The boundedness reflects the fact that beyond a certain point money ceases being useful at all, as the size of any economy at any point in time is itself bounded.
42.
As the coefficients of the discretised equation are always positive hence satisfying the requirements for boundedness and also the coefficient matrix is diagonally dominant therefore no irregularities occur in the solution.
43.
In other words, even if you only require weak boundedness on the extremes " p " and " q ", you still get regular boundedness inside.
44.
In other words, even if you only require weak boundedness on the extremes " p " and " q ", you still get regular boundedness inside.
45.
This suggests that if a sequence is o p ( 1 ), then it is O p ( 1 ), i . e . convergence in probability implies stochastic boundedness.
46.
The uniform boundedness of the difference between the transform and the truncated transform can also be seen for odd " k " using the Calder�n-Zygmund method of rotation.
47.
In the calculus of variations, where one is typically interested in infinite-dimensional function spaces, the condition is necessary because some extra notion of compactness beyond simple boundedness is needed.
48.
Informally these embeddings say that to convert an " L " " p " estimate to a boundedness estimate costs 1 / " p " derivatives per dimension.
49.
For example, together with the uniform boundedness principle, it can be used to show that the Fourier series of a continuous function may fail to converge pointwise, in rather dramatic fashion.
50.
The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum.
