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| − | The sphere in real analysis which is known as the [[Riemann sphere|Riemann sphere]] in the theory of functions of a complex variable.
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| | + | An $n$-place function defined on the set of truth values (cf. [[Truth value|Truth value]]) $\{\text T,\text F\}$ and taking values in this set. With every [[Logical operation|logical operation]] $\mathfrak A$ is associated a logical function $f_\mathfrak A$: If $V_1,\ldots,V_n$ are truth values, then $f_\mathfrak A(V_1,\ldots,V_n)$ is the truth value of the proposition $\mathfrak A(P_1,\ldots,P_n)$, where $P_1,\ldots,P_n$ are propositions such that the truth value of $P_i$ is equal to $V_i$, $i=1,\ldots,n$. |
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| − | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155401.png" /> be the unit sphere in the Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155402.png" />-space, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155404.png" /> be its north and south pole, respectively; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155406.png" /> be planes tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155407.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155409.png" /> respectively; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554011.png" /> be coordinate systems in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554013.png" /> with axes parallel to the corresponding axes of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554014.png" /> in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554015.png" /> and pointing in the same directions; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554016.png" /> be the [[Stereographic projection|stereographic projection]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554017.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554018.png" /> from the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554019.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554020.png" /> be the stereographic projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554021.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554022.png" /> from the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554023.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554024.png" /> is the Bendixson sphere with respect to any one of the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554026.png" />. It generates the bijection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554027.png" /> of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554028.png" /> (punctured at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554029.png" />) onto the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554030.png" />, which is punctured at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554031.png" />. This bijection is employed in the study of the behaviour of the trajectories of an autonomous system of real algebraic ordinary differential equations of the second order in a neighbourhood of infinity in the phase plane (the right-hand sides of the equations are polynomials of the unknown functions). This bijection reduces the problem to a similar problem in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554032.png" />. Named after I. Bendixson.
| + | A logical function is sometimes defined as an $n$-place function defined on a set $M$ and taking values in the set $\{\text T,\text F\}$. Such functions are used in mathematical logic as an analogue of the concept of a [[Predicate|predicate]]. |
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| − | ====References====
| + | [[Category:Logic and foundations]] |
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)</TD></TR></table>
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Latest revision as of 16:55, 2 November 2014
An $n$-place function defined on the set of truth values (cf. Truth value) $\{\text T,\text F\}$ and taking values in this set. With every logical operation $\mathfrak A$ is associated a logical function $f_\mathfrak A$: If $V_1,\ldots,V_n$ are truth values, then $f_\mathfrak A(V_1,\ldots,V_n)$ is the truth value of the proposition $\mathfrak A(P_1,\ldots,P_n)$, where $P_1,\ldots,P_n$ are propositions such that the truth value of $P_i$ is equal to $V_i$, $i=1,\ldots,n$.
A logical function is sometimes defined as an $n$-place function defined on a set $M$ and taking values in the set $\{\text T,\text F\}$. Such functions are used in mathematical logic as an analogue of the concept of a predicate.
How to Cite This Entry:
Bendixson sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bendixson_sphere&oldid=19212
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article