Positive Solutions Research Articles

Positive Solutions of Elliptic Systems with Superlinear Terms on the Critical Hyperbola

Positive Solutions of Elliptic Systems with Superlinear Terms on the Critical Hyperbola

Positive solutions for 2nth order p-Laplacian problem with Sturm–Liouville type boundary conditions

Positive solutions for 2nth order p-Laplacian problem with Sturm–Liouville type boundary conditions

Closed-Form Kinematics Solutions for Redundant Medical Robots with Joint Limits and Singularity Avoidance

Closed-form forward and inverse position, velocity and acceleration kinematics problems solutions are developed for a redundant medical robot family. Developed solutions allow one to choose a specific joint as a set one. Solutions work in constant time, suitable for real-time operation. Methods for joint limit and singularity avoidance are proposed. Simulation experiments are performed on simulated KUKA MED 14 r820 robot.

Schrödinger–Poisson systems with zero mass in the Sobolev limiting case

AbstractWe study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schrödinger equation driven by a weighted ‐Laplace operator and without the mass term, and a higher‐order fractional Poisson equation. Since the system is set in , the limiting case for the Sobolev embedding, we consider nonlinearities with exponential growth. Existence is proved relying on the study of a corresponding Choquard equation in which the Riesz kernel is a logarithm, hence sign‐changing and unbounded from above and below. This is in turn solved by means of a variational approximating procedure for an auxiliary Choquard equation, where the logarithm is uniformly approximated by polynomial kernels. Our results are new even in the planar case .

Positive solutions for a p-Laplacian equation arising from micro-electromechanical systems model

In the present research paper, we investigate the positive solutions of the equation ΔpV=g(|x|)V−q(x),x∈RN, where N>p>2, 0<q≠1 and g is a continuous radial and strictly positive function on [0,+∞[.We consider various assumptions on the function g and delve into the intricate analysis of sky states and singular sky states solutions through logarithmic transformations. Additionally, we apply the theory of invariant manifolds in dynamical systems to establish the uniqueness of singular sky states.

Analytical Solution of the Two-Dimensional Steady-State Seepage Field of a Seepage Anisotropy Pit Considering the Free Surface

An anisotropic foundation pit steady-state seepage field under a suspended waterproof curtain support considering the position of the free surface is studied analytically, and an analytical solution for the free surface position is given. The head distribution in the three zones is expressed as a series solution using the separation of variables method, and the explicit solution for the extent of the seepage field in each zone is obtained by combining the continuity condition between zones and the series orthogonality condition. The free surface position is determined according to the condition that the total head of the free surface is equal to the position head. A comparison of the calculation results of the analytical method and the indoor test and finite element analysis results verifies the correctness of the analytical solution, and the analytical method has more calculation efficiency than the finite element numerical method. Employing the aforementioned methods to analyze the influence parameters of the free surface position, the results show that drawdown increases as the ratio of the vertical permeability coefficient to the horizontal permeability coefficient increases; the greater the ratio of pit width to depth, the more significant the drawdown, but when the ratio continues to exceed 1.5, the drawdown is negligible.

Positive and decreasing solutions for higher order Caputo boundary valueproblems with sign-changing Green’s function

Positive and decreasing solutions for higher order Caputo boundary valueproblems with sign-changing Green’s function

Minimum Principles for Sturm–Liouville Inequalities and Applications

A minimum principle for a Sturm–Liouville (S-L) inequality is obtained, which shows that the minimum value of a nonconstant solution of a S-L inequality never occurs in the interior of the domain (a closed interval) of the solution. The minimum principle is then applied to prove that any nonconstant solutions of S-L inequalities subject to separated inequality boundary conditions (IBCs) must be strictly positive in the interiors of their domains and are increasing or decreasing for some of these IBCs. These positivity results are used to prove the uniqueness of the solutions of linear S-L equations with separated BCs. All of these results hold for the corresponding second-order differential inequalities (or equations), which are special cases of S-L inequalities (or equations). These results are applied to two models arising from the source distribution of the human head and chemical reactor theory. The first model is governed by a nonlinear S-L equation, while the second one is governed by a nonlinear second-order differential equation. For the first model, the explicit solutions are not available, and there are no results on the existence of solutions of the first model. Our results show that all the nonconstant solutions are increasing and are strictly positive solutions. For the second model, many results on the uniqueness of the solutions and the existence of multiple solutions have been obtained before. Our results are applied to prove that all the nonconstant solutions are decreasing and strictly positive.

