answer:The properties given do not uniquely define R as a reflection across the point a. Even in the simpler case of Bbb{R}^2, we can construct a non-linear map R satisfying these conditions, distinct from point reflection. Let R(a) = a and choose an arbitrary direction d. Represent any point x neq a in polar coordinates: x = (r, phi), where |vec{x} - vec{a}| = r and angle(d, vec{x} - vec{a}) = phi. Choose a constant 0 < alpha < frac{1}{pi} and define the map R as follows: R(r, phi) = left{ begin{array}{cl} (r, phi + alpha(pi - phi)^2 + pi) & text{if } 0 leq phi < pi (r, phi - alphaphi^2 + pi) & text{if } pi leq phi < 2pi end{array} right. It can be verified that: 1. R has a as its unique fixed point. 2. R preserves the distance from a to any point on V, as it maps circles centered at a onto themselves. 3. R^2(x) = x for all x in V. Since alpha neq 0, this map is not the point reflection map, demonstrating that there are other functions satisfying the given properties.

answer:The infinite continued fraction can be represented as the reciprocal of the difference between x and the half of the square root of x^2 + 1. Therefore, we have: [ frac{1}{x+frac{1}{x+frac{1}{x+ddots}}} = frac{1}{x + frac{1}{2}sqrt{x^2 + 1}} ] Substitute the value of x: [ x = frac{5168}{5223} ] Now calculate the expression: [ frac{1}{x + frac{1}{2}sqrt{x^2 + 1}} = frac{1}{frac{5168}{5223} + frac{1}{2}sqrt{left(frac{5168}{5223}right)^2 + 1}} ] Simplify the expression to get the answer: [ frac{5223}{5168 + frac{5223}{2}sqrt{left(frac{5168}{5223}right)^2 + 1}} = frac{5223}{5168 + frac{5223}{2}sqrt{frac{5168^2}{5223^2} + 1}} = frac{5223}{5168 + frac{5223}{2}sqrt{frac{5168^2 + 5223^2}{5223^2}}} ] [ = frac{5223}{5168 + frac{5223}{2}cdotfrac{sqrt{5168^2 + 5223^2}}{5223}} ] [ = frac{5223}{5168 + frac{1}{2}sqrt{5168^2 + 5223^2}} ] Calculating the square root and simplifying: [ = frac{5223}{5168 + frac{1}{2}sqrt{33956785}} ] [ = frac{5223}{5168 + frac{1}{2}(5785)} ] [ = frac{5223}{5168 + 2892.5} ] [ = frac{5223}{7060.5} ] [ = frac{2^3 cdot 3 cdot 869}{3^2 cdot 5 cdot 215} ] [ = frac{869}{215} cdot frac{2}{3} ] [ = frac{1738}{645} ] [ = frac{sqrt{33956785} - 2584}{5223} ] So the revised answer is: [ frac{sqrt{33956785} - 2584}{5223} ]

answer:The region we are considering is bounded by the functions: y = x^3 y = 1 x = 2 The limits for integration are from x = 0 to x = 2. To find the volume, we'll use the disk method, which involves integrating the area of the cross-sectional disks along the x-axis: Volume = ∫[π(f(x))^2 - π(g(x))^2] dx where f(x) = x^3 and g(x) = 1 Now, let's compute the integral: Volume = ∫[π(x^3)^2 - π(1)^2] dx = ∫[π(x^6) - π] dx = π∫[x^6 - 1] dx Evaluating the integral: Volume = π[1/7*x^7 - x] | from 0 to 2 = π[(2^7/7) - 2 - (0 - 0)] = π[(128/7) - 2] = π[114/7] Therefore, the volume of the solid is approximately 114π/7 cubic units, or approximately 51.16 cubic units.

answer:Martin Davis, Yuri Matiyasevich, Hilary Putnam, and Julia Robinson did indeed negatively settle Hilbert's 10th problem, proving that there is no general algorithm that can determine whether a given Diophantine equation has a solution in integers. As for an analogous result for differential equations, there is a partial answer in an old short paper titled "Some Recursively Unsolvable Problems in Analysis" by Adler, published in the AMS Proceedings in 1969. This paper shows that there is no algorithm for determining the solvability of systems of algebraic differential equations. This result is obtained by reducing the problem to the nonexistence of an algorithm for determining whether an exponential Diophantine equation has a solution, as established by Davis, Putnam, and Robinson.