On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern     

Ab initio study on the formation of chemically ordered Zr2Al phase by coupled replacive–displacive transformation

We performed plane wave-based first principles calculations using the projector augmented wave (PAW) potential under the generalized gradient approximation (GGA) within the density functional theory to study the formation of ordered omega (B8₂-structured) Zr₂Al phase in β-Zr₃Al alloy. The transformation involves both replacive and displacive processes. We investigated two possible paths for the transformation where steps involving replacive (diffusive) and displacive processes occur in succession with their sequence of occurrence being different in the two paths. From this study, it was possible to show that the initial chemical ordering facilitates the displacive process leading to the transformation. It was also possible to correlate instability with respect to omega-type displacements in Zr₂Al alloy with the number of Zr–Al bonds present in the unit cell. Electronic structure analysis indicated that stronger Zr–Al bonding plays an important role in the formation of chemically ordered omega phase.     

Study of yield stress anomaly of Fe2MnAl single crystal by in situ TEM straining

L2₁-structured Fe₅₉Mn₁₇Al₂₄ shows a yield stress anomaly with a peak yield stress at 700?K. The aim of the work reported here was to determine the dislocation mechanisms involved in this anomalous behaviour by performing in situ straining on Fe₅₉Mn₁₇Al₂₄ single crystals in a transmission electron microscope at 300–900?K. Cross-slip of ?111? dislocations was frequently found to occur between {110} and {211} planes at all temperatures tested. At 300?K, dislocations were four-fold dissociated and the partials moved together under stress. At 700–800?K, partial dislocations with a Burgers vector of a/2?111? dominated the deformation. These partial dislocations moved independently in the ordered matrix in a jerky manner, with rapid motion between long periods of locking. X-ray diffraction measurements showed that the degree of L2₁-order slightly decreased with increasing temperature in the range 300–700?K, and dropped rapidly above 700?K. At 900?K, the material was B2-ordered. The increased yield stress at elevated temperatures is primarily attributed to the uncoupling of a/2?111? partial dislocations trailing shear-induced anti-phase boundaries.     

Effect of impurity atoms on α2/γ lamellar interfacial misfit in Ti–Al alloy: a systematic first principles study

The effect of transition metal solutes on the lattice parameters of γ-TiAl and α₂-Ti₃Al were studied by first principles calculations to find suitable elements for controlling the α₂/γ interfacial misfit in lamellar Ti–Al alloys. Better agreement was found between the calculated and experimental phase and site preferences of impurity atoms than in a previous first principles study. The calculated lattice parameters suggest that elements in groups 6–11 of the 4th period (Cr, Mn, Fe, Co, Ni and Cu) are effective for increasing the misfit, leading to increasing density of misfit dislocation and, in turn, higher yield strength and ductility. This effect is caused by the change in the lattice parameter of the γ-TiAl phase rather than those of α₂-Ti₃Al phase. This prediction agrees qualitatively with experimental data from a previous study although the effects of temperature are not taken into account. Further improvements should be possible by considering those effects. Nevertheless, the results highlight the effects of impurity addition on interfacial misfit at a level which cannot be achieved by classical concepts such as atomic size in a hard sphere model. The results will also be valuable in further more quantitative predictions and in understanding the effects of temperature, including off-stoichiometry, thermal expansion and vibration entropy.     

Supersymmetry and the formal loop space

For any algebraic super-manifold M we define the super-ind-scheme LM of formal loops and study the transgression map (Radon transform) on differential forms in this context. Applying this to the super-manifold M=SX, the spectrum of the de Rham complex of a manifold X, we obtain, in particular, that the transgression map for X is a quasi-isomorphism between the [2,3)-truncated de Rham complex of X and the additive part of the [1,2)-truncated de Rham complex of LX. The proof uses the super-manifold SSX and the action of the Lie super-algebra sl(1|2) on this manifold. This quasi-isomorphism result provides a crucial step in the classification of sheaves of chiral differential operators in terms of geometry of the formal loop space.     

The Cuntz semigroup and comparison of open projections

We show that a number of naturally occurring comparison relations on positive elements in a C^?-algebra are equivalent to natural comparison properties of their corresponding open projections in the bidual of the C^?-algebra. In particular we show that Cuntz comparison of positive elements corresponds to a comparison relation on open projections, that we call Cuntz comparison, and which is defined in terms of—and is weaker than—a comparison notion defined by Peligrad and Zsidó. The latter corresponds to a well-known comparison relation on positive elements defined by Blackadar. We show that Murray-von Neumann comparison of open projections corresponds to tracial comparison of the corresponding positive elements of the C^?-algebra. We use these findings to give a new picture of the Cuntz semigroup.     

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L²(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L² estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.     

Transference principles for semigroups and a theorem of Peller

A general approach to transference principles for discrete and continuous operator (semi)groups is described. This allows one to recover the classical transference results of Calderón, Coifman and Weiss and of Berkson, Gillespie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) operator semigroups that need not be groups. As an application, functional calculus estimates for bounded operators with at most polynomially growing powers are derived, leading to a new proof of classical results by Peller from 1982. The method allows for a generalization of his results away from Hilbert spaces to Lp-spaces and—involving the concept of γ-boundedness—to general Banach spaces. Analogous results for strongly-continuous one-parameter (semi)groups are presented as well. Finally, an application is given to singular integrals for one-parameter semigroups.     

Asymptotically almost periodic solutions for abstract neutral integro-differential equations

We study the existence of asymptotically almost periodic classical solutions for a class of abstract neutral integro-differential equation with unbounded delay. A concrete application to partial neutral integro-differential equations which arise in the study of heat conduction in fading memory material is considered.