Integration of differential forms on manifolds with locally-finite variations

It is well known that one can integrate any compactly supported, continuous, differential n-form over n-dimensional C¹-manifolds in ℝ^m (m ≥ n). For n = 1, the integral may be defined over any locally rectifiable curve. Another generalization is the theory of currents (linear functionals on the space of compactly supported C^∞-differential forms). The theme of the article is integration of measurable differential n-forms over n-dimensional n C⁰-manifolds in ℝ^m with locally-finite n-dimensional variations (a generalization of locally rectifiable curves to dimension n > 1). The main result states that any such manifold generates an n-dimensional current in ℝ^m (i.e., any compactly supported C^∞ n-form can be integrated over a manifold with the properties mentioned above). Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 168–206.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

On the existence and approximation of zeroes

In this paper we consider the problem of finding zeroes of a continuous functionf from a convex, compact subsetU of ℝ ⁿ to ℝ ⁿ . In the first part of the paper it is proved thatf has a computable zero iff:C ⁿ →ℝ ⁿ satisfies the nonparallel condition for any two antipodal points on bdC ⁿ, i.e. if for anyx∈bdC ⁿ ,f(x)≠αf(−x), α≥0, holds. Therefore we describe a simplicial algorithm to approximate such a zero. It is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm, i.e. the number of times the algorithm switches from a face τ on bdC ⁿ to the face −τ. Therefore the index of a terminal simplex σ is defined which equals the local Brouwer degree of the function if σ is full-dimensional. In the second part of the paper the algorithm is used to generate possibly several approximate zeroes off. Two sucessive solutions may have both the same or opposite degrees, again depending on the number of reflection steps. By extendingf:U→ℝ ⁿ to a function g from a cube containingU to ℝ ⁿ , the procedure can be applied to any continuous functionf without having any information about the global and local Brouwer degrees a priori.     

Hausdorff measures of different dimensions are not Borel isomorphic

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (ℝ, B, H ^s ) and (ℝ, B, H ^t ) are not isomorphic if s ≠ t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of ℝ and H ^d is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss. To prove our result, we apply a random construction and show that for every Borel function ƒ: ℝ → ℝ and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that ƒ(C) has Hausdorff dimension ≤ d. We also prove this statement in a more general form: If A ⊂ ℝⁿ is Borel and ƒ: A → ℝ^m is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dim B = d·dim A and dim ƒ(B) ≤ d·dim ƒ (A). Partially supported by the Hungarian Scientific Research Fund grant no. T 49786.     

Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation

In this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second order neutral delay differential equation (NDDE)

Proof of the double bubble curvature conjecture

An area minimizing double bubble in ℝⁿ is given by two (not necessarily connected) regions which have two prescribed n-dimensional volumes whose combined boundary has least (n−1)-dimensional area. The double bubble theorem states that such an area minimizer is necessarily given by a standard double bubble, composed of three spherical caps. This has now been proven for n = 2, 3,4, but is, for general volumes, unknown for n ≥ 5. Here, for arbitrary n, we prove a conjectured lower bound on the mean curvature of a standard double bubble. This provides an alternative line of reasoning for part of the proof of the double bubble theorem in ℝ³, as well as some new component bounds in ℝⁿ.     

Non-oscillatory criteria for a class of second order non-linear forced neutral-delay differential equations

In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation

Mean ergodic theorems for bi-continuous semigroups

In this paper we study the main properties of the Cesàro means of bi-continuous semigroups, introduced and studied by Kühnemund (Semigroup Forum 67:205–225, 2003). We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in C _b (ℝ ^N ) and to evolution operators associated with nonautonomous second-order differential operators in C _b (ℝ ^N ) with time-periodic coefficients.     

Efficient Construction of Contact Coordinates for Partial Prolongations

Let V be a vector field distribution or Pfaffian system on manifold M. We give an efficient algorithm for the construction of local coordinates on M such that V may be locally expressed as some partial prolongation of the contact distribution C⁽¹⁾_q, on the first-order jet bundle of maps from ℝ to ℝ^q, q ≥ 1. It is proven that if V is locally equivalent to a partial prolongation of C⁽¹⁾_q, then the explicit construction of contact coordinates algorithmically depends upon the determination of certain first integrals in a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on M. The number of these first integrals that must be computed satisfies a natural minimality criterion. These results provide a full and constructive generalisation of the Goursat normal form from the theory of exterior differential systems.     

Germes de d\'etermination infinie dans des classes lisses

We give sufficient conditions of infinite determinacy (in the sense of Mather) with respect to right equivalence, in subrings of C ^∞ (ℝ ⁿ ,0) defined by estimates on successive derivatives, such as rings of Gevrey germs. In this quantitative setting, a defect (hidden in the classical C ^∞ case) appears between the regularity of equivalent germs and the regularity of local diffeomorphisms of (ℝⁿ,0) giving the equivalence. Our conditions yield precise estimates of this defect, related to some suitable Łojasiewicz exponents of critical loci. We also show that the result is sharp in general. Received: Received: 27 May 1998     

On the spaces C k (ℝ) ∩ L p (ℝ)

We show that the Fréchet-Sobolev spaces C(ℝ) ∩ L ^p (ℝ) and C ^k (ℝ) ∩ L ^p (ℝ) are not isomorphic for p ≠ 2 and k ∈ ℕ. Research supported by the Italian MURST.     

