Rectilinearization of semi-algebraic p-adic sets and Denef's rationality of Poincaré series

In [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine Angew. Math. 540 (2001) 105-114], it is shown that a p-adic semi-algebraic set can be partitioned in such a way that each part is semi-algebraically isomorphic to a Cartesian product where the sets R(k) are very basic subsets of Qp. It is suggested in [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine Angew. Math. 540 (2001) 105-114] that this result can be adapted to become useful to p-adic integration theory, by controlling the Jacobians of the occurring isomorphisms. In this paper we show that the isomorphisms can be chosen in such a way that the valuations of their Jacobians equal the valuations of products of coordinate functions, hence obtaining a kind of explicit p-adic resolution of singularities for semi-algebraic p-adic functions. We do this by restricting the used isomorphisms to a few specific types of functions, and by controlling the order in which they appear. This leads to an alternative proof of the rationality of the Poincaré series associated to the p-adic points on a variety, as proven by Denef in [J. Denef, The rationality of the Poincaré series associated to the p-adic points on a variety, Invent. Math. 77 (1984) 1-23].     

A new facility for time-resolved PIV measurements in rotating channels

A new facility to measure the time evolution of 2D velocity fields in a rotating channel is presented, and the accuracy is discussed in detail. Measurements are made by means of a time-resolved PIV system composed of a continuous laser diode, coupled by a fiber optics cable to a laser plane optical module, and a CMOS high-speed camera. Both the PIV system and divergent channel are fixed on a 2.5 m rotating disk. This allows a direct measurement of the relative velocity of flows with Reynolds numbers between 3 × 10³ and 3 × 10⁴ and Rotation numbers between 0.0 and 0.52. These values correspond to the flow conditions in small radial impellers and can be independently adjusted by a change of the relative flow velocity and RPM. It is shown that this new facility allows high signal-to-noise ratios, and that the direct acquisition of the data in a rotating frame drastically reduces the measurement error. The accuracy and high spatial and temporal resolution of the measurements allow a detailed analysis of boundary layer characteristics in stationary and rotating conditions.     

CFB cyclones at high temperature: Operational results and design assessment

Pressure drop and cut size measurements are reported for a full scale cyclone operating within a 58 MWth CFB-combustor unit at 775℃.The paper reviews the vast number of equations to calculate the pressure drop and separation efficiency of cyclones, generally for operation at ambient temperature and at low Cs[0.5]. None of the literature correlations predicts the pressure drop with a fair accuracy within the range of experimental operating conditions. The cut size d50 can be estimated using direct empirical methods or using the Stokes number, Stk50. Both methods were used to compare measured and predicted values of d50. With the exception of Muschelknautz and Krambrock, none of the equations made accurate predictions.Finally, an alternative method to determine the friction factor of the pressure drop equation (Euler number, Eu) and of the cut size is proposed. The Eu number is determined from the geometry of common cyclones, and the derived value of Stk50 defines more accurate cut sizes. The remaining discrepancy of less than 5%, when compared with the measured values, is tentatively explained in terms of a reduced cyclone diameter due to the solids layer formed near its wall. Further measurements, mostly using positron emission particle tracking, elucidate the particle motion in the cyclone and both tracking results and the influence of the particle movement on Eu and Stk50 will be discussed in a follow-up paper.     

Intermolecular Order of Poly-(2,5-dimethyl-para-phenylene vinylene) and Poly-(para-phenylene vinylene) – A Comparison

Summary.  Poly-(para-phenylene vinylene) (2b, PPV) and poly-(2,5-dimethyl-para-phenylene vinylene) (1b) were prepared via the formation of a double bond by thermal elimination of an octylsulfinyl side group at 200°C under vacuum. X-Ray diffraction experiments revealed a certain degree of long-range order within both polymers. However, considerable differences in the diffraction pattern were observed, the packing of the polymer chains within the bulk structure being responsible for these differences. Within the crystal structure of 2b, neighbouring PPV chains pack in a zig-zag arrangement (herringbone pattern). Analysis of the diffraction pattern of 1b on the basis of the crystal structure of a model compound (2,5,2′,5′-tetramethylstilbene) revealed that the molecular planes of neighbouring polymer chains pack parallel to each other. A model for the crystal structure of 1b is given. The change of packing from a herringbone to a parallel arrangement of the molecular planes is related to the introduction of methyl groups as side chains to PPV. Received December 14, 2000. Accepted (revised) December 20, 2000     

Practical Preparation of β‐Amino‐α‐Hydroxy Diesters: Key Intermediates for 1,4‐Oxazin‐2‐Ones

In the presence of a catalytic amount of amino acid hydrochloride, trans‐β‐phenylglycidic ester undergoes ring opening with high stereo‐ and regioselectivity when treated with glycine esters. Alkylation of the resulting β‐amino‐α‐hydroxy diesters with benzyl bromide, followed by cyclization to furnish the expected 1,4‐oxazin‐2‐ones, is also described.     

On tridiagonal matrices unitarily equivalent to normal matrices

In this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied.It is well-known that any matrix is unitarily equivalent to a tridiagonal matrix. In case of a normal matrix the resulting tridiagonal inherits a strong relation between its super- and subdiagonal elements. The corresponding elements of the super- and subdiagonal will have the same absolute value.In this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied. Both an iterative reduction based on Krylov sequences as a direct tridiagonalization procedure via Householder transformations are reconsidered. This equivalence transformation is then applied to the normal case and equality of the absolute value between the super- and subdiagonals is proved. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved. Properties when applying the reduction on symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices and their relations with, e.g., complex symmetric and pseudo-symmetric matrices are presented.It is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the Takagi factorization. Finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given.     

Lipschitz Continuity Properties for p−Adic Semi-Algebraic and Subanalytic Functions

We prove that a (globally) subanalytic function ${f : X \subset {\bf Q}^{n}_{p} \rightarrow {\bf Q}_{p}}We prove that a (globally) subanalytic function f : X ì Qⁿ_p ? Q_p{f : X \subset {\bf Q}^{n}_{p} \rightarrow {\bf Q}_{p}} which is locally Lipschitz continuous with some constant C is piecewise (globally on each piece) Lipschitz continuous with possibly some other constant, where the pieces can be taken to be subanalytic. We also prove the analogous result for a subanalytic family of functions f_y : X_y ì Qⁿ_p ? Q_p{f_{y} : X_{y} \subset {\bf Q}^{n}_{p} \rightarrow {\bf Q}_{p}} depending on p−adic parameters. The statements also hold in a semi-algebraic set-up and also in a finite field extension of Q _p . These results are p−adic analogues of results of K. Kurdyka over the real numbers. To encompass the total disconnectedness of p−adic fields, we need to introduce new methods adapted to the p−adic situation.     

Graded Calabi Yau algebras of dimension 3

In this paper, we prove that Graded Calabi Yau algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d, the set of good superpotentials of degree d, i.e. those that give rise to Calabi Yau algebras, is either empty or almost everything (in the measure theoretic sense). We also give some constraints on the structure of quivers that allow good superpotentials, and for the simplest quivers we give a complete list of the degrees for which good superpotentials exist.     

Analytic -adic cell decomposition and integrals

We prove a conjecture of Denef on parameterized -adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.