Weak Density of Smooth Maps in W 1, 1(M,N) for Non-Abelian π1(N)

We prove that smooth maps are dense in the sense of biting convergence in W ^{1, 1}(M, N) when M and Nare compact Riemannian manifolds and N is closed.     

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L ₂(ℝ ^d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H ^s (ℝ ^d ) and H ^−s (ℝ ^d ). In terms of masks for φ,ψ ¹,…,ψ ^L ∈H ^s (ℝ ^d ) and , simple sufficient conditions are given to ensure that (X ^s (φ;ψ ¹,…,ψ ^L ), forms a pair of dual wavelet frames in (H ^s (ℝ ^d ),H ^−s (ℝ ^d )), where

Trace results on domains with self-similar fractal boundaries

This work deals with trace theorems for a family of ramified bidimensional domains Ω with a self-similar fractal boundary Γ^∞. The fractal boundary Γ^∞ is supplied with a probability measure μ called the self-similar measure. Emphasis is put on the case when the domain is not a −δ domain and the fractal is not post-critically finite, for which classical results cannot be used. It is proven that the trace of a function in H¹(Ω) belongs to for all real numbers p1. A counterexample shows that the trace of a function in H¹(Ω) may not belong to BMO(μ) (and therefore may not belong to ). Finally, it is proven that the traces of the functions in H¹(Ω) belong to H^s(Γ^∞) for all real numbers s such that 0s<d_H/4, where d_H is the Hausdorff dimension of Γ^∞. Examples of functions whose traces do not belong to H^s(Γ^∞) for all s>d_H/4 are supplied.There is an important contrast with the case when Γ^∞ is post-critically finite, for which the functions in H¹(Ω) have their traces in H^s(Γ^∞) for all s such that 0s<d_H/2.     

On the Propagation of Regularity and Decay of Solutions to the k-Generalized Korteweg-de Vries Equation

We study special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u ₀ ∈ H ^3/4+ (?) whose restriction belongs to H ^l ((b, ∞)) for some l ∈ ?⁺ and b ∈ ? we prove that the restriction of the corresponding solution u(·, t) belongs to H ^l ((β, ∞)) for any β ∈ ? and any t ∈ (0, T). Thus, this type of regularity propagates with infinite speed to its left as time evolves.     

Integral Operators in Sobolev Spaces on Domains with Boundary

Properties of integral operators with weak singularities arc investigated. It is assumed that G ? ?ⁿ is a bounded domain. The boundary δG should be smooth concerning the Sobolev trace theorem. It will be proved that the integral operators $\int {_G \frac{{f\left(\Theta \right)}}{{x - y|^{n - 1} }}u\left(\nu \right)d\partial G_\nu }$ and $ \int {_{\partial G} \frac{{f\left(\Theta \right)}}{{|x - y|^{n - 1} }}u\left(y \right)d\partial G_y }$ maps W_p^k(G) into W_p^k+1(G) and W_p^k?1(G) into W_p^k/p(G), respectively, and are bounded. Here θ ∈ S ? ?ⁿ, where S is the unit sphere. Furthermore, f possesses bounded first order derivatives and is bounded on S. Then applications to first order systems are discussed.     

A note on extreme cases of Sobolev embeddings

We study the spaces of functions on for which the generalized partial derivatives exist and belong to different Lorentz spaces L^p_k,s_k. For this kind of functions we prove a sharp version of the extreme case of the Sobolev embedding theorem using L(∞,s) spaces.     

Computing a family of reproducing kernels for statistical applications

For an open subset of , an integer,m, and a positive real parameter , the Sobolev spacesH ^m () equipped with the norms: u²=u(t)²dt+(1/^2m u ^(m)(t)² constitute a family of reproducing kernel Hilbert spaces. When is an open interval of the real line, we describe the computation of their reproducing kernels. We derive explicit formulas for these kernels for all values ofm in the case of the whole real line, and form=1 andm=2 in the case of a bounded open interval.This research was partly supported by NSF Grant DMS-9002566.     

