Degree Sequences and Long Cycles in Graphs

Let $G$ be a graph on $n$ vertices with degree sequence $\delta=d_1\le d_2\le...\le d_n$. Let $c$ be the circumference - the order of a longest cycle and $p$ the order of a longest path in $G$. In 1952, Dirac proved: (i) every graph with $2d_1\ge n$ is hamiltonian; (ii) in every 2-connected graph, $c\ge \min\{p,2d_1\}$. Recently, the bounds $2d_1\ge n$ and $c\ge \min\{p,2d_1\}$ in (i) and (ii) are improved to $2d_\delta\ge n$ and $c\ge \min\{p,2d_\delta\}$, respectively, by Koulakzian, Mosesyan and Nikoghosyan. In this paper we present two new sharp bounds $d_\delta+d_{\delta+1}\ge n$ and $\min\{2d_{\delta+1},d_\delta+d_{\delta+2}\}\ge n$ instead of $2d_\delta\ge n$, as well as two new sharp bounds $c\ge \min\{p,d_\delta+d_{\delta+1}\}$ and $c\ge \min\{p,2d_{\delta+1},d_\delta+d_{\delta+2}\}$ instead of $c\ge \min\{p,2d_\delta\}$.Comment: 9 page

Everolimus dosing recommendations for tuberous sclerosis complex–associated refractory seizures

ObjectiveThe present analysis examined the exposure-response relationship by means of the predose everolimus concentration (C-min) and the seizure response in patients with tuberous sclerosis complex-associated seizures in the EXIST-3 study. Recommendations have been made for the target C-min range of everolimus for therapeutic drug monitoring (TDM) and the doses necessary to achieve this target C-min

Effect of heating rate on gas emissions and properties of fired clay bricks and fired clay bricks incorporated with cigarette butts

In general, the firing process of clay bricks generates a range of gas emissions into the atmosphere. At high concentrations, these volatile emissions can be a serious source of environmental pollutions. The main purpose of this study was to evaluate the effect of different heating rates on gas emissions and properties during the firing of clay bricks and clay bricks incorporated with cigarette butts (CBs). In this investigation, four different heating rates were used: 0.7 °C min−1−1, 2 °C min−1, 5 °C min−1 and 10 °C min. The samples were fired in solid form from room temperature to 1050 °C. During the firing cycles, carbon monoxide, carbon dioxide, nitrogen oxides,hydrogen cyanide and chlorine emissions were measured at different heating rates. All bricks were also tested for their physical and mechanical properties including dry density, compressive strength, tensile strength, water absorption and initial rate of absorption. Results show that gas emissions were reduced significantly with higher heating rates (10 °C min) followed by 5 °C min−1−1 and 2 °C min for both types of brick samples. Higher heating rates also decrease the compressive strength and tensile strength value but demonstrate an insignificant effect on the water absorption properties respectively. In conclusion, a higher heating rate is preferable in terms of decreasing gas emissions and it is also able to produce adequate physical and mechanical properties especially for the CB brick

Circumferences of 3-connected claw-free graphs, II

For a graph H , the circumference of H , denoted by c ( H ) , is the length of a longest cycle in H . It is proved in Chen (2016) that if H is a 3-connected claw-free graph of order n with δ ≥ 8 , then c ( H ) ≥ min { 9 δ − 3 , n } . In Li (2006), Li conjectured that every 3-connected k -regular claw-free graph H of order n has c ( H ) ≥ min { 10 k − 4 , n } . Later, Li posed an open problem in Li (2008): how long is the best possible circumference for a 3-connected regular claw-free graph? In this paper, we study the circumference of 3-connected claw-free graphs without the restriction on regularity and provide a solution to the conjecture and the open problem above. We determine five families F i ( 1 ≤ i ≤ 5 ) of 3-connected claw-free graphs which are characterized by graphs contractible to the Petersen graph and show that if H is a 3-connected claw-free graph of order n with δ ≥ 16 , then one of the following holds: (a) either c ( H ) ≥ min { 10 δ − 3 , n } or H ∈ F 1 . (b) either c ( H ) ≥ min { 11 δ − 7 , n } or H ∈ F 1 ∪ F 2 . (c) either c ( H ) ≥ min { 11 δ − 3 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 . (d) either c ( H ) ≥ min { 12 δ − 10 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 ∪ F 4 . (e) if δ ≥ 23 then either c ( H ) ≥ min { 12 δ − 7 , n } or H ∈ F 1 ∪ F 2 ∪ F 3 ∪ F 4 ∪ F 5 . This is also an improvement of the prior results in Chen (2016), Lai et al. (2016), Li et al. (2009) and Mathews and Sumner (1985)

