XC.The characteristic numbers of the Mathieu equation with purely imaginary parameter

1929 ◽  
Vol 8 (53) ◽  
pp. 834-840 ◽  
Author(s):  
H.P. Mulholland ◽  
S. Goldstein
Keyword(s):  
Mathieu Equation ◽  
Characteristic Numbers

XXVIII.—Researches into the Characteristic Numbers of the Mathieu Equation—(Second Paper)

1927 ◽  
Vol 46 ◽  
pp. 316-322 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Present Author ◽  
Mathieu Equation ◽  
London Mathematical Society ◽  
Mathematical Society ◽  
Characteristic Numbers ◽  
Image Position

In a paper recently communicated to the London Mathematical Society the present author showed that the characteristic numbers an and bn of the Mathieu equation—may be developed, for large positive values of q, in the forms

XXII.—Researches into the Characteristic Numbers of the Mathieu Equation—(Third Paper)

1928 ◽  
Vol 47 ◽  
pp. 294-301 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Bessel Functions ◽  
Mathieu Equation ◽  
Mathematical Physics ◽  
Mathieu Functions ◽  
Characteristic Numbers ◽  
Asymptotic Developments ◽  
The Way

The importance in Mathematical Physics of the Bessel functions, whose order is half an odd integer, suggests that the corresponding Mathieu functions may be worthy of a closer attention than they have yet received. At the very least it is expedient to pave the way for their computation. In the second paper bearing the above title, asymptotic developments of the characteristic numbers which correspond to these functions were given; it is here proposed, in the first place, to take the more direct line of approach.

IV.—Researches into the Characteristic Numbers of the Mathieu Equation

1927 ◽  
Vol 46 ◽  
pp. 20-29 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Fourier Series ◽  
Periodic Solutions ◽  
Mathieu Equation ◽  
Mathieu Functions ◽  
Characteristic Numbers ◽  
Image Position

The characteristic numbers of the Mathieu equationare those values of a for which, when q is given, the equation admits of a solution of period π or 2π. The periodic solutions, or Mathieu functions, may be developed as a Fourier-series convergent for all values of q,multiplied by one or other of the factors

XVII.—On the Asymptotic Expansion of the Characteristic Numbers of the Mathieu Equation

1930 ◽  
Vol 49 ◽  
pp. 210-223 ◽  
Author(s):  
Sydney Goldstein
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Mathieu Equation ◽  
Characteristic Number ◽  
Numerical Approximations ◽  
Characteristic Numbers ◽  
The Difference ◽  
Image Position ◽  
Integral Order

An asymptotic formula has recently been given for the characteristic numbers of the Mathieu equation From tabular values, it will be seen that the formula provides good numerical approximations to the characteristic numbers of integral order; but as pointed out by Ince, it provides better approximations to the characteristic numbers of order (m + ½), where m is a positive integer or zero. In this paper we shall first attempt to find out why this should be so, and then go on to show that the formula is probably an asymptotic expansion, in the Poincaré sense, for any characteristic number. A new asymptotic formula is then found for the difference between two characteristic numbers.

Characteristic numbers ofG-manifolds. I

1971 ◽  
Vol 13 (3) ◽  
pp. 213-224 ◽  
Author(s):  
Tammo Tom Dieck
Keyword(s):  
Characteristic Numbers

Mathieu equation with application to analysis of dynamic characteristics of resonant inertial sensors

2013 ◽  
Vol 18 (2) ◽  
pp. 401-410 ◽  
Author(s):  
Yan Li ◽  
Shangchun Fan ◽  
Zhanshe Guo ◽  
Jing Li ◽  
Le Cao ◽  
...  
Keyword(s):  
Dynamic Characteristics ◽  
Inertial Sensors ◽  
Mathieu Equation

AN INTENSIONAL LEIBNIZ SEMANTICS FOR ARISTOTELIAN LOGIC

2010 ◽  
Vol 3 (2) ◽  
pp. 262-272 ◽  
Author(s):  
KLAUS GLASHOFF
Keyword(s):  
Natural Deduction ◽  
Predicate Logic ◽  
Formal System ◽  
Aristotelian Logic ◽  
Categorical Logic ◽  
Characteristic Numbers ◽  
Formal Syntax ◽  
Philosophical Standpoint ◽  
Classical Philosophy

Since Frege’s predicate logical transcription of Aristotelian categorical logic, the standard semantics of Aristotelian logic considers terms as standing for sets of individuals. From a philosophical standpoint, this extensional model poses problems: There exist serious doubts that Aristotle’s terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this question—they looked at terms as standing for concepts (“Begriffe”). In 1972, Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction, while, with respect to semantics, he still made use of an extensional interpretation. In this paper we deal with a simple intensional semantics for Corcoran’s syntax—intensional in the sense that no individuals are needed for the construction of a complete Tarski model of Aristotelian syntax. Instead, we view concepts as containing or excluding other, “higher” concepts—corresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcoran’s work, furnishing his formal syntax with an adequate semantics which is free from presuppositions which have entered into modern interpretations of Aristotle’s theory via predicate logic.

Observations Regarding Numerical Results Obtained by the Floquet-Method

2013 ◽  
Author(s):  
Till J. Kniffka ◽  
Horst Ecker
Keyword(s):  
Numerical Methods ◽  
Monodromy Matrix ◽  
Mathieu Equation ◽  
Numerical Results ◽  
The Other ◽  
Benchmark Problem ◽  
Stability Studies ◽  
Floquet Method ◽  
System Equations ◽  
Time Periodic

Stability studies of parametrically excited systems are frequently carried out by numerical methods. Especially for LTP-systems, several such methods are known and in practical use. This study investigates and compares two methods that are both based on Floquet’s theorem. As an introductary benchmark problem a 1-dof system is employed, which is basically a mechanical representation of the damped Mathieu-equation. The second problem to be studied in this contribution is a time-periodic 2-dof vibrational system. The system equations are transformed into a modal representation to facilitate the application and interpretation of the results obtained by different methods. Both numerical methods are similar in the sense that a monodromy matrix for the LTP-system is calculated numerically. However, one method uses the period of the parametric excitation as the interval for establishing that matrix. The other method is based on the period of the solution, which is not known exactly. Numerical results are computed by both methods and compared in order to work out how they can be applied efficiently.

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