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The Lyapunov equation for polytopic linear inclusions over the space of Minkowski functions of nonempty compact and convex sets that contain the origin as an interior point is studied. In particular, necessary and sufficient conditions for the characterization, existence and uniqueness of its fundamental solution are derived.

In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number $\chi(M)

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen-Margulis-Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method., The first version of the paper has appeared in arXiv:1612.00909 by the second named author. This current version now includes several details which were missing. 54 pages, 2 figures

In the single-point diamond turning (SPDT) of optical spherical/aspheric surface, tool deviating from the spindle rotation center significantly deteriorates the form accuracy of the spherical/aspherical surface and its optical performance. In this study, the influence of tool deviation on the form accuracy of a convex spherical surface and the cutting force forms during the turning process were studied first, following which a force-based tool deviation model was derived to identify the tool deviation using cutting force. Finally, by analyzing the influence of tool deviation on the three-dimensional (3D) form of residual structures at the center of the convex spherical surface, the 3D form of the convex spherical surface was predicted online. Results indicate the existence of a mapping relation between the tool deviation and the cutting force form, which could be further used to online predict the 3D form of the machined convex spherical surface in SPDT through the established geometric model.

The paper investigates the existence, asymptotic boundary behavior and uniqueness of convex solutions to the Monge-Ampere equation that take infinite values on the boundary of smooth and bounded open convex sets in R n .

The article shows the features of the construction of cams with a convex profile, the method of kinematic calculation of the pusher when it moves along a convex and rounded surface is given. A technique is proposed to determine the coordinate center of the radius of the convex surface of the cam. Calculations of the stroke, speed and acceleration of the pusher depending on the angle of rotation of the cam shaft are performed using the Mathcad system. Using the Mathcad program for cams with a convex profile, graphs of the pusher lift, changes in its speed and acceleration are constructed. The proposed method of calculating the cams of a convex profile of various sizes with the determination of the lifting height, speed, acceleration of the pusher and the construction of their graphs in the Mathcad system will allow designing cams and copiers necessary for processing cam shafts.

It is a well-known fact in physics that classical mechanics describes the macro-world, and quantum mechanics describes the atomic and sub-atomic world. However, principles of quantum mechanics, such as Heisenberg’s Uncertainty Principle, can create visible real-life effects. One of the most commonly known of those effects is the stability problem, whereby a one-dimensional point base object in a gravity environment cannot remain stable beyond a time frame. This paper expands the stability question from 1- dimensional rod to 2-dimensional highly symmetrical structures, such as an even-sided polygon. Using principles of classical mechanics, and Heisenberg’s uncertainty principle, a stability equation is derived. The stability problem is discussed both quantitatively as well as qualitatively. Using the graphical analysis of the result, the relation between stability time and the number of sides of polygon is determined. In an environment with gravity forces only existing, it is determined that stability increases with the number of sides of a polygon. Using the equation to find results for circles, it was found that a circle has the highest degree of stability. These results and the numerical calculation can be utilized for architectural purposes and high-precision experiments. The result is also helpful for minimizing the perception that quantum mechanical effects have no visible effects other than in the atomic, and subatomic world. Keywords: Quantum mechanics, Heisenberg Uncertainty principle, degree of stability, polygon, the highest degree of stability

In this article we study some geometric properties of proximally smooth sets. First, we introduce a modification of the metric projection and prove its existence. Then we provide an algorithm for constructing a rectifiable curve between two sufficiently close points of a proximally smooth set in a uniformly convex and uniformly smooth Banach space, with the moduli of smoothness and convexity of power type. Our algorithm returns a reasonably short curve between two sufficiently close points of a proximally smooth set, is iterative and uses our modification of the metric projection. We estimate the length of the constructed curve and its deviation from the segment with the same endpoints. These estimates coincide up to a constant factor with those for the geodesics in a proximally smooth set in a Hilbert space.

In this paper, we consider a kind of area-preserving flow for closed convex planar curves which will decrease the length of the evolving curve and make the evolving curve more and more circular during the evolution process. And the final shape of the evolving curve will be a circle as time $$t\rightarrow +\infty $$ .

The aim of this paper is to find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in the classes SP_{p}({\alpha},\b{eta}) and UCV_{p}({\alpha},\b{eta}) of uniformly spirallike functions. Further, we consider properties of a special function related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data has been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger (increasing) convex means. We prove similar results on balls. We also prove a comparison principle on generalized cylinders with mixed boundary conditions (Neumann and Robin)., Improved notation, new mixed results added, typos corrected

We establish the existence and multiplicity of solutions for a Kirchhoff-type problem in $$\mathbb R^4$$ involving a critical and concave–convex nonlinearity. Since in dimension four, the Sobolev critical exponent is $$2^*=4$$ , there is a tie between the growth of the nonlocal term and the critical nonlinearity. This turns out to be a challenge to study our problem from the variational point of view. Some of the main tools used in this paper are the mountain-pass and Ekeland’s theorems, Lions’ Concentration Compactness Principle and an extension to $$\mathbb R^N$$ of the Struwe’s global compactness theorem.

