Fear of needles to the phobic imagining the procedure being performed on themselves. Recent neuroscience research shows that feeling a pin prick sensation and watching someone else's hand get pricked by a pin activate the same part of the brain. Fear of needles, especially in its more severe forms, is often comorbid with other phobias and psychological ailments; for example, iatrophobia, or an irrational fear of doctors, is often seen in needle phobic patients. A needle phobic patient does not need to physically be in a doctor's office to experience panic attacks or anxiety brought on by needle phobia. There are many triggers
Irrational rotation between the additive and multiplicative notations is the group isomorphism It can be shown that is an isometry. There is a strong distinction in circle rotations that depends on whether is rational or irrational. Rational rotations are less interesting examples of dynamical systems because if formula_4 and formula_5, then formula_6 when formula_7. It can also be shown that formula_8 when formula_9. Irrational rotations form a fundamental example in the theory of dynamical systems. According to the Denjoy theorem, every orientation-preserving -diffeomorphism of the circle with an irrational rotation number is topologically conjugate to . An irrational rotation is a measure-preserving