Positivity and monotonicity properties ofC 0-semigroups. I

If exp {?tH}, exp {?tK}, are self-adjoint, positivity preserving, contraction semigroups on a Hilbert space ?=L ²(X;dμ) we write (*) $$e^{ - tH} \succcurlyeq e^{ - tK} \succcurlyeq 0$$ whenever exp {?tH}-exp {?tK} is positivity preserving for allt≧0 and then we characterize the class of positive functions for which (*) always implies $$e^{ - tf(H)} \succcurlyeq e^{ - tf(K)} \succcurlyeq 0.$$ This class consists of thef∈C ^∞(0, ∞) with $$( - 1)^n f^{(n + 1)} (x) \geqq 0,x \in (0,\infty ),n = 0,1,2, \ldots .$$ In particular it contains the class of monotone operator functions. Furthermore if exp {?tH} isL ^p (X;dμ) contractive for allp∈[1, ∞] and allt>0 (or, equivalently, forp=∞ andt>0) then exp {?tf(H)} has the same property. Various applications to monotonicity properties of Green's functions are given.     

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.     

Asymptotic behavior of orbits of C 0-semigroups and solutions of linear and semilinear abstract differential equations

In this paper, we study the asymptotic behavior of solutions of semilinear abstract differential equations (*) u′(t) = Au(t) + t ⁿ f(t, u(t)), where A is the generator of a C ₀-semigroup (or group) T(·), f(·, x) ∈ A for each x ∈ X, A is the class of almost periodic, almost automorphic or Levitan almost periodic Banach space valued functions ϕ: ℝ → X and n ∈ {0, 1, 2, ...}. We investigate the linear case when T(·)x is almost periodic for each x ∈ X; and the semilinear case when T(·) is an asymptotically stable C ₀-semigroup, n = 0 and f(·, x) satisfies a Lipschitz condition. Also, in the linear case, we investigate (*) when ϕ belongs to a Stepanov class S ^p-A defined similarly to the case of S ^p-almost periodic functions. Under certain conditions, we show that the solutions of (*) belong to A _u:= A ∩ BUC(ℝ, X) if n = 0 and to t ⁿ A _u ⊕ w _n C ₀ (ℝ, X) if n ∈ ℕ, where w _n(t) = (1 + |t|)ⁿ. The results are new for the case n ∈ ℕ and extend many recent ones in the case n = 0. Dedicated to the memory of B. M. Levitan     

Positivity properties in elastic scattering of particles with arbitrary spins

We prove the positivity of the coefficients of the expansion in Legendre polynomials of $\text{(}\text{dσ}{}^{\mathrm{A}}\text{dt}\text{)}$(s, cosθ), the absorptive contribution to the elastic unpolarized differential cross-section for particles with arbitrary spins.     

Positivity of energy in five dimensional classical unified field theories, II

Five dimensional classical unified field theories as well as Yang-Mills theory with gauge group U(1), are described in terms of a Lorentzian five dimensional space V₅ with metric tensor γβ which admits a space-like Killing vector ζ. It is assumed that: (1) V₅ has the topology of V₄ x S¹, S¹ is a circle and V₄ is a four dimensional Lorentzian space that is asymptotically flat and (2) the Einstein tensor Γ_β of V₅ satisfies Γ_β Uυ^{β 0} where U and υ are future oriented time-like vectors with γ_βυζ^β = 0. The spinor approach of Witten [4], Nester [3] and Moreschi and Sparling [5] is used to show that the conserved five dimensional energymomentum vector P = ifΓ_β = 0 then V₅ must admit a time-like Killing vector. Lichnerowicz's results [1] then imply that V₅ must be flat. A lower bound for P⁴ (the mass) similar to that found by Gibbons and Hull [6] is obtained.     

Morphisms,hemimorphisms, and baer*-semigroups

The relationship between CROCs (complete orthomodular lattices) and complete Baer^*-semigroups is discussed using an explicit construction of the adjoint of a hemimorphism. Simple examples provide much insight into the structures involved.     

