Universal manifold pairings and positivity

Gluing two manifolds M_1 and M_2 with a common boundary S yields a closed manifold M. Extending to formal linear combinations x=Sum_i(a_i M_i) yields a sesquilinear pairing p= with values in (formal linear combinations of) closed manifolds. Topological quantum field theory (TQFT) represents this universal pairing p onto a finite dimensional quotient pairing q with values in C which in physically motivated cases is positive definite. To see if such a "unitary" TQFT can potentially detect any nontrivial x, we ask if is non-zero whenever x is non-zero. If this is the case, we call the pairing p positive. The question arises for each dimension d=0,1,2,.... We find p(d) positive for d=0,1, and 2 and not positive for d=4. We conjecture that p(3) is also positive. Similar questions may be phrased for (manifold, submanifold) pairs and manifolds with other additional structure. The results in dimension 4 imply that unitary TQFTs cannot distinguish homotopy equivalent simply connected 4-manifolds, nor can they distinguish smoothly s-cobordant 4-manifolds. This may illuminate the difficulties that have been met by several authors in their attempts to formulate unitary TQFTs for d=3+1. There is a further physical implication of this paper. Whereas 3-dimensional Chern-Simons theory appears to be well-encoded within 2-dimensional quantum physics, eg in the fractional quantum Hall effect, Donaldson-Seiberg-Witten theory cannot be captured by a 3-dimensional quantum system. The positivity of the physical Hilbert spaces means they cannot see null vectors of the universal pairing; such vectors must map to zero.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper53.abs.htm

Solutions of the Einstein-Dirac and Seiberg-Witten Monopole Equations

We present unique solutions of the Seiberg-Witten Monopole Equations in which the U(1) curvature is covariantly constant, the monopole Weyl spinor consists of a single constant component, and the 4-manifold is a product of two Riemann surfaces of genuses p_1 and p_2. There are p_1 -1 magnetic vortices on one surface and p_2 - 1 electric ones on the other, with p_1 + p_2 \geq 2 p_1 = p_2= 1 being excluded). When p_1 = p_2, the electromagnetic fields are self-dual and one also has a solution of the coupled euclidean Einstein-Maxwell-Dirac equations, with the monopole condensate serving as cosmological constant. The metric is decomposable and the electromagnetic fields are covariantly constant as in the Bertotti-Robinson solution. The Einstein metric can also be derived from a K\"{a}hler potential satisfying the Monge-Amp\`{e}re equations.Comment: 22 pages. Rep. no: FGI-99-

Brieskorn manifolds as contact branched covers of spheres

We show that Brieskorn manifolds with their standard contact structures are contact branched coverings of spheres. This covering maps a contact open book decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.Comment: 8 pages, 1 figur

Fake R^4's, Einstein Spaces and Seiberg-Witten Monopole Equations

We discuss the possible relevance of some recent mathematical results and techniques on four-manifolds to physics. We first suggest that the existence of uncountably many R^4's with non-equivalent smooth structures, a mathematical phenomenon unique to four dimensions, may be responsible for the observed four-dimensionality of spacetime. We then point out the remarkable fact that self-dual gauge fields and Weyl spinors can live on a manifold of Euclidean signature without affecting the metric. As a specific example, we consider solutions of the Seiberg-Witten Monopole Equations in which the U(1) fields are covariantly constant, the monopole Weyl spinor has only a single constant component, and the 4-manifold M_4 is a product of two Riemann surfaces Sigma_{p_1} and Sigma_{p_2}. There are p_{1}-1(p_{2}-1) magnetic(electric) vortices on \Sigma_{p_1}(\Sigma_{p_2}), with p_1 + p_2 \geq 2 (p_1=p_2= 1 being excluded). When the two genuses are equal, the electromagnetic fields are self-dual and one obtains the Einstein space \Sigma_p x \Sigma_p, the monopole condensate serving as the cosmological constant.Comment: 9 pages, Talk at the Second Gursey Memorial Conference, June 2000, Istanbu

