# written test – Is the following statement mathematically well written?

Supposing $${A, Y, X }$$ are matrices, and $${f (.), g (.), h (.) }$$ are functions with scalar outputs, and $$vec {x} _i$$ is the $$i$$e column of $$X$$ and $$x_ {ti}$$ denotes the $$t$$row of $$i$$e column of $$X$$.

I mean that the function $$f (Y)$$ at its minimum if for each column $$vec {y} _i$$ in $$Y$$ the correspondent $$vec {x} _i$$ (based on the relationship $$g ( vec {y} _i) about h (A, vec {x} _i)$$) has non-zero entries $$x_ {ti}$$ such as linked columns $$t$$ in $$A$$ have their non-zero entries $$a_ {st}$$
with clues $$s$$ so that these two conditions are met:

1. $$vec {h} _i = vec {h} _s$$
2. \$ | y_i-y_s | _2 ^ 2 is small enough

I mean the above in a clear and condensed mathematical way. I've tried the following, but I'm not sure of its quality!

Function $$f (Y)$$ at its minimum if
$$forall vec {y} _i$$, $$g ( vec {y} _i) about h (A, vec {x} _i)$$
such as
$$forall vec {a} _t$$, or $$vec {x} _ {ti} neq 0$$ and $$forall s$$, or $$a_ {st} neq 0$$, $$vec {h} _i = vec {h} _s$$ and $$| y_i-y_s | _2 ^ 2 leq epsilon$$ for a sufficiently small $$epsilon> 0$$.