Difference and Difference Quotient. Part III

In this article, we give some important theorems of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions.Liang Xiquan - Qingdao University of Science and Technology, ChinaTang Ling - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Bo Li, Yan Zhang, and Xiquan Liang. Difference and difference quotient. Formalized Mathematics, 14(3):115-119, 2006, doi:10.2478/v10037-006-0014-z.Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998

A study of factors related to primary grade spelling: correlations of factors studied.

Thesis (Ed.M.)--Boston Universit

Fractal fractal dimensions of deterministic transport coefficients

If a point particle moves chaotically through a periodic array of scatterers the associated transport coefficients are typically irregular functions under variation of control parameters. For a piecewise linear two-parameter map we analyze the structure of the associated irregular diffusion coefficient and current by numerically computing dimensions from box-counting and from the autocorrelation function of these graphs. We find that both dimensions are fractal for large parameter intervals and that both quantities are themselves fractal functions if computed locally on a uniform grid of small but finite subintervals. We furthermore show that there is a simple functional relationship between the structure of fractal fractal dimensions and the difference quotient defined on these subintervals.Comment: 16 pages (revtex) with 6 figures (postscript

The relationship between a method of measuring tension and two methods of measuring the attitude toward self in thirty-six electrically recorded initial social casework interviews.

Thesis (M.S.)--Boston Universit

Diagonal Coinvariants and Double Affine Hecke Algebras

We establish a q-generalization of Gordon's theorem that the space of diagonal coinvariants has a quotient identified with a perfect representation of the rational double affine Hecke algebra. It leads to a simple proof of his theorem and relates it to the Weyl algebras at roots of unity. The universal double affine Hecke algebra and the corresponding universal double Dunkl operators acting in noncommutative polynomials in terms of two sets of variables are introduced.Comment: The final variant to appear in IMR