For unsymmetric tridiagonal matrices one can compute the eigendecomposition using a similarity transformation.
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Vibronic states that are symmetric with respect to i are denoted g for (German for "even"), and unsymmetric states are denoted u for (German for "odd").
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Y. C. Hon and R. Schaback, "On unsymmetric collocation by radial basis functions," Applied Mathematics and Computation, vol. 119, pp. 177–186, 2001. In addition, the mixed boundary conditions also destroy the symmetry of its interpolation matrix. Refs.
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As symmetric dipolarophiles have one orientation for cycloaddition, only one regioisomer, but multiple stereoisomers can be obtained. On the contrary, unsymmetric dipolarophiles can have multiple regioisomers and stereoisomers. These regioisomers and stereoisomers may be predicted based on frontier molecular orbital (FMO) theory, steric interactions, and stereoelectronic interactions. Scheme 9.
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Another class of methods builds on the unsymmetric Lanczos iteration, in particular the BiCG method. These use a three-term recurrence relation, but they do not attain the minimum residual, and hence the residual does not decrease monotonically for these methods. Convergence is not even guaranteed. The third class is formed by methods like CGS and BiCGSTAB.
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Functionalized matrix detergent MALDI matrix compounds such as α-cyano-4-hydroxycinnamic acid have been linked through a linker consisting of an unsymmetric formaldehyde acetals to dodecanol. This type of cleavable detergent is inherently compatible with MALDI and does not have to be removed prior to analysis. UV light- or fluoride-cleavable surfactants have also been developed but are not in current use.
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Individual flowers have four petals that are rose- purple to pink, rarely white. The petals are free at the base rather than united in a floral tube, as in Epilobium. The lower petals are narrower than the upper ones, making the flower radially unsymmetric (zygomorphic). There are eight more-or-less equally sized stamens, and a long, four-lobed style.
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The resulting materials have rigidly planar, highly conjugated cores. They exhibit good processing characteristics for fabrication of soft electronic devices. Aside from the NDIs, other members include the diimide derivatives of perylene-3,4:9,10-tetracarboxylic dianhydride and terrylene-3,4:11,12-tetracarboxylic dianhydride. Synthesis of symmetric and unsymmetric NDIs Naphthalene diimides (NDIs) are often fluorescent, although the intensity is sensitive to substituents.
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Synthesis of symmetric and unsymmetric NDIs. NTCDI is redox-active, forming stable radical anions near -1.10 V vs. Fc/Fc+. Its ability to accept electrons reflects the presence of an extended conjugated ring system and the electron withdrawing groups (carbonyl centers). NDI is used in supramolecular chemistry owing to its tendency to form charge-transfer complexes with crown ethers, e.g.
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The first reported decarboxylative cross coupling reaction was an Ullmann reaction, in 1966 by Nilsson et al. Thermal decarboxylation of copper benzoates, in the presence of an aryl halide, was found to produce (both symmetric and unsymmetric) biaryls through aryl-Cu intermediates.Nilsson M. A new biaryl synthesis illustrating a connection between the Ullmann biaryl synthesis and copper-catalyzed decarboxylation. Acta Chem Scand, 1966, 20: 423–426.
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The Arnoldi iteration reduces to the Lanczos iteration for symmetric matrices. The corresponding Krylov subspace method is the minimal residual method (MinRes) of Paige and Saunders. Unlike the unsymmetric case, the MinRes method is given by a three-term recurrence relation. It can be shown that there is no Krylov subspace method for general matrices, which is given by a short recurrence relation and yet minimizes the norms of the residuals, as GMRES does.
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As definitive as Smith and Baer's early mechanism seems, there are several phenomena that it did not account for. The problem with their mechanism mainly focused around TDR precursors that have alkyl substituents on the ring. When said substituent is placed on the ring as to make the molecule still symmetric, one product is formed upon exposure to TDR conditions. However, if the alkyl is placed on the ring as to make the molecule unsymmetric, several products could form.
