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We show that shock polars for an ideal non-polytropic gas (thermally but not calorically perfect) have a unique velocity angle maximum, the critical shock, assuming a convex equation of state (positive fundamental derivative) and other standard conditions. We also show that the critical shock is always transonic. These properties are explained by a brief informal mass flux argument, which is then extended into a precise calculation. In the process we show that temperature, pressure, energy, enthalpy, normal mass flux and entropy are increasing along the forward Hugoniot curves, and hence along the polar from vanishing to normal shock; speed is decreasing along the entire polar, mass flux and, importantly, Mach number are decreasing on subsonic parts of the polar. If the equation of state is not convex, counterexamples can be given with multiple critical shocks, permitting more than two shocks that attain the same velocity angle, in particular, more than one shock of weak type, which would cause theoretical problems and practical risks of misprediction. For dissociating diatomic gases, numerical experiments suggest that positive fundamental derivative and uniqueness of critical shocks hold at all realistic pressures, although both fail at very low purely theoretical pressures.

Building on work of Prandtl and Alexander, we study logarithmic vortex spiral solutions of the two-dimensional incompressible Euler equations. We consider multi-branched spirals that are not symmetric, including mixtures of sheets and continuum vorticity. We find that non-trivial solutions allow only sheets, that there is a large variety of such solutions, but that only the Alexander spirals with three or more symmetric branches appear to yield convergent Biot–Savart integral.

Consider inviscid flows around any bounded simple polygon. For sufficiently small but nonzero Mach number, irrotational flows do not exist.

We consider pairs of self-similar 2d vortex sheets forming cusps, equivalently single sheets merging into slip condition walls, as in classical Mach reflection at wedges. We derive from the Birkhoff-Rott equation a reduced model yielding formulas for cusp exponents and other quantities as functions of similarity exponent and strain coefficient. Comparison to numerics shows that piecewise quadratic and higher approximation of vortex sheets agree with each other and with the model. In contrast piecewise linear schemes produce spurious results and violate conservation of mass, a problem that may have been undetected in prior work for other vortical flows where even point vortices were sufficient. We find that vortex cusps only exist if the similarity exponent is sufficiently large and if the circulation on the sheet is counterclockwise (for a sheet above the wall with cusp opening to the right), unless a sufficiently positive strain coefficient compensates. Whenever a cusp cannot exist a spiral-ends jet forms instead; we find many jets are so narrow that they appear as false cusps.

We prove existence of certain compressible subsonic inviscid flows with vortex sheets separating from a solid boundary. To leading order, any perturbation of the upstream boundary causes positive drag. We also prove that if a region of irrotational inviscid flow bounded by a vortex sheet and slip condition wall is enclosed in an angle less than $180^\circ$, then the velocity is zero.

We construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The boundaries between regions of positive and negative vorticity form algebraic spirals, similar to the Kaden spiral and as opposed to Prandtl’s logarithmic vortex spirals. Also unlike the Prandtl case, spirals are not initially present.

We construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The regions of positive and negative vorticity form algebraic spirals.

We prove that there are no nonzero uniformly subsonic potential flows around bodies with three or more protruding corners, for piecewise analytic boundary and for equation of state a $\gamma$-law with $\gamma>1$. This generalizes an earlier result limited to the low-Mach limit for nondegenerate polygons. For incompressible flows we show the velocity cannot be globally bounded.

We motivate and discuss several recent results on non-existence of irrotational inviscid flow around bounded solids that have two or more protruding corners, complementing classical results for the case of a single protruding corner. For a class of two-corner bodies including non-horizontal flat plates, compressible subsonic flows do not exist. Regarding three or more corners, bounded simple polygons do not admit compressible flows with arbitrarily small Mach number, and any incompressible flow has unbounded velocity at at least one corner. Finally, irrotational flow around smooth protruding corners with non-vanishing velocity at infinity does not exist. This can be considered vorticity generation by a slip-condition solid in absence of viscosity.

