Hölder Regularity for Solutions of Ultraparabolic Equations in Divergence Form

We consider the second order differential equation , where (x,t) ^N+1, 0<m ₀N, the coefficients a _i,j belong to a suitable space of vanishing mean oscillation functions VMO _L and B=(b _i,j ) is a constant real matrix. The aim of this paper is to study interior regularity for weak solutions to the above equation assuming that F _j belong to a function space of Morrey type.     

Uniform Morse lemma and isotopy criterion for Morse functions on surfaces

Let M be a smooth compact (orientable or not) surface with or without a boundary. Let $ \mathcal{D}_0 $ \mathcal{D}_0 ⊂ Diff(M) be the group of diffeomorphisms homotopic to id _M . Two smooth functions f, g: M → ℝ are called isotopic if f = h ₂ ℴ g ℴ h ₁ for some diffeomorphisms h ₁ ∈ $ \mathcal{D}_0 $ \mathcal{D}_0 and h ₂ ∈ Diff⁺(ℝ). Let F be the space of Morse functions on M which are constant on each boundary component and have no critical points on the boundary. A criterion for two Morse functions from F to be isotopic is proved. For each Morse function f ∈ F, a collection of Morse local coordinates in disjoint circular neighborhoods of its critical points is constructed, which continuously and Diff(M)-equivariantly depends on f in C ^∞-topology on F (“uniform Morse lemma”). Applications of these results to the problem of describing the homotopy type of the space F are formulated.     

An Extension of Polyak’s Theorem in a Hilbert Space

Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (⋅,⋅) _H . Let us consider three linear bounded operators,

On random surface area

Consider a random smooth Gaussian field G(x): F ? \mathbbR F \to \mathbb{R} , where F is a compact set in \mathbbR^d {\mathbb{R}^d} . We derive a formula for the average area of a surface determined by the equation G(x) = 0 and give some applications. As an auxiliary result, we obtain an integral expression for the area of a surface determined by zeros of a nonrandom smooth field. Bibliography: 13 titles.     

Dilation of Newton Polytope and <Emphasis Type="Italic">p</Emphasis>-adic Estimate

Let f(X) be a polynomial in n variables over the finite field  \mathbbF_q\mathbb{F}_{q}. Its Newton polytope Δ(f) is the convex closure in ℝ ⁿ of the origin and the exponent vectors (viewed as points in ℝ ⁿ ) of monomials in f(X). The minimal dilation of Δ(f) such that it contains at least one lattice point of $\mathbb{Z}_{>0}^{n}$\mathbb{Z}_{>0}^{n} plays a vital pole in the p-adic estimate of the number of zeros of f(X) in  \mathbbF_q\mathbb{F}_{q}. Using this fact, we obtain several tight and computational bounds for the dilation which unify and improve a number of previous results in this direction.     

Estimates for the real taylor coefficients in one function class

Considering the class {ie48-01} of analytic functions {ie48-02} in the unit disk with a _m,n ε ℝ and the nonvanishing nth divided difference [F(z);z ₀, ⋯, z _n ] for all z ₀, ℝ, z _n ∈ E we establish that {ie48-03}, where {ie48-04}. If n is an odd number then {ie48-05}.     

Solution blow-up for a new stationary Sobolev-type equation

A new nonlinear stationary Sobolev-type equation with a parameter η ∈ ℝ¹ is derived. For η > 0, global solvability in the weak generalized sense is proved in the entire waveguide $ \mathbb{S} $ \mathbb{S} ⊗ ℝ₊¹. For η < 0, the strong generalized solution is shown to blow up in a certain waveguide cross section z = R ₀ > 0. An upper bound for R ₀ in terms of the original parameters of the problem is obtained.     

