonlinear Science and Numerical Simulation 15, no. 11 (November 2010): 3254–62. http://dx.doi.org/10.1016/j.cnsns.2009.12.009.

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36

KĄKOL, JERZY, STEPHEN A. SAXON, and AARON R. TODD. "BARRELLED SPACES WITH(OUT) SEPARABLE QUOTIENTS." Bulletin of the Australian Mathematical Society 90, no. 2 (June 13, 2014): 295–303. http://dx.doi.org/10.1017/s0004972714000422.

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AbstractWhile the separable quotient problem is famously open for Banach spaces, in the broader context of barrelled spaces we give negative solutions. Obversely, the study of pseudocompact$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$and Warner bounded$X$allows us to expand Rosenthal’s positive solution for Banach spaces of the form$ C_{c}(X) $to barrelled spaces of the same form, and see that strong duals of arbitrary$C_{c}(X) $spaces admit separable quotients.

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37

Parovik, Roman. "Mathematical Modeling of Linear Fractional Oscillators." Mathematics 8, no. 11 (October 29, 2020): 1879. http://dx.doi.org/10.3390/math8111879.

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In this work, based on Newton’s second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov–Caputo fractional derivatives is carried out. For the resulting model equation, local initial conditions are set (the Cauchy problem). Numerical methods for solving the Cauchy problem using an explicit non-local finite-difference scheme (ENFDS) and the Adams–Bashforth–Moulton (ABM) method are considered. An analysis of the errors of the methods is carried out on specific test examples. It is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method. Forced oscillations of linear fractional oscillators (LFO) are investigated. Using the ABM method, the amplitude–frequency characteristics (AFC) were constructed, which were compared with the AFC obtained by the analytical formula. The Q-factor of the LFO is investigated. It is shown that the orders of fractional derivatives are responsible for the intensity of energy dissipation in fractional vibrational systems. Specific mathematical models of LFOs are considered: a fractional analogue of the harmonic oscillator, fractional oscillators of Mathieu and Airy. Oscillograms and phase trajectories were constructed using the ABM method for various values of the parameters included in the model equation. The interpretation of the simulation results is carried out.

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Orszaghova, J., H. Wolgamot, S. Draper, P. H. Taylor, and A. Rafiee. "Onset and limiting amplitude of yaw instability of a submerged three-tethered buoy." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2235 (March 2020): 20190762. http://dx.doi.org/10.1098/rspa.2019.0762.

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In this paper the dynamics of a submerged axi-symmetric wave energy converter are studied, through mathematical models and wave basin experiments. The device is disk-shaped and taut-moored via three inclined tethers which also act as a power take-off. We focus on parasitic yaw motion, which is excited parametrically due to coupling with heave. Assuming linear hydrodynamics throughout, but considering both linear and nonlinear tether geometry, governing equations are derived in 6 degrees of freedom (DOF). From the linearized equations, all motions, apart from yaw, are shown to be contributing to the overall power absorption. At higher orders, the yaw governing equation can be recast into a classical Mathieu equation (linear in yaw), or a nonlinear Mathieu equation with cubic damping and stiffness terms. The well-known stability diagram for the classical Mathieu equation allows prediction of onset/occurrence of yaw instability. From the nonlinear Mathieu equation, we develop an approximate analytical solution for the amplitude of the unstable motions. Comparison with regular wave experiments confirms the utility of both models for making relevant predictions. Additionally, irregular wave tests are analysed whereby yaw instability is successfully correlated to the amount of parametric excitation and linear damping. This study demonstrates the importance of considering all modes of motion in design, not just the power-producing ones. Our simplified 1 DOF yaw model provides fundamental understanding of the presence and severity of the instability. The methodology could be applied to other wave-activated devices.

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Marathe, Amol, and Anindya Chatterjee. "Asymmetric Mathieu equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2070 (February 14, 2006): 1643–59. http://dx.doi.org/10.1098/rspa.2005.1632.

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An inverted pendulum with asymmetric elastic restraints (e.g. a one-sided spring), when subjected to harmonic vertical base excitation, on linearizing trigonometric terms, is governed by an asymmetric Mathieu equation. This system is parametrically forced and strongly nonlinear (linearization for small motions is not possible). However, solutions are scaleable: if x ( t ) is a solution, then so is αx ( t ) for any real α >0. We numerically study the stability regions in the parameter plane of this system for a fixed degree of asymmetry in the elastic restraints. A Lyapunov-like exponent is defined and numerically evaluated to find these regions of stable and unstable behaviour. These numerics indicate that there are infinitely many possibilities of instabilities in this system that are missing in the usual or symmetric Mathieu equation. We find numerically that there are periodic solutions at the boundaries of stable regions in the parameter plane, analogous to the symmetric Mathieu equation. We compute and plot several of these solution branches, which provide a relatively simpler means of computing the stability transition curves of this system. We prove theoretically that such periodic solutions must exist on all stability boundaries. Our theoretical results apply to the asymmetric Hill's equation, of which the pendulum system is a special case. We demonstrate this with numerical studies of a more general asymmetric Mathieu equation.

