**The class of a torus in the Grothendieck ring of varieties**

K. Rökaeus

We establish a formula for the classes of certain tori in the Grothendieck ring of varieties, expressing them in terms of the natural lambda-structure on the Grothendieck ring. More explicitly, we will see that if L* is the torus of invertible elements in the n-dimensional separable k-algebra L, then the class of L* can be expressed as an alternating sum of the images of the spectrum of L under the lambda-operations, multiplied by powers of the Lefschetz class. This formula is suggested from the cohomology of the torus, illustrating a heuristic method that can be used in other situations. To prove the formula will require some rather explicit calculations in the Grothendieck ring. To be able to perform these we introduce a homomorphism from the Burnside ring of the absolute Galois group of k, to the Grothendieck ring of varieties over~k. In the process we obtain some information about the structure of the subring generated by zero-dimensional varieties.

**An extension of the Hardy-Ramanujan circle method and applications to partitions without sequences**

K. Bringmann, K. Mahlburg

We develop a generalized version of the Hardy-Ramanujan "circle method" in order to derive asymptotic series expansions for the products of modular forms and mock theta functions. Classical asymptotic methods (including the circle method) do not work in this situation because such products are not modular, and in fact, the "error integrals" that occur in the transformations of the mock theta functions can (and often do) make a significant contribution to the asymptotic series. The resulting series include principal part integrals of Bessel functions, whereby the main asymptotic term can also be identified.

To illustrate the application of our method, we calculate the asymptotic series expansion for the number of partitions without sequences. Andrews showed that the generating function for such partitions is the product of the third order mock theta function $\chi$ and a (modular) infinite product series. The resulting asymptotic expansion for this example is particularly interesting because the error integrals in the modular transformation of the mock theta function component appear in the exponential main term.

**Optimal Sobolev inequalities on complete Riemannian manifolds with Ricci curvature bounded below and positive injectivity radius**

E. Hebey

The concept of best constants for Sobolev embeddings appeared to be crucial for solving limiting cases of some partial differential equations. A striking example where it has played a major role is given by the very famous Yamabe problem. While the situation is well understood for compact manifolds, things are less clear when dealing with complete manifolds. Aubin proved in '76 that optimal Sobolev inequalities are valid for complete manifolds with bounded sectional curvature and positive injectivity radius. We prove here that the result still holds if the bound on the sectional curvature is replaced by a lower bound on the Ricci curvature (a much weaker assumption). We also get estimates for the remaining constants.

**On the curvature of some free boundaries in higher dimensions**

G. Bjorn, S. Makoto

Abstract. It is known that any subharmonic quadrature domain in two dimensions satisfies a natural inner ball condition, in other words there is a specific upper bound on the curvature of the boundary. This result directly applies to free boundaries appearing in obstacle type problems and in Hele- Shaw flow. In the present paper we make partial progress on the corresponding question in higher dimensions. Specifically, we prove the equivalence between several different ways to formulate the inner ball condition, and we compute the Brouwer degree for a geometrically important mapping related to the Schwarz potential of the boundary. The latter gives in particular a new proof in the two dimensional case

**Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings**

K. J. Nowak

10.4064/ap96-3-5

This paper investigates the geometry of the expansion RQ of the real field R by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusiński–Lion–Rolin). To this end, we study non-standard models R of the universal diagram T of RQ in the language L augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation theorem, upon which the majority of fundamental results in analytic geometry rely, but which is unavailable in the general quasianalytic geometry. The basic tools applied here are transformation to normal crossings and decomposition into special cubes. The latter method, developed in our earlier article [Ann. Polon. Math. 96 (2009), 65–74], combines modifications by blowing up with a suitable partitioning. Via an analysis of L-terms and infinitesimals, we prove the valuation property for functions given by L-terms, and next the exchange property for substructures of a given model R. Our proofs are based on the concepts of analytically independent as well as active and non-active infinitesimals, introduced in this article. Further, quantifier elimination for T is established through model-theoretic compactness. The universal theory T is thus complete and o-minimal, and RQ is its prime model. Under the circumstances, every definable function is piecewise given by L-terms, and therefore the previous results concerning L-terms generalize immediately to definable functions. In this fashion, we obtain the valuation property and preparation theorem for quasi-subanalytic functions. Finally, a quasi-subanalytic version of Puiseux's theorem with parameter is demonstrated.

