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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.

The paper is devoted to the behavior of volume ratios, the modified Banach–Mazur distance, and the vertex index for sums of convex bodies. It is shown that sup d ( A ⊕ K , B ⊕ L ) ≥ sup ∂ ( A ⊕ K , B ⊕ L ) ≥ c ⋅ n 1 − k + k ′ 2 n , \begin{equation*} \sup d (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq \sup \partial (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq c \cdot n^{1-\frac {k+k’}{2n}}, \end{equation*} if K ⊂ R n \mathrm {K}\subset \mathbb {R}^n and L ⊂ R k \mathrm {L}\subset \mathbb {R}^k are convex and symmetric (the supremum is taken over all symmetric convex bodies A ⊂ R n − k \mathrm {A}\subset \mathbb {R}^{n-k} and B ⊂ R n − k ′ ) \mathrm {B}\subset \mathbb {R}^{n-k’}) . Furthermore, some examples are discussed that show that the available extimates of the vertex index in terms of the volume ratio are not sharp.

In this study, an asymptotically analytical solution of elastic stress is proposed for the problem of a convex polygonal hole in an infinite plane under general loading conditions. The whole region surrounding a hole is taken as the union of a series of half-planes with each upper surface coinciding with the associated side of the hole. The elastic stresses can be calculated precisely based on Flamant’s analytical solution for stresses in a half-plane, induced by a concentrated load acting on its upper surface. To simplify the computation scheme and improve efficiency, the concept of equivalent concentrated forces is proposed, resulting in final solutions identical with those of the high-precision numerical integration method. The tractions along the two outer parts of the upper surfaces of the half-planes are initially assumed and then solved by an iterative procedure, yielding solutions within a tolerable number of iterations, which asymptotically approach the analytical ones in theory. Results of example studies show that the present method can give values of the stresses in the regions close to the corners of the hole, which are found to be essentially identical with the available analytical solutions. Values of stresses in the farther regions are simultaneously obtained, and validated by the finite element method.

In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the distance between two adjacent inclusions. In this paper, we derive upper and lower bounds of the gradient of solutions to the conductivity problem where two perfectly conducting inclusions are located very close to each other. To be specific, we extend the known results of Bao-Li-Yin (ARMA 2009) in two folds: First, we weaken the smoothness of the inclusions from C 2 , α to C 1 , α . To obtain a pointwise upper bound of the gradient, we follow an iteration technique which is first used to deal with elliptic systems in a narrow domain by Li-Li-Bao-Yin (QAM 2014). However, when the inclusions are of C 1 , α , we can not use W 2 , p estimates for elliptic equations any more. In order to overcome this new difficulty, we take advantage of De Giorgi-Nash estimates and Campanato's approach to apply an adapted version of the iteration technique with respect to the energy. A lower bound in the shortest line between two inclusions is also obtained to show the optimality of the blow-up rate. Second, when two inclusions are only convex but not strictly convex, we prove that blow-up does not occur any more. The establishment of the relationship between the blow-up rate of the gradient and the order of the convexity of the inclusions reveals the mechanism of such concentration phenomenon.

This article focuses on wave impedance calculations employed in the transmission line analogy for the low-frequency magnetic shielding problem: a planar shield against a loop current source placed parallel to the shield. We present a unified formula for wave impedance calculations that is distinct from traditional definitions of the transverse E -field to H -field ratio, which is related to the ratio of the H -field components to its longitudinal derivative. The formula is relatively simple in terms of mathematical complexity when applied to derive a concrete wave impedance expression. For a circular loop, the formula produces the same wave impedance expression as the traditional definition for field points on the central axis of the loop. For field points off the axis, the formula more accurately predicts the shielding effectiveness relative to traditional definitions. Based on the proposed formula, we also obtain wave impedance expressions for rectangular, elliptical, and regular polygon loops under quasi-static assumptions. The effectiveness of these expressions is verified through comparisons with finite element simulations.

Polarization is useful for estimating surface normals. In this paper, polarization is applied to shape estimation. Surface normal estimation from a single image is commonly performed by shape-from-shading. An alternative approach is shape-from-polarization, which usually cannot uniquely determine the surface normal and must be accompanied by additional information. The shading information is a promising option, but if the index of refraction is unknown, the surface normal is still non-uniquely determined. To resolve this problem, we assume that the object surface is locally convex. Under this assumption, we show that the surface normal can be uniquely determined from the gradients of the shading information. Our method works when the light and camera are closely arranged.

Based on the notion of plane asymptote, we introduce the new concept of cone asymptote of a set in the n-dimensional Euclidean space. We discuss the existence and describe some families of cone asymptotes.

We present a version of the Discrete Element Method considering the particles as rigid polyhedra. The Principle of Virtual Work is employed as basis for a multibody dynamics model. Each particle surface is split into sub-regions, which are tracked for contact with other sub-regions of neighboring particles. Contact interactions are modeled pointwise, considering vertex-face, edge-edge, vertex-edge and vertex-vertex interactions. General polyhedra with triangular faces are considered as particles, permitting multiple pointwise interactions which are automatically detected along the model evolution. We propose a combined interface law composed of a penalty and a barrier approach, to fulfill the contact constraints. Numerical examples demonstrate that the model can handle normal and frictional contact effects in a robust manner. These include simulations of convex and non-convex particles, showing the potential of applicability to materials with complex shaped particles such as sand and railway ballast.

In this paper, the authors consider a class of generalized curve flow for convex curves in the plane. They show that either the maximal existence time of the flow is finite and the evolving curve collapses to a round point with the enclosed area of the evolving curve tending to zero, i.e., $$\mathop {\lim}\limits_{t \to T} A(t) = 0$$ , or the maximal time is infinite, that is, the flow is a global one. In the case that the maximal existence time of the flow is finite, they also obtain a convergence theorem for rescaled curves at the maximal time.

The aim of this paper is to study the following fractional critical problem with Hardy potential and concave–convex nonlinearities (−Δ)su−μu|x|2s=λuq+u2s∗−1,u>0 in Ω,u=0in RN∖Ω, where (−Δ)s is the ...

We consider the application of the method of fundamental solutions (MFS) to certain two–dimensional second order boundary value problems (BVPs). We assume that the domain of the problem under consideration is a regular polygon and that the sources in the MFS are placed uniformly on a circle concentric to and surrounding the polygon. For certain uniform distributions of the collocation points we prove that, for general classes of regular polygons, the MFS discretization leads to singular coefficient matrices.

In this paper, we consider the weak Galerkin finite element approximations of second order linear parabolic problems in two dimensional convex polygonal domains under the low regularities of the solutions. Optimal order error estimates in L 2 ( L 2 ) and L 2 ( H 1 ) norms are shown to hold for both the spatially discrete continuous time and the discrete time weak Galerkin finite element schemes, which allow using the discontinuous piecewise polynomials on finite element partitions with the arbitrary shape of polygons with certain shape regularity. The fully discrete scheme is based on first order in time Euler method. We have derived O ( h r + 1 ) in L 2 ( L 2 ) norm and O ( h r ) in L 2 ( H 1 ) norm when the exact solution u ∈ L 2 ( 0 , T ; H r + 1 ( Ω ) ) ∩ H 1 ( 0 , T ; H r − 1 ( Ω ) ) , for some r ≥ 1 . Numerical experiments are reported for several test cases to justify our theoretical convergence results.

We characterize solutions of the mean value linear elliptic equation with constant coefficients in the complex plane in the case of regular polygon.

We prove that the Dirichlet problem for the Lane–Emden equation in a half-space has no positive solution that is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution that is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.

