Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

<p style='text-indent:20px;'>In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-<inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-<inline-formula><tex-math id="M2">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.</p>

ANALISIS PRODUK KOSMETIK IMPOR TERKAIT BRAND DAN PENGARUHNYA TERHADAP PEMBELIAN KOSMETIK IMPOR DI SAMARINDA, KALIMANTAN TIMUR

This study discusses about imported cosmetic products which are beauty products that are becoming a great choice for Indonesian women, where most of the imported products have higher quality brand than local products. This study uses a quantitative descriptive approach that is empirical inductive namely data collection, data analysis, based on facts and previous research in the field. This research uses a quantitative approach through a survey method and is carried out with an explanatory approach using a Likert scale. Respondents were asked to convey their level of agreement about the values and influences between the variables to be measured. The analytical method used in this study is the the method of multiple linear analysis with statistical methods using the Statistical Solution and Services (SPSS) version 24 software tools.The results of the analysis that have been done show that Brand Image has a significant effect on purchasing decisions, this result is proven by a significance value greater than 5%. Brand awareness also has a significant effect on purchasing decisions, this result is evidenced by the significance value greater than 5%. Perception of Price has a significant effect on purchasing decisions, this result is evidenced by the significance value greater than 5%.

Equivalence between invariant measures and statistical solutions for the 2D non-autonomous magneto-micropolar fluid equations

In this article, we aim to investigate the regularity of statistical solution for the 2D non-autonomous magneto-micropolar fluid equations as well as the relationship between invariant measures and statistical solutions. Firstly, to get the regularity of the statistical solution, we prove the existence and regularity of the pullback attractor for the equations. Then we prove the statistical solution possesses some regularity properties by using regularity of the pullback attractor. Finally, we prove the statistical solution is actual an invariant measure for the equations.

Exact statistical solution for the hopping transport of trapped charge via finite Markov jump processes

In this study, we developed a discrete theory of the charge transport in thin dielectric films by trapped electrons or holes, that is applicable both for the case of countable and a large number of traps. It was shown that Shockley–Read–Hall-like transport equations, which describe the 1D transport through dielectric layers, might incorrectly describe the charge flow through ultra-thin layers with a countable number of traps, taking into account the injection from and extraction to electrodes (contacts). A comparison with other theoretical models shows a good agreement. The developed model can be applied to one-, two- and three-dimensional systems. The model, formulated in a system of linear algebraic equations, can be implemented in the computational code using different optimized libraries. We demonstrated that analytical solutions can be found for stationary cases for any trap distribution and for the dynamics of system evolution for special cases. These solutions can be used to test the code and for studying the charge transport properties of thin dielectric films.

Statistical Solutions for a Nonautonomous Modified Swift-Hohenberg Equation

We consider the nonautonomous modified Swift-Hohenberg equation $$u_t+\Delta^2u+2\Delta u+au+b|\nabla u|^2+u^3=g(t,x)$$ on a bounded smooth domain $\Omega\subset\R^n$ with $n\leqslant 3$. We show that, if $|b|<4$ and the external force $g$ satisfies some appropriate assumption, then the associated process has a unique pullback attractor in the Sobolev space $H_0^2(\Omega)$. Based on this existence, we further prove the existence of a family of invariant Borel probability measures and a statistical solution for this equation.

Assessing heterogeneity in spatial data using the HTA index with applications to spatial transcriptomics and imaging

Tumour heterogeneity is being increasingly recognised as an important characteristic of cancer and as a determinant of prognosis and treatment outcome. Emerging spatial transcriptomics data hold the potential to further our understanding of tumour heterogeneity and its implications. However, existing statistical tools are not sufficiently powerful to capture heterogeneity in the complex setting of spatial molecular biology. We provide a statistical solution, the HeTerogeneity Average index (HTA), specifically designed to handle the multivariate nature of spatial transcriptomics. We prove that HTA has an approximately normal distribution, therefore lending itself to efficient statistical assessment and inference. We first demonstrate that HTA accurately reflects the level of heterogeneity in simulated data. We then use HTA to analyse heterogeneity in two cancer spatial transcriptomics datasets: spatial RNA sequencing by 10x Genomics and spatial transcriptomics inferred from H&E. Finally, we demonstrate that HTA also applies to 3D spatial data using brain MRI. In spatial RNA sequencing we use a known combination of molecular traits to assert that HTA aligns with the expected outcome for this combination. We also show that HTA captures immune-cell infiltration at multiple resolutions. In digital pathology we show how HTA can be used in survival analysis and demonstrate that high levels of heterogeneity may be linked to poor survival. In brain MRI we show that HTA differentiates between normal ageing, Alzheimer's disease and two tumours. HTA also extends beyond molecular biology and medical imaging, and can be applied to many domains, including GIS. Source code and python package are available at https://alonalj.github.io/HTA.html .

Using optimized statistical distances to confront distributed denial of service attacks in software defined networks

Software-defined networks (SDN) are an emerging architecture that provides promising amends to put an end to current infrastructure constraints by optimized bandwidth utilization, flexibility in network management and configuration, and pulling down operating costs in traditional network structures. Despite the advantages of this architecture, SDNs may become the victim of a distributed denial of service (DDOS) attacks as the result of potential vulnerabilities in various layers. Therefore, the rapid detection of attack traffic in the early stages is very important. In this paper, we have proposed statistical solution to detect and to mitigate distributed denial of service attack in software-defined networks utilizing the unique capabilities of the SDN architecture. Here, the exponential weighted moving average protection mechanism (EWMA) in statistical distances is exploited. The simulation results of our extensive experiments showed that our mechanism is able to quick detection of attack traffics and take amendatory actions. Moreover, the evaluations show the superiority of the proposed algorithm with respect to other statistical methods.

Active Galactic Nuclei as Factories for Eccentric Black Hole Mergers

Black hole mergers detected by LIGO and Virgo continue delivering transformational discoveries. The most recent example is the merger GW190521, which is the first detected with component masses exceeding the limit predicted by stellar models, and the first with non-zero orbital eccentricity. The large masses can be explained by build up through successive mergers, which has been suggested to occur efficiently in the gas disks of active galactic nuclei (AGN). The eccentricity, however, is a major puzzle. Here we show that AGN-disk environments naturally lead to a very high fraction of highly eccentric mergers, if interactions between binaries and singles are frequent, and the interactions are constrained to a plane representing the AGN-disk. By deriving a statistical solution to the chaotic 3-body problem with the inclusion of General Relativistic corrections, we find in our fiducial AGN-disk model that up to 70 percent of all black hole mergers could appear with an eccentricity >0.1 in LIGO/Virgo. Besides representing the most effective mechanism for producing eccentric mergers presented to date, our results have also profound implications for the origin of GW190521, and open up new lines of research on black hole scatterings in disk environments with far-reaching implications for gravitational wave astrophysics.