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1. Ring graphs and Goldberg's bound on chromatic index

   https://doi.org/10.1002/jgt.22494  

   Cao, Yan; Chen, Guantao; He, Shushan; Jing, Guangming (August 2019, Journal of Graph Theory)

   Abstract Let be a multigraph with maximum degree and chromatic index . If is bipartite then . Otherwise, by a theorem of Goldberg, , where denotes the odd girth of . Stiebitz, Scheide, Toft, and Favrholdt in their book conjectured that if then contains as a subgraph a ring graph with the same chromatic index. Vizing's characterization of graphs with chromatic index attaining the Shannon's bound showed the above conjecture holds for . Stiebitz et al verified the conjecture for graphs with and . McDonald proved the conjecture when is divisible by . In this paper, we show that the chromatic index condition alone is not sufficient to give the conclusion in the conjecture. On the positive side, we show that the conjecture holds for every with , and the maximum degree condition is best possible. This positive result leans on the positive resolution of the Goldberg‐Seymour conjecture. 

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2. Twist positivity, L-space knots, and concordance

   https://doi.org/10.1007/s00029-024-00996-6  

   Krishna, Siddhi; Morton, Hugh (February 2025, Selecta Mathematica)

   Many well studied knots can be realized as positive braid knots where the braid word contains a positive full twist; we say that such knots are twist positive. Some important families of knots are twist positive, including torus knots, 1-bridge braids, algebraic knots, and Lorenz knots. We prove that if a knot is twist positive, the braid index appears as the third exponent in its Alexander polynomial. We provide a few applications of this result. After observing that most known examples of L-space knots are twist positive, we prove: if K is a twist positive L-space knot, the braid index and bridge index of K agree. This allows us to provide evidence for Baker’s reinterpretation of the slice-ribbon conjecture: that every smooth concordance class contains at most one fibered, strongly quasipositive knot. In particular, we provide the first example of an infinite family of positive braid knots which are distinct in concordance, and where, as g tends to infinity, the number of hyperbolic knots of genus g gets arbitrarily large. Finally, we collect some evidence for a few new conjectures, including the following: the braid and bridge indices agree for any L-space knot. 

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3. Impact of the Eurasian Teleconnection on the Interannual Variability of Haze-Fog in Northern China in January

   https://doi.org/10.3390/atmos10030113  

   Li, Ye; Sheng, Lifang; Li, Chun; Wang, Yuhang (March 2019, Atmosphere)

   Using meteorological observation data and NCEP/NCAR (National Centers for Environmental Prediction/National Center for Atmospheric Research) reanalysis data, the impacts of the atmospheric circulation pattern on the interannual variability of haze-fog in northern China in January are studied by means of statistical methods. The results showed that the Eurasian teleconnection (EU) at the 500 hPa isostatic surface is the most important pattern affecting the haze-fog frequency in northern China. However, the existing EU index cannot perfectly describe this pattern. To this end, this study selects three main activity centers to define a new EU index, which are located in the Europe (10 °E, 55 °N), Siberia (80 °E, 60 °N), and Shandong, China (120 °E, 40 °N). The difference between the existing EU index and the new EU index is mainly the position of the anomaly center of the 500 hPa geopotential height. The EU is in a negative phase in higher haze-fog years but is in a positive phase in lower haze-fog years. The 500 hPa geopotential height shows negative anomalies in Europe and East Asian and a positive anomaly in Siberia in the negative EU phase. Using Plumb wave activity flux analysis, it was found that the cold wave affecting northern China is less in the negative EU phase than that in the positive EU phase, which resulted in more haze-fog days. In addition, the results also showed that the EU pattern goes through a considerable development and decay within 13 days. The visibility starts to significantly decrease at a lag of −1 to 2 days in the negative EU peak phase and is influenced by the weak north wind that is caused by the high pressure. 

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4. The Southern Annular Mode: Variability, trends, and climate impacts across the Southern Hemisphere

   https://doi.org/10.1002/wcc.652  

   Fogt, Ryan L.; Marshall, Gareth J. (May 2020, WIREs Climate Change)

   Abstract The Southern Annular Mode (SAM) is the leading mode of extratropical Southern Hemisphere climate variability, associated with changes in the strength and position of the polar jet around Antarctica. This variability in the polar jet drives large fluctuations in the Southern Hemisphere climate, from the lower stratosphere into the troposphere, and stretching from the midlatitudes across the Southern Ocean to Antarctica. Notably, the SAM index has displayed marked positive trends in the austral summer season (stronger and poleward shifted westerlies), associated with stratospheric ozone loss. Historical reconstructions demonstrate that these recent positive SAM index values are unprecedented in the last millennia, and fall outside the range of natural climate variability. Despite these advances in the understanding of the SAM behavior, several areas of active research are identified that highlight gaps in our present knowledge. This article is categorized under:Paleoclimates and Current Trends > Earth System Behavior 

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5. Law of the First Passage Triple of a Spectrally Positive Strictly Stable Process

   https://doi.org/10.1007/s10959-019-00898-w  

   Chi, Zhiyi (January 2019, Journal of Theoretical Probability)

   Law of the first passage triple of a spectrally positive strictly stable process. For a spectrally positive and strictly stable process with index in (1,2), a series representation is obtained for the joint distribution of the "first passage triple" that consists of the time of first passage and the undershoot and the overshoot at first passage. The result leads to several corollaries, including (1) the joint law of the first passage triple and the pre-passage running supremum, and (2) at a fixed time point, the joint law of the process' value, running supremum, and the time of the running supremum. The representation can be decomposed as a sum of strictly positive functions that allow exact sampling of the first passage triple. 

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