A certificate of non-membership for a Boolean function f with respect to a class C, f 6∈ C, is a set S of input strings such that the values of f on strings from S are inconsistent with any function h ∈ C. We study certificates of non-membership with respect to several classes of read-once functions, generated by their bases. For the basis {&,∨,¬}, we determine the optimal certificate size for ...

Lysosome function is essential to many physiological processes. It has been suggested that deregulation of lysosome function could contribute to cancer. Through a genetic screen in Drosophila, we have discovered that mutations disrupting lysosomal degradation pathway components contribute to tumor development and progression. Loss-of-function mutations in the Class C vacuolar protein sorting (V...

The "comparison" class The value returned by compare() is an object of class "comparison". Importantly, this is not simply a logical value. In order to test whether the comparison was successful overall, for example as the condition in an if statement, you must use the isTRUE() function. > isTRUE(compare(1:10, 1:10)) [1] TRUE > The transforms() function can be used to extract the vector of tran...

where power(z, n) is the power function for integer exponents, pow(z, p) is the power function for real or complex exponents, and root(z, n) denotes the nth root function. An interval library for the real standard functions in SF is required by CoStLy, for evaluating the compositions of real functions that make up the real and imaginary parts of the complex functions in SF . CoStLy has been pro...

Let $C_0(alpha)$ denote the class of concave univalent functions defined in the open unit disk $mathbb{D}$. Each function $f in C_{0}(alpha)$ maps the unit disk $mathbb{D}$ onto the complement of an unbounded convex set. In this paper, we study the mapping properties of this class under integral operators.

Let $G$ be a $p$-group of nilpotency class $k$ with finite exponent $exp(G)$ and let $m=lfloorlog_pk floor$. We show that $exp(M^{(c)}(G))$ divides $exp(G)p^{m(k-1)}$, for all $cgeq1$, where $M^{(c)}(G)$ denotes the c-nilpotent multiplier of $G$. This implies that $exp( M(G))$ divides $exp(G)$, for all finite $p$-groups of class at most $p-1$. Moreover, we show that our result is an improvement...