Error saturation in Gaussian radial basis functions on a finite interval

Radial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. One restriction is “error saturation” which occurs for many types of RBFs including Gaussian RBFs of the form ?(x;α,h)=exp(−α²(x/h)²): in the limit h→0 for fixed α, the error does not converge to zero, but rather to E_S(α). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases E_S(α).) We show experimentally that the saturation error on the unit interval, x∈[−1,1], is about 0.06exp(−0.47/α²)‖f‖_∞ — huge compared to the O(2π/α²)exp(−π²/[4α²]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing α?1, the “flat limit”, but the condition number of the interpolation matrix explodes as O(exp(π²/[4α²])). The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance δ, then .     

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.     

On fractional powers of generators of fractional resolvent families

We show that if −A generates a bounded α-times resolvent family for some α∈(0,2], then −Aβ generates an analytic γ-times resolvent family for and γ∈(0,2). And a generalized subordination principle is derived. In particular, if −A generates a bounded α-times resolvent family for some α∈(1,2], then −A1/α generates an analytic C₀-semigroup. Such relations are applied to study the solutions of Cauchy problems of fractional order and first order.     

An overshoot approach to recurrence and transience of Markov processes

We develop criteria for recurrence and transience of one-dimensional Markov processes which have jumps and oscillate between +∞ and −∞. The conditions are based on a Markov chain which only consists of jumps (overshoots) of the process into complementary parts of the state space.In particular, we show that a stable-like process with generator −(−Δ)^α(x)/2 such that α(x)=α for x<−R and α(x)=β for x>R for some R>0 and α,β∈(0,2) is transient if and only if α+β<2, otherwise it is recurrent.As a special case, this yields a new proof for the recurrence, point recurrence and transience of symmetric α-stable processes.     

An arithmetical property of powers of Salem numbers

Let ζ be a nonzero real number and let α be a Salem number. We show that the difference between the largest and smallest limit points of the fractional parts of the numbers ζαn, when n runs through the set of positive rational integers, can be bounded below by a positive constant depending only on α if and only if the algebraic integer α−1 is a unit.     

On the bin packing problem with a fixed number of object weights

We consider a bin packing problem where the number of different object weights is fixed to C. We analyze a simple approximate approach and show that it leads to an asymptotically exact polynomial algorithm with absolute error 0 if C = 2, at most 1 if 1 < C ? 6, and at most 1 + ⌊(C − 1)/3⌋ if C > 6. A consequence of our analysis is a new upper bound on the gap between the optimal value of the problem at hand and the round-up of the optimal value of the linear relaxation of its Gilmore–Gomory formulation.     

A nonoscillation theorem for superlinear Emden-Fowler equations with near-critical coefficients

We are interested in the oscillatory behavior of solutions of the Emden-Fowler equation y^″+a(x)|y|^γ−1y=0, γ>1, where a(x) is a positive continuous function on (0,∞). In the special case when the coefficient a(x) is a power of x, i.e. a(x)=xα for some constant α, the value α^∗=−(γ+3)/2 plays a critical role: The equation has both oscillatory and nonoscillatory solutions if α>α^∗, while all solutions are nonoscillatory if α<α^∗. When a(x) is close to the critical exponent, one of the known results is that if a(x)=x−(γ+3)/2log−σ(x), where σ>0, then all solutions are nonoscillatory. In this paper, this result is further extended to include a class of coefficients in which the above condition with log(x) can be replaced by loglog(x), or logloglog(x) and so on.     

A fiberwise analogue of the Borsuk-Ulam theorem for sphere bundles over a 2-cell complex

For a vector bundle α, let indα denote the largest integer m for which there exists a Z/2-map from Sm−1 to S(α). We prove that the equality indα=dimα holds for every vector bundle α over the complex Sn−1k_∪en, where n?2 and k≠0, if and only if either k is even and n≠2,3,4,8 or k is odd.     

Operators with rough singular kernels

For α>0, we study the singular integral operators TΩ,α and the Marcinkiewicz integral operator μΩ,α. The kernels of these operators behave like |y|^−n−α near y=0, and contain a distribution Ω on the unit sphere Sn−1. We prove that if Ω∈Hr(Sn−1)(r=(n−1)/(n−1+α)) satisfying certain cancellation condition, then both TΩ,α and μΩ,α can be extend to be the bounded operators from the Sobolev space to the Lebesgue space Lp(Rn). The result improves and extends some known results.     

