The capacity of a channel with arbitrarily varying channel probability functions and binary output alphabet

Summary Let X={1,..., a} be the input alphabet and Y={1,2} be the output alphabet. Let X ^t =X and Y ^t =Y for t=1,2,..., X _n = X ^t and Y _n = Y ^t . Let S be any set, C=={w(·¦·¦)s)¦sS} be a set of (a×2) stochastic matrices w(··¦s), and S ^t=S, t=1,..., n. For every s _n =(s ¹,...,s ⁿ ) S ^t define P(·¦·¦s _n)= w(y ^t ¦x ^t ¦s ^t ) for every x _n=x ¹, , x ⁿX _n and every y _n=(y ¹, , y ⁿ)Y _n. Consider the channel C _n ={P(·¦·¦)s _n )¦s _n S _n } with matrices (·¦·¦s), varying arbitrarily from letter to letter. The authors determine the capacity of this channel when a) neither sender nor receiver knows s _n, b) the sender knows s _n, but the receiver does not, and c) the receiver knows s _n, but the sender does not.Research of both authors supported by the U.S. Air Force under Grant AF-AFOSR-68-1472 to Cornell University.     

Three-dimensional gravity waves in a channel of variable depth

We consider existence of three-dimensional gravity waves traveling along a channel of variable depth. It is well known that the long-wave small-amplitude expansion for such waves results in the stationary Korteweg–de Vries equation, coefficients of which depend on the transverse topography of the channel. This equation has a single-humped solitary wave localized in the direction of the wave propagation. We show, however, that there exists an infinite set of resonant Fourier modes that travel at the same speed as the solitary wave does. This fact suggests that the solitary wave confined in a channel of variable depth is always surrounded by small-amplitude oscillatory disturbances in the far-field profile.     

Channels with arbitrarily varying channel probability functions in the presence of noiseless feedback

In this article we study a channel with arbitrarily varying channel probability functions in the presence of a noiseless feedback channel (a.v.ch.f.). We determine its capacity by proving a coding theorem and its strong converse. Our proof of the coding theorem is constructive; we give explicitly a coding scheme which performs at any rate below the capacity with an arbitrarily small decoding error probability. The proof makes use of a new method ([1]) to prove the coding theorem for discrete memoryless channels with noiseless feedback (d.m.c.f.). It was emphasized in [1] that the method is not based on random coding or maximal coding ideas, and it is this fact that makes it particularly suited for proving coding theorems for certain systems of channels with noiseless feedback.As a consequence of our results we obtain a formula for the zero-error capacity of a d.m.c.f., which was conjectured by Shannon ([8], p. 19).     

The capacities of certain special channels with arbitrarily varying channel probability functions

In [5] Ahlswede and Wolfowitz have obtained the capacities of a.v.ch. with binary output in a number of cases, essentially with the aid of a lemma which relates the capacity of the a.v.ch. to that of a suitable (“underlying”) d.m.c. A generalization of this lemma to a special kind of a.v.ch. with output alphabet b>2, has been given by Ahlswede (Lemma 1 of [1]) and used in [1] and [2] to prove the existence of the weak capacities of various channels under different conditions. We give a detailed proof of a weakened version of Ahlswede's lemma and show, in passing, that his lemma is incorrect. We then define certain special types of a.v.ch and, on the basis of the detailed analysis given by us earlier, we prove lemmas of a similar type for these a.v.ch. We are thus able to extend certain results given for binary output a.v.ch. in [4] and [5] to these special a.v.ch. for which b>2.     

Capillary waves past a flat plate in water of finite depth

Free-surface flow past a semi-infinite flat plate in a channelof finite depth is considered. The fluid is assumed to be inviscidand incompressible, and the flow to be two-dimensional and irrotational.Surface tension is included in the dynamic boundary conditionbut the effects of gravity are neglected. It is shown that thereis a three-parameter family of solutions with waves in the farfield and a discontinuity in slope at the separation point.This family includes as particular cases the solutions previouslycomputed by Osborn & Stump (2001, Phys. Fluids, 13, 616–623)and by Andersson & Vanden-Broeck (1996, Proc. R. Soc., 452,1985–1997).     

On the loaded elastic half-space with a depth varying Poisson's ratio

Résumé Dans cette étude, on obtient des expressions pour les contraintes et les déplacements qui se produisent dans un semi-espace élastique non-homogène, soumis à une pression sur sa surface plane. Ce milieu a un module uniforme de glissement et un module de compressiblité, ou coefficient de Poisson, qui varie de façon arbitraire en fonction de la profondeur.     

