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  • Articles  (12)
  • 06A10  (7)
  • cluster analysis  (5)
  • 2015-2019
  • 1980-1984  (12)
  • 1970-1974
  • 1950-1954
  • 1940-1944
  • 1984  (12)
  • Mathematics  (12)
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  • Articles  (12)
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  • 2015-2019
  • 1980-1984  (12)
  • 1970-1974
  • 1950-1954
  • 1940-1944
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  • 1
    Electronic Resource
    Electronic Resource
    Maximum antichains in the product of chains (1984)
    Springer
    Order 1 (1984), S. 21-28 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; 05A05 ; Partially ordered sets ; Sperner's Theorem ; LYM property ; product of chains
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let P be the poset k 1 × ... × k n , which is a product of chains, where n≥1 and k 1≥ ... ≥k n ≥2. Let $$M = k_1 - \sum\nolimits_{i = 2}^n {(k_i - 1)} $$ . P is known to have the Sperner property, which means that its maximum ranks are maximum antichains. Here we prove that its maximum ranks are its only maximum antichains if and only if either n=1 or M≤1. This is a generalization of a classical result, Sperner's Theorem, which is the case k 1= ... =k n =2. We also determine the number and location of the maximum ranks of P.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00396270
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  • 2
    Electronic Resource
    Electronic Resource
    Finite cutsets and finite antichains (1984)
    Springer
    Order 1 (1984), S. 35-46 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; Ordered set ; chain ; antichains ; width ; cutset
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An ordered set (P,≤) has the m cutset property if for each x there is a set Fx with cardinality less than m, such that each element of Fx is incomparable to x and {x} ∪ Fx meets every maximal chain of (P,≤). Let n be least, such that each element x of any P having the m cutset property belongs to some maximal antichain of cardinality less than n. We specify n for m 〈 w. Indeed, n-1=m= width P for m=1,2,n=5 if m=3 and n⩾ℵ1 if m ≥4. With the added hypothesis that every bounded chain has a supremum and infimum in P, it is shown that for 4⩽m⩽ℵ0, n=ℵ0. That is, if each element x has a finite cutset Fx, each element belongs to a finite maximal antichain.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00396272
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  • 3
    Electronic Resource
    Electronic Resource
    Minimizing the jump number for partially ordered sets: A graph-theoretic approach (1984)
    Springer
    Order 1 (1984), S. 7-19 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; 05C20 ; 68C25 ; Partial order ; linear extension ; jump number ; line digraph ; cyclomatic number ; spanning branching ; Eulerian digraph
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The purpose of this paper is to present a graph-theoretic approach to the jump number problem for N-free posets which is based on the observation that the Hasse diagram of an N-free poset is a line digraph. Therefore, to every N-free poset P we can assign another digraph which is the root digraph of the Hasse diagram of P. Using this representation we show that the jump number of an N-free poset is equal to the cyclomatic number of its root digraph and can be found (without producing any linear extension) by an algorithm which tests if a given poset is N-free. Moreover, we demonstrate that there exists a correspondence between optimal linear extensions of an N-free poset and spanning branchings of its root digraph. We provide also another proof of the fact that optimal linear extensions of N-free posets are exactly greedy linear extensions. In conclusion, we discuss some possible generalizations of these results to arbitrary posets.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00396269
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  • 4
    Electronic Resource
    Electronic Resource
    Powers of ordered sets (1984)
    Springer
    Order 1 (1984), S. 83-92 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; Ordered sets ; exponentiation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Recently there has been significant progress in the study of powers of ordered sets. Much of this work has concerned cancellation laws for powers and uses these two steps. First, logarithmic operators are introduced to transform cancellation problems for powers into questions involving direct product decompositions. Second, refinement theorems for direct product decompositions are brought to bear. Here we present two results with the aim of highlighting these steps.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00396275
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  • 5
    Electronic Resource
    Electronic Resource
    Balancing poset extensions (1984)
    Springer
    Order 1 (1984), S. 113-126 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; 68E05 ; Sorting ; comparison ; information theoretic bound ; linear extension
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We show that any finite partially ordered setP (not a total order) contains a pair of elementsx andy such that the proportion of linear extensions ofP in whichx lies belowy is between 3/11 and 8/11. A consequence is that the information theoretic lower bound for sorting under partial information is tight up to a multiplicative constant. Precisely: ifX is a totally ordered set about which we are given some partial information, and ife(X) is the number of total orderings ofX compatible with this information, then it is possible to sortX using no more thanC log2 e (X) comparisons whereC is approximately 2.17.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00565647
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  • 6
    Electronic Resource
    Electronic Resource
    Ordered sets with no chains of ideals of a given type (1984)
    Springer
    Order 1 (1984), S. 159-172 
    Details
    ISSN: 1572-9273
    Keywords: 06A05 ; 06A10 ; Ordered sets ; ideals ; better quasi-order
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the possible order types of chains of ideals in an ordered set. Our main result is this. Given an indecomposable countable order type α, there is a finite listA 1 α , ...,A n α of ordered sets such that for every ordered setP the setJ(P) of ideals ofP, ordered by inclusion, contains a chain of type α if and only ifP contains a subset isomorphic to one of theA 1 #x03B1; , ...,A n α . The finiteness of the list relies on the notion of better quasi-ordering introduced by Nash-Williams and the properties of scattered chains obtained by Laver.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00565651
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  • 7
    Electronic Resource
    Electronic Resource
    A Bayesian algorithm for global optimization (1984)
    Springer
    Annals of operations research 1 (1984), S. 111-128 
    Details
    ISSN: 1572-9338