Nets ofC*-algebras and classification of states

The concept of locality in quantum physics leads to mathematical structures in which the basic object is an operator algebra with a net of distinguished subalgebras (the local subalgebras). Such nets provide a classification of the states of this algebra in equivalence classes determined by local or asymptotic properties. The corresponding equivalence relations are natural generalizations of the (more stringent) standard quasiequivalence relation (they are also useful for classifying states by their properties with respect to automorphism groups). After discussing general nets from this point of view we investigate in the last section more specialized nets (funnels of von Neumann algebras) with special emphasis on their locally normal states.The research in this paper was supported in part by the N.S.F. and the Ministère de l'Education Nationale.     

Baser*-semigroups and the logic of quantum mechanics

The theory of orthomodular ortholattices provides mathematical constructs utilized in the quantum logic approach to the mathematical foundations of quantum physics. There exists a remarkable connection between the mathematical theories of orthomodular ortholattices and Baer*-semigroups; therefore, the question arises whether there exists a phenomenologically interpretable role for Baer *-semigroups in the context of the quantum logic approach. Arguments, involving the quantum theory of measurements, yield the result that the theory of Baer *-semigroups provides the mathematical constructs for the discussion of operations and conditional probabilities.Supported in part by the United States Atomic Energy Commission.     

Amorphous germanium II. Structural properties

The coherent X-ray scattering for momentum transfer, k, between 0·025 and 15·0 Å^?1 has been measured for a series of sputtered amorphous Ge films prepared at various substrate temperatures, T _s, between 0 and 350°C. Differences in the radial distribution function (RDF) of films of different T _s have been determined by an accurate differential scattering technique. The small angle scattering (SAS) of the films is less than 100 electron units for k < 1 Å^?1. From a combination of SAS, RDF and scanning electron microscope studies, it is concluded that an observed increase in film density with increasing T _s occurs through a reduction in the number of voids about 7 Å or less in diameter. No variation of bond length with T _s is found. With increasing T _s, there is an increase in first and second-neighbour coordination and a reduction in bond angle distortion.

The rate of change of coordination, C, with density, ρ₀, is found to be d ln C/d ln ρ₀ = 0·6±0·2. Using a new, general theory of the dependence of the RDF on the dihedral angle distribution, P(θ), it is shown that with increasing T _s there is an increased probability of dihedral angles corresponding to the staggered configuration. For all films, the experimental RDF between r = 4·5 and 6·2 Å agrees with a nearly random P(θ) distribution. Comparison of experimental RDF's of crystalline and amorphous Ge indicates the static distortion of the first-neighbour bond length has a standard deviation of only about 0·04 Å.     

Dynamical entropy ofC* algebras and von Neumann algebras

The definition of the dynamical entropy is extended for automorphism groups ofC* algebras. As an example, the dynamical entropy of the shift of a lattice algebra is studied, and it is shown that in some cases it coincides with the entropy density.     

Supersymmetric Positivity and Supersymmetric Hilbert Space

We introduce simple notions of positivity and Hilbert spaces of supersym metric functions naturally suggested by the superspace formulation of supersymmetric quantum field theory. Several applications are indicated.     

Thin carbon films: II. Structure and properties

The growth of thin carbon films of various modifications is studied. The films are applied by ion-beam sputtering of graphite, and the resulting carbon condensate is exposed to either ion or electron beams at low temperature and pressures. The phase composition, structure, surface morphology, and emissivity of the carbon films are examined by X-ray diffraction, Raman spectroscopy, and atomic-force microscopy.     

Phonons in disordered solids. II. Thermodynamic properties

For model disordered crystals we show that Debye theory yields the correct low-temperature specific heat if the measured low-frequency speed of sound is used in the calculation of the T³ term. This contrasts with the anomalous leading T term and other anomalies of amorphous or glassy materials, sugesting that thermodynamic properties might be useful in distinguishing a disordered-crystal phase from the amorphous or glassy phase. Our results make it plausible that below a threshold amount of disorder all thermodynamic properties are basically those of an elastic crystalline phase, while above this threshold value, they belong to a distinct phase, dependent on nonlinearities for its stability.     

