On quasi-contraction mappings of Ćirić and Fisher type via ω-distance

Kada, Suzuki, and Takahashi introduced and studies the concept of ω- distance in fixed point theory. In this paper, we generalize and unify ?iri?’ and Fisher fixed points results for quasi-contractions on metric space to ω-distance on complete metric spaces. We also extend some results of Kada, Suzuki and Takahashi, and Suzuki. Our methods of proofs are new and even simpler than the corresponding methods in metric spaces.     

Structural and aromatic aspects for tautomerism of (Z)-6-((4-bromophenylamino)methylene)-2,3-dihydroxycyclohexa-2,4-dienone

The molecular and crystal structure of (Z)-6-((4-bromophenylamino)methylene)-2,3-dihydroxycyclohexa-2,4-dienone were determined by single crystal X-ray diffraction and spectroscopic methods. Molecules of the compound can be regarded as a resonance hybrid of cis-keto tautomer and zwitterionic form. Pairs of molecules of the compound generate pseudocyclic centrosymmetric R₂² (10) R_{2}^{2} (10) supramolecular synthons with the aid of O–H···O type intermolecular H-bonds. Stacking of R₂² (10) R_{2}^{2} (10) synthons along b-axis is stabilized by π···π interactions. Changes in both covalent topology and molecular geometry of the compound accompanying proton transfer were monitored by a relaxed PES scan with respect to hydroxyl bond length used as redundant internal coordinate. Quantum chemical studies at 6-311 + G(d,p) level reveal that bond lengths which are indicative to tautomerization process cannot reach their expected values even if proton transfer occurs in gas phase and pseudo-aromatic chelate ring formation has primary effect on the stabilization of NH tautomer. Resonance-assisted intramolecular H-bond affects the electronic state of its neighboring aromatic fragments.     

End-labeling of peptide nucleic acid with osmium complex. Voltammetry at carbon and mercury electrodes

Peptide nucleic acid (PNA), the DNA mimic with electrically neutral pseudopeptide backbone, is intensively used in biotechnologies and particularly in single-base mismatch detection in DNA hybridization sensors. We propose a simple method of covalent end-labeling of PNA with osmium tetroxide, 2,2′-bipyridine (Os,bipy). Os,bipy-modified PNA (PNA–Os,bipy) produces voltammetric stripping peaks at carbon and mercury electrodes. Peak potential (E_p) of one of the anodic peaks of PNA–Os,bipy at the pyrolytic graphite electrode (PGE) differs from E_p of the reagent, allowing PNA–Os,bipy analysis directly in the reaction mixture. At the hanging mercury drop electrode (HMDE) the PNA–Os,bipy yields a catalytic peak Cat_p, in addition to the redox couples. Using Cat_p it is possible to detect purified PNA–Os,bipy down to 1 pM concentration at accumulation time 60 s. To our knowledge this is the highest sensitivity of the electrochemical detection of PNA.     

Platinum–tin complexes as catalysts for the anodic oxidation of borohydride

Platinum–tin complexes were prepared by the reduction of Pt(IV) with Sn(II) in HCl media and studied by light absorption spectrometry, X-ray photoelectron spectroscopy (XPS), and electron microscopy. The formation of three complexes, H₃[Pt(SnCl₃)₅], H₂[Pt(SnCl₃)₂Cl₂], and H₂[Pt₃(SnCl₃)₈], depending on HCl and SnCl₂ concentrations, has been shown. The glassy carbon (GC) electrode modified in the complexes solutions was found to be an electrocatalyst for borohydride oxidation in a 1.0-M NaOH solution. Comparison of BH₄ ⁻ electrooxidation on Pt and on GC modified with platinum–tin complexes has shown that catalytic hydrolysis of BH₄ ⁻ did not proceed in the latter case in contrast to its oxidation on the Pt electrode, and only direct BH₄ ⁻ oxidation has been observed in the positive potentials scan. The activity of Pt–Sn complexes for BH₄ ⁻ oxidation changes with time and eventually decreases due to Sn(II), bound in the complex with Pt(II), oxidation by atmospheric oxygen. The complexes may be renewed by addition of missing amounts of SnCl₂ and HCl.     

