Hilbert Functions and Generic Forms

Let A be a homogeneous K-algebra where K is a field of characteristic 0, and h A a generic form. We bound the Hilbert function H(A/(h),-) in terms of H(A,-) which extends the bound given by M.Green for generic linear forms. We apply this to some conjectures from Higher Castelnuovo Theory and Cayley-Bacharach Theory.     

Whitney property in two dimensional Sobolev spaces

For , we prove that all the functions of satisfy the Whitney property; i.e., if is such that (in the sense of capacity) on a connected set , then is constant on .

     

On the Structure of Maximal Cohen–Macaulay Modules over the Ring $\boldsymbol{K[[x,y]]/(x^n)}$

We study maximal Cohen–Macaulay modules over the hypersurface ring K being a field. Infinite families of non-isomorphic indecomposable maximal Cohen–Macaulay modules of arbitrary number of minimal generators or of arbitrary rank are constructed. The second author was supported by the CEEX Programme of the Romanian Ministry of Education and Research, contract 2-CEX 06-11-20/25.07.06, and the grant CNCSIS 1055/ 2006.     

Simultaneous Determination of As,Bi, Sb,Se, Te,Hg, Pb and Sn by Small-Sized Electrothermal Vaporization Capacitively Coupled Plasma Microtorch Optical Emission Spectrometry Using Direct Liquid Microsampling

The simultaneous determination of chemical vapor-generating elements involving derivatization is difficult even by inductively coupled plasma optical emission spectrometry or mass spectrometry. This study proposes a new direct liquid microsampling method for the simultaneous determination of As, Bi, Se, Te, Hg, Pb, and Sn, using a fully miniaturized set-up based on electrothermal vaporization capacitively coupled plasma microtorch optical emission spectrometry. The method is cost-effective, free from non-spectral interference, and easy to run by avoiding derivatization. The method involves the vaporization of analytes from the 10 µL sample and recording of episodic spectra generated in low-power (15 W) and low-Ar consumption (150 mL min⁻¹) plasma microtorch interfaced with low-resolution microspectrometers. Selective vaporization at 1300 °C ensured the avoidance of non-spectral effects and allowed the use of external calibration. Several spectral lines for each element even in the range 180–210 nm could be selected. Generally, this spectral range is examined with large-scale instrumentation. Even in the absence of derivatization, the obtained detection limits were low (0.02–0.75 mg kg⁻¹) and allowed analysis of environmental samples, such as cave and river sediments. The recovery was in the range of 86–116%, and the accuracy was better than 10%. The method is of general interest and could be implemented on any miniaturized or classical laboratory spectrometric instrumentation.     

On the Asymptotic Limit of Flows Past a Ribbed Boundary

We consider a stationary Navier–Stokes flow in a bounded domain supplemented with the complete slip boundary conditions. Assuming the boundary of the domain is formed by a family of unidirectional asperities, whose amplitude as well as frequency is proportional to a small parameter ε, we shall show that in the asymptotic limit the motion of the fluid is governed by the same system of the Navier–Stokes equations, however, the limit boundary conditions are different. Specifically, the resulting boundary conditions prevent the fluid from slipping in the direction of asperities, while the motion in the orthogonal direction is allowed without any constraint. The work of Š. N. supported by Grant IAA100190505 of GA ASCR in the framework of the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503.     

Minimization of the k-th eigenvalue of the Dirichlet Laplacian

For every ${k \in \mathbb{N}}$ , we prove the existence of a quasi-open set minimizing the k-th eigenvalue of the Dirichlet Laplacian among all sets of prescribed Lebesgue measure. Moreover, we prove that every minimizer is bounded and has a finite perimeter. The key point is the observation that such quasi-open sets are shape subsolutions for an energy minimizing free boundary problem.     

The Uniform Rugosity Effect

Relying on the effect of microscopic asperities, one can mathematically justify that viscous fluids adhere completely on the boundary of an impermeable domain. The rugosity effect accounts asymptotically for the transformation of complete slip boundary conditions on a rough surface in total adherence boundary conditions, as the amplitude of the rugosities vanishes. The decreasing rate (average velocity divided by the amplitude of the rugosities) computed on close flat layers is definitely influenced by the geometry. Recent results prove that this ratio has a uniform upper bound for certain geometries, like periodical and ??almost Lipschitz?? boundaries. The purpose of this paper is to prove that such a result holds for arbitrary (non-periodical) crystalline boundaries and general (non-smooth) periodical boundaries.     

Comparison of Different Extraction Solvents for Characterization of Antioxidant Potential and Polyphenolic Composition in Boletus edulis and Cantharellus cibarius Mushrooms from Romania

Edible mushrooms are well-known for their nutritional benefits and low energy density. In addition, mushroom extracts contain various bioactive compounds that account for their antioxidant activity; the applied extraction conditions influence the extraction efficiency of such compounds. Therefore, this study investigates the effects of four extractants on the content of polyphenols and antioxidant properties of Boletus edulis and Cantharellus cibarius mushrooms, aiming to optimize the extraction process. Powders of B. edulis and C. cibarius mushrooms were subjected to extraction with acidic water (10% CH₃COOH), ethanol/water/acetic acid (15:76.5:8.5, v/v/v), hexane, and diethyl ether to measure their total phenolic content (TPC), total flavonoid content (TFC), and Trolox equivalent antioxidant capacity (TEAC). Furthermore, the level of individual polyphenolic compounds in these extracts was quantified using an HPLC-DAD-ESI-MS method. Results showed that the type of solvent significantly influenced the TPC and TEAC of mushroom powder but insignificantly influenced the TFC. A very strong positive correlation was found between TPC and TEAC, but no correlation was found between TFC and TEAC or TPC and TFC. Acidic water extracted the highest amount of polyphenolic compounds from these mushroom powders. Therefore, the aqueous extract showed the highest TPC and strongest antioxidant activity. Thus, acidic water is recommended for polyphenol analysis in B. edulis and C. cibarius mushrooms.