Let d and k be integers with 1 ≤ k ≤ d − 1. Let Λ be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in Λ ∩ K. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional n ×...

where the sum is performed for k ∈ Z and on indices i ∈ Z such that iε ∈ Ω and (i+ k)ε ∈ Ω. If we picture the lattice εZ ∩ Ω as the reference configuration of a set of material points interacting through some forces, and ui represents the displacement of the i-th point, then ψ ε can be thought as the energy density of the interaction of points with distance kε in the reference lattice. Note tha...

The experimental design problem concerns the selection of k points from a potentially large design pool of p-dimensional vectors, so as to maximize the statistical efficiency regressed on the selected k design points. Statistical efficiency is measured by optimality criteria, including A(verage), D(eterminant), T(race), E(igen), V(ariance) and G-optimality. Except for the Toptimality, exact opt...

A graph operation that contracts edges is one of the fundamental operations in theory minors. Parameterized Complexity editing to a family graphs by contracting k has recently gained substantial scientific attention, and several new results have been obtained. Some important families graphs, namely, subfamilies chordal context edge contractions, proven be significantly difficult than might expe...

We show that, for any fixed ε > 0 and almost all primes p, the gary expansion of any fraction m/p with gcd(m, p) = 1 contains almost all g-ary strings of length k < (17/72 − ε) logg p. This complements a result of J. Bourgain, S. V. Konyagin, and I. E. Shparlinski that asserts that, for almost all primes, all g-ary strings of length k < (41/504− ε) logg p occur in the g-ary expansion of m/p.

For A, ε > 0 and any sufficiently large odd n we show that for almost all k ≤ R := n 1/5−ε there exists a representation n = p 1 + p 2 + p 3 with primes p i ≡ b i mod k for almost all admissible triplets b 1 , b 2 , b 3 of reduced residues mod k.

in the course of expansion in turbines steam nucleates to become a two-phase mixture, the liquid consisting of a very large number of extremely small droplets carried by the vapor. formation and subsequent behaviour of the liquid lowers the performance of turbine wet stages. the calculations were carried out assuming that the flow is two dimensional, compressible, turbulent and viscous. a class...

We give a new lower bound on the query complexity of any non-adaptive algorithm for testing whether an unknown Boolean function is a k-junta versus ε-far from every k-junta. Our lower bound is that any non-adaptive algorithm must make Ω ( k log k εc log(log(k)/εc) ) queries for this testing problem, where c is any absolute constant < 1. For suitable values of ε this is asymptotically larger tha...

Let r ≥ 2 and write K+ r (s1, . . . , sr) for the complete r-partite graph with parts of size s1 ≥ 2, . . . , sr with an edge added to the first part. Letting tr (n) be the number of edges of the r-partite Turán graph of order n, we prove the following theorems: (A) For all r ≥ 2, sufficiently small ε > 0 and sufficiently large n, every graph of order n with tr (n) + 1 edges contains a K + r ( ...

We consider maximizing a monotone submodular function under cardinality constraint or knapsack in the streaming setting. In particular, elements arrive sequentially and at any point of time, algorithm has access to only small fraction data stored primary memory. propose following algorithms taking O(ε− 1) passes: (1) (1 − e− 1 ε)-approximation for cardinality-constrained problem, (2) (0.5 knaps...