Fermi-Dirac and Bose-Einstein functions of negative integer order

We find simple explicit closed-form formulas for the Fermi-Dirac function $ \mathcal{F}_{ - n} (z) $ \mathcal{F}_{ - n} (z) and Bose-Einstein function $ \mathcal{B}_{ - n} (z) $ \mathcal{B}_{ - n} (z) for arbitrary n ε ℕ. The obtained formulas involve the higher tangent numbers defined by Carlitz and Scoville. We present some examples and direct consequences of applying the main results.     

Structural theorems for quasiasymptotics of distributions at the origin

An open question concerning the quasiasymptotic behavior of distributions at the origin is solved. The question is the following: Suppose that a tempered distribution has quasiasymptotic at the origin in S ′(?), then the tempered distribution has quasiasymptotic in D ′(?), does the converse implication hold? The second purpose of this article is to give complete structural theorems for quasiasymptotics at the origin. For this purpose, asymptotically homogeneous functions with respect to slowly varying functions are introduced and analyzed (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Noncommutative classical and quantum mechanics for quadratic Lagrangians (Hamiltonians)

Classical and quantum mechanics based on an extended Heisenberg algebra with additional canonical commutation relations for position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra by a linear transformation of coordinates and transferred to the Hamiltonian (Lagrangian). This linear transformation does not change the quadratic form of the Hamiltonian (Lagrangian), and Feynman’s path integral preserves its exact expression for quadratic models. The compact general formalism presented here can be easily illustrated in any particular quadratic case. As an important result of phenomenological interest, we give the path integral for a charged particle in the noncommutative plane with a perpendicular magnetic field. We also present an effective Planck constant ħ _eff which depends on additional noncommutativity.     

Separation of tryptic peptides of native and glycated BSA using open‐tubular CEC with salophene–lanthanide–Zn2+ complex as stationary phase

Open‐tubular CEC (OT‐CEC) with a new stationary phase, salophene–lanthanide–Zn²⁺ complex, has been applied to the separation of tryptic peptides of native BSA and BSA glycated by glucose and ribose. Glycation of proteins (non‐enzymatic modification by sugars) significantly affects their properties and it is of great importance from a physiological point of view. Separation of tryptic peptides of glycated BSA by CZE was poor because of their strong adsorption to the bare fused silica capillary. An improved separation of tryptic peptides of both native and glycated BSA was achieved by OT‐CEC in the fused silica capillary non‐covalently coated with salophene–lanthanide–Zn²⁺ complex, which suppressed the adsorption of peptides to the capillary and via specific interactions with some (glyco)peptides enhanced selectivity of the separation. Significant differences have been found in OT‐CEC analyses of tryptic hydrolysates of native and glycated BSA. In OT‐CEC‐UV profile of tryptic peptides of native BSA, 44 peaks could be resolved, whereas a reduced number of 38 peaks were observed in the profile of tryptic peptides of glucose‐glycated BSA and only 30 peaks were found in the case of ribose‐glycated BSA. The developed OT‐CEC can be potentially used for monitoring of protein glycation.     

Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling <Emphasis Type="Italic">ν</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, ·) → + ∞ AS <Emphasis Type="Italic">p</Emphasis> → + ∞

Over a large range of the pressure, one cannot ignore the fact that the viscosity grows significantly (even exponentially) with increasing pressure. This paper concerns long-time and large-data existence results for a generalization of the Navier-Stokes fluid whose viscosity depends on the shear rate and the pressure. The novelty of this result stems from the fact that we allow the viscosity to be an unbounded function of pressure as it becomes infinite. In order to include a large class of viscosities and in order to explain the main idea in as simple a manner as possible, we restrict ourselves to a discussion of the spatially periodic problem. K. R. Rajagopal thanks the National Science Foundation for its support.     

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov–Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.