Interaction of multiple shock waves generally produces a contact discontinuity whose circulation has previously been analyzed using “thermodynamic” arguments based on the Hugoniot relations across the shocks. We focus on “kinematic” techniques that avoid assumptions about the equation of state, using only jump relations for the conservation of mass and momentum but not energy. We give a new short proof for the nonexistence of pure (no contact) triple shocks, recovering a result of Serre. For Mach reflection with a zero-circulation but nonzero-density-jump contact, we show that the incident shock must be normal. Nonexistence without contacts generalizes to two or more incident shocks if we assume that all shocks are compressive. The sign of circulation across the contact has previously been controlled with entropy arguments, showing that the post-Mach-stem velocity is generally smaller. We give a kinematic proof assuming compressive shocks and another condition, such as backward incident shocks, or a weak form of the Lax condition. We also show that for 2 + 2 and higher interactions (multiple “upper” shocks with clockwise flow meeting multiple “lower” shocks with counterclockwise flow in a single point), the circulation sign can generally not be controlled. For γ-law pressure, we show that 2 + 2 interactions without contacts must be either symmetric or antisymmetric, with symmetry favored at low Mach numbers and low shock strengths. For full potential flow instead of the Euler equations, we surprisingly find, contrary to folklore and prior results for other models, that pure triple shocks without contacts are possible, even for γ-law pressure with 1 < γ < 3.

We consider self-similar solutions of the 2d incompressible Euler equations. We construct a class of solutions with vorticity forming algebraic spirals near the origin, in analogy to vortex sheets rolling up into algebraic spirals.

We prove non-existence of nontrivial uniformly subsonic inviscid irrotational flows around several classes of solid bodies with two protruding corners, in particular vertical and angled flat plates; horizontal plates are the only case where solutions exists. This fills the gap between classical results on bodies with a single protruding corner on one hand and recent work on bodies with three or more protruding corners. Thus even with zero viscosity and slip boundary conditions solids can generate vorticity, in the sense of having at least one rotational but no irrotational solutions. Our observation complements the commonly accepted explanation of vorticity generation based on Prandtl's theory of viscous boundary layers.

We consider shock reflection which has a well-known local non-uniqueness: the reflected shock can be either of two choices, called weak and strong. We consider cases where existence of a global solution with weak reflected shock has been proven, for compressible potential flow. If there was a global strong-shock solution as well, then potential flow would be ill-posed. However, we prove non-existence of strong-shock analogues in a natural class of candidates.

It is shown that an arbitrary function from D ⊂ R n to R n will become C0, α-continuous in almost every x ∈ D after restriction to a certain subset with limit point x. For n ≥ m differentiability can be obtained. Examples show the Holder exponent α = min { 1 , n m } is optimal.

We consider self-similar potential flow for compressible gas with polytropic pressure law. Self-similar solutions arise as large-time asymptotes of general solutions, and as exact solutions of many important special cases like Mach reflection, multidimensional Riemann problems, or flow around corners. Self-similar potential flow is a quasilinear second-order PDE of mixed type which is hyperbolic at infinity (if the velocity is globally bounded). The type in each point is determined by the local pseudo-Mach number L, with L < 1 (respectively, L > 1) corresponding to elliptic (respectively, hyperbolic) regions. We prove an ellipticity principle: the interior of a parabolic-elliptic region of a sufficiently smooth solution must be elliptic; in fact L must be bounded above away from 1 by a domain-dependent function. In particular there are no open parabolic regions. We also discuss the case of slip boundary conditions at straight solid walls.

Compressible (full) potential flow is expressed as an equivalent first-order system of conservation laws for density $\rho$ and velocity $v$. Energy $E$ is shown to be the only nontrivial entropy for that system in multiple space dimensions, and it is strictly convex in $\rho,v$ if and only if $|v

We consider solutions of two-dimensional $m \times m$ systems hyperbolic conservation laws that are constant in time and along rays starting at the origin. The solutions are assumed to be small $L^\infty$ perturbations of a constant state and entropy admissible, and the system is assumed to be non-strictly hyperbolic with eigenvalues of constant multiplicity. We show that such a solution, initially assumed bounded, must be a special function of bounded variation, and we determine the possible configuration of waves. As a corollary, we extend some regularity and uniqueness results for some one-dimensional Riemann problems.

