Narrow Results

We determine, in terms of and , the homotopy groups of certain groups of invertibles and of certain equivalence classes in the infinite Grassmann space on a Hilbert C*--module. These results provide various interpretations of .

The stick index of a knot K is defined to be the least number of line segments needed to construct a polygonal embedding of K. We define the projection stick index of K to be the least number of line segments in any projection of a polygonal embedding of K. In this paper, we establish bounds on the projection stick index for various torus knots. We then show that the stick index of a (p, 2p + 1)-torus knot is 4p, and the projection stick index is 2p + 1. This provides examples of knots such that the projection stick index is one greater than half the stick index. We show that for all other torus knots for which the stick index is known, the projection stick index is larger than this. We conjecture that a projection stick index of half the stick index is unattainable for any knot.

We study asymptotic curves on generically immersed surfaces in ℝ⁵. We characterize asymptotic directions via the contact of the surface with flat objects (k-planes, k = 1 - 4), give the equation of the asymptotic curves in terms of the coefficients of the second fundamental form and study their generic local configurations.

Recently, it has been proven in [V. Milman, A. Segal and B. Slomka, A characterization of duality through section/projection correspondence in the finite dimensional setting, J. Funct. Anal. 261(11) (2011) 3366–3389] that the well-known duality mapping on the class of closed convex sets in ℝⁿ containing the origin is the only operation, up to obvious linear modifications, that interchanges linear sections with projections. In this paper, we extend this result to the class of geometric log-concave functions (attaining 1 at the origin). As the notions of polarity and the support function were recently uniquely extended to this class by Artstein-Avidan and Milman, a natural notion of projection arises. This notion of projection is justified by our result. As a consequence of our main result, we prove that, on the class of lower semi continuous non-negative convex functions attaining 0 at the origin, the polarity operation is the only operation interchanging addition with geometric inf-convolution and the support function is the only operation interchanging addition with inf-convolution.

In recent years, many mammalian cells, especially erythrocytes because of simpleness of their membrane surfaces, were widely studied by atomic force microscopy. In our study, diseased erythrocytes were taken from patients of lung cancer, myelodisplastic syndrome (MDS), and so on. We obtained many clear topographical images of numerous erythrocytes, single erythrocyte, and ultramicrostructure of erythrocyte membrane surfaces from normal persons and patients. By studying the red cells of lung cancer patients, we found that many erythrocytes of lung cancer patient have changed into echinocytes. One erythrocyte has 10–20 short projections, most of which, with a mean width of 589.0 nm and a length of 646.7 nm, are on the edge of cell. The projections in the center of echinocytes are lodged and embedded, but in conventional model of echinocytes, the projections in the center stretch outside cell membrane, so a novel model of erythrocytes was designed in our paper. After observation of microstructure of MDS patient's erythrocyte membrane surface, we found that many apertures with different diameters of tens to hundreds nanometers appeared on the surface of cell membrane. It can be concluded that AFM may be widely applied in clinic pathological inspection.

We assess in this paper the spatiotemporal projections of extreme temperature indices over Algeria derived from the adjusted multi-model ensemble mean (MME) data derived from 11 daily historical simulations of CMIP6-GCMs models that participated in the IPCC Sixth Assessment Report (AR6). The projected spatial patterns of 12 extreme temperature indices defined by the Expert Team on Climate Change Detection and Indices are assessed for two future time periods: the mid-future 2041–2070 and the far future 2071–2100, relative to the baseline period 1985–2014 and under three Shared Socio-economic Pathways (SSP) scenarios: low emission SSP1-2.6; medium emission SSP2-4.5 and high emission scenario (SSP5-8.5). The selected climate indices reflect the intensity (TXx, TNx, TXn, TNn and DTR), frequency (TX90p and TN10p) and duration (WSDI, SU, CSDI, TR and FD) of the extreme thermal events. The MME Projections show a global heightened warming over Algeria. Future Climate features depict a continuous increase in the occurrence of hot days by the end of the century reaching 60% for SSP5-8.5 and an amplification of the intensity of the extreme temperature of about 6^∘C for SSP5-8.5 and an extension of the heat wave duration period of about 80 days in the north and 100 days in the south of the country compared to the historical period. However, the study shows a projected simultaneous decline in the cold spell duration of 7 days and in the frost days reaching 25 days. A stabilization of the upsurge trend is remarkably observed for most indices under SSP1-2.6 starting from the 2050s. The future changes depicted in this study should help to assess the distribution of the impacts across different regions of Algeria in order to enhance resilience, establish the appropriate adaptation responses and improve disaster preparedness.

