Li–Yau inequalities for general non-local diffusion equations via reduction to the heat kernel

We establish a reduction principle to derive Li–Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li–Yau inequality for positive solutions u to the fractional (in space) heat equation of the form $$(-\Delta )^{\beta /2}(\log u)\le C/t$$ ( - Δ ) β / 2 ( log u ) ≤ C / t , where $$\beta \in (0,2)$$ β ∈ ( 0 , 2 ) . We also show that this Li–Yau inequality allows to derive a Harnack inequality. We further illustrate our general result with an example in the discrete setting by proving a sharp Li–Yau inequality for diffusion on a complete graph.

A Convenient Approximation of the Projectile Motion with Quadratic Air Resistance

A convenient approximated analytic solution is proposed for the problem of the motion of a body under a resistive force, acting in the magnitude of the squared velocity of the body. This solution is an explicit function of time, that keeps a good behavior both near the initial state and far from the initial state. To obtain a general analytic solution, we firstly used a reduction principle to be able to manipulate scalar objects, and we analyzed limit behaviors, both near the initial state and far from the initial state. Secondly, we proposed an approximated analytic solution with heuristics based on the built knowledge. Finally, a robust and stable integration scheme is proposed, based on the obtained analytic solution. We compared the scheme with other standard integration schemes.

Innovative design of modern mortise and tenon structure under the concept of green reduction

In the furniture industry, some traditional Chinese mortise and tenon joints are not suitable for the current requirements for carbon reduction and environmental protection of furniture products. This article aims to explore new ideas and new methods of modern mortise and tenon structure design. The reduction principle in the green design concept is introduced for the furniture modern mortise and tenon structure design. Based on modern furniture, a systematic analysis of the modern mortise and tenon structure design is carried out. Additionally, this review discusses the method of applying the reduction principle in modern tenon and tenon structure design in the context of green design. The development status and trend of modern mortise and tenon structure design is summarized and an innovative design practice of modern mortise and tenon structure is carried out. The combination of green design and furniture design has important practical significance in a modern context.

Innovative design of modern mortise and tenon structure under the concept of green reduction

In the furniture industry, some traditional Chinese mortise and tenon joints are not suitable for the current requirements for carbon reduction and environmental protection of furniture products. This article aims to explore new ideas and new methods of modern mortise and tenon structure design. The reduction principle in the green design concept is introduced for the furniture modern mortise and tenon structure design. Based on modern furniture, a systematic analysis of the modern mortise and tenon structure design is carried out. Additionally, this review discusses the method of applying the reduction principle in modern tenon and tenon structure design in the context of green design. The development status and trend of modern mortise and tenon structure design is summarized and an innovative design practice of modern mortise and tenon structure is carried out. The combination of green design and furniture design has important practical significance in a modern context.

Application of Double Strength Reduction Factor Method in the Stability Analysis of Rock Slopes

The cohesion c and internal friction angle φ play different roles in the progressive failure process of the slope, which indicates that the reduction factors kc and kφ should be different in the calculation. Based on this, the program of double strength reduction factor method was compiled with FISH language, in order to study its application in the rock slopes under different distributions of weak interlayer, and the following conclusions were drawn: (1) the plastic zone calculated by double strength reduction factor method is generally distributed in the weak interlayer, which is basically consistent with the calculation result of the traditional method; (2) the degree to which c and φ play a role is related to the inclination angle of the bottom sliding surface of the unstable block θ. If θ < 45°, φ will play a greater role. If θ ≥ 45°, c will play a greater role; (3) according to the “Pan’s principle,” the matching reduction principle of “kc > kφ” can be adopted when θ < 45°, and the matching reduction principle of “kc < kφ” can be adopted when θ ≥ 45°; (4) the definition of the comprehensive safety factor “K2” in the text is more suitable for the application of double strength reduction factor method in the stability analysis of rock slopes. The applicability of the above conclusions is verified by an actual engineering.

Dimensionality reduction applied to logical judgments

Purpose: To study the effect of the application of the dimensionality reduction in logical judgment (or logical reasoning, logical inference) programs. Methods: Use enumeration and dimensionality reduction methods to solve logical judgment problems.The effect of the two methods is illustrated in the form of a case study. Results: For logical judgmentproblems, using enumeration method to find the best answer is a comprehensive and fundamental method, but the disadvantage is that it is computationally intensive and computationally inefficient. Compared with the ideas of parallel treatment of known conditions by enumeration method, the application of dimensionality reduction thinking was built on the basis of fully mining information for feature extraction and feature selection. Conclusions: The dimensionality reduction method was applied to the logical judgment problems, and on the basis of fully mining information, the dimensionality reduction principle of statistics were applied to stratify and merge variables with the same or similar characteristics to achieve the purpose of streamlining variables, simplifying logical judgment steps, reducing computation and improving algorithm efficiency.

Decomposition and stability of linear singularly perturbed systems with two small parameters

In the domain $\Omega =\left\{\left(t,\varepsilon _{1}, \varepsilon _{2} \right): t\in {\mathbb R},\varepsilon _{1}>0, \varepsilon _{2} >0\right\}$, we consider a linear singularly perturbed system with two small parameters \[ \left\{ \begin{array}{l} {\dot{x}_{0} =A_{00} x_{0} +A_{01} x_{1} +A_{02} x_{2},} \\ {\varepsilon _{1} \dot{x}_{1} =A_{10} x_{0} +A_{11} x_{1} +A_{12} x_{2},} \\ {\varepsilon _{1} \varepsilon _{2} \dot{x}_{2} =A_{20} x_{0} +A_{21} x_{1} +A_{22} x_{2},} \end{array}\right. \] where $x_{0} \in {\mathbb R}^{n_{0}}$, $x_{1} \in {\mathbb R}^{n_{1}}$, $x_{2} \in {\mathbb R}^{n_{2}}$. In this paper, schemes of decomposition and splitting of the system into independent subsystems by using the integral manifolds method of fast and slow variables are investigated. We give the conditions under which the reduction principle is truthful to study the stability of zero solution of the original system.

Reduction principle at work

The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.