"One man's trend is another man's range" -- Anonymous
Or, more accurately, one periodicity's trend is another periodicity's range.
What is the trend of the DJIA over the last 100 years? Up right? But what if in the next 100 years the DJIA cycles lower. Then, as far as the 200 year periodicity is concerned, the DJIA would have been in a range as of now, no? No doubt if you search for "definition of trend" you'll find enough references to fulfill your question. But you want technical, mathematical techniques to determine trend. We all wish we could create a consistent indicator of trend. For if we knew (really knew) the trend, trading would be a breeze. Alas, one can only come up with makeshift proximities to a trend indicator.
MA1 > MA2 > MA3 > MA4 == UP
:::
Sum [Period over period weighted returns] > 0 == UP
:::
Price > MA(200) == UP
There are dozens more I'm sure you might extract from the tribe here.