Prove that trace (A*A)=||A||F2 Definition. Let A be an m x n complex matrix. Its Frobenius noTm, denoted by ||A||F, is defined as follows: m ΣΣΑ Β) / . A|| F = j=1 k=1 1. (7 marks) Let A be as above. Prove that trace(AH A) || A||?
(Complex Analysis) Prove the following maximum principle for harmonic functions: Let u be harmonic in a bounded domain E and continuous in E ∪ dE. Then max(x, y)E E U dE u= sup(x, y)E E U dE u; min(x, y)E E U dE u= inf(x, y)E E U dE u. (Not the first E after the subscript (x, y) denotes element of, and the next on is the domain and the next is the derivative of the domain.)
complex analysis, cite all theorems used Let fcz) be an entire function and there exists a real number Ro such that Ifcail sizl for any complex number z 12/7RO Prove that f is of the form VZEC with f(Z)= arbe
Definition:In the complex numbers, let J denote the set, {x+y√3i :x and y are in Z}. J is an integral domain containing Z. If a is in J, then N(a) is a non-negative member of Z. If a and b are in J and a|b in J, then N(a)|N(b) in Z. The units of J are 1, -1 Question:If a and b are in J and ab = 2, then prove one of a and b is a unit. Thus,...
A1. Let M be an R-module and let I, J be ideals in FR (a) Prove that Ann(I +J) -Ann(I) n Ann(J). (b) Prove that Ann(InJ)2 Ann(I) + Ann(J). (c) Give an example where the inclusion in (b) is strict. (d) If R is commutative ald unital and I, J are cornaximal (that is, 1 +J-(1)), prove that Ann(InJ) Ann(I)+Ann(J).
complex analysis Let f(z) be continuous on S where for some real numbers 0< a < b. Define max(Re(z)Im(z and suppose f(z) dz = 0 S, for all r E (a, b). Prove or disprove that f(z) is holomorphic on S.
4. Prove that N1+ N2,+N3-1 for the Constant Strain Triangle Element (15%) 4. Prove that N1+ N2,+N3-1 for the Constant Strain Triangle Element (15%)
Let α be a complex number with α^2 = √3 − √5. Prove that Q(α)/Q is not Galois.
Please Prove. Prove 2 n > n2 by induction using a basis > 4: Basis: n 5 32> 25 Assume: Prove:
Complex Analysis (use the Liouville equation): Suppose that f(z)u, ) (, u) is an entire function such that 7u9n is bounded. Prove that fis constant Hint: Multiply f by an appropriate complex constant. Suppose that f(z)u, ) (, u) is an entire function such that 7u9n is bounded. Prove that fis constant Hint: Multiply f by an appropriate complex constant.