This is one of the most common ways to form an absolute phrase. It might be helpful for some people to imagine this pattern with a verb between the noun and the participle. For example, if you say The question was still unanswered, you have a complete sentence; if, on the other hand, you say The question unanswered and you then attach that phrase to a main sentence, then you have an absolute phrase.

Here are some examples. The absolute phrases look like this.

The question still unanswered, the teacher decided to address the confusion of her students more closely.
The train running late, we decided to get off at the next stop and take a taxi home.
There are many industries in California vital to its economy, with technology being one of the most important.

Compare these sentences with the verbs and conjunctions in them:

The question was still unanswered, and the teacher decided to address the confusion of her students more closely.
The train was running late, so we decided to get off at the next stop and take a taxi home.
There are many industries in California vital to its economy, and technology is one of the most important.

Important! Although many of these absolute phrases could be written with the word being in them, more formal English (and ETS!) tends not to use being when being is optional. If you've studied GMAT Sentence Correction for a while, then you know that the word being raises a big red flag on the test!

Here are some examples:

The movie being over, we left the theater.

This sentence could be rewritten like this:
The movie over, we left the theater.

Similarly, having + past participle is often so semantically similar to the sentence without it that many sentences are written without having + past participle.

An example would be very good here: