Plane R2 Research Articles

A log-weighted Moser inequality on the plane

We establish a sharp Moser type inequality with logarithmic weight in the nonradial mass-weighted Sobolev spaces, on the whole plane R2. We identify the sharp threshold for the uniform boundedness of the weighted Moser functional, which is still given by 4π: further, we prove the validity of the inequality also at the limiting sharp value 4π. Even if the increasing nature of the log weight prevents the application of any symmetrization tool, we prove our inequality in the general framework of Sobolev space, and not on radial subspaces, as in the available literature. The main strategy is a careful analysis of the behaviour of the normalized maximizing sequences.

Symmetric Phase Portraits of Homogeneous Polynomial Hamiltonian Systems of Degree 1, 2, 3, 4, and 5 with Finitely Many Equilibria

Roughly speaking, the Poincaré disc D2 is the closed disc centered at the origin of the coordinates of R2, where the whole of R2 is identified with the interior of D2 and the circle of the boundary of D2 is identified with the infinity of R2, because in the plane R2, we can go to infinity in as many directions as points have the circle. The phase portraits of the quadratic Hamiltonian systems in the Poincaré disc were classified in 1994. Since then, no new interesting classes of Hamiltonian systems have been classified on the Poincaré disc. In this paper, we determine the phase portraits in the Poincaré disc of five classes of homogeneous Hamiltonian polynomial differential systems of degrees 1, 2, 3, 4, and 5 with finitely many equilibria. Moreover, all these phase portraits are symmetric with respect to the origin of coordinates. We showed that these polynomial differential systems exhibit precisely 2, 2, 3, 3, and 4 topologically distinct phase portraits in the Poincaré disc. Of course, the new results are for the homogeneous Hamiltonian polynomial differential systems of degrees 3, 4, and 5. The tools used here for obtaining these phase portraits also work for obtaining any phase portrait of a homogeneous Hamiltonian polynomial differential system of arbitrary degree.

Bitangent planes of surfaces and applications to thermodynamics

The classical van der Waals equation, applied to one or two mixing fluids, and the Helmholtz (free) energy function A yield, for fixed temperature T, a curve in the plane R2 (one fluid) or a surface in 3-space R3 (binary fluid). A line tangent to this curve in two places (bitangent line), or a set of planes tangent to this surface in two places (bitangent planes) have a thermodynamic significance which is well documented in the classical literature. Points of contact of bitangent planes trace ‘binodal curves’ on the surface in R3. The study of these bitangents is also classical, starting with D.J. Korteweg and J.D. van der Waals at the end of the 19th century, but continuing into modern times. In this paper we give a summary of the thermodynamic background and of other mathematical investigations and then present a new mathematical approach which classifies a wide range of situations in R3 where bitangents occur. In particular, we are able to justify many of the details in diagrams of binodal curves observed by Korteweg and others, using techniques from singularity theory.

On periodically modulated rolls in the generalized Swift–Hohenberg equation: Galerkin’ approximations

We study in this paper the existence of periodically modulated in one variable and localized in another variable solutions to the cubic Swift–Hohenberg equation on the plane R2. In the first part we try to apply the method by Kirschgässner–Mielke to reduce the problem to the search of finite dimensional submanifolds with periodic orbits on them in some formal infinite-dimensional dynamical system generated by the stationary SH equation. It turns out that it cannot be done immediately due to properties of the spectrum for the linearized system at the localized roll (a one-dimensional pulse). In the second part we change roles of variables and formulate the problem as funding homoclinic orbits to an equilibrium of the formal infinite-dimensional system in the space of periodic functions in variable y. Staying apart the proof of the exact theorem on the existence of related center manifold, we exploit the Bubnov–Galerkin method to derive the Hamiltonian finite-dimensional ODEs with four or six degrees of freedom having the saddle type equilibrium whose homoclinic orbits correspond to approximate solution on needed type. The search for homoclinic orbits is performed by means of numerical methods.

Example of a solution for Dorodnitzyn’s limit formula

In the present paper, we show an example of a solution for Dorodnitzyn’s gaseous boundary layer limit formula. Oleinik’s no back-flow condition ensures the existence and uniqueness of solutions for the Prandtl equations in a rectangular domain R⊂R2. It also allowed us to find a limit formula for Dorodnitzyn’s stationary compressible boundary layer with constant total energy on a bounded convex domain in the plane R2. Under the same assumption, we can give an approximate solution u for the limit formula if |u|<<<1 such that: u(z)≅δ∗c∗z+625⋅12i0⋅4U23z4+o(z5),that corresponds to an approximate horizontal velocity component when a small parameter ϵ given by the quotient of the maximum height of the domain divided by its length tends to zero. Here, c>0, δ is the boundary layer’s height in Dorodnitzyn’s coordinates, U is the free-stream velocity at the upper boundary of the domain, and T0 is the absolute surface temperature.

