Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions

This paper deals with the blow-up phenomena for the following porous medium equation systems with nonlinear boundary conditions $\left\{\begin{array}{cc}{u}_t=\Delta u^m+k_1\left(t\right)f_1\left(v\right),v_t=\Delta v^n+k_2\left(t\right)f_2\left(u\right)\hfill & in\Omega \times (0,t^1),\hfill \\ \hfill \\ \hfill \\ \frac{?u}{?\nu }=g_1\left(u\right),\frac{?v}{?\nu }=g_2\left(v\right)\hfill & on?\Omega \times (0,t^1),\hfill \\ \hfill \\ \hfill \\ u(x,0)=u_0\left(x\right)\ge 0,v(x,0)=v_0\left(x\right)\ge 0\hfill & in\overline{\Omega },\hfill \end{array}\right.$ where $m,n>1$, $\Omega ?{\mathbb{R}}^N(N\ge 2)$ is bounded convex domain with smooth boundary. Using a differential inequality technique and a Sobolev inequality, we prove that under certain conditions on data, the solution blows up in finite time. We also derive an upper and a lower bound for blow-up time. In addition, as applications of the abstract results obtained in this paper, an example is given.     

Quantification of thermodynamic properties for vaporisation reactions above solid Ga2O3 and In2O3 by Knudsen Effusion Mass Spectrometry

Even though Ga₂O₃ and In₂O₃ are broadly used as semi-conductors, thermodynamic data for their vaporisation reactions exhibit a large spread. Therefore, the vaporisation behaviour of solid Ga₂O₃ and In₂O₃ was determined by means of Knudsen Effusion Mass Spectrometry (KEMS). Ga₂O₃ and In₂O₃ were studied in an iridium Knudsen cell and heated over a temperature range of 1200–1750 K in order to identify the species present in the vapour phase, and determine their partial pressures. We find that M₂O (where M = Ga or In) is the most abundant gas species above the solid oxide, followed by M and MO, in accord with tabulated data. Following the calculation of partial pressures and equilibrium constants, we propose ${\Delta }_f{H}_{\mathrm{298,}3\mathrm{rd}}^{\mathrm{o}}\left(\mathrm{G}{\mathrm{a}}_2\mathrm{O}\left(\mathrm{g}\right)\right)$ = −68966 ± 7442 Jmol^-1 and ${\Delta }_f{H}_{\mathrm{298,}3\mathrm{rd}}^{\mathrm{o}}\left(\mathrm{I}{\mathrm{n}}_2\mathrm{O}\left(\mathrm{g}\right)\right)$ = −22245 ± 964 Jmol^-1 from the 3^rd law method. Deviations in ${\Delta }_f{H}_{\mathrm{298,}3\mathrm{rd}}^{\mathrm{o}}\left(i\right)$ relative to literature KEMS measurements are generally within ∼2% relative, and can be ascribed to the use of different ionisation cross sections, Knudsen cell material, temperature calibrations, as well as tabulated Gibbs energy functions. However, comparison with ab initio studies suggests the data reported in this work is more accurate than in previous studies, given that the ${\Delta }_f{H}_{\mathrm{298,}3\mathrm{rd}}^{\mathrm{o}}\left(\mathrm{I}\mathrm{n}\mathrm{O}\left(\mathrm{g}\right)\right)$ = 157744 ± 3681 Jmol^-1 deviates by only ∼0.2% from the theoretical value.     

Existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional Kirchhoff-type systems

This paper is concerned with the following linearly coupled fractional Kirchhoff-type system

$\left\{\begin{array}{c}\left(a+b{\int }_{{\mathbb{R}}^3}|{(?△)}^{\frac{\alpha }{2}}u{|}^2\mathrm{d}x\right){(?△)}^{\alpha }u+\lambda u=f\left(u\right)+\gamma v,\text{in}{\mathbb{R}}^3,\hfill \\ \left(c+d{\int }_{{\mathbb{R}}^3}|{(?△)}^{\frac{\alpha }{2}}v{|}^2\mathrm{d}x\right){(?△)}^{\alpha }v+\mu v=g\left(v\right)+\gamma u,\text{in}{\mathbb{R}}^3,\hfill \\ u,v\in H^{\alpha }\left({\mathbb{R}}^3\right),\hfill \end{array}\right.$

where $a,c,\lambda ,\mu >0$, $b,d\ge 0$ are constants, $\alpha \in [\frac{3}{4},1)$ and $\gamma >0$ is a coupling parameter. Under the general Berestycki–Lions conditions on the nonlinear terms $f$ and $g$, we prove the existence of positive vector ground state solutions of Poho?aev type for the above system via variational methods. Moreover, the asymptotic behavior of these solutions as $\gamma \to 0^{+}$ is explored as well. Recent results from the literature are generally improved and extended.     

