Stereoselective photo-rearrangement of a pyrroline 1-oxide

Bapat, Jayant; Black, D. St. C.

doi:10.1039/c19670000073

Cited by 7 publications

(7 citation statements)

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“…Xiao and Gutman 4 determined certain properties of R ( G ), including the determinant of R ( G ). Bapat 17 generalized their work by obtaining formulae for the inverse and determinant of R ( G ) for weighted graphs.…”

Section: Introductionmentioning

confidence: 99%

Resistance distance and Kirchhoff index in circulant graphs

Zhang

Yang

2006

Int J of Quantum Chemistry

123

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ABSTRACT:The resistance distance r ij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf (G) is the sum of resistance distances between all pairs of vertices. In this work, closed-form formulae for Kirchhoff index and resistance distances of circulant graphs are derived in terms of Laplacian spectrum and eigenvectors. Special formulae are also given for four classes of circulant graphs complete graphs, complete graphs minus a perfect matching, cycles, Möbius ladders M p . In particular, the asymptotic behavior of Kf

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Section: Introductionmentioning

confidence: 99%

Resistance distance and Kirchhoff index in circulant graphs

Zhang

Yang

2006

Int J of Quantum Chemistry

123

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“…Lemma 3.4 ([ 10, 15). Let Γ be a graph with discrete Laplacian L and the resistance function r(x,y), and let H be a generalized inverse of L, i.e.…”

Section: The Discrete Laplacian L and The Pseudo Inverse L+mentioning

confidence: 99%

“…In another direction, Weinberg 8, Klein and Randić 9, Corollary C1], and Bapat 10, Lemma 2] proved identities that are equivalent to the Foster's first identity. The properties of discrete Laplacian and its pseudo inverse were used in Refs 9 and 10.…”

Section: Introductionmentioning

confidence: 99%

“…Both of the proofs rely on the properties of the discrete Laplacian and its pseudo inverse. In the first proof, we use a remarkable relation between the resistance values and the pseudo inverse L + of the discrete Laplacian 10, 15 (see also Lemma 3.4) to express the voltage values by using L + (see Lemma 3.5). Similarly, in the second proof, we use an equally interesting relation between the resistance values and some “equilibrium measures” ν i 's 14, Proposition 2.1] to express the voltage values by using ν i 's (see Proposition 4.4).…”

Section: Introductionmentioning

confidence: 99%

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Generalized Foster's identities

Cinkir

2011

Int J of Quantum Chemistry

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“…Now we use the following remarkable relation (see18 or Theorem A in1): Recall that R i is defined as r

_{Γ − italice_{italici}}

( p i , q i ), so we have Note that R i is independent of p and q if p , q , p i , and q i are four distinct vertices.…”

Section: The Deletion and The Contraction Identities For The Resismentioning

confidence: 99%

A new tool for analysis of channel membrane proteins

2010

Biotech & Bioengineering

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Objective The anatomic basis for joint disease localization in polymyalgia rheumatica (PMR) is poorly understood. This study used contrast‐enhanced and fat suppression magnetic resonance imaging (MRI) to evaluate the relationship between synovial and extracapsular inflammation in PMR and early rheumatoid arthritis (RA). Methods Ten patients with new‐onset PMR and 10 patients with early RA underwent dynamic contrast‐enhanced MRI and conventional MRI of affected metacarpophalangeal (MCP) joints. Synovitis and tenosynovitis were calculated based on the number of enhancing voxels, initial rate of enhancement, and maximal enhancement of gadolinium diethylenetriaminepentaacetic acid (Gd‐DTPA). Periarticular bone erosion and bone edema were scored according to the OMERACT (Outcome Measures in Rheumatology Clinical Trials) scoring system in both groups. The degree of extracapsular Gd‐DTPA enhancement was assessed in both conditions using semiquantitative scoring. Results No significant differences were seen in the volume of synovitis (P = 0.294), degree of flexor tenosynovitis (P = 0.532), periarticular erosions (P = 0.579), or degree of bone edema (P = 0.143) between RA and PMR joints. However, despite comparable degrees of synovitis, the proportion of MCP joints showing extracapsular enhancement was higher in the PMR group (100%) than in the RA group (50%) (P = 0.030). One PMR patient, but none of the RA patients, had bone edema at the capsular insertion. Conclusion Despite degrees of synovitis and tenosynovitis comparable with those in RA, PMR‐related hand disease is associated with prominent extracapsular changes, suggesting that inflammation in these tissues is more prominent than joint synovitis, which is common in both conditions. This suggests that the anatomic basis for joint disease localization differs between RA and PMR.

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Stereoselective photo-rearrangement of a pyrroline 1-oxide

Cited by 7 publications

References 0 publications

Resistance distance and Kirchhoff index in circulant graphs

Resistance distance and Kirchhoff index in circulant graphs

Generalized Foster's identities

A new tool for analysis of channel membrane proteins

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