1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?
While all of us regularly use basic math symbols such as those for plus, minus, and equals, few of us know that many of these symbols weren't available before the sixteenth century. What did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? In Enlightening Symbols, popular math writer Joseph Mazur explains the fascinating history behind the development of our mathematical notation system. He shows how symbols were used initially, how one symbol replaced another over time, and how written math was conveyed before and after symbols became widely adopted.
Traversing mathematical history and the foundations of numerals in different cultures, Mazur looks at how historians have disagreed over the origins of the numerical system for the past two centuries. He follows the transfigurations of algebra from a rhetorical style to a symbolic one, demonstrating that most algebra before the sixteenth century was written in prose or in verse employing the written names of numerals. Mazur also investigates the subconscious and psychological effects that mathematical symbols have had on mathematical thought, moods, meaning, communication, and comprehension. He considers how these symbols influence us (through similarity, association, identity, resemblance, and repeated imagery), how they lead to new ideas by subconscious associations, how they make connections between experience and the unknown, and how they contribute to the communication of basic mathematics.
From words to abbreviations to symbols, this book shows how math evolved to the familiar forms we use today.
http://www.amazon.com/Enlightening-Symbols-History-Mathematical-Notation/dp/0691154635/
Many descriptions (especially of future events) contain
elements, which are uncertain and not precisely known.
- For example future rainfall, or discharge from a river.
- More generally, action from surrounding environment.
- The system itself may contain only incompletely known
- parameters, processes or fields (not possible or too
- costly to measure)
- There may be small, unresolved scales in the model,
- they act as a kind of background noise.
All these introduce some uncertainty in the model.
- Uncertainty may be aleatoric, which means random and not reducible, or
- epistemic, which means due to incomplete knowledge.
In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with some subset of the integers between 0 and m inclusive, such that no two vertices share a label, and such that each edge is uniquely identified by the positive, or absolute difference between its endpoints. A graph which admits a graceful labeling is called a graceful graph.
The name "graceful labeling" is due to Solomon W. Golomb; this class of labelings was originally given the name β-labelings by Alex Rosa in a 1967 paper on graph labelings.
A major unproven conjecture in graph theory is the Graceful Tree conjecture or Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, which hypothesizes that all trees are graceful. The Ringel-Kotzig conjecture is also known as the "graceful labeling conjecture". Kotzig once called the effort to prove the conjecture a "disease".
http://en.wikipedia.org/wiki/Graceful_labeling
