The Spectra for the Conjugate Invariant Subgroups of n2 × 4 Orthogonal Arrays

1980 ◽  
Vol 32 (5) ◽  
pp. 1126-1139 ◽  
Author(s):  
C. C. Lindner ◽  
R. C. Mullin ◽  
D. G. Hoffman
Keyword(s):  
Orthogonal Array ◽  
Orthogonal Arrays

An n2 × k orthogonal array is a pair (P, B) where P = {1, 2, …, n} and B is a collection of k-tuples of elements from P (called rows) such that if i < j ∈ {1, 2, …, k} and x and y are any two elements of P (not necessarily distinct) there is exactly one row in B whose ith coordinate is x and whose jth coordinate is y. We will refer to the ith coordinate of a row r as the ith column of r. The number n is called the order (or size) of the array and k is called the strength.

scholarly journals Orthogonal Arrays with Parameters $OA(s^3,s^2+s+1,s,2)$ and 3-Dimensional Projective Geometries

10.37236/556 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Kazuaki Ishii
Keyword(s):  
Orthogonal Array ◽  
Prime Power ◽  
Orthogonal Arrays ◽  
3 Dimensional ◽  
Projective Geometries ◽  
Finite Spaces ◽  
Alpha Type

There are many nonisomorphic orthogonal arrays with parameters $OA(s^3,s^2+s+1,s,2)$ although the existence of the arrays yields many restrictions. We denote this by $OA(3,s)$ for simplicity. V. D. Tonchev showed that for even the case of $s=3$, there are at least 68 nonisomorphic orthogonal arrays. The arrays that are constructed by the $n-$dimensional finite spaces have parameters $OA(s^n, (s^n-1)/(s-1),s,2)$. They are called Rao-Hamming type. In this paper we characterize the $OA(3,s)$ of 3-dimensional Rao-Hamming type. We prove several results for a special type of $OA(3,s)$ that satisfies the following condition: For any three rows in the orthogonal array, there exists at least one column, in which the entries of the three rows equal to each other. We call this property $\alpha$-type. We prove the following. (1) An $OA(3,s)$ of $\alpha$-type exists if and only if $s$ is a prime power. (2) $OA(3,s)$s of $\alpha$-type are isomorphic to each other as orthogonal arrays. (3) An $OA(3,s)$ of $\alpha$-type yields $PG(3,s)$. (4) The 3-dimensional Rao-Hamming is an $OA(3,s)$ of $\alpha$-type. (5) A linear $OA(3,s)$ is of $\alpha $-type.

scholarly journals On Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1895 ◽  
Author(s):  
M. Higazy ◽  
A. El-Mesady ◽  
M. S. Mohamed
Keyword(s):  
Computer Science ◽  
Finite Fields ◽  
Orthogonal Array ◽  
Orthogonal Arrays ◽  
Latin Squares ◽  
Error Correcting Codes ◽  
Recursive Construction ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Definition Of

During the last two centuries, after the question asked by Euler concerning mutually orthogonal Latin squares (MOLS), essential advances have been made. MOLS are considered as a construction tool for orthogonal arrays. Although Latin squares have numerous helpful properties, for some factual applications these structures are excessively prohibitive. The more general concepts of graph squares and mutually orthogonal graph squares (MOGS) offer more flexibility. MOGS generalize MOLS in an interesting way. As such, the topic is attractive. Orthogonal arrays are essential in statistics and are related to finite fields, geometry, combinatorics and error-correcting codes. Furthermore, they are used in cryptography and computer science. In this paper, our current efforts have concentrated on the definition of the graph-orthogonal arrays and on proving that if there are k MOGS of order n, then there is a graph-orthogonal array, and we denote this array by G-OA(n2,k,n,2). In addition, several new results for the orthogonal arrays obtained from the MOGS are given. Furthermore, we introduce a recursive construction method for constructing the graph-orthogonal arrays.

