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  • 1
    Electronic Resource
    Electronic Resource
    Sextets in P1: The probabilistic theory of the six-phase structure invariant φ = φh + φk + φl + φm + φn + φp (1977)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 33 (1977), S. 697-701 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: It is assumed that a crystal structure P1 is fixed and that the 31 non-negative numbers R1, R2, . . ., R6; R12, R13, . . ., R56; R123, R124, . . ., R156 are also specified. The random variables (vectors) h, k, 1, m, n, p are assumed to be uniformly and independently distributed in the regions of reciprocal space defined by and |Eh| = R1, |Ek| = R2, . . . , |Ep| = R6; (1) |Eh + k| = R12, |Eh + l| = R13, . . . , |En + p| = R56; (2) |Eh + k + l| = R123, |Eh + k + m| = R124, . . . , |Eh + n + p| = R156; (3) and h + k + l + m + n + p = 0. (4) Then the structure invariant φ = φh + φk + φl + φm + φn + φp, as a function of the primitive random variables h, k, l, m, n, p, is itself a random variable, and its conditional probability distribution, given (1), (2) and (3), is derived from the main result of the previous paper [equation (3.1); Fortier & Hauptman (1977). Acta Cryst. A33, 694-696.] and compared with the distribution when only (1) is given. The distribution leads to estimates for cos φ in terms of the 31 magnitudes (1), (2) and (3).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S056773947700179X
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  • 2
    Electronic Resource
    Electronic Resource
    Quintets in P\overline{1}: probabilistic theory of the five-phase structure invariant in the space group P\overline{1} (1977)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 33 (1977), S. 829-833 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: It is assumed that a crystal structure in P{\bar 1} is fixed and that the 15 non-negative numbers R1, R2, R3, R4, R5; R12, R13, R14, R15, R23, R24, R25, R34, R35, R45 are also specified. The random vector (h, k, l, m, n) is assumed to be uniformly distributed over the subset of the fivefold Cartesian product W × W × W × W × W of reciprocal space W defined by |Eh| = R1, |Ek| = R2, |El| = R3, |Em| = R4, |En| = R5; (1) |Eh + k| = R12, |Eh + l| = R13, |Eh + m| = R14, |Eh + n| = R15, |Ek + l| = R23 |Ek + m| = R24, |Ek + n| = R25, |El + m| = R34, |El + m| = R35, |Em + n| = R45; (2) and h + k + l + m + n = 0. (3) Then the structure invariant (φ = φh + φk + φl + φm + φn, (4) as a function of the primitive random variables h, k, l, m, n, is itself a random variable, and its conditional probability distribution, given (1) and (2), is derived. The distribution yields reliable estimates for large numbers of quintets φ in terms of the 15 magnitudes (1) and (2).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739477002009
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  • 3
    Electronic Resource
    Electronic Resource
    A conditional probability distribution of the structure semivariant φh + φk in P\overline{1}:Effects of higher-order terms (1976)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 32 (1976), S. 940-944 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A crystal structure in P{\bar 1} consisting of N identical atoms in the whole unit cell is fixed, and the four non-negative numbers R1, R2, R12/2, R_{1{\bar 2}/2} are also specified. The random variables (vectors) h, k are assumed to be uniformaly and independently distributed in the regions of reciprocal space defined by |Eh| = R1, |Ek| = R2, |E(h + k)/2| = R12/2, |Eh - k/2| = R_{1{\bar 2}/2}, (1), and h + k ≡ 0 (mod ωs), (2), where ωs, the seminvariant modulus for P{\bar 1}, is the three-dimensional vector ωs = (2,2,2), (3). Then the components of each of h ± k)/2 are integers. In view of (2) and (3) the linear combination of the phases φ = φh + φk, (4), is a structure seminvariant which, as a function of the primitive random variables h, k, is itself a random variable. Two approximations Q±, P±, of respective orders 1/N, 1/N2, to the conditional probability distribution of φ, given the four magnitudes (1), are derived and compared. In favorable cases, i.e. when the variance of the distribution happens to be small, they yield a reliable estimate (0 or π) for the structure seminvariant φ.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739476012333
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  • 4
    Electronic Resource
    Electronic Resource
    A sequence of nested neighborhoods of the structure invariant φh + φk + φl + φm (1977)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 33 (1977), S. 553-555 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A sequence of nested neighborhoods of the structure invariant φ = φh + φk + φl + φm is derived. Each neighborhood is a subset of the succeeding ones and consists of the small number of structure factor magnitudes |E| upon which, in favorable cases, the value of φ mostly depends. Thus the stage is set for deriving reliable estimates for φ in terms of known magnitudes |E| via appropriate conditional probability distributions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739477001429
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  • 5
    Electronic Resource
    Electronic Resource
    The probabilistic theory of the structure invariants φh + φk + φl + φm in P\overline{1} (1977)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 33 (1977), S. 565-568 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Probability distributions associated with the third, fourth, fifth and sixth neighborhoods of the structure invariant φlmφh + φk + φl + φm are derived. These distributions yield estimates for φlm in terms of magnitudes |E| alone which are particularly good in the favorable case that the standard deviations of the distributions are small. The estimates for the φlm lead in turn to unique values for the individual phases φ even for very complex structures. Conditional distributions of φlm given the values of one or more structure invariants φpq, φrs, φtu and appropriate sets of magnitudes |E| are also described.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739477001442
