The Deformed Trigonometric Functions of two Variables

MSC 2010: 33B10, 33E20

Functions of real variables.

Mode of access: Internet.

Functions of real variables. Functions of a complex variable.

Mode of access: Internet.

Enumeration of threshold functions of eight variables,

Bibliography: p. 58-59.

Major and minor threshold functions of six variables /

Issued also as thesis (M.S.) University of Illinois.

Enumeration of pseudo-separable functions of five variables,

Thesis (M. Sc.)--University of Illinois.

Lectures on the theory of functions of real variables.

Mode of access: Internet.

Statnote 14: the correlation of two variables (Pearson's 'r')

Pearson's correlation coefficient (‘r’) is one of the most widely used of all statistics. Nevertheless, care needs to be used in interpreting the results because with large numbers of observations, quite small values of ‘r’ become significant and the X variable may only account for a small proportion of the variance in Y. Hence, ‘r squared’ should always be calculated and included in a discussion of the significance of ‘r’. The use of ‘r’ also assumes that the data follow a bivariate normal distribution (see Statnote 17) and this assumption should be examined prior to the study. If the data do not conform to such a distribution, the use of a non-parametric correlation coefficient should be considered. A significant correlation should not be interpreted as indicating ‘causation’ especially in observational studies, in which the two variables may be correlated because of their mutual correlations with other confounding variables.

Special functions of Weyl groups and their continuous and discrete orthogonality

Cette thèse s'intéresse à l'étude des propriétés et applications de quatre familles des fonctions spéciales associées aux groupes de Weyl et dénotées $C$, $S$, $S^s$ et $S^l$. Ces fonctions peuvent être vues comme des généralisations des polynômes de Tchebyshev. Elles sont en lien avec des polynômes orthogonaux à plusieurs variables associés aux algèbres de Lie simples, par exemple les polynômes de Jacobi et de Macdonald. Elles ont plusieurs propriétés remarquables, dont l'orthogonalité continue et discrète. En particulier, il est prouvé dans la présente thèse que les fonctions $S^s$ et $S^l$ caractérisées par certains paramètres sont mutuellement orthogonales par rapport à une mesure discrète. Leur orthogonalité discrète permet de déduire deux types de transformées discrètes analogues aux transformées de Fourier pour chaque algèbre de Lie simple avec racines des longueurs différentes. Comme les polynômes de Tchebyshev, ces quatre familles des fonctions ont des applications en analyse numérique. On obtient dans cette thèse quelques formules de <>, pour des fonctions de plusieurs variables, en liaison avec les fonctions $C$, $S^s$ et $S^l$. On fournit également une description complète des transformées en cosinus discrètes de types V--VIII à $n$ dimensions en employant les fonctions spéciales associées aux algèbres de Lie simples $B_n$ et $C_n$, appelées cosinus antisymétriques et symétriques. Enfin, on étudie quatre familles de polynômes orthogonaux à plusieurs variables, analogues aux polynômes de Tchebyshev, introduits en utilisant les cosinus (anti)symétriques.

Two prognostic factors in breast cancer: Novel functions of EGFR and beta-catenin

In this study, we demonstrated the novel functions of two important prognostic markers in breast cancer, EGFR and b -catenin in proliferation and/or other transformation phenotype. ^ First we demonstrated that EGFR could be detected in the nucleus in highly proliferating tissues, including primary breast cancer samples and a breast cancer cell line. We found that EGFR contained a strong transactivation domain, complexed with an AT-rich consensus DNA sequence and activated promoters containing this sequence, including cyclin D1 promoter. Therefore, EGFR may function as a transcription factor to activate genes required for highly proliferating activity such as cyclin D1 in breast cancer. ^ In the second part of this study, we identified b -catenin as an important prognostic factor in breast cancer. We found that cyclin D1 was one of the genes regulated by b -catenin in breast cancer cells. The transactivation activity of b -catenin correlated significantly with cyclin D1 expression in both breast cancer cell lines and in breast cancer patient samples, in which high b -catenin activity correlated with poor prognosis of the patients. Moreover, blockage of b -catenin activity significantly inhibited transformation phenotypes in breast cancer cells. Therefore, our results indicate that b -catenin can be involved in breast cancer formation and/or progression and may serve as a target for breast cancer therapy. ^

The theory of functions of a complex variable. [Lectures] New York University, Institute for Mathematics and Mechanics.

"First edition 1948. Second printing 1952."

Theory of functions of a real variable.

Mode of access: Internet.

Elements of the theory of functions of a complex variable with especial reference to the methods of Riemann,

Translation of Elemente der Theorie der Funktionen einer complexen veränderlichen Grösse.

Theory of functions of a complex variable,

Available on demand as hard copy or computer file from Cornell University Library.

On the construction of new bent functions from the max-weight and min-weight functions of old bent functions

Given a bent function f (x) of n variables, its max-weight and min-weight functions are introduced as the Boolean functions f + (x) and f − (x) whose supports are the sets {a ∈ Fn2 | w( f ⊕la) = 2n−1+2 n 2 −1} and {a ∈ Fn2 | w( f ⊕la) = 2n−1−2 n 2 −1} respectively, where w( f ⊕ la) denotes the Hamming weight of the Boolean function f (x) ⊕ la(x) and la(x) is the linear function defined by a ∈ Fn2 . f + (x) and f − (x) are proved to be bent functions. Furthermore, combining the 4 minterms of 2 variables with the max-weight or min-weight functions of a 4-tuple ( f0(x), f1(x), f2(x), f3(x)) of bent functions of n variables such that f0(x) ⊕ f1(x) ⊕ f2(x) ⊕ f3(x) = 1, a bent function of n + 2 variables is obtained. A family of 4-tuples of bent functions satisfying the above condition is introduced, and finally, the number of bent functions we can construct using the method introduced in this paper are obtained. Also, our construction is compared with other constructions of bent functions.