# $\Omega(m)$ and $O(m)$ meaning in theorem proof about dynamic array complexity

My algorithms and data structures' book states that to create a dynamic array the following procedure is followed:

Let $$d$$ be the length of an array $$a$$ and $$n$$ the number of elements stored in it. Each time an insertion operation is done, if there is enough space $$(n+1, $$n$$grows by 1; otherwise if $$n=d$$, we allocate and array $$b$$ of size $$2d$$, $$d$$ is updated to$$2d$$ and all elements are copied to it, then we do $$a=b$$. Similarly, everey time a deletion operation is performed, $$n$$ decreases by 1, when $$n=d/4$$, we allocate and array of size $$d/2$$, $$d$$ is updated to $$d/2$$ and we copy all elements of the array $$a$$ to the array $$b$$ an do $$a=b$$.

The pseudocode of the functions doing the array doubling and halving is shown in the picture then the following theorem is proved:

The execution of N insertion or deletion operations in a dynamic array requires a time $$O(n)$$, beside d=$$O(n)$$

proof: Let $$d$$ be the length of an array and n the number of elements stored in it After doubling the array's size there are $$m=d+1$$ elements in the new array of $$2d$$ positions. We need at least m-1 insertion requests for a new doubling and at least $$m/2$$ deletion requests for a halving. Similarly,after a halving, there are$$m=d/4$$ elements in the new array of $$d/2$$ elements, for wich at least$$m+1$$ insertion requests are needed for a new doubling and at least $$m/2$$ deletion requests are needed for a new halving.

In any case, the cost of $$O(m)$$time required for the resizing can be virtually distributed among the $$\Omega(m)$$ operations that caused it(starting from the last resizing) . Finally, supposing the when the array is created there are $$n=1$$ and $$d=1$$, the number of elements of the array is always one fourth of its size, so $$d=O(n)$$

I am a beginner with this $$\Omega(m)$$ and $$O(m)$$ , I know the mathematical definitions and I've been reading a lot about it but I am not able to understand it well in context. I know big O should indicate an upper bound on the time complexity of an algorithm and big omega a lower bound.

I don't understand the last paragraph when they use these symbols, Why is the time cost O(m), why are they using $$\Omega(m)$$ for the number of operations that caused the resizing (besides specifically what operations are they referring to?) and why do they write $$d=O(n)$$, what should I understand from it? Any help will be greatly appreciated

Note: I got this from : Strutture di dati e algoritmi, progettazione, analisi e visualizzazione. Crescenzi, Gambosi, Grossi-2nd edition.