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Abstract In the present article, the Hyers–Ulam stability of the following inequality is analyzed: 0.1 { d ( f ( ı + ȷ ) , ( f ( ı ) + f ( ȷ ) ) ) ≤ d ( ρ 1 ( ( f ( ı + ȷ ) + f ( ı − ȷ ) , 2 f ( ı ) ) ) d ( f ( ı + ȷ ) , ( f ( ı ) + f ( ȷ ) ) ) ≤ + d ( ρ 2 ( 2 f ( ı + ȷ 2 ) , ( f ( ı ) + f ( ȷ ) ) ) ) $$ \textstyle\begin{cases} d (f(\imath +\jmath ), \ (f(\imath )+ \ f(\jmath )) )\leq d (\rho _{1}((f(\imath +\jmath )+ f(\imath - \jmath ),\ 2f(\imath )) ) \\ \hphantom{ d (f(\imath +\jmath ), \ (f(\imath )+ \ f(\jmath )) )\leq}{}+ d (\rho _{2} (2f (\frac{\imath +\jmath}{2} ), \ (f(\imath )+ f(\jmath )) ) ) \end{cases} $$ in the setting of digital metric space, where ρ 1 $\rho _{1}$ and ρ 2 $\rho _{2}$ are fixed nonzero complex numbers with 1 > 2 | ρ 1 | + | ρ 2 | $1>\sqrt{2}|\rho _{1}|+|\rho _{2}|$ by using fixed point and direct approach.

In this paper, we study the Onsager-Machlup function and its relationship to the Freidlin-Wentzell function for measures equivalent to arbitrary infinite dimensional Gaussian measures. The Onsager-Machlup function can serve as a density on infinite dimensional spaces, where a uniform measure does not exist, and has been seen as the Lagrangian for the "most likely element". The Freidlin-Wentzell rate function is the large deviations rate function for small-noise limits and has also been identified as a Lagrangian for the "most likely element". This leads to a conundrum - what is the relationship between these two functions? We show both pointwise and G-convergence (which is essentially the convergence of minimizers) of the Onsager-Machlup function under the small-noise limit to the Freidlin-Wentzell function - and give an expression for both. That is, we show that the small-noise limit of the most likely element is the most likely element in the small noise limit for infinite dimensional measures that are equivalent to a Gaussian. Examples of measures include the law of solutions to path-dependent stochastic differential equations and the law of an infinite system of random algebraic equations. [ABSTRACT FROM AUTHOR]

In this article, we consider a nonlinear elliptic unilateral equation whose model is − ∑ i = 1 N ∂ i σ i (x , u , ∇ u) + L (x , u , ∇ u) + N (x , u , ∇ u) = μ − div ϕ (u) in Ω. We prove the existence of entropy solutions for the aforementioned equation in the anisotropic Sobolev space, under the hypotheses, μ = f − div F belongs to L 1 (Ω) + W − 1 , p ′ (Ω) . The nonlinear terms L (x , s , ∇ u) satisfy the sign and growth conditions, and N (x , s , ∇ u) verifies only the growth conditions. [ABSTRACT FROM AUTHOR]

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Aims: To let students know that packaging design teaching is both a science and an art like any other course, only when students understand this, can they teach and educate people once they are engaged in this profession. Methods: Observation, experiment, investigation, and other methods were used to carry out a relatively long time of teaching practice in the course of Packaging Design, and the teaching art was explored. Results: Practice shows that if students want to truly realize that teaching is both a science and an art, they must adhere to the purpose of instilling in teaching for a long time, and fully understand the history of teaching artistry, its characteristics, and its functions. Conclusion: Through the mathematical statistical analysis and research of teaching art practice and effect, we believe that any course is not a simple knowledge teaching, it is a process of educating people. We must cultivate the next generation to be well-rounded individuals, ensuring they are intellectually enriched and nurtured in moral values, physical health, aesthetic appreciation, and work skills. This holistic approach ensures that while they acquire knowledge, they are also exposed to and trained in the arts and practical skills. Only when people are well-trained can society be harmonious and beautiful. Therefore, the research on teaching art in packaging design has analyzed the three most important artistic effects of teaching blackboard art, language art, and teaching preparation art. Those with good teaching art effects account for 16-18%, those with medium teaching art effects account for 70-80%, and those with poor teaching art effects account for 10-16%.