ADDITIVE FAMILIES OF INVARIANTS OF FORMS OF DEGREE d

Abstract

Let d be a positive integer, and F be a field of characteristic zero. Suppose that for each positive integer n, I _n, is a GL _n,(F)- invariant of forms of degree d in x₁, …, x _n, over F. We call {I _n} an additive family of invariants if I _p+q (f ⊥ g) = I _p(f).I _q(g) whenever f; g are forms of degree d over F in x _l, …, x _p; …, x _q respectively, and where (f ⊥ g)(x _l, …, x _p+q) = f(x ₁, …, x _p,) + g (x _p+1, …, x _p+q). It is well-known that the family of discriminants of the quadratic forms is additive. We prove that in odd degree d each invariant in an additive family must be a constant. We also give an example in each even degree d of a nontrivial family of invariants of the forms of degree d. The proofs depend on the symbolic method for representing invariants of a form, which we review.     

Indecomposable forms of higher degree

All nondegenerate indecomposable forms of higher degree over a perfect field k can be realized as traces of nondegenerate absolutely indecomposable forms of higher degree over a suitable algebraic field extension of k. With the help of trace forms of certain nonassociative algebras we construct classes of indecomposable forms of degree d≥3.     

u-Invariants for forms of higher degree

Both a general and a diagonal u-invariant for forms of higher degree are defined, generalizing the u-invariant of quadratic forms. We give a survey of both old and new results on these u-invariants.     

Random Diophantine equations,I

We consider additive Diophantine equations of degree k in s   variables and establish that whenever s?3k+2$s?3k+2$ then almost all such equations satisfy the Hasse principle. The equations that are soluble form a set of positive density, and among the soluble ones almost all equations admit a small solution. Our bound for the smallest solution is nearly best possible.     

Maximal Linear Subspaces of Strong Self-Dual 2-forms and the Bonan 4-form

The notion of self-duality of 2-forms in 4-dimensions plays an eminent role in many areas of mathematics and physics, but although the 2-forms have a genuine meaning related to curvature and gauge-field-strength in higher dimensions also, their “self-duality” is something which is almost avoided above 4-dimensions. We show that self-duality of 2-forms is a very natural notion in higher (even) dimensions also and we prove the equivalence of some scattered and rarely used definitions in the literature. We demonstrate the usefulness of this higher self-duality by studying it in 8-dimensions and we derive a natural expression for the Bonan form in terms of self-dual 2-forms and we give an explicit expression of the local action of SO(8) on the Bonan form.     

Higher trace forms and essential dimension in central simple algebras

We show that the essential dimension of a finite-dimensional central simple algebra coincides with the essential dimension of its r-linear trace form, for any r ≥ 3. Received: 15 March 2006     

高维拟线性双曲型方程组的对角化问题

本文讨论了高维一阶拟线性方程组可对角化的条件，并给出其在二维等熵流方程组、三维空气动力学方程组及具有旋转对称性的守恒律组等情形的应用；然后给出了高维拟线性对角型方程组Ｃａｕｃｈｙ问题存在整体经典解的一个充要条件，并给出其应用。     

DECOMPOSITION OF A HIGHER DEGREE FORM

ABSTRACT

We give another proof of Harrison's decomposition result,[2] Harrison, D.K. 1975. A Grothendieck Ring of Higher Degree Forms. Journal of Algebra, 35: 123–138. [Crossref], [Web of Science ®] , [Google Scholar] Prop. 2.3 for higher degree forms over a noetherian ring, exploiting an earlier introduction of the centre. We generalise to higher degree forms over a noetherian scheme: we extend the notion of centre; we prove a decomposition result; we extend Harrison's result,[2] Harrison, D.K. 1975. A Grothendieck Ring of Higher Degree Forms. Journal of Algebra, 35: 123–138. [Crossref], [Web of Science ®] , [Google Scholar] Prop. 4.3 on the behaviour of the centre under a flat base extension; and we improve his result,[2] Harrison, D.K. 1975. A Grothendieck Ring of Higher Degree Forms. Journal of Algebra, 35: 123–138. [Crossref], [Web of Science ®] , [Google Scholar] Prop. 4.2, giving conditions on the base scheme under which the centre of the tensor product of two higher degree forms is isomorphic to the tensor product of their centres.     

Forms Represented by Linear Forms

This paper is the third in a series in which the author investigates the question of representation of forms by linear forms. Whereas in the first two treatments the proportion of forms F of degree 3 (resp. degree d) which can be written as a sum of two cubes (resp. d-th powers) of linear forms with algebraic coefficients is determined, the generalization now consists in allowing more general expressions of degree d in two linear forms. The main result is thus to give an asymptotic formula, in terms of their height, for the number or decomposable forms that have a representation

Pattern formation for oscillatory bulk-mode competition in a two-layer Bénard problem

Two immiscible liquids lie between parallel plates and are heated from below. The focus is on the case where the interfacial mode is strongly stabilized by surface tension and a suitable density stratification. A mechanism for a Hopf bifurcation is the competition between the least stable of the bulk modes in each fluid. The well known criterion for balancing the effective Rayleigh numbers in both fluids is augmented with a criterion for non-self-adjointness of the system, yielding a heuristic method for picking suitable fluids when Hopf modes are desired. The pattern formation problem in three dimensions is addressed for the case of doubly periodic solutions on a hexagonal lattice. Of the solutions with maximal symmetry, the travelling rolls are found to be stable.Research supported by the Newton Institute (Cambridge), NSF Grant CTS-9307238 and ONR Grant N00014-92-J-1664. The author is grateful to D. C. Andereck and M. Renardy for discussions.     