Symmetric boundary values for the Dirichlet problem for harmonic maps from the disc into the 2-sphere

Let Aut(D) denote the group of biholomorphic diffeormorphisms from the unit disc D onto itself and O(3) the group of orthogonal transformations of the unit sphere S ². The existence of multiple solutions to the Dirichlet problem for harmonic maps from D into S ² is related to the symmetries (if any) of the boundary value γ : ∂D → S ², by invariance of the Dirichlet energy under the action of Aut(D) × O(3). In this paper, we classify the stabilizers in Aut(D) × O(3) of boundary values in H ^1/2(S ¹, S ²) and . We give two applications to the Dirichlet problem for harmonic maps. This work was partially supported by the CMLA, Ecole Normale Supérieure de Cachan, Cachan, France.     

Odd‐angulated graphs and cancelling factors in box products

Under what conditions is it true that if there is a graph homomorphism G □ H → G □ T, then there is a graph homomorphism H→ T? Let G be a connected graph of odd girth 2k + 1. We say that G is (2k + 1)‐angulated if every two vertices of G are joined by a path each of whose edges lies on some (2k + 1)‐cycle. We call G strongly (2k + 1)‐angulated if every two vertices are connected by a sequence of (2k + 1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2k + 1)‐angulated, H is any graph, S, T are graphs with odd girth at least 2k + 1, and ?: G□ H→S□T is a graph homomorphism, then either ? maps G□{h} to S□{t_h} for all h∈V(H) where t_h∈V(T) depends on h; or ? maps G□{h} to {s_h}□ T for all h∈V(H) where s_h∈V(S) depends on h. This theorem allows us to prove several sufficient conditions for a cancelation law of a graph homomorphism between two box products with a common factor. We conclude the article with some open questions. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:221‐238, 2008     

Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent

We prove the boundedness of the maximal operator Mr in the spaces L^p（·）（Г,p） with variable exponent p（t） and power weight p on an arbitrary Carleson curve under the assumption that p（t） satisfies the log-condition on Г. We prove also weighted Sobolev type L^p（·）（Г, p） → L^q（·）（Г, p）-theorem for potential operators on Carleson curves.     

The valuations of the near hexagons related to the Witt designs S(5,6,12) and S(5,8,24)

Valuations of dense near polygons were introduced in 16 . In the present paper, we classify all valuations of the near hexagons ??₁ and ??₂, which are related to the respective Witt designs S(5,6,12) and S(5,8,24). Using these classifications, we prove that if a dense near polygon S contains a hex H isomorphic to ??₁ or ??₂, then H is classical in S. We will use this result to determine all dense near octagons that contain a hex isomorphic to ??₁ or ??₂. As a by‐product, we obtain a purely geometrical proof for the nonexistence of regular near 2d‐gons, d ≥ 4, whose parameters s, t, t_i (0 ≤ i ≤ d) satisfy (s, t₂, t₃) = (2, 1, 11) or (2, 2, 14). The nonexistence of these regular near polygons can also be shown with the aid of eigenvalue techniques. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 214–228, 2006     

Constrained Ramsey numbers of graphs

Given two graphs G and H, let f(G,H) denote the minimum integer n such that in every coloring of the edges of K_n, there is either a copy of G with all edges having the same color or a copy of H with all edges having different colors. We show that f(G,H) is finite iff G is a star or H is acyclic. If S and T are trees with s and t edges, respectively, we show that 1+s(t?2)/2≤f(S,T)≤(s?1)(t²+3t). Using constructions from design theory, we establish the exact values, lying near (s?1)(t?1), for f(S,T) when S and T are certain paths or star‐like trees. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 1–16, 2003     

Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers

We study continuity envelopes in spaces of generalised smoothness B_pq^(s,Ψ) and F_pq^(s,Ψ) and give some new characterisations for spaces B_pq^(s,Ψ). The results are applied to obtain sharp asymptotic estimates for approximation numbers of compact embeddings of type id:B_pq^(s₁,Ψ)(U)→B_∞∞^s₂(U), where and U stands for the unit ball in . In case of entropy numbers we can prove two-sided estimates.     

A Boundedness Criterion via Atoms for Linear Operators in Hardy Spaces

Let p∈(0,1] and s≥[n(1/p−1)], where [n(1/p−1)] denotes the maximal integer no more than n(1/p−1). In this paper, the authors prove that a linear operator T extends to a bounded linear operator from the Hardy space H ^p (ℝ ⁿ ) to some quasi-Banach space ℬ if and only if T maps all (p,2,s)-atoms into uniformly bounded elements of ℬ.      