    Keywords: Global optimization ; Bayesian nonparametric inference ; random distributions ; cluster analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract A crucial step in global optimization algorithms based on random sampling in the search domain is decision about the achievement of a prescribed accuracy. In order to overcome the difficulties related to such a decision, the Bayesian Nonparametric Approach has been introduced. The aim of this paper is to show the effectiveness of the approach when an ad hoc clustering technique is used for obtaining promising starting points for a local search algorithm. Several test problems are considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01876142
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  • 8
    Electronic Resource
    Electronic Resource
    Compactness and subsets of ordered sets that meet all maximal chains (1984)
    Springer
    Order 1 (1984), S. 147-157 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; 54D30 ; Maximal chain ; cutset ; chain complete ; antichain ; cofinal subset
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract LetP be a chain complete ordered set. By considering subsets which meet all maximal chains, we describe conditions which imply that the space of maximal chains ofP is compact. The symbolsP 1 andP 2 refer to two particular ordered sets considered below. It is shown that the space of maximal chains ℳ (P) is compact ifP satisfies any of the following conditions: (i)P contains no copy ofP 1 or its dual and all antichains inP are finite. (ii)P contains no properN and every element ofP belongs to a finite maximal antichain ofP. (iii)P contains no copy ofP 1 orP 2 and for everyx inP there is a finite subset ofP which is coinitial abovex. We also describe an example of an ordered set which is complete and densely ordered and in which no antichain meets every maximal chain.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00565650
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  • 9
    Electronic Resource
    Electronic Resource
    A new cluster algorithm for orientation data (1984)
    Springer
    Mathematical geology 16 (1984), S. 139-153 
    Details
    ISSN: 1573-8868
    Keywords: cluster analysis ; numerical taxonomy ; natural classification ; multimodality ; density estimation ; orientation data ; fabric diagrams
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract An algorithm to classify data points on the sphere in distinct cluster groups is defined. The characteristics of the cluster groups and the rule for assigning data to the groups are related to a continuous differentiable density estimation. The modes of the estimated density are assumed to be representative of the groups; data points are then assigned to the mode reached by the steepest ascent. The major advantage of this procedure is its sensitivity in detecting cluster groups independently of their geometry and configuration. As a consequence, the procedure is capable of handling orientation data that may be arranged in girdles.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01032213
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  • 10
    Electronic Resource
    Electronic Resource
    An example of cluster analysis applied to a large geologic data set: Aerial radiometric data from Copper Mountain, Wyoming (1984)
    Springer
    Mathematical geology 16 (1984), S. 479-498 
    Details
    ISSN: 1573-8868
    Keywords: aerial radiometrics ; cluster analysis ; k-means analysis ; principal components analysis ; Copper Mountain ; Wyoming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract One objective of the aerial radiometric surveys flown as part of the U.S. Department of Energy's National Uranium Resource Evaluation (NURE) program was to ascertain the spatial distribution of near-surface radioelement abundances on a regional scale. Some method for identifying groups of observations with similar γ-ray spectral signatures and radioelement concentration values was therefore required. It is shown in this paper that cluster analysis can identify such groups with or without a priori knowledge of the geology of an area. An approach that combines principal components analysis with convergentk-means cluster analysis is used to classify 6991 observations (each observation comprising three radiometric variables) from the Precambrian rocks of the Copper Mountain, Wyoming area. This method is compared with a convergentk-means analysis that utilizes available geologic knowledge. Both methods identify four clusters. Three of the clusters represent background values for the Precambrian rocks of the area, and the fourth represents outliers (anomalously high214Bi). A segmentation of the data corresponding to “geologic reality” as interpreted by other methods has been achieved by perceptive quantitative analysis of aerial radiometric data. The techniques employed are composites of classical clustering methods designed to handle the special problems presented by large data sets.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01886328
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  • 11
    Electronic Resource
    Electronic Resource
    Application of fuzzy clustering and fuzzy classification to evaluate the provenance of glacial till (1984)
    Springer
    Mathematical geology 16 (1984), S. 283-301 
    Details
    ISSN: 1573-8868
    Keywords: Bayesian classifiers ; classification ; cluster analysis ; cluster validity ; fuzzyc means ; fuzzyc varieties ; fuzzy covariances ; geochemical prospecting ; glacial till ; kriging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract To evaluate the provenance of glacial till, the trace element content of magnetite was used. Magnetite was present in all known rock types and all till samples in the area investigated. By using fuzzy-set theory it was possible to group samples of magnetite taken from bedrock into relatively homogeneous and geologically meaningful groups and also, by fuzzy classification, to relate the till samples to the rocks in such a way that the relative contribution of each rock type to the till is estimated. Each rock and till sample is assigned a membership value between 0 and 1 for each rock type. The membership values, for a certain rock type in the till, are then interpolated by kriging onto maps. Magnetites from skarns associated with sulfide ores especially are rather distinct, and so a map of such membership values for till unveils all known ore deposits some 1–5 km “downstream”in the general direction of the ice flow. Other anomalies show up which cannot be related to hitherto known ores or mineralizations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01032692
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  • 12
    Electronic Resource
    Electronic Resource
    A method for stratigraphic correlation of several boreholes (1984)
    Springer
    Mathematical geology 16 (1984), S. 393-406 
    Details
    ISSN: 1573-8868
    Keywords: geophysical logs ; multivariate statistics ; cluster analysis ; linear programming ; dynamic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The borehole slotting method previously proposed by Gordon and Reyment for two boreholes may be extended to three or more boreholes, but soon becomes computationally intractable. We propose another method based on the mixture model of cluster analysis, the computational labor of which increases linearly with the number of boreholes. This method produces profiles of the unknown strata to which the individual boreholes are easily matched by the two-borehole slotting method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01029888
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