Sentence examples for functions of balls from inspiring English sources

A quantitative version of the standard Sobolev inequality, with sharp constant, for functions u in W1,1(Rn) (or BV(Rn)) is established in terms of a distance of u from the manifold of all multiples of characteristic functions of balls.

The equality holds in (1.2) if and only if f is the characteristic function of balls.

In order to assess the function of ball in proliferating tissue and in stem cells, we generated a null allele of ball (ball ; supplementary material Fig. S1).

Here we report that the BALL kinase is required to maintain the proliferative potential of NBs and that this function of BALL is a prerequisite for self-renewal.

In order to address the function of BALL for neuronal development, we used the ball null allele (Herzig et al., 2014).

To discriminate between a cell autonomous function of BALL in stem cells and a systemic non-autonomous function of BALL, we generated genetic mosaics in adult ovaries.

In order to directly address the function of ball for GSC self-renewal and germline differentiation, we generated genetic mosaics identifying ball mutant cell clones by the absence of GFP expression (Fig. 2C).

To assess whether the function of BALL for self-renewal of GSC is specific for the female germline, we examined the function of BALL in GSCs of adult testes.

Here, we show that this function of BALL is not restricted to niche-controlled stem cells but is also required in pNBs, which depend on asymmetric distribution of cell fate determinants for self-renewal.

Thus, the function of BALL for stem cell self-renewal is not limited by the factors and mechanisms that mediate cell fate decisions in the different stem cell systems.

We consider the Fourier series of the indicator functions of several dimensional balls.