Cold sensitivity test for individuals with non-freezing cold injury: the effect of prior exercise

BACKGROUND: One of the chronic symptoms of non-freezing cold injury (NFCI) is cold sensitivity. This study examined the effects of prior exercise on the response to a cold sensitivity test (CST) in NFCI patients with the aim of improving diagnostic accuracy. METHODS: Twenty three participants, previously diagnosed with NFCI by a Cold Injuries Clinic, undertook two CSTs. Participants either rested (air temperature 31°C) for approximately 80 min (prior rest condition (REST)) or rested for 30 min before exercising gently for 12 min (prior exercise condition (EX)). Following REST and EX, the participants placed their injured foot, covered in a plastic bag, into 15°C water for 2 min; this was followed by spontaneous rewarming in 31°C air for 10 min. RESULTS: The great toe skin temperature (T(sk)) before immersion averaged 32.5 (3.4)°C in both conditions. Following immersion, the rate of rewarming of the great toe T(sk) was faster in EX compared to REST and was higher 5 min (31.7 (3.4)°C vs. 29.8 (3.4)°C) and 10 min (33.8 (4.0)°C vs. 32.0 (4.0)°C) post-immersion. Over the first 5 min of rewarming, changes in the great toe T(sk) correlated with the changes in skin blood flow (SkBF) in EX but not the REST condition. No relationship was observed between T(sk) in either CST and the severity of NFCI as independently clinically assessed. CONCLUSIONS: Exercise prior to the CST increased the rate of the toe T(sk) rewarming, and this correlated with the changes in SkBF. However, the CST cannot be used in isolation in the diagnosis of NFCI, although the EX CST may prove useful in assessing the severity of post-injury cold sensitivity for prognostic and medico-legal purposes

The Search for Maximal Values of min(A,B,C) / gcd(A,B,C) for A^x + B^y = C^z

This paper answers a question asked by Ed Pegg Jr. in 2001: "What is the maximal value of min(A,B,C)/ gcd(A,B,C) for A^x + B^y = C^z with A,B,C >= 1; x,y,z >= 3?" Equations of this form are analyzed, showing how they map to exponential Diophantine equations with coprime bases. A search algorithm is provided to find the largest min/gcd value within a given equation range. The algorithm precalculates a multi-gigabyte lookup table of power residue information that is used to eliminate over 99% of inputs with a single array lookup and without any further calculations. On inputs that pass this test, the algorithm then performs further power residue tests, avoiding modular powering by using lookups into precalculated tables, and avoiding division by using multiplicative inverses. This algorithm is used to show the largest min/gcd value for all equations with C^z <= 2^100.Comment: Body: 16 pages, Appendices: 11 pages, 5 tables, 1 figur

Field-induced magnetic ordering in the Haldane system PbNi2V2O8

The Haldane system PbNi2V2O8 was investigated by the temperature dependent magnetization M(T) measurements at fields higher than H_c, with H_c the critical fields necessary to close the Haldane gap. It is revealed that M(T) for H > H_c exhibits a cusp-like minimum at T_{min}, below which M(T) increases with decreasing T having a convex curve. These features have been observed for both $H \parallel c$ and $H \perp c$, with c-axis being parallel to the chain. These data indicate the occurrence of field-induced magnetic ordering around T_{min}. Phase boundaries for $H \parallel c$ and $H \perp c$ do not cross each other, consistent with the theoretical calculation for negative single-ion anisotropy D.Comment: 3 figures, submitted to Phys. Rev.