(7) In the ring (Z, +,.), we get n {P: P non trival prime ideal in Z} %3D ....... (a) o (b) (Z, +,.) (c) ({0}, +,.) (d) No Choice

Q: Consider P, with the standard inner product. Then which one of the following pair is orthogonal a)…

A: for orthogonality of the two polynomial f and g is given by <f(x),g(x)>=∫-11f(x)g(x)dx=0

Q: 1. Consider the formula A = (vx ₁(x, a₁)) → (y (p2(y) ^ P3(y, α2))). (a) Create a formula tree for…

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Q: The complex numbers an, n > 0, are defined by the condition that Σan (z-2i)" = 1+ 2z+3z² + Σ(n+1)z"…

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Q: 7. Give a non-combinatorial proof of the following identities: (a) n - k = n k - 1 (b) EOC)-)- n-k m…

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Q: Consider the following polynomials in P • P4(x)=x+1; Ps(x)=²+1; • Pu(x)=2²³ +1. P₁(x) = x³ + x² +1;…

A: The given polynomials are p1x=x3+x2+1p2x=x3+xp3x=x2+xp4x=x+1p5x=x2+1p6x=x3+1 (a) We have to find a…

Q: . On the other hand, any ideal I satisfies V(√I) = V(I). Prove that, using ideals and varietes…

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Q: We assign a complex number Am to each n-tuple of non-negative integers m = (m₁, ..., mn)…

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Q: 3. Let r(r) r*...*r(x) be the p-fold convolution of r(x) by itself. Verify by induction that 0 2 be…

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Q: (8) In the ring (Z, +o6), we get n {(a): a prime element in Z,}: (a) o (b) ((0), +66) (c) (Z6, +66)…

A: Prime Ideal: Let R, +, · be a commutative ring. An ideal A of R is said to be a prime ideal of R if…

Q: 1 2 3 3) Consider A = 0 1 2 2 31 teristic polynomial of A? a) 23 – 2:2 – 9z +3. Which one of the…

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Q: [1 2 [1 0 3 and C = 2 1 [1 1 31 1 a) Let A =|-1 2), B = |2 2 1 If possible, 1 3 -1] compute (A" –…

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Q: 6) The ving of integers(2) is an ideal ot veal number (R ) OTrue D Fu lse

A: Given:   Statement :  The Ring of integer (Z) is an ideal of the real (R) number

Q: (7) In the ring (Z, +, .), we get n {P:P non trival prime ideal in Z}: (a) o (b) (Z, +, .) (c)…

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Q: Prove that (n – 1) +n° + (n + 1)° is divisible by 9 for n E {1,2,3, . }

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Q: Show (x – a,)²(x – az)² .. (x – an)? + 1| is irreducible in Z[x] and Q[x]

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Q: deal with Hermite’s equation: y′′−2xy′+2αy =0, −∞< x <∞.  ...........(11.3.13) Use some form…

A: note : Since you have posted multiple questions, we will provide the solution only to the first…

Q: Suppose X = (X1, X2, . .. , Xk) ~ Multinomial(n, p) , with p = (p1, P2, ..., Pk), prove that Cov(X;,…

A: Introduction :-  Given that X is a vector of random variable and it follows Multinomial(n,p) Where p…

Q: In Z[i], show that 3 is irreducible but 2 and 5 are not

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Q: a.) Ker(p) and p(24) for p : Z → Z7 such that Ф(1) %3D 4 b.) Ker(p) and p(16) for p : Z → Z10 such…

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Q: (" * ?) Prove: 1n + 2(n – 1) + 3(n – 2) + …+ (n – 1)(2) + n(1) = using lattice paths.

A: As per the question we have to prove the following summation identity : 1n + 2(n-1) + 3(n-2) + ... +…

Q: (8) In the ring (Z,, +6,•6 ), we get n {(a): a prime element in Zg} ...... (а) ф (b) ((0), +6, ·6 )…

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Q: solve b and c to Show the, for fo)- y The Bernstein Polynom iall of The Bernstein Poly nom ial of…

A: Use the following formulae, to prove the required result.…

Q: Prove the following identity by interpreting both sides in terms of partitions: zkak(k+1)/2 I[(1+…

A: We will assume both sides of the equation as an polynomial in z  Then we will use algebraic…

Q: (7.8) Lemma. Let K, L be simplicial complexes which intersect in a common subcomplex, then z(K U L)…

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Q: (1) Let I be ideal and a does not exist in I, then I is a subset of (a, I). T O F

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Q: 19.1.3. Recall that if a is any complex number, then Z[a] = {p(a) | p(x) = Z[x]}. Let R = Z[]. (a)…

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Q: Q1) For f(x), g(x), and Z„[x] given in Exercises 7-10, find the greatest common divisor d(x) of f(x)…

A: Given - f(x)=x3+x2+2x+2 and g(x)=x4+2x2+x+1  We have to find greatest common divisor d(x) for the…

Q: 12) The solution of 2lnx-In(x+ 2)=In2 is a) 0 b) 1 c) {-1,4) d) 4 e) поп 13) sin( 2 csc"1(2)) = b) -…

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Q: - Let R=(Z,t9 ,.9). Find 1. Char(R) 2. Nilpotent ideals of R 3. Prime ideals of R.

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Q: (8) In the ring (Z,, +6, •6 ), we get n{(a): a prime element in Z6} = ...... (a) $ (b) ((0), +6, 6 )…

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Q: A. Verify that the Manhattan metric (from page 71) is really a metric on M = R². (The interesting…

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Q: Consider P2 with the standard inner product. Then which one of the following pair is orthogonal a)…

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Q: Let z = x + yi, z, = 1- 0.5i, and z, + z, = 3-1.5i. Find In(z,) - In(z2) C) In(50) + 2nin A) In(25)…

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Q: 17. (a) If p is prime and n > 1, then (p") is a primary, but not a prime ideal of Z.

A: We'll answer the first question since the exact one wasn't specified. Please submit a new question…

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.