An analysis of the solution set to a homotopy equation between polynomials with real coefficients

We investigate the structure of the solution setS to a homotopy equationH(Z,t)=0 between two polynomialsF andG with real coefficients in one complex variableZ. The mapH is represented asH(x+iy, t)=h ₁(x, y, t)+ih ₂(x, y, t), whereh ₁ andh ₂ are polynomials from ℝ² × [0,1] into ℝ and i is the imaginary unit. Since all the coefficients ofF andG are real, there is a polynomialh ₃ such thath ₂(x, y, t)=yh₃(x, y, t). Then the solution setS is divided into two sets {(x, t)∶h ₁(x, 0, t)=0} and {(x+iy, t)∶h ₁(x, y, t)=0,h ₃(x, y, t)=0}. Using this division, we make the structure ofS clear. Finally we briefly explain the structure of the solution set to a homotopy equation between polynomial systems with real coefficients in several variables.     

Small resolutions and non-liftable Calabi-Yau threefolds

We use properties of small resolutions of the ordinary double point in dimension three to construct smooth non-liftable Calabi-Yau threefolds. In particular, we construct a smooth projective Calabi-Yau threefold over \mathbbF₃{\mathbb{F}_3} that does not lift to characteristic zero and a smooth projective Calabi-Yau threefold over \mathbbF₅{\mathbb{F}_5} having an obstructed deformation. We also construct many examples of smooth Calabi-Yau algebraic spaces over \mathbbF_p{\mathbb{F}_p} that do not lift to algebraic spaces in characteristic zero.     

Equality of two variable weighted means: reduction to differential equations

Summary. Let F, Y \Phi, \Psi be strictly monotonic continuous functions, F,G be positive functions on an interval I and let n ? \Bbb N \{1} n \in {\Bbb N} \setminus \{1\} . The functional equation¶¶F^-1 ([(?_i=1ⁿF(x_i)F(x_i))/(?_i=1ⁿ F(x_i)]) Y^-1 ([(?_i=1ⁿY(x_i)G(x_i))/(?_i=1ⁿ G(x_i))])  (x₁,?,x_n ? I) \Phi^{-1}\,\left({\sum\limits_{i=1}^{n}\Phi(x_{i})F(x_{i})\over\sum\limits_{i=1}^{n} F(x_{i}}\right) \Psi^{-1}\,\left({\sum\limits_{i=1}^{n}\Psi(x_{i})G(x_{i})\over\sum\limits_{i=1}^{n} G(x_{i})}\right)\,\,(x_{1},\ldots,x_{n} \in I) ¶was solved by Bajraktarevi' [3] for a fixed n 3 3 n\ge 3 . Assuming that the functions involved are twice differentiable he proved that the above functional equation holds if and only if¶¶Y(x) = [(aF(x) + b)/(cF(x) + d)],       G(x) = kF(x)(cF(x) + d) \Psi(x) = {a\Phi(x)\,+\,b\over c\Phi(x)\,+\,d},\qquad G(x) = kF(x)(c\Phi(x) + d) ¶where a,b,c,d,k are arbitrary constants with k(c²+d²)(ad-bc) 1 0 k(c^2+d^2)(ad-bc)\ne 0 . Supposing the functional equation for all n = 2,3,... n = 2,3,\dots Aczél and Daróczy [2] obtained the same result without differentiability conditions.¶The case of fixed n = 2 is, as in many similar problems, much more difficult and allows considerably more solutions. Here we assume only that the same functional equation is satisfied for n = 2 and solve it under the supposition that the functions involved are six times differentiable. Our main tool is the deduction of a sixth order differential equation for the function j = F°Y^-1 \varphi = \Phi\circ\Psi^{-1} . We get 32 new families of solutions.     

Holomorphic Cliffordian functions

The aim of this paper is to put the foundations of a new theory of functions, called holomorphic Cliffordian, which should play an essential role in the generalization of holomorphic functions to higher dimensions. Let ℝ_0,2m+1 be the Clifford algebra of ℝ^2m+1 with a quadratic form of negative signature, be the usual operator for monogenic functions and Δ the ordinary Laplacian. The holomorphic Cliffordian functions are functionsf: ℝ^2m+2 → ℝ_0,2m+1, which are solutions ofDδ ^m f = 0. Here, we will study polynomial and singular solutions of this equation, we will obtain integral representation formulas and deduce the analogous of the Taylor and Laurent expansions for holomorphic Cliffordian functions. In a following paper, we will put the foundations of the Cliffordian elliptic function theory.     

