On a class of stochastic partial differential equations related to turbulent transport

Summary. We consider the Cauchy problem for the mass density ρ of particles which diffuse in an incompressible fluid. The dynamical behaviour of ρ is modeled by a linear, uniformly parabolic differential equation containing a stochastic vector field. This vector field is interpreted as the velocity field of the fluid in a state of turbulence. Combining a contraction method with techniques from white noise analysis we prove an existence and uniqueness result for the solution ρ∈C ^1,2([0,T]×ℝ ^d ,(S)^*), which is a generalized random field. For a subclass of Cauchy problems we show that ρ actually is a classical random field, i.e. ρ(t,x) is an L ²-random variable for all time and space parameters (t,x)∈[0,T]×ℝ ^d . Received: 27 March 1995 / In revised form: 15 May 1997     

Instability of a planar liquid layer in an alternating longitudinal magnetic field with non-zero mean

The conducting liquid interface is found to undulate in an alternating magnetic field. It was shown earlier that ifM =B ₀ ²/μηω, B₀, ω, μ andη being the amplitude (complex) of the alternating longitudinal magnetic field imposed at the interface, the angular frequency of the field, the magnetic permeability and the viscosity respectively, and ifM _c was the critical value ofM then the planar layer was stable or unstable according asM < M _c orM > M _c. In this paper we have determined the stability criterion when in addition to the alternating longitudinal field there acts a uniform field in the same direction. After comparing our results with those obtained earlier, in the absence of the uniform field, we find that the additional uniform field has a significant destabilizing effect.     

Models of Peano Arithmetic and a question of Sikorski on ordered fields

Using models of Peano Arithmetic, we solve a problem of Sikorski by showing that the existence of an ordered field of cardinalityλ with the Bolzano-Weierstrass property forκ-sequences is equivalent to the existence of aκ-tree with exactlyλ branches and with noκ-Aronszajn subtrees. Supported in part by NSF Grant MCS-8301603.     

Stability of the Reeb vector field of H-contact manifolds

It is well known that a Hopf vector field on the unit sphere S ²ⁿ⁺¹ is the Reeb vector field of a natural Sasakian structure on S ²ⁿ⁺¹. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction. Supported by funds of the University of Lecce and M.I.U.R.(PRIN).     

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\p ^ℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall. Received: 28 May 1998 / Revised version: 20 December 1998     

Graded Poisson structures on the algebra of differential forms

We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplectic forms on ΩM) for which the exterior derivative is a Hamiltomian graded vector field. Finally, we determine the topological obstructions to the possibility of obtaining all odd symplectic forms with this property as the image by the pullback of an automorphism of Ω(M) of a graded symplectic form of degree 1 with respect to which the exterior derivative is a Hamiltonian graded vector field.     

Σ-Definability of countable structures over real numbers, complex numbers, and quaternions

We study Σ-definability of countable models over hereditarily finite {ie193-01} superstructures over the field ℝ of reals, the field ℂ of complex numbers, and over the skew field ℍ of quaternions. In particular, it is shown that each at most countable structure of a finite signature, which is Σ-definable over {ie193-02} with at most countable equivalence classes and without parameters, has a computable isomorphic copy. Moreover, if we lift the requirement on the cardinalities of the classes in a definition then such a model can have an arbitrary hyperarithmetical complexity, but it will be hyperarithmetical in any case. Also it is proved that any countable structure Σ-definable over {ie193-03}, possibly with parameters, has a computable isomorphic copy and that being Σ-definable over {ie193-04} is equivalent to being Σ-definable over {ie193-05}. Supported by RFBR-DFG, grant No. 06-01-04002 DFGa. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 335–363, May–June, 2008.     

A generalization of maillet and demyanenko determinants for the Cyclotomic Zp-Extension

LetK be an imaginary abelian number field. By means of a generalization of Maillet and Demyanenko determinants we give a relative class number formula for an intermediate field of the cyclotomic ℤp-extension ofK. The degree of the generalized determinant is a half of the degree ofK over ℚ.     

Euler–Poincaré formula in equal characteristic¶under ordinariness assumptions

Let X be an irreducible smooth projective curve over an algebraically closed field of characteristic p>0. Let ? be either a finite field of characteristic p or a local field of residue characteristic p. Let F be a constructible étale sheaf of $\BF$-vector spaces on X. Suppose that there exists a finite Galois covering π:Y→X such that the generic monodromy of π^* F is pro-p and Y is ordinary. Under these assumptions we derive an explicit formula for the Euler–Poincaré characteristic χ(X,F) in terms of easy local and global numerical invariants, much like the formula of Grothendieck–Ogg–Shafarevich in the case of different characteristic. Although the ordinariness assumption imposes severe restrictions on the local ramification of the covering π, it is satisfied in interesting cases such as Drinfeld modular curves. Received: 7 December 1999 / Revised version: 28 January 2000     

Einstein–Weyl structures on contact metric manifolds

In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W ^± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, f η) are presented.      

