On a problem proposed by H. Braß concerning the remainder term in quadrature for convex functions

Summary We prove that the error inn-point Gaussian quadrature, with respect to the standard weight functionw1, is of best possible orderO(n ^–2) for every bounded convex function. This result solves an open problem proposed by H. Braß and published in the problem section of the proceedings of the 2. Conference on Numerical Integration held in 1981 at the Mathematisches Forschungsinstitut Oberwolfach (Hämmerlin 1982; Problem 2). Furthermore, we investigate this problem for positive quadrature rules and for general product quadrature. In particular, for the special class of Jacobian weight functionsw _, (x)=(1–x)(1+x), we show that the above result for Gaussian quadrature is not valid precisely ifw _, is unbounded.Dedicated to Prof. H. Braß on the occasion of his 55th birthday     

Phototransformations of 5-aminolevulinic acid-induced protoporphyrin IX in vitro: a spectroscopic study

Human adenocarcinoma cells of the line WiDr were incubated with 5-aminolevulinic acid to induce protoporphyrin IX (PpIX) and then exposed to laser light of wavelength 635 nm. The PpIX fluorescence decreased with increasing exposure. The decay rate was slightly dependent on the initial PpIX concentration. The PpIX fluorescence was halved by a fluence of about 40 J/cm2. Several fluorescing photoproducts were formed. The main one, supposedly the chlorine-type photoprotoporphyrin (Ppp), had a fluorescence excitation spectrum stretching out to about 680 nm with a maximum at around 668 nm. The formation kinetics of this product was dependent on the initial PpIX concentration. Moreover, it was selectively bleached by exposure to light at 670 nm. A photoproduct with an emission maximum at 652 nm, different from Ppp, remained after this exposure. Traces of a photoproduct(s) with fluorescence emission slightly blue-shifted compared with that of PpIX, supposedly water-soluble porphyrins, were also detected after light exposure.     

Asymptotics for the remainder of a class of positive quadratures for integrands with an interior singularity

Summary For functions with an interior singularity, the errors of a class of positive quadrature formulae with high algebraic degree are reduced to those of the much simpler Euler-Maclaurin type formulae. Applying this method to certain classes of functions, such as, for example,f(x)=h(x)|x-u| , where >–1, with a sufficiently smooth functionh, we obtain the main term of the error expansion for quadrature rules of ultraspherical type.     

Spectroscopic diagnosis of colonic dysplasia.

We have developed a method for defining diagnostic algorithms for pathologic conditions based on fluorescence spectroscopy. We apply this method to human colon tissue and show that fluorescence can be used to diagnose the presence or absence of colonic adenoma. This method uses fluorescence excitation-emission matrices (EEM) to identify optimal excitation regions for obtaining fluorescence emission spectra which can be used to differentiate normal and pathologic tissues. In the case of normal and adenomatous colon tissue, these were found to be: 330, 370, and 430 nm +/- 10 nm. At these excitation wavelengths, emission wavelengths for use in diagnostic algorithms are identified from average difference and ratio of the spectra from normal and pathologic tissues. In colon tissue, at 370 nm excitation, 404, 480, and 680 nm were found to be useful emission wavelengths for diagnosing the presence of adenoma in vitro. The basis of colon tissue autofluorescence was investigated using EEM of pure molecules and relevant excitation-emission maxima in the literature.     

Iron-Catalyzed Halogen Exchange of Trifluoromethyl Arenes**

The facile production of ArCF₂X and ArCX₃ from ArCF₃ using catalytic iron(III)halides is reported, which constitutes the first iron-catalyzed halogen exchange for non-aromatic C−F bonds. Theoretical calculations suggest direct activation of C−F bonds by iron coordination. ArCX₃ and ArCF₂X products of the reaction are synthetically valuable due to their diversification potential. In particular, chloro- and bromodifluoromethyl arenes (ArCF₂Cl, ArCF₂Br respectively) provide access to a myriad of difluoromethyl arene derivatives (ArCF₂R). To optimize for mono-halogen exchange, a statistical method called Design of Experiments was used. Optimized parameters were successfully applied to electron rich and electron deficient aromatic substrates, and to the late stage diversification of flufenoxuron, a commercial insecticide. These methods are highly practical, being run at convenient temperatures and using inexpensive common reagents.     

Functional equations of the dilogarithm in motivic cohomology

We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH²(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generating the integral motivic cohomology groups of some number fields.     

Estimating the ground state energy of the Schrödinger equation for convex potentials

In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error ε$\epsilon$. Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom d$d$. We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$ and ε⁻¹${\epsilon }^{-1}$, while the number of qubits is polynomial in d$d$ and logε⁻¹$log{\epsilon }^{-1}$. In addition, we present an algorithm for preparing a quantum state that overlaps within 1−δ,δ∈(0,1)$1-\delta ,\delta \in (0,1)$, with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$, ε⁻¹${\epsilon }^{-1}$ and δ⁻¹${\delta }^{-1}$, while the number of qubits is polynomial in d$d$, logε⁻¹$log{\epsilon }^{-1}$ and logδ⁻¹$log{\delta }^{-1}$.     

On Proinov’s bound for the quadrature error of functions having a given modulus of continuity

Summary We discuss the question, under which conditions Proinovs upper bound for the quadrature error of functions having a given modulus of continuity is not improvable. Improvability usually holds if the quadrature formula is not a Riemannian sum. We derive a condition on the second Peano kernel yielding always the sharpness of Proinovs result. This condition is applicable to Gaussian quadrature.