Math 502
Written Assignment No. 3

due Wednesday, April 5, 2006

Directions: Use Maple for assistance in responding to the following problems. Please typeset your solutions. Explain what you have done. Maple session details are not necessary unless you think it important to include them. Accuracy is important.

Although you may refer to books and notes, you may not seek help from others on this written assignment.

1. Determine polynomials $f_0\left(x\right),f_1\left(x\right),f_2\left(x\right),\dots$ such that $${\left(x{y}^3+1+x+x^{2}y\right)}^3=\sum _{k\ge 0}{f}_k\left(x\right)y^k\text{.}$$

2. Find the least non-negative integer $x$ that solves the congruence equation $${10}^x\equiv 2\left(mod{40487}^7\right)\text{.}$$

3. How can the length 15 string represented by the vector of ASCII codes

   
     [97, 39, 98, 96, 99, 34, 100, 10, 101, 39, 102, 34, 103, 96, 104]
   

   -- which contains a newline and various quoting characters -- be entered as an input string in Maple without reference to the ASCII codes?

4. The vector

   
     [1374123367812492240294819411451017240070232394572901767016863846,
      839316502484928463600482820810738376821725250432320253174346182,
      1239738327596678208340985633338387227858526317346603640791773335,
      1786140405373535737086895440853230782772901282581414488521399167,
      500665432788843543145295210921142588622600876314918005209142550,
      1134791724566968592421020750873161158581657388017051854867699847,
      2289042576082270612059353048240349198604028364564225150215042759]
   

   represents a sequence of ASCII codes (range $0$ to $127$) that has been squeezed into blocks of 30 and then encrypted by taking a certain power modulo $$m=2468256835981809063232453777824906406995466865716803865301401733\text{.}$$ It may be decrypted by using the exponent $5^{15}$. What string was encrypted?

5. Write code for a Maple procedure that takes an expression $e$ in two variables $x,y$ as first argument and the variable names as second and third arguments and returns the anonymous function of two arguments $$\left(x,y\right)\mapsto x\frac{\partial e}{\partial x}+y\frac{\partial e}{\partial y}\text{.}$$ Make sure that the code you submit works correctly.