Approximation Schemes for Capacitated Vehicle Routing on Graphs of Bounded Treewidth, Bounded Doubling, or Highway Dimension

In this article, we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser in 1959 [ 14 ], we are given a graph G=(V,E) with metric edges costs, a depot r ∈ V , and a vehicle of bounded capacity Q . The goal is to find a minimum cost collection of tours for the vehicle that returns to the depot, each visiting at most Q nodes, such that they cover all the nodes. This generalizes classic TSP and has been studied extensively. In the more general setting, each node v has a demand d v and the total demand of each tour must be no more than Q . Either the demand of each node must be served by one tour (unsplittable) or can be served by multiple tours (splittable). The best-known approximation algorithm for general graphs has ratio α +2(1-ε) (for the unsplittable) and α +1-ε (for the splittable) for some fixed \(&amp;#x03B5; \gt \frac{1}{3000}\) , where α is the best approximation for TSP. Even for the case of trees, the best approximation ratio is 4/3 [ 5 ] and it has been an open question if there is an approximation scheme for this simple class of graphs. Das and Mathieu [ 15 ] presented an approximation scheme with time n log O(1/ε) n for Euclidean plane ℝ 2 . No other approximation scheme is known for any other class of metrics (without further restrictions on Q ). In this article, we make significant progress on this classic problem by presenting Quasi-Polynomial Time Approximation Schemes (QPTAS) for graphs of bounded treewidth, graphs of bounded highway dimensions, and graphs of bounded doubling dimensions. For comparison, our result implies an approximation scheme for the Euclidean plane with run time n O(log 6 n/ε 5 ) .

Properties of Coordinated h1,h2-Convex Functions of Two Variables Related to the Hermite–Hadamard–Fejér Type Inequalities

In this paper, we prove the Hermite–Hadamard–Fejér type inequalities for coordinated h1,h2-convex functions on the rectangle from the plane R2. Some generalizations of the Hermite–Hadamard-type inequalities of two variables are also obtained as a consequence. Some properties of two functionals which are connected with the coordinated h1,h2-convex functions are provided as well. Finally, we give applications of the acquired results to special means of positive real numbers.

Some classifications of 2-dimensional self-shrinkers

It is well-known that self-shrinkers play an important role in the study of mean curvature flows. In this paper, we develop new techniques to study the rigidity of self-shrinkers. We prove that any self-shrinker X:M→R3 with nonzero constant Gauss curvature is the round sphere S2(2). Moreover, we prove that any flat self-shrinker X:M→R3 is a plane R2, a cylinder S1(1)×R, a generalized cylinder Γ×R, where Γ is an Abresch-Langer curve. At last, we show that any self-shrinker X:M→R3 with constant mean curvature is a plane R2, a cylinder S1(1)×R or the round sphere S2(2).

Taming closed subsets of the Cantor fence

Let C be a Cantor set in the real line. Tymchatyn and Walker in [3] proved that every embedding h of C×[0,1] into the plane R2 is tame, i.e., there is a homeomorphism H:R2→R2 such that H(h(C×[0,1]))⊂C×[0,1]. In this paper we answer a question of L. Oversteegen [2] by proving that every embedding of a closed subset of C×[0,1] into the plane is tame.

Existence of the stationary Navier-Stokes flow in [formula omitted] around a radial flow

We consider the stationary Navier-Stokes equations on the whole plane R2. We show that for a given small and smooth external force around a radial flow, there exists a classical solution decaying like |x|−1. In our result, it is not necessary to impose any symmetric conditions on external forces.

On inequalities of Simpson type for co-ordinated convex functions via generalized fractional integrals

In this study, we prove equality for twice partially differentiable mappings involving the double generalized fractional integral. Using the established identity, we offer some Simpson?s type inequalities for differentiable co-ordinated convex functions in a rectangle from the plane R2.

The solution of the Poincaré problem on the rational first integral for the Liénard polynomial differential equations

In this work we classify the polynomial Liénard differential equations ẍ+f(x)ẋ+x=0, having a rational first integral. Such classification was asked by Poincaré in 1891 for any general polynomial differential systems in the plane R2. As far as we know it is the first time that the complete classification is given for a relevant class of polynomial differential equations of arbitrary degree.

