Jump discontinuities of semilinear,strictly hyperbolic systems in two variables: Creation and propagation

The creation and propagation of jump discontinuities in the solutions of semilinear strictly hyperbolic systems is studied in the case where the initial data has a discrete set, {x _i } _i ⁼¹ⁿ , of jump discontinuities. LetS be the smallest closed set which satisfies:

1. S is a union of forward characteristics.
2. S contains all the forward characteristics from the points {x _i } _i ⁼¹ⁿ .
3. if two forward characteristics inS intersect, then all forward characteristics from the point of intersection lie inS.

We prove that the singular support of the solution lies inS. We derive a sum law which gives a lower bound on the smoothness of the solution across forward characteristics from an intersection point. We prove a sufficient condition which guarantees that in many cases the lower bound is also an upper bound.     

Small-angle X-ray scattering on spinodal,nucleotic or coarsening processes in two-phase systems containing spherical,fibrillar or lamellar inhomogeneities

We study the effects of some of the most important and typical structural changes in two-phase systems on selected structural parameters obtained from small-angle x-ray scattering (SAXS) measurements. To limit the present study, it was assumed that the Phase, 1, embedded in the matrix

1. is monodispersed and homogeneous,
2. possesses one of the three most extreme shapes (spherical, fibrillar or lamellar) and
3. changes its behaviour

1. through type change (spinodal or nucleotic or coarsening), without changing the shape,
2. through a change of the shape only, or
3. through a) (type change) and b) (shape change) simultaneously.

To find the type of change for three basically different shapes of Phase 1 and to calculate its intensity (amount of the change) the following three SAXS parameters must be compared before and after the treatment of the system:

1. chord lengthl ₁ (and/or radius of gyrationR),
2. volume partw ₁ of the Phase 1, and
3. relative inner surfaceS _v of the system.

It is shown by this comparison that by the pure type change in the case of

1. spinodal change, all three SAXRS parameters are increasing or decreasing simultaneously and proportional to a power of the intensity of the change,
2. nucleotic change,l ₁ (and/orR) is unchanged, the other two (w ₁ andS _v ) are increasing or decreasing simultaneously and directly proportional to the intensity,
3. coarsening change,w ₁ is unchanged and anincreasing ofl ₁ is always accompanied by adecreasing ofS _v and vice versa.

In addition to this type change, the cases of mere changes of the shape (“shape change”) and finally of simultaneous type and/or shape change are studied. For the case of pure shape change the alteration of the dimensions (chord lengthl ₁ and/or radius of gyrationR) must be followed. The limitations of the analyses of the simultaneous type and/or shape change are discussed in detail.     

The use of unitarity bounds for a stable extrapolation of low energy data

The properties of the scattering amplitude allow to define a functionf(z) satisfying the following conditions:

1. f(z) is holomorphic in a simply connected domain , which can be mapped conformally onto the unit disk;
2. |Imf(z)| is bounded by some constantM in ;
3. |Ref(z)| is known not to exceed some constantm on a certain part Γ₁ of the boundary Γ of ;f(z) is continuously extensible onto Γ.

Using these properties, constraints are derived on the real part off(z) valid at any point . The result is used for performing a stable extrapolation of low energy pion-pion scattering data to any finite energy. We derive a bound on energy averaged values of the real part of the scattering amplitude. The bound depends onm, M, on the energy variabless and on the energy average intervals ₂?s ₁. Generalizations of the method are discussed.     

Magnetostriction measurement by means of strain modulated ferromagnetic resonance (SMFMR)

A novel method for measuring magnetostriction constants is presented. A strain, periodic in time, applied to the sample, causes a modulation of the ferromagnetic resonance line position. The height of the signal obtained after phase-sensitive detection is proportional to the strain modulation depth. The appropriate magnetostriction constant λ is obtained by comparing the height of the SMFMR signal with that of the FMR line, as recorded by means of magnetic field modulation. Features of the new technique are:

1. high sensitivity: λ_min? 10_?9 forM=100 Oe and linewidth ΔH _d=1 Oe;
2. λ's belonging to distinct precession modes are separately determined;
3. applicable to thin layers for which strain gauge techniques cannot be used;
4. wide temperature range: 1.2 K<T<300 K;
5. uniform stress.

An illustrative example (YIG layer on GGG substrate) is given.     

The moment of inertia at the saddle point of low energy fission of even-even nuclei

We use the molecular model of low energy fission, which describes the nucleus by two interacting fragments, to calculate the moment of inertia for U²³⁶ in the cranking approximation including BCS theory. We show that the moment of inertia at the saddle point:

1. depends almost linearly on the fragment distance.
2. is influenced only very weakly by the pairing constant and by the fragment deformations.
3. shows, as a function of the distribution of mass between the two fragments (A ₁ ,A ₂ ), a minimum near the magic configurationA ₁=132,Z ₁=50 and depends in this mass region strongly on the term structure near the Fermi energy.
4. is approximately that of a rigid body.

