Question on meaning of some symbols


by yungman
Tags: meaning, symbols
yungman
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#1
Sep26-10, 12:32 PM
P: 3,843
I don't know the meaning of these:


1) [tex]sup_{B_\delta}|f(x,y)| [/tex]

Where [itex]B_\delta [/itex] is the ball of radius [itex]\delta[/itex].

2) [tex]\int \int _{R^2 \B _{\delta} } f(xy)dxdy[/tex]

I don't know what is [itex]R^2 [/itex]\B[itex] _{\delta} [/itex]

Please read my latex because the symbol really don't show correctly.
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Landau
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#2
Sep26-10, 03:06 PM
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1) The supremum of {|f(x,y)|} where (x,y) ranges over the ball centered at 0 of radius delta: |(x,y)|=sqrt(x^2+y^2)<delta.

2) the plane R^2 without the ball centered at 0 of radius delta, i.e. \ (in Latex: "\backslash") means 'complement' or 'set difference'. So it consists of pairs (x,y) of real numbers such that |(x,y)|=sqrt(x^2+y^2)>=delta.
yungman
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#3
Sep27-10, 12:20 AM
P: 3,843
Quote Quote by Landau View Post
1) The supremum of {|f(x,y)|} where (x,y) ranges over the ball centered at 0 of radius delta: |(x,y)|=sqrt(x^2+y^2)<delta.

2) the plane R^2 without the ball centered at 0 of radius delta, i.e. \ (in Latex: "\backslash") means 'complement' or 'set difference'. So it consists of pairs (x,y) of real numbers such that |(x,y)|=sqrt(x^2+y^2)>=delta.
Thanks for you reply, so for

1) Is the upper bound of |f(x,y)| in the ball.

2) Is the whole 2D plane minus the circle center at 0 with radius [itex]\delta[/itex]

Landau
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#4
Sep27-10, 06:43 AM
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Question on meaning of some symbols


Quote Quote by yungman View Post
1) Is the upper bound of |f(x,y)| in the ball.
The least upper bound, a.k.a. the supremum ;)
2) Is the whole 2D plane minus the circle center at 0 with radius [itex]\delta[/itex]
Yes.
HallsofIvy
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#5
Sep27-10, 08:09 AM
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Thanks
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Quote Quote by yungman View Post
Thanks for you reply, so for

1) Is the upper bound of |f(x,y)| in the ball.
No. There is no such thing as "the" upper bound of a set of numbers. If a set has an upper bound, then it has an infinite number of upper bounds. This is the least upper bound- the smallest number in the set of all upper bounds.

2) Is the whole 2D plane minus the circle center at 0 with radius [itex]\delta[/itex]


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