Mean Convex Research Articles

Non-orthogonal unicast and multicast transmission with IRS-assisted Rate-Splitting Multiple Access in multiuser MISO systems

Abstract. In MISO systems, novel communication technologies such as NOMA and multicast transmission have attracted widespread attention to meet the diverse needs of users. However, traditional NOMA and multicast transmission schemes have certain limitations in dynamic interference management and resource allocation. To resolve this issue, this paper proposes a RSMA scheme assisted by phase quantized Intelligent Reflecting Surface (IRS) for downlink MISO systems with nonorthogonal transmission of unicast and multicast signals. In this scheme, through reconfiguring unicast and multicast signals and combining them into a super-common stream and partial unicast messages, RSMA achieves dynamic interference management of multi-user while effectively separating unicast and multicast streams. To maximize the weighted sum rate(WSR), we put forward an alternating optimization(AO) algorithm that utilizes Weighted Minimum Mean Square Error (WMMSE) estimation and convex set intersection representation, which jointly optimizes the equivalent beamforming matrix,rate allocation and phase shift matrix. Simulation results illustrate that the proposed single-layer RSMA exhibits performance closer to that of DPC encoding compared to traditional NOMA and MULP schemes. Furthermore, we extensively investigate the relationship between the WSR of RSMA and system parameters, providing important guidance for practical engineering applications.

The first Dirichlet eigenvalue and the width

For a geodesic ball with non-negative Ricci curvature and mean convex boundary, it is known that the first Dirichlet eigenvalue of this geodesic ball has a sharp lower bound in term of its radius. We show a quantitative explicit inequality, which bounds the width of geodesic ball in terms of the spectral gap between the first Dirichlet eigenvalue and the corresponding sharp lower bound.

A Heintze-Karcher inequality with free boundaries and applications to capillarity theory

In this paper we analyze the shape of a droplet inside a smooth container. To characterize their shape in the capillarity regime, we obtain a new form of the Heintze–Karcher inequality for mean convex hypersurfaces with boundary lying on curved substrates.

Rigidity of Free Boundary Minimal Disks in Mean Convex Three-Manifolds

Rigidity of Free Boundary Minimal Disks in Mean Convex Three-Manifolds

A Note on Alexandrov Immersed Mean Curvature Flow

We demonstrate that the property of being Alexandrov immersed is preserved along mean curvature flow. Furthermore, we demonstrate that mean curvature flow techniques for mean convex embedded flows such as noncollapsing and gradient estimates also hold in this setting. We also indicate the necessary modifications to the work of Brendle–Huisken to allow for mean curvature flow with surgery in the Alexandrov immersed, 2-dimensional setting.

A Minkowski-type inequality for capillary hypersurfaces in a half-space

In this article, we investigate a flow of inverse mean curvature type for capillary hypersurfaces in the half-space. We establish the global existence of solutions for this flow and demonstrate that it converges smoothly to a spherical cap as time tends to infinity. As a result, we derive a new Minkowski-type inequality for star-shaped and mean convex capillary hypersurfaces for the whole range of contact angle θ∈(0,π).

Abstract PO2-02-11: Radiomics and an Erosion Model to Predict Post-Neoadjuvant Therapy Tumor Characteristics for Personalized Breast Cancer Treatment