scholarly journals A geometrical representation theory for orthogonal arrays

1994 ◽  
Vol 49 (2) ◽  
pp. 311-324 ◽  
Author(s):  
David G. Glynn
Keyword(s):  
Representation Theory ◽  
Projective Space ◽  
Orthogonal Array ◽  
Prime Power ◽  
Infinite Order ◽  
Orthogonal Arrays ◽  
Geometrical Representation

Every orthogonal array of strength s and of prime-power (or perhaps infinite) order q, has a well-defined collection of ranks r. Having rank r means that it can be constructed as a cone cut by qs hyperplanes in projective space of dimension r over a field of order q.

scholarly journals Constructions of irredundant orthogonal arrays

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guangzhou Chen ◽  
Xiaotong Zhang
Keyword(s):  
Finite Fields ◽  
Orthogonal Array ◽  
Orthogonal Arrays ◽  
Physical Review ◽  
Local Dimension ◽  
Difference Matrices ◽  
Uniform State ◽  
Definition Of

<p style='text-indent:20px;'>An <inline-formula><tex-math id="M1">\begin{document}$ N \times k $\end{document}</tex-math></inline-formula> array <inline-formula><tex-math id="M2">\begin{document}$ A $\end{document}</tex-math></inline-formula> with entries from <inline-formula><tex-math id="M3">\begin{document}$ v $\end{document}</tex-math></inline-formula>-set <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{V} $\end{document}</tex-math></inline-formula> is said to be an <i>orthogonal array</i> with <inline-formula><tex-math id="M5">\begin{document}$ v $\end{document}</tex-math></inline-formula> levels, strength <inline-formula><tex-math id="M6">\begin{document}$ t $\end{document}</tex-math></inline-formula> and index <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, denoted by OA<inline-formula><tex-math id="M8">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula>, if every <inline-formula><tex-math id="M9">\begin{document}$ N\times t $\end{document}</tex-math></inline-formula> sub-array of <inline-formula><tex-math id="M10">\begin{document}$ A $\end{document}</tex-math></inline-formula> contains each <inline-formula><tex-math id="M11">\begin{document}$ t $\end{document}</tex-math></inline-formula>-tuple based on <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{V} $\end{document}</tex-math></inline-formula> exactly <inline-formula><tex-math id="M13">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> times as a row. An OA<inline-formula><tex-math id="M14">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula> is called <i>irredundant</i>, denoted by IrOA<inline-formula><tex-math id="M15">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula>, if in any <inline-formula><tex-math id="M16">\begin{document}$ N\times (k-t ) $\end{document}</tex-math></inline-formula> sub-array, all of its rows are different. Goyeneche and <inline-formula><tex-math id="M17">\begin{document}$ \dot{Z} $\end{document}</tex-math></inline-formula>yczkowski firstly introduced the definition of an IrOA and showed that an IrOA<inline-formula><tex-math id="M18">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula> corresponds to a <inline-formula><tex-math id="M19">\begin{document}$ t $\end{document}</tex-math></inline-formula>-uniform state of <inline-formula><tex-math id="M20">\begin{document}$ k $\end{document}</tex-math></inline-formula> subsystems with local dimension <inline-formula><tex-math id="M21">\begin{document}$ v $\end{document}</tex-math></inline-formula> (Physical Review A. 90 (2014), 022316). In this paper, we present some new constructions of irredundant orthogonal arrays by using difference matrices and some special matrices over finite fields, respectively, as a consequence, many infinite families of irredundant orthogonal arrays are obtained. Furthermore, several infinite classes of <inline-formula><tex-math id="M22">\begin{document}$ t $\end{document}</tex-math></inline-formula>-uniform states arise from these irredundant orthogonal arrays.</p>

Mixed Variable Optimization Using Taguchi’s Orthogonal Arrays

1995 ◽  
Author(s):  
H.-W. Chi ◽  
C. L. Bloebaum
Keyword(s):  
Orthogonal Array ◽  
Optimization Problem ◽  
Orthogonal Arrays ◽  
Paper Analysis ◽  
Optimal Designs ◽  
Number Of Factors ◽  
The Mean ◽  
Optimal Material ◽  
Structural Optimization Problem ◽  
Mixed Variable