Bound states and the Szeg? condition for Jacobi matrices and Schrödinger operators

For Jacobi matrices with a_n=1+(−1)ⁿαn^−γ, b_n=(−1)ⁿβn^−γ, we study bound states and the Szeg? condition. We provide a new proof of Nevai's result that if , the Szeg? condition holds, which works also if one replaces (−1)ⁿ by . We show that if α=0, β≠0, and , the Szeg? condition fails. We also show that if γ=1, α and β are small enough ( will do), then the Jacobi matrix has finitely many bound states (for α=0, β large, it has infinitely many).     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

Gallai colorings of non-complete graphs

Gallai-colorings of complete graphs-edge colorings such that no triangle is colored with three distinct colors-occur in various contexts such as the theory of partially ordered sets (in Gallai’s original paper), information theory and the theory of perfect graphs. We extend here Gallai-colorings to non-complete graphs and study the analogue of a basic result-any Gallai-colored complete graph has a monochromatic spanning tree-in this more general setting. We show that edge colorings of a graph H without multicolored triangles contain monochromatic connected subgraphs with at least (α(H)²+α(H)−1)⁻¹|V(H)| vertices, where α(H) is the independence number of H. In general, we show that if the edges of an r-uniform hypergraph H are colored so that there is no multicolored copy of a fixed F then there is a monochromatic connected subhypergraph H₁⊆H such that |V(H₁)|≥c|V(H)| where c depends only on F, r, and α(H).     

Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions

In this paper we show that b∈Lipβ,μ if and only if the commutator [b,T] of the multiplication operator by b and the singular integral operator T is bounded from Lp(μ) to Lq(μ1−q), where 1<p<q<∞, 0<β<1 and 1/q=1/p−β/n. Also we will obtain that b∈Lipβ,μ if and only if the commutator [b,Iα] of the multiplication operator by b and the fractional integral operator Iα is bounded from Lp(μ) to Lr(μ1−(1−α/n)r), where 1<p<∞, 0<β<1 and 1/r=1/p−(β+α)/n with 1/p>(β+α)/n.     

Trotter-Kato product formula and fractional powers of self-adjoint generators

Let A and B be non-negative self-adjoint operators in a Hilbert space such that their densely defined form sum obeys dom(H^α)⊆dom(A^α)∩dom(B^α) for some α∈(1/2,1). It is proved that if, in addition, A and B satisfy dom(A^1/2)⊆dom(B^1/2), then the symmetric and non-symmetric Trotter-Kato product formula converges in the operator norm:

||(e−tB/2ne−tA/ne−tB/2n)n−e−tH||=O(n−(2α−1))||(e−tA/ne−tB/n)n−e−tH||=O(n−(2α−1))     

Eigenfunction expansions and spectral projections for isotropic elasticity outside an obstacle

We consider the operator −Δ−αgraddiv acting on an exterior domain Ω in Rn (with α>0 and n=2,3) subject to Dirichlet boundary conditions. The spectral resolution for the operator is written in terms of an expansion of generalized eigenfunctions.     

Error bounds for high-resolution quantization with Rényi-α-entropy constraints

We consider the problem of optimal quantization with norm exponent r > 0 for Borel probability measures on ? ^d under constrained Rényi-α-entropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the special cases α = 0 (memory-constrained quantization) and α = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter α ∈ [1+r/d,∞]. For α ∈ [0,1 + r/d] we specify the asymptotical order of the optimal quantization error under large entropy bound. The optimal quantization error is decreasing exponentially fast with the entropy bound and the exact rate is determined for all α ∈ [0, ∞].     

Uniqueness of positive radial solutions for Dirichlet problems on annular domains

This paper deals with the uniqueness of positive radial solutions to Dirichlet problems on annular domains in Rn, n?3. As an application we can obtain the results to equation Δu+up−α₁u−α₀=0, where p>1, α₁?0, α₀?0 and α₁+α₀>0.     

Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations

In this paper, existence and attractiveness of solutions for quadratic Urysohn fractional integral equations on an unbounded interval are obtained by virtue of Tichonov fixed point theorem and suitable conjunction of the well known measure ω₀(X) and the spaces C(R⁺). Further, three certain solutions sets X_L,γ, X_1,α and X_{1,(1−(α+v))}, which tending to zero at an appropriate rate t^−ν (ν > 0), ν = γ (or α or 1 − (α + v)) as t → ∞, are introduced and stability of solutions for quadratic Urysohn fractional integral equations are obtained based on these solutions sets respectively by applying Schauder fixed point theorem via some easy checked conditions. An example is given to illustrate the results.     

Fully discrete finite element scheme for Maxwell's equations with non-linear boundary condition

We study a full Maxwell's system accompanied with a non-linear degenerate boundary condition, which represents a generalization of the classical Silver-Müller condition for a non-perfect conductor. The relationship between the normal components of electric E and magnetic H field obeys the following power law ν×H=ν×(|E×ν|^α−1E×ν) for some α∈(0,1]. We establish the existence and uniqueness of a weak solution in a suitable function spaces under the minimal regularity assumptions on the boundary Γ and the initial data E₀ and H₀. We design a non-linear time discrete approximation scheme and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time discretization. As a next step we study the fully discrete problem using curl-conforming edge elements and derive the corresponding error estimates. Finally we present some numerical experiments.