Weight distribution of some reducible cyclic codes

Let q=p^m where p is an odd prime, m3, k1 and gcd(k,m)=1. Let Tr be the trace mapping from to and . In this paper we determine the value distribution of following two kinds of exponential sums

and

where is the canonical additive character of . As an application, we determine the weight distribution of the cyclic codes and over with parity-check polynomial h₂(x)h₃(x) and h₁(x)h₂(x)h₃(x), respectively, where h₁(x), h₂(x) and h₃(x) are the minimal polynomials of π⁻¹, π⁻² and π^−(pk+1) over , respectively, for a primitive element π of .     

Uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball

The following Dirichlet problem

(1.1)

is considered, where , N≥2, KC²[0,1] and K(r)>0 for 0≤r≤1, , sf(s)>0 for s≠0. Assume moreover that f satisfies the following sublinear condition: f(s)/s>f^′(s) for s≠0. A sufficient condition is derived for the uniqueness of radial solutions of (1.1) possessing exactly k−1 nodes, where . It is also shown that there exists KC^∞[0,1] such that (1.1) has three radial solutions having exactly one node in the case N=3.     

On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian

Let λ be a positive number, and let be a fixed Riesz-basis sequence, namely, (x_j) is strictly increasing, and the set of functions is a Riesz basis (i.e., unconditional basis) for L₂[−π,π]. Given a function whose Fourier transform is zero almost everywhere outside the interval [−π,π], there is a unique sequence in , depending on λ and f, such that the function

is continuous and square integrable on (−∞,∞), and satisfies the interpolatory conditions I_λ(f)(x_j)=f(x_j), . It is shown that I_λ(f)converges to f in , and also uniformly on , as λ→0⁺. In addition, the fundamental functions for the univariate interpolation process are defined, and some of their basic properties, including their exponential decay for large argument, are established. It is further shown that the associated interpolation operators are bounded on for every p[1,∞].     

Regularity results for the gradient of solutions of a class of linear elliptic systems with data

We prove that if a vector-function f belongs to the Morrey space , with , n≥3, N≥2, λ]0,n−2], and u is the solution of the system

then Du belongs to the space , for any , provided the matrix of bounded measurable coefficients (A_ij) has sufficiently small dispersion of the eigenvalues.     

Nonradial large solutions of sublinear elliptic problems

Let p be a nonnegative locally bounded function on , N3, and 0<γ<1. Assuming that the oscillation sup_|x|=rp(x)−inf_|x|=rp(x) tends to zero as r→∞ at a specified rate, it is shown that the equation Δu=p(x)u^γ admits a positive solution in satisfying lim_|x|→∞u(x)=∞ if and only if

     

Eigenvalue problems of nonhomogeneous semilinear elliptic equations in Esteban–Lions domains with holes

In this article, we consider the following eigenvalue problems

('∗

λ' render=n">

where λ>0, N2 and is the upper semi-strip domain with a hole in . Under some suitable conditions on f and h, we show that there exists a positive constant λ^* such that Eq. (*)_λ has at least two solutions if λ(0,λ^*), a unique positive solution if λ=λ^*, and no positive solution if λ>λ^*. We also obtain some further properties of the positive solutions of (*)_λ.     

Asymptotic behavior of Solutions for Hénon systems with nearly critical exponent

We consider in this paper the problem

(0.1)

where Ω is the unit ball in centered at the origin, 0α<pN, β>0, N8, p>1, q_ε>1. Suppose q_ε→q>1 as ε→0⁺ and q_ε,q satisfy respectively

we investigate the asymptotic behavior of the ground state solutions (u_ε,v_ε) of (0.1) as ε→0⁺. We show that the ground state solutions concentrate at a point, which is located at the boundary. In addition, the ground state solution is non-radial provided that ε>0 is small.     

Second-order boundary value problems with nonhomogeneous boundary conditions (II)

Sufficient conditions are given for the existence of solutions of the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition

We prove that the whole plane is divided by a “continuous decreasing curve” Γ into two disjoint connected regions Λ^E and Λ^N such that the above problem has at least one solution for (λ₁,λ₂)Γ, has at least two solutions for (λ₁,λ₂)Λ^EΓ, and has no solution for (λ₁,λ₂)Λ^N. We also find explicit subregions of Λ^E where the above problem has at least two solutions and two positive solutions, respectively.     

On a generalization of Jentzsch’s theorem

Let E be a compact subset of with connected, regular complement and let G(z) denote Green’s function of Ω with pole at ∞. For a sequence (p_n)_nΛ of polynomials with degp_n=n, we investigate the value-distribution of p_n in a neighbourhood U of a boundary point z₀ of E if G(z) is an exact harmonic majorant of the subharmonic functions

in . The result holds for partial sums of power series, best polynomial approximations, maximally convergent polynomials and can be extended to rational functions with a bounded number of poles.     

