Let L(¦Á1, ¦Á2) be a lattice with minimal base ¦Á1 and ¦Á2. Consider a
sublattice L(¦Â1, ¦Â2) of
L(¦Á1, ¦Á2) with minimal base ¦Â1 and ¦Â2. Let ¦Ñ be the index of
L(¦Â1, ¦Â2) and
R = {x¦Â1 + y¦Â1 : 0 ¡Ü x, y < 1} ,
i.e. the half-closed half open parallelogram spanned by ¦Â1 and ¦Â2.
Show that
(a) the area of R is ¦Ñ |¦Á1 ¡Á ¦Á2|,
(b) the area of R is equal to the number of lattice points in L(¦Á1,
¦Á2) contained in R times
|¦Á1 ¡Á ¦Á2|,
(c) the total number of lattice points in L(¦Á1, ¦Á2) contained in R
is exactly ¦Ñ. |
Request for Question Clarification by
haversian-ga
on
24 Oct 2002 11:54 PDT
You appear to be using a non-ASCII character coding for some of your
characters. I've tried several likely character codings and none
appear to be right. Could you let us know what coding you are using
so we can properly view your question?
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Request for Question Clarification by
haversian-ga
on
24 Oct 2002 11:55 PDT
Also, the topic of your question appears to be something I covered in
my elementary number theory class - did you intend the title of your
question to be "Wireless networking" or was there some mix-up between
two questions?
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Clarification of Question by
lijo999-ga
on
30 Oct 2002 13:57 PST
Let L(¦Á1, ¦Á2) be a lattice with minimal base ¦Á1 and ¦Á2. Consider a
sublattice L(¦Â1, ¦Â2) of
L(¦Á1, ¦Á2) with minimal base ¦Â1 and ¦Â2. Let ¦Ñ be the index of
L(¦Â1, ¦Â2) and
R = {x¦Â1 + y¦Â1 : 0 ¡Ü x, y < 1} ,
i.e. the half-closed half open parallelogram spanned by ¦Â1 and ¦Â2.
Show that
(a) the area of R is ¦Ñ |¦Á1 ¡Á ¦Á2|,
(b) the area of R is equal to the number of lattice points in L(¦Á1,
¦Á2) contained in R times
|¦Á1 ¡Á ¦Á2|,
(c) the total number of lattice points in L(¦Á1, ¦Á2) contained in R
is exactly ¦Ñ.
|
Clarification of Question by
lijo999-ga
on
30 Oct 2002 14:01 PST
i am really sorry. i do not know which character coding i am using . i
would rewrite the entire question for you.
Let L(alpha1, alpha2) be a lattice with minimal base alpha1 and
alpha2. Consider a sublattice L(beta1, beta2) of
L(alpha1, alpha2) with minimal base beta1 and beta2. Let ro be the
index of L(beta1, beta2) and
R = {xbeta1 + ybeta1 : 0 less than or equal to x, y < 1} ,
i.e. the half-closed half open parallelogram spanned by beta1 and
beta2. Show that
(a) the area of R is ro |alpha1 cross product alpha2|,
(b) the area of R is equal to the number of lattice points in
L(alpha1, alpha2) contained in R times
|alpha1 cross product alpha2|,
(c) the total number of lattice points in L(alpha1, alpha2) contained
in R is exactly ro.
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