Some remarks on derivations in Banach algebras and related results

Summary In the first section of this paper we consider some functional equations which are closely connected to derivations (i.e. additive mappings with the propertyD(ab) = aD(b) + D(a)b) on Banach algebras. IfD is a derivation on some algebraA, then the equationD(a) = – aD(a ^–1 )a holds for all invertible elementsa A. It seems natural to ask whether this functional equation characterizes derivations among all additive mappings. It is too much to expect an affirmative answer to this question in arbitrary algebras, since it may happen that even in normed algebras the group of all invertible elements contains only scalar multiples of the identity. We try to answer the question above in Banach algebras, since in Banach algebras invertible elements exist in abundance. In the second section of the paper we prove some results concerning representability of quadratic forms by bilinear forms.     

Magnetism in very thin films of permalloy measured by spin polarized cascade electrons

The spin polarizationP of the low energy cascade electrons excited with a primary unpolarized electron beam is measured with ultrathin films of permalloy (Ni₈₀Fe₂₀) as a function of film thickness, external magnetic field, and temperatureT. Surface adsorbates of small concentrations of less than 10% of a monolayer can change the Curie point and the saturation value ofP ₀(T0) by as much as 30%. The Ta-substrate induces a magnetically dead region in permalloy. Conventional spin wave theory cannot account for the observed smallT-dependence of the magnetizationM. Films on a nonmagnetic substrate are compared to similar films coupled to bulk permalloy over an interface of Ta. TheT-dependence ofM with the coupled films can be explained by spin wave theory. At lowT, the films coupled to the bulk exhibit a faster decrease ofM than the uncoupled films. We propose that this thermal stabilization of the magnetization in very thin ferromagnetic films is due to quenching of the long wavelength spin modes.     

Ferromagnetic order and critical behavior at surfaces of ultrathin epitaxial films

The critical behavior, ferromagnetic order and magnetic anisotropies of ultrathin, epitaxial, magnetic films is studied using electron capture spectroscopy (ECS), which is capable of probing the long-ranged and short-ranged electron spin polarization (ESP) at the topmost surface layer of uncoated and coated magnetic structures. For all systems [Ni(100)/Cu(100), Ni(100)/NaCl(100), fcc Fe(111)/Cu(111), Fe(100)/Ag(100), Tb/Fe(100)/Ag(100), Fe(100)/Au(100), hcp Tb(0001)/W(110), Fe(110)/W(110), V(100)/Ag(100), Pd(100)/Ag(100), Pd/W(110)] investigated so far, ferromagnetic order is detected. It is found that the surface Curie temperatureT _Cs depends on film thicknessd. ECS data obtained at the surface of various systems reveal the existence ofT- andd-dependent magnetic anisotropies. Although for V(100)/Ag(100) the measured critical exponent=0.128 agrees very well with=1/8 predicted for the two-dimensional Ising model, for other systems, such as Fe(100)/Au(100), the measured value (0.25) is in disagreement with theoretical predictions. The experimental results are discussed within the framework of presently available experimental and theoretical data.     

An extremal problem related to probability

Summary We consider a walk from a stateA ₁ to a stateA _n+1 in which the probability of remaining atA _i isp _i , and the probability of progressing fromA _i toA _i+1 is 1 –p _i . The probabilityW _nk of reachingA _n+1 fromA ₁ in exactlyn + k steps can then be expressed as a polynomial of degreen + k in then variablesp ₁,,p _n . We determine the maximum value ofW _nk and the (unique) choice (p ₁,,p _n ) for which this extremum occurs.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Measures of inset information on the open domain — I: Inset entropies and information functions of all degrees

This is one in a series of papers studying measures of information in the so-called mixed theory of information (i.e. considering the events as well as their probabilities) on the open domain (i.e. without empty sets and zero probabilities). In this paper we find all-recursive, 3-semisymmetric inset entropies on the open domain. We do this by solving the fundamental equation of inset information of degree () on the open domain.Dedicated to Professor János Aczél on his 60th birthday.     

Synthetische und NMR-spektroskopische Untersuchungen der Benzylaminaddition an N-Maleyl-aminosäurederivate

Summary During the addition of amines to maleylamino acids in general -amino derivatives are formed. In order to change the reactivity within the vinylogous maleyl system we introduced the -benzylester group into the starting compound N-(cis--carboxyacryloyl)-phenylalanine methylester1. The obtained benzylester2 is adding benzylamine in -position yielding N-benzyl--aspartyl(-benzyl)-phenylalanine methylester4. The addition reaction was investigated by¹H-NMR-spectroscopy. The structures of the compounds have been confirmed by elemental analysis, IR,¹H-NMR and¹³C-NMR spectroscopy.

     

Determinations of combustion and formation enthalpies of C60 and C70

By using a micro-bomb calorimeter, the standard enthalpies of combustion of C₆₀ and C₇₀ have been determined to be - (25 947.1±8.5) and - (29 956.1 ± 8.9) kJ/mol respectively. A g. 1. c. analysis indicated that the amounts of residual organic solvents in the samples were very small, and their effects on the final results were negligible. The energy of combustion ofC60 determined in this work is in agreement in the uncertainty interval with that determined by means of traditional calorimeter using macro amount of sample. The enthalpies of formation of these two substances have been derived. The strain energies in the molecules of C₆₀ and C₇₀ were estimated by a bond energy scheme and by using corannulene as the model compound, the results estimated from different methods are very close. Project supported by the National Natural Science Foundation of China (Grant No. 29473143)     

Studies on novel cyclodextrans: Inclusion of C60 and C70

Cycloisomalto-heptaose (CI-7) and cycloisomalto-octaose (CI-8) are two novel cyclodextrans. Treatment with C₆₀ or C₇₀ by kneading leads to the formation of four distinct water-soluble inclusion complexes: CI-7/C₆₀ (2: 1), CI-8/C₆₀ (2: 1), CI-7/C₇₀ (2: 1) and CI-8/C₇₀ (2: 1). Their formation and structures have been examined by UV vis spectroscopy, X-ray powder diffraction and fluorescence spectral studies. The reaction is a reversible process.     

Bilimits are bifinal objects

We prove that a (lax) bilimit of a 2-functor is characterized by the existence of a limiting contraction in the 2-category of (lax) cones over the diagram. We also investigate the notion of bifinal object and prove that a (lax) bilimit is a limiting bifinal object in the 2-category of (lax) cones. Everything is developed in the context of marked 2-categories, so that the machinery can be applied to different levels of laxity, including pseudo-limits.     

Packing directed circuits fractionally

LetG be a digraph, and letk1, such that no fractional packing of directed circuits ofG has value >k, when every vertex is given capacity 1. We prove there is a set ofO (k logk logk) vertices meeting all directed circuits ofG.