Naturally Ordered Research Articles

Conditional generalized estimating equations of mean-variance-correlation for clustered data

Joint modelling of mean-covariance structure is an important topic in clustered data analysis. Existing methods, such as those based on modified Cholesky decomposition (MCD), alternative Cholesky decomposition (ACD) and hyperspherical coordinates decomposition (HPC), have two main restrictions. First, they often assume that responses in the same cluster are naturally ordered, for example, by time in longitudinal studies. Second, the existing methods model transformed parameters, for instance, the generalized autoregressive parameters and innovation variances in MCD/ACD, and the hyperspherical coordinates in HPC, making the dependence of correlation coefficients or variances on covariates hardly understandable. As an alternative, a data-driven method that models directly the mean, variances and correlation coefficients for clustered data is proposed. Comparing to the existing methods, the proposed approach not only has no need of natural order in responses but also works on original correlation coefficients and variances. The proposed models are flexible and interpretable, and the parameter estimators in joint generalized estimating equations (GEE) are shown to be consistent and asymptotically normally distributed. Consistent model selection criteria in spirit of quasi-likelihood under independence model criterion (QIC) are considered. The use of the proposed approach is demonstrated by intensive simulation studies and real data analysis.

Naturally ordered strong endomorphisms on graphs

In this paper, we study the natural partial order $$\le $$ on SEnd(G), the strong endomorphism monoid of a finite graph G and characterize minimal elements and maximal elements of $$(SEnd(G),\le )$$. Then we introduce the concept of connectedness on $$(SEnd(G),\le )$$ by using the natural partial order $$\le $$ and determine the number of connected components of SEnd(G) under certain conditions.

Naturally Ordered Semigroups of Partial Transformations Preserving an Equivalence Relation

Let X be a set with |X| ≥3, P(X) the semigroup of all partial transformations on the set X, E be an equivalence relation on X, and Then P E (X) forms a subsemigroup of P(X). In this article, we consider the natural order ≤on P E (X) and determine when two elements of P E (X) are related under this order. Then we characterize those elements of P E (X) which are left (right) compatible with ≤on P E (X). Besides, the minimal (maximal) elements are described.

Naturally Ordered Transformation Semigroups Preserving an Equivalence Relation and a Cross-section

The paper is concerned with the so-called natural order on the semigroup [Formula: see text], where [Formula: see text] is the full transformation semigroup on a set X, E is a non-trivial equivalence on X and R is a cross-section of the partition X/E induced by E. We determine when two elements of TE(X,R) are related under this order, find elements of TE(X,R) which are compatible with ≤ on TE(X,R), and observe the maximal and minimal elements and the covering elements.

Das Alphabet der Natur und das Alphabet der Kultur im 18. Jahrhundert

In the middle of the eighteenth century, Carl von Linné, Johann Christoph Gatterer, and Christian Wilhelm Büttner attempted to realize the old idea of deciphering the alphabet of the world, which Francis Bacon had raised as a general postulate of science. This article describes these attempts and their interrelations. Linné used the model of the alphabet to classify plants according to the characters of this fruiting body. Gatterer, one of the leading German historians during the Enlightenment, adopted the botanical method of classification by genus and species to classify the history of scripts. He used the forms of the alphabetic characters to measure the age of manuscripts and to map the process of history as a genealogy of culture. Gatterer collaborated closely with Büttner, the first Göttingen professor of natural history. Büttner constructed a general alphabet of languages which connected the phonetics of language with the historically known alphabets. Early on, diplomatics and ethnography combined the natural order of natural history and the cultural order of the alphabet with the attempt to register development and to document development by the evolution of forms. Based on the shared model of the alphabet and on the common necessity to classify their empirical material, natural history and the description of culture were related attempts in the middle of the eighteenth century to comprehend the alphabetically organized nature and a naturally ordered culture.