Linear stability of radially-heated circular Couette flow with simulated radial gravity

The stability of circular Couette flow between vertical concentric cylinders in the presence of a radial temperature gradient is considered with an effective “radial gravity.” In addition to terrestrial buoyancy − ρg e _z we include the term − ρg _m f(r)e _r where g _m f(r) is the effective gravitational acceleration directed radially inward across the gap. Physically, this body force arises in experiments using ferrofluid in the annular gap of a Taylor–Couette cell whose inner cylinder surrounds a vertical stack of equally spaced disk magnets. The radial dependence f(r) of this force is proportional to the modified Bessel function K ₁(κr), where 2π/κ is the spatial period of the magnetic stack and r is the radial coordinate. Linear stability calculations made to compare with conditions reported by Ali and Weidman (J. Fluid Mech., 220, 1990) show strong destabilization effects, measured by the onset Rayleigh number R, when the inner wall is warmer, and strong stabilization effects when the outer wall is warmer, with increasing values of the dimensionless radial gravity γ = g _m /g. Further calculations presented for the geometry and fluid properties of a terrestrial laboratory experiment reveal a hitherto unappreciated structure of the stability problem for differentially-heated cylinders: multiple wavenumber minima exist in the marginal stability curves. Transitions in global minima among these curves give rise to a competition between differing instabilities of the same spiral mode number, but widely separated axial wavenumbers.     

Kato–Milne cohomology and polynomial forms

Given a prime number p, a field F with $\mathrm{char}(F)=p$ and a positive integer n, we study the class-preserving modifications of Kato–Milne classes of decomposable differential forms. These modifications demonstrate a natural connection between differential forms and p-regular forms. A p-regular form is defined to be a homogeneous polynomial form of degree p for which there is no nonzero point where all the order $p?1$ partial derivatives vanish simultaneously. We define a ${\stackrel{?}{C}}_{p,m}$ field to be a field over which every p-regular form of dimension greater than $p^m$ is isotropic. The main results are that for a ${\stackrel{?}{C}}_{p,m}$ field F, the symbol length of $H_p^2(F)$ is bounded from above by $p^{m?1}?1$ and for any $n??(m?1){\log }_2?(p)?+1$, $H_p^{n+1}(F)=0$.     

ON HOMOGENEOUS RIEMANN BOUNDARY VALUE PROBLEMS OF HIGHER DEGREE

0IntroductionThesolutionforlinearmemannboundaryvalueproblemsiswellkn...[1'2].Fornonlinearsimilarproblems,onlythoseofveryparticularformwereconsideredinliterature,e.g.,[3,4,5].Here,thefollowinghomogeneousRiemannboundaryvalueproblemofdegreenwillbediscussed:whereLisasmoothclosedcontourinthecomplexplane,orientedcounter-clockwiselywithitsinterior(eXterior)regionS (S--),aj(t)EH(L)(H5ldercontinuousonL)and.(z)issectionallyholomorphicwithjumpcurveL.Weshallrestrictourselvestothenormalcase,i.e.,ac(t)…     

TRIPLE INTERACTIONS OF CONORMAL WAVES FOR HIGHER ORDER SEMILINEAR HYPERBOLIC EQUATIONS

The interaction of three conormal waves for send-linear strictly hyperbolic equations of third oeder is considered. Let ∑i:i = 1,2,3, be smooth characteristic surfaces for P = Di(D^2i-△) intersecting transversally at the origin. Suppose that the solution u to Pu = f(t, x, y, D^αu), |α| ≤2 is conormal to ∑i:i = 1, 2, 3, for t < 0. The author uses Bony‘s second microlocalization techniques and commutator arguments to conclude that the new singularities a short time after the triple interaction lie on the surface of the light cone Γ over the origin plus the surfaces obtainedby flow-outs of the lines of intersection Γ （交集） ∑i and ∑i（交集） ∑j，i, j = 1, 2, 3.     

TRIPLE INTERACTIONS OF CONORMAL WAVES FOR HIGHER ORDER SEMILINEAR HYPERBOLIC EQUMIONS

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A note on number of orderings of higher level

The main aim of the paper is to find an explicit formula for the number of orderings of higher level. Among others, we discuss the relationship between the number of orderings of higher level and the number of orderings of level 1. We also construct a field with a given possible number of orderings of higher level. Received: 10 December 2004     

The asymptotics of neutral curve crossing in Taylor–Dean flow

The fluid flow between a pair of coaxial circular cylinders generated by the uniform rotation of the inner cylinder and an azimuthal pressure gradient is susceptible to both Taylor and Dean type instabilities. The flow can be characterised by two parameters: a measure of the relative magnitude of the rotation and pressure effects and a non-dimensional Taylor number. This work considers the small gap, large wavenumber limit for linear perturbations when the onset of the Taylor and Dean instabilities is concurrent. A consistent, matched asymptotic solution is found across the whole annular domain and identifies five regions of interest: two boundary adjustment regions and three internal critical points. Necessary conditions for the Taylor number and wavenumber are found and interpreted with reference to the suggestion of neutral curve kinks occurring at moderate wavelengths. Received: October 21, 2003; revised: November 11, 2004     

几类无解的高次不定方程

本文目的是证明几类高次不定方程的无解性。