**Arthur Paul Mattuck** (born June 13, 1930) is a professor of mathematics at the Massachusetts Institute of Technology.^{[1]} He may be best known for his 1998 book, *Introduction to Analysis* (ISBN 013-0-81-1327) and his differential equations video lectures featured on MIT's OpenCourseWare. Inside the department, he is well known to graduate students and instructors, as he watches the videotapes of new recitation teachers (an MIT-wide program in which the department participates). From 1959 to 1977 Mattuck was married to chemist Joan Berkowitz.^{[2]} Mattuck is quoted extensively in Sylvia Nasar's biography of John Nash A Beautiful Mind.

In 2012 he became a fellow of the American Mathematical Society.^{[3]}

Arthur Paul Mattuck | |
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Born | June 13, 1930 Brooklyn, New York, U.S. |

Scientific career | |

Institutions | MIT |

Website | math |

**^**Debus, A.G.; Marquis Who's Who, Inc (1968).*World Who's Who in Science: A Biographical Dictionary of Notable Scientists from Antiquity to the Present*.**2**. Marquis-Who's Who, Incorporated. Retrieved 2015-08-27.**^**"Sylvia M. Stoesser Lectures in Chemistry: Joan B. Berkowitz". Department of Chemistry at the University of Illinois at Urbana-Champaign. 2010. Retrieved 2011-07-07.**^**List of Fellows of the American Mathematical Society, retrieved 2013-02-02.

- Differential Equations video lectures by Prof. Arthur Mattuck
- Prof. Arthur Mattuck Home Page
- "The Unofficial 18.02/18.03 Quote Book"

In mathematics, the Dirac delta function (δ function) is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.

In engineering and signal processing, the delta function, also known as the unit impulse symbol, may be regarded through its Laplace transform, as coming from the boundary values of a complex analytic function of a complex variable. The formal rules obeyed by this function are part of the operational calculus, a standard tool kit of physics and engineering. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin (in theory of distributions, this is a true limit). The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.

Heaviside cover-up methodThe Heaviside cover-up method, named after Oliver Heaviside, is one possible approach in determining the coefficients when performing the partial-fraction expansion of a rational function.

Joan BerkowitzJoan B. Berkowitz (born March 13, 1931) is an American chemist.

Juliet Popper ShafferJuliet Popper Shaffer (born 1932) is an American psychologist, statistician and statistics educator known for her research on multiple hypothesis testing. She is a teaching professor emerita at the University of California, Berkeley.

Limit cycleIn mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).

List of people by Erdős numberPaul Erdős (1913–1996) was the most prolifically published mathematician of all time. He considered mathematics to be a social activity and often collaborated on his papers, having 511 joint authors, many of whom also have their own collaborators. The Erdős number measures the "collaborative distance" between an author and Erdős. Thus, his direct co-authors have Erdős number one, theirs have number two, and so forth. Erdős himself has Erdős number zero.

There are more than 11,000 people with an Erdős number of two. This list is intended to include only those authors with an Erdős number of three or less who are notable in their own right and have existing articles. For more complete listings of Erdős numbers, see the databases maintained by the Erdős Number Project or the collaboration distance calculator maintained by the American Mathematical Society.

Phase portraitA phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different curve, or point.

Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the state space. This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value. The concept of topological equivalence is important in classifying the behaviour of systems by specifying when two different phase portraits represent the same qualitative dynamic behavior. An attractor is a stable point which is also called 'sink'. The repellor is considered as an unstable point, which is also known as 'source'.

A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stable steady states (with dots) and unstable steady states (with circles) in a state space. The axes are of state variables.

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