On a class of elliptic equations with critical perturbations in the hyperbolic space

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a ( x ) u p − 1 + ε u 2 ∗ − 1 in B N , u ∈ H 1 ( B N ) , where B N denotes the hyperbolic space, 2 &lt; p &lt; 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 &lt; p &lt; + ∞, if N = 2, λ &lt; ( N − 1 ) 2 4 , and 0 &lt; a ∈ L ∞ ( B N ). We first prove the existence of a positive radially symmetric ground-state solution for a ( x ) ≡ 1. Next, we prove that for a ( x ) ⩾ 1, there exists a ground-state solution for ε small. For proof, we employ “conformal change of metric” which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a ( x ) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.

Multiple high energy solutions of a nonlinear Hardy–Sobolev critical elliptic equation arising in astrophysics

In this article, we study the existence and multiplicity of high energy solutions to the problem proposed as a model for the dynamics of galaxies: −Δu+V(x)u=|u|2∗−2u|y|,x=(y,z)∈Rm×Rn−m,where n>4, 2≤m<n, 2∗≔2(n−1)n−2 and potential function V(x):Rn→R. Benefiting from a global compactness result, we show that there exist at least two positive high energy solutions. Our proofs are based on barycenter function, quantitative deformation lemma and Brouwer degree theory.

Analysis of High-Performance Parallel Computing using TOPSIS Method

High-performance parallel computing involves the simultaneous execution of multiple tasks or processes, orchestrated to achieve improved computational speed and efficiency. This approach leverages the power of parallelism, exploiting both multi-core CPUs and GPUs, distributed computing clusters, and specialized hardware accelerators. The fundamental idea is to divide a task into smaller sub-tasks that can be executed concurrently, thereby reducing processing time and enhancing overall performance. High-performance parallel computing is a transformative approach that enables us to tackle computationally intensive tasks efficiently. This abstract highlight its significance in contemporary computing and sets the stage for further exploration of the intricacies and innovations within this dynamic field. Researchers and practitioners continue to push the boundaries of what is achievable, making high-performance parallel computing a cornerstone of modern computational science and technology. High-performance parallel computing research is of paramount significance due to its transformative impact across diverse fields. It empowers scientists to tackle complex problems that were once computationally intractable, unlocking new frontiers in scientific discovery. It drives innovation in engineering and design, optimizing product development and manufacturing processes across industries. In healthcare, it accelerates genomics research and drug discovery, offering hope for improved medical treatments. Financial institutions rely on it for data analysis and risk assessment, shaping the global economy. Weather forecasting and environmental modelling are enhanced, aiding disaster preparedness and conservation efforts. In the digital age, parallel computing underpins artificial intelligence, enabling advancements in natural language processing and machine learning. Furthermore, it has vital applications in national security, space exploration, and materials science. In essence, high-performance parallel computing research serves as the backbone of technological progress, fostering innovation, efficiency, and problem-solving across a wide spectrum of disciplines, ultimately shaping the future of our world. TOPSIS, this method involves evaluating the geometric distance between each alternative solution and two reference solutions: the positive ideal solution and the negative ideal solution. The underlying principle of TOPSIS assumes that the criteria being assessed are of an ascending nature, where larger values represent better performance. To account for disparate dimensions or scales among the criteria, normalization is often employed within the TOPSIS framework. From the result Scatter- free imaging is got the first rank and object-scatter imaging is having the lowest rank.

Nonexistence of nontrivial solutions to Kirchhoff-like equations

Subject to given boundary data, nonexistence of solution to the one-dimensional Kirchhoff-like equation − M ( ( a ∗ | u | q ) ( 1 ) ) u ( t ) = λ f ( t , u ( t ) ) , 0 &gt; t &gt; 1 \begin{equation*} -M\Big (\big (a*|u|^q\big )(1)\Big )u(t)=\lambda f\big (t,u(t)\big ),\ 0&gt;t&gt;1 \end{equation*} is considered. In particular, a condition is provided on the parameter λ \lambda such that for each λ &gt; λ 0 \lambda &gt;\lambda _0 , where λ 0 \lambda _0 is defined in terms of initial data, the boundary value problem has no nontrivial positive solution.