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  • 6
    Electronic Resource
    Electronic Resource
    Quintets: a joint probability distribution of fifteen structure factors (1977)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 33 (1977), S. 572-575 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: It is assumed that a crystal structure in P1 is fixed and that the random variables (vectors) h, k, l, m, n are uniformly and independently distributed over the subset of reciprocal space defined by h + k + l + m + n = 0. Then the 15 structure factors Eh, Ek, El, Em, En, Eh + k, Eh + l, Eh + m, Eh + n, Ek + l, Ek + m, Ek + n, El + m, El + n, Em + n as functions of the primitive random variables h, k, l, m, n, are themselves random variables, and their joint probability distribution is found. This distribution plays the central role in the theory and estimation of the five-phase structure invariants φh + φk + φl + φm + φn.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739477001466
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  • 7
    Electronic Resource
    Electronic Resource
    Sextets in P1: The joint probability distribution of thirty-one structure factors (1977)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 33 (1977), S. 694-696 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: It is assumed that a crystal structure in P1 is fixed and that the random variables (vectors) h, k, l, m, n, p are uniformly and independently distributed over the subset of reciprocal space defined by h + k +l + m + n + p = 0. Then the 31 structure factors Eh, Ek, . . . ., Ep; Eh + k, Eh + l, . . ., En + p; Eh + k + l, Eh + k + m, . . ., Eh + n + p, as functions of the primitive random variables h, k, l, m, n, p, are themselves random variables, and their joint probability distribution is found. This distribution plays the central role in the theory and estimation of the six-phase structure invariant φh + φk + φl + φm + φn + φp
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739477001788
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  • 8
    Electronic Resource
    Electronic Resource
    Conditional probability distributions of the four-phase structure invariant φh + φk + φl + φm in P\overline{1} (1976)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 32 (1976), S. 45-49 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A crystal structure in P{\bar 1} is assumed to be fixed and the seven non-negative numbers R1, R2, R3, R4, R12, R23, R31 are also given. It is assumed that h, k, l, m are random variables uniformly and independently distributed over the subsets of reciprocal space defined by |Eh| = R1, |Ek| = R2, |El| = R3, |Em| = R4, (1) |Eh + k| = R12, |Ek + l| = R23, |El + h| = R31, (2) and h + k + l + m = 0. (3) Then the structure invariant φ = φh + φk + φl + φm (4) is a function of the primitive random variables h, k, l, m. The conditional probability distribution of φ, given (1) and (2), is obtained and compared with the conditional probability distribution of φ when only (1) is given. Some calculations are presented which show the usefulness of the distribution, given (1) and (2), in estimating the value of φ.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739476000077
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  • 9
    Electronic Resource
    Electronic Resource
    A heuristic study of neighborhoods of the structure seminvariants in the space group P\overline{1} (1976)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 32 (1976), S. 934-940 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: The first two neighborhoods of each of the structure seminvariants φh, φh + φk, φh + φk + φl, φh + φk + φl + φm in the space group P{\bar 1} are found. Not only do these neighborhoods identify the magnitudes of the normalized structure factors |E| most intimately related to the value of the associated structure seminvariant, but the analysis suggests in a qualitative way what the nature of this relation must be. Thus the stage is set for determining the conditional probability distribution of a structure seminvariant, given the magnitudes |E| of a corresponding neighborhood (or an appropriate subset), and this leads in turn to a probabilistic estimate for the seminvariant in terms of a suitably chosen small set of magnitudes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739476001964
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  • 10
    Electronic Resource
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    Pairs in P21: probability distributions which lead to unique estimates of the two-phase structure seminvariants in the interval (–π,π) (1978)
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 34 (1978), S. 230-241 
    Details
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Conditional probability distributions of the two-phase structure seminvariant φ12 = φh1kl1 - φh2kl2 in P21, given the values of one or more two-phase structure seminvariants, as well as appropriately chosen sets of structure factor magnitudes |E|, are derived. These distributions yield unique and reliable estimates, in the whole interval (-π,π), of the values of these seminvariants. In the case that a given structure seminvariant specifies the enantiomorph, the values of the remaining seminvariants are consistent with this choice of enantiomorph. In any event, the variance of any distribution of the kind described here is much smaller, in favorable cases, than that of a distribution which assumes as known magnitudes |E| alone, reflecting the fact that previously derived phase information severely limits the possible values of other phases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1107/S0567739478000455
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