Basic Hoops: an Algebraic Study of Continuous t-norms

A continuous t-norm is a continuous map * from [0, 1]2 into [0, 1] such that $$\langle [0, 1], *, 1 \rangle$$ is a commutative totally ordered monoid. Since the natural ordering on [0, 1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and $$\langle [0, 1], *,\rightarrow, 1\rangle$$ becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras $$\langle [0, 1], *,\rightarrow, 1\rangle$$ , where * is a continuous t-norm. In this paper we investigate the structure of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hajek’s BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [0, 1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL-algebras, of Godel algebras and of product algebras.

Naturally ordered regular semigroups with an inverse monoid transversal

The notion of an inverse transversal of a regular semigroup is well-known. Here we investigate naturally ordered regular semigroups that have an inverse transversal. Such semigroups are necessarily locally inverse and the inverse transversal is a quasi-ideal. After considering various general properties that relate the imposed order to the natural order, we highlight the situation in which the inverse transversal is a monoid. The regularity of Green’s relations is also characterised. Finally, we determine the structure of a naturally ordered regular semigroup with an inverse monoid transversal.

A graphical chain model for inferring regulatory system networks from gene expression profiles

A procedure for graphical chain modeling has been designed for analyzing the expression profiles of genes that can be classified into several blocks in a natural order. Since the gene expression profiles often share similar patterns, the genes within a block are grouped into some clusters, as a prerequisite for the modeling. Then, the clusters in the naturally ordered blocks are regarded as variables. Finally, the associations of the variables within and between blocks are inferred by covariance selection in graphical Gaussian modeling. The newly designed procedure for graphical chain modeling was applied to 619 expression profiles of cell cycle related genes in yeast, which were selected from 792 genes experimentally identified as being transcribed in the order of four cell cycle phases, G 1 , S , G 2 , and M . By the application of the procedure, the 619 genes were classified into 50 clusters, and a chain graph was fitted for 50 clusters in the four phases. On focusing on the clusters, including transcription factors, characteristic relationships between the clusters emerged from the associations of the clusters within and between the four phases; one of the remarkable features is the distinctive relationships for the clusters between neighboring and non-neighboring phases. The merits and pitfalls of the graphical chain model are discussed in terms of its application to the field of molecular biology.

Christianity, Englishness and the southern English countryside: A study of the work of H.J. Massingham

This paper explores the relationships between Christianity, Englishness, and ideas about the southern English landscape in the writings of the 1930s and 1940s rural commentator, H.J. Massingham. The paper begins by looking in general terms at the conjunction of religious and national identities in the context of national landscapes before moving on to consider in more detail one particular instance of this in the writing of H.J. Massingham. Massingham's understanding of a divine natural order, his construction of a kind of 'divine Englishness' and the way in which he relates this to particular English landscapes is explored. In particular, the paper investigates the natural, social and political power relationships which are embedded in Massingham's work, and suggests that his writing provides an interesting example of one way in which theological reasoning can reflect and reinforce concepts of a naturally ordered national identity. Cet article explore la relation entre le Christianisme, l'identité anglaise, et la notion de paysage anglais méridional dans les écrits du commentateur rural H.J. Massingham entre 1930 et 1940. L'analyse débute par des considérations générales sur la conjonction des identités religieuse et nationale et se poursuit par un regard plus détaillé sur un cas spécifique de cette conjonction dans les écrits de H.J. Massingham. Sa conception d'un ordre naturel divin, sa construction d'une sorte d'identité anglaise divine' et la façon dont il élabore celle-ci en rapport avec le paysage anglais sont explorés. En particulier, cet article examine les relations de pouvoir naturelles, sociales et politiques au centre de l'oeuvre de Massingham et suggère que ses écrits fournissent un exemple intéressant de la façon dont le raisonnement théologique peut refléter et renforcer l'idée d'une identiténationale procédant d'un ordre naturel. Este papel examina las relaciones entre el cristianismo, el carácter ingles y el paisaje del sur de Inglaterra en las obras del comentarista rural de los años 30 y 40 del siglo veinte, H.J. Massingham. El papel empieza por examinar, de manera general, la conjunción de identidades religiosas y nacionales en el contexto de paisajes nacionales. Luego considera en más detalle un ejemplo de ésto en las escrituras de H.J. Massingham. Examino su entendimiento de un orden natural divino, su construcción de una especie de 'carácter inglés divino' y también examino la manera en que establece una conexion entre ésto y paisajes determinados de Inglaterra. En particular, el papel investiga las relaciones de poder naturales, sociales y políticas empotradas en las obras de Massingham y sugiere que su escritura constituye un ejemplo interesante de una manera en que el razonamiento teológico puede reflejar y reforzar conceptos de una identidad nacional de orden natural.

