answer:Yes, there is a way to calculate the number and sum of elements in A cap B without manually comparing each element. First, note that 360 = 2^3 times 3^2 times 5^1. Therefore, the divisors of 360 are of the form 2^a times 3^b times 5^c, where 0 leq a leq 3, 0 leq b leq 2, and 0 leq c leq 1. Next, observe that the elements of A are of the form 6 + 4k, where k is a non-negative integer. Therefore, the elements of A cap B are those elements of A that are also divisors of 360. To find the number of elements in A cap B, we need to count the number of ways to choose a, b, and c such that 2^a times 3^b times 5^c is a divisor of 6 + 4k. Since 2 is a factor of 6 + 4k, we must have a geq 1. Since 3 is not a factor of 6 + 4k, we must have b = 0. Since 5 is not a factor of 6 + 4k, we must have c = 0. Therefore, the number of elements in A cap B is equal to the number of ways to choose a such that 2^a is a divisor of 6 + 4k. Since 2^1 is a divisor of 6 + 4k, we have a = 1. Since 2^2 is not a divisor of 6 + 4k, we have a neq 2. Since 2^3 is not a divisor of 6 + 4k, we have a neq 3. Therefore, the number of elements in A cap B is equal to boxed{1}. To find the sum of the elements in A cap B, we need to add up the elements of A that are also divisors of 360. Since the only element of A that is also a divisor of 360 is 6, the sum of the elements in A cap B is equal to boxed{6}.

answer:The conversion of 1409 to base 19 is represented as text{3h3}_{19}.

answer:The inequality is not necessarily true. A counterexample to demonstrate this is given by the operator A in mathcal{B}(mathcal{H}) defined as follows: A = begin{bmatrix}0&10&0end{bmatrix} Here, we have: supleft{|langle Ax;|;yrangle| : |x|=|y|=1right} = 1 This can be achieved, for instance, when x = (1, 0)^T and y = (0, 1)^T. On the other hand, supleft{|langle Ax;|;xrangle| : |x|=1right} = frac{1}{2} This can be seen by considering the unit vector x = left(frac{1}{sqrt{2}}, frac{1}{sqrt{2}}right)^T. Hence, the inequality does not hold in general.

answer:The function f(x) = x^2 + 1 is a parabola that opens upwards because the coefficient of x^2 is positive. The vertex of the parabola is (0, 1), which is the point where the parabola changes direction.