Abstract Taguchi’s orthogonal arrays for Robust Design are used in this paper in a non-taditional way to solve a mixed continuous-discrete structural optimization problem. The factors of an orthogonal array correspond to the members of a structure and the levels of each factor correspond to the material choices of each member. Based on the number of factors to be studied and the number of levels of each factor, an appropriate orthogonal array is selected for each specific problem. The number of rows of the orthogonal array correspond to the number of experiments (i.e. continuous sizing optimizations) to be conducted. The response of these experiments, which are the weight of the optimal designs corresponding to different material settings, are then used to calculate the mean effect of each factor level. Some possible optimal material settings can then be determined. Three examples are presented in this paper. Analysis using Taguchi’s orthogonal arrays was able to isolate several near optimal or optimal designs. The accuracy and efficiency of the proposed method compared to more traditinoal solution methodologies are also discussed.

Percentiles of LINMAP Conjoint Indices of Fit for Various Orthogonal Arrays: A Simulation Study

1986 ◽  
Vol 23 (3) ◽  
pp. 286-290 ◽  
Author(s):  
Gary M. Mullet ◽  
Marvin J. Karson
Keyword(s):  
Conjoint Analysis ◽  
Simulation Study ◽  
Orthogonal Array ◽  
Probability Distributions ◽  
Orthogonal Arrays ◽  
Estimation Algorithm ◽  
Experimental Designs ◽  
Practical Applications ◽  
Index Of Fit

The authors present and examine simulated probability distributions for the index of fit of the LINMAP estimation algorithm as used in conjoint analysis. Important percentiles in both tails of the distributions are shown in tables for different orthogonal array experimental designs. Practical applications are suggested.

scholarly journals Use of Orthogonal Arrays for Efficient Evaluation of Geometric Designs for Reducing Vibration of a Non-Pneumatic Wheel during High-Speed Rolling

2010 ◽  
Vol 38 (4) ◽  
pp. 246-275 ◽  
Author(s):  
William Rutherford ◽  
Shashank Bezgam ◽  
Amarnath Proddaturi ◽  
Lonny Thompson ◽  
John C. Ziegert ◽  
...  
Keyword(s):  
Orthogonal Array ◽  
High Speed ◽  
Contact Region ◽  
Orthogonal Arrays ◽  
Nonlinear Effects ◽  
Element Model ◽  
Geometric Design ◽  
Design Parameters ◽  
Design Variables ◽  
Wheel Vibration

Abstract During high speed rolling of a nonpneumatic wheel, vibration may be produced by the interaction of collapsible spokes with a shear deformable ring as they enter the contact region, buckle, and then snap back into a state of tension. In the present work, a systematic study of the effects of six key geometric design parameters is presented using Orthogonal Arrays. Orthogonal Arrays are part of a design process method developed by Taguchi which provides an efficient way to determine optimal combinations of design variables. In the present work, a two-dimensional planar finite element model with geometric nonlinearity and explicit time-stepping is used to simulate rolling of the nonpneumatic wheel. Vibration characteristics are measured from the FFT frequency spectrum of the time signals of perpendicular distance of marker nodes from the virtual plane of the spoke, and ground reaction forces. Both maximum peak amplitudes and RMS measures are considered. Two complementary Orthogonal Arrays are evaluated. The first is the L8 orthogonal array which considers the six geometric design variables evaluated at lower and higher limiting values for a total of eight experiments defined by statistically efficient variable combinations. Based on the results from the L8 orthogonal array, a second L9 orthogonal array experiment evaluates the nonlinear effects in the four parameters of greatest interest, (a) spoke length, (b) spoke curvature, (c) spoke thickness, and (d) shear beam thickness. The L9 array consists of nine experiments with efficient combinations of low, intermediate, and high value levels. Results from use of the Orthogonal Array experiments were used to find combinations of parameters which significantly reduce peak and RMS amplitudes, and suggest that spoke length has the greatest effect on vibration amplitudes.