Averaging shifted histograms

A method which consists in shifting different histograms of the same spectrum and then taking their average is presented in order to smooth the data and to increase the localization accuracy and separation of the peaks. The statistical properties of this method are investigated. The average of two histograms with shifted bin limits is studied. It is shown that for histograms with random bin limits, distributed according to

$\text{F}{}_{\mathrm{i}}\text{(x)=}\text{∫}\text{?∞}\text{x}\text{?}{}_{\mathrm{i}}\text{(ξ, μ}{}_{\mathrm{i}}\text{, σ)}\text{d}\text{ξ}$

; where the standard deviation σ is very small compared to the difference of the means (μ_i+1 ? μ_i) for ll i the zero order approximation to the variance of this histogram is given by:

$\text{var}\text{∽}\text{(H)=}\text{∑}\text{i=0}\text{m}\text{(A}{}_{\mathrm{i+1}}\text{?a}{}_{\mathrm{i}}\text{)}{}^2\text{F}{}_{\mathrm{i+1}}\text{(x)(1?F}{}_{\mathrm{i+1}}\text{(x))}$

, where

$\text{a}{}_{\mathrm{i}}\text{=}\text{1}\text{x}{}_{\mathrm{i=1}}\text{?x}{}_{\mathrm{i}}\text{∫}\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i+1}}\text{g(ξ)}\text{d}\text{ξ}$

and g is an unknown function fitted by the histogram. Formula (1) gives also the relation:

$\text{v}\text{a}\text{?}\text{r}\text{(}\text{(H}{}_1\text{+ H}{}_2\text{)}\text{2}\text{) =}\text{1}\text{4}\text{(}\text{v}\text{a}\text{?}\text{r}\text{(H}{}_1\text{(x)) +}\text{v}\text{a}\text{?}\text{r}\text{(H}{}_2\text{(x))}$

, when H₁ and H₂ have stochastically independent bin limits.When the histogram H is considered as a spline function S of order one it is shown that for the minimization criterion with respect to the coefficient of the spline:

$\text{M}{}_1\text{=}\text{min}\text{∫}\text{x}{}_1\text{x}{}_{\mathrm{m+1}}\text{(g(x) ? S}{}_1\text{(x))}{}^2\text{d}\text{x}$

, the following result holds: $\text{M}{}_{\mathrm{a}}\text{?}{}^1{}_2\text{(M}{}_1\text{+ M}{}_2\text{)}$, where $\text{S}{}_{\mathrm{a}}\text{(x) =}{}^1{}_2\text{(S}{}_1\text{(x) + S}{}_2\text{(x))}$. If the number of shifted histograms tends to infinity, then

$\text{S}{}_{\mathrm{\infty }}\text{(x) = [Γ(x + h) + Γ(x ? h) ? 2Γ(x)]/h}{}^2$

, where $\text{Γ(x) =}\text{∫}\text{?∞}\text{x}\text{∫}\text{?∞}\text{η}\text{g(ξ)}\text{d}\text{ξ}\text{d}\text{η}$, and h is a constant bin size. Then

$\text{M}{}_{\mathrm{\infty }}\text{≈}\text{h}{}^4\text{144}\text{∫}\text{x}{}_1\text{x}{}_{\mathrm{m+1}}\text{g″}{}^2\text{(x)}\text{D}\text{x}$

. Extensions to two-dimensional histograms and to higher order (empirical distributions) are presented.     

Fault-Tolerant Embedding of Pairwise Independent Hamiltonian Paths on a Faulty Hypercube with Edge Faults

A Hamiltonian path in G is a path which contains every vertex of G exactly once. Two Hamiltonian paths P ₁=〈u ₁,u ₂,…,u _n 〉 and P ₂=〈v ₁,v ₂,…,v _n 〉 of G are said to be independent if u ₁=v ₁, u _n =v _n , and u _i ≠v _i for all 1<i<n; and both are full-independent if u _i ≠v _i for all 1≤i≤n. Moreover, P ₁ and P ₂ are independent starting at u ₁, if u ₁=v ₁ and u _i ≠v _i for all 1<i≤n. A set of Hamiltonian paths {P ₁,P ₂,…,P _k } of G are pairwise independent (respectively, pairwise full-independent, pairwise independent starting at u ₁) if any two different Hamiltonian paths in the set are independent (respectively, full-independent, independent starting at u ₁). A bipartite graph G is Hamiltonian-laceable if there exists a Hamiltonian path between any two vertices from different partite sets. It is well known that an n-dimensional hypercube Q _n is bipartite with two partite sets of equal size. Let F be the set of faulty edges of Q _n . In this paper, we show the following results:

Real-time estimation scheme for the spot cross volatility of jump diffusion processes

