Existence of local strong solutions to fluid–beam and fluid–rod interaction systems

We study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. Considering three different structure models, we prove existence of a unique local-in-time strong solution, for which there is no gap between the regularity of the initial data and the regularity of the solution enabling to obtain a blow up alternative. In the case of a damped beam this is an alternative proof (and a generalization to non zero initial displacement) of the result that can be found in [20]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. The key points consist in studying the coupled system without decoupling the fluid from the structure and to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.     

Finding a representative nondominated set for multi-objective mixed integer programs

In this paper, we develop algorithms to find small representative sets of nondominated points that are well spread over the nondominated frontiers for multi-objective mixed integer programs. We evaluate the quality of representations of the sets by a Tchebycheff distance-based coverage gap measure. The first algorithm aims to substantially improve the computational efficiency of an existing algorithm that is designed to continue generating new points until the decision maker (DM) finds the generated set satisfactory. The algorithm improves the coverage gap value in each iteration by including the worst represented point into the set. The second algorithm, on the other hand, guarantees to achieve a desired coverage gap value imposed by the DM at the outset. In generating a new point, the algorithm constructs territories around the previously generated points that are inadmissible for the new point based on the desired coverage gap value. The third algorithm brings a holistic approach considering the solution space and the number of representative points that will be generated together. The algorithm first approximates the nondominated set by a hypersurface and uses it to plan the locations of the representative points. We conduct computational experiments on randomly generated instances of multi-objective knapsack, assignment, and mixed integer knapsack problems and show that the algorithms work well.     

The density and minimal gap of visible points in some planar quasicrystals

It has previously been observed that the limiting gap distribution of the directions to visible points of planar quasicrystals may vanish near zero, that is, there exist planar quasicrystals with a positive limiting minimal normalised gap between the angles of visible points. The exact values of these limiting minimal normalised gaps have not been determined. In this paper we give explicit formulas for the densities of visible points for planar quasicrystals from several families, which include the Ammann–Beenker point set and the vertex sets of some rhombic Penrose tilings. Combining these results with a known characterisation of the limiting minimal gap in terms of a probability measure on an associated homogeneous space of quasicrystals, we give explicit values of the limiting minimal normalised gap between the angles of visible points for several families of planar quasicrystals, in particular, for the Ammann–Beenker point set and for the vertex sets of some rhombic Penrose tilings. We also compare our results with numerical observations.     

Fluid models of integrated traffic and multipath routing

In this paper we consider a stochastic model describing the varying number of flows in a network. This model features flows of two types, namely file transfers (with fixed volume) and streaming traffic (with fixed duration), and extends the model of Key, Massoulié, Bain and Kelly [27] by allowing more general bandwidth allocation criteria. We analyse the dynamics of the system under a fluid scaling, and show Lyapunov stability of the fluid limits under a natural stability condition. We provide natural interpretations of the fixed points of these fluid limits. We then compare the fluid dynamics of file transfers under (i) balanced multipath routing and (ii) parallel, uncoordinated routing. We show that for identical traffic demands, parallel uncoordinated routing can be unstable while balanced multipath routing is stable. Finally, we identify multi-dimensional Ornstein-Uhlenbeck processes as second-order approximations to the first-order fluid limit dynamics.     

Log-weight scheduling in switched networks

We consider switched queueing networks in which there are constraints on which queues may be served simultaneously. The scheduling policy for such a network specifies which queues to serve at any point in time. We introduce and study a variant of the popular maximum weight or backpressure policy which chooses the collection of queues to serve that has maximum weight. Unlike the maximum weight policies studied in the literature, the weight of a queue depends on logarithm of its queue-size in this paper. For any multihop switched network operating under such maximum log-weighted policy, we establish that the network Markov process is positive recurrent as long as it is underloaded. As the main result of this paper, a meaningful fluid model is established as the formal functional law of large numbers approximation. The fluid model is shown to be work-conserving. That is, work (or total queue-size) is nonincreasing as long as the network is underloaded or critically loaded. We identify invariant states or fixed points of the fluid model. When underloaded, null state is the unique invariant state. For a critically loaded fluid model, the space of invariant states is characterized as the solution space of an optimization problem whose objective is lexicographic ordering of total queue-size and the negative entropy of the queue state. An important contribution of this work is in overcoming the challenge presented by the log-weight function in establishing meaningful fluid model. Specifically, the known approaches in the literature primarily relied on the “scale invariance” property of the weight function that log-function does not possess.     