**Lyapunov type inequalities for a second order differential equation with a damping term**

S. H. Saker

10.4064/ap103-1-4

Abstract: For a second order differential equation with a damping term, we establish some new inequalities of Lyapunov type. These inequalities give implicit lower bounds on the distance between zeros of a nontrivial solution and also lower bounds for the spacing between zeros of a solution and/or its derivative. We also obtain a lower bound for the first eigenvalue of a boundary value problem. The main results are proved by applying the Hölder inequality and some generalizations of Opial and Wirtinger type inequalities. The results yield conditions for disfocality and disconjugacy. An example is considered to illustrate the main results.

**Coexistence of antiferromagnetism and superconductivity within t-J model with strong correlations and nonzero spin polarization**

J. Kaczmarczyk, J.Spałek

The coexistence of antiferromagnetism with superconductivity is studied theoretically within the t-J model with the Zeeman term included. The strong electron correlations are accounted for by means of the extended Gutzwiller projection method within a statistically consistent approach proposed recently. The phase diagram on the band filling-magnetic field plane is shown, and subsequently the system properties are analyzed for the fixed band filling n=0.97. In this regime, the results reflect principal qualitative features observed recently in selected heavy-fermion systems, namely, (i) with the increasing magnetic field the system evolves from coexisting antiferromagnetic-superconducting phase through antiferromagnetic phase toward polarized paramagnetic state and (ii) the onset of superconducting order suppresses partly the staggered moment. The superconducting gap has both the spin-singlet and the staggered-triplet components, a direct consequence of a coexistence of the superconducting state with antiferromagnetism.

**Magnetoelectric correlations in BiMnO$_3$ whithin Landau theory: comparison with experiment**

O.Howczak, J.Spalek

Here we present a simple model formulated in the spirit of Landau theory in order to qualitatively describe magnetoelectric multiferroics. The system consists of two coupled single-component order parameters P and M which represent ferroelectric and ferromagnetic phases, respectively. We show that the magnetoelectric coupling strongly renormalizes the original magnetic transition temperature, TM → TRM (with TRM TM), as well as generates an additional anomaly in ferroelectric subsystem at TRM. Under the inﬂuence of both magnetic and electric ﬁelds the concept of Arrot plot is reproduced by the Arrot planes. We compare obtained results with available experimental data for the BiMnO3 which undergoes ferroelectric transition below 700 K (Tf ) and ferromagnetic transition at 100 K (TRM). The results are in good overall agreement with experimental data for the ferroelectromagnetic BiMnO3. We also estimate the contribution of Gaussian ﬂuctuations of both order parameters, that lead to corrections to the mean-ﬁeld speciﬁc heat. Those corrections are still insuﬃcient even though other quantities agree quite well with experiment. We calculate the temperature dependence of the coherence length for both types of order as well.

**Information from Complexity: Challenges of TOF-SIMS Data Interpretation**

Graham DJ, Wagner MS, Castner DG

Time-of-flight secondary ion mass spectrometry (TOF-SIMS) data are complex, even for the simplest systems. Yet it is within this complexity that information about sample composition, molecular orientation, surface order, chemical bonding, sample purity, etc., is contained. The challenge is how to easily extract this information from the spectra and images. Multivariate analysis (MVA) has shown promise in taming the complexity challenges presented by TOF-SIMS data while using all the information in the entire spectrum. The recent success of MVA methods such as principal component analysis (PCA) and partial least squares (PLS) in the spectroscopic and imaging analysis of organic and biological materials has led to a great increase in the interest of MVA processing of TOF-SIMS data. However, there is still a need to better understand what data to use to answer a given question, how to optimally process the data before applying MVA, and how to correctly interpret the MVA results. The challenges of TOF-SIMS data interpretation will only get more complex, especially for biological samples, further increasing the need for well-controlled MVA methodologies.

