英语翻译(e) (0 Fig.3.The zero-crossing points may be of the foll
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英语翻译
(e) (0 Fig.3.The zero-crossing points may be of the following types.(a) Ellip-tic.(b) Hyperbolic.(c) Parabolic.The zero-crossing lines may be of the
following types.(d) Elliptic umbilic.(e) Hyperbolic umbilic.(f) Para-bolic umbilic.
where the coefficients a,
{3,
j,and 0
are obtained by the Taylor expansion.It is easy to see that the set of points
The ZC lines may be a) an elliptic umbilic,if RA consists of three lines [see Fig.2(d)],b) a hyperbolic.umbilic,ifRA consists of a single real line [see Fig.2(e)] ,c) a parabolic umbilic,if RA consists of three lines,two of which are coincident [see Fig.2(f)],d) a symbolic umbilic,if RA consists of three coin-cident lines.5) If h(x,y) is a smooth function and in P
hex,y) depends on the fourth-order terms,the ZC lines have a complex shape that can be analyzed using results of [42].
Bifurcations of Zero Crossings:The isotopy theorem [50],[1] shows that transversal intersections are structur-ally stable,i.e.,that "transversal zero-crossings" are structurally stable:their topological properties do not change if the size (and thus the scale) of the filter is slightly changed.
If f(x,y) is a Morse function then Sf may meet So non-
elliptic Z.c.
(b) Fig.4.The two types of bifurcations that can occur for incre~sing
and
decreasing a in the case of Morse functions.(a) Left to right.(b) Right to left.
transverally,and these intersections are not structurally stable (observe that Morse functions are structurally sta-ble but not their intersections with So),If f is a Morse function,then Sf may meet So nontransversally at elliptic points and hyperbolic points.These intersections are not structurally stable and may change their topological prop-erties for small perturbations of f.More specifically,we may have two bifurcations:
a) Elliptic ZC:At elliptic ZC,a small perturbation of f may lead to the disappearance of the ZC or to the appearance of a contour of ZC constituted by a closed curve.
(e) (0 Fig.3.The zero-crossing points may be of the following types.(a) Ellip-tic.(b) Hyperbolic.(c) Parabolic.The zero-crossing lines may be of the
following types.(d) Elliptic umbilic.(e) Hyperbolic umbilic.(f) Para-bolic umbilic.
where the coefficients a,
{3,
j,and 0
are obtained by the Taylor expansion.It is easy to see that the set of points
The ZC lines may be a) an elliptic umbilic,if RA consists of three lines [see Fig.2(d)],b) a hyperbolic.umbilic,ifRA consists of a single real line [see Fig.2(e)] ,c) a parabolic umbilic,if RA consists of three lines,two of which are coincident [see Fig.2(f)],d) a symbolic umbilic,if RA consists of three coin-cident lines.5) If h(x,y) is a smooth function and in P
hex,y) depends on the fourth-order terms,the ZC lines have a complex shape that can be analyzed using results of [42].
Bifurcations of Zero Crossings:The isotopy theorem [50],[1] shows that transversal intersections are structur-ally stable,i.e.,that "transversal zero-crossings" are structurally stable:their topological properties do not change if the size (and thus the scale) of the filter is slightly changed.
If f(x,y) is a Morse function then Sf may meet So non-
elliptic Z.c.
(b) Fig.4.The two types of bifurcations that can occur for incre~sing
and
decreasing a in the case of Morse functions.(a) Left to right.(b) Right to left.
transverally,and these intersections are not structurally stable (observe that Morse functions are structurally sta-ble but not their intersections with So),If f is a Morse function,then Sf may meet So nontransversally at elliptic points and hyperbolic points.These intersections are not structurally stable and may change their topological prop-erties for small perturbations of f.More specifically,we may have two bifurcations:
a) Elliptic ZC:At elliptic ZC,a small perturbation of f may lead to the disappearance of the ZC or to the appearance of a contour of ZC constituted by a closed curve.
(e) (0.3.零相交点也许是以下类型.(a)椭圆.(b)双曲线.(c)抛物面.零相交线也许是以下类型.(d)椭圆umbilic.(e)双曲线中心的.(f)抛物面中心的.那里系数a,{3,j和0由泰勒扩展获得.看套点是容易的 ZC线也许是a)一椭圆中心的,如果镭包括三条线[看见.2 (d)]b) 双曲线的.中心的,ifRA包括一条唯一真正的线[看见.2 (e)] c)一抛物面中心的,如果镭包括三条线,二,其中是一致的[看见.2 (f)]d)一符号中心的,如果镭包括三条一致线.5) 如果h (x,y)是一个光滑的作用和在P不吉利的东西,y)取决于四秩序期限,ZC线有能使用结果被分析的复杂形状[42].零相交的叉路:同位定理[50],[1]表示,横截交叉点结构地是槽枥,即,“横截零相交”结构地是槽枥:他们的拓扑学物产不改变,如果轻微地改变大小(和因而标度)过滤器.如果f (x,y)是莫尔斯作用那么Sf也许遇见那么非椭圆Z.c.(b).4.可能为增加和减少a发生在莫尔斯情况下叉路的二个类型起作用.(a)左到右.(b)右到左.横向地和这些交叉点不结构地是槽枥(观察莫尔斯作用结构地是槽枥,但没有他们的交叉点与如此),如果f是莫尔斯作用,然后Sf也许见面那么非横向地在椭圆点和双曲线点.这些交叉点不结构地是槽枥,并且也许改变他们的拓扑学物产为f.的小扰动.更加具体地,我们也许有二叉路:a) 椭圆ZC :在椭圆ZC,f的小扰动也许导致ZC的失踪或ZC等高的出现闭合的曲线构成的.
英语翻译(e) (0 Fig.3.The zero-crossing points may be of the foll
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