     

Bounds on the number of eigenvalues of the Schrödinger operator

Inequalities on eigenvalues of the Schrödinger operator are re-examined in the case of spherically symmetric potentials. In particular, we obtain:

1. A connection between the moments of order (n ? 1)/2 of the eigenvalues of a one-dimensional problem and the total number of bound statesN _n, inn space dimensions;
2. optimal bounds on the total number of bound states below a given energy in one dimension;
3. alower bound onN ₂;
4. a self-contained proof of the inequality for α ≧ 0,n ≧ 3, leading to the optimalC ₀₄,C ₃;
5. solutions of non-linear variation equations which lead, forn ≧ 7, to counter examples to the conjecture thatC _0n is given either by the one-bound state case or by the classic limit; at the same time a conjecture on the nodal structure of the wave functions is disproved.

     

Dynamical enhanced electron emission and discharges at contaminated surfaces

Broad-area electrodes show electron emission already at electric field strengthsF≈10⁷ V/m. This enhanced field emission (EFE) occurs only for contaminated surfaces. EFE is accompanied by photon emission and gas desorption yielding finally discharges. EFE is caused by dust and contaminants initiating the following effects:

an electron is stochastically emitted in a trigger zone

the electron gains energyΔE?eΔxF ^*

which excites electronic states

which relax by the emission of electrons, photons, and atoms

where the positive charges left behind enhanceF ^*=βF (β?1) initiating so an electron avalanche, i.e., a high conductivity channel. Because of charge migration and neutralization, this avalanche has a life time. This pulsating EFE is accompanied by light emission and gas desorption yielding finally a gas cloud and a discharge.

The pulsating, self-sustained EFE has the same root as:

the enhanced secondary emission found first by Malter

the conductivity switching exhibited by thin (≈ 1 μm) layers of semiconductors or insulators

the normal cathode fall and

the firing-wave instability in neurodynamics.

     

Photoemission on the highT c superconductors Y−Ba−Cu−O

XPS and UPS photoemission experiments on the highT _c superconductors (T _c ≈90 K) with nominal composition YBa₂Cu₃O_9-y (y≈2) show the following:

1. The density of electronic states at the Fermi energy is very small, much smaller than in pure Cu.
2. The Cu 2p spectra show only a Cu²⁺ contribution.
3. The Ba core levels show a structure with two components of nearly equal magnitude, which leads to the suggestion that these compounds have large O^2? vacancies coordinated to Ba²⁺ sites.
4. Annealing at 400°C under UHV conditions leads possibly to a partial reduction of Cu²⁺ to lower Cu valence states and to a small increase of the O^2? vacancy component of the Ba²⁺ line.

     

The possibility of using a semiconductor thermal probe to investigate the power flow distribution in waveguides

Conclusions

1. The temperature profile of an absorbing film for a given microwave power flux distribution has been considered.
2. The conditions under which agreement is found between the temperature relief and the flux distribution has been clarified.
3. An experimental investigation has been carried out of the temperature profile in a matched film for the TE₁₀ mode.
4. The possibility of using a semiconductor thermal probe to investigate the microwave power flux distribution has been demonstrated.

     

Absence of singular continuous spectrum for certain self-adjoint operators

We give a sufficient condition for a self-adjoint operator to have the following properties in a neighborhood of a pointE of its spectrum:

1. its point spectrum is finite;
2. its singular continuous spectrum is empty;
3. its resolvent satisfies a class of a priori estimates.

     

Spatial representation of groups of automorphisms of von Neumann algebras with properly infinite commutant

Theorem. Let a topological groupG be represented (a→φ _a ) by *-automorphisms of a von Neumann algebraR acting on a separable Hilbert spaceH. Suppose that

1. G is locally compact and separable,
2. R′ is properly infinite,
3. for anyT∈R,x,y∈H the function

$$a \to \left\langle {\phi _a (T)x,y} \right\rangle _H $$ is measurable onG. Then there exists a strongly continuous unitary representation ofG onH,a→U _a , such that forT∈R,a∈G, $$\phi _\alpha (T) = U_a TU_a *.$$ .     

Continuous measures in one dimension

Families of unimodal maps satisfying

1. T _λ: [?1,1]?[?1,1] withT(±1)=?1 and |T _λ ^′ (1)|>1.
2. T _λ(x) isC ² inx ² and λ, and symmetric inx.
3. T ₀(0)=0,T ₁(0)=1 with \(\frac{d}{{d\lambda }}\) T _λ(0)>0

are considered. The results of Guckenheimer (1982) are extended to show that there is a positive measure of λ for whichT _λ has a finite absolutely continuous invariant measure. The appendix contains general theorems for the existence of such measures for some markov maps of the interval.     

