scholarly journals A NOTE ON THE HU–HWANG–WANG CONJECTURE FOR GROUP TESTING

2008 ◽  
Vol 49 (4) ◽  
pp. 561-571
Author(s):  
MING-GUANG LEU
Keyword(s):  
Boundary Problem ◽  
Upper Bound ◽  
Group Testing ◽  
Combinatorial Problem ◽  
Minimax Test ◽  
Test Number

AbstractHu et al. [“A boundary problem for group testing”, SIAM J. Algebraic Discrete Meth.2 (1981), 81–87] conjectured that the minimax test number to find d defectives in 3d items is 3d−1, a surprisingly difficult combinatorial problem about which very little is known. In this article we state three more conjectures and prove that they are all equivalent to the conjecture of Hu et al. Notably, as a byproduct, we also obtain an interesting upper bound for M(d,n).

Meniscus growth during the wiping stage of intaglio (gravure) printing

2016 ◽  
Vol 807 ◽  
pp. 419-440
Author(s):  
Umut Ceyhan ◽  
S. J. S. Morris
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Upper Bound ◽  
Capillary Number ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Gravure Printing ◽  
Tip Radius

During intaglio (gravure) printing, a blade wipes excess ink from the engraved plate with the object of leaving ink-filled cells defining the image to be printed. That objective is not completely attained. Capillarity draws some ink from the cell into a meniscus connecting the blade to the substrate, and the continuing motion of the engraved plate smears that ink over its surface. By examining the limit of vanishing capillary number ($Ca$, based on substrate speed), we reduce the problem of determining smear volume to one of hydrostatics. Using numerical solutions of the corresponding free-boundary problem for the Stokes equations of motion, we show that the hydrostatic theory provides an upper bound to smear volume for finite $Ca$. The theory explains why polishing to reduce the tip radius of the blade is an effective way to control smearing.

scholarly journals A tight upper bound for group testing in graphs

1994 ◽  
Vol 48 (2) ◽  
pp. 101-109 ◽  
Author(s):  
Peter Damaschke
Keyword(s):  
Upper Bound ◽  
Group Testing

BOUNDS FOR NONADAPTIVE GROUP TESTS TO ESTIMATE THE AMOUNT OF DEFECTIVES

2011 ◽  
Vol 03 (04) ◽  
pp. 517-536 ◽  
Author(s):  
PETER DAMASCHKE ◽  
AZAM SHEIKH MUHAMMAD
Keyword(s):  
Error Probability ◽  
Group Testing ◽  
Constant Factor ◽  
Constant Number ◽  
Constant Ratio ◽  
Information Theoretic ◽  
Small Constant ◽  
Test Number ◽  
Linear Programming Formulation ◽  
Two Stages

Group testing is the problem of finding d defectives in a set of n elements, by asking carefully chosen subsets (pools) whether they contain defectives. Strategies are preferred that use both a small number of tests close to the information-theoretic lower bound d log 2 n, and a small constant number of stages, where tests in every stage are done in parallel, in order to save time. They should even work if d is not known in advance. In fact, one can succeed with O(d log n) queries in two stages, if certain tests are randomized and a constant failure probability is allowed. An essential ingredient of such strategies is to get an estimate of d within a constant factor. This problem is also interesting in its own right. It can be solved with O( log n) randomized group tests of a certain type. We prove that Ω( log n) tests are also necessary, if elements for the pools are chosen independently. The proof builds upon an analysis of the influence of tests on the searcher's ability to distinguish between any two candidate numbers with a constant ratio. The next challenge is to get optimal constant factors in the O( log n) test number, depending on the prescribed error probability and the accuracy of d. We give practical methods to derive upper bound tradeoffs and conjecture that they are already close to optimal. One of them uses a linear programming formulation.

scholarly journals Improving a method of constructing finite time blow-up solutions and its an application

2020 ◽  
Author(s):  
QF Long
Keyword(s):  
Boundary Problem ◽  
Weak Solutions ◽  
Upper Bound ◽  
Finite Time ◽  
Blow Up ◽  
Initial Boundary ◽  
Sufficient Condition ◽  
Improved Method ◽  
Initial Boundary Problem ◽  
Blow Up Rate

