An integer polynomial for even parity modulo a power of two
For integer ell>=1 define p_ell(Y)=(1-Y)^ell times sum_{j=0}^{ell-1} binom(ell+j-1,j)Y^j. The finite sum is the inverse of (1-Y)^ell modulo Y^ell: multiplying leaves constant term one and all degrees 1 through ell-1 zero. Thus p_ell is congruent to 1 modulo Y^ell and is divisible by (1-Y)^ell. At an even integer y, y^ell is divisible by 2^ell, so p_ell(y) is congruent to 1 modulo 2^ell. At an odd y, (1-y)^ell is divisible by 2^ell, giving residue zero. These are integer congruences, not the assertion that p_ell(y) always equals zero or one.
0
1
Tags
Math
Related
Using to Model the Height of a Dropped Stone
Using to Model the Height of a Dropped Ball
Using to Model the Height of a Dropped Ball
Evaluating for and
Evaluating for and
Remainder Theorem
Polynomial Division Check in Function Notation
Imagine your company uses a polynomial function, , to model projected weekly profit based on , the number of products manufactured. Your manager asks you to evaluate this profit function for an input of 500 products. Which sequence of steps correctly describes the mathematical procedure you must follow?
A facility manager uses a mathematical expression to estimate the weight capacity of a storage floor based on its thickness. To find the capacity for a specific thickness, the manager must evaluate the expression. Arrange the steps of this evaluation process in the correct sequence.
Procedural Steps for Evaluating Models
An operations manager at a logistics company uses polynomial functions to predict fuel consumption based on vehicle weight. To analyze these models correctly, the manager must understand the specific components of the mathematical process. Match each term with the description of its role in this analysis.
A logistics company uses a polynomial function to model fuel consumption based on vehicle speed. True or False: To evaluate this function for a specific speed, that value must be substituted for the variable in every term of the polynomial expression before the order of operations is used to simplify the result.
At a manufacturing plant, an industrial engineer uses a polynomial function to model the daily production cost based on units produced. To find the cost of producing 150 units, the engineer must first substitute 150 for the variable throughout the polynomial expression, and then apply the ________ to simplify the expression and obtain the final cost.
Evaluating a Cost-Analysis Polynomial Model
A polynomial that indicates whether a clause is satisfied
Restoring unused variables and reading an exact count from the clause network
Related
Using to Model the Height of a Dropped Stone
Using to Model the Height of a Dropped Ball
Using to Model the Height of a Dropped Ball
Evaluating for and
Evaluating for and
Remainder Theorem
Polynomial Division Check in Function Notation
Imagine your company uses a polynomial function, , to model projected weekly profit based on , the number of products manufactured. Your manager asks you to evaluate this profit function for an input of 500 products. Which sequence of steps correctly describes the mathematical procedure you must follow?
A facility manager uses a mathematical expression to estimate the weight capacity of a storage floor based on its thickness. To find the capacity for a specific thickness, the manager must evaluate the expression. Arrange the steps of this evaluation process in the correct sequence.
Procedural Steps for Evaluating Models
An operations manager at a logistics company uses polynomial functions to predict fuel consumption based on vehicle weight. To analyze these models correctly, the manager must understand the specific components of the mathematical process. Match each term with the description of its role in this analysis.
A logistics company uses a polynomial function to model fuel consumption based on vehicle speed. True or False: To evaluate this function for a specific speed, that value must be substituted for the variable in every term of the polynomial expression before the order of operations is used to simplify the result.
At a manufacturing plant, an industrial engineer uses a polynomial function to model the daily production cost based on units produced. To find the cost of producing 150 units, the engineer must first substitute 150 for the variable throughout the polynomial expression, and then apply the ________ to simplify the expression and obtain the final cost.
Evaluating a Cost-Analysis Polynomial Model
A polynomial that indicates whether a clause is satisfied
Restoring unused variables and reading an exact count from the clause network