91Ó°ÊÓ

Using the geometric interpretation of complex numbers a) explain the inequalities \(\left|z_{1}+z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|\) and \(\left|z_{1}\right|+\cdots+\left|z_{n}\right| \leq\left|z_{1}\right|+\) \(\cdots+\left|z_{n}\right| ;\) b) exhibit the locus of points in the plane \(\mathbb{C}\) satisfying the relation \(|z-1|+\) \(|z+1| \leq 3\) c) describe all the \(n\)th roots of unity and find their sum; d) explain the action of the transformation of the plane \(\mathbb{C}\) defined by the formula \(z \mapsto \bar{z}\).

Efficiency in rocket propulsion. a) Let \(Q\) be the chemical energy of a unit mass of rocket fuel and \(\omega\) the outflow speed of the fuel. Then \(\frac{1}{2} \omega^{2}\) is the kinetic energy of a unit mass of fuel when ejected. The coefficient \(\alpha\) in the equation \(\frac{1}{2} \omega^{2}=\alpha Q\) is the efficiency of the processes of burning and outflow of the fuel. For engines of solid fuel (smokeless powder) \(\omega=2 \mathrm{~km} / \mathrm{s}\) and \(Q=1000 \mathrm{kcal} / \mathrm{kg}\), and for engines of liquid fuel (gasoline with oxygen) \(\omega=3 \mathrm{~km} / \mathrm{s}\) and \(Q=2500 \mathrm{kcal} / \mathrm{kg} .\) Determine the efficiency \(\alpha\) for these cases. b) The efficiency of a rocket is defined as the ratio of its final kinetic energy \(m_{\mathrm{R}} \frac{v^{2}}{2}\) to the chemical energy of the fuel burned \(m_{\mathrm{F}} Q .\) Using formula \((5.139)\), obtain a formula for the efficiency of a rocket in terms of \(m_{\mathrm{R}}, m_{\mathrm{F}}, Q\), and \(\alpha\) (see part a)). c) Evaluate the efficiency of an automobile with a liquid-fuel jet engine, if the automobile is accelerated to the usual city speed limit of \(60 \mathrm{~km} / \mathrm{h}\). d) Evaluate the efficiency of a liquid-fuel rocket carrying a satellite into low orbit around the earth. e) Determine the final speed for which rocket propulsion using liquid fuel is maximally efficient. f) Which ratio of masses \(m_{\mathrm{F}} / m_{\mathrm{R}}\) yields the highest possible efficiency for any kind of fuel?

Let \(f\) be a function that is infinitely differentiable at 0 . Show that a) if \(f\) is even, then its Taylor series at 0 contains only even powers of \(x\); b) if \(f\) is odd, then its Taylor series at 0 contains only odd powers of \(x\).

Show that a) if a convex function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is bounded, it is constant; b) if $$ \lim _{x \rightarrow-\infty} \frac{f(x)}{x}=\lim _{x \rightarrow+\infty} \frac{f(x)}{x}=0 $$ for a convex function \(f: \mathbb{R} \rightarrow \mathbb{R}\), then \(f\) is constant. c) for any convex function \(f\) defined on an open interval \(a

The Legendre transform \(^{21}\) of a function \(f: I \rightarrow \mathbb{R}\) defined on an interval \(I \subset \mathbb{R}\) is the function $$ f^{*}(t)=\sup _{x \in I}(t x-f(x)) $$ Show that a) The set \(I^{*}\) of values of \(t \in \mathbb{R}\) for which \(f^{*}(t) \in \mathbb{R}\) (that is, \(\left.f^{*}(t) \neq \infty\right)\) is either empty or consists of a single point, or is an interval of the line, and in this last case the function \(f^{*}(t)\) is convex on \(I^{*}\). b) If \(f\) is a convex function, then \(I^{*} \neq \varnothing\), and for \(f^{*} \in C\left(I^{*}\right)\) $$ \left(f^{*}\right)^{*}=\sup _{t \in I^{*}}\left(x t-f^{*}(t)\right)=f(x) $$ for any \(x \in I .\) Thus the Legendre transform of a convex function is involutive, (its square is the identity transform). c) The following inequality holds: $$ x t \leq f(x)+f^{*}(t) \quad \text { for } x \in I \text { and } t \in I^{*} $$ d) When \(f\) is a convex differentiable function, \(f^{*}(t)=t x_{t}-f\left(x_{t}\right)\), where \(x_{t}\) is determined from the equation \(t=f^{\prime}(x) .\) Use this relation to obtain a geometric interpretation of the Legendre transform \(f^{*}\) and its argument \(t\), showing that the Legendre transform is a function defined on the set of tangents to the graph of \(f\). e) The Legendre transform of the function \(f(x)=\frac{1}{\alpha} x^{\alpha}\) for \(\alpha>1\) and \(x \geq 0\) is the function \(f^{*}(t)=\frac{1}{\beta} t^{\beta}\), where \(t \geq 0\) and \(\frac{1}{\alpha}+\frac{1}{\beta}=1 .\) Taking account of \(\left.\mathrm{c}\right)\), use this fact to obtain Young's inequality, which we already know: $$ x t \leq \frac{1}{\alpha} x^{\alpha}+\frac{1}{\beta} t^{\beta} $$ f) The Legendre transform of the function \(f(x)=\mathrm{e}^{x}\) is the function \(f^{*}(t)=\) \(t \ln \frac{t}{e}, t>0\), and the inequality $$ x t \leq \mathrm{e}^{x}+t \ln \frac{t}{\mathrm{e}} $$ holds for \(x \in \mathbb{R}\) and \(t>0\)