A generalized mean value inequality for subharmonic functions

If u ≥ 0 is subharmonic on a domain Ω in ⁿ and 0 < p < 1, then it is well-known that there is a constant C(n,p) ≥ 1 such u(x)^p ≤ C)n,p) MV )u^p,B(x,r)) for each ball B(x,r)) Ω. We show more generally that a similar result holds for functions ψ : ₊ → ₊ may be any surjective, concave function whose inverse ψ⁻¹ satisfies the Δ₂-condition.     

Product representations of polynomials

For a fixed polyomial , let ρ_k(N) denote the maximum size of a set A{1,2,…,N} such that no product of k distinct elements of A is in the value set of f. In this paper, we determine the asymptotic behaviour of ρ_k(N) for a wide class of polynomials. Our results generalize earlier theorems of Erdős, Sós and Sárközy.     

A Pair in a Crowd of Unit Balls

Let F be a family of mutually nonoverlapping unit balls in the n -dimensional Euclidean space Rⁿ. The distance between the centres of A,B   F is denoted by d(A, B). We prove, among others, that if d(A, B)  <  4 and n ≥  5, then A andB are always visible from each other, that is, a light ray emanating from the surface of A reaches B without being blocked by other unit balls. Furthermore, if d(A, B)  < 2n / 2, then any small “shake’ of F can make A, B visible from each other.     

A pointwise convergence for Sobolev space functions

Let 1<p<∞, and k,m be positive integers such that 0(k−2m)pn. Suppose ΩRⁿ is an open set, and Δ is the Laplacian operator. We will show that there is a sequence of positive constants c_j such that for every f in the Sobolev space W^k,p(Ω), for all xΩ except on a set whose Bessel capacity B_k−2m,p is zero.     

An Inequality on the Size of a Set in a Cartesian Product

Let Ω be a finite subset of the Cartesian productW₁  ×   × W_nof n sets. ForA    {1, 2, , n }, denote by Ω_Athe projection ofΩ onto the Cartesian product of W_i, i   A. Generalizing an inequality given in an article by Shen, we prove that | Ω |² ≤  |Ω_A1 || Ω_Ak| provided that { A₁, , A_k} is a double cover of {1, 2, , n }. This inequality is applied to give some bounds on the numbers of special subgraphs of a graph.     

The uniform closure of non-dense rational spaces on the unit interval

Let P_n denote the set of all algebraic polynomials of degree at most n with real coefficients. Associated with a set of poles a₁,a₂,…,a_n R[-1,1] we define the rational function spaces Associated with a set of poles a₁,a₂,… R[-1,1], we define the rational function spacesIt is an interesting problem to characterize sets a₁,a₂,… R[-1,1] for which P(a₁,a₂,…) is not dense in C[-1,1], where C[-1,1] denotes the space of all continuous functions equipped with the uniform norm on [-1,1]. Akhieser showed that the density of P(a₁,a₂,…) is characterized by the divergence of the series .In this paper, we show that the so-called Clarkson–Erdős–Schwartz phenomenon occurs in the non-dense case. Namely, if P(a₁,a₂,…) is not dense in C[-1,1], then it is “very much not so”. More precisely, we prove the following result.Theorem Let a₁,a₂,… R[-1,1]. Suppose P(a₁,a₂,…) is not dense in C[-1,1], that is,Then every function in the uniform closure of P(a₁,a₂,…) in C[-1,1] can be extended analytically throughout the set C -1,1,a₁,a₂,… .     

Polyhedral results on single node variable upper-bound flow models with allowed configurations

In this paper we investigate the convex hull of single node variable upper-bound flow models with allowed configurations. Such a model is defined by a set , where ρ is one of , = or , and Z{0,1}ⁿ consists of the allowed configurations. We consider the case when Z consists of affinely independent vectors. Under this assumption, a characterization of the non-trivial facets of the convex hull of X_ρ(Z) for each relation ρ is provided, along with polynomial time separation algorithms. Applications in scheduling and network design are also discussed.     

LOWER BOUNDS FOR SUP＋INF AND SUP＊INF AND AN EXTENSION OF CHEN-LIN RESULT IN DIMENSION 3

We give two results about Harnack type inequalities. First, on Riemannian surfaces, we have an estimate of type sup ＋ inf. The second result concern the solutions of prescribed scalar curvature equation on the unit ball of Rn with Dirichlet condition. Next, we give an inequality of type （supK ^u）^2s-1 × infπu ≤ c for positive solutions of △u = V u^5 on Ω belong toR^3, where K is a compact set of Ω and V is s-Holderian, s ∈] - 1/2, 1]. For the case s = 1/2 and Ω = S3, we prove that, if minΩ u 〉 m 〉 0 （for some particular constant m 〉 0）, and the H¨olderian constant A of V tends to 0 （in certain meaning）, we have the uniform boundedness of the supremum of the solutions of the previous equation on any compact set of Ω.     

On a generalization of a theorem of Erdős and Fuchs

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers, let k≥2 be a fixed integer and denote by r_k(A,n) the number of solutions of a_i1+a_i2++a_ik≤n. Montgomery and Vaughan proved that r₂(A,n)=cn+o(n^1/4) cannot hold for any constant c>0. In this paper, we extend this result to k>2.     

