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-constants and equivariant Arakelov-Euler characteristics
Get PDFLet be a tame -cover of regular arithmetic varieties over with a finite group. Assuming that and have “tame” reduction we show how to determine the -constant in the conjectural functional equation of the Artin–Hasse–Weil function for a symplectic representation of from a suitably refined equivariant Arakelov–de Rham–Euler characteristic of . Our result may be viewed firstly as a higher dimensional version of the Cassou-Noguès–Taylor characterization of tame symplectic Artin root numbers in term of rings of integers with their trace form, and secondly as a signed equivariant version of Bloch's conductor formula
The cumulant process and Esscher's change of measure
Get PDFIn this paper two kinds of cumulant processes are studied in a general setting. These processes generalize the cumulant of an infinitely divisible random variable and they appear as the exponential compensator of a semimartingale. In a financial context cumulant processes lead to a generalized Esscher transform. We also provide some new criteria for uniform integrability of exponential martingales
A collection of benchmark examples for model reduction of linear time invariant dynamical systems.
Get PDFIn order to test the numerical methods for model reduction we present here a benchmark collection, which contain some useful real world examples reflecting current problems in applications. All simulations were obtained via Matlab and some SLICOT programs of Niconet
Continua of states in boundary-layer flows
No full textWe consider a class of three-dimensional boundary-layer flows, which may be viewed
as an extension of the Falkner�Skan similarity form, to include a cross-flow velocity
component, about a plane of symmetry. In general, this provides a range of three-
dimensional boundary-layer solutions, parameterized by a Falkner�Skan similarity
parameter, n, together with a further parameter, Ψ� , which is associated with a
cross-flow velocity component in the external flow. In this work two particular cases
are of special interest: for n = 0 the similarity equations possess a family of solutions
related to the Blasius boundary layer; for n = 1 the similarity solution provides an
exact reduction of the Navier�Stokes equations corresponding to the flow near a
saddle point of attachment. It is known from the work of Davey (1961) that in
this latter class of flow, a continuum of solutions can be found. The continuum
arises (in general) because it is possible to find states with an algebraic, rather than
exponential, behaviour in the far field. In this work we provide a detailed overview
of the continuum states, and show that a discrete infinity of �exponential modes� are
smoothly embedded within the �algebraic modes� of the continuum. At a critical value
of the cross-flow, these exponential modes appear as a cascade of eigensolutions to
the far-field equations, which arise in a manner analogous to the energy eigenstates
found in quantum mechanical problems described by the Schr �dinger equation.
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The presence of a discrete infinity of exponential modes is shown to be a generic
property of the similarity equations derived for a general n. Furthermore, we show
that there may also exist non-uniqueness of the continuum; that is, more than one
continuum of states can exist, that are isolated for fixed n and Ψ� , but which are
connected through an unfolded transcritical bifurcation at a critical value of the
cross-flow parameter, Ψ� .
The multiplicity of states raises the question of solution selection, which is addressed
using two stability analyses that assume the same basic symmetry properties as the
base flow. In one case we consider a steady, algebraic form in the �streamwise�
direction, whilst in the other a temporal form is assumed. In both cases it is possible
to extend the analysis to consider a continuous spectrum of disturbances that decay
algebraically in the wall-normal direction. We note some obvious parallels that exist
between such stability analyses and the approach to the continua of states described
earlier in the paper.
