and this is the better and "really" complete PDF, Dr., I am talking about an abstract structure named "curve" which derived from geometrics curves.Needed? Perhaps not. But it is important in introducing a new idea, and especially in convincing others that it is worth pursuing. To show why we bother defining fields, for example, we show that the real numbers and the rational numbers are fields. Anything that there are two of seems worth naming and studying ...
I still see a lot missing. What does it mean to add two curves? And why do you vary between f(x, y), f(x), and [imath]\gamma(t)[/imath]?
Then you indicate that you are aware that there is already a specific theory of algebraic curves, to which you don't seem to be adding anything. Why do you claim to be discussing "curves", but only define algebraic curves?
I feel as if I were looking at AI-generated material, since AI is much better at looking meaningful than at actually being valid.
I meant Algebraic Curve.if by this you mean a curve of kind of my theory, yeah pretty much.
no algebraic curve is defined as equations like this [imath]y^2 = x^3 -ax + b[/imath] but my curves are algebraic but abstract.I meant Algebraic Curve.
Do you meant that [imath]f(x,y)[/imath] must equal zero, otherwise it is not algebraic curve?no algebraic curve is defined as equations like this [imath]y^2 = x^3 -ax + b[/imath] but my curves are algebraic but abstract.
yeah its on definition of a my curves.Do you meant that [imath]f(x,y)[/imath] must equal zero, otherwise it is not algebraic curve?
Beautiful. Continue. Explain 2.2 Closure with an example.yeah its on definition of a my curves.
Yeah?guys ??
okay, Closure is like this imagine a curve with points P, and Q if the addition of x and y pair of these points is Y and defined on the curve we say "point Y is closure on the curve".Beautiful. Continue. Explain 2.2 Closure with an example.
Yeah?
You mean like [imath]P = (p_1,p_2)[/imath] and [imath]Q = (q_1,q_2)[/imath]. If [imath]P + Q = (p_1 + q_1, p_2 + q_2)[/imath] is on the curve, the curve is called closure?okay, Closure is like this imagine a curve with points P, and Q if the addition of x and y pair of these points is Y and defined on the curve we say "point Y is closure on the curve".
yes.You mean like [imath]P = (p_1,p_2)[/imath] and [imath]Q = (q_1,q_2)[/imath]. If [imath]P + Q = (p_1 + q_1, p_2 + q_2)[/imath] is on the curve, the curve is called closure?
hey, do you copy ? its been 16 minutes you goneYou mean like [imath]P = (p_1,p_2)[/imath] and [imath]Q = (q_1,q_2)[/imath]. If [imath]P + Q = (p_1 + q_1, p_2 + q_2)[/imath] is on the curve, the curve is called closure?
Copy what?hey, do you copy ? its been 16 minutes you gone
Curves exhibit abstraction where the underlying representation of elements does not affect the validity of algebraic operations defined on them. For example, if f(x) and g(x) are curves, the operation f(x) +g(x) is defined without reference to the specific form of f and g.Copy what?
Please, continue. I am learning new things because of your theory. What about 2.3 Abstraction?
I thought that we have already defined the curve as a function of two variables, but now you are telling me one variable. I am confused. Do you mean [imath]f(x,y)[/imath] and [imath]g(x,y)[/imath] are two curves. Do you mean that their sum [imath]f(x,y) + g(x,y)[/imath] will still be a curve and this operation is called abstraction?Curves exhibit abstraction where the underlying representation of elements does not affect the validity of algebraic operations defined on them. For example, if f(x) and g(x) are curves, the operation f(x) +g(x) is defined without reference to the specific form of f and g.
I thought that we have already defined the curve as a function of two variables, but now you are telling me one variable. I am confused. Do you mean [imath]f(x,y)[/imath] and [imath]g(x,y)[/imath] are two curves. Do you mean that their sum [imath]f(x,y) + g(x,y)[/imath] will still be a curve and this operation is called abstraction?
I thought that we have already defined the curve as a function of two variables, but now you are telling me one variable. I am confused. Do you mean [imath]f(x,y)[/imath] and [imath]g(x,y)[/imath] are two curves. Do you mean that their sum [imath]f(x,y) + g(x,y)[/imath] will still be a curve and this operation is called abstraction?
An identity element in curve theory behaves such that any operation with this element leaves other elements unchanged. For a curve C and operation ⊕, there exists an element e ∈ C such that for any P ∈ C:[math]P \oplus e = e \oplus P = P[/math]I still did not understand abstraction, but it's ok. You can proceed to 2.4 Identity.
Is the identity here meant to be [imath]e_1 = (1,0,0)[/imath], [imath]e_2 = (0,1,0)[/imath], and [imath]e_3 = (0,0,1)[/imath] or something else?An identity element in curve theory behaves such that any operation with thiselement leaves other elements unchanged. For a curve C and operation ⊕, thereexists an element e ∈ C such that for any P ∈ C:[math]P \oplus e = e \oplus P = P[/math]
again we did abstraction here to avoid confusing examples and there is an identity example here : [imath]1 \cdot 2 = 2[/imath] if let 1 be e and let 2 be P we see the idea.Is the identity here meant to be [imath]e_1 = (1,0,0)[/imath], [imath]e_2 = (0,1,0)[/imath], and [imath]e_3 = (0,0,1)[/imath] or something else?