Asymptotics of 3-Stack-Sortable Permutations

We derive a simple functional equation with two catalytic variables characterising the generating function of 3-stack-sortable permutations. Using this functional equation, we extend the 174-term series to 1000 terms. From this series, we conjecture that the generating function behaves as $$W(t) \sim C_0(1-\mu_3 t)^\alpha \cdot \log^\beta(1-\mu_3 t), $$ so that $$[t^n]W(t)=w_n \sim \frac{c_0\mu_3^n}{ n^{(\alpha+1)}\cdot \log^\lambda{n}} ,$$ where $\mu_3 = 9.69963634535(30),$ $\alpha = 2.0 \pm 0.25.$ If $\alpha = 2$ exactly, then $\lambda = -\beta+1$, and we estimate $\beta \approx -2,$ but with a wide uncertainty of $\pm 1.$ If $\alpha$ is not an integer, then $\lambda=-\beta$, but we cannot give a useful estimate of $\beta$. The growth constant estimate (just) contradicts a conjecture of the first author that $$9.702 < \mu_3 \le 9.704.$$ We also prove a new rigorous lower bound of $\mu_3\geq 9.4854$, allowing us to disprove a conjecture of Bóna. We then further extend the series using differential-approximants to obtain approximate coefficients $O(t^{2000}),$ expected to be accurate to $20$ significant digits, and use the approximate coefficients to provide additional evidence supporting the results obtained from the exact coefficients.

Exponential growth constants for spanning forests on Archimedean lattices: Values and comparisons of upper bounds

We compare our upper bounds on the exponential growth constant [Formula: see text] characterizing the asymptotic behavior of spanning forests on Archimedean lattices [Formula: see text] with recently derived upper bounds.Our upper bounds on [Formula: see text], which are very close to the respective values of [Formula: see text] that we have calculated, are shown to be significantly better for these lattices than the new upper bounds.

Passivation of chromium disilicide in acidic media

The anodic behavior of CrSi2 electrode in 0,5 M H2SO4, 0,5 M HClO4, 0,5 M HNO3 and 0,5 M HCl solutions has been studied by the methods of polarization, capacitance, and impedance measurements. It has been concluded that in the process of anodic oxidation at potentials from corrosion E to transpassivation E inclusive, an oxide film is formed on the surface of chromium disilicide in the studied media, which is close in composition to SiO2 (with a small content of chromium oxides). The presence of this film on the silicide surface determines its high chemical resistance in the studied solutions. The thickness of the oxide film on CrSi2 has been calculated depending on the potential and composition of the electrolyte. The growth constant of the oxide film has been determined.

The Enhanced Mechanism of 0.05 wt. % Nd Addition on High Temperature Reliability of Sn-3.8Ag-0.7Cu/Cu Solder Joint

In this study, the effect of appropriate Nd addition on improving the high-temperature reliability of Sn-3.8Ag-0.7Cu (SAC387)/Cu solder joint after aging treatment was investigated. The interfacial microstructure of solder joint was refined with proper addition of Nd. This phenomenon could be explained as the adsorbing-hindering effect of surface-active Nd atoms which blocked the growth of brittle intermetallic compounds (IMCs) in the solder joint. Theoretical analysis indicated that 0.05 wt. % addition of Nd could distinctly decrease the growth constant of Cu6Sn5 IMCs and slightly decrease the growth constant of Cu3Sn IMCs respectively. The shear force of SAC387-0.05Nd/Cu solder joint was evidently improved compared with the origin solder joint. In addition, SAC387-0.05Nd/Cu solder joint maintained excellent mechanical property compared with SAC387/Cu solder joint even after 1440 h aging treatment.

Stieltjes Moment Sequences for Pattern-Avoiding Permutations

A small set of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on $[0, \infty)$. Such sequences are known as Stieltjes moment sequences. They have a number of nice properties, such as log-convexity, which are useful to rigorously bound their growth constant from below. This article focuses on some classical sequences in enumerative combinatorics, denoted $Av(\mathcal{P})$, and counting permutations of $\{1, 2, \ldots, n \}$ that avoid some given pattern $\mathcal{P}$. For increasing patterns $\mathcal{P}=(12\ldots k)$, we recall that the corresponding sequences, $Av(123\ldots k)$, are Stieltjes moment sequences, and we explicitly find the underlying density function, either exactly or numerically, by using the Stieltjes inversion formula as a fundamental tool. We first illustrate our approach on two basic examples, $Av(123)$ and $Av(1342)$, whose generating functions are algebraic. We next investigate the general (transcendental) case of $Av(123\ldots k)$, which counts permutations whose longest increasing subsequences have length at most $k-1$. We show that the generating functions of the sequences $\, Av(1234)$ and $\, Av(12345)$ correspond, up to simple rational functions, to an order-one linear differential operator acting on a classical modular form given as a pullback of a Gaussian $\, _2F_1$ hypergeometric function, respectively to an order-two linear differential operator acting on the square of a classical modular form given as a pullback of a $\, _2F_1$ hypergeometric function. We demonstrate that the density function for the Stieltjes moment sequence $Av(123\ldots k)$ is closely, but non-trivially, related to the density attached to the distance traveled by a walk in the plane with $k-1$ unit steps in random directions. Finally, we study the challenging case of the $Av(1324)$ sequence and give compelling numerical evidence that this too is a Stieltjes moment sequence. Accepting this, we show how rigorous lower bounds on the growth constant of this sequence can be constructed, which are stronger than existing bounds. A further unproven assumption leads to even better bounds, which can be extrapolated to give an estimate of the (unknown) growth constant.