We also discuss the appearance of analogous non-unique states to the Falkner�Skan
equation in the presence of an adverse pressure gradient (i.e. n < 0) in an appendi
Global attractors of pinched skew products
Get PDFA class of skew products over irrational rotations of the circle is defined which contains some systems which have strange nonchaotic attractors. The global attractor of these systems is characterized: it lies between an upper semi-continuous curve and a lower semi-continuous curve. With additional assumptions on the class of maps considered more detail of the attractor can be given
Special transverse slices and their enveloping algebras
Get PDFLet G be a simple, simply connected algebraic group over Image, Image=Lie G, Image (Image) the nilpotent cone of Image, and (E,H,F) an ImageImage2-triple in Image. Let S=E+Ker ad F, the special transverse slice to the adjoint orbit Ω of E, and S0=S∩Image (Image). The coordinate ring Image[S0] is naturally graded (See Slodowy, "Simple Singularities and Simple Algebraic Groups," Lecture Notes in Mathematics, Vol. 815, Springer-Verlag, Berlin/Heidelberg/New York, 1980). Let Z(Image) be the centre of the enveloping algebra U(Image) and η:Z(Image)→Image an algebra homomorphism. Identify Image with Image* via a Killing isomorphism and let χ denote the linear function on Image corresponding to E. Following Kawanaka (Generalized Gelfand–Graev representations and Ennola duality, in "Algebraic Groups and Related Topics" Advanced Studies in Pure Mathematics, Vol. 6, pp. 175–206, North-Holland, Amsterdam/New York/Oxford, 1985), Moeglin (C.R. Acad. Sci. Paris, Ser. I 303 No. 17 (1986), 845–848), and Premet (Invent. Math.121 (1995), 79–117), we attach to χ a nilpotent subalgebra Imageχsubset ofImage of dimension (dim Ω)/2 and a 1-dimensional Imageχ-module Imageχ. Let ImageH" height="14" width="14">χ denote the algebra opposite to EndImage(U(Image)multiply sign in circleU(Imageχ)Imageχ) and ImageH" height="14" width="14">χ,η=ImageH" height="14" width="14">χmultiply sign in circleZ(Image)Imageη. It is proved in the paper that the algebra ImageH" height="14" width="14">χ,η has a natural filtration such that gr(ImageH" height="14" width="14">χ,η), the associated graded algebra, is isomorphic to Image[S0]. This construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of Image
Some observations on induction in predicate probabilistic reasoning
No full textWe consider the desirability, or otherwise, of various forms of induction in the light of certain principles and inductive methods within predicate uncertain reasoning. Our general conclusion is that there remain conflicts within the area whose resolution will require a deeper understanding of the fundamental relationship between individuals and properties
On the scattering of baroclinic Rossby waves by a ridge in a continuoisly stratified ocean
Get PDFIn global ocean dynamics Rossby waves play a vital rôle in the long-term distribution of vorticity; knowledge of the interaction between these waves and topography is crucial to a full understanding of this process, and hence to the transportation of energy, mixing and ocean circulation. The interaction of baroclinic Rossby waves with abrupt topography is the focus of this study. In this paper we model the ocean as a continuously stratified fluid for which the linear theory predicts a qualitatively different structure for the wave modes than that predicted by barotropic or simple layered models, even if most of the density variation is confined to the thermocline. We consider the scattering of a westward-propagating baroclinic Rossby wave by a narrow ridge on the ocean floor, modelled by a line barrier of infinite extent, orientated at an arbitrary angle to the incident wave. Transmission and reflection coefficients for the propagating modes are found using both an algebraic method and, in the case where this breaks down, matched asymptotic expansions. The results are compared with recent analyses of satellite altimetry data
Double-elliptic dynamical systems from generalized Mukai-Sklyanin algebras
No full textWe consider the double-elliptic generalisation of dynamical systems of Calogero–Toda–Ruijsenaars type using finite-dimensional Mukai–Sklyanin algebras. The two-body system, which involves an elliptic dependence both on coordinates and momenta, is investigated in detail and the relation with Nambu dynamics is mentioned. We identify the 2D complex manifold associated with the double elliptic system as an elliptically fibered rational ("1/2K3") surface. Some generalisations are suggested which provide the ground for a description of the N-body systems. Possible applications to SUSY gauge theories with adjoint matter in d=6 with two compact dimensions are discussed
Dual forms on supermanifolds and Cartan calculus
No full textWe introduce and study the complex of "stable forms" on supermanifolds. Stable forms on a supermanifold M are represented by Lagrangians of "copaths" (formal systems of equations, which may or may not specify actual surfaces) on M 2 ÂD. Changes of D give rise to stability isomorphisms. The resulting (direct limit) {Cartan-de Rham} complex made of stable forms extends both in positive and negative degree. Its positive half is isomorphic to the complex of forms defined as Lagrangians of paths, studied earlier. Including the negative half is crucial, in particular, for homotopy invariance. For stable forms we introduce (non-obvious) analogs of exterior multiplication by covectors and contraction with vectors and find the anticommutation relations that they obey. Remarkably, the version of the Clifford algebra so obtained is based on the super anticommutators rather than the commutators and (before stabilization) it includes some central element †. An analog of Cartan's homotopy identity is proved, which also contains this "stability operator" †