We consider self-similar (pseudo-steady) shock reflection at an oblique wall. There are three parameters: wall corner angle, Mach number, angle of incident shock. Ever since Ernst Mach discovered the irregular reflection named after him, it has been an open problem to predict precisely for what parameters the reflection is regular. Three conflicting proposals, the detachment, sonic and von Neumann criteria, have been studied extensively without a clear result. We demonstrate that the sonic criterion is not correct. We consider polytropic potential flow and prove that there is an open nonempty set of parameters that admit a global regular reflection with a reflected shock that is \emph{transonic}. We also provide a clear physical reason: the flow type (sub- or supersonic) is not decisive; instead the reflected shock type (weak or strong) determines whether structural perturbations decay towards the reflection point.

A particular case of initial data for the two-dimensional Euler equations is studied numerically. The results show that the Godunov method does not always converge to the physical solution, at least not on feasible grids. Moreover, they suggest that entropy solutions (in the weak entropy inequality sense) are not well posed.

High performance applications and the underlying hardware platforms are becoming increasingly dynamic; runtime changes in the behavior of both are likely to result in inappropriate mappings of tasks to parallel machines during application execution. This fact is prompting new research on mapping and scheduling the dataflow graphs that represent parallel applications. In contrast to recent research which focuses on critical paths in dataflow graphs, this paper presents new mapping methods that compute near-min-cut partitions of the dataflow graph. Our methods deliver mappings that are an order of magnitude more efficient than those of DSC, a state-of-the-art critical-path algorithm, for sample high performance applications.

We study the classical problem of self-similar reflection of shocks at a ramp, modeled by potential flow with γ-law pressure. Depending on corner angle θ and upstream Mach number MI , either regular (RR) or Mach reflections occur. There are several conflicting transition criteria predicting the corner angle at which the type of reflection changes. We show that in some cases, in particular MI =1 and γ =5 /3, an exact RR solution exists for all θ specified by the sonic criterion. Thus all weaker criteria are false.

One of the most forbidding chapters in UNIX, as with computers in general, is error localization. However, often the difficulties arise simply from a lack of information—the various programs take different approaches to outputting error messages, and often a highly interesting message lands in a quiet grave in some file deep in the Linux directory tree. This need not be the case: in the following we describe the various log files and mechanisms as well as frequent errors and error messages.

This command reference presents the most important Linux commands. Any Linux command comes with a variety of command line options. We only present the most important options; any others can be looked up in the corresponding Manual pages. Consequently, there may be more options than the command syntax—as presented in this reference—contains.

As the propagation of graphical workstations began, there were scarcely any standards for programming graphical user interfaces (GUIs). Most manufacturers provided their own GUIs GUIs with their machines. If an application that was intended to exploit the graphical possibilities of the new machines was to run on different platforms, it had to be developed and maintained in multiple platforms variants. Large institutions that worked with systems from various manufacturers particularly felt the brunt of these problems.

In recent years the problem of computer security has increasingly awakened public agencies and various organizations, and sensitive data of all kinds traverse it: personal letters, credit card numbers, private data etc. The Internet has developed from a university playground to a communication network for private persons, companies, government agencies, and various organizations, and sensitive data of all kinds traverse it: personal letters, credit card numbers and bank account PINs, medical histories and examination results, management data, military secrets, and development plans of high-tech firms. Since ever more computers are networked, increasingly many computers are the targets of hackers. There are sporting hackers who want to hackers demonstrate their intelligence by intruding into others’ computer systems as well as professional hackers who target systems for destruction or to copy and sell classified information.

One argument that is frequently posed, especially in commercial settings, to reject free or public domain software is the lack of a licensing company that provides support and a hotline. In the hotline USA some companies have recognized this market niche and offer commercial support for free software. A similar trend is developing free software in Europe as well.

One particular feature of Linux is the large number of freeware programming languages in binary format available from Linux FTP servers and contained in the various Linux distributions. These freeware tools seldom take a back seat to proprietary systems. This chapter introduces the various compilers, interpreters, and software development tools that are available under Linux and identifies their most important features.