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# Finite Element Method

### Determine which of the given strings are recognized by the given deterministic finite-state automaton.

Determine whether each of these strings is recognized by the given deterministic finite-state automaton (which is displayed in an attached .doc file): (a) 010 (b) 1101 (c) 1111110 (d) 010101010

### 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)

### Repeat squaring method

What is the remainder if.... (See attached file for full problem description)

### Secant ( or Chord ) Method and Newton-Raphson Formula

Let f(x) = x^2 - x - 3 (a) Find the Newton-Raphson formula pk = g(pk-1). (b) Start with p0=1.6 and find p1,p2 and p3. (c) Start with p0=0.0 and find p1,p2, p3 and p4. What do you conjecture about this sequence.

### First Order Initial Value Problem: RK4 Method

Use the RK4 method, with a step size of h=0.5, to obtain an approximation... Please see the attached file for the fully formatted problem.

### 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?

### Wilson's Theorem : Cyclic Groups and Order of an Element

13.a) If G={g1,g1,....,gr} is an abelian group, show that g1,g2....gr equals the product of the elements of order 2. b) Prove Wilson's Theorem: If p is a prime then (p-1)! R (-1)(modp) note: R is a equivalence relation