Reflected brownian motion in a wedge: Semimartingale property

In this paper, the object of study is reflected Brownian motion in a two-dimensional wedge with constant direction of reflection on each side of the wedge. The basic question considered here is “When is this process a semimartingale?”. It is first shown that a related process, defined by specifying the corner of the wedge to be an absorbing state, rather than an instantaneous one, is a semimartingale. Conditions for the existence and uniqueness of the process for which the corner is an instantaneous state were given by Vardhan and Williams (“Brownian motion in a wedge with oblique reflection”, Comm. Pure Appl. Math., to appear). Under these conditions, it is shown that starting away from the corner, the process is a semimartingale if and only if there is a convex combination of the directions of reflection that points into the wedge. This equivalence is also shown to hold starting from the corner, except in one unresolved case for which the wedge angle exceeds π and the directions of reflection are exactly opposed.     

Remarks on minimal annuli in a slab

We show that the two closed boundary curves of a minimal annulus in a slab are both convex if one of them is convex and along the other curve the surface meets the plane at a constant angle. And therefore, under the same condition, the minimal annulus is foliated by convex planar curves all of which are parallel to the boundary. In particular, if the convex curve is a circle, then the annulus is part of a catenoid.     

ASYMPTOTIC BEHAVIOUR OF SUPERSONIC FLOW PAST ACONVEX COMBINED WEDGE

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Steady supersonic flow past an almost straight wedge with large vertex angle

This paper studies the problem on the steady supersonic flow at the constant speed past an almost straight wedge with a piecewise smooth boundary. It is well known that if each vertex angle of the straight wedge is less than an extreme angle determined by the shock polar, the shock wave is attached to the tip of the wedge and constant states on both side of the shock are supersonic. This paper is devoted to generalizing this result. Under the hypotheses that each vertex angle is less than the extreme angle and the total variation of tangent angle along each edge is sufficiently small, a sequence of approximate solutions constructed by a modified Glimm scheme is proved to be convergent to a global weak solution of the steady problem. A sequence of the corresponding approximate leading shock fronts issuing from the tip is shown to be convergent to the leading shock front of the obtained solution. The regularity of the leading shock front is established and the asymptotic behaviour of the obtained solution at infinity is also studied.     

On evolutoids of planar convex curves

There are many interesting curves which we can associate with a given convex curve, however, in this work we are especially interested in studying the relations between the given curve and its evolutoids: that is, the curve obtained as the envelope of lines making a fixed angle with the normal line at every point of the curve. The first result is an inequality between the area enclosed by the given curve and the area enclosed by its evolutoid. Also, we proved that a convex curve is of constant width (centrally symmetric) if and only if its evolutoid for a fixed angle is of constant width (centrally symmetric).     

Brownian motion in a wedge with oblique reflection

This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties:

• (i) The state space is an infinite two-dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion.
• (ii) The process reflects instantaneously at the boundary of the wedge, the angle of reflection being constant along each side.
• (iii) The amount of time that the process spends at the comer of the wedge is zero (i.e., the set of times for which the process is at the comer has Lebesgue measure zero).

Hereafter, let ξ be the angle of the wedge (0 < ξ < 2π), let θ₁ and θ₂ be the angles of reflection on the two sides of the wedge, measured from the inward normals, the positive angles being toward the corner (-½π < θ₁, θ₂ ½π), and set α = (θ₁ + θ₂)/ξ. The question of existence and uniqueness is recast as a submartingale problem in the style used by Stroock and Varadhan (Diffusion processes with boundary conditions, Comm. Pure Appl. Math. 24, 1971, pp. 147-225), for diffusions on smooth domains with smooth boundary conditions. It is shown that no solution exists if α ≧ 2. In this case, there is a unique continuous strong Markov process satisfying (i)-(ii) above; it reaches the corner of the wedge almost surely and it remains there. If α < 2, however, then there is a unique continuous strong Markov process statisfying (i)-(iii). It is shown that starting away from the corner this process does not reach the corner of the wedge if α ≦ 0, and does reach the corner if 0 < α < 2. The general theory of multi-dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner. For some values of α, the process arises from diffusion approximations to storage systems and queueing networks. (i) The state space is an infinite two-dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the wedge, and the angle of reflection being constant along each side. (iii) The amount of time that the process spends at the corner of the wedge is zero (i.e., the set of times for which the process is at the corner has Lebesgue measure zero).     

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 Rational Basis for Function Approximation

A method is developed for constructing a convenient basis ofrational wedge functions for collocation on convex polygon boundaries.     

Global existence of shock for the supersonic Euler flow past a curved 2-D wedge

In this paper, we study the global existence of the supersonic shock for the steady supersonic Euler flow past a curved 2-D wedge. By using the method of characteristic, we show that the shock exists globally and the flow between the shock and wedge is continuous provided the wedge is a small perturbation of a straight wedge under a weighted global Sobolev norm and the vertex angle is less than the extreme angle.     

Global Existence of a Shock for the Supersonic Flow Past a Curved Wedge

This note is devoted to the study of the global existence of a shock wave for the supersonic flow past a curved wedge. When the curved wedge is a small perturbation of a straight wedge and the angle of the wedge is less than some critical value, we show that a shock attached at the wedge will exist globally.     

Almost convex sets

The concept of a wedgoid, a generalized wedge, is introduced and studied. Almost all convex sets are characterized by the fact that balls can be separated from such a set by means of wedgoids.     

Measure of segments which intersect a convex body from rotational formulae

Classical problems in integral geometry and geometric probability involve the kinematic measure of congruent segments of fixed length within a convex body in R³. We give this measure from rotational formulae; that is, from isotropic plane sections through a fixed point. From this result we also obtain a new rotational formula for the volume of a convex body; which is proved to be equivalent to the wedge formula for the volume.     

Study of the stability in the problem on flowing around a wedge. The case of strong wave

We consider the flow of an inviscid nonheatconducting gas in the thermodynamical equilibrium state around a plane infinite wedge and study the stationary solution to this problem, the so-called strong shock wave; the flow behind the shock front is subsonic.We find a solution to a mixed problem for a linear analog of the initial problem, prove that the solution trace on the shock wave is the superposition of direct and reflected waves, and, the main point, justify the Lyapunov asymptotical stability of the strong shock wave provided that the angle at the wedge vertex is small, the uniform Lopatinsky condition is fulfilled, the initial data have a compact support, and the solvability conditions take place if needed (their number depends on the class in which the generalized solution is found).     

Unified unsteady supersonic/hypersonic theory of flow past double wedge airfoils

A unified supersonic/hypersonic theory is given of flow past a pitching oscillating double wedge airfoil at arbitrary mean angle of attack. The amplitude and the (reduced) frequency parameter of the oscillation are assumed small and a perturbation method is employed. Closed form formulae are obtained for the stiffness and damping-in-pitch derivatives. They are exact with respect to the free stream Mach number, angle of attack and body thickness etc., provided only that the bow shock is attached to the leading edge. The theory predicts negative damping (instability) if the angle of attack, or the airfoil thickness is sufficiently large, or if the free stream Mach number is sufficiently low. It is shown to be in good agreement with experiments of Scruton et. al. Comparisons with Van Dyke second order potential theory and with Lighthill piston theory are also given. Finally the theory may easily be extended to arbitrary smooth airfoils.     

Characterization of inner product spaces in V- and L-spaces

In [4], Freese and Murphy introduce a new class of spaces, the V-spaces, which include Banach spaces, hyperbolic spaces, and other metric spaces. In this class of spaces they investigate conditions which are equivalent to strict convexity in Banach spaces, and extend some of the Banach space results to this new class of spaces. It is natural to ask if known characterizations of real inner product spaces among Banach spaces can also be extended to this larger class of spaces. In the present paper it will be shown that a metrization of an angle bisector property used in [3] to characterize real inner product spaces among Banach spaces also characterizes real inner product spaces among V-spaces, and among another class of spaces, the L-spaces, which include hyperbolic spaces and strictly convex Banach spaces. In the process it is shown that in a complete, convex, externally convex metric space M, if the foot of a point on a metric line is unique, then M satisfies the monotone property, thus answering a question raised in [4].     

An isoperimetric inequality for convex polygons and convex sets with the same symmetrals

Suppose that two distinct plane convex bodies have the same Steiner symmetrals about a finite number n of given lines. Then we obtain an upper bound for the measure of their symmetric difference. The bound is attained if, and only if, the directions of the lines are equally spaced and the bodies are two regular concentric polygons, with n sides, each obtained from the other by rotation through an angle /n. This result follows from a new isoperimetric inequality for convex polygons.     

Classes of analytic functions subordinate to convex functions and extreme points

In this paper we prove two sufficient conditions for an analytic function f to be an extreme point of the set of functions subordinate to a given convex mapping F when the image of the unit disk under F is a convex domain other than a half-plane, a strip or an infinite wedge.     

Unsteady thermal boundary-layer on an infinite yawed wedge whose temperature gradient is prescribed

In this paper, the temperature distribution at the surface of an infinite yawed wedge, when temperature of the main-stream is constant, is studied under two cases; one when temperature gradient of the wall is steady and the velocity of the main-stream is unsteady, and the other when temperature gradient of the wall is unsteady and the velocity of the main-stream is steady. It is found that the heat transfer depends on the wedge angle, the angle parameter and the yaw of the wedge. The behaviour with these parameters are studied and are illustrated graphically.     

Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge

For a supersonic Euler flow past a straight-sided wedge whose vertex angle is less than the extreme angle, there exists a shock-front emanating from the wedge vertex, and the shock-front is usually strong especially when the vertex angle of the wedge is large. In this paper, we establish the L¹ well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge whose boundary slope function has small total variation, when the total variation of the incoming flow is small. In this case, the Lipschitz wedge perturbs the flow, and the waves reflect after interacting with the strong shock-front and the wedge boundary. We first obtain the existence of solutions in BV when the incoming flow has small total variation by the wave front tracking method and then establish the L¹ stability of the solutions with respect to the incoming flows. In particular, we incorporate the nonlinear waves generated from the wedge boundary to develop a Lyapunov functional between two solutions containing strong shock-fronts, which is equivalent to the L¹ norm, and prove that the functional decreases in the flow direction. Then the L¹ stability is established, so is the uniqueness of the solutions by the wave front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also obtained.     

关于半连续函数与凸函数的注记

在半连续前提下,给出凸函数和严格凸函数的不等式刻划．指出非空凸集上的半连续函数满足中间点凸性时,成为凸函数,满足中间点严格凸性时,成为严格凸函数．最后定义F—G广义凸函数和条件p1,p2等概念,列举若干满足条件p1,p2的数量函数和向量函数,并指出,对于F—G广义凸函数,在条件p1,p2及一定连续性条件下,可以得到类似结果．