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  • Articles  (15)
  • CR: 5.14  (8)
  • 06A10  (7)
  • 2015-2019
  • 1980-1984  (15)
  • 1970-1974
  • 1950-1954
  • 1940-1944
  • 1984  (15)
  • Mathematics  (15)
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  • Articles  (15)
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  • 2015-2019
  • 1980-1984  (15)
  • 1970-1974
  • 1950-1954
  • 1940-1944
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  • Mathematics  (15)
  • 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
    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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  • 8
    Electronic Resource
    Electronic Resource
    An alternative givens ordering (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 83-90 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F05 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A new Givens ordering is shown, empirically and by an approximate theoretical analysis, to take appreciably fewer stages than the standard scheme. Sharper error bounds than Gentleman's ensue, and the scheme is better suited to parallel computation. Other schemes, less efficient but more easily analysed, are discussed. The effect of a possible limit in practice on the number of simultaneous computations is considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389639
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  • 9
    Electronic Resource
    Electronic Resource
    An accelerated bisection method for the calculation of eigenvalues of a symmetric tridiagonal matrix (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 153-160 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 15A18 ; 15-04 ; 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present a method for the determination of eigenvalues of a symmetric tridiagonal matrix which combines Givens' Sturm bisection [4, 5] with interpolation, to accelerate convergence in high precision cases. By using an appropriate root of the absolute value of the determinant to derive the interpolation weight, results are obtained which compare favorably with the Barth, Martin, Wilkinson algorithm [1].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389644
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  • 10
    Electronic Resource
    Electronic Resource
    AB-Algorithm and its modifications for the spectral problems of linear pencils of matrices (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 329-342 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F15 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper describes and algorithm and its modifications for solving spectral problems for linear pencils of matrices both regular as well as singular.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390177
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  • 11
    Electronic Resource
    Electronic Resource
    Applications of shortest path algorithms to matrix scalings (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 111-126 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F35 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A symmetric scaling of a nonnegative, square matrixA is a matrixXAX −1, whereX is a nonsingular, nonnegative diagonal matrix. By associating a family of weighted directed graphs with the matrixA we are able to adapt the shortest path algorithms to compute an optimal scaling ofA, where we call a symmetric scalingA′ ofA optimal if it minimizes the maximum of the ratio of non-zero elements.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389759
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  • 12
    Electronic Resource
    Electronic Resource
    Zur Kondition des linearen Ausgleichsproblems mit linearen Gleichungen als Nebenbedingungen (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 139-152 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F20 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary F.L. Bauer has treated in several papers [1, 3, 4] the condition related to the solution of linear equations and to the algebraic eigenvalue problem. We study the condition for the linear least squares problem with linear equality constraints (problem LSE). A perturbation theory of problem LSE is presented and three condition numbers are defined. Problem LSE includes the linear least squares problem (problem LS). There are examples with identical solution of problem LSE and of problem LS. Sometimes the condition of problem LSE is better and sometimes the condition of problem LS is better. Several numerical tests illustrate the theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389762
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  • 13
    Electronic Resource
    Electronic Resource
    Eigenvector matrices of symmetric tridiagonals (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 103-110 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) 6540 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A simple test is given for determining whether a given matrix is the eigenvector matrix of an (unknown) unreduced symmetric tridiagonal matrix. A list of known necessary conditions is also provided. A lower bound on the separation between eigenvalues of tridiagonals follows from our Theorem 3.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389758
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  • 14
    Electronic Resource
    Electronic Resource
    On neighbouring matrices with quadratic elementary divisors (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 1-21 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F 15 ; 65F20 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents new results on matrices with quadratic elementary divisors neighbouring a given matrixA and relates them to those given by Kakan. It shows that all results of this kind may give severe overestimates of the distance to the nearest defective matrix and that matrices with elementary divisors of degrees higher than two are often at a much greater distance than those with quadratic divisors.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389751
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  • 15
    Electronic Resource
    Electronic Resource
    Comparisons of regular splittings of matrices (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 23-35 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this article, new comparison theorems for regular splittings of matrices are derived. In so doing, the initial results of Varga in 1960 on regular splittings of matrices, and the subsequent unpublished results of Woźnicki in 1973 on regular splittings of matrices, will be seen to be special cases of these new comparison theorems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389752
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