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The unsymmetric nature of the ligand and square planar coordination thereof gives rise to two possible geometric isomers: a cis and a trans form. Multiple ways of differentiating the geometric isomers exist, an easily accesible one being IR spectroscopy with the characteristic number of C–N, C–O, and CuII–N identifying the ligand configuration. Crystal appearance may also be of some value for isomer indication, though the ultimate diagnostic technique is X-ray crystallography. All forms of the complex have been characterized crystallographically, the most commonly isolated one being the cis monohydrate (x = 1).
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A transformation that reduces a general matrix to Hessenberg form will reduce a Hermitian matrix to tridiagonal form. So, many eigenvalue algorithms, when applied to a Hermitian matrix, reduce the input Hermitian matrix to (symmetric real) tridiagonal form as a first step. A tridiagonal matrix can also be stored more efficiently than a general matrix by using a special storage scheme. For instance, the LAPACK Fortran package stores an unsymmetric tridiagonal matrix of order n in three one-dimensional arrays, one of length n containing the diagonal elements, and two of length n − 1 containing the subdiagonal and superdiagonal elements.
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These magnetic susceptibility studies are consistent with the presence of a mononuclear Cr3+ center and an unsymmetric trinuclear Cr3+ assembly with bridging oxo ligands. In chromodulin isolated from bovine liver, x-ray absorption spectroscopy studies have shown that the chromium (III) atoms are surrounded by 6 oxygen atoms with an average Cr—O distance of 1.98 Å, while the distance between 2 chromium (III) atoms is 2.79 Å. These results are indicative of a multinuclear assembly. No sulfur ligands coordinate to chromium and instead, it has been proposed that a disulfide linkage between 2 cysteine residues occurs owing to a characteristic peak at 260 nm.
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The Paternò–Büchi reaction, named after Emanuele Paternò and George Büchi who established its basic utility and form, is a photochemical reaction that forms four-membered oxetane rings from an excited carbonyl and reacting with an alkene. Here an electronically excited carbonyl group is added to a ground state olefin yielding an oxetane. With substrates benzaldehyde and 2-methyl-2-butene the reaction product is a mixture of structural isomers: :unsymmetric reaction Another substrate set is benzaldehyde and furan The alternative strategy for the above reaction is called the Transposed Paternò−Büchi reaction. The aza-equivalent of the above reactoin is the Aza Paternò−Büchi reaction.
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In the general case of rotation of an unsymmetric body, which has different values of the moment of inertia about the three principal axes, the rotational motion can be quite complex unless the body is rotating around a principal axis. As described in the tennis racket theorem, rotation of an object around its first or third principal axis is stable, while rotation around its second principal axis (or intermediate axis) is not. The motion is simplified in the case of an axisymmetric body, in which the moment of inertia is the same about two of the principal axes. These cases include rotation of a prolate spheroid (the shape of an American football), or rotation of an oblate spheroid (the shape of a pancake).
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The first EMACs with three metal atoms were synthesized in the early 1990s independently by the groups of Shie-Ming Peng (NTU) and F. Albert Cotton (Texas A&M;), who coined the term extended metal atom chains. The cobalt-containing molecule Co3(dpa)4Cl2 (dpa = 2,2'-dipyridylamide) was synthesized by both research groups, but each proposed a different structure: the group from Taiwan reported an unsymmetric structure with a long and a short Co-Co bond, whereas the Texas group identified a symmetric structure with equal Co-Co bond lengths. This disagreement sparked a controversy that lasted for years, until it was realized that both forms of the molecule actually exist simultaneously. While this debate led to the realization that the compound can be used as a molecular switch, it also created a new problem since none of the recognized types of isomerism could explain the existence of a molecule in two structural forms that differ only in the length of one or more bonds (and not in their stereochemistry or connectivity of the atoms).
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