Ordinal Classification in Medical Prognosis

Medical prognosis is commonly expressed in terms of ordered outcome categories. This paper provides simple statistical procedures to judge whether the predictor variables reflect this natural ordering. The concept of stochastic ordering in logistic regression and discrimination models is applied to naturally ordered outcome scales in medical prognosis. The ordering stage is assessed by a data-generated choice between ordered, partially ordered, and unordered models. The ordinal structure of the outcome is particularly taken into consideration in the construction of allocation rules and in the assessment of their performance. The specialized models are compared to the unordered model with respect to the classification efficiency in a clinical prognostic study. It is concluded that our approach offers more flexibility than the widely used cumulative-odds model and more stability than the multinomial logistic model. The procedure described in this paper is strongly recommended for practical applications to support medical decision-making.

Parallel Compact FFTs for Real Sequences

Eight algorithms for the in-place, compact, fast Fourier transform (FFT) of real and conjugate-symmetric sequences are presented for single instruction multiple data (SIMD) and multiple instruction multiple data (MIMD) distributed-memory multiprocessors. The “conditional ordering” is introduced for conjugate-symmetric sequences as the natural ordering that leads to minimal communication costs. For a transform of a real, naturally ordered, input sequence of length N distributed evenly across P processors, $2 + \log _2 P$ nearest neighbor communication exchanges of $\tfrac{1}{2}N/P$ contiguous elements (i-cycles) are necessary. For a transform of a conjugate-symmetric, conditionally ordered input sequence, $1 + \log _2 P$i-cycles are necessary. In all cases the computational complexity is $\tfrac{5}{2}N/ P\log _2 N$, and there are no extra memory requirements. The algorithms require about half the communication of a parallel FFT of a real sequence that involves a complex FFT and pre- or postprocessing. Communication complexity is based on the hypercube topology; the algorithms, however, can be implemented on general connection topologies. Some of the schemes are implemented on an nCUBE/2 MIMD multiprocessor, and timings are given.

Parallel elliptic preconditioners: Fourier analysis and performance on the connection machine

We study the performance of several widely used preconditioners for 2D elliptic partial differential equations (SSOR, ILU, MILU and polynomial preconditioners) with the natural and red-black orderings implemented on the Connection Machine (CM). Their performance is primarily influenced by two factors: the rate of convergence and the ease of parallelization. The convergence rate is analyzed by Fourier analysis and confirmed with experimental results. Although the naturally ordered SSOR and MILU preconditioners have convergence rates one order faster than the other preconditioners, the experiments show that the red-black ordered SSOR, ILU, MILU, polynomial preconditioners takes less execution time than their naturally ordered counterparts. This is partially due to the fact that the red-blavk ordering provides more parallelism than the natural ordering.

Further Symmetries of in-Place FFTs

It has long been known that an in-place version of the Fast Fourier Transform (FFT) exists for real sequences of data. More recently, in-place FFTs have been devised for real sequences with even, odd, or quarter wave symmetries. All of these symmetric FFTs take the input sequence in scrambled (bit-reversed) order and produce the transform sequence in natural order. For many applications, this is the opposite of what is needed, i.e., one would like to provide the input sequence in natural order. In this paper, an in-place version of the FFT is presented which takes a real sequence in natural order and produces the transform in scrambled order. The algorithm requires half of the operations and storage of the complex algorithms. Analogous in-place algorithms are also given for naturally ordered even, odd, quarter wave even and quarter wave odd sequences.