Active set algorithms for isotonic regression; A unifying framework

In this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods. The active set approach provides a unifying framework for studying algorithms for isotonic regression, simplifies the exposition of existing algorithms and leads to several new efficient algorithms. We also investigate the computational complexity of several algorithms.In this paper we consider the isotonic regression problem with respect to a complete order where eachw _i is strictly positive and eachy _i is an arbitrary real number. We show that the Pool Adjacent Violators algorithm (due to Ayer et al., 1955; Miles, 1959; Kruskal, 1964), is a dual feasible active set method and that the Minimum Lower Set algorithm (due to Brunk et al., 1957) is a primal feasible active set method of computational complexity O(n ²). We present a new O(n) primal feasible active set algorithm. Finally we discuss Van Eeden's method and show that it is of worst-case exponential time complexity.This work was supported by the National Science and Engineering Research Council of Canada under Research Grant A8189 and an Ontario Graduate Scholarship.     

A vanishing theorem for differential operators in positive characteristic

Let X be a smooth variety over an algebraically closed field k of characteristic p, and let F: X → X be the Frobenius morphism. We prove that if X is an incidence variety (a partial flag variety in type A _n ) or a smooth quadric (in this case p is supposed to be odd) then Hⁱ( X,End( \sfF_*O_X ) ) = 0 {H^i}\left( {X,\mathcal{E}nd\left( {{\sf{F}_*}{\mathcal{O}_X}} \right)} \right) = 0 for i > 0. Using this vanishing result and the derived localization theorem for crystalline differential operators [3], we show that the Frobenius direct image \sfF_*O_X {\sf{F}_*}{\mathcal{O}_X} is a tilting bundle on these varieties provided that p > h, the Coxeter number of the corresponding group.     

Type II Hermite�CPad�� Approximations of Generalized Hypergeometric Series

We shall present short proofs for type II (simultaneous) Hermite–Padé approximations of the generalized hypergeometric and q-hypergeometric series

The Besov Space B^{{0,1}}_{1} on Domains

Let Ω be a bounded Lipschitz domain. Define B ^0,1 _{1, r} (Ω) = {f∈L ¹ (Ω): there is an F∈B ^0,1 ₁ (ℝ ⁿ ) such that F|_Ω = f} and B ^0,1 _{1 z} (Ω) = {f∈B ^0,1 ₁ (ℝ ⁿ ) : f = 0 on ℝ ⁿ \}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation of ℝ ⁿ ₊. Received June 8, 2000, Accepted October 24, 2000     

Piercing Numbers for Balanced and Unbalanced Families

Given a finite family F\mathcal{F} of convex sets in ℝ ^d , we say that F\mathcal{F} has the (p,q) _r property if for any p convex sets in F\mathcal{F} there are at least r q-tuples that have nonempty intersection. The piercing number of F\mathcal{F} is the minimum number of points we need to intersect all the sets in F\mathcal{F}. In this paper we will find some bounds for the piercing number of families of convex sets with (p,q) _r properties.     

Global existence of small radially symmetric solutions to quadratic nonlinear wave equations in an exterior domain

We study the initial boundary value problem for the nonlinear wave equation: (*) $$\left\{ \begin{gathered} \partial _t^2 u - (\partial _r^2 + \frac{{n - 1}}{r}\partial _r )u = F(\partial _t u,\partial _t^2 u),t \in \mathbb{R}^ + ,R< r< \infty , \hfill \\ u(0,r) = \in _0 u_0 (r),\partial _t u(0,r) = \in _0 u_1 (r),R< r< \infty , \hfill \\ u(t,R) = 0,t \in \mathbb{R}^ + , \hfill \\ \end{gathered} \right.$$ wheren=4,5,u ₀,u ₁ are real valued functions and ∈₀ is a sufficiently small positive constant. In this paper we shall show small solutions to (*) exist globally in time under the condition that the nonlinear termF:?²→? is quadratic with respect to ? _t u and ? _t ² u.