Linkless and Flat Embeddings in 3-Space

We consider piecewise linear embeddings of graphs in 3-space ℝ³. Such an embedding is linkless if every pair of disjoint cycles forms a trivial link (in the sense of knot theory). Robertson, Seymour and Thomas (J. Comb. Theory, Ser. B 64:185–227, 1995) showed that a graph has a linkless embedding in ℝ³ if and only if it does not contain as a minor any of seven graphs in Petersen’s family (graphs obtained from K ₆ by a series of YΔ and ΔY operations). They also showed that a graph is linklessly embeddable in ℝ³ if and only if it admits a flat embedding into ℝ³, i.e. an embedding such that for every cycle C of G there exists a closed 2-disk D⊆ℝ³ with D∩G=∂D=C. Clearly, every flat embedding is linkless, but the converse is not true. We consider the following algorithmic problem associated with embeddings in ℝ³:     

Éclatement des coefficients des séries entières et deux théorèmes de Gabrielov

Given a non trivial power series in ℝ ^m × ℝ ^k , it is in general not possible to choose a good direction in ℝ ^k in order to apply Weierstrass Preparation Theorem. Now, one can make it possible by blowing-up coefficients in ℝ ^m . This enables e. g. to prove in some natural way Gabrielov’s complement theorem, as well as Gabrielov’s fiber components theorem in subanalytic geometry.      

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.     

Instanton counting on blowup. I. 4-dimensional pure gauge theory

We give a mathematically rigorous proof of Nekrasov’s conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on ℝ⁴ gives a deformation of the Seiberg-Witten prepotential for N=2 SUSY Yang-Mills theory. Through a study of moduli spaces on the blowup of ℝ⁴, we derive a differential equation for the Nekrasov’s partition function. It is a deformation of the equation for the Seiberg-Witten prepotential, found by Losev et al., and further studied by Gorsky et al. Mathematics Subject Classification (2000) 14D21, 57R57, 81T13, 81T60     

Schmidt’s game on fractals

We construct (α, β) and α-winning sets in the sense of Schmidt’s game, played on the support of certain measures (absolutely friendly) and show how to compute the Hausdorff dimension for some. In particular, we prove that if K is the attractor of an irreducible finite family of contracting similarity maps of ℝ ^N satisfying the open set condition, (the Cantor’s ternary set, Koch’s curve and Sierpinski’s gasket to name a few known examples), then for any countable collection of non-singular affine transformations, Δ _i : ℝ ^N → ℝ ^N ,

Darboux property of Gâteaux derivatives of functions on ℝ n

D. Preiss proved that the graph of the derivative of a continuous Gateaux-differentiable function f : ℝ² → ℝ is always connected. We show that this is no longer true in higher dimensions: we construct a continuous, Gateaux-differentiable function f : ℝ³ → ℝ for which the range of its gradient mapping {∇ f(x) : x ∈ ℝ³} is disconnected. We also give an example of an approximately differentiable continuous function on ℝ² such that the range of its gradient mapping is disconnected. The work is a part of the research project MSM 0021620839 financed by MSMT and it was also partly supported by GAČR 201/06/0198 and GAČR 201/06/0018.     

Duality Questions for Operators,Spectrum and Measures

We explore spectral duality in the context of measures in ℝ ⁿ , starting with partial differential operators and Fuglede’s question (1974) about the relationship between orthogonal bases of complex exponentials in L ²(Ω) and tiling properties of Ω, then continuing with affine iterated function systems. We review results in the literature from 1974 up to the present, and we relate them to a general framework for spectral duality for pairs of Borel measures in ℝ ⁿ , formulated first by Jorgensen and Pedersen.     

<Emphasis Type="Italic">C</Emphasis><Superscript>1+<Emphasis Type="Italic">α</Emphasis></Superscript>-Regularity for Two-Dimensional Almost-Minimal Sets in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We give a new proof and a partial generalization of Jean Taylor’s result (Ann. Math. (2) 103(3), 489–539, 1976) that says that Almgren almost-minimal sets of dimension 2 in ℝ³ are locally C ^1+α -equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in Ann. Fac. Sci. Toulouse 18(1), 65–246, 2009 and an extension of Reifenberg’s parameterization theorem (David et al. in Geom. Funct. Anal. 18, 1168–1235, 2008). The key idea is still that if X is the cone over an arc of small Lipschitz graph in the unit sphere, but X is not contained in a disk, we can use the graph of a harmonic function to deform X and substantially diminish its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in ℝ ⁿ , but in this setting our final regularity result on E may depend on the list of minimal cones obtained as blow-up limits of E at a point.     

Positive Solutions to Singular Semilinear Elliptic Problems

We obtain a characterization of all locally bounded functions p ≥ 0 for which the equation (E) Δu +p(x)ψ(u) = 0 has a positive solution in Ω vanishing on the boundary, where Ω is a domain of ℝ^N and ψ > 0 is a nonincreasing continuous function on ]0,∞[. In particular, for Ω = ℝ^N with N ≥ 3, it is shown that (E) has a (unique) positive solution in ℝ^N which decays to zero at infinity if and only if the set {p > 0} has positive Lebesgue measure and This condition can be replaced by if p is radial.     

A complete analogue of Hardy’s theorem on SL2(ℝ) and characterization of the heat kernel

A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on ℝ from estimates on the function and its Fourier transform. In this article we establisha full group version of the theorem for SL₂(ℝ) which can accommodate functions with arbitraryK-types. We also consider the ‘heat equation’ of the Casimir operator, which plays the role of the Laplacian for the group. We show that despite the structural difference of the Casimir with the Laplacian on ℝⁿ or the Laplace—Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel for the full group can also be characterized by Hardy-like estimates.     