Generalized F statistics are the quotients of convex combinations of central chi-squares divided by their degrees of freedom. Exact expressions are obtained for the distribution of these statistics when the degrees of freedom either in the numerator or in the denominator are even. An example is given to show how these expressions may be used to check the accuracy of Monte-Carlo methods in tabling these distributions. Moreover, when carrying out adaptative tests, these expressions enable us to estimate the p-values whenever they are available.

We improve the linear tetrahedron method to overcome systematic errors due to overestimations (underestimations) in integrals for convex (concave) functions, respectively. Our method is applicable to various types of calculations such as the total energy, the harge (spin) density, response functions, and the phonon frequency, in contrast with the Bl\"ochl correction, which is applicable to only the first two. We demonstrate the ability of our method by calculating phonons in MgB$_2$ and fcc lithium., Comment: 9pages, 8 figures

We give an alternative proof to the well known fact that each convex compact centrally symmetric subset of R 2containing the origin is a zonoid, i.e., the range of a two dimensional vector measure, and we prove that a two dimensional zonoid whose boundary contains the origin is strictly convex if and only if it is the range of a Chebyshev measure. We give a condition under which a two dimensional vector measure admits a decomposition as the difference of two Chebyshev measures, a necessary condition on the density function for the strict convexity of the range of a measure and a characterization of two dimensional Chebyshev measures.

We consider the evolution of sets by nonlocal mean curvature and we discuss the preservation along the flow of two geometric properties, which are the mean convexity and the outward minimality. The main tools in our analysis are the level set formulation and the minimizing movement scheme for the nonlocal flow. When the initial set is outward minimizing, we also show the convergence of the (time integrated) nonlocal perimeters of the discrete evolutions to the nonlocal perimeter of the limit flow.

In this article, the measurement method of convex hyperboloid using Hindle sphere is further studied. Firstly, the precise analytical formulas for solving radius of curvature, the diameters of outer edge and inner hole of Hindle sphere mirror are developed. The formula can be degenerated to the common approximate formula under certain conditions. In addition, the semi exact analytical formulas for solving the parameters of auxiliary Hindle sphere mirror are derived under some conditions, and the approximate formula is further extended to other occasions of aspheric mirror test, and a feasible stitching detection scheme for the secondary mirror of the CFGT (China Future Giant Telescope) is given. Then, based on the aberration theory, using Hindle auxiliary sphere, the relationships and formulas of the measurement error of vertex radius of curvature of convex hyperboloid mirror caused by the adjustment of interferometer position, the mirrors spacing, and machining error of radius of curvature of the Hindle sphere mirror are obtained, and the Zemax optical software is used for simulation verification.

In this paper, we present two stable virtual element methods for the Brinkman equations robust in the Stokes and Darcy limits. We use a pair of stable virtual elements for the velocity and pressure. Firstly, we define an $$\varvec{H}({\mathrm{div}})$$ -conforming velocity reconstruction operator for the velocity test function. By employing it in the discretizations of the zero-order term and the right-hand-side source term, we propose the first virtual element method on convex polygonal meshes. Secondly, we construct the enhanced virtual element space in place of the original virtual element space such that one can exactly compute the $$\varvec{L}^2$$ -projection of virtual function onto linear polynomial space. We make full use of this projection to introduce the second virtual element method on general polygonal meshes. The well-posedness and uniform energy-error estimates of each method are strictly established. Finally, numerical experiments are provided to validate the theoretical results and illustrate the good performance of our methods.

In this paper, we prove that convolution of two half-plane harmonic mappings with suitable dilatation is convex in some particular direction. We also prove that convolution of slanted half-plane mappings and slanted strip mapping with fixed dilatation is convex in a particular direction. We give some examples for support of our results.

We construct a compact, convex ancient solution of mean curvature flow in $\mathbb{R}^{n+1}$ with $O(1) \times O(n)$ symmetry that lies in a slab of width $\pi$. We provide detailed asymptotics for this solution and show that, up to rigid motions, it is the only compact, convex, $O(n)$-invariant ancient solution that lies in a slab of width $\pi$ and in no smaller slab.

A construction is given for the recovery of a disjoint union of strictly convex smooth planar obstacles from travelling-time information. The obstacles are required to be such that no Euclidean line meets more than two of them., 14 pages, 3 figures

In this paper, we mainly study the stability of Riemann problem for the improved Aw–Rascle–Zhang model which describes the formation and dynamics of traffic jams. First of all, we construct the classical Riemann solutions by elementary waves with the method of characteristic analysis. With the generalized Rankine–Hugoniot and entropy conditions, we prove the existence and uniqueness of δ -shock wave for arbitrary convex F ( u ) in this model. Then through a small perturbation, we analyze the wave interactions of different kinds of waves. As a result, we get the stability for the Riemann problem by letting the perturbed parameter e → 0 .

