Schrödinger operators and unique continuation. Towards an optimal result

In this article we prove the property of unique continuation (also known for C^∞ functions as quasianalyticity) for solutions of the differential inequality |Δu|?|Vu| for V from a wide class of potentials (including class) and u in a space of solutions YV containing all eigenfunctions of the corresponding self-adjoint Schrödinger operator. Motivating question: is it true that for potentials V, for which self-adjoint Schrödinger operator is well defined, the property of unique continuation holds?     

On Wiener‐type polynomials of thorn graphs

We derive the expressions of the ordinary, the vertex‐weighted and the doubly vertex‐weighted Wiener polynomials of a type of thorn graph, for which the number of pendant edges attached to any vertex of the underlying parent graph is a linear function of its degree. We also define variable vertex‐weighted Wiener polynomials and calculate them for the same type of thorn graphs. Copyright © 2009 John Wiley & Sons, Ltd.     

On the Anti-forcing Number of Benzenoids

The anti-forcing number is introduced as the smallest number of edges that have to be removed that any benzenoid remains with a single Kekulé structure. Similarly, the anti- Kekulé number is discussed as the smallest number of edges that have to be removed that any benzenoid remains connected but without any Kekulé structure. These concepts have been exemplified on damaged benzenoid parallelograms.     

On the anti-Kekulé number and anti-forcing number of cata-condensed benzenoids

Previously introduced concepts the anti-Kekulé number and anti-forcing number (Vukicevic et al., J. Math. chem., in press) are applied to cata-condensed benzenoids. It is shown that all cata-benzenoids have anti-Kekulé number either 2 or 3 and both classes are characterized. The explicit formula for anti-forcing number of chain (unbranched) cata-benzenoids is given. It is also shown that anti-forcing number of any cata-benzenoid goes up to h/2 where h is the number of hexagons in a cata-benzenoid.      

Synthesis and magnetic properties of NiFe2−xAlxO4 nanoparticles

Nanocrystalline Al-doped nickel ferrite powders have been synthesized by sol–gel auto-ignition method and the effect of non-magnetic aluminum content on the structural and magnetic properties has been studied. The X-ray diffraction (XRD) revealed that the powders obtained are single phase with inverse spinel structure. The calculated grain sizes from XRD data have been verified using transmission electron microscopy (TEM). TEM photographs show that the powders consist of nanometer-sized grains. It was observed that the characteristic grain size decreases from 29 to 6 nm as the non-magnetic Al content increases, which was attributed to the influence of non-magnetic Al concentration on the grain size. Magnetic hysteresis loops were measured at room temperature with a maximum applied magnetic field of ≈1 T. As aluminum content increases, the measured magnetic hysteresis curves become more and more narrow and the saturation magnetization and remanent magnetization both decreased. The reduction of magnetization compared to bulk is a consequence of spin non-collinearity. Further reduction of magnetization with increase of aluminum content is caused by non-magnetic Al³⁺ ions and weakened interaction between sublattices. This, as well as the decrease in hysteresis was understood in terms of the decrease in particle size.     

Wigner's theorem in Hilbert -algebras of compact operators

Let be a Hilbert -module over the -algebra of all compact operators on a Hilbert space. It is proved that any function which preserves the absolute value of the -valued inner product is of the form , where is a phase function and is an -linear isometry. The result generalizes Molnár's extension of Wigner's classical unitary-antiunitary theorem.

     

J-Inner matrix functions, interpolation and inverse problems for canonical systems, II: The inverse monodromy problem

This is the second of a planned sequence of papers on inverse problems for canonical systems of differential equations. It is devoted to the inverse monodromy problem for canonical integral and differential systems. In this part, which focuses on the case of a diagonal signature matrixJ, a parametrization is obtained of the set of all solutionsM (t) for the inverse problem for integral systems in terms of two chains of entire matrix valued inner functions. Special classes of solutions correspond to special choices of these chains. This theme will be elaborated upon further in a third part of this paper which will be published in a subsequent issue of this journal. There the emphasis will be on symmetries and growth conditions all of which serve to specify or restrict the chains alluded to above, from the outside, so to speak.     

Infinite Saturated Orders

We generalize the notion of saturated orders to infinite partial orders and give both a set-theoretic and an algebraic characterization of such orders. We then study the proof theoretic strength of the equivalence of these characterizations in the context of reverse mathematics, showing that depending on one’s choice of definitions, this equivalence is either provable in RCA ₀ or equivalent to ACA ₀.     

D?→Dπ and D?→Dγ decays: axial coupling and magnetic moment of D? meson

The axial coupling and the magnetic moment of D ^∗-meson or, more specifically, the couplings g_D^*Dpg_{D^{\ast}D\pi} and g_D^*Dgg_{D^{\ast}D\gamma }, encode the non-perturbative QCD effects describing the decays D ^∗→Dπ and D ^∗→Dγ. We compute these quantities by means of lattice QCD with N _f=2 dynamical quarks, by employing the Wilson (“clover”) action. On our finer lattice (a≈0.065 fm) we obtain g_D^*Dp⁺=20±2g_{D^{\ast}D\pi^{+}}=20\pm2, and g_{D^*0 D⁰g}=2.0±0.6 GeV^-1g_{D^{\ast0} D^{0}\gamma}=2.0\pm 0.6~{\rm GeV}^{-1}. This is the first determination of g_{D^*0 D⁰g}g_{D^{\ast0} D^{0}\gamma} on the lattice. We also provide a short phenomenological discussion and the comparison of our result with experiment and with the results quoted in the literature.