**Tof-SIMS analysis of myocardial infracted tissue**

Park JW, Cha MJ, Shon HK, Kim SH, Lee TG, Moon DW, Hwang KCh

Heart tissues from a normal rat, myocardial infarcted rat, and mesenchymal stem cell (MSC)-treated rat after myocardial infarction (MI) were studied using ToF-SIMS, with the assistance of principal component analysis (PCA). To our knowledge, this is the first such published study. In PCA results obtained from both positive and negative ion spectra of normal (control), MI, and MSC-treated MI tissues, the normal and MSC-treated tissues could be clearly distinguished from the MI tissue in terms of chemical composition. For the MI tissue, fatty acid signals from inside the cell and phosphocholine signals from the cell membrane decreased while glucose signals increased. In the MSC-treated MI tissue, the signal levels of phosphocholine and fatty acids returned to those for normal tissue. Our observations for myocardium tissues are consistent with previous biological studies, which show that the loss of reversibility of cell injury associated with MI could cause degradation to the cell membrane, and that MI could slow fatty acid uptake and oxidation and increase glucose uptake and utilization for ATP synthesis in myocardium. This is, of course, because fatty acids need oxygen for β-oxidation. We anticipate that this label-free ToF-SIMS method partnered with PCA will be helpful to the investigation of myocardial disease and MSC therapy since it is based on chemical information at the tissue level.

**Top-down approach to studying biological components using ToF-SIMS**

Piwowar A, Fletcher J, Lockyer N, Vickerman J

We present in this article a novel approach to identifying subcellular components from spectral images collected with ToF-SIMS. The method utilizes a separation technique to isolate and enrich cellular fractions which are then examined with ToF-SIMS to identify peak or peak ratio markers to assign reference spectra. The ultimate aim of the project is to utilize these reference spectra to identify cellular components from spectral images. We present data from three such extracts (nuclei, cytosol and membrane-bound organelles) which confirms that characteristic spectral differences do exist between these systems. These results indicate that such a ‘top-down' approach to subcellular analysis can aid cellular imaging with ToF-SIMS.

**Surface characterization of collage/elastin based biomateriale for tissue regeneration.**

Skopinska-Wisniewska J, Sionkowska A, Kamińska A, Kalnica A, Jachimiak R

Collagen and elastin are the main proteins of extracellular matrix. Collagen plays a crucial role in tensile strength of tissues, whereas elastin provides resilience to many organs. Both biopolymers are readily available and biocompatible. These properties point out that collagen and elastin are good components of materials for many potential medical applications. The surface properties of biomaterials play an important role in biomedicine as the majority of biological reactions occur on the surface of implanted materials. One of the methods of surface modification is UV-irradiation. The exposition of the biomaterial on ultraviolet light can alterate surface properties of the materials, their chemical stability, swelling properties and mechanical properties as well.The aim of our work was to study the surface properties and biocompatibility of new collagen/elastin based biomaterials and consideration of the influence of ultraviolet light on these properties.The surface properties of collagen/elastin based biomaterials modified by UV-irradiation were studied using the technique of atomic force microscopy (AFM) and contact angle measurements. On the basis of the results the surface free energy and its polar component was calculated using Owens–Wendt method. To assess the biological performance of films based on collagen, elastin and their blends, the response of 3T3 cell was investigated.It was found that the surface of collagen/elastin film is enriched in less polar component – collagen. Exposition on UV light increases polarity of collagen/elastin based films, due to photooxidation process. The AFM images have shown that topography and roughness of the materials had been also affected by UV-irradiation. The changes in surface properties influence on interaction between the material's surface and cells. The investigation of 3T3 cells grown on films based on collagen, elastin and their blends, leads to the conclusion that higher content of elastin in biomaterial promotes the cell adhesion and their viability on the surface. Also the suitable dose of UV light (1, 2 h) improves the biocompatibility of the materials.

