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    Finite Element Method

    Finite Element Method Problem

    Let F be a finite field. Show that every element of F is the sum of two squares. (hint: given , show that and each have more than elements. (See attached file for full problem description with proper symbols) ---

    Basis and Finite and Infinite Field Extensions

    Find a basis for the extension of and also calculate We know already that is infinite. Give an example of fields and (with neither nor equal to ) such that: a) and are both infinite b) is infinite and is finite. (See attached file for full problem description with proper symbols) All gaps are Q,

    Newton's method approximation

    Use Newton's method to approximate the x value of the point near x=3 of 2 functions 1. f(x) = 3 - x 2. g(x) = 1/(x^2) + 1 Do this problem for complete iterations to get an answer of about .001 of the real value hint let H(x) = f(x) - g(x)

    Conjugacy Classes Non-Identity Elements

    G is a finite group with elements a and b. Let the conjugacy classes of these elements be A and B respectively and suppose |A|^2, |B|^2 < |G|. Prove that there is a non-identity element x in G s.t. x commutes with both a and b.

    Test Binary Relations

    Test the binary relations on S for reflexivity, symmetry ,antisymmetry, and transitivity A) S = Q X p Y <-> ABS(X) <= ABS(Y) B) S = Z X p Y <-> x -y is an integral multiple of 3 C) S = N X P Y <-> X is odd D) S = Set of all squares in the place S1 p S2 <-> length of side of S1 = length of side S2 E)

    Finite-Element Method Description

    Please see the attached files for full problem description. Using Finite-Element Methods, assuming that stiffness of each element is equal to f.

    It is dealing with finite automatons

    (a)For each of the following languages over the unary alphabet {a}, construct a finite automaton accepting it. i. {a^2} ii. {a^2, a^3, a^4} (b) Let A be any finite nonempty subset of {a, a^2, a^3, a^4,...}. Is there always a finite automaton that accepts A?

    Golden Search Method

    Please help provide the details of how to arrive at the answer. The attached file has the problem, and an example.