STRONG LIMIT THEOREMS FOR EXTENDED INDEPENDENT RANDOM VARIABLES AND EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES UNDER SUB-LINEAR EXPECTATIONS

Limit theorems for non-additive probabilities or non-linear expectations are challenging issues which have attracted a lot of interest recently.The purpose of this paper is to study the strong law of large numbers and the law of the iterated logarithm for a sequence of random variables in a sub-linear expectation space under a concept of extended independence which is much weaker and easier to verify than the independence proposed by Peng[20].We introduce a concept of extended negative dependence which is an extension of the kind of weak independence and the extended negative independence relative to classical probability that has appeared in the recent literature.Powerful tools such as moment inequality and Kolmogorov’s exponential inequality are established for these kinds of extended negatively independent random variables,and these tools improve a lot upon those of Chen,Chen and Ng[1].The strong law of large numbers and the law of iterated logarithm are also obtained by applying these inequalities.

Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications

Classical Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of random variables are basic tools for studying the strong laws of large numbers.In this paper,motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng(2008),we introduce the concept of negative dependence of random variables and establish Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of negatively dependent random variables under the sub-linear expectations.As an application,we show that Kolmogorov's strong law of larger numbers holds for independent and identically distributed random variables under a continuous sub-linear expectation if and only if the corresponding Choquet integral is finite.

Laser induced photo-detachment of O2 in DC discharge

Determination of the negative ion number density of O2 and O in a DC discharge of oxygen plasma was made employing Langmuir probe in conjunction with eclipse laser photo- detachment technique. The temporal evolution of the extra electrons resulting from the photo- detachment of O2- and O- were used to evaluate the negative ion number density. The ratio of O2 number density to O varied from 0.03 to 0.22. Number density of both O~ and O increased with increasing power and decreased as the pressure was increased. Electron number density was evaluated from the electron energy distribution function （EEDF） using the I-V recorded characteristic curves. Electron temperature between 2 and 2.7 eV were obtained. Influence of the 02（al△g） metastable state is discussed.

On the convergence for PNQD sequences with general moment conditions

Let fX;Xn;n≥1g be a sequence of identically distributed pairwise negative quadrant dependent(PNQD)random variables and fan;n1g be a sequence of positive constants with an=f(n)and f(θ^k)=f(θ^k-1)for all large positive integers k,where 1<θ≤βand f(x)>0(x≥1)is a non-decreasing function on[b;+1)for some b≥1:In this paper,we obtain the strong law of large numbers and complete convergence for the sequence fX;Xn;n≥1g,which are equivalent to the general moment conditionΣ∞n=1P(|X|>an)<1.Our results extend and improve the related known works in Baum and Katz[1],Chen at al.[3],and Sung[14].

Apropos 1+2+3+4+5+...=<img src="http://latex.codecogs.com/gif.latex?-\frac{1}{12}"title="-\frac{1}{12}"/>: Mapping Infinity in Light of the Number Circle (or Cycle), in L. Euler’s Footsteps and with the Aid of Two Dimensional Infinite Series, and Replacing Negative Infinity and Positive Infinity with Just Infinity

The number circle—that is, the notion that the largest possible positive numbers are followed by infinity and then by the smallest possible negative numbers—is not new. L. Euler defended it in the eighteenth century and, before him, J. Wallis considered something vaguely similar. However, in the nineteenth century, the number circle was for the most part abandoned—even if something similar is on occasion accepted in geometry, in the sense that space is circular. The design of the present paper is to present positive proof of the veracity of the number circle and therefore, at the same time, to falsify the number line. Verifying the number circle implies falsifying negative infinity and positive infinity—infinity instead being neither negative nor positive, just like 0. Part of said proof involves showing that infinity can be defined both as 1+1+1+1+1+1+... and as -1-1-1-1-1-... and that the following Equation applies: 1+1+1+1+1+1+...=-1-1-1-1-1-... The principal mathematical technique that will be used to provide said proof is introduced here for the first time. It is called the two dimensional infinite series. It is an infinite series of infinite series. Some additional observations regarding the geography of infinity will be made. A more detailed description of the geography of infinity will be reserved for other papers. The Equation is discussed in this paper only to the extent that the attention that has been paid to it has necessitated the construction of a theory of infinity that, upon closer inspection, makes the Equation more self-evident and intuitively apparent;a fuller discussion will take place in a later paper.

Laws of Large Numbers for Cesàro alpha-integrable Random Variables under Dependence Condition AANA or AQSI

Both residual Cesaro alpha-integrability （RCI（α） and strongly residual Cesaro alpha- integrability （SRCI（α）） are two special kinds of extensions to uniform integrability, and both asymptotically almost negative association （AANA） and asymptotically quadrant sub-independence （AQSI） are two special kinds of dependence structures. By relating the RCI（α） property as well as the SRCI（α） property with dependence condition AANA or AQSI, we formulate some tail-integrability conditions under which for appropriate α the RCI（（α） property yields Ll-convergence results and the SRCI（α） property yields strong laws of large numbers, which is the continuation of the corresponding literature.

A GmNINa-miR172c-NNC1 Regulatory Network Coordinates the Nodulation and Autoregulation of Nodulation Pathways in Soybean

Symbiotic root nodules are root lateral organs of plants in which nitrogen-fixing bacteria(rhizobia)convert atmospheric nitrogen to ammonia.The formation and number of nodules in legumes are precisely controlled by a rhizobia-induced signal cascade and host-controlled autoregulation of nodulation(AON).However,how these pathways are integrated and their underlying mechanisms are unclear.Here,we report that microRNA172c(miR172c)activates soybean(Glycine max)R hizobia-induced CLE1(GmRICI)and GmRIC2 by removing the transcriptional repression of these genes by Nodule Number Control 1(NNC1),leading to the activation of the AON pathway.NNC1 interacts with GmNINa,the soybean ortholog of Lotus NODULE INCEPTION(NIN),and hampers its transcriptional activation o i G m RICI and GmRIC2.Importantly,GmNINa acts as a transcriptional activator of miR172c.Intriguingly,NNC1 can transcriptionally repress miR172c expression,adding a negative feedback loop into the NNC1 regulatory network.Moreover,GmNINa interacts with NNC1 and can relieve the NNC1-mediated repression of miR172c transcription.Thus,the GmNINa-miR172c-NNC1 network is a master switch that coordinately regulates and optimizes NF and AON signaling,supporting the balance between nodulation and AON in soybean.