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Flame ball with thermally sensitive intermediate kinetics
No full textSpherical flame balls are studied using a model for the chemical kinetics which involves a non-exothermic autocatalytic reaction, describing the chain-branching generation of a chemical radical and an exothermic completion reaction, the rate of which does not depend on temperature. When the chain-branching reaction has a large activation temperature, an asymptotic structure emerges in which the branching reaction generates radicals and consumes fuel at a thin flame interface, although heat is produced and radicals are consumed on a more distributed scale. Another model, based more simply, but less realistically, on the generation of radicals by decomposition of the fuel, provides exactly the same leading order matching conditions. These can be expressed in terms of jump conditions across a reaction sheet that are linear in the dependent variables and their normal gradients. Using these jump conditions, a reactive–diffusive model with linear heat loss then leads to analytical solutions that are multivalued for small enough levels of heat loss, having either a larger or a smaller radius of the interface where fuel is consumed. The same properties are found, numerically, to persist as the activation temperature of the branching reaction is reduced to values that seem to be typical for hydrocarbon chemistry. Part of the solution branch with larger radius is shown to become stable for low enough values of the Lewis number of the fuel
Common tranversals and tangents to two lines and two quadrics in P
No full textWe solve the following geometric problem, which arises in several {\mbox three-dimensional} applications in computational geometry: For which arrangements of two lines and two spheres in are there infinitely many lines simultaneously transversal to the two lines and tangent to the two spheres? We also treat a generalization of this problem to projective quadrics. Replacing the spheres in by quadrics in projective space , and fixing the lines and one general quadric, we give the following complete geometric description of the set of (second) quadrics for which the two lines and two quadrics have infinitely many transversals and tangents: in the nine-dimensional projective space of quadrics, this is a curve of degree 24 consisting of 12 plane conics, a remarkably reducible variety
Generating H*(BO(3), F2) as a module over the Steenrod algebra
No full textThis paper continues the investigation of the hit problem, started in [5], for the algebra of symmetric polynomials B(n) viewed as a left -module graded by degree, where denotes the Steenrod algebra over the field of two elements {\bb F}_2. We recall that a homogeneous element f of grading d in a graded left -module M is hit if there is a hit equation in the form of a finite sum , where the homogeneous elements hk in M have grading less than d and the Sqk are the Steenrod squares, which generate . We denote by Q = Q(M) = {\bb F}_2\otimes_{\cal A} M the quotient of the module M by the hit elements, where {\bb F}_2 is here viewed as a right -module concentrated in grading 0. Then Q is a graded vector space over {\bb F}_2 and a basis for Q lifts to a minimal generating set for M as a module over . The hit problem is to find minimal generating sets for M and criteria for elements to be hit. We recall that B(n) = {\bb F}_{2}\[\sigma_1,\ldots,\sigma_n\] is the polynomial subalgebra of P(n) = {\bb F}_{2}\[x_1,\ldots,x_n\] generated by the elementary symmetric functions in the variables xj . In particular, . The algebras P(n) and B(n) realize respectively the cohomology of the product of n copies of infinite real projective space and the cohomology of the classifying space BO(n) of the orthogonal group O(n) over {\bb F}_2, where the usual grading in cohomology corresponds to degree in the polynomial algebra. The ideal M(n) in B(n), generated by , can be identified with the cohomology H*(MO(n), {\bb F}_2) of the Thom space MO(n) in positive dimensions. It is also convenient to introduce the notation L(n) for the polynomials in P(n) divisible by . Topologically, L(n) corresponds in positive degrees to the cohomology of the n-fold smash product of infinite real projective space. From the topological point of view, is the algebra of universal stable operations in ordinary cohomology with {\bb F}_2 coefficients and this explains the action of on P(n), L(n), B(n) and M(n). However, the whole subject may be treated in a purely algebraic fashion [5, 11]
Extensions to three-dimensional flow in a porous channel
No full textWe consider the ow of a viscous, incompressible uid contained between two parallel, porous walls. The
ow is driven by a spatially uniform injection/suction of uid through the bounding walls. We extend the
solution structure of previous investigations to a more general three-dimensional stagnation-point form which
can capture a whole range of phenomena in a single class of states. In particular, we show that this form of
solution contains states previously discussed under more restrictive assumptions on the ow ÿeld. We show
that a range of two- and three-dimensional states exist, together with symmetry-broken solutions and periodic
states. We discuss the stability of these states and relate the previous results of Drazin, Banks, Zaturska and
co-workers to those of Goldshtik and Javorsky on the �bifurcation to swirl� and of Hewitt and Duck on
non-axisymmetric von Karman fows
Openness of momentum maps and persistence of extremal relative equilibria
No full textWe prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e. images of G-invariant open sets are open). As an application we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries, extremal relative equilibria persist for every perturbation of the value of the momentum map, provided the isotropy subgroup of this value is compact. We also demonstrate how this persistence result applies to an example of ellipsoidal figures of rotating fluid. We also provide an example with plane point vortices which shows how the compactness assumption is related to persistence
Nilpotent commuting varieties of reductive Lie algebras
Get PDFLet G be a connected reductive algebraic group over an algebraically closed field k of characteristic pge0, and gfr=LiethinspG. In positive characteristic, suppose in addition that p is good for G and the derived subgroup of G is simply connected. Let Nscr=Nscr(gfr) denote the nilpotent variety of gfr, and Cfrnil(gfr):={(x,y)isinNscr×Nscrthinsp|thinspthinsp[x,y]=0}, the nilpotent commuting variety of gfr. Our main goal in this paper is to show that the variety Cfrnil(gfr) is equidimensional. In characteristic 0, this confirms a conjecture of Vladimir Baranovsky; see [2]. When applied to GL(n), our result in conjunction with an observation in [2] shows that the punctual (local) Hilbert scheme hamilt n subHilb n (Popf2) is irreducibl
Duality and Hermitian Galois Module Structure
No full textSuppose is either the ring of integers of a number field, the ring of integers of a -adic local field, or a field of characteristic . Let be a regular projective scheme which is flat and equidimensional over of relative dimension . Suppose is a finite group acting tamely on . Define to be the Hermitian class group of . Using the duality pairings on the de Rham cohomology groups of the fiber of over , we define a canonical invariant in . When and is either , or , we determine the image of in the adelic Hermitian classgroup by means of -constants. We also show that in this case, the image in of a closely related Hermitian Euler characteristic both determines and is determined by the -constants of the symplectic representations of
Spectra of Bernoulli convolutions as multipliers in Lp on the circle
No full textIt is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution μθ parameterized by a Pisot number θ is countable. Combined with results of R. Salem and P. Sarnak, this proves that for every fixed θ>1 the spectrum of the convolution operator f\mapsto \mu\sb \theta\ast f in Lp(S1) (where S1 is the circle group) is countable and is the same for all , namely, \overline{\{\widehat {\mu\sb \theta}(n) : n\in \mathbb {Z}\}}. Our result answers the question raised by Sarnak in [8]. We also consider the sets \overline{\{\widehat {\mu\sb \theta}(rn) : n\in \mathbb {Z}\}} for r >0 which correspond to a linear change of variable for the measure. We show that such a set is still countable for all but uncountable (a nonempty interval) for Lebesgue-a.e. r>0