Given a finite set of observed data {_Xtk(_ω0),_Ytk(_ω0)}$\{X_{t_k}({\omega }_0)\text{,}{Y}_{t_k}({\omega }_0)\}$ of just one sample path at n$n$ regularly spaced time of the processes _Xt$X_t$ and _Yt$Y_t$ satisfying d_Xt=_a0(t)dt+_a1(t)d_W1(t)+_a2(t)d_W2(t)+d_J1(t),d_Yt=_b0(t)dt+_b1(t)d_W1(t)+_b2(t)d_W2(t)+d_J2(t),t∈[0,T]$d{X}_t=a_0(t)dt+a_1(t)d{W}_1(t)+a_2(t)d{W}_2(t)+d{J}_1(t)\text{,}d{Y}_t=b_0(t)dt+b_1(t)d{W}_1(t)+b_2(t)d{W}_2(t)+d{J}_2(t)\text{,}t\in [0\text{,}T]$, where _J1,_J2$J_1\text{,}{J}_2$ are jump process, we are to investigate a numerical scheme for the estimation of the value _νX,_Y(t)=_a1(t)_b1(t)+_a2(t)_b2(t)${\nu }_{X\text{,}Y}(t)=a_1(t)b_1(t)+a_2(t)b_2(t)$ called cross volatility. Our framework also contains the volatility estimation problem as a special case. We will show that our scheme works under mild assumptions on the activity of the jump process _Jt$J_t$.     

Validation of SeaWiFS chlorophyll a concentrations in the Southern Ocean: A revisit

Surface chlorophyll a concentrations (C_a, mg m^− 3) in the Southern Ocean estimated from SeaWiFS satellite data have been reported in the literature to be significantly lower than those measured from in situ water samples using fluorometric methods. However, we found that high-resolution (∼ 1 km²/pixel) daily SeaWiFS C_a (C_a^SWF) data (SeaDAS4.8, OC4v4 algorithm) was an accurate measure of in situ C_a during January-February of 1998-2002 if concurrent in situ data measured by HPLC (C_a^HPLC) instead of fluorometric (C_a^Fluor) measurements were used as ground truth. Our analyses indicate that C_a^Fluor is 2.48 ± 2.23 (n = 647) times greater than C_a^HPLC between 0.05 and 1.5 mg m^− 3 and that the percentage overestimation of in situ C_a by fluorometric measurements increases with decreasing concentrations. The ratio of C_a^SWF/C_a^HPLC is 1.12 ± 0.91 (n = 96), whereas the ratio of C_a^SWF/C_a^Fluor is 0.55 ± 0.63 (n = 307). Furthermore, there is no significant bias in C_a^SWF (12% and − 0.07 in linear and log-transformed C_a, respectively) when C_a^HPLC is used as ground truth instead of C_a^Fluor. The high C_a^Fluor/C_a^HPLC ratio may be attributed to the relatively low concentrations of chlorophyll b (C_b/C_a = 0.023 ± 0.034, n = 482) and relatively high concentrations of chlorophyll c (C_c/C_a = 0.25 ± 0.59, n = 482) in the phytoplankton pigment composition when compared to values from other regions. Because more than 90% of the waters in the study area, as well as in the entire Southern Ocean (south of 60° S), have C_a^SWF between 0.05 and 1.5 mg m^− 3, we consider that the SeaWiFS performance of C_a retrieval is satisfactory and for this C_a range there is no need to further develop a “regional” bio-optical algorithm to account for the previous SeaWiFS “underestimation”.     

Linguistic Z-number fuzzy soft sets and its application on multiple attribute group decision making problems

In this study, the concept of linguistic Z-number fuzzy soft set ($\text{◂⋅▸}LZnFSS$) is proposed to describe multiple uncertainties in practical decision making problems. $\text{◂⋅▸}LZnFSS$ combines the concepts of fuzzy soft set, linguistic Z-number, and soft set, which could reflect both of the uncertainty in structure and the uncertainty in detailed evaluations. As an initial idea, the set operations on $\text{◂⋅▸}LZnFSSs$ are put forward, the properties of such operations are also discussed. With traditional soft set based decision procedure and fuzzy soft set based decision procedure, a novel linguistic Z-number fuzzy soft set based group decision procedure is developed to solve multiattribute group decision making with linguistic Z-numbers. Wherein an extended technique for order preference by similarity to ideal solution is also developed. Finally, a numerical example is shown to illustrate the practicality and effectiveness of the given method.     

Local noncuppability in R/M

Given any [c],[a],[d]∈R/M such that [d]≤[a]≤[c], [a] is locally noncuppable between [c] and [d] if [d]<[a]≤[c]and [a] ∨ [b] < [c] for any [b]∈R/M such that [d]≤ [ b ] < [ c ]. It will be shown that given any nonzero [ c ] ∈ R/M, there are [ a ], [ d ] ∈R/M such that [d]<[a]≤[c] and[a] is locally noncuppable between [c] and[d].     

Fault diameter of Cartesian product graphs

The (k−1)-fault diameter Dk(G) of a k-connected graph G is the maximum diameter of G−F for any F⊂V(G) with |F|<k. Krishnamoorthy and Krishnamurthy first proposed this concept and gave Dκ(G₁)+κ(G₂)(G₁×G₂)?Dκ(G₁)(G₁)+Dκ(G₂)(G₂) when κ(G₁×G₂)=κ(G₁)+κ(G₂), where κ(G) is the connectivity of G. This paper gives a counterexample to this bound and establishes Dk₁+k₂(G₁×G₂)?Dk₁(G₁)+Dk₂(G₂)+1 for any ki-connected graph Gi and ki?1 for i=1,2.