We propose an algorithm to compute upper and lower bounds for the star discrepancy of an arbitrary sequence of points in the s-dimensional unit cube. The method is based on a particular partition of the unit cube into subintervals and on a specialized procedure for orthogonal range counting. The cardinality of the partition depends on the dimension and on an accuracy parameter that has to be sp...

Laczkovich proved that if bounded subsets A and B of R have the same non-zero Lebesgue measure and the upper box dimension of the boundary of each set is less than k, then there is a partition of A into finitely many parts that can be translated to form a partition of B. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, thi...

1. The Darboux definition of the Riemann Integral Let f : [a, b]→ R be a it bounded function. Definition 1.1. A partition P of [a, b] is collection of points {x0, · · · , xn} which satisfiy a = x0 < x1 < · · · < xn = b. Denote by ∆xk the length of the interval [xk−1, xk], i.e., ∆xk = xk−xk−1. Then the mesh |P| is by definition equal to max1≤k≤n∆xk. Given a bounded function f and a partition P o...

A piecewise algebraic curve is a curve defined by the zero set of a bivariate spline function. Given two bivariate spline spaces Sm Δ and S t n Δ over a domain D with a partition Δ, the Bezout number BN m,r;n,t;Δ is defined as the maximum finite number of the common intersection points of two arbitrary piecewise algebraic curves f x, y 0 and g x, y 0, where f x, y ∈ Sm Δ and g x, y ∈ Sn Δ . In ...

We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-toNeumann map at selected frequencies as the data. A conditional Lipschitz stability estimate for the inverse problem holds in the case of wavespeeds that are a linear combination of piecewise constant functions (following a domain partition) and gives a framework in which the scheme converges. The stabil...

Consider a graph and a collection of (not necessarily edge-disjoint) connected spanning subgraphs (factors) of the graph. We consider the problem of coloring the vertices of the graph so that each color class of the vertices dominates each factor. We find upper and lower bounds on α(t, k), which we define as the minimum radius of domination d such that every graph with a collection of k factors...

The set R of relevant cycles of a graph G is the union of its minimum cycle bases. We introduce a partition of R such that each cycle in a class W can be expressed as a sum of other cycles in W and shorter cycles. It is shown that each minimum cycle basis contains the same number of representatives of a given class W. This result is used to derive upper and lower bounds on the number of distinc...

Aiming at the power balance control of multi-source energy storage grid in case a high proportion new connection. In this article, dynamic partition method based on Markov field principle and priori knowledge model is proposed. Combined with coordinated dispatching source-load, two-layer topological characteristics nodes established. First, upper-layer model, homogenization studied, node intera...

The computational complexity of the partition problem, which concerns the partitioning of a set of n vectors in d-space into p parts so as to maximize an objective function which is convex on the sum of vectors in each part, is determined by the number of vertices of the corresponding p-partition polytope defined to be the convex hull in (d × p)-space of all solutions. In this article, introduc...