as a function of s at the k+ 1 points ti, . . . , ti+k. The elements of $k,t are called polynomial splines of order k with knot sequence t. Let τ := (τi) n 1 be a strictly increasing real sequence. As is shown in [12], there exists, for given f , exactly one s ∈ $k,t, such that s(τi) = f(τi), i = 1, . . . , n, if and only if Ni,k(τi) 6= 0, i = 1, . . . , n, i.e., if and only if ti < τi < ti+k, ...

Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability p, independently of each other, and a deterministic spreading rule with a fixed parameter k: if a vacant site has at least k occupied neighbors at a certain time step, then it becomes occupied in the next step. This process is well-studied on Z; here we investigate it o...

Theory and algorithms are presented for the following smoothing problem. We are given n measurements of a real-valued function that have been altered by random errors caused by the deriving process. For a given integer k , some efficient algorithms are developed that approximate the data by minimizing the sum of strictly convex functions of the errors in such a way that the approximated values ...

The KdpB polypeptides in the cyanobacterium Anabaena torulosa were shown to be two membrane-bound proteins of about 78 kDa, expressed strictly under K(+) deficiency and repressed or degraded upon readdition of K(+). In both Anabaena and Escherichia coli strain MC4100, osmotic and ionic stresses caused no significant induction of steady-state KdpB levels during extreme potassium starvation.

Let T be a tent map with the slope strictly between √ 2 and 2. Suppose that the critical point of T is not recurrent. Let K denote the inverse limit space obtained by using T repeatedly as the bonding map. We prove that every homeomorphism of K to itself is isotopic to some power of the natural shift homeomorphism.

The Mabuchi K-energy map is exhibited as a singular metric on the refined CM polarization of any equivariant family X p → S. Consequently we show that the generalized Futaki invariant is the leading term in the asymptotics of the reduced K-energy of the generic fiber of the map p. Properness of the K-energy implies that the generalized Futaki invariant is strictly negative.

For strictly ergodic systems, we introduce the class of CF-Nil(k) systems: systems for which maximal measurable and topological k-step pronilfactors coincide as measure-preserving systems. Weiss’ theorem implies that such are abundant in a precise sense. We show precisely minimal nilsequence version Wiener—Wintner average converges everywhere. As part proof establish pronilsystems coalescent bo...

A fixed point index is defined for mappings defined on a cone K which do not necessarily take their values in K but satisfy a weak type of boundary condition called generalized inward. This class strictly includes the well-known weakly inward class. New results for existence of multiple fixed points are established.

Assume that C1, . . . , Ck : [0, 1]d → [0, 1] are copulas. Let gji : [0, 1] → [0, 1] for j = 1, . . . , k, i = 1, . . . , d be functions with the property that each of them is strictly increasing or is identically equal to 1. Suppose that ∏k j=1 gji(v) = v for v ∈ [0, 1], i = 1, . . . , d, and limv→0+0 gij(v) = gji(0) for j = 1, . . . , k, i = 1, . . . , d. Here we consider the two-dimensional ...