This study presents a novel Arcs-based discrete element method (ArcDEM) for efficient simulation of realistic granules with arbitrary convex and concave 2D particle outlines. In the proposed ArcDEM, a series of computational geometry algorithms are first developed to identify the convex corners and concave troughs of an arbitrary-shaped particle outline. Then, the circle-growing technique and the least squares method are combined to establish the Arcs-based particle that can represent the whole particle outline with multi-connected inward and outward arcs. Next, a new algorithm for efficient overlapping detection and precise contact resolution is developed for the Arcs-based particles. Finally, the ArcDEM is developed as a simulation tool with several implemented contact force laws and particle motions solved by an explicit time integration. To validate the feasibility and efficiency of the proposed ArcDEM, several numerical examples are performed, including (1) random allocation of non-overlapping irregular particles, (2) random packing of super-elliptical particles of different aspect ratios and blockiness, and (3) simulation of repose angle and biaxial compression tests of realistic rock particles with various roundness . The developed ArcDEM tool shows a powerful capability of numerically investigating the macro- and micromechanical properties of realistic convex and/or concave granular particles.

We consider the critical points (equilibria) of a planar potential generated by $n$ Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered previously in studies of the circular restricted $n$-body problem. We show that the number of equilibria is finite for a generic set of parameters, and we establish estimates for the number of equilibria. We prove that the number of equilibria is bounded below by $n+1$, and we provide examples to show that this lower bound is sharp. We prove an upper bound on the number of equilibria that grows exponentially in $n$. In order to establish a lower bound on the maximum number of equilibria, we analyze a class of examples, referred to as ``ring configurations'', consisting of $n-1$ equal masses positioned at vertices of a regular polygon with an additional mass located at the center. Previous numerical observations indicate that these configurations can produce as many as $5n-5$ equilibria. We verify analytically that the ring configuration has at least $5n-5$ equilibria when the central mass is sufficiently small. We conjecture that the maximum number of equilibria grows linearly with the number of point masses. We also discuss some mathematical similarities to other equilibrium problems in mathematical physics, namely, Maxwell's problem from electrostatics and the image counting problem from gravitational lensing., 16 pages, 1 figure

We use results of [6] to enlarge our knowledge of the approximate fixed point property (AFPP) for digital images in Z 2 . In particular, we study conditions under which the union of two convex digital disks has the AFPP.

The gear effect arises during the collision of a ball and an extended object due to rotation of the object across the surface of the ball. The effect is well known in golf, drivers being designed with a convex face to help reduce the problem of a mis-hit. An experiment is described to examine the gear effect for two different balls, one with a high tangential coefficient of restitution and one with a low coefficient. The outgoing ball spins were different, indicating that a driver designed for one type of golf ball is unlikely to work well with a different type of golf ball.

The Stokes eigenmodes on two-dimensional regular polygons of N apexes, 3 ≤ N ≤ 40 , are studied numerically using two different solvers: the lattice Boltzmann equation and the Legendre–Galerkin spectral element method. In particular, the lowest 55 eigenmodes on regular N -polygons have been computed and investigated for the following properties including (a) symmetries, (b) the asymptotic behaviour of the Stokes eigenvalues λ ( N ) in the limit of the apex number N → ∞ , i.e., in the limit of a regular N -polygon becoming its circumcircle, (c) the splitting doublet modes due to boundary geometry of N -polygons, and (d) the one-to-one correspondence between the Stokes modes on regular N -polygons and on the disc.

The aim of this paper is to introduce an orignal coupling procedure between surface integral equation formulations and on-surface radiation condition (OSRC) methods for solving two-dimensional scattering problems for non convex structures. The key point is that the use of the OSRC introduces a sparse block in the surface operator representation of the wave field while the integral part leads to an improved accuracy of the OSRC method in the non convex part of the scattering structure. The procedure is given for both the Dirichlet and Neumann scattering problems. Some numerical simulations show the improvement induced by the coupling method.

In order to deeply study the static and dynamic behavior of hemispherical convex-concave composite plate (HCCP) under different conditions and avoid structural damage, the equivalent model of HCCP was developed based on variational asymptotic method. The constitutive modeling of unit cell was carried out to obtain the constitutive relations that can be used as effective plate properties for the 2-D equivalent plate model (2D-EPM). The accuracy of the equivalent model and the obtained equivalent stiffness are verified by comparing with the static displacement, global buckling and free-vibration of 3D finite element model (3D-FEM). The influence of different layup configurations of laminates and geometric parameters on the equivalent stiffness, vibration characteristics and global buckling of HCCP are discussed. The results show that the buckling loads and natural frequencies obtained by using equivalent model are in good agreement with 3D-FEM. The vibration and buckling of HCCP are closely related to the geometric sizes and layup configurations of the laminate, and the local field distributions within the unit cell of HCCP are well captured. Most importantly, the calculation efficiency is greatly improved. The research results provide a reference for the practical application of HCCP in the civil engineering.