Involving Landau's 'big oh' notation

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In summary, Landau's "big oh" notation is a mathematical notation commonly used in scientific research to describe the asymptotic behavior of a function. It is different from other asymptotic notations in that it only provides an upper bound on the growth rate of a function. It is not suitable for describing extremely fast-growing or irregular functions. "Big oh" notation is related to the concept of algorithmic complexity and is often used to compare the efficiency of different algorithms. However, it has limitations in accurately describing functions with large or infinite input sizes and does not provide information about specific behavior at any given point.
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We say that two sequences f,g are f=O(g) if-f there is a c>0 such that |f(n)|<c|g(n)| uniformly as n tends to infinity.

If g(n)>2, does f=O(g) imply lnf=O(ln(g))?
 
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|f(n)| < c|g(n)| => ln(|f(n)|) < ln(c) + ln(|g(n)|). You should have |f(n)|, c > 1 to avoid problems with abs. value of logs.
 

FAQ: Involving Landau's 'big oh' notation

What is Landau's "big oh" notation and how is it used in scientific research?

Landau's "big oh" notation is a mathematical notation used to describe the asymptotic behavior of a function. It is commonly used in scientific research to analyze the time or space complexity of algorithms, as well as to describe how a quantity changes as the input size approaches infinity.

How is "big oh" notation different from other asymptotic notations?

"Big oh" notation is a type of "big theta" notation, which is a more general notation used to describe the asymptotic behavior of a function. Unlike other notations such as "big omega" or "little oh", "big oh" notation only provides an upper bound on the growth rate of a function.

Can "big oh" notation be used for all types of functions?

No, "big oh" notation is typically used for functions that grow at a relatively slow rate. It is not suitable for describing the behavior of extremely fast-growing functions or functions with irregular behavior.

How is "big oh" notation related to the concept of algorithmic complexity?

"Big oh" notation is often used to analyze the time or space complexity of algorithms. It provides a way to compare the efficiency of different algorithms by looking at how their run time or memory usage changes as the input size increases.

Are there any limitations to using "big oh" notation in scientific research?

While "big oh" notation is a useful tool for analyzing the behavior of functions, it has some limitations. It may not accurately describe the behavior of functions with large or infinite input sizes, and it does not provide information about the specific behavior of a function at any given point.

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