Roots of Characteristic Polynomial Sequences in Iterative Block Cyclic Reductions

The block cyclic reduction method is a finite-step direct method used for solving linear systems with block tridiagonal coefficient matrices. It iteratively uses transformations to reduce the number of non-zero blocks in coefficient matrices. With repeated block cyclic reductions, non-zero off-diagonal blocks in coefficient matrices incrementally leave the diagonal blocks and eventually vanish after a finite number of block cyclic reductions. In this paper, we focus on the roots of characteristic polynomials of coefficient matrices that are repeatedly transformed by block cyclic reductions. We regard each block cyclic reduction as a composition of two types of matrix transformations, and then attempt to examine changes in the existence range of roots. This is a block extension of the idea presented in our previous papers on simple cyclic reductions. The property that the roots are not very scattered is a key to accurately solve linear systems in floating-point arithmetic. We clarify that block cyclic reductions do not disperse roots, but rather narrow their distribution, if the original coefficient matrix is symmetric positive or negative definite.

Hadamard matrices from Goethals — Seidel difference families with a repeated block

Purpose: To construct Hadamard matrices by using Goethals — Seidel difference families having a repeated block, generalizingthe so called propus construction. In particular we construct the first examples of symmetric Hadamard matrices of order 236.Methods: The main ingredient of the propus construction is a difference family in a finite abelian group of order v consisting offour blocks (X1, X2, X3, X4) where X1 is symmetric and X2 X3. The parameters (v; k1, k2, k3, k4; λ) of such family must satisfythe additional condition ki  λ  v. We modify this construction by imposing different symmetry conditions on some of theblocks and construct many examples of Hadamard matrices of this kind. In this paper we work with the cyclic group Zv of order v.For larger values of v we build the blocks Xi by using the orbits of a suitable small cyclic subgroup of the automorphism groupof Zv. Results: We continue the systematic search for symmetric Hadamard matrices of order 4v by using the propus construction.Such searches were carried out previously for odd v  51. We extend it to cover the case v53. Moreover we construct thefirst examples of symmetric Hadamard matrices of order 236. A wide collection of symmetric and skew-symmetric Hadamardmatrices was obtained and the corresponding difference families tabulated by using the symmetry properties of their blocks.Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, compression and masking ofvideo information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in themathematical network Internet together with executable on line algorithms.

Effect of Creatine Supplementation Jumping Performance in Elite Volleyball Players

Jump height is a critical aspect of volleyball players’ blocking and attacking performance. Although previous studies demonstrated that creatine monohydrate supplementation (CrMS) improves jumping performance, none have yet evaluated its effect among volleyball players with proficient jumping skills. We examined the effect of 4 wk of CrMS on 1 RM spike jump (SJ) and repeated block jump (BJ) performance among 12 elite males of the Sherbrooke University volleyball team. Using a parallel, randomized, double-blind protocol, participants were supplemented with a placebo or creatine solution for 28 d, at a dose of 20 g/d in days 1–4, 10 g/d on days 5-6, and 5 g/d on days 7-28. Pre- and postsupplementation, subjects performed the 1 RM SJ test, followed by the repeated BJ test (10 series of 10 BJs; 3 s interval between jumps; 2 min recovery between series). Due to injuries (N = 2) and outlier data (N = 2), results are reported for eight subjects. Following supplementation, both groups improved SJ and repeated BJ performance. The change in performance during the 1 RM SJ test and over the first two repeated BJ series was unclear between groups. For series 3-6 and 7-10, respectively, CrMS further improved repeated BJ performance by 2.8% (likely beneficial change) and 1.9% (possibly beneficial change), compared with the placebo. Percent repeated BJ decline in performance across the 10 series did not differ between groups pre- and postsupplementation. In conclusion, CrMS likely improved repeated BJ height capability without influencing the magnitude of muscular fatigue in these elite, university-level volleyball players.

The Linus Sequence

Define the Linus sequence Ln for n ≥ 1 as a 0–1 sequence with L1 = 0, and Ln chosen so as to minimize the length of the longest immediately repeated block Ln−2r+1 ⋅⋅⋅ Ln−r = Ln−r+1 ⋅⋅⋅ Ln. Define the Sally sequence Sn as the length r of the longest repeated block that was avoided by the choice of Ln. We prove several results about these sequences, such as exponential decay of the frequency of highly periodic subwords of the Linus sequence, zero entropy of any stationary process obtained as a limit of word frequencies in the Linus sequence and infinite average value of the Sally sequence. In addition we make a number of conjectures about both sequences.

Investigation Into Cumulative Damage Rules to Predict Fretting Fatigue Life of Ti-6Al-4V Under Two-Level Block Loading Condition1

The effects of variable amplitude loading on fretting fatigue behavior of titanium alloy, Ti-6Al-4V were examined. Fretting fatigue tests were carried out under constant stress amplitude and three different two-level block loading conditions: high-low (Hi-Lo), low-high (Lo-Hi), and repeated block of high and low stress amplitudes. The damage fractions and fretting fatigue lives were estimated by linear and non-linear cumulative damage rules. Damage curve analysis (DCA) and double linear damage rule (DLDR) were capable to account for the loading order effects in Hi-Lo and Lo-Hi loadings. In addition, the predictions by DCA and DLDR were better than that by linear damage rule (LDR). Besides its simplicity of implementation, LDR was also capable of estimating failure lives reasonably well. Repeated two-level block loading resulted in shorter lives and lower fretting fatigue limit compared to those under constant amplitude loading. The degree of reduction in fretting fatigue lives and fatigue strength depended on the ratio of cycles at lower stress amplitude to that at higher stress amplitude. Fracture surface of specimens subjected to Hi-Lo and repeated block loading showed the clear evidence of change in stress amplitude of applied load. Especially, the repeated two-level block loading resulted in characteristic markers which reflected change in crack growth rates corresponding to different stress amplitudes.

MESH ALGORITHMS FOR FINDING REPETITIONS AND PARTIAL SYMMETRIES IN ARRAYS

The two-dimensional mesh computer architecture has proven to be an appropriate means to apply parallel computation to problems in image processing. However, this is most often done using local-neighbourhood operations to accomplish image filtering and morphological transformations. The discovery of structures in an image such as repetitions and symmetries is another form of visual analysis, and yet relatively little has been done to apply mesh computers to this problem. In this paper, we apply the primitive operations of prefix scanning and sorting to efficiently implement a repetition finding algorithm for arrays. The computational complexity of the algorithm on a n×n mesh is O(n log k) where k is the width of the largest repeated block in the array. The algorithm was implemented on a MasPar MP-1 computer. We describe variations of the algorithm for solving several related problems including the detection of partial symmetries in an image and repetitions in images modulo pixel-value transformations.