Unbounded derivations ofC*-algebras II

It is demonstrated that a closed symmetric derivation δ of aC?-algebra \(\mathfrak{A}\) generates a strongly continuous one-parameter group of automorphisms of aC?-algebra \(\mathfrak{A}\) if and only if, it satisfies one of the following three conditions

1. (αδ+1)(D(δ))= \(\mathfrak{A}\) , α∈?\{0}.
2. δ possesses a dense set of analytic elements.
3. δ possesses a dense set of geometric elements.

Together with one of the following two conditions

1. ∥(αδ+1)(A)∥≧∥A∥, α∈IR,A∈D(δ).
2. If α∈IR andA∈D(δ) then (αδ+1)(A)≧0 impliesA≧0.

Other characterizations are given in terms of invariant states and the invariance ofD(δ) under the square root operation of positive elements.     

Probing of the Higgs-fermion coupling atγγ-colliders

Three possibilities to observe the Higgs-top interation at future γγ-colliders are discussed:

1. associated Higgs production via the \(\gamma \gamma \to t\bar tH\) reaction,
2. Higgs obliged radiative correction to the \(\gamma \gamma \to t\bar t\) channel,
3. Higgs resonance production via γγ→H→ZZ.

The results obtained show windows of the Higgs mass where the Yukawa interaction of the Higgs with the top quark can be studied at γγ-colliders.     

A proof of the unitarity ofS-matrix in a nonlocal quantum field theory

Using the formfactors which are entire analytic functions in a momentum space, nonlocality is introduced for a wide class of interaction Lagrangians in the quantum theory of one-component scalar field φ(x). We point out a regularization procedure which possesses the following features:

1. The regularizedS ^δ matrix is defined and there exists the limit $$\mathop {\lim }\limits_{\delta \to 0} S^\delta = S.$$
2. The Green positive-frequency functions which determine the operation of multiplication in \(S \cdot S^ + \mathop = \limits_{Df} S \circledast S^ + \) can be also regularized ?^δ and there exists the limit $$\mathop {\lim }\limits_{\delta \to 0} \circledast ^\delta = \circledast \equiv .$$
3. The operator \(J(\delta _1 ,\delta _2 ,\delta _3 ) = S^{\delta _1 } \circledast ^{\delta _2 } S^{\delta _3 + } \) is continuous at the point δ₁=δ₂=δ₃=0.
4. $$S^\delta \circledast ^\delta S^{\delta + } \equiv 1at\delta > 0.$$ Consequently, theS-matrix is unitary, i.e. $$S \circledast S^ + = S \cdot S^ + = 1.$$

     

Spin-order and magnetic properties of BaZn2Fe16O27 (Zn2-W) hexagonal ferrite

We have observed large peaks in the ultrasonic attenuation of a pure and two doped LaAl₂ crystals in the temperature region of about 45 K which are possibly of the Bordoni type. Their most interesting characteristics are:

1. There is only one peak in each sample.
2. The peaks are so narrow that they can be described by the original Seeger model with only one actication energy and one attempt frequency.

     

An investigation of the axial mode behaviour of a helical TEA-CO2 laser

The frequency behaviour of axial modes was investigated during the initial phase of mode competition in case of a helical TEA-CO₂ laser. With the help of a homodyne technique single-shot and multi-shot beat spectra were measured. Analysing these under various aspects and combining the results of an earlier investigation it was found that

1. inferior modes exist only for 100–200 ns.
2. their spectral width is less than 1 MHz and is determined by lifetime broadening, with the dominant mode narrower than 0.77 MHz,
3. in the average over many shots the spectral envelope of modes does not follow a Lorentzian shape as expected for the Lorentzian gain curve,
4. the beat powers change widely from shot to shot, whereas the total laser power remains constant,
5. no specific phase structures are likely to govern the laser emission, although the maximum emission principle appears to be obeyed with every individual shot. In an appendix relations are derived and summarized which are required for the evaluation of beat mode spectra and for the determination of line width as they apply to the actual time dependence of the laser emission.

     

On continuity of Bowen-Ruelle-Sinai measures in families of one dimensional maps

Let us consider a family of mapsQ _a (x)=ax(1?x) from the unit interval [0,1] to itself, wherea∈[0,4] is the parameter. We show that, for any β<2, there exists a subsetE?4 in [0,4] with the properties

1. Leb([4??,4]?E) < ?^β for sufficiently small ?>0,
2. Q _a admits an absolutely continuous BRS measure µ_a whena∈E, and
3. µ_a converges to the measure µ₄ asa tends to 4 on the setE. Also we give some generalization of this results.

     

Phases coexistence and surface tensions for the potts model

Theq states Potts model exhibits a first order phase transition at some inverse temperature β _t between “ordered” and “disordered” phases forq large as proved in [1]. In space dimension 2 we use theduality transformation as aninternal symmetry of the partition function at β _t to derive an estimate on the probability of a contour. This enables us to prove the preceding result and the following new results:

1. The discontinuity of the mass gap at β _t .
2. The existence of astrictly positive surface tension between two ordered phases up to β _t .
3. The existence of a non-zero surface tension between an “ordered” and the “disordered” phase at β _t .