Abstract Background: Neoadjuvant therapy (NAT) has become a standard-of-care (SOC) for patients diagnosed with locally advanced or early-stage breast cancer. A critical step towards optimizing treatment selection and coordinating surgical care is to estimate the tumor’s response to NAT. While pathological complete response (pCR) is commonly used to measure NAT effectiveness, patients who do not achieve pCR still benefit from NAT depending on the tumor’s response. A reduction in tumor extent could alter the surgical approach from mastectomy to breast conserving therapy and improve recurrence and overall survival. In this study we used pre-treatment imaging data to provide a comprehensive prediction of clinically relevant metrics including tumor volume, tumor convex hull volume, and spatially derived metrics; this research represents an important advancement in predicting NAT effectiveness. Methods: 262 breast cancer patients from 5 institutions who underwent NAT with pre- and post-NAT dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) were included in this study. Each MRI was automatically segmented using a convolutional neural network with manual edits and verification. Radiomic features were extracted from the pre-NAT MRI and tumor segmentation. Significant radiomic features were selected based on their correlation with post-NAT volume, feature-to-feature correlation, and LASSO regression. Altogether, 12 and 17 radiomic features were selected for volume and convex hull volume prediction, respectively. The selected features were inputted into a Huber loss regression model to predict either the post-NAT tumor volume or the convex hull volume (surgical excision volume). To enable spatial and morphological comparison between the predicted and post-NAT tumor, an erosion model was developed in which the pre-NAT and post-NAT MRIs were co-registered, and the pre-NAT tumor was symmetrically eroded until its volume matched the predicted volume. Clinically relevant metrics such as the distance from the tumor to the nipple, skin, and chest were computed to assess the morphological and spatial prediction capabilities of the model. All models were tested on patients from cohort holdout sets (n=110) unseen during training. Results: The volume prediction analysis yielded promising results, with an R^2 value of 0.62 for post-NAT tumor volume prediction and an R^2 value of 0.56 for post-NAT tumor convex hull volume prediction. The mean volume and convex hull volume error was 1.1 cc and 1.3 cc, respectively. Table 1 provides an overview of the morphological and spatial comparisons between the predicted and post-NAT tumors. Overall, the predicted tumors demonstrated close agreement with their post-NAT counterparts. Conclusion: This study demonstrates a successful approach towards estimating post-NAT tumor volume, convex hull volume, and morphological and spatial characteristics. By providing reliable estimates of post-NAT tumor characteristics using SOC pre-NAT data, our predictive models enable personalized treatment planning and patient stratification to optimize patient care. Table 1. Predicted vs. post-NAT tumor comparison. Results demonstrate accurate predictions for clinically relevant distance metrics. Citation Format: Bradley Feiger, John Pfeiffer, John Cole, Anuja Antony, Joseph Peterson. Radiomics and an Erosion Model to Predict Post-Neoadjuvant Therapy Tumor Characteristics for Personalized Breast Cancer Treatment [abstract]. In: Proceedings of the 2023 San Antonio Breast Cancer Symposium; 2023 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2024;84(9 Suppl):Abstract nr PO2-02-11.

The Dirichlet problem of translating mean curvature equations

In this paper, we study the Dirichlet problem of translating mean curvature equations over a domain via topological restrictions. Its main difficulty is the non-existence of a C^{0} a priori estimate for their classical solutions except few cases. Inspired by the work of Miranda and Giusti, we define a generalized solution of these Dirichlet problems and establish its general existence. We propose a non-closed-minimal (NCM) condition on the underlying domain. When the domain is mean convex and NCM, the generalized solution with continuous boundary data is the classical smooth solution. Moreover, the NCM assumption cannot be removed by a hemisphere example.

Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p-Laplace operator, namely:λ1(β,Ω)=minψ∈W1,p(Ω)∖{0}⁡∫ΩF(∇ψ)pdx+β∫∂Ω|ψ|pF(νΩ)dHN−1∫Ω|ψ|pdx, where p∈]1,+∞[, Ω is a bounded, anisotropic mean convex domain in RN, νΩ is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on RN. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ1 in the case β>0, and a upper bound in the case β<0. As a consequence, we prove, for β>0, a lower bound for λ1(β,Ω) in terms of the anisotropic inradius of Ω and, for β<0, an upper bound of λ1(β,Ω) in terms of β.

Morphological changes in the interosseous membrane of the forearm during forearm rotation.

[Purpose] Changes in forearm interosseous membrane dynamics during forearm rotation relative to the shoulder joint position remain poorly understood. The purpose of this study was to clarify interosseous membrane dynamics during forearm rotation in shoulder abduction and external rotation positions. [Participants and Methods] We conducted open magnetic resonance imaging on 17 healthy forearms in the prone position. Three limb positions were set for measuring the forearm rotation angle: intermediate, maximum pronation, and maximum supination. Images were obtained with the shoulder joint abducted at 90°and externally rotated at 90°. The forearm interosseous membrane angle was measured at three points: the apex of the forearm interosseous membrane, the radius, and the ulna. The measurement of the interosseous angle was repeated thrice. [Results] Sufficient intra-rater reliability was confirmed for the forearm interosseous membrane angle measurement. The interosseous membrane of the forearm showed a mean dorsal convex shape during forearm pronation (141.7° ± 0.83°), and the mean palmar convex shape during forearm supination (-141.6° ± 0.64°). [Conclusion] This study provides useful information for future research by quantifying the dynamics of the interosseous membrane of the forearm, which is an important soft tissue for forearm rotation. The establishment of a quantitative evaluation method for forearm interosseous morphological changes will help further elucidate forearm rotation movements during sports activities.

Schur-power convexity of integral mean for convex functions on the coordinates

Abstract In this article, we investigate the concepts of monotonicity, Schur-geometric convexity, Schur-harmonic convexity, and Schur-power convexity for the lower and upper limits of the integral mean, focusing on convex functions on coordinate axes. Furthermore, we introduce novel and fascinating inequalities for binary means as a practical application.

