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Quantitative aspects of systems like consumption of resources, output of goods, or reliability can be modeled by weighted automata. Recently, objectives like the average cost or the longtime peak power consumption of a system have been modeled by weighted automata which are not semiring weighted anymore. Instead, operations like limit superior, limit average, or discounting are used to determine the behavior of these automata. Here, we introduce a new class of weight structures subsuming a range of these models as well as semirings. Our main result shows that such weighted automata and Kleene-type regular expressions are expressively equivalent both for finite and infinite words.

With the application of the dynamic control system, Cellular Automata model has become a valued tool for the simulation of human behavior and traffic flow. As an integrated kind of railway signal-control pattern, the four-aspect color light automatic block signaling has accounted for 50% in the signal-control system in China. Thus, it is extremely important to calculate correctly its carrying capacity under the automatic block signaling. Based on this fact the paper proposes a new kind of "cellular automata model" for the four-aspect color light automatic block signaling under different speed states. It also presents rational rules for the express trains with higher speed overtaking trains with lower speed in a same or adjacent section and the departing rules in some intermediate stations. In it, the state of mixed-speed trains running in the section composed of many stations is simulated with CA model, and the train-running diagram is acquired accordingly. After analyzing the relevant simulation results, the needed data are achieved herewith for the variation of section carrying capacity, the average train delay, the train speed with the change of mixed proportion, as well as the distance between the adjacent stations.

In vitro incubation of tissues; in particular, skeletal muscles from rodents, is a widely-used experimental method in diabetes research. This experimental method has previously been validated, both experimentally and theoretically. However, much of the method's experimental data remains unclear, including the high-rate of lactate production and the lack of an observable increase in glycogen content, within a given time. The predominant hypothesis explaining the high-rate of lactate production is that this phenomenon is dependent on a mechanism in glycolysis that works as a safety valve, producing lactate when glucose uptake is super-physiological. Another hypothesis is that existing anoxia forces more ATP to be produced from glycolysis, leading to an increased lactate concentration. The lack of an observable increase in glycogen content is assumed to be dependent on limitations in sensitivity of the measuring method used. We derived a mathematical model to investigate which of these hypotheses is most likely to be correct. Using our model, data analysis indicates that the in vitro incubated muscle specimens, most likely are sensing the presence of existing anoxia, rather than an overflow in glycolysis. The anoxic milieu causes the high lactate production. The model also predicts an increased glycogenolysis. After mathematical analyses, an estimation of the glycogen concentration could be made with a reduced model. In conclusion, central anoxia is likely to cause spatial differences in glycogen concentrations throughout the entire muscle. Thus, data regarding total glycogen levels in the incubated muscle do not accurately represent the entire organ. The presented model allows for an estimation of total glycogen, despite spatial differences present in the muscle specimen.

In this paper, the authors characterize the Sobolev spaces $W^{\alpha ,p}({ℝ}^n)$ with $\alpha \in (0,2]$ and $p\in (max\{1,\frac{2n}{2\alpha +n}\},\infty )$ via a generalized Lusin area function and its corresponding Littlewood–Paley $g_{\lambda }^{\ast }$-function. The range $p\in (max\{1,\frac{2n}{2\alpha +n}\},\infty )$ is also proved to be nearly sharp in the sense that these new characterizations are not true when $\frac{2n}{2\alpha +n}>1$ and $p\in (1,\frac{2n}{2\alpha +n})$. Moreover, in the endpoint case $p=\frac{2n}{2\alpha +n}$, the authors also obtain some weak type estimates. Since these generalized Littlewood–Paley functions are of wide generality, these results provide some new choices for introducing the notions of fractional Sobolev spaces on metric measure spaces.

Let G be a finite group, we define the average codegree of the irreducible characters of G as $\mathrm{\text{acod}}(G)=\frac{1}{|\mathrm{\text{Irr}}(G)|}{\sum }_{\chi \in \mathrm{\text{Irr}}(G)}\mathrm{\text{cod}}(\chi )$, where $\mathrm{\text{cod}}(\chi )=\frac{|G:ker\chi |}{\chi (1)}$. We prove that if $G$ is non-solvable, then $\mathrm{\text{acod}}(G)\ge 68/5$, and the equality holds if and only if $G\cong A_5$. Also, we show that if $G$ is non-supersolvable, then $\mathrm{\text{acod}}(G)\ge 11/4$, and the equality holds if and only if $G\cong A_4$. In addition, we obtain that if $p$ is the smallest prime divisor of $\left|G\right|$, then $\mathrm{\text{acod}}(G)<p$ if and only if $G$ is an elementary abelian $p$-group.

Let $a>1$ be an integer. Denote by $l_a(n)$ the multiplicative order of $a$ modulo integer $n\ge 1$. We prove that there is a positive constant $\delta$ such that if $x^{1-\delta }=o(y)$, then

The teaching through problem solving approach uses problem solving as a means to stimulate students’ learning of mathematics. Students work either individually or in groups on an open problem before the concept is taught to the class. Teachers act as facilitators, asking probing questions to help students reflect on their thinking. This approach of teaching allows students to explore and think about a problem with a realistic context and discover knowledge in the process and is in contrast to a didactic teaching approach, where the teacher explicitly tells students what they are to learn. The opportunity to explore and think provided by the teaching through problem solving approach may better prepare students to thrive in a rapidly changing world, as opposed to students receiving information directly from teachers. This chapter discusses the teaching through problem solving approach, its benefits and its challenges. A lesson carried out by the author using this approach will be discussed.

This paper deals with traditional algorithms, Newton's method and a higher order generalization due to Euler. These iterations schemes and their Modifications have had a great success in solving nonlinear systems of equations. We give some understanding of this phenomenon by giving estimates of efficiency. The problem we focus on is that of finding a zero of a complex polynomial.