An Explicit Construction for a Ramsey Problem

An explicit coloring of the edges of K_n is constructed such that every copy of K₄ has at least four colors on its edges. As n , the number of colors used is n^1/2+o(1). This improves upon the previous bound of O(n^2/3) due to Erds and Gyárfás obtained by probabilistic methods. The exponent 1/2 is optimal, since it is known that at least (n^1/2) colors are required in such a coloring.The coloring is related to constructions giving lower bounds for the multicolor Ramsey number r_k(C₄). It is more complicated however, because of restrictions imposed on interactions between color classes.* Research supported in part by NSF Grant No. DMS–9970325.     

On Graphs With Small Ramsey Numbers,II

There exists a constant C such that for every d-degenerate graphs G ₁ and G ₂ on n vertices, Ramsey number R(G ₁,G ₂) is at most Cn, where is the minimum of the maximum degrees of G ₁ and G ₂.* The work of this author was supported by the grants 99-01-00581 and 00-01-00916 of the Russian Foundation for Fundamental Research and the Dutch-Russian Grant NWO-047-008-006. The work of this author was supported by the NSF grant DMS-9704114.     

Formulas for the Inverses of Toeplitz Matrices with Polynomially Singular Symbols

We consider large finite Toeplitz matrices with symbols of the form (1– cos )^p f() where p is a natural number and f is a sufficiently smooth positive function. By employing techniques based on the use of predictor polynomials, we derive exact and asymptotic formulas for the entries of the inverses of these matrices. We show in particular that asymptotically the inverse matrix mimics the Green kernel of a boundary value problem for the differential operator Submitted: June 20, 2003     

Commutants of Analytic Toeplitz Operators on the Harmonic Bergman Space

We study the commuting problem for Toeplitz operators on the harmonic Bergman space of the unit disk. We show that an analytic Toeplitz operator and a co-analytic Toeplitz operator with certain noncyclicity hypothesis can commute only when one of their symbols is constant. We also obtain analogous results for semi-commutants.     

Regularity of Integral Operators

In this paper we consider a class of integral operators whose kernels satisfy certain generalizations of Hörmander condition and establish their regularity results.     

On the volume of unit vector fields on spaces of constant sectional curvature

A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields.     

Epitaxial Growth Without Slope Selection: Energetics,Coarsening, and Dynamic Scaling

We study a continuum model for epitaxial growth of thin films in which the slope of mound structure of film surface increases. This model is a diffusion equation for the surface height profile h which is assumed to satisfy the periodic boundary condition. The equation happens to possess a Liapunov or free-energy functional. This functional consists of the term | h|², which represents the surface diffusion, and - log (1 + | h|²), which describes the effect of kinetic asymmetry in the adatom attachment-detachment. We first prove for large time t that the interface width---the standard deviation of the height profile---is bounded above by O(t^1/2), the averaged gradient is bounded above by O(t^1/4), and the averaged energy is bounded below by O(- log t). We then consider a small coefficient ² of | h|² with = 1/L and L the linear size of the underlying system, and study the energy asymptotics in the large system limit 0. We show that global minimizers of the free-energy functional exist for each > 0, the L²-norm of the gradient of any global minimizer scales as O(1/), and the global minimum energy scales as O( log ). The existence of global energy minimizers and a scaling argument are used to construct a sequence of equilibrium solutions with different wavelengths. Finally, we apply our minimum energy estimates to derive bounds in terms of the linear system size L for the saturation interface width and the corresponding saturation time.     

Singularities of positive supersolutions in elliptic PDEs

We present several extensions of the Brezis–Lions Lemma on removable singularities. We also give a positive answer to a question raised by H. Brezis and M. Marcus about an inverse maximum principle for the Laplacian.     

Representations of twisted Yangians associated with skew Young diagrams

Let $G_M$ be either the orthogonal group $O_M$ or the symplectic group $Sp_M$ over the complex field; in the latter case the non-negative integer $M$ has to be even. Classically, the irreducible polynomial representations of the group $G_M$ are labeled by partitions $\mu=(\mu_{1},\mu_{2},\,\ldots)$ such that $\mu^{\prime}_1+\mu^{\prime}_2\le M$ in the case $G_M=O_M$, or $2\mu^{\prime}_1\le M$ in the case $G_M=Sp_M$. Here $\mu^{\prime}=(\mu^{\prime}_{1},\mu^{\prime}_{2},\,\ldots)$ is the partition conjugate to $\mu$. Let $W_\mu$ be the irreducible polynomial representation of the group $G_M$ corresponding to $\mu$. Regard $G_N\times G_M$ as a subgroup of $G_{N+M}$. Then take any irreducible polynomial representation $W_\lambda$ of the group $G_{N+M}$. The vector space $W_{\lambda}(\mu)={\rm Hom}_{\,G_M}( W_\mu, W_\lambda)$ comes with a natural action of the group $G_N$. Put $n=\lambda_1-\mu_1+\lambda_2-\mu_2+\ldots\,$. In this article, for any standard Young tableau $\varOmega$ of skew shape $\lm$ we give a realization of $W_{\lambda}(\mu)$ as a subspace in the $n$-fold tensor product $(\mathbb{C}^N)^{\bigotimes n}$, compatible with the action of the group $G_N$. This subspace is determined as the image of a certain linear operator $F_\varOmega (M)$ on $(\mathbb{C}^N)^{\bigotimes n}$, given by an explicit formula. When $M=0$ and $W_{\lambda}(\mu)=W_\lambda$ is an irreducible representation of the group $G_N$, we recover the classical realization of $W_\lambda$ as a subspace in the space of all traceless tensors in $(\mathbb{C}^N)^{\bigotimes n}$. Then the operator $F_\varOmega\(0)$ may be regarded as the analogue for $G_N$ of the Young symmetrizer, corresponding to the standard tableau $\varOmega$ of shape $\lambda$. This symmetrizer is a certain linear operator on $\CNn$$(\mathbb{C}^N)^{\bigotimes n} $ with the image equivalent to the irreducible polynomial representation of the complex general linear group $GL_N$, corresponding to the partition $\lambda$. Even in the case $M=0$, our formula for the operator $F_\varOmega(M)$ is new. Our results are applications of the representation theory of the twisted Yangian, corresponding to the subgroup $G_N$ of $GL_N$. This twisted Yangian is a certain one-sided coideal subalgebra of the Yangian corresponding to $GL_N$. In particular, $F_\varOmega(M)$ is an intertwining operator between certain representations of the twisted Yangian in $(\mathbb{C}^N)^{\bigotimes n}$.