Onde di discontinuità ed equazioni costitutive nei corpi elastici isotropi sottoposti a deformazioni finite

Summary We consider an elastic isotropic material with finite deformations. The existence of a double wave (therefore exceptional because the field equations are in the conservative form) for all the deformations and the discontinuity propagation direction is required. This because in the linear theory there exists a double wave, furthermore, because in many non-linear theories of Mathematical physics there exists at least one exceptional wave (this wave doesn’t produce shocks). This request implies conditions for the response function in the constitutive equations. Furthermore, under these assumptions, we can determine explicitly all the possible propagation speeds. Therefore we can find theorems generalizing (in the case of the imposed conditions) those ones obtained by Truesdell and Green for the principal waves (whose unit normal has the direction of the eigenvectors of the deformation matrix). In the last part of this work we examine the case of a hyperelastic material and we determine some classes of possible thermodynamic potentials.

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Entrata in Redazione il 18 febbraio 1976.

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Lavoro eseguito nell’ambito dei contratti del C.N.R. - Gruppo Nazionale per la Fisica Matematica.     

Shape Analysis of Bounded Traveling Wave Solutions and Solution to the Generalized Whitham-Broer-Kaup Equation with Dissipation Terms

This paper deals with the problem of the bounded traveling wave solutions'shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equati...     

Evolution of the long wave finite perturbations during nonsteady combustion of powder

The problem of nonsteady combustion of powder is considered within the framework of the modified Zel'dovich-Novozhilov model /1,2/. The development of the finite surface perturbations and the initial temperature profile and studied under the assumptions that the characteristic distance at which the perturbations change significantly is much greater than the characteristic dimension of relaxation of the stationary temperature profile. The method of multiple scales is used to derive the equations describing the evolution in time of the combustion surface and temperature field in the three-dimensional space. Only the long wave structure is considered, and solutions discontinuous in the given approximation are constructed. The internal structure of the discontinuities is not studied.     

Nilpotent algebras and affinely homogeneous surfaces

To every nilpotent commutative algebra N{\mathcal{N}} of finite dimension over an arbitrary base field of characteristic zero a smooth algebraic subvariety S ì N{S\subset\mathcal{N}} can be associated in a canonical way whose degree is the nil-index and whose codimension is the dimension of the annihilator A{\mathcal{A}} of N{\mathcal{N}}. In case N{\mathcal{N}} admits a grading, the surface S is affinely homogeneous. More can be said if A{\mathcal{A}} has dimension 1, that is, if N{\mathcal{N}} is the maximal ideal of a Gorenstein algebra. In this case two such algebras N{\mathcal{N}}, [(N)\tilde]{\tilde{\mathcal{N}}} are isomorphic if and only if the associated hypersurfaces S, [(S)\tilde]{\tilde S} are affinely equivalent. If one of S, [(S)\tilde]{\tilde S} even is affinely homogeneous, ‘affinely equivalent’ can be replaced by ‘linearly equivalent’. In case the nil-index of N{\mathcal{N}} does not exceed 4 the hypersurface S is always affinely homogeneous. Contrary to the expectation, in case nil-index 5 there exists an example (in dimension 23) where S is not affinely homogeneous.     

A characterization of wavelet families arising from biorthogonal MRA’s of multiplicityd

In this article we give a necessary and sufficient condition for a pair of wavelet families in L²(ℝ ⁿ ), to arise from a pair of biorthogonal MRA’s. The condition is given in terms of simple equations involving the functions ψ^ℓ and . To work in greater generality, we allow multiresolution analyses of arbitrary multiplicity, based on lattice translations and matrix dilations. Our result extends the characterization theorem of G. Gripenberg and X. Wang for dyadic orthonormal wavelets in L²(ℝ),and includes, as particular cases, the sufficient conditions of P. Auscher and P.G. Lemarié in the biorthogonal situation.     

Hochschild （Co）homology of Galois Coverings of Grassmann Algebras

Let ∧ be the Z2-Galois covering of the Grassmann algebra A over a field k of characteristic not equal to 2. In this paper, the dimensional formulae of Hochschild homology and cohomology groups of ∧ are calculated explicitly. And the cyclic homology of∧ can also be calculated when the underlying field is of characteristic zero. As a result, we prove that there is an isomorphism from i≥1 HH^i（∧） to i≥1 HH^i（∧）.     

Three-dimensional Tollmien-Schlichting waves generated by sound in the boundary layer on an elastic surface at transonic free-stream velocities

Within the framework of the triple-deck theory, the effect of surface elasticity on three-dimensional packets of Tollmien-Schlichting waves generated by acoustic disturbances induced near the boundary layer at transonic free-stream velocities is investigated. It is shown that the elasticity of the surface considerably weakens the most unstable oblique waves but does not change the characteristic horseshoe shape of wave packets with two disturbance peaks propagating at an angle to the incoming flow.     

Mixtures of Euler’s fluids and second sound propagation in superfluid helium

We study the second sound propagation in superfluid helium using the theoretical background developed in a recent paper [1], through which this phenomenon, both in crystals and in Helium II, can be inserted in a unique framework. In particular, considering an initial rectangular heat pulse and exploiting the selection rules of the physical shocks, a wide phenomenology of travelling shock waves, including the very special case of double shocks, is explained. (Received: February 17, 2004; revised: October 10, 2004)     

Mixtures of Euler’s fluids and second sound propagation in superfluid helium

We study the second sound propagation in superfluid helium using the theoretical background developed in a recent paper [1], through which this phenomenon, both in crystals and in Helium II, can be inserted in a unique framework. In particular, considering an initial rectangular heat pulse and exploiting the selection rules of the physical shocks, a wide phenomenology of travelling shock waves, including the very special case of double shocks, is explained.     

