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Aerobic treatment is a biological wastewater treatment process that takes place in the presence of oxygen. Aerobic biomass converts organics in the wastewater into carbon dioxide and new biomass.

Subscripts and superscripts can be made using the underscore _ and carats symbols.

We can apply both a subscript and a superscript at the same time. You must surround what you want to be the exponent/superscript with curly braces for subscripts or superscripts with more than one character. For example, x10 produces, while x10 does not.

Here are some math commands that are commonly used.

We can group the numerator and the denominator together as one number with fractions with a 1-digit numerator and a 1-digit denominator. For fractions with a numerator or a denominator that requires more than one character, you need to surround everything in curly brackets.

For continued fractions, use cfrac.

The commands sum, prod, lim, and log are used. Lower and upper bounds are the base of the logarithm and can be used in the same way as subscripts and superscripts. The upper and lower bounds for integrals are the same.

Some of these are in a display mode.

We can use sums, products, and logarithms.

These are often better in display mode.

The trigonometric function is just the abbreviation of the function with a backslash added.

There are a couple examples.

There are some examples of expressions. Most of these commands have been used before. The difference between regular math and display mode for definite integrals can be seen.

The makes a small space before thedx.

Some of the commands take subscripts in the same way as sums, products, and logarithms do. Some render differently in display mode.

The beginarray...endarray commands can be used to build an array or matrix.

We can use the matrix environments in the amsmath package.

You can read more about how the environment works.

This environment can be used to typeset any mathematics that calls for multiple columns.

It would be better to use the cases environment and text command.

In math mode, you can render letters in a variety of styles. You should be able to use these with any letters.

The amsfonts package is required to be included in the preamble. Do not attempt to do mathbbyear. You'll get, and it looks like nothing.

You can say a few more out of this document if you're persistent.

The text command can be used to drop a little text in the middle of math mode. In the math mode, the text command is most useful because it will force the output to a new line. SW.

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