224 research outputs found
On the virial theorem for the relativistic operator of Brown and Ravenhall, and the absence of embedded eigenvalues
A virial theorem is established for the operator proposed by Brown and
Ravenhall as a model for relativistic one-electron atoms. As a consequence, it
is proved that the operator has no eigenvalues greater than , where is the fine structure constant, for
all values of the nuclear charge below the critical value : in
particular there are no eigenvalues embedded in the essential spectrum when . Implications for the operators in the partial wave
decomposition are also described.Comment: To appear in Letters in Math. Physic
Dirac-Sobolev inequalities and estimates for the zero modes of massless Dirac operators
The paper analyses the decay of any zero modes that might exist for a
massless Dirac operator H:= \ba \cdot (1/i) \bgrad + Q, where is -matrix-valued and of order O(|\x|^{-1}) at infinity. The approach
is based on inversion with respect to the unit sphere in and
establishing embedding theorems for Dirac-Sobolev spaces of spinors which
are such that and lie in Comment: 11 page
Hydrodynamic chains and a classification of their Poisson brackets
Necessary and sufficient conditions for an existence of the Poisson brackets
significantly simplify in the Liouville coordinates. The corresponding
equations can be integrated. Thus, a description of local Hamiltonian
structures is a first step in a description of integrable hydrodynamic chains.
The concept of Poisson bracket is introduced. Several new Poisson brackets
are presented
Reduced pre-Lie algebraic structures, the weak and weakly deformed Balinsky-Novikov type symmetry algebras and related Hamiltonian operators
The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures
Quadratic Poisson brackets compatible with an algebra structure
Quadratic Poisson brackets on a vector space equipped with a bilinear
multiplication are studied. A notion of a bracket compatible with the
multiplication is introduced and an effective criterion of such compatibility
is given. Among compatible brackets, a subclass of coboundary brackets is
described, and such brackets are enumerated in a number of examples.Comment: 6 page
Block-Diagonalization of Operators with Gaps, with Applications to Dirac Operators
We present new results on the block-diagonalization of Dirac operators on
three-dimensional Euclidean space with unbounded potentials. Classes of
admissible potentials include electromagnetic potentials with strong Coulomb
singularities and more general matrix-valued potentials, even non-self-adjoint
ones. For the Coulomb potential, we achieve an exact diagonalization up to
nuclear charge Z=124 and prove the convergence of the Douglas-Kroll-He\ss\
approximation up to Z=62, thus improving the upper bounds Z=93 and Z=51,
respectively, by H.\ Siedentop and E.\ Stockmeyer considerably. These results
follow from abstract theorems on perturbations of spectral subspaces of
operators with gaps, which are based on a method of H.\ Langer and C.\ Tretter
and are also of independent interest
Fin development in a cartilaginous fish and the origin of vertebrate limbs
Recent fossil finds and experimental analysis of chick and mouse embryos highlighted the lateral fin fold theory, which suggests that two pairs of limbs in tetrapods evolved by subdivision of an elongated single fin1. Here we examine fin development in embryos of the primitive cartilaginous fish, Scyliorhinus canicula (dogfish) using scanning electron microscopy and investigate expression of genes known to be involved in limb positioning, identity and patterning in higher vertebrates. Although we did not detect lateral fin folds in dogfish embryos, Engrailed-1 expression suggests that the body is compartmentalized dorso-ventrally. Furthermore, specification of limb identity occurs through the Tbx4 and Tbx5 genes, as in higher vertebrates. In contrast, unlike higher vertebrates, we did not detect Shh transcripts in dogfish fin-buds, although dHand (a gene involved in establishing Shh) is expressed. In S. canicula, the main fin axis seems to lie parallel to the body axis. 'Freeing' fins from the body axis and establishing a separate 'limb' axis has been proposed to be a crucial step in evolution of tetrapod limbs2, 3. We suggest that Shh plays a critical role in this process
Mechanism of gallic acid biosynthesis in bacteria (Escherichia coli) and walnut (Juglans regia)
Gallic acid (GA), a key intermediate in the synthesis of plant hydrolysable tannins, is also a primary anti-inflammatory, cardio-protective agent found in wine, tea, and cocoa. In this publication, we reveal the identity of a gene and encoded protein essential for GA synthesis. Although it has long been recognized that plants, bacteria, and fungi synthesize and accumulate GA, the pathway leading to its synthesis was largely unknown. Here we provide evidence that shikimate dehydrogenase (SDH), a shikimate pathway enzyme essential for aromatic amino acid synthesis, is also required for GA production. Escherichia coli (E. coli) aroE mutants lacking a functional SDH can be complemented with the plant enzyme such that they grew on media lacking aromatic amino acids and produced GA in vitro. Transgenic Nicotianatabacum lines expressing a Juglans regia SDH exhibited a 500% increase in GA accumulation. The J. regia and E. coli SDH was purified via overexpression in E. coli and used to measure substrate and cofactor kinetics, following reduction of NADP+ to NADPH. Reversed-phase liquid chromatography coupled to electrospray mass spectrometry (RP-LC/ESI–MS) was used to quantify and validate GA production through dehydrogenation of 3-dehydroshikimate (3-DHS) by purified E. coli and J. regia SDH when shikimic acid (SA) or 3-DHS were used as substrates and NADP+ as cofactor. Finally, we show that purified E. coli and J. regia SDH produced GA in vitro
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