Direct problem and inverse problem for the supersonic plane flow past a curved wedge

In this paper, we consider the isentropic irrotational steady plane flow past a curved wedge. First, for a uniform supersonic oncoming flow, we study the direct problem: For a given curved wedge y = f(x), how to globally determine the corresponding shock y = g(x) and the solution behind the shock? Then, we solve the corresponding inverse problem: How to globally determine the curved wedge y = f(x) under the hypothesis that the position of the shock y = g(x) and the uniform supersonic oncoming flow are given? This kind of problems plays an important role in the aviation industry. Under suitable assumptions, we obtain the global existence and uniqueness for both problems. Copyright © 2011 John Wiley & Sons, Ltd.     

A strongly polynomial algorithm for the minimum maximum flow degree problem

Let us consider a network flow respecting arc capacities and flow conservation constraints. The flow degree of a node is sum of the flow entering and leaving it. We study the problem of determining a flow that minimizes the maximum flow degree of a node. We show how to solve it in strongly polynomial time with linear programming.     

The steady two-dimensional flow over a rectangular obstacle lying on the bottom

We study a plane problem with mixed boundary conditions for a harmonic function in an unbounded Lipschitz domain contained in a strip. The problem is obtained by linearizing the hydrodynamic equations which describe the steady flow of a heavy ideal fluid over an obstacle lying on the flat bottom of a channel. In the case of obstacles of rectangular shape we prove unique solvability for all velocities of the (unperturbed) flow above a critical value depending on the obstacle depth. We also discuss regularity and asymptotic properties of the solutions.     

一种保持周长的平面曲线流

The purpose of this paper is to investigate a new type of evolution problem for closedconvex plane curves which will preserves the perimeter of the curve but expands the enclosedarea and the final limiting curve is a circle in the Hausdorff metric in the plane.     

On a “stability” in the linear complementarity problem

In this work we re-wrote the Linear Complementarity Problem in a formulation based on unknown projector operators. In particular, this formulation allows the introduction of a concept of “stability” that, in a certain way, might explain the way block pivotal algorithm performs.     

On a first boundary value problem for hyperbolic equations in the plane

In the paper we study a boundary value problem for a hyperbolic equation with two independent variables; this problem is a generalization of the well-known Darboux problem. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 294–306, February, 1999.     

Leray's problem for a viscous incompressible micropolar fluid

This work concerns the steady motion of a viscous incompressible micropolar fluid in unbounded domains having cylindrical outlets to infinity. We prove the existence of a solution that approaches prescribed parallel solutions along the outlets of the domain. We also study the uniqueness, the regularity and the asymptotic behavior of the solution.     

On the combinatorial structure of the permutation flow shop problem

In this paper we investigate the combinatorial structure of then/m/P/C _max permutation flow shop problem. To each solution (a permutation ofn elements) we introduce a network which represents the job and machine orders. Based on the introduced generalized distance between a permutation and a path of a network we derive combinatorial properties with respect to special neighbourhoods which are important for developing iterative heuristics for the considered problem.For several neighbourhood structures we calculate the mean cardinality of reduced neighbourhoods where each neighbour satisfies a necessary condition for an objective improvement.

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Zusammenfassung In der vorliegenden Arbeit wird die kombinatorische Struktur desn/m/P/C _max Permutationsflußproblems untersucht. Zu jeder zulässigen Lösung, beschreibbar durch eine Permutation vonn Elementen, wird ein Netzplan eingeführt, der die Bearbeitungsreihenfolge der Operationen charakterisiert. Aufbauend auf einem eingeführten verallgemeinerten Abstand zwischen einer Permutation und einem Weg in einem Netzplan werden danach einige kombinatorische Eigenschaften bezüglich spezieller Nachbarschaftsstrukturen abgeleitet, die im Hinblick auf die Entwicklung von iterativen Heuristiken für das betrachtete Problem von Bedeutung sind. So werden u. a. für ausgewählte Nachbarschaftsstrukturen mittlere Mächtigkeiten reduzierter Nachbarschaften, bei denen jeder Nachbar eine notwendige Bedingung für eine Zielfunktionswertverbesserung erfüllt, hergeleitet.

     

On a free piston problem for potential ideal fluid flow

Our goal was to model and analyze a stationary and evolutionary potential ideal fluid flow through the junction of two pipes in the gravity field. Inside the ‘vertical’ pipe, there is a heavy piston that can freely move along the pipe. In the stationary case, we are interested in the equilibrium position of the piston in dependence on the geometry of junction, and in the evolutionary case, we study motion of the piston also in dependence on geometry. We formulate corresponding initial and boundary value problems and prove the existence results. The problem is nonlinear because the domain is unknown. Furthermore, we study some qualitative properties of the solutions and compare them with the qualitative properties of a free piston problem for Newtonian fluid flow. All theoretical results are illustrated with numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

On the Frobenius problem for Beatty sequences

We study the solvability of linear diophantine equations in two variables within the context of Beatty sequences.     

On a generalization of the Neumann problem for the Laplace equation outside cuts in a plane

A boundary value problem for harmonic functions outside cuts in a plane is considered. The jump of the normal derivative is specified on the cuts as well as a linear combination of the normal derivative on one side of the cut and the jump of the unknown function. The problem is studied with three different conditions at infinity, which lead to different results on existence and number of solutions. The integral representation for a solution is obtained in the form of potentials density in which satisfies the uniquely solvable Fredholm integral equation of the 2nd kind. Copyright © 2004 John Wiley & Sons, Ltd.     

On the Dirichlet problem for a nonlinear elastic beam equation

Using a direct variational approach with no global growth conditions on the nonlinear term, we consider the existence of solutions and their dependence on a functional parameter for the fourth order Dirichlet problem connected with the elastic beam equation. We investigate also the existence of an optimal process for such an optimal control problem in which the dynamics is described by the beam equation.     

On the adjustment problem for linear programs

We propose a generalization of the inverse problem which we will call the adjustment problem. For an optimization problem with linear objective function and its restriction defined by a given subset of feasible solutions, the adjustment problem consists in finding the least costly perturbations of the original objective function coefficients, which guarantee that an optimal solution of the perturbed problem is also feasible for the considered restriction. We describe a method of solving the adjustment problem for continuous linear programming problems when variables in the restriction are required to be binary.     

A new formulation of the water wave problem for Stokes waves of constant vorticity

We consider steady symmetric gravity water waves on finite depth with constant vorticity and a monotone surface profile between crests and troughs. The problem is transformed into one concerning the vertical velocity. A representation formula for the stream function in terms of the surface and the vorticity is presented, and we show that the surface can be determined from the vertical velocity.     

An inverse problem for supersonic flow past a curved wedge

This paper studies an inverse problem for supersonic potential flow past a curved wedge, in which we design a suitable curved wedge such that the shock produced by the curved wedge can be controlled to the given position. Under suitable conditions, by characteristic method, we prove the existence of the global classical solution to this inverse problem and develop an optimal decay rate on the given shock’s second order derivatives. We finally construct a specific wedge such that the shock generated by the wedge is a convex combined one.     

A flow method to the Orlicz-Aleksandrov problem

In this paper, we obtain an existence result of smooth solutions to the Orlicz-Aleksandrov problem from the perspective of geometric flow. Furthermore, a special uniqueness result of solutions to this problem shall be discussed.     

The minimum speed for a blocking problem on the half plane

We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ>1. The proof relies on a geometric lemma of independent interest. Namely, let K⊂R² be a compact, simply connected set with smooth boundary. We define dK(x,y) as the minimum length among all paths connecting x with y and remaining inside K. Then dK attains its maximum at a pair of points both on the boundary of K.