Online Disjoint Set Cover Without Prior Knowledge

The disjoint set cover (DSC) problem is a fundamental combinatorial optimization problem concerned with partitioning the (hyper)edges of a hypergraph into (pairwise disjoint) clusters so that the number of clusters that cover all nodes is maximized. In its online version, the edges arrive one-by-one and should be assigned to clusters in an irrevocable fashion without knowing the future edges. This paper investigates the competitiveness of online DSC algorithms. Specifically, we develop the first (randomized) online DSC algorithm that guarantees a poly-logarithmic (O(log^{2} n)) competitive ratio without prior knowledge of the hypergraph\u27s minimum degree. On the negative side, we prove that the competitive ratio of any randomized online DSC algorithm must be at least Omega((log n)/(log log n)) (even if the online algorithm does know the minimum degree in advance), thus establishing the first lower bound on the competitive ratio of randomized online DSC algorithms

Detecting Wave Function Collapse Without Prior Knowledge

We are concerned with the problem of detecting with high probability whether a wave function has collapsed or not, in the following framework: A quantum system with a $d$-dimensional Hilbert space is initially in state $\psi$; with probability $0<p<1$, the state collapses relative to the orthonormal basis $b_1,...,b_d$. That is, the final state $\psi'$ is random; it is $\psi$ with probability $1-p$ and $b_k$ (up to a phase) with $p$ times Born's probability $|\langle b_k|\psi \rangle|^2$. Now an experiment on the system in state $\psi'$ is desired that provides information about whether or not a collapse has occurred. Elsewhere, we identify and discuss the optimal experiment in case that $\psi$ is either known or random with a known probability distribution. Here we present results about the case that no a priori information about $\psi$ is available, while we regard $p$ and $b_1,...,b_d$ as known. For certain values of $p$, we show that the set of $\psi$s for which any experiment E is more reliable than blind guessing is at most half the unit sphere; thus, in this regime, any experiment is of questionable use, if any at all. Remarkably, however, there are other values of $p$ and experiments E such that the set of $\psi$s for which E is more reliable than blind guessing has measure greater than half the sphere, though with a conjectured maximum of 64% of the sphere.Comment: 16 pages LaTeX, 1 figure; v2: more detail added to the proof of Thm. 1 (half a page added on page 12) and minor improvements. A previous version of this paper was included as chapters 6 and 7 in arXiv:1307.0810v1, but it will not be contained in subsequent, revised versions of arXiv:1307.081

Contribution of Prior Knowledge, Appreciation of Mathematics and Logical-mathematical Intelligence to the Ability of Solving Mathematical Problems

The objectives of this research were to figure out the contribution of prior knowledge, appreciation of mathematics and logical-mathematical intelligence toward the ability to solve mathematical problems as well as to explore the errors made by students in solving mathematical problems concerning polyhedron. The population of this research consisted of 3,583 students of grade IX of all state middle schools across over Denpasar City. The sampling technique we used was a stratified cluster random sampling technique with samples number of 553 students. The type of this research is ex-post facto research with path analysis technique. The data were collected by using questionnaires and carrying out a mathematical ability test. Furthermore, the kinds of students answers on the ability to solve mathematical problems were analyzed to study the errors made by the students. The results of the research show two regression relationships, namely X3 = 0.523X1 + 0.636X2 + 0.506ɛ3 and Y = 0.640X1 + 0.264X2 + 0.280X3 + 0.311ɛY. The first regression relationship indicates that (1) the contribution of mathematical appreciation towards prior knowledge is of 52.3 percent, and (2) the contribution of logical-mathematical intelligence towards prior knowledge is of 63.3 percent. Whereas the second regression relationship describes that (1) the direct contribution of mathematical appreciation towards the ability of solving mathematical problems is of 64 percent, and the indirect contribution is of 14.6 percent, (2) the direct contribution of logical-mathematical intelligence to the ability of solving mathematical problems was is of 26.4 percent, and the indirect contribution is of 17.8 percent, (3) the direct contribution of prior knowledge towards the ability solving mathematical problems is of 28 percent, (4) the mathematical appreciation and logical-mathematical intelligence contributed simultaneously towards prior knowledge is of 74.4 percent, (5) the mathematical appreciation, logical-mathematical intelligence, and prior knowledge contributed simultaneously towards the ability to solve mathematical problems is 90.3 percent. Furthermore, based on the analysis of students answers in mathematical ability test showed that the students still made errors in the concept of prior knowledge, in interpreting questions and weaknesses in arithmetic skills related to logical-mathematical intelligence

Universal quantum information compression and degrees of prior knowledge

We describe a universal information compression scheme that compresses any pure quantum i.i.d. source asymptotically to its von Neumann entropy, with no prior knowledge of the structure of the source. We introduce a diagonalisation procedure that enables any classical compression algorithm to be utilised in a quantum context. Our scheme is then based on the corresponding quantum translation of the classical Lempel-Ziv algorithm. Our methods lead to a conceptually simple way of estimating the entropy of a source in terms of the measurement of an associated length parameter while maintaining high fidelity for long blocks. As a by-product we also estimate the eigenbasis of the source. Since our scheme is based on the Lempel-Ziv method, it can be applied also to target sequences that are not i.i.d.Comment: 17 pages, no figures. A preliminary version of this work was presented at EQIS '02, Tokyo, September 200