Integrability of det△↓u and Evolutionary Wente＇s Problem Associated to Heat Operator

In this note, the authors resolve an evolutionary Wente's problem associated to heat equation, where the special integrability of det▽u for u ∈ H1(R2,R2) is used.     

Positive definite functions and stable random vectors

We say that a random vector X = (X ₁, …, X _n ) in ℝ ⁿ is an n-dimensional version of a random variable Y if, for any a ∈ ℝ ⁿ , the random variables Σa _i X _i and γ(a)Y are identically distributed, where γ: ℝ ⁿ → [0,∞) is called the standard of X. An old problem is to characterize those functions γ that can appear as the standard of an n-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in L ₀. This result is almost optimal, as the norm of any finite-dimensional subspace of L _p with p ∈ (0, 2] is the standard of an n-dimensional version (p-stable random vector) by the classical result of P. Lèvy. An equivalent formulation is that if a function of the form f(‖ · ‖ _K ) is positive definite on ℝ ⁿ , where K is an origin symmetric star body in ℝ ⁿ and f: ℝ → ℝ is an even continuous function, then either the space (ℝ ⁿ , ‖·‖ _K ) embeds in L ₀ or f is a constant function. Combined with known facts about embedding in L ₀, this result leads to several generalizations of the solution of Schoenberg’s problem on positive definite functions.     

Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

Existence of T-solution for degenerated problem via Minty’s lemma

We study the existence result of solutions for the nonlinear degenerated elliptic problem of the form, -div（a（x, u,△↓u）） = F in Ω, where Ω is a bounded domain of R^N, N≥2, a ：Ω×R×R^N→R^N is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but they verify only the large monotonicity. The second term F belongs to W^-1,p′（Ω, w^＊）. The existence result is proved by using the L^1-version of Minty＇s lemma.     

Equidistribution of expanding translates of curves and Dirichlet’s theorem on diophantine approximation

We show that for almost all points on any analytic curve on ℝ ^k which is not contained in a proper affine subspace, the Dirichlet’s theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in ℝ ^k+1. The proof involves Ratner’s theorem on ergodic properties of unipotent flows on homogeneous spaces. Dedicated to my inspiring teacher Professor A.R. Rao (VASCSC, Ahmedabad) on his 100th birthday. Research supported in part by Swarnajayanti Fellowship.     

A characterization of the higher dimensional groups associated with continuous wavelets

A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ ⁿ , considered as a subgroup of the affine group on ℝ ⁿ , admits wavelets ψ ∈ L²(ℝ ⁿ ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D_x for the transpose action of D on ℝ ⁿ must be compact for a. e. x. ∈ ℝ ⁿ ; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ ⁿ there exists an ε > 0 for which the ε-stabilizer D _x ^ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.     

Mirror configurations of points and lines and algebraic surfaces of degree four

We prove that mirror nonsingular configurations of m points and n lines in ℝP ³ exist only for m≤3, n≡0 or 1 (mod 4) and for m=0 or 1 (mod 4), n≡0 (mod 2). In addition, we give an elementary proof of V. M. Kharlamov’s well-known result saying that if a nonsingular surface of degree four in ℝP ³ is noncontractible and has M≥5 components, then it is nonmirror. For the cases M=5, 6, 7 and 8, Kharlamov suggested an elementary proof using an analogy between such surfaces and configurations of M−1 points and a line. Our proof covers the remaining cases M=9, 10. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 299–308. Translated by N. Yu. Netsvetaev.     

Gibbsianness of fermion random point fields

We consider fermion (or determinantal) random point fields on Euclidean space ℝ^d. Given a bounded, translation invariant, and positive definite integral operator J on L²(ℝ^d), we introduce a determinantal interaction for a system of particles moving on ℝ^d as follows: the n points located at x₁,· · ·,x_n ∈ ℝ^d have the potential energy given by where j(x−y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d≥2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I+J)⁻¹ is a Gibbs measure admitted to the specification.     

On certain new inequalities in information theory

We prove a global assertion on logarithmic convexity of Csiszár’s ƒ-divergence. It follows that the relative s-information measure is log-convex for s ∈ ℝ, wherefrom some new inequalities connecting Kullback-Leibler divergence and χ² and Hellinger distances arise.     

Pointwise characterization of Sobolev classes

We prove that a function f is in the Sobolev class W _loc ^m,p (ℝ ⁿ ) or W ^m,p (Q) for some cube Q ⊂ ℝ ⁿ if and only if the formal (m − 1)-Taylor remainder R ^m−1 f(x,y) of f satisfies the pointwise inequality |R ^m−1 f(x,y)| ≤ |x − y| ^m [a(x) + a(y)] for some a ε L ^p (Q) outside a set N ⊂ Q of null Lebesgue measure. This is analogous to H. Whitney’s Taylor remainder condition characterizing the traces of smooth functions on closed subsets of ℝ ⁿ . Dedicated to S.M. Nikol’skiĭ on the occasion of his 100th birthday The main results and ideas of this paper were presented in the plenary lecture of the author at the International Conference and Workshop Function Spaces, Approximation Theory and Nonlinear Analysis dedicated to the centennial of Sergei Mikhailovich Nikol’skii, Moscow, May 24–28, 2005.