On weak and monotone σ-closures ofC*-algebras

It is proved that the monotone -closure of the self-adjoint part of anyC*-algebraA is the self-adjoint part of aC*-algebra . IfA is of type I it is proved that is weakly -closed, i.e. is a*-algebra. The physical importance of*-algebras was explained in [1] and [7].     

Characterization of commercial polybutylenes. part II. Crystallinity and tensile properties

A series of previously characterized polybutylenes were compression molded using various cooling conditions. Crystallinity decreased as cooling rate increased. The samples exhibited three types of tensile behavior. The crystallinity of samples containing around 5% ethylene was about 40%. These behaved as elastomers, exhibiting typical entropic elasticity. For the remaining samples, crystallinities between 55% and 70% were observed, depending on grade, and more particularly on cooling conditions. For these, lower crystallinity levels favor sample necking and stress-induced orientation, producing high elongation at break and high tensile strength values, while the higher crystallinity samples show high moduli and elasticity values coupled with uniform deformation. In the former case, significant crystalline deformation—causing alignment of the c axes of the crystallites towards the stretching direction, and thereby reinforcing the sample—is responsible for the high elongations and tensile strength encountered. In the higher-crystallinity case much of the deformation is taken up by interlamellar void generation, with little or no c-axis alignment, this elastic mechanism having been termed “hard elasticity.”     

Statistical properties of many-particle spectra. II. Two-point correlations and fluctuations

The two-point correlation function for complex spectra described by the Gaussian Orthogonal Ensemble (GOE) is calculated, and its essential simplicity displayed, by an elementary procedure which derives from orthogonal invariance and the dominance of intrinsic binary correlations. The resultant function is used for an approximate calculation of the standard fluctuation measures. Good agreements are found with exact results where these are available, this incidentally demonstrating that the measures are, for the most part, two-point measures. It is shown that they vary slowly over the spectrum, a result which is in agreement both with experiment and with Monte Carlo calculations. The same technique can be used for higher-order correlation functions, and possibly also for more complicated ensembles in which case the results would be relevant to the question why GOE fluctuations give a good account of experimental results.     

Generalized Robertson-Walker metrics and some of their properties. II

The Robertson-Walker (RW) metrics, of dimensionality four and signature –2, are generalized to metrics of dimensionality (n+1) and of arbitrary signature,n (> 1) being an arbitrary integer. In canonical coordinates (t, x ¹,x ², ...,x ⁿ ) these generalized Robertson-Walker (GRW) metrics are functions of the coordinatet. The following statements are proved to be equivalent: The GRW metrics are (a) expressible int-independent form, (b) of constant curvature, (c) Einstein spaces. Furthermore, there are six, and only six, classes of GRW metrics satisfying these three statements. The coordinate transformations which transform these metrics to theirt-independent form are given explicitly. Two of these classes of GRW metrics reduce, in theirt-independent form, to the same flat (generalized Minkowski) metrics, three reduce to the samet-independent metrics which are generalizations of the de Sitter space-time metric, and the last class tot-independent metrics which are generalizations of the anti-de Sitter space-time metric.     

Growth of the Eigensolutions of Laplacians on Riemannian Manifolds II: Positivity of the Initial Energy

In this paper, the energy function is used to investigate the eigen-solutions of ?Δu + V u = λu on the Riemannian manifolds. We give a new way to prove the positivity of the initial energy of energy function, which leads to a simple way to obtain the growth of eigen-solutions.     

The luminescence properties of pesticides. II. Non-fluorescing carbamates

The phosphorescence emission and excitation spectra of four pesticidal carbamates have been recorded. The phosphorescence decay times were determined over several half-lives. The studies were carried out in three different solvents — EPA, isopropyl alcohol, and 3-methyl pentane. The effect of solvent on the spectral properties of carbamates is discussed.