An index theorem for Toeplitz operators on totally ordered groups

We show that for every totally ordered group and invertible function which does not have a logarithm, there is a representation in which the Toeplitz operator is a Breuer-Fredholm operator with nonzero index; this representation is the GNS-representation associated to a natural unbounded trace on the Toeplitz algebra .

     

Dualities and reciprocities on graphs on surfaces

We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally bi-walkable orientations. In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations. Furthermore, we define the balanced flow polynomial which is connected with tension polynomial by duality and with totally bi-walkable orientations by reciprocity.     

Issues with the Smith–Wilson method

We analyse various features of the Smith–Wilson method used for discounting under the EU regulation Solvency II, with special attention to hedging. In particular, we show that all key rate duration hedges of liabilities beyond the Last Liquid Point will be peculiar. Moreover, we show that there is a connection between the occurrence of negative discount factors and singularities in the convergence criterion used to calibrate the model. The main tool used for analysing hedges is a novel stochastic representation of the Smith–Wilson method.     

Shellable graphs and sequentially Cohen-Macaulay bipartite graphs

Associated to a simple undirected graph G is a simplicial complex ΔG whose faces correspond to the independent sets of G. We call a graph G shellable if ΔG is a shellable simplicial complex in the non-pure sense of Björner-Wachs. We are then interested in determining what families of graphs have the property that G is shellable. We show that all chordal graphs are shellable. Furthermore, we classify all the shellable bipartite graphs; they are precisely the sequentially Cohen-Macaulay bipartite graphs. We also give a recursive procedure to verify if a bipartite graph is shellable. Because shellable implies that the associated Stanley-Reisner ring is sequentially Cohen-Macaulay, our results complement and extend recent work on the problem of determining when the edge ideal of a graph is (sequentially) Cohen-Macaulay. We also give a new proof for a result of Faridi on the sequentially Cohen-Macaulayness of simplicial forests.     

A protocol for the measurement of all the parameters of the mass transfer kinetics in columns used in liquid chromatography

Band broadening in chromatography results from the combination of the dispersive effects that are associated with the different steps involved in the migration of compound bands along the column. These steps include longitudinal diffusion, trans-particle mass transfer, external film mass transfer, overall eddy diffusion, including trans-column, short-range inter-channel, trans-channel eddy diffusion, and the possible, additional mass transfer contributions arising from heat friction and the thermal heterogeneity of the column. We describe a series of experiments that provide the data needed to determine the coefficients of the contributions to band broadening of each one of these individual mass transfer steps. This specifically designed protocol can provide key information regarding the kinetic performance of columns used in liquid chromatography and explain why different columns behave so differently. The limitations, accuracy and precision of these methods are discussed. Further avenues of research that could improve the characterization of the mass transfer mechanisms in chromatographic columns, possibly contributing to the development of better columns, are suggested.     

Embeddings with multiple regularity

We introduce (k,l)-regular maps, which generalize two previously studied classes of maps: affinely k-regular maps and totally skew embeddings. We exhibit some explicit examples and obtain bounds on the least dimension of a Euclidean space into which a manifold can be embedded by a (k,l)-regular map. The problem can be regarded as an extension of embedding theory to embeddings with certain non-degeneracy conditions imposed, and is related to approximation theory.     

GEVREY-HYPOELLIPTICITY FOR A CLASS OF TOTALLY CHARACTERISTIC OPERATORS

1IntroductionWeconsidertlietangentialhypoellipticityforaclassoftota-llycl1aracteristicoperatorsinGevreyclass-wherezERndenotesthetangentialvariableandt3oistl1enormalvariable(withtl1eboundaryatt=o),theoperator(1.1)isnoteutirelyhypoelliptic(seeI1]),themaintroublearisesfromthenormalvariablet,becausetheopterator(1.1)l1assinguIarityatf=o.Somanyauthorscon8iderthetangentialhypoellipticity(tl1ati8,thehypoellipticityinx).Therearemanyresultsabouttheellipticoperatorinthisaspect(8ee[2-6]).ChenHuastudied…     

Embedding partial totally symmetric quasigroups

This paper concerns the embedding problem for partial totally symmetric quasigroups. For all n?9, it is shown that any partial totally symmetric quasigroup of order n can be embedded in a totally symmetric quasigroup of order v if v is even and v?2n+4, and this is the best possible such inequality.     

New families of atomic Latin squares and perfect 1-factorisations

A perfect 1-factorisation of a graph G is a decomposition of G into edge disjoint 1-factors such that the union of any two of the factors is a Hamiltonian cycle. Let p?11 be prime. We demonstrate the existence of two non-isomorphic perfect 1-factorisations of K_p+1 (one of which is well known) and five non-isomorphic perfect 1-factorisations of K_p,p. If 2 is a primitive root modulo p, then we show the existence of 11 non-isomorphic perfect 1-factorisations of K_p,p and 5 main classes of atomic Latin squares of order p. Only three of these main classes were previously known. One of the two new main classes has a trivial autotopy group.