Naturally ordered transformation semigroups

The full transformation semigroup (Tx o) on the setX is studied with respect to the natural partial order which is defined on any regular semigruop. A characterization of this ordering in terms of images and kernels is given, the maximal and minimal elements are determined and the covering elements are described. Lower and upper bounds for two transformations are studied and necessary as well as sufficient conditions for their existence are gieven.

Maximum idempotents in naturally ordered regular semigroups

We shall denote by ω the natural partial order on the idempotents E = E(S) of a regular semigroup S, so that in E,A partially ordered semigroup S(≦) is called naturally partially ordered [9] if the imposed partial order ≦ extends ω in the sense thatNo assumption is made about the reverse implication.

Naturally ordered regular semigroups with a greatest idempotent

SynopsisWe consider ordered regular semigroups in which the order extends the natural order on the idempotents, and which are graced with the presence of a greatest idempotent. This implies that every element has a greatest inverse. An investigation into the properties ofthese special elements allows a description of Green's relations on the subsemigroup generated by the idempotents. This in turn leads to a complete description of the structure of idempotent-generated naturally ordered regular semigroups having a greatest idempotent. The smallest such semigroup that is not orthodox is also described. These results lead us to obtain structure theorems in the general case with the added condition that Green's relations be regular. Finally, necessary and sufficient conditions for such a semigroup to be a Dubreil-Jacotin semigroup are found.

Natural, Dyadic, and Sequency Order Algorithms and Processors for the Walsh-Hadamard Transform

The Walsh-Hadamard transform has recently received increasing attention in engineering applications due to the simplicity of its implementation and to its properties which are similar to the familiar Fourier transform. The transform matrices found so far to possess fast algorithms are the naturally ordered and dyadically ordered matrices, whose algorithms are similar to the Cooley-Tukey algorithm, and to the machine-oriented algorithm of Corinthios [2], respectively.

Naturally ordered bands

In the terminology of Clifford and Preston [2], a band B is a semigroup in which every element is idempotent. On such a semigroup there is a natural (partial) order relation defined by the ruleIf the order relation ≧ is compatible with the multiplication in B, in the sense that e ≧ f and g ≧ h together imply that eg ≧ fh, we shall say that B is a naturally ordered band. The object of this note is to describe the structure of naturally ordered bands.

On the convergence of ordered sets of projections

E. R. Lorch has shown [3, p. 223 ] that a uniformly bounded, naturally ordered sequence of projections in a reflexive Banach space converges to its supremum in the strong topology of operators. In this note it is shown that a uniformly bounded, naturally ordered set of projections in a Banach space X having a weak x-cluster point for each x CX converges to its supremum in the strong topology of operators. The method of proof is substantially different from that of Lorch. A bounded operator E, acting in a Banach space X, is a projection if E2 =E. Letting I denote the identity projection, E(X) and (I-E) (X) are strongly closed manifolds. The projections in X have a natural order: E?<E2 if ElE2=E2E, -EB. If { a } is a directed set, and a, < a2 implies Ea,, <Ea2, then BEa} is naturally ordered. If Y is a subset of X, w( Y) will denote the weak closure of Y. The span of Y, denoted by sp(Y), is the strongly closed linear manifold generated by Y. Suppose {EBa} is a set of projections in X. A projection E such that E(X) =sp(UaEa(X)) and (I-E)(X) =fna(I-Ea)(X) is the supremum of {Ea, }, and is denoted by UaEa. If { Ea } is naturally ordered, y, is a weak x-cluster point of { Ea} if yx G faw( { E:(x) I j _ a }). The central result of this note follows: