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  • 1
    Electronic Resource
    Electronic Resource
    The structure and Gel'fand patterns inherent in the Heisenberg supergenerator algebra of dual Liouville-space ||kq{K˜.}(k1−kn)B: [λ˜],Sn〉〉 tensors and the Cayley algebra of scalar invariants over a (k1−kn) field (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 1638-1650 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: For many-body spin cluster problems, dual-symmetry recoupled tensors over Liouville space provide suitable bases for a generalized torque formalism using the Sn-adapted density operator in which to discuss NMR and related techniques. The explicit structure of such tensors is considered in the context of the Cayley algebra of scalar invariants over a field, specified by the inner ki rank labels of the Tkq(kl−kn)s. The pertinence of both lexical combinatorial architectures over inner rank sets and SU2 propagative topologies in specifying the structure of dual recoupling tensors is considered in the context of the Sn partitional aspects of spin clusters. The form of Heisenberg superoperator generators whose algebra underlies the Gel'fand pattern algebra of SU(2) and SU(2)×Sn tensor bases over Liouville space is presented together with both the related s-boson algebras and a description of the associated {||2k 0〉〉} pattern sets of CF29H carrier space under the appropriate symmetry. These concepts are correlated with recent work on SU(2)×Sn induced symmetry hierarchies over Liouville spin space. The pertinence of this theoretical work to an understanding of multiquantum NMR in Liouville space formalisms is stressed in a discussion of the nature of pathways for intracluster J coupling, which also gives a valuable physical insight into the nature of coherence transfer in more general spin-1/2 systems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529276
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  • 2
    Electronic Resource
    Electronic Resource
    Demonstration of the origins of cooperativity within SU2×L 6 mapping over $$\left\{ {I\tilde H_\upsilon } \right\}$$ Liouvillian carrier subspaces characterising the MQ-NMR cluster [A]6(L 6) and its $$\{ |kq\upsilon :[\tilde \lambda ] 〉 〉 \} $$ tensorial bases (1992)
    Springer
    The European physical journal 89 (1992), S. 335-342 
    Details
    ISSN: 1434-6036
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The fundamental mappings over carrier subspace and substructures associated with $$\{ |kq\upsilon 〉 〉 \} $$ augmented spin algebras of Liouville space, and their mapping onto a subduced symmetry, are derived for [A]6(L 6) spin clusters within the combinatorial context of Rota-Cayley algebra over a field. Use of suitable lexical sets of combinatorialp-tuples (number partitions) over {|IM(M 1−M n )〉}M, followed by the subsequent use ofL n inner tensor product (ITP) algebra, allows the substructure of Liouville space to be derived. For SU2×L 6 mapping over the simply-reducible $$\left\{ {I\tilde H_\upsilon } \right\}$$ carrier subspaces, the $$D^k \left( {\tilde U} \right) \times \tilde \Gamma ^{\left[ {\tilde \lambda } \right]} \left( \upsilon \right)$$ (L 6) dual irreps, also arise as a consequence of the Liouville space recoupling termsv≡{k 1−k n } being distinct labels for $$\left\{ {I\tilde H_\upsilon } \right\}$$ which are themselves amenible to combinatorial analysis within the concept of Rota-Cayley algebra. Hence, theL n -induced symmetry aspects of multiquantum NMR density matrix formalisms and their dual $$\{ |kq\upsilon :[\tilde \lambda ] 〉 〉 \} $$ tensorial bases of spin cluster problems are derived and the nature of the cooperative, aspect between the individual symmetries comprising the duality is demonstrated, i.e. in the context of the operator bases of Liouville space. These practical arguments correlate, well with those based on an augmented boson pattern algebra derived from a Heisenburg algebra for superoperators, ℐ±,ℐ0. An earlier, treatment of conventional Hilbert space SU2×L 6 dualitycould only be realised in terms of standard SU2 boson algebra. Since the recoupling Rota-‘field’v for Liouville space is an explicit aspect of the dual mapping, a direct demonstration of cooperativity exists.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01318165
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  • 3
    Electronic Resource
    Electronic Resource
    Cooperative Liouvillian mapping over carrier subspaces and induced of NMR spin clusters as a consequence of simple reducibility being applicable to tensorial operator bases, (1992)
    Springer
    The European physical journal 88 (1992), S. 83-92 
    Details
    ISSN: 1434-6036
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The significance of simple-reducibility under groups being applicable to Liouville spin space {|k q v»} is considered in the context of induced symmetry and of mappings arising under cooperative symmetries. Some consequences of the realization of a further Heisenberg algebra, now under the {ℐ±, ℐ0} superoperators, are considered in terms of mappings and Gel'fand pattern algebras over the distinct carrier space, . The cooperative aspects of induced symmetry are shown to be a consequence of fundamental Wigner superoperators and the transformational properties of an additional Yamanouchi permutation index being applicable over subspaces. The inherent structure of Liouville space under a more general symmetry is summarized from the viewpoint of lexical combinatorial treatments of the (k 1−k n ) -fields inherent in (Rota)-Cayley algebra.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01573841
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  • 4
    Electronic Resource
    Electronic Resource
    On general forms for structure of some [n−1, 1] ⊗ [λ]L n n inner tensor products with 6 ≤n≤ 20, (60) forn even, in the context of spin cluster problems of multiquantum NMR (1993)
    Springer
    Journal of mathematical chemistry 13 (1993), S. 153-165 
    Details
    ISSN: 1572-8897
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Mathematics
    Notes: Abstract Consideration ofL n groups for n even and between 6 ≤n:≤ 20, (60) is consistent with the existence (over a simply reducible (SR) space) of certain similarities in the inner tensor product (ITP) structures associated with -p ≤ 2 [n -1, 1] ⊗ [λ.], or -⊗ [(n/2)2] products, and for general (even)n for [n -1, 1] ⊗ [n - 2, 12] ITPs, but not for ITPs involving higher generalm (p ≤ 3) components, as in [n-1, 1] ⊗ [n-m, m-1, 1].These observations provide considerable insight into the nature of van der Waals (3 ≤ n ≤ 20), metallic-, or “met-carb-” clusters andn = 12, 20 cage molecules analogous to dodecahedrane (a cage,n = 20 molecule) or13C buckminsterfullerene(ane) (n = 60) [A] n , [AX] n clusters, besides allowing for further combinatorial views on higherL n group characters and their associated group algebra. Mathematical insight to date into the nature of general ITPs involvingnon-SR direct sums has proved less fruitful on account of the number of component partitions spanned by specific TTP maps and their associated multiplicities.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01165561
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  • 5
    Electronic Resource
    Electronic Resource
    Summary ofS n mapping properties of some lexicalp-tuples, or wordlengths, over (M 1-M 2) fields of Rota-Cayley algebra in generating Hilbert space SO(3) xS n irreps of higherI i identical spin clusters, [A] n (S n ), forn ≤ 7 (1992)
    Springer
    Journal of mathematical chemistry 11 (1992), S. 341-352 
    Details
    ISSN: 1572-8897
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Mathematics
    Notes: Abstract Subdomain partitioning of n-fold fields associated withS n -scalar invariants of Rota-Cayley algebra and enumerative combinatorias are shown to provide direct access to the hierarchical substructure (inventory) of higher identical spin clusters. Such lexical methods based onp-tuples are especially valuable in treating higherI i spin cluster bases. On examining dimensionality criteria, in the context of inner tensor product (ITP) algebra, they clarify the nature of such spin bases under SO(3) ×S n dualities involving extended symmetric groups and their relatedS s ↓G subduced symmetries for 5 ≤ n ≤ 7 in the context of multiquantum NMR, using simple combinatorial arguments. The present work reports the conceptual aspects of monocluster substructures over Hilbert space underlying our recent papers on bicluster spin dynamics overS n -partitioned Liouville space. The substructure of [2D] n (S n ) spin systems over ([λ]) sets derived here may be utilized in direct product subsets ([λ]) q pertinent to the partitioned Liouvillians describing the spin dynamics of [11BD] 6 2− and [11BD] 7 2− cage ions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01164213
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  • 6
    Electronic Resource
    Electronic Resource
    Book reviews (1999)
    Springer
    Journal of mathematical chemistry 26 (1999), S. 267-269 
    Details
    ISSN: 1572-8897
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019142129746
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  • 7
    Electronic Resource
    Electronic Resource
    On mixed vs. exclusively-bosonic $$\left\{ {\left[ {\tilde \lambda } \right]:p \leqslant 2^2 } \right\}$$ (SU2×S n ) irrep sets over $$\left\{ \tilde {\mathbb{H}}_v \right\}$$ carrier subspaces, in the structure of (SU2 ×S n ⊗(SU2×S n )†)-derived Liouville space of NMR spin dynamics (1995)
    Springer
    Journal of mathematical chemistry 18 (1995), S. 91-95 
    Details
    ISSN: 1572-8897
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Mathematics
    Notes: Abstract The SU2 ×S n dual group algebra underlying Liouville NMR formalisms, which applies to extended spin cage-clusters [A] n of [AX] NMR spin systems, is briefly presented in terms of its mapping and superbosonic properties over the $$\left\{ \tilde {\mathbb{H}}_v \right\}$$ specialised carrier subspaces. By considering further both the origins of Liouville space as an augmented space based on inner tensor products of simple Hilbert space, and also the λSA-self-associate forms in terms of the contrasting structural aspects of the 8,12,16 ⩽n and 17, 20, ⩽n S n groups, it is shown that mixed boson/fermion sets of component irreps should only exist up to n 〈, 16. This property is traced to the need for the symmetric group irreps to span a set of $$\left\{ {\left[ {\tilde \lambda } \right]:p \leqslant 2^2 } \right\}$$ S n ) irreps corresponding to SU2 branching over the augmented SU2 × ς n , spin space. Parallels are drawn with the more general quonic algebras over simple Hilbert space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01166605
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  • 8
    Electronic Resource
    Electronic Resource
    Determinable partitional modelling of Cayleyan $$SU(m \leqslant n/2) \times S_n \downarrow \mathcal{G}NMR$$ spin symmetries of isotopomeric cage-clusters: SpecificS n≥12⊃...⊃D 3 group chains, forS n -modules in high-n weak-branching limit (1996)
    Springer
    Journal of mathematical chemistry 20 (1996), S. 311-329 
    Details
    ISSN: 1572-8897
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Mathematics
    Notes: Abstract The use over certain modestly branched (λ ⊢n) partitional models of Young'sS n -module decomposition algorithm in the high-n limit is considered for SU(m) x $$SU(m) \times S_n ( \downarrow \mathcal{G})$$ nuclear spin algebras associated with both NMR and ro-vibrational (R-V) aspects of specific cluster isotopomers. This approach allows additional dual-group projective mapping over simple Hilbert spaces to be derived from the natural embedding of higher finite groups in specificS n groups, for either the original simply-reducible (SR) SU(2)-, or various related higher non-SR SU(m) xS n , forms. The work arises from earlier interests in the NMR spin symmetry of the [11B1H] 12 2− borohydride anions and the nature of analogous ro-vibrational (R-V) spin statistical problems for highern ≥ 12-fold clusters. Here, the role of the scalar invariants is shown to be critical in determining the spin algebras of isotopomeric clusters within Cayley's theorem for some particular depth of SU(m) branching in the SU(m) xS n dual-group algebra. Certain additional quasi-geometric models for the full λ (≥ λSA (self-associate) dominant-sector set of (λ ⊢n) partitions-of-n are discussed, in the context of specific determinacy of natural $$S_n \supset \mathcal{G}$$ group embeddings at a given branching level.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01165351