On permanent and breaking waves in hyperelastic rods and rings

We prove that the only global strong solution of the periodic rod equation vanishing in at least one point (_t0,_x0)∈R⁺×S¹$(t_0,x_0)\in {\mathbb{R}}^{+}\times {\mathbb{S}}^1$ is the identically zero solution. Such conclusion holds provided the physical parameter γ   of the model (related to the Finger deformation tensor) is outside some neighborhood of the origin and applies in particular for the Camassa–Holm equation, corresponding to γ=1$\gamma =1$. We also establish the analogue of this unique continuation result in the case of non-periodic solutions defined on the whole real line with vanishing boundary conditions at infinity. Our analysis relies on the application of new local-in-space blowup criteria and involves the computation of several best constants in convolution estimates and weighted Poincaré inequalities.     

Shoaling of nonlinear wave-groups on water of slowly varying depth

An approximate analytical solution describing the shoaling of modulated wave-trains is presented. This solution provides new information about the wave field evolution as well as about the wave-induced mean current and set down.

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Zusammenfassung Eine angenäherte analytische Lösung wird gegeben, die das Brechen von modulierten Wellenzügen beschreibt. Die Lösung gibt neue Informationen über die Entwicklung des Wellenfeldes wie auch über die mittlere Strömung und mittlere Höhenänderung, die von den Wellen induziert wird.

     

Periodic waves with constant vorticity in water of infinite depth

Periodic waves propagating at a constant velocity at the surfaceof a fluid with constant vorticity in water of infinite depthare considered. The problem is solved numerically by a boundary-integral-equationmethod. Simmen & Saffman (Stud. Appl. Maths 75, 35, 1985)showed that there are families of solutions which have limitingconfigurations with a 120 angle at their crests or a trappedbubble at their troughs. It is shown that there are additionalfamilies of solutions. These families have limiting configurationswith trapped bubbles at their crests. Each bubble is circularand contains fluid in rigid-body rotation. The results are consistentwith previous calculations for solitary waves in water of finitedepth.     

A note on the effect of submerged obstacles on water waves in a channel

Summary By combining the results due to Jeffrey and Mvungi [1], with the work of Jeffrey and Saw Tin [2, 3], the transmission and reflection properties of an acceleration wave propagating on the surface of water at rest in a vertical walled channel of arbitrary continuously varying width and piecewise continuously varying depth are determined. An explicit expression derived for the amplitude of the transmitted wave as a function of position by Jeffrey and Mvungi is used to find the effect of a discontinuous change of depth on the wave amplitude and to derive a criterion for the breaking of the wave.

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Zusammenfassung Durch Kombination der Resultate von Jeffrey und Mvungi [1] mit denen von Jeffrey and Saw Tin [2, 3] wird die Transmission und die Reflexion einer Beschleunigungswelle an der Oberfläche von ruhendem Wasser berechnet, in einem Kanal mit vertikalen Wänden, dessen Weite beliebig kontinuierlich und dessen Tiefe abschnittsweise kontinuierlich variiert. Ein expliziter Ausdruck für die Amplitude der durchgelassenen Welle in Funktion der Lage nach Jeffrey und Mvungi wird dazu benützt, um den Effekt einer Diskontinuität in der Tiefe zu finden und ein Kriterium für das Brechen der Welle herzuleiten.

     

On the incoming water waves against a vertical cliff

The problems of obliquely incident surface water waves against a vertical cliff have been handled in both the cases of water of infinite as well as finite depth by straight-forward uses of appropriate Havelock-type expansion theorems. The logarithmic singularity along the shore-line has been incorporated in a direct manner, by suitably representing the Dirac’s delta function.     

The breaking of waves on a sloping beach

Zusammenfassung In dieser Arbeit wird eine neue Methode für das Studium der Wellenbrandung an einem geneigten Strand beschrieben. Sie liefert einen Ausdruck für den Ort und die Zeit des Brechens der Wellen. Die Ergebnisse für ein typisches Beispiel werden graphisch dargestellt.     

Levi-Civita theory for irrotational water waves in a one-dimensional channel and the complex Korteweg-de Vries equation

We review the Levi-Civita theory, which reduces the study of the irrotational flow in a one-dimensional channel or the solution of a non-linear differential-functional partial differential equatin for the velocity potential. We show how, by considering small perturbations in a shallow water channel, we can reduce the non-linear differential-functional equation to a complex Korteweg-de Vries equation which, for almost horizontal flow and for initial conditions indepedent of the vertical variable, reduces to the usual one.Dipartimento di Fisica, Università di Roma III, Instituto Nazionale di Fisica Nucleare, Sezione di Roma, P.zale A. Moro 2, 00185 Roma, Italy. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 435–440, June, 1994.     

On a kinetic model for shallow water waves

The system of shallow water waves is one of the classical examples for non-linear, two-dimensional conservation laws. The paper investigates a simple kinetic equation depending on a parameter ? which leads for ? → 0 to the system of shallow water waves. The corresponding ‘equilibrium’ distribution function has a compact support which depends on the eigenvalues of the hyperbolic system. It is shown that this kind of kinetic approach is restricted to a special class of non-linear conservation laws. The kinetic model is used to develop a simple particle method for the numerical solution of shallow water waves. The particle method can be implemented in a straightforward way and produces in test examples sufficiently accurate results.