**ToF-SIMS and laser-SNMS analysis of apatite formation in extracellular protein matrix of osteoblasts in vitro**

Dambach S, Fartmann M, Kriegeskotte C, Bruning C, Hellweg S, Wiesmann HP, Lipinsky D, Arlinghaus HF

We have applied laser secondary neutral mass spectrometry (laser-SNMS) for examining different states of biomineralization in vitro. Additionally, time-of-flight secondary ion mass spectrometry (ToF-SIMS) was used to analyse fibronectin and collagen type I, the main matrix proteins involved in the mineralization process, in a model system. Primary osteoblast-like cells derived from bovine metacarpals were cultured for 5 weeks on clean smooth silicon substrates. For mass spectrometric investigations, cells and newly-formed minerals were cryofixed, freeze-fractured, and freeze-dried. The results indicate that extracellular enrichment of potassium typically occurs in the vicinity of single osteoblasts during the initial stages of mineralization. This interaction of potassium with matrix macromolecules may prevent uncontrolled apatite nucleation and control the transformation of the matrix into a nucleating system. Thus, monovalent cations seem to be involved in the regulation of mineral nucleation of the extracellular matrix. From the data obtained it can be concluded that ToF-SIMS and laser-SNMS are well suited for imaging trace element and molecule concentrations in biological samples as well as for identifying molecular fragments characteristic of matrix proteins.

**Improving the interpretation of ToF- SIMS measurements on adsorbed proteins using PCA**

Bruning C, Hellweg S, Dambach S, Lipinsky D, Arlinghaus HF

Analyzing different proteins with time-of-flight secondary ion mass spectrometry (ToF-SIMS) is a big challenge owing to the absence of unique, identifying peaks between the protein spectra. One possibility for improving the interpretation of the protein spectra is the application of principal component analysis (PCA). The proteins fibronectin and collagen and three fibronectin–collagen mixtures, adsorbed on smooth silicon substrates, were analyzed with ToF-SIMS. The intensities of molecular protein fragment signals were used to perform a PCA calculation. It was shown that PCA could clearly distinguish the various protein mixtures. From the loadings, it was possible to draw conclusions about the peak intensity distribution. Furthermore, PCA was found to be an appropriate statistical tool to identify outlier.

**Mass Spectral Imaging of Glycophospholipids, Cholesterol, and Glycophorin A in Model Cell Membranes**

Baker MJ, Zheng L, Winograd N, Lockyer NP, Vickerman JC

Time of flight secondary ion mass spectrometry (ToF-SIMS) and the Langmuir−Blodgett (LB) technique have been used to create and analyze reproducible membrane mimics of the inner and outer leaflets of a cellular membrane to investigate lipid−protein and lipid−lipid interactions. Films composed of phospholipids, cholesterol and an integral membrane protein were utilized. The results show the outer membrane leaflet mimic (DPPC/cholesterol/glycophorin A LB film) consisting of a single homogeneous phase whereas the inner membrane leaflet mimic (DPPE/cholesterol/glycophorin A LB film) displays heterogeneity in the form of two separate phases. A DPPE/cholesterol phase and a glycophorin A phase. This points to differences in membrane domain formation based upon the different chemical composition of the leaflets of a cell membrane. The reliability of the measurements was enhanced by establishing the influence of the matrix effect upon the measurement and by utlilizing PCA to enhance the contrast of the images.

