Relations between optimum biased spring balance weighing designs and optimum chemical balance weighing designs with non-homogeneity of the variances of errors

This paper deals with the problem of estimating the individual weights of objects with minimum variances by using a weighing design with non-homogeneity of the variances of errors in the model. The necessary and sufficient conditions for optimum biased spring balance weighing designs with non-homogeneity of the variances of errors and for optimum chemical balance weighing designs with non-homogeneity of the variances of errors are given and the relations between these designs are investigated. New optimum weighing designs are also found. Department of Mathematical and Statistical Methods, Agricultural University, 60-637 Poznan, Poland; Institute of Mathematics, Adam Mickiewicz University, 60-769 Poznan, Poland. Published in Lietuvos Matematikos Rinkinys, Vol. 34, No. 3, pp. 267–273, July–September, 1994.     

A relation between BIB designs and chemical balance weighing designs

The paper gives certain new construction method for optimum chemical balance weighing designs. It utilizes a relation between the incidence matrices of a set of BIB designs and the design matrix of a chemical balance weighing design.     

Circulant weighing designs

Algebraic techniques are employed to obtain necessary conditions for the existence of certain families of circulant weighing designs. As an application we rule out the existence of many circulant weighing designs. In particular, we show that there does not exist a circulant weighing matrix of order 43 for any weight. We also prove two conjectures of Yosef Strassler. © 1996 John Wiley & Sons, Inc.     

A note on weighing designs

Summary Banerjee [1], [2] has shown that the arrangements afforded by a Balanced Incomplete Block Design can be used as an efficient spring balance design. Such designs suffer from one drawback viz., there are only a few or no degrees of freedom left for the estimation of error-variance,σ ². To overcome this difficulty, it has been suggested that the whole design may be repeated a certain number of times to get an estimate of the error variance. In the present note an attempt has been made to give an alternative design where there is no necessity of such repetition. It has been also shown that these designs give a lesser variance of the estimated weights than the repeated design. Institute of Agricultural Research Statistics     

Balanced bipartite weighing designs

A block B denotes a set of k = k₁ + k₂ elements which are divided into two subsets B¹ and B², where |Bⁱ| = k_i, i = 1 or 2. Two elements of B are said to be linked or n-linked in B if and only if they belong to different subsets or the same subset of B respectively. A balanced bipartite weighing design, (briefly BBWD (υ, k₁, k₂, λ₁)) is an arrangement of υ elements into b blocks, each containing k elements, such that each element occurs in exactly r blocks, any two distinct elements are linked in exactly λ₁ blocks and n-linked in exactly λ₂ blocks.Given fixed k₁ and k₂, there is always a minimal value of λ₁ such that the necessary conditions for the existence of a BBWD are satisfied for same υ. It is proved that in many cases, the necessary conditions are also sufficient. Some general methods for constructing BBWD's as well as a table of all designs with υ ? 13 are obtained.     

Inner balance of symmetric designs

A triple array is a row-column design which carries two balanced incomplete block designs (BIBDs) as substructures. McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005), Section 8, gave one example of a triple array that also carries a third BIBD, formed by its row-column intersections. This triple array was said to be balanced for intersection, and they made a search for more such triple arrays among all potential parameter sets up to some limit. No more examples were found, but some candidates with suitable parameters were suggested. We define the notion of an inner design with respect to a block for a symmetric BIBD and present criteria for when this inner design can be balanced. As triple arrays in the canonical case correspond to SBIBDs, this in turn yields new existence criteria for triple arrays balanced for intersection. In particular, we prove that the residual design of the related SBIBD with respect to the defining block must be quasi-symmetric, and give necessary and sufficient conditions on the intersection numbers. This, together with our parameter bounds enable us to exclude the suggested triple array candidates in McSorley et al. (Des Codes Cryptogr 35: 21–45, 2005) and many others in a wide search. Further we investigate the existence of SBIBDs whose inner designs are balanced with respect to every block. We show as a key result that such SBIBDs must possess the quasi-3 property, and we answer the existence question for all known classes of these designs.     