We consider a biharmonic problem Δ2uω = fω with Navier type boundary conditions uω = Δuω = 0, on a family of truncated sectors Ωω in ℝ2 of radius r, 0 < r < 1 and opening angle ω, ω ∈ (2π/3, π] when ω is close to π. The family of right-hand sides (fω)ω∈(2π/3, π] is assumed to depend smoothly on ω in L2(Ωω). The main result is that uω converges to uπ when ω → π with respect to the H2-norm. We can also show that the H2-topology is optimal for such a convergence result.

The solution to the problem of the shape of the lateral surface of a vertical 3D catenoidal liquid bridge of small volume between two arbitrary convex solid surfaces in the axisymmetric case taking into account the gravity force is presented. A variational formulation of the initial problem is given. The solution is found by the iteration method under the assumption of a small Bond number. An algorithm of the iterative process is proposed. We have detected domains of variation of parameters in which the uniqueness of the solution is not observed. It is found that the maximal number of different lateral surface profiles of the liquid bridge, which correspond to a single chosen set of parameters, is four. By way of example, the problem of the liquid bridge shape between two spheres is solved.

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

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 study complex geodesics and complex Monge–Ampere equations on bounded strongly linearly convex domains in $$\mathbb C^n$$ . More specifically, we prove the uniqueness of complex geodesics with prescribed boundary value and direction in such a domain, when its boundary is of minimal regularity. The existence of such complex geodesics was proved by the first author in the early 1990s, but the uniqueness was left open. Based on the existence and the uniqueness proved here, as well as other previously obtained results, we solve a homogeneous complex Monge–Ampere equation with prescribed boundary singularity, which was first considered by Bracci et al. on smoothly bounded strongly convex domains in $${\mathbb {C}}^n$$ .

This paper presents a new general methodology to obtain an approximate analytical expression of the Saint-Venant's torsion. The shear stress in each of the principal Cartesian directions is obtained by the derivation of the stress function, whose analytical expression is obtained from the Prandtl analogy. The proposed methodology uses two variables quadratic piecewise functions to define the Prandtl membrane. This document shows that the approximate procedure is especially suitable for steel shapes, giving values very close to those obtained with more precise methods. The main advantage of the presented methodology is its simplicity, which makes it useful for both pedagogic and practical applications. Several examples are developed.

In the present infrared target detection system, simultaneously achieving both the target detection performance, as well as, computing efficiency is considered as a big task. To, address the above said issue, a non‐convex triple tensor factorisation is incorporated into the existing infrared patch tensor model in the proposed work. In the proposed model (triple tensor factorisation‐infrared patch tensor), local prior information using linear structure tensor and corner strength is incorporated so that the strong clutters in the background can be easily suppressed and the target can be detected correctly. Finally, the method proposed, was solved by alternating direction method of the multiplier. Large number of experiments were conducted and the results presented in the experiments suggest that the method proposed has shown good performance for background suppression, as well as, for the target detection in the clutter environment, when compared with the other baseline methods.

The representation of particles of complex shapes is one of the key challenges of numerical simulations based on the discrete element method (DEM). A novel algorithm has been developed by the authors to accurately represent 2D arbitrary particles for DEM modelling. In this paper, the algorithm is extended from 2D to 3D to model convex polyhedral particles based on multisphere methods, which includes three steps: the placement of spheres at the corners, along the edges, and on the facets in sequence. To give a good representation of a polyhedral particle, the spheres are placed tangent to the particle surface in each step. All spheres placed in the three steps are clumped together into a clump in DEM. In addition, the mass properties of the clump are determined based on the corresponding polyhedral particle to obtain accurate simulation results. Finally, an example is used to validate the robust and automatic performance of the algorithm in generating a sphere clump model for an assembly of polyhedral particles. A current FORTRAN version of the algorithm is available by contacting the authors.

We investigate under the notion of Large Deviation Principle & Concentration of Measure as a technique,the ability of estimating the probability density function of any random vector in the space Rn. We found that an appropriate probability distribution for any convex body in the space is sub – Gaussian.