The Virtual Poincaré Polynomials of Homogeneous Spaces

We factor the virtual Poincaré polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t^2u (t²–1)^r Q_G/H(t²) for a polynomial Q_G/H with nonnegative integer coefficients. Moreover, we show that Q_G/H(t²) divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.     

The minimum degree of Ramsey‐minimal graphs

We write H → G if every 2‐coloring of the edges of graph H contains a monochromatic copy of graph G. A graph H is G‐minimal if H → G, but for every proper subgraph H′ of H, H′ ? G. We define s(G) to be the minimum s such that there exists a G‐minimal graph with a vertex of degree s. We prove that s(K_k) = (k ? 1)² and s(K_a,b) = 2 min(a,b) ? 1. We also pose several related open problems. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 167–177, 2007     

A rough hypersingular integral operator with an oscillating factor

We study certain hypersingular integrals TΩ,α,βf defined on all test functions f∈S(Rn), where the kernel of the operator TΩ,α,β has a strong singularity |y|^−n−α(α>0) at the origin, an oscillating factor ei|y|^−β(β>0) and a distribution Ω∈Hr(Sn−1), 0<r<1. We show that TΩ,α,β extends to a bounded linear operator from the Sobolev space to the Lebesgue space Lp for β/(β−α)<p<β/α, if the distribution Ω is in the Hardy space Hr(Sn−1) with 0<r=(n−1)/(n−1+γ)(0<γ?α) and β>2α>0.     

The Relative Brauer Group of an Affine Double Plane

We study algebra classes and divisor classes on a normal affine surface of the form z ² = f(x, y). The affine coordinate ring is T = k[x, y, z]/(z ² ? f), and if R = k[x, y][f ^?1] and S = R[z]/(z ² ? f), then S is a quadratic Galois extension of R. If the Galois group is G, we show that the natural map H¹(G, Cl(T)) → H¹(G, Pic(S)) factors through the relative Brauer group B(S/R) and that all of the maps are onto. Sufficient conditions are given for H¹(G, Cl(T)) to be isomorphic to B(S/R). The groups and maps are computed for several examples.     

Cubature over the sphere in Sobolev spaces of arbitrary order

This paper studies numerical integration (or cubature) over the unit sphere for functions in arbitrary Sobolev spaces H^s(S²), s>1. We discuss sequences of cubature rules, where (i) the rule Q_m(n) uses m(n) points and is assumed to integrate exactly all (spherical) polynomials of degree ≤n and (ii) the sequence (Q_m(n)) satisfies a certain local regularity property. This local regularity property is automatically satisfied if each Q_m(n) has positive weights. It is shown that for functions in the unit ball of the Sobolev space H^s(S²), s>1, the worst-case cubature error has the order of convergence O(n^-s), a result previously known only for the particular case . The crucial step in the extension to general s>1 is a novel representation of , where P_ℓ is the Legendre polynomial of degree ℓ, in which the dominant term is a polynomial of degree n, which is therefore integrated exactly by the rule Q_m(n). The order of convergence O(n^-s) is optimal for sequences (Q_m(n)) of cubature rules with properties (i) and (ii) if Q_m(n) uses m(n)=O(n²) points.     

Nonstationary Stokes system in weighted Sobolev spaces

We consider an initial‐boundary value problem for nonstationary Stokes system in a bounded domain Omega??³ with slip boundary conditions. We assume that Ω is crossed by an axis L. Let us introduce the following weighted Sobolev spaces with finite norms: and where ?(x) = dist{x, L}. We proved the result. Given the external force f∈L_{2, ?µ}(Ω^T), initial velocity v₀∈H(Ω), µ∈?₊\? there exist velocity v∈H(Ω^T) and the pressure p, ?p∈L_{2, ?µ}(Ω^T) and a constant c, independent of v, p, f, such that As we consider the Stokes system in weighted Sobolev spaces the following two things must be used:

• 1. the slip boundary condition and
• 2. the Helmholtz–Weyl decomposition.

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