On the structure of semi-invariant polynomials in Ore extensions

Let D be an X-outer S-derivation of a prime ring R, where S is an automorphism of R. The following is proved among other things: The degree of the minimal semi-invariant polynomial of the Ore extension R[X;S,D] is ν if charR=0, and is p^kν for some k0 if charR=p2, where ν is the least integer ν1 such that S^νDS^−ν−D is X-inner. A similar result holds for cv-polynomials. These are done by introducing the new notion of k-basic polynomials for each integer k0, which enable us to analyze semi-invariant polynomials inductively.     

Representation of Reproducing Kernels and the Lebesgue Constants on the Ball

For the weight function (1−x²)^μ−1/2 on the unit ball, a closed formula of the reproducing kernel is modified to include the case −1/2<μ<0. The new formula is used to study the orthogonal projection of the weighted L² space onto the space of polynomials of degree at most n, and it is proved that the uniform norm of the projection operator has the growth rate of n^(d−1)/2 for μ<0, which is the smallest possible growth rate among all projections, while the rate for μ0 is n^μ+(d−1)/2.     

Refined blowup criteria and nonsymmetric blowup of an aggregation equation

We consider an aggregation equation in , d2, with fractional dissipation: u_t+(uK*u)=−νΛ^γu, where ν0, 0<γ<1, and K(x)=e^−|x|. We prove a refined blowup criteria by which the global existence of solutions is controlled by its norm, for any . We prove the finite time blowup of solutions for a general class of nonsymmetric initial data. The argument presented works for both the inviscid case ν=0 and the supercritical case ν>0 and 0<γ<1. Additionally, we present new proofs of blowup which does not use free energy arguments.     

Variations on a theme of Jost and Pais

We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set , , n2, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants associated with operators in L²(Ω;dⁿx), , to modified Fredholm determinants associated with operators in L²(∂Ω;dⁿ⁻¹σ), n2. Applications involving the Birman–Schwinger principle and eigenvalue counting functions are discussed.     

Existence of nonstationary bubbles in higher dimensions

We are interested in the existence of travelling-waves for the nonlinear Schrödinger equation in R^N with “ψ³−ψ⁵”-type nonlinearity. First, we prove an abstract result in critical point theory (a local variant of the classical saddle-point theorem). Using this result, we get the existence of travelling-waves moving with sufficiently small velocity in space dimension N4.     

Orthogonal graphs of odd characteristic and their automorphisms

The (isotropic) orthogonal graph O(2ν+δ,q) over of odd characteristic, where ν1 and δ=0,1 or 2 is introduced. When ν=1, O(21+δ,q) is a complete graph. When ν2, O(2ν+δ,q) is strongly regular and its parameters are computed, as well as its chromatic number. The automorphism groups of orthogonal graphs are also determined.     

A supplement to the Davis–Gut law

Let {X,X_n;n1} be a sequence of i.i.d. real-valued random variables and set , n1. Let h() be a positive nondecreasing function such that . Define Lt=log_emax{e,t} for t0. In this note we prove that

if and only if E(X)=0 and E(X²)=1, where , t1. When h(t)≡1, this result yields what is called the Davis–Gut law. Specializing our result to h(t)=(Lt)^r, 0<r1, we obtain an analog of the Davis–Gut law.     

Rational Approximation with Varying Weights III

Approximation by weighted rationals of the form wⁿr_n, where r_n=p_n/q_n, p_n and q_n are polynomials of degree at most [αn] and [βn], respectively, and w is an admissible weight, is investigated on compact subsets of the real line for a general class of weights and given α0, β0, with α+β>0. Conditions that characterize the largest sets on which such approximation is possible are given. We apply the general theorems to Laguerre and Freud weights.     

Global well-posedness and scattering for the defocusing -subcritical Hartree equation in

We prove the global well-posedness and scattering for the defocusing -subcritical (that is, 2<γ<3) Hartree equation with low regularity data in , d3. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space with s>4(γ−2)/(3γ−4), which also scatters in both time directions. This improves the result in [M. Chae, S. Hong, J. Kim, C.W. Yang, Scattering theory below energy for a class of Hartree type equations, Comm. Partial Differential Equations 33 (2008) 321–348], where the global well-posedness was established for any s>max(1/2,4(γ−2)/(3γ−4)). The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solution Iu, instead of an interaction Morawetz estimate for the solution u, and that we make careful analysis of the monotonicity property of the multiplier m(ξ)ξ^p. As a byproduct of our proof, we obtain that the H^s norm of the solution obeys the uniform-in-time bounds.     

Fast adaptive algorithms in the non-standard form for multidimensional problems

We present a fast, adaptive multiresolution algorithm for applying integral operators with a wide class of radially symmetric kernels in dimensions one, two and three. This algorithm is made efficient by the use of separated representations of the kernel. We discuss operators of the class (−Δ+μ²I)^−α, where μ0 and 0<α<3/2, and illustrate the algorithm for the Poisson and Schrödinger equations in dimension three. The same algorithm may be used for all operators with radially symmetric kernels approximated as a weighted sum of Gaussians, making it applicable across multiple fields by reusing a single implementation. This fast algorithm provides controllable accuracy at a reasonable cost, comparable to that of the Fast Multipole Method (FMM). It differs from the FMM by the type of approximation used to represent kernels and has the advantage of being easily extendable to higher dimensions.     

Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity

We consider the semilinear elliptic equation Δu=h(u) in Ω{0}, where Ω is an open subset of (N2) containing the origin and h is locally Lipschitz continuous on [0,∞), positive in (0,∞). We give a complete classification of isolated singularities of positive solutions when h varies regularly at infinity of index q(1,C_N) (that is, lim_u→∞h(λu)/h(u)=λ^q, for every λ>0), where C_N denotes either N/(N−2) if N3 or ∞ if N=2. Our result extends a well-known theorem of Véron for the case h(u)=u^q.     

Viscosity approximation to common fixed points of a nonexpansive semigroup with a generalized contraction mapping

The purpose of this paper is to introduce and construct the implicit and explicit viscosity iterative processes by a generalized contraction mapping f and a nonexpansive semigroup {T(t):t0}, and to prove that under suitable conditions these iterative processes converge strongly to a unique common fixed point of {T(t):t0} in reflexive Banach spaces which admits a weakly sequentially continuous duality mapping.     

On Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}ⁿ_j=1 be n distinct points in a compact set K and letL_n[·] denote the corresponding Lagrange interpolation operator. Let v be a suitably restricted function on K. What conditions on the array {x_jn}_1jn, n1 ensure the existence of p>0 such that lim_n→∞ (f−L_n[f]) v_Lp(K)=0 for very continuous f: K→ ? We show that it is necessary and sufficient that there exists r>0 with sup_n1 π_nv_Lr(K) ∑ⁿ_j=1 (1/|π′_n| (x_jn))<∞. Here for n1, π_n is a polynomial of degree n having {x_jn}ⁿ_j=1 as zeros. The necessity of this condition is due to Ying Guang Shi.     

On triconnected and cubic plane graphs on given point sets

Let S be a set of n4 points in general position in the plane, and let h<n be the number of extreme points of S. We show how to construct a 3-connected plane graph with vertex set S, having max{3n/2,n+h−1} edges, and we prove that there is no 3-connected plane graph on top of S with a smaller number of edges. In particular, this implies that S admits a 3-connected cubic plane graph if and only if n4 is even and hn/2+1. The same bounds also hold when 3-edge-connectivity is considered. We also give a partial characterization of the point sets in the plane that can be the vertex set of a cubic plane graph.     

On a Conjecture of Ditzian and Runovskii

Let M_θ be the mean operator on the unit sphere in ⁿ, n3, which is an analogue of the Steklov operator for functions of single variable. Denote by D the Laplace–Beltrami operator on the sphere which is an analogue of second derivative for functions of single variable. Ditzian and Runovskii have a conjecture on the norm of the operator θ²D(M_θ)^m, m2 from X=L^p (1p∞) to itself which can be expressed as

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. We give a proof of this conjecture.     

Uniform Kazhdan constant for some families of linear groups

Let R be a ring generated by l elements with stable range r. Assume that the group EL_d(R) has Kazhdan constant ₀>0 for some dr+1. We prove that there exist (₀,l)>0 and , s.t. for every nd, EL_n(R) has a generating set of order k and a Kazhdan constant larger than . As a consequence, we obtain for where n3, a Kazhdan constant which is independent of n w.r.t. generating set of a fixed size.     

Best Harmonic and Superharmonic L-Approximants in Strips

Let Ω denote the open strip (−1, 1)× ⁿ⁻¹, where n2. We completely solve the problem of characterizing a best harmonic L¹-approximant to a subharmonic function s on Ω (all functions are assumed to be continuous and integrable on Ω). This characterization was previously known only under highly restrictive hypotheses on s. The approach of this paper is based, in part, on ideas used recently to solve the corresponding problem for the unit ball. However, the unboundedness of Ω presents difficulties which require the use of new techniques and recent results from other branches of harmonic approximation theory. Superharmonic L¹-approximation of subharmonic functions is also treated.     

Primitive normal polynomials with a prescribed coefficient

In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let n15 be a positive integer and q a prime power. We prove that for any aF_q and any 1m<n, there exists a primitive normal polynomial f(x)=xⁿ−σ₁xⁿ⁻¹++(−1)ⁿ⁻¹σ_n−1x+(−1)ⁿσ_n such that σ_m=a, with the only exceptions σ₁≠0. The theory can be extended to polynomials of smaller degree too.     

Existence of HSOLSSOMs of type

In this paper we investigate the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 2ⁿu¹ (HSOLSSOM(2ⁿu¹)). For u2, necessary conditions for existence of such an HSOLSSOM are that u must be even and n3u/2+1. Xu Yunqing and Hu Yuwang have shown that these HSOLSSOMs exist whenever either (1) n9 and n3u/2+1 or (2) n263 and n2(u-2). In this paper we show that in (1) the condition n9 can be extended to n30 and that in (2), the condition n263 can be improved to n4, except possibly for 19 pairs (n,u), the largest of which is (53,28).     

Regularity of smooth curves in biprojective spaces

Maclagan and Smith [D. Maclagan, G.G. Smith, Multigraded Castelnuovo–Mumford regularity, J. Reine Angew. Math. 571 (2004) 179–212] developed a multigraded version of Castelnuovo–Mumford regularity. Based on their definition we will prove in this paper that for a smooth curve (a,b2) of bidegree (d₁,d₂) with nondegenerate birational projections the ideal sheaf is (d₂−b+1,d₁−a+1)-regular. We also give an example showing that in some cases this bound is the best possible.