Partitioned difference families: the storm has not yet passed

Two years ago, we alarmed the scientific community about the large number of bad papers in the literature on {\it zero difference balanced functions}, where direct proofs of seemingly new results are presented in an unnecessarily lengthy and convoluted way. Indeed, these results had been proved long before and very easily in terms of difference families. In spite of our report, papers of the same kind continue to proliferate. Regrettably, a further attempt to put the topic in order seems unavoidable. While some authors now follow our recommendation of using the terminology of {\it partitioned difference families}, their methods are still the same and their results are often trivial or even wrong. In this note, we show how a very recent paper of this type can be easily dealt with

Partitioned difference families: the storm has not yet passed

Two years ago, we alarmed the scientific community about the large number of bad papers in the literature on zero difference balanced functions, where direct proofs of seemingly new results are presented in an unnecessarily lengthy and convoluted way. Indeed, these results had been proved long before and very easily in terms of difference families. In spite of our report, papers of the same kind continue to proliferate. Regrettably, a further attempt to put the topic in order seems unavoidable. While some authors now follow our recommendation of using the terminology of partitioned difference families, their methods are still the same and their results are often trivial or even wrong. In this note, we show how a very recent paper of this type can be easily dealt with

Probabilistic Analysis of the Median Rule: Asymptotics and Applications

The solution of integer optimization problems by relaxation methods consists of three parts. First, the discrete problem is converted into a continuous optimization problem, which is generally more tractable. Second, the relaxed problem is solved efficiently, yielding a optimal solution in the continuous space. Finally, an assignment procedure is used to map this solution to a suitable discrete solution. One heuristic - we call it the relaxation heuristic - that often guides the choice and design of assignment algorithms is: given a continuous optimal solution, the corresponding integer optimal solution is likely to be nearby (with respect to some well defined metric). Intuitively, this heuristic is reasonable for objective functions that are, say, Lipschitz functions. For such functions, an assignment algorithm might map the continuous optimal solution to the nearest feasible solution in the discrete space, in the hope that the discrete solution will be optimal as well. In this paper, we consider properties of a particular assignment algorithm known as the median rule. Define a binary vector to be balanced when the numbers of its 1 \u27s and 0\u27s differ at most by one. The median rule used to assign n-dimensional real vectors to n-dimensional balanced binary vectors, may be loosely described as follows: map the ith component of a real vector to a 0 or 1, depending on whether that component is smaller or greater than the median value of the vector components. We address two aspects of the median rule. The first result is that given a real vector, the median rule produces the closest balanced binary vector, with respect to any Schur-convex distance criteria. This includes several Minkowski norms, entropy measures, gauge functions etc. In this sense, the median rule optimally implements the relaxation heuristic. The second result addresses the issue of relaxation error. Though the median rule produces the nearest balanced integer solution to a given real vector, it is possible that this solution is sub-optimal, and the actual optimal solution is located elsewhere. The difference between the actual optimal cost and the cost of the solution obtained by the median rule is called the relaxation error. We consider the optimization of real valued, parametrized, multivariable Lipschitz functions where domains are the set of balanced binary vectors. Varying the parameters over the range of their values, we obtain an ensemble of such problems. Each problem instance in the ensemble has an optimal real cost, an integer cost, and an associated relaxation error. We establish upper bounds on the probability that the relaxation error is greater than a given threshold t. In general, these bounds depend on the random model being considered. These results have an immediate bearing on the important graph bisection width problem, which involves the minimization of a certain semidefinite quadratic cost function over balanced binary domains. This important problem arises in a variety of areas including load balancing, [11,16], storage management [22], distributed directories [15], and VLSI design [10]. The results obtained indicate that the median rule in a certain precise sense, is an optimal assignment procedure for this problem. The rest of the paper is organized as follows: In section 3, we prove the shortest distance properties of the median rule. In section 4, we introduce the concept of relaxation error and the Lipschitz bisection problem. Upper bounds on the relaxation error are obtained in Section 5. A discussion on these results is given in Section 6

Difference Balanced Functions and Their Generalized Difference Sets

Difference balanced functions from $F_{q^n}^*$ to $F_q$ are closely related to combinatorial designs and naturally define $p$-ary sequences with the ideal two-level autocorrelation. In the literature, all existing such functions are associated with the $d$-homogeneous property, and it was conjectured by Gong and Song that difference balanced functions must be $d$-homogeneous. First we characterize difference balanced functions by generalized difference sets with respect to two exceptional subgroups. We then derive several necessary and sufficient conditions for $d$-homogeneous difference balanced functions. In particular, we reveal an unexpected equivalence between the $d$-homogeneous property and multipliers of generalized difference sets. By determining these multipliers, we prove the Gong-Song conjecture for $q$ prime. Furthermore, we show that every difference balanced function must be balanced or an affine shift of a balanced function.Comment: 17 page

Nuclear magnetic resonance implementation of the Deutsch-Jozsa algorithm using different initial states

The Deutsch-Jozsa algorithm distinguishes constant functions from balanced functions with a single evaluation. In the first part of this work, we present simulations of the nuclear magnetic resonance (NMR) application of the Deutsch-Jozsa algorithm to a 3-spin system for all possible balanced functions. Three different kinds of initial states are considered: a thermal state, a pseudopure state, and a pair (difference) of pseudopure states. Then, simulations of several balanced functions and the two constant functions of a 5-spin system are described. Finally, corresponding experimental spectra obtained by using a 16-frequency pulse to create an input equivalent to either a constant function or a balanced function are presented, and the results are compared with those obtained from computer simulations.Comment: accepted for publication in the Journal of Chemical Physic

Commercial video games and cognitive functions: video game genres and modulating factors of cognitive enhancement

Background Unlike the emphasis on negative results of video games such as the impulsive engagement in video games, cognitive training studies in individuals with cognitive deficits showed that characteristics of video game elements were helpful to train cognitive functions. Thus, this study aimed to have a more balanced view toward the video game playing by reviewing genres of commercial video games and the association of video games with cognitive functions and modulating factors. Literatures were searched with search terms (e.g. genres of video games, cognitive training) on database and Google scholar. Results video games, of which purpose is players entertainment, were found to be positively associated with cognitive functions (e.g. attention, problem solving skills) despite some discrepancy between studies. However, the enhancement of cognitive functions through video gaming was limited to the task or performance requiring the same cognitive functions. Moreover, as several factors (e.g. age, gender) were identified to modulate cognitive enhancement, the individual difference in the association between video game playing and cognitive function was found. Conclusion Commercial video games are suggested to have the potential for cognitive function enhancement. As understanding the association between video gaming and cognitive function in a more balanced view is essential to evaluate the potential outcomes of commercial video games that more people reported to engage, this review contributes to provide more objective evidence for commercial video gaming.It was supported by a grant from Game Science Forum in South Kore