Establishment and analysis of a unified mathematical model of kinematics and dynamics of flexible parallel mechanisms

Abstract The conventional positive solution approach for determining the forward motion of the kinematic mechanism relies primarily on the outcomes of the inverse solution of motion, which is not universally applicable to flexible parallel mechanisms. Furthermore, the process of analyzing the natural frequency of parallel mechanisms is complex, cumbersome, and challenging. More importantly, the two models are independent, and there is no unified mathematical analysis model for flexible parallel mechanisms. Therefore, this paper establishes a unified mathematical model for kinematics and dynamics analysis of flexible parallel mechanisms through an improved multi-body system transfer matrix method. Taking the flexible 3-RPU planar mechanism and the space 2-RPU (2-UPS) space mechanism as examples, respectively, the forward motion solution of the mechanism was analyzed and its working space range and resonance frequency were solved. Finally, the calculation results are compared with the simulation results, and the feasibility and accuracy of the method are proved. Through this method, we can quickly solve the motion characteristics of the mechanism without relying on the results of the inverse solution of the mechanism, which provides suggestions for the precise control of flexible parallel mechanisms and also provides corresponding reference basis for the practical application of flexible parallel mechanisms.

Long-Term Behavior of Positive Solutions of a Certain Nonlinear System of Difference Equations

Long-Term Behavior of Positive Solutions of a Certain Nonlinear System of Difference Equations

Single‐peak solution for a fractional slightly subcritical problem with non‐power nonlinearity

AbstractWe consider the following fractional problem involving slightly subcritical non‐power nonlinearity, where is a smooth bounded domain in , , , is the fractional critical Sobolev exponent and is a small parameter, is the spectral fractional Laplacian operator. We construct a positive bubbling solution, which concentrates at a nondegenerate critical point of the Robin function by Lyapunov–Schmidt reduction procedure.

Penerapan Metode TOPSIS Terhadap Skala Prioritas Aspirasi Civitas Kampus UKRI

The campus community at the National University of the Republic of Indonesia has a fairly high level of criticism of the development of the campus itself, however, it turns out that the campus does not have sufficient media to accommodate the aspirations expressed by the campus community. Aspirations can be conveyed, but with a limited scale and time that also makes it difficult for campuses to make decisions that will later have an impact on the development of the National University of the Republic of Indonesia campus, therefore the TOPSIS method is here to make it easier for campuses to make decisions. The TOPSIS calculation refers to the alternative distances to the positive ideal solution and the negative ideal solution. From the results of calculations that have been normalized, the highest value has the right to get priority as the best solution, and conversely, the lowest value is not prioritized to be used as a reference in decision making.

Exposure Of Positive Integer Solutions to an Equation Comprising Kaprekar Numbers

Exposure Of Positive Integer Solutions to an Equation Comprising Kaprekar Numbers

Instabilities of standing waves and positivity in traveling waves to a higher‐order Schrödinger equation

The aim of this paper is to explore a Schrödinger equation that incorporates a higher‐order operator. Traditional models for electron dynamics have utilized a second‐order diffusion Schrödinger equation, where oscillatory behavior is achieved through complex domain formulations. Incorporating a higher‐order operator enables the induction of oscillatory spatial patterns in solutions. Our analysis initiates with a variational formulation within generalized spaces, facilitating the examination of solution boundedness. Subsequently, we delve into the oscillatory characteristics of solutions, drawing upon a series of lemmas originally applied to the Kuramoto–Sivashinsky equation, the Cahn–Hilliard equation, and other equations that employ higher‐order operators. Specific solution types, such as standing waves, are numerically investigated to illustrate the oscillatory spatial patterns. The discussion then extends to the theory of traveling waves to establish general conditions for positive solutions. A contribution of this work is the precise evaluation of a critical traveling wave speed, denoted as , above which the first minimum remains positive. For values of the traveling wave speed significantly greater than , the solutions can be entirely positive.

Negative News about the Catholic Church as A Challenge for a More Active and Critical Reading of the News

Media content is not the facts that happened, but the media's representation of these facts. The media report and narrate reality in a way that is as interesting as possible for the audience. Because media content contains exaggeration, drama and speculation and frames facts in different contexts, consumers of media content rightly feel that the media distort the truth. We find that there is a very large discrepancy between the life of the Catholic Church (and Catholics) and the news we find in the media, and that the line between fake news and convincing news is very thin. In order to ascertain the facts in media news (and to distinguish between facts, exaggerations and representations), it is necessary to critically evaluate media content. This requires a high level of personal engagement and knowledge of the topic, but also professional expertise in media content. As a result, most users do not analyse media content, but interpret it intuitively or take it literally. Our 2019 survey shows that the media in Slovenia most often associate the Catholic Church with sexual abuse, money and politics. In an analysis of 3 735 articles, we examined these topics in detail and found that the connotations of these articles are polarised: half are positive, half negative. Furthermore, in these articles we find no reports on serious socio-political problems and the positive solutions offered by the Catholic Church.

Existence of positive solutions for an elliptic system with sign-changing weights

Using the method of sub-super solutions and comparison principle, we study the existence of positive solutions for a class of quasilinear elliptic systems with sign-changing weights.