Class notes, rings and modules : some of 23/03/2017 and 04/04/2017
... Theorem 0.1. Let R be a PID and M a finitely generated R−module. Then we can write M∼ = Rr ⊕ R/(pα1 1 ) ⊕ · · · ⊕ R/(pαnn ), for some r, n ≥ 0 and pi prime elements (not necessarily distinct) of R. Moreover, this decomposition is unique, i.e. if there is another decomposition β βm M∼ ...
... Theorem 0.1. Let R be a PID and M a finitely generated R−module. Then we can write M∼ = Rr ⊕ R/(pα1 1 ) ⊕ · · · ⊕ R/(pαnn ), for some r, n ≥ 0 and pi prime elements (not necessarily distinct) of R. Moreover, this decomposition is unique, i.e. if there is another decomposition β βm M∼ ...
algebra 31 - Fairfield Public Schools
... their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set o ...
... their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set o ...
Geometry - San Ramon Valley High School
... since if two lines are cut by a transversal and same-side interior angles are supplementary the lines are parallel. But this contradicts the given information that 3) a is not // to b. ...
... since if two lines are cut by a transversal and same-side interior angles are supplementary the lines are parallel. But this contradicts the given information that 3) a is not // to b. ...
Section 5.1: Polynomial Functions as Mathematical Models
... 3. Definition: The degree of a monomial is the sum of the exponents of its variables. The degree of a nonzero constant is zero. 4. Definition: The degree of a polynomial is the highest degree of any monomial in it. 5. Definition: A polynomial in one variable is a function of the form P(x) = anxn + a ...
... 3. Definition: The degree of a monomial is the sum of the exponents of its variables. The degree of a nonzero constant is zero. 4. Definition: The degree of a polynomial is the highest degree of any monomial in it. 5. Definition: A polynomial in one variable is a function of the form P(x) = anxn + a ...
Math/Stat 2300 Smoothing (4.3): Low
... without the disadvantages. One technique is to choose a low-order polynomial regardless of the number of data points (then the number of data points exceeds the number of constants). The polynomial will not pass through all data points, so a choice with regards to best fit must be decided. For examp ...
... without the disadvantages. One technique is to choose a low-order polynomial regardless of the number of data points (then the number of data points exceeds the number of constants). The polynomial will not pass through all data points, so a choice with regards to best fit must be decided. For examp ...
