Invariant symmetric restrictions of a self-adjoint operator. II

We give a criterion of invariance and symmetry of the restriction of an arbitrary unbounded self-adjoint operator in the space L ₂(ℝⁿ, dx) by using the introduced notion of support of an arbitrary vector and the notion of capacity of a subspace N ⊂ ℝⁿ. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 781–791, June, 1998. This work was partially supported by the Foundation for Fundamental Research of the Ministry of Science and Technology of the Ukraine (grant No. 1/238 “Operator”).     

An integral estimate of Bessel function and its application

In this paper the authors give a new integral estimate of the Bessel function,which is an extension of Calder(?)n-Zygmund's result.As an application of this result,we prove that the parameterized Marcinkiewicz integralμ_Ω~p with variable kernels is of type (2,2),where the kernel functionΩdoes not have any smoothness on the unit sphere in R~n.     

Asymptotic Smoothing Effect of Solutions to Davey-Stewartson Systems on the Whole Plane

This paper discusses the Cauchy problem of elliptic-elliptic-type Davey-Stewartson systems with zero-order dissipation on R^2. Making use of the Fourier spectral projector, together with a long time comparison between the solutions to the Davey-Stewartson systems and to an auxiliary problem, we prove that the global attractor in H^1 （R^2） for the addressed Davey-Stewartson systems is a compact subset of H^2（R^2）, and thus reveal the asymptotic smoothing effect of the solutions for the systems.     

On the connected components of the space of line transversals to a family of convex sets

Let ℒ be the space of line transversals to a finite family of pairwise disjoint compact convex sets in ℝ³. We prove that each connected component of ℒ can itself be represented as the space of transversals to some finite family of pairwise disjoint compact convex sets. The research of J. E. Goodman was supported in part by NSF Grant DMS91-22065 and by NSA Grant MDA904-92-H-3069. R. Pollack's research was supported in part by NSF Grant CCR91-22103 and by NSA Grant MDA904-92-H-3075. The research of R. Wenger was supported in part by NSA Grant MDA904-93-H-3026 and by the NSF Regional Geometry Institute (Smith College, July 1993) Grant DMS90-13220.     

Extension by zero of functions of several variables

We obtain sufficient conditions on a domainG ⊂ ℝn for functions defined onG to be extendable by zero to the entire space ℝn with smoothness preserved in an integral norm. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 351–365, September, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00243 and by program “Leading Science Schools” under grant No. 96-15-96102.     

On acyclic colorings of graphs on surfaces

A properk-coloring of a graph is acyclic if every 2-chromatic subgraph is acyclic. Borodin showed that every planar graph has an acyclic 5-coloring. This paper shows that the acyclic chromatic number of the projective plane is at most 7. The acyclic chromatic number of an arbitrary surface with Euler characteristic η=−γ is at mostO(γ^4/7). This is nearly tight; for every γ>0 there are graphs embeddable on surfaces of Euler characteristic −γ whose acyclic chromatic number is at least Ω(γ^4/7/(logγ)^1/7). Therefore, the conjecture of Borodin that the acyclic chromatic number of any surface but the plane is the same as its chromatic number is false for all surfaces with large γ (and may very well be false for all surfaces). This author's research was supported in part by a United States Israeli BSF grant. This author's research was supported by the Ministry of Research and Technology of Slovenia, Research Project P1-0210-101-92. This author's research was supported by the Office of Naval Research, grant number N00014-92-J-1965.     