Some remarks on permutation designs

A definition of isomorphism of two permutation designs is proposed, which differs from the definition in Bandt [J. Combinatorial Theory Ser. A21 (1976), 384–392]. The proposed definition has the (generally required) property that the allowed permutations always transform a permutation design into a permutation design. It is shown that the n permutation designs coming from the partitioning of S_n into permutation designs, as constructed in Bandt [J. Combinatorial Theory Ser. A21 (1976), 384–392] are all isomorphic. Further we find that this modified definition does not increase the number of nonisomorphic (6, 4) permutation designs. The same investigation showed that one of the designs, claimed to be a (6, 4) permutation design in [J. Combinatorial Theory Ser. A21 (1976), 384–392], is actually not a (6, 4) permutation design.     

Circular balanced uniform repeated measurements designs

Circular balanced uniform repeated measurements designs, which have been shown to be universally optimal by Kunert (1984), are constructed whenever the number of treatments is a prime or an even number.     

A-optimal chemical balance weighing design with correlated errors

In this paper we study the estimation problem of individual weights of objects using an A-optimal chemical balance weighing design. We assume that in this model errors are correlated and they have the same variances. The lower bound oftr (X′G ^?1 X)^?1 is obtained and a necessary and sufficient condition for this lower bound to be attained is given. There is given new construction method of A-optimal chemical balance weighing design.     

Some remarks on Euclidean tight designs

In this paper we present a new 4-dimensional tight Euclidean 5-design on 3 concentric spheres, together with a list of all known tight Euclidean designs which has been updated since the last survey paper by Bannai and Bannai (2009) [6]. We also examine whether each of all known tight Euclidean designs has the structure of a coherent configuration.     

Some new symmetric designs

For q, an odd prime power, we construct symmetric (2q²+2q+1,q²,½q(q-1)) designs having an automorphism group of order q that fixes 2q+1 points. The construction indicates that for each q the number of such designs that are pairwise non-isomorphic is very large.     

Some results on resolvable incomplete block designs

This paper is concerned with the uniformity of a certain kind of resolvable incomplete block (RIB for simplicity) design which is called the PRIB design here. A sufficient and necessary condition is obtained, under which a PRIB design is the most uniform in the sense of a discrete discrepancy measure, and the uniform PRIB design is shown to be connected. A construction method for such designs via a kind of U-type designs is proposed, and an existence result of these designs is given. This method sets up an important bridge between PRIB designs and U-type designs.     

Some observations on cohomologicallyp-ample bundles

Sunto Usando un recente risultato di Deligne e Illusie si prova un teorema di annullamento per la coomologia di un fibrato coomologicamente p-ampio. Si studiano le proprietà dei gruppi di coomologia dei fibrati vettoriali nel caso in cui il morfismo di Frobenius si estenda a un sollevamento della varietà su W₂(k), anello dei vettori di Witt di lunghezza 2 del campo. Si studiano, infine, a titolo di esempio, i fibrati coomologicamente p-ampi su P^d.

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Partially supported by the MRST 40% and 60% funds.     

Some observations on weighted GMRES

We investigate the convergence of the weighted GMRES method for solving linear systems. Two different weighting variants are compared with unweighted GMRES for three model problems, giving a phenomenological explanation of cases where weighting improves convergence, and a case where weighting has no effect on the convergence. We also present a new alternative implementation of the weighted Arnoldi algorithm which under known circumstances will be favourable in terms of computational complexity. These implementations of weighted GMRES are compared for a large number of examples. We find that weighted GMRES may outperform unweighted GMRES for some problems, but more often this method is not competitive with other Krylov subspace methods like GMRES with deflated restarting or BICGSTAB, in particular when a preconditioner is used.     

Some observations on HC-128

In this paper, we study HC-128 in detail from cryptanalytic point of view. First, we use linear approximation of the addition modulo 2 ⁿ of three n-bit integers to identify linear approximations of g ₁, g ₂, the feedback functions of HC-128. This, in turn, shows that the process of keystream output generation of HC-128 can be well approximated by linear functions. In this direction, we show that the ??least significant bit?? based distinguisher (presented by the designer himself) of HC-128 works for the complete 32-bit word. Using the above linear approximations of g ₁, g ₂, we present a new distinguisher for HC-128 which is slightly weaker than Wu??s distinguisher. Finally, in the line of Dunkelman??s observation, we also study how HC-128 keystream words leak secret state information of the cipher due to the properties of the functions h ₁, h ₂ and present improved results.