Explore the vast range of titles available.

Consider the family of generalized parabolas y=−axr+c|a,r,c,x>0,risafixedconstant that pass through a given point in the first quadrant (and hence, depend on one parameter only). Find the parameter values for which the piece of the corresponding parabola in the first quadrant either encloses a minimum area, or has a minimum length. We find a sufficient condition under which given the fixed point, the area minimizing curve and the length minimizing curve coincide. The problem led us to a certain implicit function and we explored its asymptotic behavior and convexity.

This study considers the statistical estimation of relations presented by implicit functions. Such structures define mutual interconnections of variables rather than outcome variable dependence by predictor variables considered in regular regression analysis. For a simple case of two variables, pairwise regression modeling produces two different lines of each variable dependence using another variable, but building an implicit relation yields one invertible model composed of two simple regressions. Modeling an implicit linear relation for multiple variables can be expressed as a generalized eigenproblem of the covariance matrix of the variables in the metric of the covariance matrix of their errors. For unknown errors, this work describes their estimation by the residual errors of each variable in its regression by the other predictors. Then, the generalized eigenproblem can be reduced to the diagonalization of a special matrix built from the variables’ covariance matrix and its inversion. Numerical examples demonstrate the eigenvector solution’s good properties for building a unique equation of the relations between all variables. The proposed approach can be useful in practical regression modeling with all variables containing unobserved errors, which is a common situation for the applied problems.

In this article, we formalize differentiability of implicit function theorem in the Mizar system [3], [1]. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here. In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in [6] is cited. We referred to [10], [11], and [2] in the formalization.

AbstractWe consider smooth mappings acting from one Banach space to another and depending on a parameter belonging to a topological space. Under various regularity assumptions, sufficient conditions for the existence of global and semilocal continuous inverse and implicit functions are obtained. We consider applications of these results to the problem of continuous extension of implicit functions and to the problem of coincidence points of smooth and continuous compact mappings.

A graph with n vertices is called an n-graph. A spanning tree with at most k leaves is referred to as a spanning k-ended tree. Spanning k-ended trees are important in various fields such as network design, graph theory, and communication networks. They provide a structured way to connect all the nodes in a network while ensuring efficient communication and minimizing unnecessary connections. In addition, they serve as fundamental components for algorithms in routing, broadcasting, and spanning tree protocols. However, determining whether a connected graph has a spanning k-ended tree or not is NP-complete. Therefore, it is important to identify sufficient conditions for the existence of such trees. The implicit-degree proposed by Zhu, Li, and Deng is an important indicator for the Hamiltonian problem and the spanning k-ended tree problem. In this article, we provide two sufficient conditions for K[sub.1,4] -free connected graphs to have spanning k-ended trees for k = 2, 3. We prove the following: Let G be a K[sub.1,4] -free connected n-graph. For k = 2, 3, if the implicit-degree sum of any k + 1 independent vertices of G is at least n − k + 2, then G has a spanning k-ended tree. Moreover, we give two examples to show that the lower bounds n and n − 1 are the best possible.

A geometric flow on (2, 2)-forms is introduced which preserves the balanced condition of metrics, and whose stationary points satisfy the anomaly equation in Strominger systems. The existence of solutions for a short time is established, using Hamilton’s version of the Nash–Moser implicit function theorem.

In this paper, we introduce the notion of α-implicit contractive mapping of integral type in the context of Branciari metric spaces. The results of this paper, generalize and improve several results on the topic in literature. We give an example to illustrate our results.

We prove that constant scalar curvature Kähler metric “adjacent” to a fixed Kähler class is unique up to isomorphism. The proof is based on the study of a fourth order evolution equation, namely, the Calabi flow, from a new geometric perspective, and on the geometry of the space of Kähler metrics.

In this paper, we investigate the existence and Ulam–Hyers–Rassias stability results for a class of boundary value problems for implicit ψ-Caputo fractional differential equations with non-instantaneous impulses involving both retarded and advanced arguments. The results are based on the Banach contraction principle and Krasnoselskii’s fixed point theorem. In addition, the Ulam–Hyers–Rassias stability result is proved using the nonlinear functional analysis technique. Finally, illustrative examples are given to validate our main results.

In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results.