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NORRIS, J. W. "THE NONLINEAR MATHIEU EQUATION." International Journal of Bifurcation and Chaos 04, no. 01 (February 1994): 71–86. http://dx.doi.org/10.1142/s021812749400006x.

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The purpose of this paper is to classify the different sequences of bifurcation that can occur for small amplitude solutions to the nonlinear Mathieu equation near to the Mathieu regions of instability. We do this by using the Lindstedt-Poincare perturbation method to construct a vector field which interpolates the successive iterations of the Poincare map. These vector fields are then analysed to determine the sequence of bifurcations.

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Mehrez, Khaled, and Zivorad Tomovski. "On a new (p,q)-Mathieu-type power series and its applications." Applicable Analysis and Discrete Mathematics 13, no. 1 (2019): 309–24. http://dx.doi.org/10.2298/aadm190427005m.

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Our aim in this paper, is to establish certain new integral representations for the (p,q)-Mathieu-type power series. In particular, we investigate the Mellin-Barnes type integral representations for a particular case of these special function. Moreover, we introduce the notion of the (p, q)-Mittag- Leffler functions and we present a relationships between these two functions. Some other applications are proved, in particular two Tur?n type inequalities for the (p,q)-Mathieu-type series are derived.

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Pogány, Tibor K., and Živorad Tomovski. "On multiple generalized Mathieu series." Integral Transforms and Special Functions 17, no. 4 (April 2006): 285–93. http://dx.doi.org/10.1080/10652460500431966.

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Milovanovic, Gradimir, and Tibor Pogány. "New integral forms of generalized Mathieu series and related applications." Applicable Analysis and Discrete Mathematics 7, no. 1 (2013): 180–92. http://dx.doi.org/10.2298/aadm121227028m.

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The main object of this article is to present a systematic study of integral representations for generalized Mathieu series and its alternating variant, and to derive a new integral expression for these special functions by contour integration using rectangular integration path. Also, by virtue of newly established integral form of generalized Mathieu series, we obtain a new integral expression for the Bessel function of the first kind of half integer order, solving a related Fredholm integral equation of the first kind with nondegenerate kernel.

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HU, PEI-CHU, and CHUNG-CHUN YANG. "THE TUMURA–CLUNIE THEOREM IN SEVERAL COMPLEX VARIABLES." Bulletin of the Australian Mathematical Society 90, no. 3 (June 13, 2014): 444–56. http://dx.doi.org/10.1017/s0004972714000446.

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AbstractIt is a well-known result that if a nonconstant meromorphic function $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f$ on $\mathbb{C}$ and its $l$th derivative $f^{(l)}$ have no zeros for some $l\geq 2$, then $f$ is of the form $f(z)=\exp (Az+B)$ or $f(z)=(Az+B)^{-n}$ for some constants $A$, $B$. We extend this result to meromorphic functions of several variables, by first extending the classic Tumura–Clunie theorem for meromorphic functions of one complex variable to that of meromorphic functions of several complex variables using Nevanlinna theory.

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Gerhold, Stefan, Friedrich Hubalek, and Živorad Tomovski. "Asymptotics of some generalized Mathieu series." MATHEMATICA SCANDINAVICA 126, no. 3 (September 3, 2020): 424–50. http://dx.doi.org/10.7146/math.scand.a-121106.

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We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics. As a byproduct, we prove an expansion of the functional inverse of the gamma function at infinity.

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Koike, Masao. "Mathieu group M24 and modular forms." Nagoya Mathematical Journal 99 (September 1985): 147–57. http://dx.doi.org/10.1017/s0027763000021541.

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In [6], Mason reported some connections between sporadic simple group M24 and certain cusp forms which appear in the ‘denominator’ of Thompson series assigned to Fisher-Griess’s group F1. In this paper, we discuss the ‘numerator’ of these Thompson series.

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Conway, John H., Noam D. Elkies, and Jeremy L. Martin. "The Mathieu GroupM12and Its Pseudogroup ExtensionM13." Experimental Mathematics 15, no. 2 (January 2006): 223–36. http://dx.doi.org/10.1080/10586458.2006.10128958.

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Yie, Ik-Kwon. "MATHIEU GROUP COVERINGS AND GOLAY CODES." Journal of the Korean Mathematical Society 39, no. 2 (March 1, 2002): 289–317. http://dx.doi.org/10.4134/jkms.2002.39.2.289.

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Veliev, O. A. "Isospectral Mathieu–Hill Operators." Letters in Mathematical Physics 103, no. 8 (May 30, 2013): 919–25. http://dx.doi.org/10.1007/s11005-013-0627-4.

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Gaberdiel, Matthias R., Daniel Persson, Henrik Ronellenfitsch, and Roberto Volpato. "Generalized Mathieu Moonshine." Communications in Number Theory and Physics 7, no. 1 (2013): 145–223. http://dx.doi.org/10.4310/cntp.2013.v7.n1.a5.

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