On the Shortest Distance in the Plane R2 π3

In this paper, we study the shortest distance of a point to the line and area of a triangle in Iso-taxicab Geometry

Noncommutative Reduction of Nonlinear Schrödinger Equation on Lie Groups

We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the method of noncommutative integration to the linear part of a nonlinear equation, which allows one to find bases in the space of solutions of linear partial differential equations with a set of noncommuting symmetry operators. The approach is implemented for the generalized nonlinear Schrödinger equation on a Lie group in curved space with local cubic nonlinearity. General formalism is illustrated by the example of the noncommutative reduction of the nonstationary nonlinear Schrödinger equation on the motion group E(2) of the two-dimensional plane R2. In this particular case, we come to the usual (1+1)-dimensional nonlinear Schrödinger equation with the soliton solution. Another example provides the noncommutative reduction of the stationary multidimensional nonlinear Schrödinger equation on the four-dimensional exponential solvable group.

Limit cycles of generic piecewise center-type vector fields in [formula omitted] separated by either one plane or by two parallel planes

While the limit cycles of the piecewise differential systems in the plane R2 have been studied intensively during these last twenty years, this is not the case for the limit cycles of the piecewise differential systems in the space R3.The goal of this article is to study the continuous and discontinuous piecewise differential systems in R3, formed by linear vector fields similar to planar centers separated by one or two parallel planes. We call those “center-type” differential systems, which have two pure imaginary numbers and zero as eigenvalues. When these kinds of piecewise differential systems are continuous or discontinuous separated by one plane, then they have no limit cycles. Also, if they are continuous separated by two planes, then generically they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel planes, we show that generically they can have at most four limit cycles, and that there exist such systems with four limit cycles. The genericity here means that the statements hold in a residual set of the space of parameters associated to the differential system.We recall that the same problem but for discontinuous piecewise differential systems in R2 formed by linear differential centers separated by two parallel straight lines has at most one limit cycle.

On the fractional double integral inclusion relations having exponential kernels via interval-valued co-ordinated convex mappings

In the present study, over a rectangle from the plane R2, we define and develop the conceptions of the interval-valued fractional double integrals having exponential kernels, from which we exploit Hermite–Hadamard, Fejér–Hermite–Hadamard, as well as Pachpatte type inclusion relations regarding the interval-valued co-ordinated convex mappings. These inclusion relations can be viewed as certain substantial generalizations of the previously reported findings. To identify the correctness of the inclusion relations constructed in this work, we also provide three examples regarding the interval-valued co-ordinated convex mappings.

Hermite-Hadamard-Fejér inequalities for double integrals

In this paper, we first obtain Hermite-Hadamard-Fejer inequalities for co-ordinated convex functions in a rectangle from the plane R2. Moreover, we give the some refinement of these obtained Hermite-Hadamard-Fejer inequalities utilizing two mapping. The inequalities obtained in this study provide generalizations of some result given in earlier works.

Weighted estimates for maximal functions associated with finite type curves in [formula omitted

In this paper we establish weighted estimates for the maximal function associated with the finite type curve in the plane R2. We follow an approach used by M. Lacey to obtain sparse bounds for the maximal function. Further, using a different approach we obtain a characterisation of power weights for the weighted Lp boundedness of the maximal function. We also obtain analogous results for the lacunary maximal function associated with the finite type curve in the plane R2.

The mollification method based on a modified operator to the ill-posed problem for 3D Helmholtz equation with mixed boundary

The paper is considered a Cauchy problem for Helmholtz equation with mixed boundary in three-dimensional case. The problem is severely ill-posed (small deviation in the data may produce large errors in the computed solution), to solve this problem, a mollification method based on modified bivariate de la Vallée Poussin operator is proposed. The stable error estimates are obtained in Sobolev space under the suitable choice of mollification parameters and the a priori bounds assumption, the major approach to obtain the convergence estimates is to divide the real plane R2 into 25 regions. The numerical experiments support the feasible, stable and convergent of our algorithm.

Localization effects for Dirichlet problems in domains surrounded by thin stiff and heavy bands

We consider a Dirichlet spectral problem for a second order differential operator, with piecewise constant coefficients, in a domain Ωε in the plane R2. Here Ωε is Ω∪ωε∪Γ, where Ω is a fixed bounded domain with boundary Γ, ωε is a curvilinear band of width O(ε), and Γ=Ω‾∩ω‾ε. The density and stiffness constants are of order ε−m−t and ε−t respectively in this band, while they are of order 1 in Ω; t≥1, m>2, and ε is a small positive parameter. We address the asymptotic behavior, as ε→0, for the eigenvalues and the corresponding eigenfunctions. In particular, we show certain localization effects for eigenfunctions associated with low frequencies. This is deeply involved with the extrema of the curvature of Γ.