A Statistical Analysis of the Most Affected Countries by COVID-19 (Preprint)

BACKGROUND It is almost more than six months elapsed since the crisis of COVID-19 pandemic has hit the globe. It becomes utterly crucial for policy formulation to know up to which period and up to what magnitude the disease will impact us. OBJECTIVE Aim of this study is to predict the saturation level of COVID-19 cases in USA, Brazil, India, Russia, Italy and UK. METHODS Trend analysis is used to fit exponential trend line to the weekly cumulative positive cases. Forecasted growth constants obtained by fitting power series trend to the observed values and prediction of weekly cumulative cases done. Growth constant value of 0.0039 (Doubling time 180 days), observed mean, median and maximum reported COVID-19 cases per million for countries with more than 90% recovery is used to obtain saturation levels. RESULTS Italy and UK reached saturation with growth constant values 0.0007 and 0.0025. Brazil and USA projected to achieve saturation by 26 September 2020 and 19 December 2020 with 8.24 and 9.17 million cases. India and Russia expected to reach the saturation by 16 January 2021and 27 February 2021 with 56.72 and 3.86 million cases. CONCLUSIONS The USA and Russia are predicted to reach saturation due to decrease in growth constant value. Growth constant values for Brazil and India are high and these countries are estimated to reach saturation by infecting the maximum reported per million population. It is therefore eternally essential to take more proactive decisions and to test more and isolate patients effectively to contain COVID-19. CLINICALTRIAL Nil

Mathematical Analysis, Model and Prediction of COVID-19 Data

A simple and effective mathematical procedure for the description of observed COVID-19 data and calculation of future projections is presented. An exponential function E(t) with a time-varying Growth Constant k(t) is used. E(t) closely approximates observed COVID-19 Daily Confirmed Cases with NRMSDs of 1 to 2%. An example of prediction of future cases is presented. The Effective Growth Rates of a discrete SIR model were estimated on the basis of k(t) for COVID-19 data for Germany, and were found to be consistent with those reported in a previous study (1). The proposed procedure, which involves less than ten basic algebraic, logarithm and exponentiation operations for each data point, is suitable for use in promoting interdisciplinary research, exchange and sharing of information.

Growth, mortality and length-weight relationship of the Bombay duck Harpadon nehereus (Ham. 1822) from North-eastern tip of the Bay of Bengal

The present study was conducted to investigate the growth parameters, length-weight relationshipand diverse mortality rates of Bombay duck Harpadon nehereus, from North-eastern tip of the Bay of Bengal.The length weight data were collected from January to December 2017 where a total number of 2054individuals of both sex were analyzed. The von Bertalanffy growth parameters, the asymptotic length L?(cm), the growth constant K (year-1) and t0 were estimated as 38.50, 0.88, and -0.0181 years respectively.The growth performance index (Ø?) was found to be 3.115. The total length and body weight (L-W)relationship was found as W=0.004 L3.021, indicating that the growth rate showed a positive allometricpattern (b=3.021>3). The annual fishing morality rate (F) was 1.89, whereas the natural mortality rate (M)was 1.51. The obtained value of the exploitation rate (E= 0.56) indicated that the said population was inover-exploited condition.

Growth Kinetics of Interfacial Intermetallic Compound in Al(AA4343)/Steel(SUS316) Clad Strip

The composition and microstructure of intermetallic compounds (IMC) at the interface of aluminum(AA4343)-stainless steel(SUS316) were studied upon annealing at 550°C for 1h to 20h and at 610°C for 15min to 10h by means of optical microscope(OM) , scanning electron microscope (SEM) with energy dispersive system(EDS) and transmission Electron Microscopy (TEM). The results showed that the IMC was of 4.3μm to 36.1μm thick during heat treatment at 550°C for 1h to 20h, and the IMC contained Al-Fe-Si-Cr-Ni-Mo and Al-Fe-Si -Ni. During annealing at 610°C for 15min to 5h, the thickness of IMC was 31.2 μm to 208 μm, and the IMC were mainly of η-Fe2Al5 and τ10- Al4Fe1.7Si at 550°C for 10h. As the annealing time extended to 10h, natural delamination occurred at the interface between the aluminum alloy layer and IMC layer. The growth kinetics analysis showed that the relationship between the thickness of IMC “X” and time “t” followed the relational equation X=(kt)n. For AA4343(solid) - SUS316(solid), n was 1/2, and the growth constant k = 1.9×10-13m2/s at annealing temperature of 550 °C. When the temperature was 610°C, AA4343 - SUS316 was a liquid-solid contact reaction, n was 1, the growth constant k=1.45×10-8m/s.

Effect of 0.05 wt.% Pr Addition on Microstructure and Shear Strength of Sn-0.3Ag-0.7Cu/Cu Solder Joint during the Thermal Aging Process

The evolution of interfacial morphology and shear strengths of the joints soldered with Sn-0.3Ag-0.7Cu (SAC0307) and SAC0307-0.05Pr aged at 150 °C for different times (h; up to 840 h) were investigated. The experiments showed the electronic joint soldered with SAC0307-0.05Pr has a much higher shear strength than that soldered with SAC0307 after each period of the aging process. This contributes to the doping of Pr atoms, “vitamins in alloys”, which tend to be adsorbed on the grain surface of interfacial Cu6Sn5 IMCs, inhibiting the growth of IMCs. Theoretical analysis indicates that doping 0.05 wt.% Pr can evidently lower the growth constant of Cu6Sn5 (DCu6), while the growth constant of Cu3Sn (DCu3) decreased slightly. In addition, the electronic joint soldered with SAC0307-0.05Pr still has better ductility than that soldered with SAC0307, even after a 840-h aging process.