Existence and non-existence of minimal graphs

We study the Dirichlet problem for minimal surface systems in arbitrary dimension and codimension via mean curvature flow, and obtain the existence of minimal graphs over arbitrary mean convex bounded C2 domains for a large class of prescribed boundary data. This result can be seen as a natural generalization of the classical sharp criterion for solvability of the minimal surface equation by Jenkins-Serrin. In contrast, we also construct a class of prescribed boundary data on just mean convex domains for which the Dirichlet problem in codimension 2 is not solvable. Moreover, we study existence and the uniqueness of minimal graphs by perturbation.

Rigidity of mean convex subsets in non-negatively curved RCD spaces and stability of mean curvature bounds

We prove splitting theorems for mean convex open subsets in Riemannian curvature-dimension (RCD) spaces that extend results by Kasue et al. for Riemannian manifolds with boundary to a non-smooth setting. A corollary is for instance Frankel’s theorem. Then, we prove that our notion of mean curvature bounded from below for the boundary of an open subset is stable with respect to uniform convergence of the corresponding boundary distance function. We apply this to prove almost rigidity theorems for uniform domains whose boundary has a lower mean curvature bound.

Rigidity results for mean curvature flow graphical translators moving in non-graphical direction

In this paper, we study the rigidity results of complete graphical translating hypersurfaces when the translating direction is not in the graphical direction. We proved that any entire graphical translating surface in the translating direction not parallel to the graphical one is flat if either the translating surface is mean convex or the entropy of the translating surface is smaller than 2 2 . For higher dimensional case, we show that the same conclusion holds if the graphical translating hypersurface satisfies certain growth condition.

Schur Convexity of Complementary Geometric Mean

Schur Convexity of Complementary Geometric Mean

Geometric inequalities involving three quantities in warped product manifolds

In this paper, we establish two families of sharp geometric inequalities for closed hypersurfaces in space forms or other warped product manifolds. Both families of inequalities compare three distinct geometric quantities. The first family concerns the k-th boundary momentum, area and weighted volume, and has applications to Weinstock-type inequalities for Steklov or Wentzell eigenvalues on star-shaped mean convex domains. This generalizes the main results of [12] and confirms a conjecture posed therein. The second family involves a weighted k-th mean curvature integral and two distinct quermassintegrals. It generalizes the authors' recent work [33] with G. Wheeler and V.-M. Wheeler, and the second author's recent work [43] with T. Zhou, in two different aspects.

On the Jensen convexity of quasideviation and Bajraktarević means

AbstractMotivated by the characterization theorem about the Jensen convexity of quasiarithmetic means obtained by the authors in 2021, our main goal is to establish a characterization of the Jensen convexity of quasideviation as well as of Bajraktarević means without any additional and unnatural regularity assumptions.

Uryson width of three dimensional mean convex domain with non-negative Ricci curvature

We prove that for every three dimensional manifold with nonnegative Ricci curvature and strictly mean convex boundary (non-empty), there exists a Morse function so that each connected component of its level sets has a uniform diameter bound, which depends only on the lower bound of mean curvature. This gives an upper bound of Uryson 1-width for those three manifolds with boundary.

Convexity of 2-Convex Translating and Expanding Solitons to the Mean Curvature Flow in $${{\mathbb {R}}}^{n+1}$$

In this paper, inspired by the work of Spruck–Xiao (Am J Math 142(3):993–1015, 2020) and based partly on a result of Derdziński (Math Z 172(3):273–280, 1980), we prove the convexity of complete 2-convex translating and expanding solitons to the mean curvature flow in $$\mathbb {R}^{n+1}$$ . More precisely, for $$n\ge 3$$ , we show that any n-dimensional complete 2-convex translating solitons are convex, and any n-dimensional complete 2-convex self-expanders asymptotic to (strictly) mean convex cones are convex.

Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets

Abstract In this work, we analyze Merriman, Bence and Osher’s thresholding scheme, a time discretization for mean curvature flow. We restrict to the two-phase setting and mean convex initial conditions. In the sense of the minimizing movements interpretation of Esedoğlu and Otto, we show the time-integrated energy of the approximation to converge to the time-integrated energy of the limit. As a corollary, the conditional convergence results of Otto and one of the authors become unconditional in the two-phase mean convex case. Our results are general enough to handle the extension of the scheme to anisotropic flows for which a non-negative kernel can be chosen.