A survey of our results in the diagrammatic approaches to knotted surfaces is given in part I of this paper. Results obtained by knotted surface diagrams and by movie descriptions will be reviewed. Diagrammatic ways of solving generalizations of the Yang-Baxter equation are explained. In part II, we give smoothings of higher dimensional knots that topologically generalize the classical knot smoothings of crossings and Kauffman's bracket trick for resolving the Yang-Baxter equation.

In recent years, many mammalian cells, especially erythrocytes because of simpleness of their membrane surfaces, were widely studied by atomic force microscopy. In our study, diseased erythrocytes were taken from patients of lung cancer, myelodisplastic syndrome (MDS), and so on. We obtained many clear topographical images of numerous erythrocytes, single erythrocyte, and ultramicrostructure of erythrocyte membrane surfaces from normal persons and patients. By studying the red cells of lung cancer patients, we found that many erythrocytes of lung cancer patient have changed into echinocytes. One erythrocyte has 10–20 short projections, most of which, with a mean width of 589.0 nm and a length of 646.7 nm, are on the edge of cell. The projections in the center of echinocytes are lodged and embedded, but in conventional model of echinocytes, the projections in the center stretch outside cell membrane, so a novel model of erythrocytes was designed in our paper. After observation of microstructure of MDS patient's erythrocyte membrane surface, we found that many apertures with different diameters of tens to hundreds nanometers appeared on the surface of cell membrane. It can be concluded that AFM may be widely applied in clinic pathological inspection.

Two significant problems in content based retrieval methods are (1) Accuracy: most of the current content based image retrieval methods have not been quantitatively compared nor benchmarked with respect to accuracy and (2) Efficiency: image database search methods must be analyzed for their computational efficiency and interrelationships. We assert that the accuracy problem is due to the generality of the applications involved. In the current systems, the goal of the user is not clear, which results in difficulties in creating ground truth. In this paper, we quantitatively compare and evaluate four fundamentally different methods for image copy location, namely, optimal keys, texture, projection, and template methods in a large portrait database. We discuss some important theoretical interrelationships, computational efficiency, and accuracy with respect to real noise experiments.

A sum of exponentials of the form f(x) = exp(2πiN₁x) + exp(2πiN₂x) + … + exp (2πiN_mx), where the N_k are distinct integers is called an idempotent trigonometric polynomial or, simply, an idempotent. It is known that for every p > 1, and every set S of the torus 𝕋 = ℝ/ℤ with │S│ > 0, there are idempotents concentrated on S in the L^p sense. We sketch how this concentration phenomenon originated as a reformulation of a functional analysis problem, and, in turn, studying concentration led to some interesting questions about L^p norms of Dirichlet kernels associated with multiple trigonometric series. Some counterexamples involving linear operators not of convolution type are given.

This article explains the statistical and computational methodology used to analyze species abundances collected using the LNBL Phylochip in a study of Irritable Bowel Syndrome (IBS) in rats.

Some tools already available for the analysis of ordinary microarray data are useful in this type of statistical analysis. For instance in correcting for multiple testing we use Family Wise Error rate control and step-down tests (available in the multtest package). Once the most significant species are chosen we use the hypergeometric tests familiar for testing GO categories to test specific phyla and families.

We provide examples of normalization, multivariate projections, batch effect detection and integration of phylogenetic covariation, as well as tree equalization and robustification methods.