For every integer k > 2 let [k] 1 is a subset of [k]<N of the form {w} ∪ { ww0(a0) ...wn(an) : n ∈ {0, ...,m− 1} and a0, ..., an ∈ [k] } where w is a word over k and (wn) m−1 n=0 is a finite sequence of left variable words over k. We study the behavior of a family of measurable events in a probability space index...

An external memory data structure is presented for maintaining a dynamic set ofN two-dimensional points under the insertion and deletion of points, and supporting unsorted 3-sided range reporting queries and top-k queries, where top-k queries report the k points with highest y-value within a given x-range. For any constant 0 < ε ≤ 1 2 , a data structure is constructed that supports updates in a...

Submodular-function maximization is a central problem in combinatorial optimization, generalizing many important NP-hard problems including Max Cut in digraphs, graphs and hypergraphs, certain constraint satisfaction problems, maximum-entropy sampling, and maximum facility-location problems. Our main result is that for any k ≥ 2 and any ε > 0, there is a natural local-search algorithm which has...

As in the paper, suppose that allies use a threshold strategy according to which ally i supports the ruler if her signal ki is below some threshold k , ki ≤ k , and she rebels if ki > k . Since the signal ki is distributed uniformly on the interval [κ t − ε, κ + ε], a threshold signal k implies the existence of a threshold regime strength κ such that a rebellion fails if κ ≤ κ and succeeds if κ...

For any natural number d and positive number ε, we present a point set in the d-dimensional unit cube [0, 1] that intersects every axis-aligned box of volume greater than ε. These point sets are very easy to handle and in a vast range for ε and d, we do not know any smaller set with this property. 1 The Result The dispersion of a point set P in [0, 1] is the volume of the largest axis-aligned b...

and Applied Analysis 3 which vanishes on K except perhaps for a pluripolar subset and satisfies the complex Monge-Ampère equation (see [12]): (dd c V K ) n = 0 on Cn \ K. (16) If n = 1, the Monge-Ampère equation reduces to the classical Laplace equation. For this reason, the functionV K is considered as a natural counterpart of the classical Green function with logarithmic pole at infinity and ...

Let f : Sn → Rk be a continuous map where Sn is the unit n-sphere. Then there exists a point z ∈ Rk such that the spherical n-volumes of the ε-neighbours of the level Yz = f−1(z) ⊂ Sn, denoted Yz + ε ⊂ Sn, satisfy Vol(Yz + ε) ≥ Vol(Sn−k + ε) ( )Sn for all ε > 0, where Sn−k ⊂ Sn denotes an equatorial (n− k)-sphere. This is proven in §5.9. Remarks. (a) If k = n, and card f−1(z) ≤ 2, z ∈ Rn, then ...

Wireless sensors rely on battery power, and in many applications it is difficult or prohibitive to replace them. Hence, in order to prolongate the system’s lifetime, some sensors can be kept inactive while others perform all the tasks. In this paper, we study the k-coverage problem of activating the minimum number of sensors to ensure that every point in the area is covered by at least k sensor...

We deduce the most general kinetic equation that describe the low density limit of general Feller processes for the systems of random number of particles with interaction, collisions, fragmentation and coagulation. This is done by studying the limiting as ε → 0 evolution of Feller processes on ∪∞n Xn with X = Rd or X = Zd described by the generators of the form ε−1 ∑K k=0 ε kB(k), K ∈ N , where...

and Applied Analysis 3 of positive real numbers. By S 2 − X , we denote the space of all sequences defined over X, ‖·, ·‖ . Now we define the following sequence spaces: W ( M,Δ, p, ‖, ·, ‖ ⎧ ⎪⎨ ⎪⎩ x ∈ S 2 −X : ∀ε > 0 { n ∈ : ∞ ∑ k 1 ank [ M (∥ ∥ ∥ ∥ Δxk − L ρ , z ∥ ∥ ∥ ∥ )]pk ≥ ε } ∈ I for some ρ > 0, L ∈ X and each z ∈ X ⎫ ⎪⎬ ⎪⎭ , W 0 ( A,M,Δ, p, ‖, ·, ‖ ⎧ ⎪⎨ ⎪⎩ x ∈ S 2 −X : ∀ε > 0 { n ∈ : ∞ ∑...