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Droplet collision on surface is widely existed in nature and industrial production. In our research, two-dimensional rotational models and three-dimensional symmetric models based on the phase field method have been developed to simulate the collisions of continuous droplets on spherical surfaces. Using parametric dimensionless, the spreading diameter of the coalescing droplets, the width of the liquid bridge between the droplets, and the moving velocity of the three-phase contact line are obtained. When the two droplets are coaxial, the collision velocity of the droplets increases, and the radial velocity of the liquid bridge also increases. Due to the increase of droplet energy, both the first and second maximum spreading are increased, but the characteristic spreading ts time is reduced. When using the modified capillary inertia time τ i ′ normalized spreading time ts, it is found that it fits well with the Weber number (We) by the curve 1.505 We−0.478. Increasing the ratio of curved surface to droplet diameter λ can reduce maximum spreading time and maximum rebound height of droplet. When there is a deviation between the centers of the droplets, the spread of the droplets no longer shows symmetry and the center of the condensed droplets moves towards the offset side of the tail droplets. These findings will provide insight into the dynamics of continuous droplet collisions. [ABSTRACT FROM AUTHOR]

The inception and harnessing of excitons are paramount for the electroluminescence performance of organic light-emitting devices (OLEDs). Through-space charge transfer (TSCT) via intramolecular interaction has proved to be one of the most potent techniques employed to achieve 100% internal quantum efficiency. However, molecular strategies utilized to comprehensively enhance the electroluminescent performance of TSCT emitters regarding improving the photoluminescence quantum yield (PLQY) and elevating the light out-coupling efficiency remain arduous. To surmount this challenge, we deliberately designed and synthesized a proof-of-concept TSCT emitter called CzO-TRZ by incorporating an extra carbazole donor into spiro-heterocyclic architecture. The introduction of rigid spiral fragments can immensely boost the horizontal orientation dipole ratio and establish an extra through-bond charge transfer (TBCT) radiative decay channel. As a result, a very high PLQY of 98.7%, fast kRISC of 2.2×105 s−1 and high krs of 2.2×107 s−1, and an ultrahigh horizontal dipolar ratio of 90% were concurrently achieved for CzO-TRZ blended films. Furthermore, corresponding thermally activated delayed fluorescence (TADF)- and TADF-sensitized fluorescence (TSF)-OLEDs based on CzO-TRZ demonstrated external quantum efficiencies (EQEs) of 33.4% and 30.3%, respectively, highlighting its versatile applications as both an emitter and sensitized host. [ABSTRACT FROM AUTHOR]

Fast spin-flipping is the key to exploit the triplet excitons in thermally activated delayed fluorescence based organic light-emitting diodes toward high efficiency, low efficiency roll-off and long operating lifetime. In common donor-acceptor type thermally activated delayed fluorescence molecules, the distribution of dihedral angles in the film state would have significant influence on the photo-physical properties, which are usually neglected by researches. Herein, we find that the excited state lifetimes of thermally activated delayed fluorescence emitters are subjected to conformation distributions in the host-guest system. Acridine-type flexible donors have a broad conformation distribution or bimodal distribution, in which some conformers feature large singlet-triplet energy gap, leading to long excited state lifetime. Utilization of rigid donors with steric hindrance can restrict the conformation distributions in the film to achieve degenerate singlet and triplet states, which is beneficial to efficient reverse intersystem crossing. Based on this principle, three prototype thermally activated delayed fluorescence emitters with confined conformation distributions are developed, achieving high reverse intersystem crossing rate constants greater than 106 s−1, which enable highly efficient solution-processed organic light-emitting diodes with suppressed efficiency roll-off.The distribution of dihedral angles in film state has significant influence on excited state lifetimes of thermally activated delayed fluorescence emitters. Here authors report conformation distribution confinement strategy to achieve fast spin-flipping for efficient organic light-emitting diodes. [ABSTRACT FROM AUTHOR]

Let { X , X n ; n ≥ 1 } be a sequence of independent and identically distributed Banach space valued random variables. This paper is devoted to providing a divergence criterion for a class of random series of the form ∑ n = 1 ∞ f n S n where S n = X 1 + ⋯ + X n , n ≥ 1 and f n (·) ; n ≥ 1 is a sequence of nonnegative nondecreasing functions defined on [ 0 , ∞) . More specifically, it is shown that (i) the above random series diverges almost surely if ∑ n = 1 ∞ f n c n 1 / 2 = ∞ for some c > 0 and (ii) the above random series converges almost surely if ∑ n = 1 ∞ f n c n 1 / 2 < ∞ for some c > 0 provided additional conditions are imposed involving X, the sequences S n ; n ≥ 1 and f n (·) ; n ≥ 1 , and c. A special case of this criterion is a divergence/convergence criterion for the random series ∑ n = 1 ∞ a n S n q based on the series ∑ n = 1 ∞ a n n q / 2 where a n ; n ≥ 1 is a sequence of nonnegative numbers and q > 0 . [ABSTRACT FROM AUTHOR]

