---
title: Images and Math Test
date: 2026-10-14
summary: Test local images, inline math, display equations, matrices, tables, code blocks, and PDF links.
tags: Markdown, Math, Images, PDF
---

This note tests images and mathematical notation alongside regular text. It uses the same anonymous header as the rest of the website.

## 1. Local Images

The following image loads from the website's `assets/images` folder and scales to fit the available width.

![Illustration of the function y=x²](assets/images/note-demo.svg "SVG image")

*Figure 1. An SVG illustration with x on the horizontal axis and y on the vertical axis.*

Here is the same illustration in PNG format:

![Sample PNG illustration](assets/images/note-demo.png "PNG image")

*Figure 2. A PNG illustration. Both formats use standard Markdown image syntax.*

Image paths start at the website root:

```markdown
![Image description](assets/images/your-image.png "Optional image title")
```

## 2. Inline Math

The linear elastic relation is $\sigma = E\varepsilon$, where $E$ is Young's modulus. Subscripts, powers, and Greek letters also work: $x_i^2$ and $\alpha + \beta$.

```markdown
The linear elastic relation is $\sigma = E\varepsilon$.
```

## 3. Display Equations

Place an equation between two lines containing `$$` to display it on a separate, centered line:

$$
\int_0^1 x^2\,\mathrm{d}x = \left[\frac{x^3}{3}\right]_0^1 = \frac{1}{3}
$$

```markdown
$$
\int_0^1 x^2\,\mathrm{d}x = \frac{1}{3}
$$
```

## 4. Multiline Equations and Matrices

Use the `aligned` environment to align the steps of a calculation:

$$
\begin{aligned}
f(x) &= x^2 + 2x + 1 \\
     &= (x + 1)^2 \\
f'(x) &= 2x + 2
\end{aligned}
$$

Matrices and vectors can appear in the same equation:

$$
\begin{bmatrix}
2 & -1 \\
-1 & 2
\end{bmatrix}
\begin{bmatrix}
u_1 \\ u_2
\end{bmatrix}
=
\begin{bmatrix}
F_1 \\ F_2
\end{bmatrix}
$$

## 5. Tables, Lists, and Code

Inline math also works inside tables:

| Feature | Example | Purpose |
| --- | --- | --- |
| Subscript | $x_i$ | Identify the i-th element |
| Summation | $\sum_{i=1}^{n} x_i$ | Sum a series of values |
| Fraction | $\frac{a}{b}$ | Express a ratio |

- Use **bold** and *italics* to highlight key terms.
- Use blockquotes for excerpts or reminders.
- Code blocks preserve indentation and line breaks. Dollar signs inside them are not rendered as math.

> This is a blockquote. Use it for reading excerpts, observations, or questions to revisit.

```python
def square(x):
    return x ** 2

print(square(3))
```

## 6. Literal Text and Long Equations

To show math as literal text, put it inside backticks, as in `$E = mc^2$`. Escape a regular dollar sign with `\$`: for example, the price is \$20.

On narrow screens, long equations scroll horizontally within their own area while the rest of the page keeps its normal width:

$$
\mathcal{L}(\theta) = \frac{1}{N}\sum_{i=1}^{N}\left(y_i - \sum_{j=1}^{M}\theta_j x_{ij}\right)^2 + \lambda\sum_{j=1}^{M}\theta_j^2
$$

This note checks image loading, equation layout, and page overflow on mobile screens.

## 7. PDF Files

PDF attachments use ordinary Markdown links:

- [Open the local PDF sample](assets/documents/pdf-link-test.pdf)
- [Open the W3C PDF sample](https://www.w3.org/WAI/ER/tests/xhtml/testfiles/resources/pdf/dummy.pdf)

The local file is stored in `assets/documents/` and is included when the site is built. Both PDF links open in a new tab, keeping this note available. Your browser may open its PDF viewer or offer to save the file, depending on its settings.

```markdown
[Open the local PDF sample](assets/documents/pdf-link-test.pdf)

[Open an external PDF](https://www.w3.org/WAI/ER/tests/xhtml/testfiles/resources/pdf/dummy.pdf)
```

## 8. Code Block Controls

The language is shown above each code block. Hover over a block or focus its copy button with the keyboard to reveal Copy. Touch devices keep the button visible. Copy always includes the complete code, even while the block is folded.

Blocks longer than 20 lines initially show the first 12 lines. Use the button below this example to expand or collapse it. Printing includes all lines.

```python
from statistics import fmean


def moving_average(values, window=3):
    if window < 1:
        raise ValueError("window must be positive")
    if window > len(values):
        return []

    result = []
    for start in range(len(values) - window + 1):
        sample = values[start:start + window]
        result.append(fmean(sample))
    return result


def normalize(values):
    maximum = max(values, default=0)
    if maximum == 0:
        return [0 for value in values]
    return [value / maximum for value in values]


measurements = [12, 14, 13, 18, 17, 21, 20]
smoothed = moving_average(measurements)
scaled = normalize(smoothed)

for index, value in enumerate(scaled, start=1):
    print(f"Sample {index}: {value:.3f}")
```