Symmetries and First Integrals of Differential Equations

It is known that an n-dimensional system of ordinary differential equations with Lie symmetry which involves a divergence-free Liouville vector field possesses n−1 independent first integrals (i.e., it is algebraically integrable) (ünal in Phys. Lett. A 260:352–359, [1999]). In the present paper, we show that if an n-dimensional system of ordinary differential equations admits a C ^∞-symmetry vector field which satisfies some special conditions, then it also possesses n−1 independent first integrals. Several examples are given to illustrate our result. Y. Li’s research was partially supported by NSFC Grants 10531050, 10225107, SRFDP Grant 20040183030, and the 985 program of Jilin University.     

The abc-conjecture for Algebraic Numbers

The abe-conjecture for the ring of integers states that, for every ε 〉 0 and every triple of relatively prime nonzero integers （a, b, c） satisfying a ＋ b = c, we have max（｜a｜, ｜b｜, ｜c｜） 〈 rad（abc）^1＋ε with a finite number of exceptions. Here the radical rad（m） is the product of all distinct prime factors of m. In the present paper we propose an abe-conjecture for the field of all algebraic numbers. It is based on the definition of the radical （in Section 1） and of the height （in Section 2） of an algebraic number. From this abc-conjecture we deduce some versions of Fermat＇s last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat＇s equation in fields of small degrees over Q.     

Quotient rings induced via fuzzy ideals

This note we give a construction of a quotient ringR/μ induced via a fuzzy idealμ ina ringR. The Fuzzy First, Second and Third Isomorphism Theorems are established. For some applications of this construction of quotient rings, we show that ifμ is a fuzzy ideal of a commutative ringR, thenμ is prime (resp. maximal, primary) if and only ifR/μ is an integral domain (resp.R/μ is a field, every zero divisor inR/μ is nilpotent). Moreover we give a simpler characterization of fuzzy maximal ideal of a ring.     

Valuations on arithmetic surfaces

In this paper, we give the definition of the height of a valuation and the definition of the big field Cp,G, where p is a prime and GR is an additive subgroup containing 1. We conclude that Cp,G is a field and Cp,G is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m ≤n∈ Z, let Vm,n be an R-vector space of dimension n-m + 1, whose coordinates are indexed from m to n. We generalize the definition of Cp,G, where p i...     

On log canonical divisors that are log quasi-numerically positive

Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK _X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.     

Conjectures de Greenberg et extensions¶pro-p-libres d'un corps de nombres

For an algebraic number field k and a prime number p (if p=2, we assume that μ₄⊂k), we study the maximal rank ρ _p of a free pro-p-extension of k. This problem is related to deep conjectures of Greenberg in Iwasawa theory. We give different equivalent formulations of these conjectures and we apply them to show that, essentially, ρ _k =r ₂(k)+1 if and only if k is a so-called p-rational field. Received: 29 April 1999 / Revised version: 31 January 2000     

A characterization of hyperelliptic Jacobians

In this paper we will prove a criterion for hyperelliptic Jacobians. LetD be a translation invariant vector field on an indecompssable principally polarized abelian variety (i.p.p.a.v.) (X, Θ), letDΘ be the divisor of the sectionDΘ∈H ⁰ (Θ,O(Θ)|Θ). We have that (X, Θ) is the Jacobian of an hyperelliptic curve iff (Theorem 1) all the component ofDΘ are non reduced and the singular locus of Θ has dimension less thang-2. We will prove our theorem by showing that under the above geometrical condition it is possible to construct a Kodomcev-Petviashvili (K.P.) equation which is satisfied by the theta function corresponding to the principal polarization onX.     

A Note on Mascarenhas' Counterexample about Global Convergence of the Affine Scaling Algorithm

Mascarenhas gave an instance of linear programming problems to show that the long-step affine scaling algorithm can fail to converge to an optimal solution with the step-size λ=0.999 . In this note, we give a simple and clear geometrical explanation for this phenomenon in terms of the Newton barrier flow induced by projecting the homogeneous affine scaling vector field conically onto a hyperplane where the objective function is constant. Based on this interpretation, we show that the algorithm can fail for "any" λ greater than about 0.91 (a more precise value is 0.91071), which is considerably shorter than λ = 0.95 and 0.99 recommended for efficient implementations. Accepted 17 February 1998     

Numerical evaluation of gauge invariants for <Emphasis Type="Italic">a</Emphasis>-gauge solutions in open string field theory

We explore the vacuum structure in the bosonic open string field theory expanded near an identity-based solution parameterized by a (≥ −1/2). Analyzing the expanded theory using the level-truncation approximation up to the level 20, we find that the theory has the tachyon vacuum solution for a ≥ −1/2. We also find that at a = −1/2, there exists an unstable vacuum solution in the expanded theory and the solution is expected to be the perturbative open string vacuum. These results reasonably support the hypothesis that the identity-based solution is a trivial pure gauge configuration for a > −1/2, but it can be regarded as the tachyon vacuum solution at a = −1/2.