An elementary approach to lower bounds in geometric discrepancy

For eachd≥2, it is possible to placen points ind-space so that, given any two-coloring of the points, a half-space exists within which one color outnumbers the other by as much ascn ^1/2−1/2d , for some constantc>0 depending ond. This result was proven in a slightly weaker form by Beck and the bound was later tightened by Alexander. It was recently shown to be asymptotically optimal by Matoušek. We present a proof of the lower bound, which is based on Alexander's technique but is technically simpler and more accessible. We present three variants of the proof, for three diffrent cases, to provide more intuitive insight into the “large-discrepancy” phenomenon. We also give geometric and probabilistic interpretations of the technique. Work by Bernard Chazelle has been supported in part by NSF Grant CCR-90-02352 and The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc. Work by Jiří Matoušek has been supported by Charles University Grant No. 351, by Czech Republic Grant GAČR 201/93/2167 and in part by DIMACS. Work by Micha Sharir has been supported by NSF Grant CCR-91-22103, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

Improved bounds on weak ε-nets for convex sets

LetS be a set ofn points in ℝ ^d . A setW is aweak ε-net for (convex ranges of)S if, for anyT⊆S containing εn points, the convex hull ofT intersectsW. We show the existence of weak ε-nets of size , whereβ ₂=0,β ₃=1, andβ _d ≈0.149·2 ^d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/ε) log^1.6(1/ε)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/ε), which improves a previous bound of Capoyleas. Work by Bernard Chazelle has been supported by NSF Grant CCR-90-02352 and the Geometry Center. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-89-21421. Work by Michelangelo Grigni has been supported by NSERC Operating Grants and NSF Grant DMS-9206251. Work by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Emo Welzl and Micha Sharir has been supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Work by Micha Sharir has also been supported by NSF Grant CCR-91-22103, and by a grant from the Fund for Basic Research administered by the Israeli Academy of Sciences.     

A Besov Estimate for Multilinear Singular Integrals

Abstract The main purpose of this paper is to prove a good λ inequality for a multilinear singular integral in ℝ ⁿ . The research is supported by Beijing Education Commission Foundation and Beijing Natural Science Foundation (Project No. 1982005)     

On continuous extension of locally homeomorphic simplicial maps of ℝ2 into itself byσ-processes

It is shown that the Klein bottle with two points removed can be embedded in the compactification of ℝ² by a finite tree. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 821–831, June, 1996. I wish to express sincere gratitude to A. G. Vitushkin for posing the problem and guidance. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00225 and by the Ministry of Scientific and Technological Research in the Mathematical Sciences of Italy.     

Mutual isomorphisms of translations of a homogeneous flow

Ergodic one-parameter flows (G/Γ,g _ℝ) induced by the left action of a subgroupg _ℝ ⊂G on homogeneous spaces of finite volume are considered. Letℳ ⊂ ℝ⁺ be the set of allt>0 such that the cascade (G/Γ,g _t ℤ) is metrically isomorphic to the cascade (G/Γ,g _ℤ). We prove that eitherℳ is at most countable or the subgroupg _ℝ is horocyclic andℳ=ℝ⁺. We prove that a metric isomorphism of ergodic quasi-unipotent cascades (or flows) is affine on almost all fibers of a certain natural bundle. The result generalizes Witte's theorem on the affinity of such isomorphisms of cascades with the mixing property; this is applied to the study of the structure of the setℳ ⊂ ℝ⁺. The proof is based on the fundamental Ratner theorem stating that the ergodic measures of unipotent cascades are algebraic. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 98–110, July, 1995. This research was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-00082a and by the International Science Foundation.     

On the concentration of eigenvalues of random symmetric matrices

It is shown that for every 1≤s≤n, the probability that thes-th largest eigenvalue of a random symmetricn-by-n matrix with independent random entries of absolute value at most 1 deviates from its median by more thant is at most 4e ⁻ t ²32 ^s2. The main ingredient in the proof is Talagrand’s Inequality for concentration of measure in product spaces. Research supported in part by a USA — Israel BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research supported in part by a USA — Israel BSF grant and by a Bergmann Memorial Grant.     

Decompositions of the Hardy space Hz^1（Ω）

We give a decomposition of the Hardy space Hz^1（Ω） into ＂div-curl＂ quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1（Ω） into Jacobians det Du, u ∈ W0^1,2 （Ω,R^2） for Ω in R^2. This partially answers a well-known open problem.     

