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How do you deal with other people bullying or just looking down on you?... Looking for "gifted" repliers since I've read over a dozen articles by now only repeating the same information. Although practical, the conventional advice like simply not caring doesn't always work without an underlying understanding or possibly just a calmer mind.
There are many insightful books/articles on bullying which may help with this; Experts distill many experiences into advice. Every situation is different. Here are a few links:

Davidson Database - How to help keep your kid from being bullied:
http://www.davidsongifted.org/db/Articles_id_10335.aspx

Hoagies Gifted Education Page - Bullies and Bullying: http://www.hoagiesgifted.org/bullies.htm

SENG article - Teasing and Gifted Children: https://www.sengifted.org/archives/articles/teasing-and-gifted-children

Davidson Database - Tips for Parents: Friendship Issues of the Gifted and Talented Elementary School Child: http://www.davidsongifted.org/db/Articles_id_10686.aspx

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Or how do you deal with persistent negative emotions in general? Do you have a way of viewing these situations that brings peace of mind?
Some people may handle negative emotions by bullying others, according this article by Mental Health America (MHA).

The book Living with Intensity gives comfort to many gifted people as they learn they are not alone in the intensity of their perceptions and responses.

Lots of good topics dealing with social/emotional and interpersonal issues have been covered by well-known psychologists specializing in gifted kids. Their work is available in books through Great Potential Press, Prufrock Press, and free spirit publishing*. Magination Press, imprint of American Psychological Association (APA), also offers children's books which may be of interest.

*Interested readers may wish to make note of the "Words Wound" student video contest closing April 15, 2014.