P2-type layered Na0.67MnO2 has been considered to be a promising candidate cathode material for sodium ion batteries. Nevertheless, the undesired phase transitions during operation and the large Na+ radius induced sluggish ion diffusion remain the stumbling blocks to realize its high performance. Herein, we propose a Zn/Mg co-doping strategy, which is proved to have bifunctional effects. First, relative to the pristine P2-Na0.67MnO2 and the single-ion (Zn/Mg) doped samples, the Zn/Mg dual-doped P2-Na0.67MnO2 demonstrates a lower Mn3+/Mn4+ ratio and a higher lattice O content, which facilitate the structural stability of the cathode material. More intriguingly, the Zn/Mg co-doping gives rise to enlarged interplanar spacing, which provides spacious ion diffusion channels for fast Na+ intercalation/extraction. As a result, the Zn/Mg dual-doped sample exhibits a high Na+ diffusion coefficient and a solid-solution reaction during charge/discharge, with a cell volume change determined to be only 0.55%. Taking advantages of the above favorable features, the Zn/Mg dual-doped P2-Na0.67MnO2 demonstrates a high rate performance with 67.2 mAh·g−1 delivered at 10 C and a decent cycling stability with a capacity retention of 93.8% achieved at 1 C after 100 cycles. This work introduces the Zn/Mg co-doping strategy to simultaneously improve the cycling stability and rate capability of P2-Na0.67MnO2, which may offer a promising avenue for further performance enhancement of the layered Na-ion batteries cathode materials. [ABSTRACT FROM AUTHOR]

Two probability inequalities are established and each of them is applied to obtain a probability limit theorem. For a random variable and continuous nonnegative strictly increasing functions and defined on with , , and , where is a sequence of positive constants with as , the first inequality provides an upper bound for assuming that either or . For a sequence of independent nonnegative random variables with partial sums , , the second inequality provides an upper bound for where and are nonnegative real numbers. [ABSTRACT FROM AUTHOR]

This paper deals with the conditional quantile estimation based on left-truncated and right-censored data. Assuming that the observations with multivariate covariates form a stationary α-mixing sequence, the authors derive the strong convergence with rate, strong representation as well as asymptotic normality of the conditional quantile estimator. Also, a Berry-Esseen-type bound for the estimator is established. In addition, the finite sample behavior of the estimator is investigated via simulations. [ABSTRACT FROM AUTHOR]

Let $$\mathbb{N } = \{1, 2, 3, \ldots \}$$ . Let $$\{X, X_{n}; n \in \mathbb N \}$$ be a sequence of i.i.d. random variables, and let $$S_{n} = \sum _{i=1}^{n}X_{i}, n \in \mathbb N $$ . Then $$ S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})$$ for some $$\sigma ^{2} < \infty $$ whenever, for a subsequence $$\{n_{k}; k \in \mathbb N \}$$ of $$\mathbb N $$ , $$ S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})$$ . Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when $$\{\sqrt{n}; n \in \mathbb N \}$$ is replaced by $$\{\sqrt{na_{n}};n \in \mathbb N \}$$ with $$\lim _{n \rightarrow \infty } a_{n} = \infty $$ . We show that, for given positive nondecreasing sequence $$\{a_{n}; n \in \mathbb N \}$$ with $$\lim _{n \rightarrow \infty } a_{n} = \infty $$ and $$\lim _{n \rightarrow \infty } a_{n+1}/a_{n} = 1$$ and given nondecreasing function $$h(\cdot ): (0, \infty ) \rightarrow (0, \infty )$$ with $$\lim _{x \rightarrow \infty } h(x) = \infty $$ , there exists a sequence $$\{X, X_{n}; n \in \mathbb N \}$$ of symmetric i.i.d. random variables such that $$\mathbb E h(|X|) = \infty $$ and, for some subsequence $$\{n_{k}; k \in \mathbb N \}$$ of $$\mathbb N $$ , $$ S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)$$ . In particular, for given $$0 < p < 2$$ and given nondecreasing function $$h(\cdot ): (0, \infty ) \rightarrow (0, \infty )$$ with $$\lim _{x \rightarrow \infty } h(x) = \infty $$ , there exists a sequence $$\{X, X_{n}; n \in \mathbb N \}$$ of symmetric i.i.d. random variables such that $$\mathbb E h(|X|) = \infty $$ and, for some subsequence $$\{n_{k}; k \in \mathbb N \}$$ of $$\mathbb N $$ , $$ S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)$$ . [ABSTRACT FROM AUTHOR]

For an array of rowwise pairwise negative quadrant dependent, mean 0 random variables, Chandra, Li, and Rosalsky provided conditions under which weighted averages converge in L1 to 0. The Chandra, Li, and Rosalsky result is extended to Lr convergence (1≤r<2) and is shown to hold under weaker conditions by applying a mean convergence result of Sung and an inequality of Adler, Rosalsky, and Taylor. [ABSTRACT FROM AUTHOR]

For arrays of rowwise pairwise negative quadrant dependent random variables, conditions are provided under which weighted averages converge in mean to 0 thereby extending a result of Chandra, and conditions are also provided under which normed and centered row sums converge in mean to 0. These results are new even if the random variables in each row of the array are independent. Examples are provided showing (i) that the results can fail if the rowwise pairwise negative quadrant dependent hypotheses are dispensed with, and (ii) that almost sure convergence does not necessarily hold. [ABSTRACT FROM AUTHOR]

Let X be an observable random variable with unknown distribution function F(x)=P(X≤x), −∞, and let θ=sup{r≥0:E|X|r<∞}. We call θ the power of moments of the random variable X. Let X1,X2,…,Xn be a random sample of size n drawn from F(⋅). In this paper we propose the following simple point estimator of θ and investigate its asymptotic properties: θˆn=lognlogmax1≤k≤n|Xk|, where logx=ln(e∨x), −∞. In particular, we show that θˆn→Pθif and only iflimx→∞xrP(|X|>x)=∞∀r>θ. This means that, under very reasonable conditions on F(⋅), θˆn is actually a consistent estimator of θ. [ABSTRACT FROM AUTHOR]