Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

In this paper, we investigate closed strictly convex hypersurfaces in ℝ n + 1 {\mathbb{R}^{n+1}} which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree 1 and inverse concave function of the principal curvatures with power greater than 1, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere 𝕊 + n + 1 {\mathbb{S}^{n+1}_{+}} with power greater than or equal to 1.

Fair Cake Division Under Monotone Likelihood Ratios

This work develops algorithmic results for the classic cake-cutting problem in which a divisible, heterogeneous resource (modeled as a cake) needs to be partitioned among agents with distinct preferences. We focus on a standard formulation of cake cutting wherein each agent must receive a contiguous piece of the cake. Although multiple hardness results exist in this setup for finding fair/efficient cake divisions, we show that, if the value densities of the agents satisfy the monotone likelihood ratio property (MLRP), then strong algorithmic results hold for various notions of fairness and economic efficiency. Addressing cake-cutting instances with MLRP, first we develop an algorithm that finds cake divisions (with connected pieces) that are envy free, up to an arbitrary precision. The time complexity of our algorithm is polynomial in the number of agents and the bit complexity of an underlying Lipschitz constant. We obtain similar positive results for maximizing social, egalitarian, and Nash social welfare. Many distribution families bear MLRP. In particular, this property holds if all the value densities belong to any one of the following families: Gaussian (with the same variance), linear, Poisson, and exponential distributions, linear translations of any log-concave function. Hence, through MLRP, the current work obtains novel cake-cutting algorithms for multiple distribution families.

Determination of Bounds for the Jensen Gap and Its Applications

The Jensen inequality has been reported as one of the most consequential inequalities that has a lot of applications in diverse fields of science. For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature on mathematical inequalities. The main intention of this article is to find some novel bounds for the Jensen difference while using some classes of twice differentiable convex functions. We obtain the proposed bounds by utilizing the power mean and Höilder inequalities, the notion of convexity and the prominent Jensen inequality for concave function. We deduce several inequalities for power and quasi-arithmetic means as a consequence of main results. Furthermore, we also establish different improvements for Hölder inequality with the help of obtained results. Moreover, we present some applications of the main results in information theory.

Degree estimates of one class cut-offs for improper integrals

The paper considers a normalized non-integral integral of the first kind with a variable lower bound. In this case the integrand is a generalization of the standard Gaussian distribution density. Such integrals are often called cutoffs or incomplete functions. The purpose of this paper is to obtain power inequalities for this kind of integrals. The necessity of obtaining this type of estimations is due to the fact that incomplete functions have become widespread in applications and theoretical studies. The peculiarity of the results established in the article consists in the fact that arbitrary degrees of a given integral for any value of an argument are evaluated from above not by means, of the value of integrable functions at a certain point, but by the value of the integral in question at some point proportional to this argument. The coefficient of proportionality, a parameter, can take any value from some closed interval. The main difficulty in obtaining these inequalities is that the integrand is a logarithmically concave function, that is, its logarithm is a concave function. The paper also proves that both limits of the closed interval for the parameter cannot be extended. This shows that the obtained estimates are unimprovable.

A Generalization of the Concavity of Rényi Entropy Power

Recently, Savaré-Toscani proved that the Rényi entropy power of general probability densities solving the p-nonlinear heat equation in Rn is a concave function of time under certain conditions of three parameters n,p,μ, which extends Costa’s concavity inequality for Shannon’s entropy power to the Rényi entropy power. In this paper, we give a condition Φ(n,p,μ) of n,p,μ under which the concavity of the Rényi entropy power is valid. The condition Φ(n,p,μ) contains Savaré-Toscani’s condition as a special case and much more cases. Precisely, the points (n,p,μ) satisfying Savaré-Toscani’s condition consist of a two-dimensional subset of R3, and the points satisfying the condition Φ(n,p,μ) consist a three-dimensional subset of R3. Furthermore, Φ(n,p,μ) gives the necessary and sufficient condition in a certain sense. Finally, the conditions are obtained with a systematic approach.

Evaluation of axial length to identify the effects of monocular 0.125% atropine treatment for pediatric anisometropia

The aim of the study is to determine the effects of monocular 0.125% atropine daily treatment on the longer axial length (AL) eyes in children with pediatric anisometropia. This was a retrospective cohort study. The charts of children with anisometropia (aged 6–15 years) who had a > 0.2-mm difference in AL between the two eyes were reviewed. Children who received monocular treatment of 0.125% atropine in the eye with longer AL were included for final analysis. The main outcome measure was the difference in AL between the two eyes after treatment. Regression analysis was used to model the changes in AL according to the time of treatment in both eyes. Finally, forty eyes in 20 patients (mean age 10.2 years) were included in the analyses. During the treatment period, AL was controlled in the treated eyes (p = 0.389) but elongated significantly in the untreated eyes (p < 0.001). The difference in AL between the treated and untreated eyes decreased from 0.57 to 0.22 mm (p < 0.001) after the 1-year treatment period. In the regression model, the best fit for the relationship between changes in AL and time during the treatment period in the treated eyes was the quadratic regression model with a concave function. In conclusion, these data suggest that 0.125% atropine daily is an effective treatment to reduce the interocular difference of AL in eyes with axial anisometropia. This pilot study provides useful information for future prospective and larger studies of atropine for the treatment of pediatric axial anisometropia.

Evaluation of Primary User Power Impact for Joint Optimization of Energy Efficiency in Cognitive Radio Networks

Energy efficiency (EE) is of great concern in cognitive radio networks since the throughput and energy consumption of secondary users (SUs) vary with the sensing time. However, the conditions of the detection probability and false alarm probability should be respected to better protect primary users (PUs) and to improve the sensing performance of SUs. Additionally, the PUs’ minimum averaged power provision should also be regarded as a key problem of interactive linking to SUs. Therefore, an integrated design between the PU and SUs is desired for the coordination of the whole cognitive radio system, especially regarding the satisfaction of EE and performance metrics. This study formulates sensing constraints in a unified way and calculates the minimum SNR of SUs, based on which the essential PU power provision is computed. Furthermore, EE is proved as a decreasing function with the PU’s active ratio, where the maximum EE is obtained corresponding to the minimum QoS requirements of the sensing process. Hence, a bisection-based method is proposed to maximize EE, which is considered as a concave function of SUs’ sensing time and has only one unique optimum. EE’s optimization was analyzed under different fusion rules for diverse SNR conditions. The optimum was also studied with sensing performance targets for various cases of PU power provision.

Properties and Bounds of Jensen-Type Functionals via Harmonic Convex Functions

Dragomir introduced the Jensen-type inequality for harmonic convex functions (HCF) and Baloch et al. studied its different variants, such as Jensen-type inequality for harmonic h -convex functions. In this paper, we aim to establish the functional form of inequalities presented by Baloch et al. and prove the superadditivity and monotonicity properties of these functionals. Furthermore, we derive the bound for these functionals under certain conditions. Furthermore, we define more generalized functionals involving monotonic nondecreasing concave function as well as evince superadditivity and monotonicity properties of these generalized functionals.

Influence of pressure pulsation on the performance of compression system and turbocharged engine

Pressure pulsation widely exists in power machinery combining compression components and pipelines. It has substantial effects on the performance of compressor as part of the compression system, as well as the engine. In this paper, the pressure pulsations under different excitation frequencies are measured and analyzed in the intake system of a turbo-charged and inter-cooled internal combustion engine. It is pointed out that the amplification of the pressure pulsation at the compressor inlet and outlet is caused by the coupling effects within the compression system, which consequently lead to the formation of a stable standing wave. Further research indicates that the pulsation at the compressor boundary will cause its pressure ratio to fluctuate. Additionally, because the compressor characteristic curve resembles a concave function, the fluctuation of transient pressure ratio will further cause the time-averaged pressure ratio to decline. Finally, the impact on engine performance is evaluated based on a well-validated simulation model.

lra: A long read aligner for sequences and contigs

It is computationally challenging to detect variation by aligning single-molecule sequencing (SMS) reads, or contigs from SMS assemblies. One approach to efficiently align SMS reads is sparse dynamic programming (SDP), where optimal chains of exact matches are found between the sequence and the genome. While straightforward implementations of SDP penalize gaps with a cost that is a linear function of gap length, biological variation is more accurately represented when gap cost is a concave function of gap length. We have developed a method, lra, that uses SDP with a concave-cost gap penalty, and used lra to align long-read sequences from PacBio and Oxford Nanopore (ONT) instruments as well as de novo assembly contigs. This alignment approach increases sensitivity and specificity for SV discovery, particularly for variants above 1kb and when discovering variation from ONT reads, while having runtime that are comparable (1.05-3.76×) to current methods. When applied to calling variation from de novo assembly contigs, there is a 3.2% increase in Truvari F1 score compared to minimap2+htsbox. lra is available in bioconda (https://anaconda.org/bioconda/lra) and github (https://github.com/ChaissonLab/LRA).