The deBranges theorem and uniqueness statements

Recently deBranges gave a very remarkable proof of Milin’s conjecture. This implies both the conjectures of Robertson and Bieberbach. It is our aim to show that the statement of equality case in both Robertson’s and Bierbach’s conjectures may be proved by a different and easier method.     

Effect of N-terminal glutamic acid and glutamine on fragmentation of peptide ions

A prominent dissociation path for electrospray generated tryptic peptide ions is the dissociation of the peptide bond linking the second and third residues from the ammo-terminus. The formation of the resulting b₂ and y _n−2 fragments has been rationalized by specific facile mechanisms. An examination of spectral libraries shows that this path predominates in diprotonated peptides composed of 12 or fewer residues, with the notable exception of peptides containing glutamine or glutamic acid at the N-terminus. To elucidate the mechanism by which these amino acids affect peptide fragmentation, we synthesized peptides of varying size and composition and examined their MS/MS spectra as a function of collision voltage in a triple quadrupole mass spectrometer. Loss of water from N-terminal glutamic acid and glutamine is observed at a lower voltage than any other fragmentation, leading to cyclization of the terminal residue. This cyclization results in the conversion of the terminal amine group to an imide, which has a lower proton affinity. As a result, the second proton is not localized at the N-terminus but is readily transferred to other sites, leading to fragmentation near the center of the peptide. Further confirmation was obtained by examining peptides with N-terminal pyroglutamic acid and N-acetyl peptides. Peptides with N-terminal proline maintain the trend of forming b₂ and y _n−2 because their ring contains an imine rather than imide and has sufficient proton affinity to retain the proton at the N-terminus.     

Dynamic Contrast-Enhanced Imaging and Analysis at High Spatial Resolution of MCF7 Human Breast Tumors

High resolution, dynamic GdDTPA-enhanced images of MCF7 human breast tumors in immunodeficient mice were analyzed at pixel resolution. The analysis, based on a physiological model, was performed by applying a nonlinear least-square algorithm using a color coded scale. The final output mapped at pixel resolution capillary permeability times surface area and fraction of extracellular volume, for each tumor slice. In addition, the output included assessment of the fit to the model by determining the proportion of variability (R²) for each pixel. The spatial variation in theR²values served to identify regions where the predominant mechanism of enhancement was leakage from the intravascular volume to the extracellular volume (R²close to 1). In regions with lowR²other mechanisms of enhancement appear to be dominating presumably diffusion within the extracellular space. As expected, in necrotic regions lacking microcapillaries and identified by analyzingT₂-weighted images of the same tumors, the model failed to fit the dynamic contrast enhanced data. The heterogeneous distribution of the determined pathophysiological features demonstrates the importance of recording and analyzing breast tumor images at high spatial resolution.     

From crack deflection to lattice vibrations—macro to atomistic examination of dynamic cleavage fracture

A set of cleavage experiments with strip-shaped single-crystal silicon specimens subjected to three-point bending is reported. The experiments enabled examination of the relationships between the dynamic energy release rate, the velocity, the orientation-dependent cleavage energy, and the cleavage plane of propagation.Dynamic crack propagation experiments show that when a [0 0 1] silicon single crystal is fractured under three-point bending at ‘parallel’ velocity (directly measured at the bottom surface of the specimen) of up to , it prefers to cleave along the vertical (1 1 0) plane, while when the specimen is fractured under the same conditions but at a velocity higher than , it cleaves along the inclined (1 1 1) plane. At intermediate velocities, the crack will deflect from the (1 1 0) plane to the (1 1 1) plane. Crack velocity was determined by the initial notch length. The local (calculated) velocity of deflection between the cleavage planes ranges from , for a crack propagating on the (1 1 0) plane in the direction, to about , for a crack on the (1 1 0) plane, but in the [0 0 1] direction.It is suggested that the cause of the deflection phenomenon is the anisotropic, velocity-dependent cleavage energy, resulted phonon radiation caused by anisotropic, velocity-dependent lattice vibrations. We have studied the effect of material properties and propose selection criteria to explain the deflection phenomenon: the crack will deflect to the plane of least-energy, for which G−Γ_i(V)=max, or to the plane with maximum crack tip velocity, V_i(Γ)=max.     

On the Bieberbach conjecture for functions with a small second coefficient

In the following we prove that for a given univalent function such that |a ₂|<1.05, |a _n|<n for eachn. This is an improvement of the result in [1].     

Note on error analysis for the spectral synthesis method with interfaces

Summary In a recent paper Meyer and Nelson [1] obtained error estimates for the Galerkin type spectral synthesis method relative to the continuous-energy, continuous-space, time independent neutron diffusion equation. These estimates were obtained under a coercivity condition which is not easily verifiable. The aim of this note is to replace that condition by a readily verifiable one.

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Zusammenfassung In einer kürzlichen Publikation haben Meyer und Nelson [1] Fehlerabschätzungen für eine Galerkin-ähnliche Spektralsynthesemethode, angewandt auf die zeitunabhängige Neutronen-diffusionsgleichung mit kontinuierlicher Abhängigkeit von Energie und Ort, diskutiert. Die Abschätzungen wurden unter einer nicht leicht überprüfbaren Zwangsbedingung erhalten. Das Ziel dieser Arbeit ist, diese Bedingungen durch eine bequem verifizierbare zu ersetzen.

     

Families of methods for ordinary differential equations based on trigonometric polynomials

We consider the construction of methods based on trigonometric polynomials for the initial value problems whose solutions are known to be periodic. It is assumed that the frequency w can be estimated in advance. The resulting methods depend on a parameter ν = hw, where h is the step size, and reduce to classical multistep methods if ν → 0. Gautschi [4] developed Adams and Störmer type methods. In our paper we construct Nyström's and Milne-Simpson's type methods. Numerical experiments show that these methods are not sensitive to changes in w, but require the Jacobian matrix to have purely imaginary eigenvalues.