Construction of Gibbs measures for 1-dimensional continuum fields

We study 1-dimensional continuum fields of Ginzburg-Landau type under the presence of an external and a long-range pair interaction potentials. The corresponding Gibbs states are formulated as Gibbs measures relative to Brownian motion [17]. In this context we prove the existence of Gibbs measures for a wide class of potentials including a singular external potential as hard-wall ones, as well as a non-convex interaction. Our basic methods are: (i) to derive moment estimates via integration by parts; and (ii) in its finite-volume construction, to represent the hard-wall Gibbs measure on C(ℝ;ℝ⁺) in terms of a certain rotationally invariant Gibbs measure on C(ℝ;ℝ³).     

Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four

A class of degree four differential systems that have an invariant conic x ² + Cy ² = 1, C ∈ ℝ, is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems.     

Properties of <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth mappings with one-dimensional gradient range

We find necessary and sufficient conditions for a curve in ℝ ^m×n to be the gradient range of a C ¹-smooth function υ: Ω ⊂ ℝ ⁿ → ℝ ^m . We show that this curve has tangents in a weak sense; these tangents are rank 1 matrices and their directions constitute a function of bounded variation. We prove also that in this case v satisfies an analog of Sard’s theorem, while the level sets of the gradient mapping ▿υ: Ω → ℝ ^m×n are hyperplanes.     

Some properties of the distance function and a conjecture of De Giorgi

In the article [2] Ennio De Giorgi conjectured that any compact n-dimensional regular submanifold M of ℝ ^n+m ,moving by the gradient of the functional

Dini Set-Valued Directional Derivative in Locally Lipschitz Vector Optimization

The present paper studies the following constrained vector optimization problem: min  _C f(x), g(x)∈−K, h(x)=0, where f:ℝ ⁿ →ℝ ^m , g:ℝ ⁿ →ℝ ^p and h:ℝ ⁿ →ℝ ^q are locally Lipschitz functions and C⊂ℝ ^m , K⊂ℝ ^p are closed convex cones. In terms of the Dini set-valued directional derivative, first-order necessary and first-order sufficient conditions are obtained for a point x ⁰ to be a w-minimizer (weakly efficient point) or an i-minimizer (isolated minimizer of order 1). It is shown that, under natural assumptions (given by a nonsmooth variant of the implicit function theorem for the equality constraints), the obtained conditions improve some given by Clarke and Craven. Further comparison is done with some recent results of Khanh, Tuan and of Jiiménez, Novo.     

Adjoints of semigroups acting on vector-valued function spaces

LetT(t) be the translation group onY=C ₀(ℝ×K)=C ₀(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY ^⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y ^⊙=L ¹(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT ₀(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT ₀(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ ^⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C ₀(Ω;X) andL ^p (μ;X). This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful to the CWI and the Dutch National Science Foundation NWO for financial support.     

Injectivity of differentiable maps ℝ2 → ℝ2 at infinity

The main result given in Theorem 1.1 is a condition for a map X, defined on the complement of a disk D in ℝ² with values in ℝ², to be extended to a topological embedding of ℝ², not necessarily surjective. The map X is supposed to be just differentiable with the condition that, for some ε > 0, at each point the eigenvalues of the differential do not belong to the real interval (-ε,∞). The extension is obtained by restricting X to the complement of some larger disc. The result has important connections with the property of asymptotic stability at infinity for differentiable vector fields.     

Measured creatures

We prove that two basic questions on outer measure are undecidable. First we show that consistently every sup-measurable functionf: ℝ² → ℝ is measurable. The interest in sup-measurable functions comes from differential equations and the question for which functionsf: ℝ² → ℝ the Cauchy problemy′=f(x,y), y(x₀)=y₀ has a unique almost-everywhere solution in the classAC _t(ℝ) of locally absolutely continuous functions on ℝ. Next we prove that consistently every functionf: ℝ → ℝ is continuous on some set of positive outer Lebesgue measure. This says that in a strong sense the family of continuous functions (from the reals to the reals) is dense in the space of arbitrary such functions. For the proofs we discover and investigate a new family of nicely definable forcing notions (so indirectly we deal with nice ideals of subsets of the reals—the two classical ones being the ideal of null sets and the ideal of meagre ones). Concerning the method, i.e., the development of a family of forcing notions, the point is that whereas there are many such objects close to the Cohen forcing (corresponding to the ideal of meagre sets), little has been known on the existence of relatives of the random real forcing (corresponding to the ideal of null sets), and we look exactly at such forcing notions. The first author thanks The Hebrew University of Jerusalem for support during his visits to Jerusalem and the KBN (Polish Committee of Scientific Research) for partial support through grant 2P03A03114. The research of the second author was partially supported by the Israel Science Foundation. Publication 736.     

On a method to transform algebraic differential equations into universal equations

The paper introduces an algorithm which transforms homogeneous algebraic differential equations into universal differential equations (in the sense of L. A. Rubel) havingC ⁿ (ℝ)-solutions. By applications of the algorithm to different initial equations some new universal differential equations are found, and all the known equations due to R. J. Duffin are rediscovered with this method. Assuming weak conditions one can find Cⁿ(ℝ)-solutionsy of the differential equation close to any continuous function such that 1, with 0 ≤k ₁ <k ₂ < .... <k _s ≤n are linearly independent over the field of real algebraic numbers at the rational points q1,...,qs.