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Quasi-lisse Vertex Algebras and Modular Linear Differential Equations
, 2016We introduce the notion of quasi-lisse vertex algebras, which generalizes admissible affine vertex algebras. We show that the normalized character of an ordinary module over a quasi-lisse vertex operator algebra has a modular invariance property, in the ...
T. Arakawa, Kazuya Kawasetsu
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1966
Publisher Summary This chapter focuses on higher-order linear equations. Even for second-order linear equations, no general method of solution is available as there was for first-order equations. Formulas for general solutions can be found for certain special classes of higher-order equations.
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Publisher Summary This chapter focuses on higher-order linear equations. Even for second-order linear equations, no general method of solution is available as there was for first-order equations. Formulas for general solutions can be found for certain special classes of higher-order equations.
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The solution of linear differential equations
Mathematical Proceedings of the Cambridge Philosophical Society, 1955A number of methods are at present available for the numerical solution of linear ordinary differential equations over a restricted range of the independent variable. Of these perhaps the most commonly used are methods based on finite differences which have as their aim the construction of a table of values of the dependent variable at stated intervals
S. C. R. Dennis, G. Poots
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1993
Consider the linear differential equation $$ u^{(n)} = p(t)u $$ (1.1) where. As a rule we assume that either $$ p(t) \leqslant 0 for t \in R_ + $$ (1.2) or $$ p(t) \geqslant 0 for t \in R_ + . $$ (1.3) A solution of (1.1) is said to be oscillatory if it has infinitely many zeros.
Ivan Kiguradze, T. A. Chanturia
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Consider the linear differential equation $$ u^{(n)} = p(t)u $$ (1.1) where. As a rule we assume that either $$ p(t) \leqslant 0 for t \in R_ + $$ (1.2) or $$ p(t) \geqslant 0 for t \in R_ + . $$ (1.3) A solution of (1.1) is said to be oscillatory if it has infinitely many zeros.
Ivan Kiguradze, T. A. Chanturia
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Linear Differential Equations [PDF]
, 1999 In this chapter, we shall first explain the existence of solutions of initial value problems for differential equations and then fundamental theorems for linear differential equations in the complex domain. We explain the definition of regular and irregular singularities of linear differential equations, and the behavior of of local solutions near ...
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Linear Partial Differential Equations
1957Although we shall mainly be concerned in this Part with differential equations, the methods we use here for their discussion and solution are intimately connected with the geometry of the rest of the volume. In particular, the results obtained depend to a great extent on the theory of modules and the intersections of a set of algebraic varieties ...
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2001
Here we consider a so called (scalar) normal system of n ordinary linear differential equations which is a system of the form $$\left\{ {\begin{array}{*{20}{c}} {{{{x'}}_{1}} = {{a}_{{11}}}(t){{x}_{1}} + {{a}_{{12}}}(t){{x}_{2}} + ... + {{a}_{{1n}}}(t){{x}_{n}} + {{f}_{1}}(t),} \\ {{{{x'}}_{2}} = {{a}_{{21}}}(t){{x}_{1}} + {{a}_{{22}}}(t){{x}_{2}} +
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Here we consider a so called (scalar) normal system of n ordinary linear differential equations which is a system of the form $$\left\{ {\begin{array}{*{20}{c}} {{{{x'}}_{1}} = {{a}_{{11}}}(t){{x}_{1}} + {{a}_{{12}}}(t){{x}_{2}} + ... + {{a}_{{1n}}}(t){{x}_{n}} + {{f}_{1}}(t),} \\ {{{{x'}}_{2}} = {{a}_{{21}}}(t){{x}_{1}} + {{a}_{{22}}}(t){{x}_{2}} +
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Linear difference-differential equations
Mathematical Proceedings of the Cambridge Philosophical Society, 1948The general linear difference-differential equation takes the formwhere x is a real variable, ν(x) and Aμν(x) are known functions ...
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1967
In this section we consider the vector equation $$ \mathop x\limits^. {\rm{ }} = {\rm{ }}A(t){\rm{ }}. $$ (58.1) We assume that the elements a i k (t) of the matrix A are continuous functions of t defined for t ≥ t0. Equations with a constant A, treated in sec. 4, are special cases of (58.1). On the other hand, (58.1) is a special case of the
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In this section we consider the vector equation $$ \mathop x\limits^. {\rm{ }} = {\rm{ }}A(t){\rm{ }}. $$ (58.1) We assume that the elements a i k (t) of the matrix A are continuous functions of t defined for t ≥ t0. Equations with a constant A, treated in sec. 4, are special cases of (58.1). On the other hand, (58.1) is a special case of the
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1998
We denote ri x n matrices by uppercase italic letters, $$ A = \left( {\begin{array}{*{20}{c}} {{a_{11}} \ldots {a_{1n}}} \\ { \vdots \ddots \vdots } \\ {{a_{n1}} \cdots {a_{nn}}} \end{array}} \right) = ({a_{ij}}), $$ where aij E R or C. With the usual definitions of addition and scalar multiplication of matrices, $$ A + B = ({a_{ij}} + {b_ ...
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We denote ri x n matrices by uppercase italic letters, $$ A = \left( {\begin{array}{*{20}{c}} {{a_{11}} \ldots {a_{1n}}} \\ { \vdots \ddots \vdots } \\ {{a_{n1}} \cdots {a_{nn}}} \end{array}} \right) = ({a_{ij}}), $$ where aij E R or C. With the usual definitions of addition and scalar multiplication of matrices, $$ A + B = ({a_{ij}} + {b_ ...
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