A wave method of suppressing multiple waves for any complex subsurface geometry

A wave method of suppression of multiple waves in which no a priori data about the medium’s structure are required and the depth-velocity medium’s model is considered unknown, is presented. The method is constructed to completely suppress the multiple waves of a layer in a half-space. The efficiency of the method for arbitrary 3D plane-layered media is demonstrated both theoretically and numerically. Applications of the method to real media with a substantial decrease in the multiple wave amplitudes without distortion of the dynamics of useful reflections are given.     

Propagation of non-planar weak and strong shocks in a non-ideal relaxing gas

In this article, the kinematics of one-dimensional motion have been applied to construct evolution equations for non-planar weak and strong shocks propagating into a non-ideal relaxing gas. The approximate value of exponent of shock velocity, at the instant of shock collapse, obtained from systematic approximation method is compared with those obtained from characteristic rule and Guderley’s scheme. Computation of exponent is carried out for different values of van der Waals excluded volume. Effects of non-ideal and relaxation parameters on the wave evolution, governed by the evolution equations, are analyzed.

     

A modal theory and recursion RTCM algorithm for gratings of deep grooves and arbitrary profile

A modal theory for gratings of deep grooves and arbitrary shape is developed. The theory is implemented with an algorithm of recursion reflection-transmission coefficient matrix (RTCM). Several cases of plane wave with arbitrary polarization and arbitrary angle of incidence diffracted from gratings of deep grooves and arbitrary profile are calculated. It is found that this method is more accurate, more stable and faster than others. Project supported by the National Natural Science Foundation of China (No. 69888002) and the fund for the State Educational Committee’s Returned Scholars.     

On macroscopic dimension of rationally inessential manifolds

We show that, for a rationally inessential orientable closed n-manifold M whose fundamental group is a duality group, the macroscopic dimension of its universal cover [(M)\tilde]\tilde M is strictly less than n: dim _MC [(M)\tilde] < n\tilde M < n. As a corollary, we obtain the following partial result towards Gromov’s conjecture     

An Explicit Formula and its Application to the Selberg Trace Formula

We prove that the explicit formula in a symmetric case for a triple (Z, [(Z)\tilde]\tilde{Z} , Φ) in Jorgenson-Lang’s fundamental class of functions holds for a larger class of (not necessarily differentiable or even continuous) test functions. As one of the most important applications, we show that the Selberg trace formula for a strictly hyperbolic cocompact Fuchsian group Γ is valid for a larger class of test functions. Further applications to growth estimates for the logarithmic derivative of the Selberg zeta function are considered.     

The propagation of travelling waves in an open cubic autocatalytic chemical system

The reaction-diffusion travelling waves that can be initiatedin an open isothermal chemical system governed by cubic autocatalytickinetics are discussed. The system is shown to be capable ofsustaining up to three spatially uniform steady states, the(trivial) unreacted state, which is always stable (a node),and two nontrivial states, one of which is always unstable (asaddle point). The third state can change its stability throughHopf bifurcation (both subcritical and supercritical). Thisallows the possibility of two sorts of travelling wave beingestablished; there are wave profiles which connect the unreactedstate ahead to the nontrivial state at the rear, and wave profiles(pulse waves) which have the unreacted state at both the frontand rear. The conditions under which a particular wave is initiatedare considered by both a discussion of the (ordinary) differentialequations governing the travelling waves and by numerical integrationsof an initial-value problem. This treatment also reveals thepossibility of a stable travelling wave propagating throughthe system, leaving behind a temporally unstable stationarystate. Under these conditions, spatiotemporal chaotic behaviouris seen to develop after the passage of the wave.     

Wave Breaking of the Camassa–Holm Equation

This paper gives a new and direct proof for McKean’s theorem (McKean in Asian J. Math. 2:867–874, 1998) on wave breaking of the Camassa–Holm equation. The blow-up profile is also analyzed.     

Character formulas for Feigin–Stoyanovsky’s type subspaces of standard $\mathfrak{sl}(3, \mathbb{C})^{\widetilde{}}$-modules

Exact sequences of Feigin–Stoyanovsky’s type subspaces for affine Lie algebra \mathfraksl(l+1,\mathbbC)^[\tilde]\mathfrak{sl}(l+1,\mathbb{C})^{\widetilde{}} lead to systems of recurrence relations for formal characters of those subspaces. By solving the corresponding system for \mathfraksl(3,\mathbbC)^[\tilde]\mathfrak{sl}(3,\mathbb{C})^{\widetilde{}}, we obtain a new family of character formulas for all Feigin–Stoyanovsky’s type subspaces at the general level.     

Unique determination of stratified steady water waves from pressure

Consider a two-dimensional stratified solitary wave propagating through a body of water that is bounded below by an impermeable ocean bed. In this work, we study how such a wave can be recovered from data consisting of the wave speed, upstream and downstream density and velocity profile, and the trace of the pressure on the bed. In particular, we prove that this data uniquely determines the wave, both in the (real) analytic and Sobolev regimes.     

Developing the eikonal method in nonlinear electrodynamics based on the geometrization principle

We derive the eikonal equation for an electromagnetic wave propagating in an external electromagnetic field according to the laws of nonlinear electrodynamics. Based on Logunov’s geometrization principle, we determine the metric tensors of the effective Riemannian spaces for the Born-Infeld electrodynamics, nonlinear Heisenberg-Euler electrodynamics, and a parameterized post-Maxwellian electrodynamics. We analyze the main properties of these nonlinear electrodynamics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 112–117, January, 2007.