Invariant symmetric restrictions of a self-adjoint operator. I

We prove necessary and sufficient conditions of the S-invariance of a subset dense in a separable Hilbert space H. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 623–631, May, 1998. This work was partially supported by the Foundation for Fundamental Research of the Ministry of Science and Technology of the Ukraine (grant No. 1/238).     

On visibility and covering by convex sets

A setX⊆ℝ ^d isn-convex if among anyn of its points there exist two such that the segment connecting them is contained inX. Perles and Shelah have shown that any closed (n+1)-convex set in the plane is the union of at mostn ⁶ convex sets. We improve their bound to 18n ³, and show a lower bound of order Ω(n ²). We also show that ifX⊆ℝ² is ann-convex set such that its complement has λ one-point path-connectivity components, λ<∞, thenX is the union ofO(n ⁴+n ²λ) convex sets. Two other results onn-convex sets are stated in the introduction (Corollary 1.2 and Proposition 1.4). Research supported by Charles University grants GAUK 99/158 and 99/159, and by U.S.-Czechoslovak Science and Technology Program Grant No. 94051. Part of the work by J. Matoušek was done during the author’s visits at Tel Aviv University and The Hebrew University of Jerusalem. Part of the work by P. Valtr was done during his visit at the University of Cambridge supported by EC Network DIMANET/PECO Contract No. ERBCIPDCT 94-0623.     

Partial convexity

We consider a generalization of the classical notion of convexity, which is calledpartial convexity. LetV ∋ ℝ ⁿ be some set of directions. A setX ∋ ℝ ⁿ is calledV- convex if the intersection of any line parallel to a vector inV withX is connected. Semispaces and the problem of the least intersection base for partial convexity is investigated. The cone of convexity directions is described for a closed set in ℝ ⁿ . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 406–413, September, 1996. The authors wish to express their gratitude to V. V. Gorokhovik and E. A. Barabanov for useful remarks and discussions. This research was supported in part by the Belorussian Foundation for Basic Research under grant No. F95-016.     

A Note on Hermite and Subelliptic Operators

In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 and the sub-Laplacian in the Heisenberg group Hn.     

Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

We prove Helly-type theorems for line transversals to disjoint unit balls in ℝ ^d . In particular, we show that a family of n≥2d disjoint unit balls in ℝ ^d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n≥4d−1 disjoint unit balls in ℝ ^d admits a line transversal if any subfamily of size 4d−1 admits a transversal. Andreas Holmsen was supported by the Research Council of Norway, prosjektnummer 166618/V30. Otfried Cheong and Xavier Goaoc acknowledge support from the French-Korean Science and Technology Amicable Relationships program (STAR).     

On the generalized Busemann-Petty problem

The generalized Busemann-Petty problem asks whether the origin-symmetric convex bodies in ℝ ⁿ with a larger volume of all i-dimensional sections necessarily have a larger volume. As proved by Bourgain and Zhang, the answer to this question is negative if i > 3. The problem is still open for i = 2, 3. In this article we prove two specific affirmative answers to the generalized Busemann-Petty problem if the body with a smaller i-dimensional volume belongs to given classes. Our results generalize Zhang’s specific affirmative answer to the generalized Busemann-Petty problem. This work was supported, in part, by the National Natural Science Foundation of China (Grant No. 10671117)     

Branched two-sheeted covers

In this paper we explore the connection between Weierstrass points of subspaces of the holomorphic differentials and the geometry of the canonical curve inPC ^g−1. In particular, we consider non-hyperelliptic Riemann surfaces with involution and the Weierstrass points of the −1 eigenspace of the holomorphic differentials. The case of coverings of a torus is considered in detail. Research of the first author supported in part by the Paul and Gabriella Rosenbaum Foundation, the Landau Center for Research in Mathematical Analysis (supported by Minerva Foundation-Germany) and a US-Israel BSF grant. Research by the second author supported in part by NSF Grant DMS 9003361 and a Lady Davis Visiting Professorship at the Hebrew University.