The minimal polynomials of the images of unipotent elements in irreducible rational representations of the classical algebraic groups over fields of odd characteristic are found. These polynomials have the form $(t-1)^d$ and hence are completely determined by their degrees. In positive characteristic the degree of such polynomial cannot exceed the order of a relevant element. It occurs that for each unipotent element the degree of its minimal polynomial in an irreducible representation is equal to the order of this element provided the highest weight of the representation is large enough with respect to the ground field characteristic. On the other hand, classes of unipotent elements for which in every nontrivial representation the degree of the minimal polynomial is equal to the order of the element are indicated. In the general case the problem of computing the minimal polynomial of the image of a given element of order $p^s$ in a fixed irreducible representation of a classical group over a field of characteristic $p>2$ can be reduced to a similar problem for certain $s$ unipotent elements and a certain irreducible representation of some semisimple group over the field of complex numbers. For the latter problem an explicit algorithm is given. Results of explicit computations for groups of small ranks are contained in Tables I-XII. The article may be regarded as a contribution to the programme of extending the fundamental results of Hall and Higman (1956) on the minimal polynomials from $p$-solvable linear groups to semisimple groups.

The story of a young couple whose world in turned upside down by an unexpected pregnancy.

On a distant, stormy planet-filled with strange tech, long buried secrets, and mysterious creatures-three teens' lives collide amidst tragedy and betrayal, irreparably changing their futures forever. 3 illustrations, Young Adult Science Fiction.

The sixth edition includes new developments, as well as new experiments in quantum entanglement, Schrödingers cat, the quantum computer, quantum information, the atom laser, and much more. Many experiments and problems are included.

While widely studied, the capacity of the human mind remains largely unexplored. As such, researchers are continually seeking ways to understand the brain, its function, and its impact on human behavior. Exploring Implicit Cognition: Learning, Memory, and Social Cognitive Processes explores research surrounding the ways in which an individual’s unconscious is able to influence and impact that person’s behavior without their awareness. Focusing on topics pertaining to social cognition and the unconscious process, this title is ideal for use by students, researchers, psychologists, and academicians interested in the latest insights into implicit cognition.

Starting from first principles, this graduate-level monograph discusses turbulent flow in a wide range of geophysical and astrophysical settings.