Clustering Effect on the Statistical Estimation Accuracy of Distribution Density

The paper is devoted to statistical nonparametric estimation of multivariate distribution density. The influence of data pre-clustering on the estimation accuracy of multimodal density is analyzed by means of the Monte Carlo method. It is shown that the soft clustering is more advantageous than the hard one. While a moderate increase in the number of clusters also increases the calculation time, it considerably reduces the estimation error.     

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     

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$log{\delta }^{-1}$.     

Electrochemically Deposited Ni-Sn Alloys: A 119Sn Mössbauer Study

Equiatomic electrodeposition of Ni-Sn alloys is known for its corrosion resistance. Mössbauer spectra of Ni-Sn layers and Ni-Sn intermetallics are best interpreted by the coexistence of Ni₃Sn₂, Ni₃Sn₄ and SnO in varying amounts. The presence of an intermetallic &";NiSn&'; has not yet been verified. XRD and STM exhibit the nanostructure of the electrodeposited substrate.     

Resonant gravimetric immunosensing based on capacitive micromachined ultrasound transducers

High-frequency (40 MHz) and low-frequency (7 MHz) capacitive micromachined ultrasound transducers (CMUT) were fabricated and tested for use in gravimetric detection of biomolecules. The low-frequency CMUT sensors have a gold-coated surface, while the high-frequency sensors have a silicon nitride surface. Both surfaces were functionalized with bovine leukemia virus antigen gp51 acting as the antigen. On addition of an a specific antibody labeled with horseradish peroxidase (HRP), the antigen/antibody complex is formed on the surface and quantified by HRP-catalyzed oxidation of tetramethylbenzidine. It has been found that a considerably smaller quantity of immuno complex is formed on the high frequency sensor surface. In parallel, the loading of the surface of the CMUT was determined via resonance frequency and electromechanical resistance readings. Following the formation of the immuno complexes, the resonance frequencies of the low-frequency and high-frequency sensors decrease by up to 420 and 440 kHz, respectively. Finite element analysis reveals that the loading of the (gold-coated) low frequency sensors is several times larger than that on high frequency sensors. The formation of the protein film with pronounced elasticity and stress on the gold surface case is discussed. We also discuss the adoption of this method for the detection of DNA using a hybridization assay following polymerase chain reaction. Figure

Resonant gravimetric immunosensing with six-channel, 7 MHz capacitive micromachined ultrasound transducer (CMUT). Initial modification by antigen gp51 builds the protein layer with 13.2 GPa elasticity modulus. This increases the resonance frequency of CMUT, which is decreasing after immune complex with antibody is established.