Absolute constructions can be translated:
By a Russian adverbial clause of time, cause, condition or attendant circumstances.
The weather being cold the children stayed at home.
Поскольку погода была холодная, дети остались дома.By part of a compound sentence (сложносочиненное предложение) separated from the rest of sentence by ';'.
They walked in the park, with John smoking his pipe.
Они гуляли в парке; Джон курил трубку.By Russian деепричастие.
She comes into the room, the revolver in her hand.
Она вошла в комнату, держа в руке револьвер.By a noun with a preposition.
He comes into the room, his revolver in his hand.
Он вошел в комнату с револьвером в руках.
Note: Sometimes we find a combination of the Nominative Absolute Participial and the structure there is.
Поскольку в аудитории не было доски, мы не могли в ней заниматься.
There being no blackboard in the classroom, we could not have a lesson there.
The participle of the verb to be is never omitted here.