The tau constant and the discrete Laplacian matrix of a metrized graph

Cinkir, Zubeyir

doi:10.1016/j.ejc.2011.01.008

Cited by 14 publications

(43 citation statements)

References 6 publications

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“…On the other hand, by [C3,Theorem 6.9] , we have 1 ≥ x + y, x ≥ 0, y ≥ 0, x ≥ (Λ(Γ) − 1)y and y ≥ v+6 4v (x + y) 2 . Therefore, it will be enough to find c = c(g, v) such that…”

Section: Simple Polarized Metrized Graphsmentioning

confidence: 99%

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Zhang’s Conjecture and the Effective Bogomolov Conjecture over function fields

Cinkir

2010

Invent. math.

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Abstract. We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang's Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we obtain a new proof of the slope inequality for Faltings heights on moduli space of curves.

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Section: Simple Polarized Metrized Graphsmentioning

confidence: 99%

“…Note that by [C3,Theorem 6.9 part (4)] we have gy ≥ x ≥ (Λ(Γ) − 1)y > 0, and recall that y ≥ v+6 4v (x + y) 2 , and x + y < 1 for any bridgeless metrized graph Γ with ℓ(Γ) = 1. In general, y can be arbitrarily small.…”

Section: Simple Polarized Metrized Graphsmentioning

confidence: 99%

Zhang’s Conjecture and the Effective Bogomolov Conjecture over function fields

Cinkir

2010

Invent. math.

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“…In particular, it implies that the tau constant is positive. One can find more detailed information on τ (Γ) in articles [3], [4], [5] and [7]. For more information about the resistance function r(x, y) on a metrized graph, one can consult to the articles [2], [1] and [4].…”

Section: Pm-graphs and Their Invariantsmentioning

confidence: 99%

“…This is what we did in this paper. Alternatively, we can compute these invariants by using the algorithms given in [7] and [8]. For example, how we compute the tau constant for the metrized graph in part IX of Figure 4 is illustrated in [7,Example 5.2], and computation of invariants of the pm-graph in part XIV of Figure 4 are done in [8,Example 1].…”

Section: Pm-graphs and Their Invariantsmentioning

confidence: 99%

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Admissible invariants of genus 3 curves

Cinkir

2015

manuscripta math.

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Abstract. Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we explicitly calculated these admissible invariants for all curves of genus 3. We find the sharp lower bound for the invariants ϕ, λ and ǫ for all polarized metrized graphs of genus 3. This improves the lower bound given for Effective Bogomolov Conjecture for such curves.

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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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The tau constant and the discrete Laplacian matrix of a metrized graph

Abstract: We express the tau constant of a metrized graph in terms of the discrete Laplacian matrix and its pseudo inverse.

Cited by 14 publications

References 6 publications

Zhang’s Conjecture and the Effective Bogomolov Conjecture over function fields

Zhang’s Conjecture and the Effective Bogomolov Conjecture over function fields

Admissible invariants of genus 3 curves

A new tool for analysis of channel membrane proteins

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