# Mathematics

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Solution Library **

#### Converting Trigonometric Expressions Into Algebraic Expressions

QuestionShow that the simple extension Q (cos π/9): Q is an algebraic extension and also normal extension.SummaryThe question belongs to Mathematics and it is about simple extension of trigonometric expressions into algebraic expressions.Note: The solution is in handwritten format.Total Word Coun ... Read More

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#### Calculation Of Rational Coefficients

QuestionThe number 1 + √2 is a root of a cubic polynomial t3 + at2 + bt + c with rational coefficients. Prove that this polynomial has three real roots and one root is a rational number.SummaryThe question belongs to Mathematics and it is about rational coefficients and calculating that the po ... Read More

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#### Calculation Of Galios Group Extension

QuestionProve that the order of the Galios group of the extension Q(e2iπ/17) is equal to 16.SummaryThe question belongs to Mathematics and it is about calculation of Galios group extension.Note: The solution is in handwritten format.Total Word Count NA ... Read More

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#### Calculation Of Extension Of Fields

QuestionLet L:K be a extension of fields s.t . [L:K] = 2. Show that if K does not have characteristic 2, then there exists θEL such that L = K(θ) and θ2EK. SummaryThe question belongs to Mathematics and it is about calculation of extension of fields. Note: The solution is in handwr ... Read More

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#### Conversion of Galios Group Into Algebraic Form

QuestionSuppose L: K is a Galios extension whose Galios group G is isomorphic to the Klein four group. Assume that K does not have characteristic 2. Show that there exists x, y ε k such that every element of L can be expressed in the form a + b√x + c√y + d√xy with a, b, c, ... Read More

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#### Calculation Of Minimal Polynomial

QuestionLet α be the real number √(5 + √21)1. Express α in the form b√x + c√y with b, c, x, y ε Q.2. Calculate the minimal polynomial of α over Q.SummaryThe question belongs to Mathematics and it is about calculation of minimal polynomial.Note: The s ... Read More

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#### Submartingale Convergence Theorem

QuestionLet (Xn) be a submartingale such that supn Xn < ∞ a.s. and E supn (Xn−Xn−1) + < ∞. Show that Xn converges a.s.SummaryThe question belongs to Mathematics and it is about submartingale converging.Note: The solution is in handwritten format.Total Word Count NA ... Read More

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#### Fixed Filtration With Respect To Martingales

QuestionFor a sequence (An) of events, show that SummaryThe question belongs to Mathematics and it is about fixed filtration with respect to martingales defined.Note: The solution is in handwritten format.Total Word Count NA ... Read More

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#### Calculation Of Martingale

QuestionLet (Xn) be a martingale and write Ã¢Ë†â€ n = Xn − Xn−1, Suppose that bm ↑ ∞ and Σ∞m=1 b −2m EÃ¢Ë†â€ 2m < ∞. Prove that Xn/bn → 0 a.s.SummaryThe question belongs to Mathematics and it is about martingale calculationNote: The solution is ... Read More

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#### Calculation Of Non-Negative Martingale

Question Let (Xn) be a martingale with supnE|Xn| < ∞. Show that there is a representation Xn = Yn – Zn where (Yn) and (Zn) are non-negative martingales such that supn EYn < ∞ and supn EZn < ∞ SummaryThe question belongs to Mathematics and it is about no ... Read More

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#### Martingale Calculation

QuestionLet (Xn) be adapted to (Fn) with 0 ≤ Xn ≤ 1. Let α, β > 0 with α + β = 1. Suppose X0 = x0 and P(Xn+1 = α + βXn|Fn) = Xn, P(Xn+1 = βXn|Fn) = 1 − XnShow that Xn → X∞ a.s., where P(X∞ = 1) = x0 and P(X∞ = 0) = 1− ... Read More

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#### Calculate The Value of Martingale

QuestionSuppose Fn ↑ F∞ and Yn → Y∞ in L1. Show that E(Yn|Fn) →E(Y∞|F∞) in L1SummaryThe question belongs to Mathematics and it deals with calculation of martingaleNote: The solution is in handwritten format.Total Word Count NA ... Read More

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