uniqueness

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11: 28.7 Analytic Continuation of Eigenvalues

… ►In consequence, the functions can be defined uniquely by introducing suitable cuts in the $q$-plane. …

12: 18.40 Methods of Computation

… ►Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. …

13: 23.3 Differential Equations

… ►Given $g_2$ and $g_3$ there is a unique lattice $𝕃$ such that (23.3.1) and (23.3.2) are satisfied. … ►Conversely, $g_2$, $g_3$, and the set $\{e_1,e_2,e_3\}$ are determined uniquely by the lattice $𝕃$ independently of the choice of generators. …

14: 23.20 Mathematical Applications

… ►There is a unique point $z_0\in [{\omega }_1,{\omega }_1+{\omega }_3]\cup [{\omega }_1+{\omega }_3,{\omega }_3]$ such that $\mathrm{\wp }\left(z_0\right)=0$. … ►For each pair of edges there is a unique point $z_0$ such that $\mathrm{\wp }\left(z_0\right)=0$. …

15: 20.1 Special Notation

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$m$, $n$	integers.
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$q$ $(\in ℂ)$	the nome, $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, . Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\mathrm{\Re }\tau =0$, $\mathrm{\Im }\tau >0$, so that and $\tau$ and $q$ are uniquely related.
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16: 28.2 Definitions and Basic Properties

… ►If $q\ne 0$, then for a given value of $\nu$ the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)). … ►For simple roots $q$ of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations …

17: 32.4 Isomonodromy Problems

… ►Isomonodromy problems for Painlevé equations are not unique. …

18: Bibliography O

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On the uniqueness of asymptotic solutions of linear differential equations. Methods Appl. Anal. 6 (2), pp. 165–174.

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External Links:

ISSN 1073-2772, FullText, MathReview (P. K. Palamides), ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

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19: 18.2 General Orthogonal Polynomials

… ►The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials $p_n(x)$ uniquely up to constant factors, which may be fixed by suitable standardizations. … ►

§18.2(viii) Uniqueness of Orthogonality Measure and Completeness

… ►, of the form $w(x)dx$) nor is it necessarily unique, up to a positive constant factor. However, if OP’s have an orthogonality relation on a bounded interval, then their orthogonality measure is unique, up to a positive constant factor. … ►A system of OP’s with unique orthogonality measure is always complete, see Shohat and Tamarkin (1970, Theorem 2.14). …

20: 3.11 Approximation Techniques

… ►Then there exists a unique $n$th degree polynomial $p_n(x)$, called the minimax (or best uniform) polynomial approximation to $f(x)$ on $[a,b]$, that minimizes ${\max }_{a\le x\le b}\left|{ϵ}_n(x)\right|$, where ${ϵ}_n(x)=f(x)-p_n(x)$. … ►There exists a unique solution of this minimax problem and there are at least $k+\mathrm{\ell }+2$ values $x_j$, , such that $m_j=m$, where … ►is $c_0=c_1=\mathrm{\cdots }=c_n=0$, then the approximation ${\mathrm{\Phi }}_n(x)$ is determined uniquely. …