Solution
... [a, b]. Assume also that g(v) = v 6= u. Then 0 < |u − v| = |g(u) − g(v)| < λ|u − v| < |u − v|, a contradiction. Thus, u = v and we have proved uniqueness. Convergence holds as follows: |u − xn+1 | = |g(u) − g(xn )| ≤ λ|u − xn |, which, by induction, implies convergence of xn to u according to |u − x ...
... [a, b]. Assume also that g(v) = v 6= u. Then 0 < |u − v| = |g(u) − g(v)| < λ|u − v| < |u − v|, a contradiction. Thus, u = v and we have proved uniqueness. Convergence holds as follows: |u − xn+1 | = |g(u) − g(xn )| ≤ λ|u − xn |, which, by induction, implies convergence of xn to u according to |u − x ...
TRANSCENDENCE BASES AND N
... {y1, y2, …, yn} K has the property that k(yj) K is an algebraic extension. Then m n and we may reorder the y’s so that k(x1, …, xm, ym+1, …, yn) K is an algebraic extension. Reversing the roles of {xi} and {yj} in the lemma, you see that any two finite transcendence bases have the same cardi ...
... {y1, y2, …, yn} K has the property that k(yj) K is an algebraic extension. Then m n and we may reorder the y’s so that k(x1, …, xm, ym+1, …, yn) K is an algebraic extension. Reversing the roles of {xi} and {yj} in the lemma, you see that any two finite transcendence bases have the same cardi ...
Full text
... Since the \\)n do not satisfy a three-term recursion formula, they, by the converse of Favard?s Theorem, are not an orthogonal set, no matter what w(x) or [a, b] is selected. Favard!s Theorem and converse are as follows. T/ieo/iem: If the tyn(x) are a set of simple polynomials which satisfy a threet ...
... Since the \\)n do not satisfy a three-term recursion formula, they, by the converse of Favard?s Theorem, are not an orthogonal set, no matter what w(x) or [a, b] is selected. Favard!s Theorem and converse are as follows. T/ieo/iem: If the tyn(x) are a set of simple polynomials which satisfy a threet ...
Warm up
... Your teacher will pass out a note card to each of you. Once you have your note card, write down an equation in 2 variables (Ex. y =-3x +9). The equation can be anything you want, but please keep the numbers in front of the variables between -6 and 6. When everyone has their equation break into 2 lin ...
... Your teacher will pass out a note card to each of you. Once you have your note card, write down an equation in 2 variables (Ex. y =-3x +9). The equation can be anything you want, but please keep the numbers in front of the variables between -6 and 6. When everyone has their equation break into 2 lin ...