**Characterisation of a type-I collagen trimeric cross-linked peptide from calf aorta and its cross-linked structure**

Henkel W, Glanville RW, Greifendorf D

10.1111/j.1432-1033.1987.tb11456.x

A collageneous trimeric cross-linked peptide has been isolated from the insoluble matrix of calf aorta, using trypsin solubilisation, and purified by gel filtration, cation-exchange chromatography and reversed-phase HPLC. Molecular mass and amino acid composition indicated that the C-terminal, non-helical region of type I collagen in its dimer form, designated as [Colc(I)]2, is cross-linked to a tryptic peptide TN(I) from the N-terminal helical cross-link region of an adjacent type I molecule, forming the cross-linked peptide [Colc(I)]2× TN(I). Amino acid sequence analysis of the peptide yielded a series of sequences corresponding to the cross-linking domains ColC(I) and TN(I) and furnished the first direct chemical evidence for the 4D staggered arrangement of type I molecules within native fibers. The trifunctional cross-linking amino acid pyridinoline was shown to occur in the peptide, confirming the peptides three-chain structure. Pyridinoline was isolated from the cross-linked peptide by preparative amino acid analysis and reversed-phase HPLC and identified by (a) its ultraviolet absorption spectra, (b) its fluorescence excitation and emission spectra and, for the first time, (c) its time-of-flight secondary ion-mass spectrum. The high sensitivity of the latter method, exceeding that of fast-atom-bombardement mass spectroscopy by three orders of magnitude, allowed detection of pyridinoline in the picomole range. The occurrence of pyridinoline in non-stoichiometric amounts, the presence of hydroxylysine in hydrolysates of all cross-linked peptides and the finding that hydrolysates also contained an unidentified component indicated that there is at least one cross-link form that is different from pyridinoline and is hydrolysable.

**High-resolution, imaging TOF-SIMS: novel applications in medical research**

Malmberg P, Jennische E, Nilsson D, Nygren H

The use of high-resolution, imaging TOF-SIMS is described and examples are made to demonstrate the application of the method in medical research. Cytochemistry by TOF-SIMS is shown by localization of diacylglycerol (DG) in cryostat sections of hyaline cartilage and by localization of corticosterone in cryostat sections of the adrenal gland cortex. Quantitative measurements and comparison of groups is shown by comparing the lipid content of adipose tissue from two mouse strains, transgenic mouse expressing the FOXC2 gene and wild-type controls. Finally, biopsies made for histopathological diagnosis of infantile reversible cytochrome c oxidase deficiency myopathy were analyzed in order to define the chemical content of areas showing a pathological structure in the light microscope. The use of high-resolution, imaging TOF-SIMS in medical research allows analysis of intact tissue and probe-free localization of specific target molecules in cells and tissues. The TOF-SIMS analysis is not dependent on penetration of reagents into the sample and also independent of probe reactivity such as cross-reactivity or background staining. The TOF-SIMS method can be made quantitative and allows for analysis of specific target molecules in defined tissue compartments.

**Silver deposition on freeze-dried cells allows subcellular localization of cholesterol with imaging TOF-SIMS**

Nygren H, Malmberg P

10.1111/j.0022-2720.2004.01374.x

Imaging time-of-flight secondary ion mass spectrometry (TOF-SIMS) was used for characterization and subcellular localization of organic ions in leucocytes adhering to glass surfaces. The cells were fixed by freeze drying in 0.15 m ammonium formate buffer at pH 7.2–7.4. The freeze-dried cells were sputter-coated with silver, and the silver surface was analysed with imaging TOF-SIMS. TOF-SIMS spectra were recorded by scanning the primary ion beam over the analysis area and acquiring positive mass spectra of the ions leaving the surface. The relative brightness of each pixel within the analysis area reflects the signal intensity of a selected ion in that pixel. Data were collected separately at high mass resolution m/Δm > 7000 and at high lateral resolution (= 0.5 µm). The images were analysed by principal component analysis (PCA). The glass-adhering cells showed a well defined attachment area with a diameter of up to 20 µm, and an equally well defined cell body, containing the nucleus, with a diameter of 8–10 µm. On the raw data images, the obtained cholesterol distributions were consistent with a higher cholesterol content of the cell membrane in the attachment area than in the cell body. Using PCA analysis, silver-cationized molecular cholesterol was found localized mainly in the attachment area of the cells. Cholesterol was also seen at higher concentration in circular spots of ≤ 1 µm in diameter, probably representing caveolae.