Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in $R^{n}\backslash\{0\}$

We consider the following singular semilinear problem $$ \textstyle\begin{cases} \Delta u(x)+p(x)u^{\gamma }=0,\quad x\in D ~(\text{in the distributional sense}), \\ u>0,\quad \text{in }D, \\ \lim_{ \vert x \vert \rightarrow 0} \vert x \vert ^{n-2}u(x)=0, \\ \lim_{ \vert x \vert \rightarrow \infty }u(x)=0,\end{cases} $$ { Δ u ( x ) + p ( x ) u γ = 0 , x ∈ D ( in the distributional sense ) , u > 0 , in D , lim | x | → 0 | x | n − 2 u ( x ) = 0 , lim | x | → ∞ u ( x ) = 0 , where $\gamma <1$ γ < 1 , $D=\mathbb{R}^{n}\backslash \{0\}$ D = R n ∖ { 0 } ($n\geq 3$ n ≥ 3 ) and p is a positive continuous function in D, which may be singular at $x=0$ x = 0 . Under sufficient conditions for the weighted function $p(x)$ p ( x ) , we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.

Optimal Port Design Minimizing Standing Waves with A Posteriori Long Term Shoreline Sustainability Analysis

Optimization theory is applied to a coastal engineering problem that is the design of a port. This approach was applied to the redesign of La Turballe Port in order to increase the exploitable surface area and simultaneously reduce the occurrence of long waves within the port. Having defined the cost function as a weighted function of wave amplitude and with the chosen parameterization of the port, results show that an extended jetty and a widened mole yield a unique optimal solution. This work demonstrates that numerical optimization may be quick and efficient in the identification of port solutions consistent with classic engineering even in the context of complex problems.

Steffensen-Type Inequalities with Weighted Function via (γ, a)-Nabla-Conformable Integral on Time Scales

The primary goal of this our research is to prove several new ∇-conformable dynamic Steffensen inequalities that were demonstrated in recent works. Our results generalize and extend existing results in the literature. Many special cases of the proposed results are obtained and analyzed such as new conformable fractional sum inequalities and new classical conformable fractional integral inequalities.

L p Smoothness on Weighted Besov–Triebel–Lizorkin Spaces in terms of Sharp Maximal Functions

It is known, in harmonic analysis theory, that maximal operators measure local smoothness of L p functions. These operators are used to study many important problems of function theory such as the embedding theorems of Sobolev type and description of Sobolev space in terms of the metric and measure. We study the Sobolev-type embedding results on weighted Besov–Triebel–Lizorkin spaces via the sharp maximal functions. The purpose of this paper is to study the extent of smoothness on weighted function spaces under the condition M α # f ∈ L p , μ , where μ is a lower doubling measure, M α # f stands for the sharp maximal function of f , and 0 ≤ α ≤ 1 is the degree of smoothness.

Malliavin Calculus and the Optimal Weighting Function in a Pure Jump Lévy Setting

The paper is devoted to the problem of obtaining weighting functions for the Greeks of an option price written on a stock whose dynamics are of pure jump type. The problem is motivated by the work of Fourni\'e et al. [8, 9], who considered the price sensitivities of a frictionless market and proved that Greeks can be computed as the expectation of the product of the discounted payoff $\Phi$ and a suitable weighted function, i.e.Greek = E[Φ(XT)weight]. Since the weighting functions are random variables that need to be explicitly computed on each specific case, we establish necessary and sufficient conditions to be satisfied. The method used relied on the Malliavin calculus for Levy processes.

The utility of concentration duration curve in CH4 and CO2 risk assessment

Current CH4 and CO2 risk assessment of comparing the single occurrence of worst-case concentration with trigger values of 5% and 1% respectively is often of low resolution but could be improved by the application of the concept of Concentration Duration Curve (CDC). With the aid of the Gasclam (In-borehole continuous gas monitor), four sites were monitored for CH4 and CO2 concentrations, and the time-series datasets used to construct CDC. The result shows that a 5% CH4 concentration is exceeded for 17, 41, 0, and 0% of the monitoring period in sites 1-4 respectively, whilst a 1% CO2 concentration was exceeded for 75, 75.5, 100, and 93% of the time in sites 1-4 respectively. The recorded worst case CH4 concentration are 11.5, 22.1, 2.7, and 1.56% in sites 1-4 respectively while that of CO2 concentration are 8.2, 15.5, 3.3, and 6.71% in sites 1-4 respectively This implies that treating risk in terms of a single occurrence of the worst-case ground-gas concentration rather than any sort of time-weighted function can be defective. While the concept of CDC can be useful in improving risk assessment due to CH4 and CO2, the worst-case ground-gas concentration may not occur during the monitoring period, therefore prediction is required. To predict the worst-case ground-gases concentration requires an understanding of the processes responsible for controlling gas concentration.