We in this paper improve a method of establishing the existence of finite time blow-up solutions, and then apply it to study the finite time blow-up, the blow-up time and the blow-up rate of the weak solutions on the initial boundary problem of u_t - \Delta u_{t} - \Delta u_{t} = |u|^{p - 1}u. By applying this improved method, we prove that I(u_{0}) < 0 is a sufficient condition of the existence of the finite time blow-up solutions and \frac{2(p - 1)^{-1}\|u_{0}\|_{H_{0}^{1}}^{2}}{(p - 1) \|\nabla u_{0}\|_{2}^{2} - 2(p + 1)J(u_{0})} is an upper bound for the blow-up time, which generalize the blow-up results of the predecessors in the sense of the variation. Moreover, we estimate the upper blow-up rate of the blow-up solutions, too.

A Boundary Problem for Group Testing

1981 ◽  
Vol 2 (2) ◽  
pp. 81-87 ◽  
Author(s):  
M. C. Hu ◽  
F. K. Hwang ◽  
Ju Kwei Wang
Keyword(s):  
Boundary Problem ◽  
Group Testing

scholarly journals On Tensor-Factorisation Problems,I: The Combinatorial Problem

2004 ◽  
Vol 7 ◽  
pp. 73-100
Author(s):  
Peter M. Neumann ◽  
Cheryl E. Praeger
Keyword(s):  
Abelian Group ◽  
Tensor Product ◽  
Polynomial Time ◽  
Upper Bound ◽  
Combinatorial Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Probability Of Failure ◽  
Division Problem ◽  
Group A

AbstractA k-multiset is an unordered k-tuple, perhaps with repetitions. If x is an r-multiset {x1, …, xr} and y is an s-multiset {y1, …, ys} with elements from an abelian group A the tensor product x ⊗ y is defined as the rs-multiset {xi yj | 1 ≤ i ≤ r, 1 ≤ j ≤ s}. The main focus of this paper is a polynomial-time algorithm to discover whether a given rs-multiset from A can be factorised. The algorithm is not guaranteed to succeed, but there is an acceptably small upper bound for the probability of failure. The paper also contains a description of the context of this factorisation problem, and the beginnings of an attack on the following division-problem: is a given rs-multiset divisible by a given r-multiset, and if so, how can division be achieved in polynomially bounded time?

scholarly journals A combinatorial problem that arose in integer B3 Sets

2019 ◽  
Vol 53 (2) ◽  
pp. 195-203
Author(s):  
Irene Erazo ◽  
John López ◽  
Carlos Trujillo
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Combinatorial Problem ◽  
Positive Integers

Let A = {a1, a2, …, ak} be a set of positive integers with k ≥ 3,such that a1 ≤ a2 ≤ a3 … ak = N. Our problem is to investigate thenumber of triplets (ar, as, at) ∈ A3 with ar < as < at, satisfyingar + as - at < 0 y -ar + as + at > N.In this paper we give an upper bound for the maximum number of such a triplets in an arbitrary set of integers with k elements. We also find the number of triplets satisfying () for some families of sets in order to determine lower bounds for the maximum number of such a triplets that a set with k elements can have.

scholarly journals Revisiting Counting Solutions for the Global Cardinality Constraint

2019 ◽  
Vol 66 ◽  
pp. 411-441
Author(s):  
Giovanni Lo Bianco ◽  
Xavier Lorca ◽  
Charlotte Truchet ◽  
Gilles Pesant
Keyword(s):  
Artificial Intelligence ◽  
Upper Bound ◽  
Solution Space ◽  
Combinatorial Problem ◽  
Cardinality Constraint ◽  
Number Of Solutions ◽  
Important Concern ◽  
Computation Process ◽  
Counting Solutions

Counting solutions for a combinatorial problem has been identified as an important concern within the Artificial Intelligence field. It is indeed very helpful when exploring the structure of the solution space. In this context, this paper revisits the computation process to count solutions for the global cardinality constraint in the context of counting-based search. It first highlights an error and then presents a way to correct the upper bound on the number of solutions for this constraint.

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