Estimating ranks, quantiles, and distributions over streaming data is a central task in analysis monitoring. Given stream of n items from universe equipped with total order, the to compute sketch (data structure) size polylogarithmic . query item y , one should be able approximate its rank stream, i.e., number elements smaller than or equal Most works date focused on additive ε error approximat...

In this paper, we study the bound on three kinds of hash family using the Singleton bound. To ε−U(N ;n,m) hash family, in the caes of n > m > 1 and 1 ≥ ε ≥ ε1(n,m), we get that the new bound is better. To ε − △U(N ;n,m) hash family , in the case of n > m > 1 and 1 ≥ ε ≥ ε3(n,m), the new bound is better. To ε − SU(N ;n,m) hash family, in the case of n > 2m > 2 and 1 ≥ ε ≥ ε4(n,m), we get that th...

We study the complexity of approximating the Stieltjes integral R 1 0 f (x)dg(x) for functions f having r continuous derivatives and functions g whose sth derivative has bounded variation. Let r(n) denote the nth minimal error attainable by approximations using at most n evaluations of f and g, and let comp(ε) denote the ε-complexity (the minimal cost of computing an ε-approximation). We show t...

A set of integers S is called ε-Kronecker if every function on S of modulus one can be approximated uniformly to within ε by a character. The least such ε is called the ε-Kronecker constant. We transform the problem of calculating ε-Kronecker constants for finite sets of d elements into a geometric optimization problem. Using this approach we can explicitly determine the ε-Kronecker constant fo...

We consider the following instance of projective clustering, known as the 2-line-center problem: Given a set S of n points in R2, cover S by two congruent strips of minimum width. Algorithms that find the optimal solution for this problem have near-quadratic running time. In this paper we present an algorithm that, for any ε > 0, computes in time O(n(logn+ ε−2 log(1/ε))+ ε−7/2 log(1/ε)) a cover...

We consider the linearized elasticity system in a multidomain of R 3. This multidomain is the union of a horizontal plate with fixed cross section and small thickness ε, and of a vertical beam with fixed height and small cross section of radius r ε. The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and r ε tend to zero simultaneously, with r ε ≫ ε 2 , w...

An N -channel spinless p-wave superconducting wire is known to go through a series of N topological phase transitions upon increasing the disorder strength. Here, we show that at each of those transitions the density of states shows a Dyson singularity ν(ε) ∝ ε−1| ln ε|−3, whereas ν(ε) ∝ ε|α|−1 has a power-law singularity for small energies ε away from the critical points. Using the concept of ...

Let us consider the differential equation ẋ = (A+ εQ(t, ε))x, |ε| ≤ ε0, where A is an elliptic constant matrix and Q depends on time in a quasi-periodic (and analytic) way. It is also assumed that the eigenvalues of A and the basic frequencies of Q satisfy a diophantine condition. Then it is proved that this system can be reduced to ẏ = (A∗(ε) + εR∗(t, ε))y, |ε| ≤ ε0, where R∗ is exponentially ...

We study a Dirichlet spectral problem for second-order elliptic operator with locally periodic coefficients in thin cylinder. The lateral boundary of the cylinder is assumed to be periodic. When thickness ε tends zero, eigenvalues are order ε−2 and described terms first eigenvalue μ(x1) an auxiliary cell parametrized by x1, while eigenfunctions localize rate ε.

Given a finite point set P ⊂ R, we call a multiset A a one-sided weak ε-approximant for P (with respect to convex sets), if |P ∩ C|/|P | − |A ∩C|/|A| ≤ ε for every convex set C. We show that, in contrast with the usual (two-sided) weak ε-approximants, for every set P ⊂ R there exists a one-sided weak ε-approximant of size bounded by a function of ε and d.