Toeplitz quantization on Fock space

For Toeplitz operators $T_f^{(t)}$ acting on the weighted Fock space $H_t^2$, we consider the semi-commutator $T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}$, where $t>0$ is a certain weight parameter that may be interpreted as Planck's constant ? in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit

($?$)$\underset{t\to 0}{\lim }?{6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t.$

It is well-known that ${6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t$ tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to $f,g\in \mathrm{BUC}({\mathbb{C}}^n)$ by Bauer and Coburn. We now further generalize (?) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra $\mathrm{VMO}\cap L^{\infty }$ of bounded functions having vanishing mean oscillation on ${\mathbb{C}}^n$. Our approach is based on the algebraic identity $T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}=?{(H_{\overline{f}}^{(t)})}^{?}{H}_g^{(t)}$, where $H_g^{(t)}$ denotes the Hankel operator corresponding to the symbol g, and norm estimates in terms of the (weighted) heat transform. As a consequence, only f (or likewise only g) has to be contained in one of the above classes for (?) to vanish. For g we only have to impose ${\lim \sup }_{t\to 0}{6H_g^{(t)}6}_t<\infty$, e.g. $g\in L^{\infty }({\mathbb{C}}^n)$. We prove that the set of all symbols $f\in L^{\infty }({\mathbb{C}}^n)$ with the property that ${\lim }_{t\to 0}?{6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t={\lim }_{t\to 0}?{6T_g^{(t)}{T}_f^{(t)}?T_{gf}^{(t)}6}_t=0$ for all $g\in L^{\infty }({\mathbb{C}}^n)$ coincides with $\mathrm{VMO}\cap L^{\infty }$. Additionally, we show that $\underset{t\to 0}{\lim }?{6T_f^{(t)}6}_t={6f6}_{\infty }$ holds for all $f\in L^{\infty }({\mathbb{C}}^n)$. Finally, we present new examples, including bounded smooth functions, where (?) does not vanish.     

On the homology groups of the Brauer complex for a triquadratic field extension

The homology groups , , and of the Brauer complex for a triquadratic field extension are studied. In particular, given , we find equivalent conditions for the image of D in to be zero. We consider as well the second divided power operation , and show that there are nonstandard elements with respect to γ₂. Further, a natural transformation , which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension such that the cohomology group of the Brauer complex contains the images of prescribed elements of , provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions with nontrivial are given. As a consequence we show that the homology group can be arbitrarily big.     

Semicircular limits on the free Poisson chaos: Counterexamples to a transfer principle

We establish a class of sufficient conditions ensuring that a sequence of multiple integrals with respect to a free Poisson measure converges to a semicircular limit. We use this result to construct a set of explicit counterexamples showing that the transfer principle between classical and free Brownian motions (recently proved by Kemp, Nourdin, Peccati and Speicher (2012)) does not extend to the framework of Poisson measures. Our counterexamples implicitly use kernels appearing in the classical theory of random geometric graphs. Several new results of independent interest are obtained as necessary steps in our analysis, in particular: (i) a multiplication formula for free Poisson multiple integrals, (ii) diagram formulae and spectral bounds for these objects, and (iii) a counterexample to the general universality of the Gaussian Wiener chaos in a classical setting.     

On simple modules over conformal Galilei algebras

We study irreducible representations of two classes of conformal Galilei algebras in 1-spatial dimension. We construct a functor which transforms simple modules with nonzero central charge over the Heisenberg subalgebra into simple modules over the conformal Galilei algebras. This can be viewed as an analogue of oscillator representations. We use oscillator representations to describe the structure of simple highest weight modules over conformal Galilei algebras. We classify simple weight modules with finite dimensional weight spaces over finite dimensional Heisenberg algebras and use this classification and properties of oscillator representations to classify simple weight modules with finite dimensional weight spaces over conformal Galilei algebras.     

Orthogonal Pfister involutions in characteristic two

We show that over a field of characteristic 2 a central simple algebra with orthogonal involution that decomposes into a product of quaternion algebras with involution is either anisotropic or metabolic. We use this to define an invariant of such orthogonal involutions that completely determines the isotropy behaviour of the involution. We also give an example of a non-totally decomposable algebra with orthogonal involution that becomes totally decomposable over every splitting field of the algebra.     

The behavior of differential forms under purely inseparable extensions

Let F be a field of characteristic 2. In this paper we give a complete computation of the kernel of the homomorphism $H_2^{m+1}(F)?H_2^{m+1}(L)$ induced by scalar extension, where $L/F$ is a purely inseparable extension (of any degree), $H_2^{m+1}(F)$ is the cokernel of the Artin–Schreier operator $?:{\mathrm{\Omega }}_F^m?{\mathrm{\Omega }}_F^m/\mathrm{d}{\mathrm{\Omega }}_F^{m?1}$ given by: $x\frac{\mathrm{d}{x}_1}{x_1}\wedge ?\wedge \frac{\mathrm{d}{x}_m}{x_m}?(x^2?x)\frac{\mathrm{d}{x}_1}{x_1}\wedge ?\wedge \frac{\mathrm{d}{x}_m}{x_m}+\mathrm{d}{\mathrm{\Omega }}_F^{m?1}$, where ${\mathrm{\Omega }}_F^m$ is the space of absolute m-differential forms over F and d is the differential operator. Other related results are included.     

Formation of hexagonal shaped wurtzite zinc sulphide nano rods

Zinc sulfide nanorods with good photoluminescence have been successfully fabricated using a simple sol-gel process via ultrasonication, with mercaptoethanol as capping agent. The formation of ZnS nucleation, followed by subsequent growth, is significant in obtaining highly oriented ZnS nanorods. Temperature, time, and capping agent also proved to be significant factors in the growth of ZnS nanorods and greatly affect their photo luminescent properties. X-ray diffraction (XRD) analysis, low and high-resolution transmission electron microscopy (TEM & HRTEM), selected-area electron diffraction (SAED) pattern, and scanning electron microscopy (SEM) indicated that the ZnS nanorods were single crystal in nature and that they had grown up preferentially along the [0001] direction. This simple method of nucleation, followed by their successive growth, resulted in the development of an effective and low-cost fabrication process for high-quality ZnS nanorods with good photo luminescent properties that can be applied to luminescent sensors and optoelectronic devices.     

On the concepts of intertwining operator and tensor product module in vertex operator algebra theory

We produce counterexamples to show that in the definition of the notion of intertwining operator for modules for a vertex operator algebra, the commutator formula cannot in general be used as a replacement axiom for the Jacobi identity. We further give a sufficient condition for the commutator formula to imply the Jacobi identity in this definition. Using these results we illuminate the crucial role of the condition called the “compatibility condition” in the construction of the tensor product module in vertex operator algebra theory, as carried out in work of Huang and Lepowsky. In particular, we prove by means of suitable counterexamples that the compatibility condition was indeed needed in this theory.     

States on systems of sets that are closed under symmetric difference

We consider extensions of certain states. The states are defined on the systems of sets that are closed under the formation of the symmetric difference (concrete quantum logics). These systems can be viewed as certain set‐representable quantum logics enriched with the symmetric difference. We first show how the compactness argument allows us to extend states on Boolean algebras over such systems of sets. We then observe that the extensions are sometimes possible even for non‐Boolean situations. On the other hand, a difference‐closed system can be constructed such that even two‐valued states do not allow for extensions. Finally, we consider these questions in a σ‐complete setup and find a large class of such systems with rather interesting state properties.