There exists a
countable set E in [epsilon] such that A = [E.sup.[perpendicular to].sub.w] + A where,
For each d [member of] D, define A(d) to be a
countable set of balls with center at d, with radius r(d, n) [down arrow] 0 when n [right arrow] [infinity] such that P([partial derivative][B.sub.r(d,n)](d)) = 0, for n = 1,2, ..., where x [member of] D, [B.sub.r](x) = {y | d(y, x) < r}, and [partial derivative]F denotes the boundary of given set F.
We shall show that the
countable set {d.[x.sub.o]; d [member of] D} is dense in [OMEGA]* in the weak topology.
Let B = {[u.sub.n]} [subset] PC(J, E) be a bounded and
countable set. Then [alpha](B(t)) is Lebesgue integral on J, and
Let us consider a qso V (5) defined on
countable set X.
(2) except for a
countable set of values [alpha](p, A), one has
The symmetry of KGE on the chosen orbit discloses existence of a new
countable set of the modal amplitudes oscillating with the same cut-off frequency.
Then X-C is [zeta]-open and hence for every x [member of] X - C, there exists an open set U containing x and a
countable set B such that U - B [subset not equal to] sInt(X - C) [subset not equal to] X - [Cl.sub.T] (C).
Then, the set of all jump discontinuity points of f is at most a
countable set in R.
First, if sampling a function f on the
countable set X leads to unique and stable reconstruction of f, then when does sampling on the set X' = {[x.sub.j] + [[delta].sub.j] : j [member of] J} also lead to unique and stable reconstruction?
Since B is separable we can find a
countable set {[h.sub.n]} in B such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all nand the subspace H generated by {[h.sub.n]} is dense in B.