The planar Ramsey number PR(H_{1}, H_{2}) is the smallest integer n such that any planar graph on n vertices contains a copy of H_{1} or its complement contains a copy of H_{2}.
It is known that the Ramsey number R(K_{4 }-e, K_{6})
= 21, and the planar Ramsey numbers PR(K_{4 }- e, K_{l})
for l ≤ 5 are known. In this paper,
we give the lower bounds on PR(K_{4 }? e, K_{l}) and determine
the exact value of PR(K_{4 }- e, K_{6}).

Abstract:
This paper presents a novel mathematical model with multidrug-resistant (MDR) and undetected TB cases. The theoretical analysis indicates that the disease-free equilibrium is globally asymptotically stable if ; otherwise, the system may exist a locally asymptotically stable endemic equilibrium. The model is also used to simulate and predict TB epidemic in Guangdong. The results imply that our model is in agreement with actual data and the undetected rate plays vital role in the TB trend. Our model also implies that TB cannot be eradicated from population if it continues to implement current TB control strategies. 1. Introduction China is among the highest TB burden countries in the world and is second only to India. The number of MDR-TB cases is about one third of that of the world [1, 2]. For an MDR-TB patient, the cost for the treatment is usually 10–100 times higher than that of a common TB patient [3]. Moreover, undetected TB patients pose threat to others, which plays a major role in mycobacterium transmission among the general population [4]. It is no doubt that the earlier the diagnosis and treatment, the better in attenuating the transmissions. Clearly, the understanding of the mechanism and contribution of undetected patients in transmission is important to conduct targeted interventions and control TB decease. Blower et al. proposed a transmission dynamics among the cases infected by drug-sensitive and the drug-resistant strains, respectively [5]. Many subsequent investigations focused on similar topics with various perspectives about the role of drug-resistant strain [3, 6–8]. However, these works failed to independently identify the role of the undetected patients in TB transmission. motivated by the previous work, we carefully divide the infectious cases into undetected cases and timely detected cases in this paper. This improved classification enables us to interpret the real situation in a better manner. To analyze the stability of the disease-free equilibrium in our transmission model, we introduce a novel Lyapunov function. We prove that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than one, and a forward (transcritical) bifurcation exists otherwise. While Guangdong is a developed province of China, the number of TB cases in Guangdong is about 10% of that of national TB cases. Mass migrants, MDR, and HIV/AIDS coinfections make the situation of TB transmission very complex in this region. We apply our model to describe the TB situation in Guangdong and simulate the distribution

Abstract:
Objectives Oral diseases are associated with adverse pregnancy outcomes. The routine utilization of dental care (RUDC) during pregnancy is an effective way to improve pregnant women’s oral health, and thus safeguard the health of their babies. As China has one fifth of the world’s population, it is especially meaningful to encourage RUDC there. However, the status of RUDC in China and the key underlying factors are largely unknown. Methods This cross-sectional survey investigated the current status of RUDC during pregnancy and the key underlying factors in Hangzhou City, Zhejiang Province, eastern China. We collected participants’ demographics, individual oral-hygiene behaviors, individual lifestyle, oral-health conditions and attitudes, and also their RUDC during pregnancy. Binary Logistic Regression Analysis was used to analyze the key underlying factors. Results Only 16.70% of the participants reported RUDC during pregnancy. The percentage of RUDC was significantly lower among pregnant women with the following characteristics: aged 30 or less, an annual household income under $8,000, brushing once a day or less, never flossing or rinsing the mouth, paying no attention to pregnancy-related oral-health knowledge, and being dissatisfied with one’s individual dental hygiene behavior. Conclusions RUDC during pregnancy is very low in eastern China and is greatly influenced not only by a woman’s age, annual income, individual hygiene behavior, but also by her attention and attitudes to oral health. To improve this population’s access to and use of dental care during pregnancy, appropriate programs and policies are urgently needed.

Abstract:
Let $X$ be a smooth projective variety over a number field, fibered over a curve, with geometrically integral fibers. We prove that, supposing the finiteness of $\sha(Jac(C))$, if the fibers over a generalised Hilbertian subset satisfy the Hasse principle (resp. weak approximation), then the Brauer-Manin obstruction coming from the base curve is the only obstruction to the Hasse principle (resp. to weak approximation) for zero-cycles of degree 1 on $X$.

Abstract:
We study the local-global principle for zero-cycles of degree 1 on certain varieties fibered over the projective space. Among other applications, we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation for zero-cycles of degree 1 on Severi-Brauer-variety bundles or Ch\^atelet-surface bundles over the projective space.

Abstract:
Let $X$ be a rationally connected algebraic variety, defined over a number field $k$. We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for $K$-rational points on $X_K$ for all finite extensions $K/k$; (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles of degree 1 on $X_K$ for all finite extensions $K/k$; (3) a certain sequence of local-global type for Chow groups of 0-cycles on $X_K$ is exact for all finite extensions $K/k$. We prove that (1) implies (2), and that (2) and (3) are equivalent. We also prove a similar implication for the Hasse principle. As an application, we prove the exactness of the sequence mentioned above for smooth compactifications of certain homogeneous spaces of linear algebraic groups.

Abstract:
Recently Dasheng Wei proved that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for 0-cycles of degree 1 on some fibrations over the projective line defined by bi-cyclic normic equations. In the present paper, we prove the exactness of the global-to-local sequence for Chow groups of 0-cycles of such varieties, which signifies that the Brauer-Manin obstruction is also the only obstruction to weak approximation for 0-cycles of arbitrary degree. Our main theorem also generalizes several existing results.

Abstract:
We study the Brauer-Manin obstruction to the Hasse principle and to weak approximation for 0-cycles on algebraic varieties that possess a fibration structure. The exactness of the local-to-global sequence $(E)$ of Chow groups of 0-cycles was known only for a fibration whose base is either a curve or the projective space. In the present paper, we prove the exactness of $(E)$ for fibrations whose bases are Ch\^{a}telet surfaces or projective models of homogeneous spaces of connected linear algebraic groups with connected stabilizers. We require that either all fibres are split and most fibres satisfy weak approximation for 0-cycles, or the generic fibre has a 0-cycle of degree $1$ and $(E)$ is exact for most fibres.

Abstract:
Consider weak approximation for 0-cycles on a smooth proper variety defined over a number field, it is conjectured to be controlled by its Brauer group. Let $X$ be a Ch\^atelet surface or a smooth compactification of a homogeneous space of a connected linear algebraic group with connected stabilizer. Let $Y$ be a rationally connected variety. We prove that weak approximation for 0-cycles on the product $X\times Y$ is controlled by its Brauer group if it is the case for $Y$ after every finite extension of the base field. We do not suppose the existence of 0-cycles of degree $1$ neither on $X$ nor on $Y$.

Abstract:
We prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and to the weak approximation for zero-cycles on certain fibrations over a smooth curve or over the projective space. The principal novelty is that the arithmetic hypotheses are supposed only on the fibres over a generalized Hilbertian subset.