Exotic Smoothness and Quantum Gravity

Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. Thus, the negative result of Duston (arXiv:0911.4068) was a surprise. A closer look into the semi-classical approach uncovered the implicit assumption of a close connection between geometry and smoothness structure. But both structures, geometry and smoothness, are independent of each other. In this paper we calculate the "smoothness structure" part of the path integral in quantum gravity assuming that the "sum over geometries" is already given. For that purpose we use the knot surgery of Fintushel and Stern applied to the class E(n) of elliptic surfaces. We mainly focus our attention to the K3 surfaces E(2). Then we assume that every exotic smoothness structure of the K3 surface can be generated by knot or link surgery a la Fintushel and Stern. The results are applied to the calculation of expectation values. Here we discuss the two observables, volume and Wilson loop, for the construction of an exotic 4-manifold using the knot $5_{2}$ and the Whitehead link $Wh$. By using Mostow rigidity, we obtain a topological contribution to the expectation value of the volume. Furthermore we obtain a justification of area quantization.Comment: 16 pages, 1 Figure, 1 Table subm. Class. Quant. Grav

A novel algorithm for dynamic student profile adaptation based on learning styles

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.E-learning recommendation systems are used to enhance student performance and knowledge by providing tailor- made services based on the students’ preferences and learning styles, which are typically stored in student profiles. For such systems to remain effective, the profiles need to be able to adapt and reflect the students’ changing behaviour. In this paper, we introduce new algorithms that are designed to track student learning behaviour patterns, capture their learning styles, and maintain dynamic student profiles within a recommendation system (RS). This paper also proposes a new method to extract features that characterise student behaviour to identify students’ learning styles with respect to the Felder-Silverman learning style model (FSLSM). In order to test the efficiency of the proposed algorithm, we present a series of experiments that use a dataset of real students to demonstrate how our proposed algorithm can effectively model a dynamic student profile and adapt to different student learning behaviour. The results revealed that the students could effectively increase their learning efficiency and quality for the courses when the learning styles are identified, and proper recommendations are made by using our method

Altered thymic differentiation and modulation of arthritis by invariant NKT cells expressing mutant ZAP70

Various subsets of invariant natural killer T (iNKT) cells with different cytokine productions develop in the mouse thymus, but the factors driving their differentiation remain unclear. Here we show that hypomorphic alleles of Zap70 or chemical inhibition of Zap70 catalysis leads to an increase of IFN-gamma-producing iNKT cells (NKT1 cells), suggesting that NKT1 cells may require a lower TCR signal threshold. Zap70 mutant mice develop IL-17-dependent arthritis. In a mouse experimental arthritis model, NKT17 cells are increased as the disease progresses, while NKT1 numbers negatively correlates with disease severity, with this protective effect of NKT1 linked to their IFN-gamma expression. NKT1 cells are also present in the synovial fluid of arthritis patients. Our data therefore suggest that TCR signal strength during thymic differentiation may influence not only IFN-gamma production, but also the protective function of iNKT cells in arthritis

Effect of Sun, Oven and Freeze-Drying on Anthocyanins, Phenolic Compounds and Antioxidant Activity of Black Grape (Ekşikara) (Vitis vinifera L.)

The aim of this study was to determine whether a change occurs in the phenolic compounds and antioxidant activity of grapes after drying. Grapes pre-treated with potassium hydroxide solution were dried using three different drying methods, namely freeze drying, oven drying and sun drying. The effectiveness of the drying methods was evaluated in terms of total phenolic content, antioxidant activity (ABTS, FRAP and DPPH), individual phenolics and anthocyanins. Losses in total phenolic content of the grapes were found to be 1.89, 20.26 and 46.79% for freeze-, oven- and sun-dried grapes respectively. The DPPH and ABTS antioxidant activities of the grapes decreased after drying by all three methods, while an increase was observed in the FRAP value of freeze-dried grapes compared to the fresh sample. No significant effect of drying methods was observed on the gallic acid, 2,5-dihydroxybenzoic acid and (-)-epigallocatechin gallate contents of the grapes. The highest levels of procyanidin B1, (+)-catechin, (-)-epicatechin, (-)-epicatechin gallate, chlorogenic acid, trans-resveratrol and rutin were determined in freeze-dried grapes. Sun and oven drying caused drastic decreases in all anthocyanins, while no loss of anthocyanin was observed in freeze-dried grapes. Sun drying was found as the most detrimental drying method for grapes in terms of phenolic compounds and antioxidant activity when compared to the other drying methods.