Linear Differential Equations on Some Classes of Weighted Function Spaces

Some weighted-type classes of holomorphic function spaces were introduced in the current study. Moreover, as an application of the new defined classes, the specific growth of certain entire-solutions of a linear-type differential equation by the use of concerned coefficients of certain analytic-type functions, that is the equation h(k)+Kk−1(υ)h(k−1)+…+K1(υ)h′+K0(υ)h=0, will be discussed in this current research, whereas the considered coefficients K0(υ),…,Kk−1(υ) are holomorphic in the disc ΓR={υ∈C:|υ|<R},0<R≤∞. In addition, some non-trivial specific examples are illustrated to clear the roles of the obtained results with some sharpness sense. Hence, the obtained results are strengthen to some previous interesting results from the literature.

A comprehensive G-Lz-ADC effectiveness evaluation model for the single communication satellite system in the context of poor information

PurposeDuring the service period of communication satellite systems, their performance is often degraded due to the depletion mechanism. In this paper, the grey system theory is applied to the multi-state system effectiveness evaluation and the grey Lz-transformation ADC (availability, dependability and capability) effectiveness evaluation model is constructed to address the characteristics of the communication satellite system such as different constituent subsystems, numerous states and the inaccuracy and insufficiency of data.Design/methodology/approachThe model is based on the ADC effectiveness evaluation method, combined with the Lz transformation and uses the definite weighted function of the three-parameter interval grey number as a bridge to incorporate the possibility of system performance being greater than the task demand into the effectiveness solution algorithm. At the same time, using MATLAB (Matrix laboratory) to solve each state probability, the same performance level in the Lz transform is combined. Then, the system effectiveness is obtained by Python.FindingsThe results show that the G-Lz-ADC model constructed in this paper can accurately evaluate the effectiveness of static/dynamic systems and certain/uncertain system and also has better applicability in evaluating the effectiveness of the multi-state complex system.Practical implicationsThe G-Lz-ADC effectiveness evaluation model constructed in this paper can effectively reduce the complexity of traditional effectiveness evaluation models by combining the same performance levels in the Lz-transform and solving the effectiveness of the system with the help of computer programming, providing a new method for the effectiveness evaluation of the complex MSS. At the same time, the weaknesses of the system can be identified, providing a theoretical basis for improving the system’s effectiveness.Originality/valueThe possibility solution method based on the definite weighted function comparing the two three-parameter interval grey numbers is constructed, which compensates for the traditional calculation of the probability based on numerical values and subjective preferences of decision-makers. Meanwhile, the effectiveness evaluation model integrates the basic theories of three-parameter interval grey number and its definite weighted function, Grey−Markov, grey universal generating function (GUGF), grey multi-state system (GMSS), etc., which is an innovative method to solve the effectiveness of a multi-state instantaneous communication satellite system.

Combined Effects of Wildfire and Vegetation Cover Type on Volcanic Soil (Functions and Properties) in a Mediterranean Region: Comparison of Two Soil Quality Indices

Mediterranean regions are the most impacted by fire in Europe. The effects of fire on soil greatly vary according to several factors such as vegetation cover type, but they are scarcely studied. Therefore, this research aimed at evaluating the combined impacts of fire and vegetation on single soil characteristics and on the overall soil quality and functionality through two soil quality indices, simple additive (SQI) and a weighted function (SQIFUNCT). In order to reach the aims, burnt and unburnt soils were collected under different vegetation cover types (herbs and shrubs, black locust, pine and holm oak) within the Vesuvius National Park. The soils were analyzed for the main abiotic (water and organic matter content, total C, N, Ca, K, Cu and Pb concentrations, C/N ratio) and biotic (microbial and fungal biomasses, basal respiration, β-glucosidase activity) characteristics. On the basis of the investigated soil characteristics, several soil functions (water retention, nutrient supply, contamination content, microorganism habitat and activities), and the soil quality indices were calculated. The results showed that the impact of fire on soil quality and functionality was mediated by the vegetation cover type. In fact, fire occurrence led to a decrease in water and C/N ratio under herbs, a decrease in C concentration under holm oak and a decrease in Cu and Pb concentrations under pine. Although the soil characteristics showed significant changes according to vegetation cover types and fire occurrence, both the additive and weighted function soil quality indices did not significantly vary according to both fire occurrence and the vegetation cover type. Among the different vegetation cover types, pine was the most impacted one.