Positive solutions for one-dimensional -Laplacian boundary value problems with sign changing nonlinearity

This paper deals with the existence of positive solutions for the one-dimensional p-Laplacian

subject to the boundary value conditions:

where _p(s)=|s|^p−2s,p>1. We show that it has at least one or two positive solutions under some assumptions by applying the fixed point theorem. The interesting points are that the nonlinear term f is involved with the first-order derivative explicitly and f may change sign.     

Christoffel-type functions for -orthogonal polynomials for Freud weights

This paper gives upper and lower bounds of the Christoffel-type functions , for the m-orthogonal polynomials for a Freud weight W=e^-Q, which are given as follows. Let a_n=a_n(Q) be the nth Mhaskar–Rahmanov–Saff number, φ_n(x)=max{n^-2/3,1-|x|/a_n}, and d>0. Assume that QC(R) is even, , and for some A,B>1

Then for xR

and for |x|a_n(1+dn^-2/3)

     

Monotone Jacobi parameters and non-Szegő weights

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for a_n≡1, b_n=−Cn^−β (), one has on (−2,2), and near x=2, where

     

RELATIONS BETWEEN PACKING PREMEASURE AND MEASURE ON METRIC SPACE

Let X be a metric space andμa finite Borel measure on X. Let pμq,t and pμq,t be the packing premeasure and the packing measure on X, respectively, defined by the gauge (μB(x,r))q(2r)t, where q, t∈R. For any compact set E of finite packing premeasure the authors prove: (1) if q≤0 then pμq,t(E)=pμq,t(E);(2)if q>0 andμis doubling on E then pμq,t(E) and pμq,t(E) are both zero or neither.     

Linear interpolation and Sobolev orthogonality

There is a strong connection between Sobolev orthogonality and Simultaneous Best Approximation and Interpolation. In particular, we consider very general interpolatory constraints , defined by

where f belongs to a certain Sobolev space, a_ij() are piecewise continuous functions over [a,b], b_ijk are real numbers, and the points t_k belong to [a,b] (the nonnegative integer m depends on each concrete interpolation scheme). For each f in this Sobolev space and for each integer l greater than or equal to the number of constraints considered, we compute the unique best approximation of f in , denoted by p_f, which fulfills the interpolatory data , and also the condition that best approximates f⁽ⁿ⁾ in (with respect to the norm induced by the continuous part of the original discrete–continuous bilinear form considered).     

Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u_t=Δu and v_t=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions

We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with

for p,q>0, 0≤α<1 and 0≤β<p.     

On value sets of polynomials over a field

Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+∞ otherwise. Let A₁,…,A_n be finite nonempty subsets of F, and let

with k{1,2,3,…}, a₁,…,a_nF{0} and degg<k. We show that

When kn and |A_i|i for i=1,…,n, we also have

consequently, if nk then for any finite subset A of F we have

In the case n>k, we propose a further conjecture which extends the Erdős–Heilbronn conjecture in a new direction.     

Quadratic and cubic invariants of unipotent affine automorphisms

Let K be an arbitrary field of characteristic zero, P_n:=K[x₁,…,x_n] be a polynomial algebra, and , for n2. Let σ^′Aut_K(P_n) be given by

It is proved that the algebra of invariants, , is a polynomial algebra in n−1 variables which is generated by quadratic and cubic (free) generators that are given explicitly.Let σAut_K(P_n) be given by

It is well known that the algebra of invariants, , is finitely generated (theorem of Weitzenböck [R. Weitzenböck, Über die invarianten Gruppen, Acta Math. 58 (1932) 453–494]), has transcendence degree n−1, and that one can give an explicit transcendence basis in which the elements have degrees 1,2,3,…,n−1. However, it is an old open problem to find explicit generators for F_n. We find an explicit vector space basis for the quadratic invariants, and prove that the algebra of invariants is a polynomial algebra over in n−2 variables which is generated by quadratic and cubic (free) generators that are given explicitly.The coefficients of these quadratic and cubic invariants throw light on the ‘unpredictable combinatorics’ of invariants of affine automorphisms and of SL₂-invariants.     

Galerkin solution of a singular integral equation with constant coefficients

Galerkin methods are used to approximate the singular integral equation

with solution φ having weak singularity at the endpoint −1, where a, b≠0 are constants. In this case φ is decomposed as φ(x)=(1−x)^α(1+x)^βu(x), where β=−α, 0<α<1. Jacobi polynomials are used in the discussions. Under the conditions fH^μ[−1,1] and k(t,x)H^μ,μ[−1,1]×[−1,1], 0<μ<1, the error estimate under a weighted L² norm is O(n^−μ). Under the strengthened conditions f^″H^μ[−1,1] and , 2α−<μ<1, the error estimate under maximum norm is proved to be O(n^2α−−μ+), where , >0 is a small enough constant.     

Composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps

Let E be a real Banach space and let K be a nonempty closed convex subset of E. Let be N strictly pseudocontractive self-maps of K such that , where F(T_i)={xK:T_ix=x} and let be two real sequence satisfying the conditions:

where L1 is common Lipschitz constant of . For x₀K, let be new implicit process defined by

x_n=α_nx_n−1+(1−α_n)T_ny_n,