Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).

A splitting result for the algebraic K-theory of projective toric schemes

Suppose X is a projective toric scheme defined over a ring R and equipped with an ample line bundle L . We prove that its K-theory has a direct summand of the form K(R)(k+1) where k = 0 is minimal such that L?(-k-1) is not acyclic. Using a combinatorial description of quasi-coherent sheaves we interpret and prove this result for a ring R which is either commutative, or else left noetherian.

An Investigation of the Flow Field through a Variable Geometry Turbine Stator with Vane Endwall Clearance

Variable geometry turbines provide an extra degree of flexibility in air management in turbocharged engines. The pivoting stator vanes used to achieve the variable turbine geometry necessitate the inclusion of stator vane endwall clearances. The consequent leakage flow through the endwall clearances impacts the flow in the stator vane passages and an understanding of the impact of the leakage flow on stator loss is required. A numerical model of a typical variable geometry turbine was developed using the commercial CFX-10 computational fluid dynamics software, and validated using laser doppler velocimetry and static pressure measurements from a variable geometry turbine with stator vane endwall clearance. Two different stator vane positions were investigated, each at three different operating conditions representing different vane loadings. The vane endwall leakage was found to have a significant impact on the stator loss and on the uniformity of flow entering the turbine rotor. The leakage flow changed considerably at different vane positions and flow incidence at vane inlet was found to have a significant impact.

Triply differential single ionization cross sections in coplanar and non-coplanar geometry for fast heavy ion-atom collisions

We have performed a kinematically complete experiment and calculations on single ionization in 100 MeV/amu C6+ + He collisions. For electrons ejected into the scattering plane (defined by the initial and final projectile momentum vectors) our first- and higher-order calculations are in good agreement with the data. In the plane perpendicular to the scattering plane and containing the initial projectile axis a strong forward-backward asymmetry is observed. In this plane both the first-order and the higher-order calculations do not provide good agreement neither with the data nor amongst each other.

An analytic geometry-variant approach to line ratio enhancement above the optically thin limit

We describe a simple theoretical model to investigate the anomalous effects of opacity on spectral line ratios, as previously studied in elements such as Fe XV and Fe XVII. The model developed is general: it is not specific to a particular atomic system, thus giving applicability to a number of coronal and chromospheric plasmas; furthermore, it may be applied to a variety of astrophysically relevant geometries. The analysis is underpinned by geometrical arguments, and we outline a technique for it to be used as a tool for the explicit diagnosis of plasma geometry in distant astrophysical objects.

Geometry- and diffraction-independent ionization probabilities in intense laser fields: Probing atomic ionization mechanisms with effective intensity matching

We report an experimental technique for the comparison of ionization processes in ultrafast laser pulses irrespective of pulse ellipticity. Multiple ionization of xenon by 50 fs 790 nm, linearly and circularly polarized laser pulses is observed over the intensity range 10 TW/cm(2) to 10 PW/cm(2) using effective intensity matching (EIM), which is coupled with intensity selective scanning (ISS) to recover the geometry-independent probability of ionization. Such measurements, made possible by quantifying diffraction effects in the laser focus, are compared directly to theoretical predictions of multiphoton, tunnel and field ionization, and a remarkable agreement demonstrated. EIM-ISS allows the straightforward quantification of the probability of recollision ionization in a linearly polarized laser pulse. Furthermore, the probability of ionization is discussed in terms of the Keldysh adiabaticity parameter gamma, and the influence of the precursor ionic states present in recollision ionization is observed.