We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by divergence-free vector fields. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians ( \begin{document}$ G $\end{document} below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense and establish an averaging result for Hamilton-Jacobi equations with relatively general Hamiltonian \begin{document}$ G $\end{document} .

This paper focuses on the problem of noise removal. First, we propose a new convex–nonconvex variation model for noise removal and consider the nonexistence of solutions of the variation model. Based on the new variation method, we propose a class of singular diffusion equations and prove the of solutions and comparison rule for the new equations. Finally, experimental results illustrate the effectiveness of the model in noise reduction.

Let D := {z ? C : |z| < 1} be the open unit disk, and h and 1 be two analytic functions in D. Suppose that f = h + ?g is a harmonic mapping in D with the usual normalization h(0) = 0 = g(0) and h'(0) = 1. In this paper, we consider harmonic mappings f by restricting its analytic part to a family of functions convex in one direction and, in particular, starlike. Some sharp and optimal estimates for coefficient bounds, growth, covering and area bounds are investigated for the class of functions under consideration. Also, we obtain optimal radii of fully convexity, fully starlikeness, uniformly convexity, and uniformly starlikeness of functions belonging to those family.

Inspired by the classical Poincar\'e criterion about the instability of orientation preserving minimizing closed geodesics on surfaces, we investigate the relation intertwining the instability and the variational properties of periodic solutions of a non-autonomous Lagrangian on a finite dimensional Riemannian manifold. We establish a general criterion for a priori detecting the linear instability of a periodic orbit on a Riemannian manifold for a (maybe not Legendre convex) non-autonomous Lagrangian simply by looking at the parity of the spectral index, which is the right substitute of the Morse index in the framework of strongly indefinite variational problems and defined in terms of the spectral flow of a path of Fredholm quadratic forms on a Hilbert bundle., Comment: 32 pages, no figures

Parabolic equation methods are a robust and efficient tool for modeling long-range acoustic propagation in range-dependent waveguides. A lesser known, but equally effective, application of parabolic equations is to the scattering problem. In this paper, we study the applicability and accuracy of the multisector parabolic equation approach to scattering from impenetrable objects – first developed approximately two decades ago for convex objects and recently extended to more complex scatterers – in waveguides in two and three dimensions. We benchmark the method in two dimensions in a range-dependent waveguide consisting of two different fluid media and find very good agreement with finite element methods for both forward scattering and backscattering, including the case where the scatterer is located at the interface between the media. Finally, we provide an example of scattering in a large three-dimensional waveguide, which would be extremely computationally intensive using alternate approaches, but is practical using the parabolic equation method.

In this paper, a decoupled finite element method for a modified Cahn-Hilliard-Hele-Shaw system with double well potential is constructed. The proposed scheme is based on the convex splitting of the energy functional in time and uses the mixed finite element discretzation in space. The computation of the velocity $ \mathbf{u} $ is separated from the computation of the pressure $ p $ by using an operator-splitting strategy. A Possion equation is solved to update the pressure at each time step. Unconditional stability and error estimates are analyzed in detail theoretically. Furthermore, the theoretical part is verified by several numerical examples, whose results show that the numerical examples are consistent with the results of the theoretical part.

We consider the minimization of an energy functional given by the sum of a crystalline perimeter and a nonlocal interaction of Riesz type, under volume constraint. We show that, in the small mass regime, if the Wulff shape of the anisotropic perimeter has certain symmetry properties, then it is the unique global minimizer of the total energy. In dimension two this applies to convex polygons which are reflection symmetric with respect to the bisectors of the angles. We further prove a rigidity result for the structure of (local) minimizers in two dimensions.

Nonunique weak solutions in Leray--Hopf class are constructed for the three-dimensional magneto-hydrodynamics with Hall effect. We adapt the widely appreciated convex integration framework develope...