Existence of Majorana Bound States in A Superconducting Nanowire Near an Impurity

We consider a nanowire with the s-wave superconducting order induced as a result of the proximity effect in the presence of the Zeeman field and the Rashba interaction. For a small superconducting gap and small momenta, we analytically prove the existence of Majorana bound states for a certain local change in the Zeeman field or the superconducting order and also obtain explicit expressions for the corresponding wave functions. We study the scattering of excited states with energies that are close to boundary gap points in the case of propagation through an impurity for local changes in the indicated system parameters near this impurity and show that the transmission probability is equal to unity.     

A gap theorem for hypersurfaces of the sphere with constant scalar curvature one

We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for a modified second fundamental form and determine the hypersurfaces that are at the end points of the gap. As an application we characterize the closed, two-sided index one hypersurfaces with scalar curvature one in the real projective space. Received: October 12, 2001     

Simple analysis of a fluid queue driven by an M/M/1 queue

In this note we consider the fluid queue driven by anM/M/1 queue as analysed by Virtamo and Norros [Queueing Systems 16 (1994) 373–386]. We show that the stationary buffer content in this model can be easily analysed by looking at embedded time points. This approach gives the stationary buffer content distribution in terms of the modified Bessel function of the first kind of order one. By using a suitable integral representation for this Bessel function we show that our results coincide with the ones of Virtamo and Norros.     

On the Bifurcation and Stability of Rigidly Rotating Inviscid Liquid Bridges

Summary. We consider a mathematical model that describes the motion of an ideal fluid of finite volume that forms a bridge between two fixed parallel plates. Most importantly, this model includes capillarity effects at the plates and surface tension at the free surface of the liquid bridge. We point out that the liquid can stick to the plates due to the inner pressure even in the absence of adhesion forces. We use both the Hamiltonian structure and the symmetry group of this model to perform a bifurcation and stability analysis for relative equilibrium solutions. Starting from rigidly rotating, circularly cylindrical fluid bridges, which exist for arbitrary values of the angular velocity and vanishing adhesion forces, we find various symmetry-breaking bifurcations and prove corresponding stability results. Either the angular velocity or the angular momentum can be used as a bifurcation parameter. This analysis reduces to find critical points and corresponding definiteness properties of a potential function involving the respective bifurcation parameter. Received June 21, 1996; revision received October 2, 1997, and accepted for publication October 9, 1997     

Weierstrass Pairs and Minimum Distance of Goppa Codes

We prove that elements of the Weierstrassgap set of a pair of points may be used to define a geometricGoppa code which has minimum distance greater than the usuallower bound. We determine the Weierstrass gap set of a pair ofany two Weierstrass points on a Hermitian curve and use thisto increase the lower bound on the minimum distance of particularcodes defined using a linear combination of the two points.     

Circle action with prescribed number of fixed points

Given any two positive integers k and n, this paper is concerned with the existence of a circle action on a closed, smooth orientable n-dimensional manifold with precisely k isolated fixed points. We first show that this existence problem can be reduced to that of an n-dimensional manifold with exactly three fixed points. Then by using a rigidity result, we determine possible weights on these three fixed points when n = 4.     

Converting triangulations to quadrangulations

We study the problem of converting triangulated domains to quadrangulations, under a variety of constraints. We obtain a variety of characterizations for when a triangulation (of some structure such as a polygon, set of points, line segments or planar subdivision) admits a quadrangulation without the use of Steiner points, or with a bounded number of Steiner points. We also investigate the effect of demanding that the Steiner points be added in the interior or exterior of a triangulated simple polygon and propose efficient algorithms for accomplishing these tasks. For example, we give a linear-time method that quadrangulates a triangulated simple polygon with the minimum number of outer Steiner points required for that triangulation. We show that this minimum can be at most n/3, and that there exist polygons that require this many such Steiner points. We also show that a triangulated simple n-gon may be quadrangulated with at most n/4 Steiner points inside the polygon and at most one outside. This algorithm also allows us to obtain, in linear time, quadrangulations from general triangulated domains (such as triangulations of polygons with holes, a set of points or line segments) with a bounded number of Steiner points.     