Calmodulin-mediated gating of lens gap junction channels in vesicles

1984 ◽  
Vol 42 ◽  
pp. 134-137
Author(s):  
Camillo Peracchia ◽  
Stephen J. Girsch
Keyword(s):  
Gap Junctions ◽  
Freeze Fracture ◽  
Orthogonal Arrays ◽  
Eye Lens ◽  
Lens Fiber ◽  
Cell Coupling ◽  
Particle Spacing ◽  
Fiber Cells ◽  
Gap Junction Channels ◽  
Communicating Junctions

The fiber cells of eye lens communicate directly with each other by exchanging ions, dyes and metabolites. In most tissues this type of communication (cell coupling) is mediated by gap junctions. In the lens, the fiber cells are extensively interconnected by junctions. However, lens junctions, although morphologically similar to gap junctions, differ from them in a number of structural, biochemical and immunological features. Like gap junctions, lens junctions are regions of close cell-to-cell apposition. Unlike gap junctions, however, the extracellular gap is apparently absent in lens junctions, such that their thickness is approximately 2 nm smaller than that of typical gap junctions (Fig. 1,c). In freeze-fracture replicas, the particles of control lens junctions are more loosely packed than those of typical gap junctions (Fig. 1,a) and crystallize, when exposed to uncoupling agents such as Ca++, or H+, into pseudo-hexagonal, rhombic (Fig. 1,b) and orthogonal arrays with a particle-to-particle spacing of 6.5 nm. Because of these differences, questions have been raised about the interpretation of the lens junctions as communicating junctions, in spite of the fact that they are the only junctions interlinking lens fiber cells.

Formulation Development of Tamoxifen Loaded Lipid Nanoparticle by Taguchi (L12 (2 11)) Orthogonal Array Design

2020 ◽  
Vol 16 ◽  
Author(s):  
Poovi Ganesan ◽  
N Damodharan
Keyword(s):  
Orthogonal Array ◽  
Drug Loading ◽  
Optimization Technique ◽  
Pharmaceutical Products ◽  
Water Soluble ◽  
Nanostructured Lipid Carrier ◽  
Orthogonal Array Design ◽  
Formulation Development ◽  
Array Design ◽  
Lipid Nanoparticle

Background: A better understanding of the biopharmaceutical and physicochemical properties of drugs and the pharmaco-technical factors would be of great help for developing pharmaceutical products. But, it is extremely difficult to study the effect of each variable and interaction among them through the conventional approach Objective: To screen the most influential factors affecting the particle size (PS) of lipid nanoparticle (LNPs) (solid lipid nanoparticle (SLN) and nanostructured lipid carrier (NLC)) for poorly water-soluble BCS class-II drug like tamoxifen (TMX) to improve its oral bioavailability and to reduce its toxicity to tolerable limits using Taguchi (L12 (2 11)) orthogonal array design by applying computer optimization technique. Results: The size of all LNPs formulations prepared as per the experimental design varied between 172 nm and 3880 μm, polydispersity index between 0.033 and 1.00, encapsulation efficiency between 70.8% and 75.7%, and drug loading between 5.84% and 9.68%. The study showed spherical and non-spherical as well as aggregated and non-aggregated LNPs. Besides, it showed no interaction and amorphous form of the drug in LNPs formulation. The Blank NLCs exhibited no cytotoxicity on MCF-7 cells as compared to TMX solution, SLNs (F5) and NLCs (F12) suggests that the cause of cell death is primarily from the effect of TMX present in NLCs. Conclusions: The screening study clearly showed the importance of different individual factors significant effect for the LNPs formulation development and its overall performance in an in-vitro study with minimum experimentation thus saving considerable time, efforts, and resources for further in-depth study.

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