## 2018年2月16日金曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分の性質 - 不等式(累乗と累乗根、絶対値)

1. $\begin{array}{}{\alpha }^{\frac{1}{p}}{\beta }^{\frac{1}{q}}\le \frac{\alpha }{p}+\frac{\beta }{q}\\ \frac{\left|f\right|}{{∥f∥}_{p}}·\frac{\left|g\right|}{{∥g∥}_{q}}\le \frac{{\left|f\right|}^{p}}{p{∥f∥}_{p}^{p}}+\frac{{\left|g\right|}^{q}}{q{∥g∥}_{q}^{q}}\\ {\int }_{a}^{b}\left|f\left(x\right)\right|\left|g\left(x\right)\right|\mathrm{dx}\le {∥f∥}_{p}{∥g∥}_{q}\left(\frac{1}{p}·\frac{{∥f∥}_{p}^{p}}{{∥f∥}_{p}^{p}}+\frac{1}{q}·\frac{{∥g∥}_{q}^{h}}{{∥g∥}_{q}^{q}}\right)\\ \le ||f|{|}_{p}||g||q\left(\frac{1}{p}+\frac{1}{q}\right)\\ ={∥f∥}_{p}{∥g∥}_{q}\\ \left|\underset{a}{\overset{b}{\int }}f\left(x\right)g\left(x\right)\mathrm{dx}\right|\le \underset{a}{\overset{b}{\int }}\left|f\left(x\right)\right|\left|g\left(x\right)\right|\mathrm{dx}\end{array}$

よって、

$\left|\underset{a}{\overset{b}{\int }}f\left(x\right)g\left(x\right)\right|\le {∥f∥}_{p}{∥g∥}_{q}$

ゆえに、

$\left|⟨f,g⟩\right|\le {∥f∥}_{p}{∥g∥}_{q}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, root, Function

x = symbols('x')
a, b, p, q = symbols('a, b, p, q')

def dot(f, g):
return Integral(f * g, (x, a, b))

def norm(f, p):
return root(Integral(abs(f) ** p, (x, a, b)), p)
f = Function('f')(x)
g = Function('g')(x)

l = abs(dot(f, g))
r = norm(f, p) * norm(g, q)

for t in [l, r, l <= r]:
pprint(t)
print()


$./sample2.py │b │ │⌠ │ │⎮ f(x)⋅g(x) dx│ │⌡ │ │a │ ______________ ______________ ╱ b ╱ b ╱ ⌠ ╱ ⌠ ╱ ⎮ p ╱ ⎮ q ╱ ⎮ │f(x)│ dx ⋅ ╱ ⎮ │g(x)│ dx p ╱ ⌡ q ╱ ⌡ ╲╱ a ╲╱ a ______________ ______________ ╱ b ╱ b │b │ ╱ ⌠ ╱ ⌠ │⌠ │ ╱ ⎮ p ╱ ⎮ q │⎮ f(x)⋅g(x) dx│ ≤ ╱ ⎮ │f(x)│ dx ⋅ ╱ ⎮ │g(x)│ dx │⌡ │ p ╱ ⌡ q ╱ ⌡ │a │ ╲╱ a ╲╱ a$


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">

<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="sample2.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'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
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 f = (x) => Math.exp(x),
g = (x) => 2 * x,
fns = [[f, 'red'],
[g, 'green'],
[(x) => f(x) * g(x), 'blue'],
[(x) => Math.abs(f(x) * g(x)), 'orange']];

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);

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

let points = [],
lines = [],
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();


#### 1 コメント :

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