The Maxey–Riley equation: Existence,uniqueness and regularity of solutions

The Maxey–Riley equation describes the motion of an inertial (i.e., finite-size) spherical particle in an ambient fluid flow. The equation is a second-order, implicit integro-differential equation with a singular kernel, and with a forcing term that blows up at the initial time. Despite the widespread use of the equation in applications, the basic properties of its solutions have remained unexplored. Here we fill this gap by proving local existence and uniqueness of mild solutions. For certain initial velocities between the particle and the fluid, the results extend to strong solutions. We also prove continuous differentiability of the mild and strong solutions with respect to their initial conditions. This justifies the search for coherent structures in inertial flows using the Cauchy–Green strain tensor.     

A convex-like duality scheme for quasi-convex programs

This paper describes a symmetric duality relation for quasi-convex programs. We are able to strengthen previous results and to define necessary and sufficient conditions for the absence of duality gap. In the present scheme one can generate quasi-convex quasi-concave Lagrangians and discuss the correspondence between saddle points of the Lagrangians and the solutions to the dual and primal programs. The present scheme is very similar to Rockafellar's scheme for convex programs and in this sense it may be viewed as a unified approach. Several examples are also given.     

Real algebraic curves of constant width

Rabinowitz constructed a parametric curve of constant width and expressed it as a plane algebraic curve; however, the algebraic curve also contains isolated points separate from the original curve. We show how to modify his example in order to produce a curve with no isolated points. We then conjecture a method for producing a family of such curves and prove the conjecture in several cases.     

On mappings whose sum is monotone

Proposition 1 of this article points at a gap in the proof of Kuhlmann’s characterization of algebraically maximal valued fields [2]. The author [1] showed how to fill this gap. His arguments involved tools of mathematical logic and infinite combinatorics. Proposition 2 of this article provides us with a simple proof of the key fact of Kuhlmann’s characterization.     

Witness Rectangle Graphs

In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x, y in P are adjacent whenever the open isothetic rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a connected co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. In addition, we prove that a WRG with no isolated vertices has domination number at most four. Moreover, we show that any combinatorial graph can be drawn as a WRG using a combination of positive and negative witnesses. Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.     

Vibrational mixing of a fluid between two quasiconcentric cylinders

We consider the possibility of intense mixing of a viscous fluid in the gap between two quasiconcentric cylinders, with one of the cylinders performing high-frequency vibrations about its axis. The motion of the fluid is described by Navier-Stokes equations for the axisymmetric case. The stream function is represented by a generalized Fourier series. The small parameter is the ratio of the vibration amplitude to the radius of the external cylinder. Calculations carried out in the zeroth approximation produced the pattern of stream lines for various Reynolds numbers, vibration amplitudes, and ratios of external and internal radii. The mixing intensity was found to increase substantially with the reduction of the gap between the cylinders, whereas variation of the ratio of the vibration amplitude to the Reynolds number did not produce marked qualitative changes. The fluid flow in this system generates a contraction semigroup, which makes it possible to derive the ergodicity criterion for the stream function.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 35–39, 1986.     

Real plane algebraic curves

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi-definite nondefinite or definite. We present a discussion about isolated points. By means of the P operator, these points can be easily identified for curves defined by minimal polynomials of order bigger than one. We also discuss the conditions that a curve must satisfy in order to have a minimal polynomial. Finally, we list the most relevant topological properties of affine and projective, complex and real plane algebraic curves.     

Isolated and essentially isolated Volterra-type integral operators on generalized Fock spaces

We studied some topological properties of the space of bounded Volterra-type integral operators on generalized Fock spaces with weight function growing faster than the classical Gaussian function. In particular, the connected and path-connected components, isolated and essentially isolated points of the space are described. The result shows that while the space contains all the non-compact operators as its isolated points, it fails to admit essentially isolated points.