Article
Mathematics, Applied
Andrea Malchiodi, Martin Mayer
Summary: This paper addresses the problem of prescribing conformally the scalar curvature on compact manifolds of positive Yamabe class in dimensions n >= 5. New existence results are proven using Morse theory and analysis on blowing-up solutions under suitable pinching conditions on the curvature function. Moreover, new nonexistence results are provided to demonstrate the sharpness of some assumptions, in terms of both dimension and Morse structure of the prescribed function.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Andrea Malchiodi
Summary: In this paper, we examine the classical Kazdan-Warner problem, which involves the conformal prescription of the scalar curvature of a Riemannian manifold. We demonstrate how the existence problem can be addressed using a combination of variational theory, asymptotic analysis, and Morse-theoretical tools.
MILAN JOURNAL OF MATHEMATICS
(2023)
Article
Mathematics
Andrea Malchiodi, Martin Mayer
Summary: This paper studies finite-energy blow-ups for prescribed Morse scalar curvatures in both the subcritical and critical regime. After general considerations on Palais-Smale sequences, precise blow-up rates for subcritical solutions are determined, and the possibility of tower bubbles is excluded in all dimensions. Subsequent research aims to establish the sharpness of this result and apply the analysis to deduce new existence results for the geometric problem.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics, Applied
Martin Mayer
Summary: This paper investigates the problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function. By comparing two different approaches, it is shown that under certain assumptions, these two methods are equivalent in the zero weak limit.
ADVANCES IN CALCULUS OF VARIATIONS
(2022)
Article
Mathematics, Applied
Gianmichele Di Matteo, Andrea Malchiodi
Summary: We establish the existence of double bubbles with large and constant mean curvatures in Riemannian manifolds. These double bubbles are perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor. Additionally, we obtain general results of multiplicity through Lusternik-Schnirelman theory, and extra results in the case of double bubbles with opposite boundaries having the same mean curvature.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Physics, Multidisciplinary
Dan-Feng Wang, Yu-Han Wang, Kuo-Chih Chuang
Summary: This study proposes an adjustable phononic crystal lens for focusing A0 mode Lamb waves, utilizing gradient-index and Luneburg PCLs to address configuration adjustability, and validates that focusing spots can still be achieved even when the refractive index configuration is not perfectly matched. The proposed PCLs are realized with commercially available steel balls, but can be extended to other designs using additive manufacturing as long as the contact area is minimized.
Article
Mathematics, Applied
Hang Chen, Zhida Guan
Summary: This paper investigates critical points of k-harmonic maps on the space of smooth maps and properties of CMC triharmonic hypersurfaces. It proves that under certain conditions, hypersurfaces have constant scalar curvature. The results support the generalized Chen's conjecture and provide optimal upper bounds and complete classification in specific cases.
RESULTS IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Aligadzhi Rustanov
Summary: The article focuses on nearly Sasakian manifolds of a constant type. It proves that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the nearly Sasakian manifold is a nearly Kahler manifold. It also proves that the class of nearly Sasakian manifolds of the zero constant type is equivalent to the class of Sasakian manifolds. The concept of constancy of the type of an almost contact metric manifold is introduced and its criterion is proved. The article demonstrates the coincidence of both concepts of type constancy for the nearly Sasakian manifold. It further proves that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the almost contact metric manifold of the zero constant type is the Hermitian structure.
Article
Mathematics
M. K. Gupta, Suman Sharma, Fatemah Mofarreh, Sudhakar Kumar Chaubey
Summary: This paper investigates the curvature characteristics of specific classes of Finsler spaces, focusing on homogeneous Finsler spaces. The expression for S-curvature in a homogeneous Finsler space with a generalized Matsumoto metric is derived, and it is demonstrated that the isotropic S-curvature of the homogeneous generalized Matsumoto space must be zero. Furthermore, the expression for the mean Berwald curvature is obtained using the formula of S-curvature.
Article
Mathematics
Siraj Uddin, Esmaeil Peyghan, Leila Nourmohammadifar, Rawan Bossly
Summary: In this paper, the concepts of nearly Sasakian and nearly Kahler statistical structures are introduced and a non-trivial example is given. The conditions for a real hypersurface in a nearly Kahler statistical manifold to have a nearly Sasakian statistical structure are provided. Invariant and anti-invariant statistical submanifolds of nearly Sasakian statistical manifolds are also studied. Finally, conditions under which a submanifold of a nearly Sasakian statistical manifold is itself a nearly Sasakian statistical manifold are given.
Article
Engineering, Mechanical
Jiaxiang Zhu, Guangbo Hao, Tinghao Liu, Haiyang Li
Summary: A nearly-constant amplification ratio compliant mechanism (OCARCM) is proposed to address the issue of variable amplification ratio. Closed-form solutions are obtained using the generic beam constraint model (BCM) method combined with the free-body diagram (FBD), accurately describing the nonlinear kinetostatic characteristics of the OCARCM. Comparative analysis is conducted between the OCARCM and the widely-used bridge-type compliant amplifier, with and without external payloads. Experimental results show a maximum error of 3.4% compared with analytical or FEA results.
MECHANISM AND MACHINE THEORY
(2023)
Article
Computer Science, Software Engineering
Kazuki Hayashi, Yoshiki Jikumaru, Makoto Ohsaki, Takashi Kagaya, Yohei Yokosuka
Summary: P-CMC surfaces are obtained through the mean curvature flow (MCF) and can be seen as the stationary point of an energy functional of multiple patch surfaces and auxiliary surfaces. The MCF is formulated as the negative gradient flow of the energy functional for continuous surfaces, and then discretized to determine the change in vertex positions of triangular meshes on the surface and along the internal boundaries between patches. Numerical examples demonstrate that the proposed method enables multiple patch surfaces to approximately achieve the desired mean curvatures, which expands the options for shape design using CMC surfaces.
COMPUTER AIDED GEOMETRIC DESIGN
(2023)
Article
Mathematics
Rafael Lopez-soriano, Andrea Malchiodi, David Ruiz
Summary: This paper investigates the problem of prescribing the Gaussian and geodesic curvatures of a compact surface with boundary by conformally deforming the metric. The authors derive existence results using variational methods, specifically by minimizing the Euler-Lagrange energy or using min-max techniques. A key aspect of their approach is the blow-up analysis of solutions, which can have diverging volume even with uniform bounds on their Morse index in the given setting. This paper is the first to address this aspect, and key ingredients include blow-up analysis around points different from local maxima, the use of holomorphic domain variations, and Morse index estimates.
ANNALES SCIENTIFIQUES DE L ECOLE NORMALE SUPERIEURE
(2022)
Article
Multidisciplinary Sciences
Yuankang Fu, Qi Liu, Yongjin Li
Summary: Two new geometric constants JL(X) and YJ(X) in Banach spaces are introduced, with their upper and lower bounds calculated for Hilbert spaces and common Banach spaces. Inequalities for JL(X), YJ(X), and other significant geometric constants are presented, along with conditions for uniformly non-square and normal structure, as well as uniformly non-square and uniformly convex.
Article
Computer Science, Software Engineering
Man-Chung Yue, Daniel Kuhn, Wolfram Wiesemann
Summary: In this technical note, the authors prove that the Wasserstein ball is weakly compact under mild conditions and offer necessary and sufficient conditions for the existence of optimal solutions. They also analyze the sparsity of solutions when the Wasserstein ball is centered at a discrete reference measure. The proofs are self-contained, shorter, mathematically rigorous, and the conditions for optimal solutions are easily verifiable in practice.
MATHEMATICAL PROGRAMMING
(2022)
