## 2017年11月27日月曜日

### 数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 指数関数と対数関数 - 一般の指数関数(対数関数、上に凸、不等号)

1. $\begin{array}{}f\left(t\right)=\mathrm{log}\left({a}^{t}{b}^{1-t}\right)-\mathrm{log}\left(ta+\left(1-t\right)b\right)\\ =\mathrm{log}{a}^{t}+\mathrm{log}{b}^{1-t}-\mathrm{log}\left(ta+\left(1-t\right)b\right)\\ =t\mathrm{log}a+\left(1-t\right)\mathrm{log}b-\mathrm{log}\left(ta+\left(1-t\right)b\right)\end{array}$

対数関数は上に凸な関数なので、

$t\mathrm{log}a+\left(1-t\right)\mathrm{log}b\le \mathrm{log}\left(ta+\left(1-t\right)b\right)$

よって、

$\begin{array}{}t\mathrm{log}a+\left(1-t\right)\mathrm{log}b-\mathrm{log}\left(ta+\left(1-t\right)b\right)\le 0\\ \mathrm{log}\left({a}^{t}{b}^{1-t}\right)\le \mathrm{log}\left(ta+\left(1-t\right)b\right)\end{array}$

ゆえに、

${a}^{t}{b}^{1-t}\le ta+\left(1-t\right)b$

（証明終）

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Rational, plot

x = symbols('x')
a = 2
b = 3
f = a ** x * b ** (1 - x)
g = a * x + (1 - x) * b
fs = [f, g]
for t in fs:
pprint(t)
print()

p = plot(*fs, (x, -0.5, 1.5), show=False, legend=True)
for i, color in enumerate(['red', 'green']):
p[i].line_color = color

p.save('sample15.svg')


$./sample14.py x 2 5 ⋅x -x 2 2/5 ⋅x -x 2 3/5 ⋅x -x 2 4/5 ⋅x 2 x -x 2 6/5 ⋅x -x 2 7/5 ⋅x -x 2 8/5 ⋅x -x 2 9/5 ⋅x -x 2 2 ⋅x$


HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" min="0" step="0.1" value="2">
<label for="b0">b = </label>
<input id="b0" type="number" min="0" step="0.1" value="3">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample15.js"></script>


JavaScript

let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
btn0 = document.querySelector('#draw0'),
btn1 = document.querySelector('#clear0'),
input_r = document.querySelector('#r0'),
input_dx = document.querySelector('#dx'),
input_x1 = document.querySelector('#x1'),
input_x2 = document.querySelector('#x2'),
input_y1 = document.querySelector('#y1'),
input_y2 = document.querySelector('#y2'),
input_a0 = document.querySelector('#a0'),
input_b0 = document.querySelector('#b0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_a0, input_b0],
p = (x) => pre0.textContent += x + '\n',
range = (start, end, step=1) => {
let res = [];
for (let i = start; i < end; i += step) {
res.push(i);
}
return res;
};

let draw = () => {
pre0.textContent = '';

let r = parseFloat(input_r.value),
dx = parseFloat(input_dx.value),
x1 = parseFloat(input_x1.value),
x2 = parseFloat(input_x2.value),
y1 = parseFloat(input_y1.value),
y2 = parseFloat(input_y2.value),
a0 = parseFloat(input_a0.value),
b0 = parseFloat(input_b0.value);

if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}

let points = [],
lines = [[0, y1, 0, y2, 'red'],
[1, y1, 1, y2, 'red']],
f = (x) => a0 ** x * b0 ** (1 - x),
g = (x) => x * a0 + (1 - x) * b0,
fns = [[f, 'green'],
[g, 'blue']],
fns1 = [],
fns2 = [];

fns
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = f(x);

points.push([x, y, color]);
}
});

fns1
.forEach((o) => {
let [f, color] = o;

lines.push([x1, f(x1), x2, f(x2), color]);
});

fns2
.forEach((o) => {
let [f, color] = o;

for (let x = x1; x <= x2; x += dx0) {
let g = f(x);
lines.push([x1, g(x1), x2, g(x2), color]);
}
});

let xscale = d3.scaleLinear()
.domain([x1, x2])
let yscale = d3.scaleLinear()
.domain([y1, y2])

let xaxis = d3.axisBottom().scale(xscale);
let yaxis = d3.axisLeft().scale(yscale);
div0.innerHTML = '';
let svg = d3.select('#graph0')
.append('svg')
.attr('width', width)
.attr('height', height);

svg.selectAll('line')
.data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
.enter()
.append('line')
.attr('x1', (d) => xscale(d[0]))
.attr('y1', (d) => yscale(d[1]))
.attr('x2', (d) => xscale(d[2]))
.attr('y2', (d) => yscale(d[3]))
.attr('stroke', (d) => d[4] || 'black');

svg.selectAll('circle')
.data(points)
.enter()
.append('circle')
.attr('cx', (d) => xscale(d[0]))
.attr('cy', (d) => yscale(d[1]))
.attr('r', r)
.attr('fill', (d) => d[2] || 'green');

svg.append('g')
.attr('transform', translate(0, ${height - padding})) .call(xaxis); svg.append('g') .attr('transform', translate(${padding}, 0))
.call(yaxis);

[fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();