The blow up analysis of the general curve shortening flow

It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.     

The Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at mnT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m + n − 2 containing all but one point of them also contains the last point.     

Seshadri constants and the spaces of curves with prescribed singularities and positions

We will give optimal bounds for Seshadri constants of an ample line bundle at multiple points on a complex projective surface X.We also present a solution to the long-studied classical problem on the existence of curves on X with given topological singularities at r arbitrary points p1,...,pr.Namely,we obtain a universal lower bound on the degree of curves for the existence.It is independent of the position of the singular points.     

Indirect dark matter searches in our Galaxy in comparison with FERMI data

The first 2-year data on the diffuse gamma rays from the FERMI satellite are analyzed and compared with expectations from dark matter annihilation. Although the published results at intermediate latitudes do not confirm the EGRET GeV excess, the data towards the Galactic center are hard to explain with standard propagation models. Redoing a template shape fit to the data in about 900 independent sky directions reveals that the FERMI data are best described by including a DMA contribution from an NFW DM profile with a ringlike DM substructure, as required to describe the new precise data on the gas flaring and rotation curve.     

Application of the Stoner-Wohlfarth model with interaction for the determination of the saturation magnetisation, anisotropy field, and mean field interaction in bulk amorphous ferromagnets

Magnetic hysteresis curves of bulk amorphous ferromagnet alloys of composition Nd₆₀Fe₃₀Al₁₀, Nd₆₀Fe₂₀Co₁₀Al₁₀ and Pr₅₈Fe₂₄Al₁₈ have been measured in applied magnetic fields up to 9 T at temperatures in the range 10-350 K. The behaviour of the demagnetisation curve in the first quadrant is interpreted using a mean field interaction model as proposed by Callen et al. [Phys. Rev. B 16 (1977) 263], which extends the Stoner-Wohlfarth model [Philos. Trans. Roy. Soc. A 240 (1948) 599] for a random distribution of non-interacting uniaxial grains. Application of the mean field interaction model enables the determination of the saturation magnetisation M_s, anisotropy field H_a, and interaction parameter d, and from these other magnetic parameters, such as the anisotropy constant, K, are deduced. For the three alloys, the temperature dependent behaviour of M_s, H_a, d and K over the range 20-350 K are found to be qualitatively similar, though there are quantitative differences. In all cases M_s increases with decreasing temperature, both H_a and K increase with decreasing temperature, reaching a peak in the range 75-120 K, and then decreasing, and d decreases approximately linearly as the temperature decreases. The physical mechanisms responsible for coercivity in these materials are discussed in the context of random anisotropy and a strong pinning model of domain walls.     

Study of the Complexation of Pb(II) with meso‐2,3‐ Dimercaptosuccinic Acid (DMSA) and 2,3‐Dimercapto‐1‐propanesulfonic acid (DMPS) Using a Bismuth‐Bulk Rotating Disk Electrode

For the first time, the suitability of bismuth bulk rotating disk electrode (BiB‐RDE) for the study of metal complexation has been tested. Cyclic (CV) and differential pulse (DPV) voltammetry have been used to study the complexation of Pb(II) with two of the most effective chelating agents for the treatment of Pb(II) poisoning (meso‐2,3‐dimercaptosuccinic acid, DMSA, and 2,3‐dimercapto‐1‐propanesulfonic acid, DMPS). Multivariate curve resolution has been applied to voltammetric data to obtain the stoichiometries and stability constants of the complexes formed. In both systems, the ML₂ complex was predominant, with log β₂ values of 10.13 and 8.80 for DMSA‐Pb(II) and DMPS‐Pb(II), respectively.     

Characterisation of hydrogen bond perturbations in aqueous systems using aquaphotomics and multivariate curve resolution-alternating least squares

Aquaphotomics is a new discipline that provides a framework for understanding changes in the structure of water caused by various perturbations, such as variations in temperature or the addition of solutes, using near infrared spectroscopy (NIRS). One of the main purposes of aquaphotomics is to identify water bands as main coordinates of future absorbance patterns to be used as biomarkers. These bands appear as consequence of perturbations in the NIR spectra. Curve resolution techniques may help to resolve and find new water bands or confirm already known bands. The aim of this study is to investigate the application of multivariate curve resolution-alternating least squares (MCR-ALS) to characterise the effects of various perturbations on the NIR spectra of water in terms of hydrogen bonding. For this purpose, the perturbations created by temperature change and the addition of four solutions of different ionic strength and Lewis acidity were studied (NaCl, KCl, MgCl₂ and AlCl₃, with concentrations ranging from 0.2 to 1 mol L⁻¹ in steps of 0.2 mol L⁻¹). Transmission spectra of all salt solutions and pure water were obtained at temperatures ranging from 28 to 45 °C. We have found that three distinct components with varying temperature dependence are present in water perturbed by temperature. The salt solutions studied exhibited similar trends with respect to the temperature perturbation, while the peak locations of their MCR-ALS pure components varied according to the ionic strength of the salt used.     

On the Ree curve

We point out a characterization of the Ree curve which involves the number of rational points, the genus, and the shape of two elements of the Weierstrass semigroup at a rational point.     

New version of Bäcklund transformations in Euclidean 3‐space

In this paper, we study the Bäcklund transformations for the adjoint curve in the Euclidean 3‐space. Firstly, it is obtained some essential equations of the Bäcklund transformation. After this, we give a new theorem, the Bäcklund transformations for the adjoint curve in Euclidean 3‐space.     

On elliptic curves y=x−nx with rank zero

In this paper we determine all elliptic curves E_n:y²=x³−n²x with the smallest 2-Selmer groups S_n=Sel₂(E_n(Q))={1} and S_n′=Sel₂(E_n′(Q))={±1,±n}(E_n′:y²=x³+4n²x) based on the 2-descent method. The values of n for such curves E_n are described in terms of graph-theory language. It is well known that the rank of the group E_n(Q) for such curves E_n is zero, the order of its Tate-Shafarevich group is odd, and such integers n are non-congruent numbers.