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  • 9
    Electronic Resource
    Electronic Resource
    Symmetry substructures for MQ-NMR of [A]20 spin clusters of dodecahedranes, [13CH]20 or [13CD]20, metallic M @M20 and [H2O]+ @[H2O]20 exo-cage cluster molecules and their higher-n SO(3) ×l n spin algebras, within the context of mapping,l n -ITP algebras and number partitions (1993)
    Springer
    The European physical journal 25 (1993), S. 275-284 
    Details
    ISSN: 1434-6079
    Keywords: 76.60 ; 03.65.F ; 36.40
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract For both highern andI i ≧1 spin clusters, combinatorics provides powerful arguments with which to investigate the substructural forms of cluster spin algebras; this is especially so for SO(3) ×l n symmetries for 12≦n≦60 wherex (.) [λ] character tabulations become extensive. Bijective enumerative mappings over the combinatorialp-tuples (number partitions) afford insight into the general functionf(p,n) as well as into {|IM(I 1−I n [λ]〉} M -structure of spin algebras, even where the full details of the explicitx () [λ] (l n ) characters are not readily available. Both simply-reducible and higher aspects ofl n -inner tensor product (ITP) algebras are derived from dimensionality considerations, as part of combinatorial hooklength formalisms for $$\chi _{(1^n )}^{[\lambda ]} $$ . TheI i ≦3/2 forms of [A] n clusters forn≦20, (forp≦3, 4) of multiple-quantum NMR (MQ-NMR) are considered here as part of current interest in giant cage-clusters. In addition, the SU2 substructural hierarchy over Liouville space is derived for [A]20(l n ) (I i =1/2) spin cluster of the cage-cluster molecule dodecahedrane; aspects ofI i =1 spin cluster over {|IM (...)〉} space are derived as high temperature model of the exo-cage of [H2O]+ @[H2O]20 cluster ion; 20-fold higher-I i lusters provide models for M @M20 metal-clusters and further applications ofl 20-number partitions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01426891
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  • 10
    Electronic Resource
    Electronic Resource
    Spin symmetry and gel'fand patterns of higher SU2×ℓ n groups (1992)
    Springer
    The European physical journal 23 (1992), S. 187-193 
    Details
    ISSN: 1434-6079
    Keywords: 76.60.f ; 03.65 ; 02.10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The nature ofℓ 5 spin algebra is considered together withℓ 5↓ℓ 5↓C 4v mapping in order to specify the dual spin symmetry for MQ-NMR ofnido-11B5H9. The forms of Gel'fand shapes forℓ 5 spin symmetry are presented to show how they specify thefull range of multiplicities found in higher-n ℓ n -partitions. Tuples, or number-partitions and theirℓ n ↓G invariance sets provide models for both the five-foldI i =3/2 component, and the two distinct types of spin-1/2 subsystems. The full spin symmetry is derived in terms of the direct product ((ℓ 5↓C 4v )⊗(ℓ 4↓C 4v ))1/2⊗(ℓ 5↓C 4v )3/2. The concepts used are implicit in the substructure ofp-tuple model invariances over the subduced symmetry, or derive from the inner tensor product algebras under theℓ n group. Both as a check on the combinatorially derived multiplicities of [λ]s and for insight into (non-simply-reducible) substructure of number-partitions, the study of mapping from :hrr'.:-tuples onto theℓ n -partitional set for higherI i is invaluable. The motivation for this work lies in its pertinence to the MQ-NMR spin dynamics of clusterlike molecules. The accessible information content of a spin algebra over either form of spin space is bound up with a suitable symmetry partitioning of the problem, as implied by the use of {T kq (v:[λ])} bases within higherq subspaces of the Liouville formalism.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01436743
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