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 constants in some inequalities for intermediate derivatives on a finite interval

Let L_q (1q<∞) be the space of functions f measurable on I=[−1,1] and integrable to the power q, with normL_∞ is the space of functions measurable on I with normWe denote by AC the set of all functions absolutely continuous on I. For nN, q[1,∞] we setW_n,q={f:f⁽ⁿ⁻¹⁾AC, f⁽ⁿ⁾L_q}.In this paper, we consider the problem of accuracy of constants A, B in the inequalities (1)|| f^(m)||_qA|| f||_p+B|| f^(m+k+1)||_r, mN, kW; p,q,r[1,∞], fW_m+k+1,r.     

Markov-type inequalities on certain irrational arcs and domains

Let denote the set of real algebraic polynomials of d variables and of total degree at most n. For a compact set KR^d set Then the Markov factors on K are defined by (Here, as usual, S^d-1 stands for the Euclidean unit sphere in R^d.) Furthermore, given a smooth curve ΓR^d, we denote by D_TP the tangential derivative of P along Γ (T is the unit tangent to Γ). Correspondingly, consider the tangential Markov factor of Γ given by Let . We prove that for every irrational number α>0 there are constants A,B>1 depending only on α such that for every sufficiently large n.Our second result presents some new bounds for M_n(Ω_α), where (d=2,α>1). We show that for every α>1 there exists a constant c>0 depending only on α such that M_n(Ω_α)n^clogn.     

Intersective polynomials and the polynomial Szemerédi theorem

Let be a family of polynomials such that , i=1,…,r. We say that the family P has the PSZ property if for any set with there exist infinitely many such that E contains a polynomial progression of the form {a,a+p₁(n),…,a+p_r(n)}. We prove that a polynomial family P={p₁,…,p_r} has the PSZ property if and only if the polynomials p₁,…,p_r are jointly intersective, meaning that for any there exists such that the integers p₁(n),…,p_r(n) are all divisible by k. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If are jointly intersective integral polynomials, then for any finite partition of , there exist i{1,…,k} and a,nE_i such that {a,a+p₁(n),…,a+p_r(n)}E_i.     

On the continuation of solutions for elliptic equations in two variables

We consider nonlinear elliptic differential equations of second order in two variables

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. Supposing analyticity of F, we prove analyticity of the real solution z=z(x,y) in the open set Ω. Furthermore, we show that z may be continued as a real analytic solution for F=0 across the real analytic boundary arc Γ∂Ω, if z satisfies one of the boundary conditions z= or z_n=ψ(x,y,z,z_t) on Γ with real analytic functions and ψ, respectively (z_n denotes the derivative of z w.r.t. the outer normal n on Γ and z_t its derivative w.r.t. the tangent). The proof is based on ideas of H. Lewy combined with a uniformization method. Studying quasilinear equations, we get somewhat better results concerning the initial regularity of the given solution and a little more insight.     

-Linked planar graphs

A graph G is (m,n)-linked if for any two disjoint subsets R,BV(G) with |R|m and |B|n, G has two disjoint connected subgraphs containing R and B, respectively. We shall prove that a planar graph with at least six vertices is (3,3)-linked if and only if G is 4-connected and maximal.     

Characterisations of Chebyshev sets in c0

A subset MX of a normed linear space X is a Chebyshev set if, for every xX, the set of all nearest points from M to x is a singleton. We obtain a geometrical characterisation of approximatively compact Chebyshev sets in c₀. Also, given an approximatively compact Chebyshev set M in c₀ and a coordinate affine subspace Hc₀ of finite codimension, if M∩H≠, then M∩H is a Chebyshev set in H, where the norm on H is induced from c₀.     

More on the revised GCH and the black box

We strengthen the revised GCH theorem by showing, e.g., that for , for all but finitely many regular κ<_ω, it holds that “λ is accessible on cofinality κ” in some weak sense (see below).As a corollary, λ=2^μ=μ⁺>_ω implies that the diamond holds on λ when restricted to cofinality κ for all but finitely many .We strengthen previous results on the black box and the middle diamond: previously it was established that these principles hold on for sufficiently large n; here we succeed in replacing a sufficiently large _n with a sufficiently large _n.The main theorem, concerning the accessibility of λ on cofinality κ, Theorem 3.1, implies as a special case that for every regular λ>_ω, for some κ<_ω, we can find a sequence such that , , and we can fix a finite set of “exceptional” regular cardinals θ<_ω so that if Aλ satisfies |A|<_ω, there is a pair-coloring so that for every -monochromatic BA with no last element, letting δ:=supB it holds that —provided that is not one of the finitely many “exceptional” members of .     

Values of the Pukánszky invariant in free group factors and the hyperfinite factor

Let AMB(L²(M)) be a maximal abelian self-adjoint subalgebra (masa) in a type II₁ factor M in its standard representation. The abelian von Neumann algebra generated by A and JAJ has a type I commutant which contains the projection onto L²(A). Then decomposes into a direct sum of type I_n algebras for n{1,2,…,∞}, and those n's which occur in the direct sum form a set called the Pukánszky invariant, Puk(A), also denoted Puk_M(A) when the containing factor is ambiguous. In this paper we show that this invariant can take on the values S{∞} when M is both a free group factor and the hyperfinite factor, and where S is an arbitrary subset of . The only previously known values for masas in free group factors were {∞} and {1,∞}, and some values of the form S{∞} are new also for the hyperfinite factor.We also consider a more refined invariant (that we will call the measure-multiplicity invariant), which was considered recently by Neshveyev and Størmer and has been known to experts for a long time. We use the measure-multiplicity invariant to distinguish two masas in a free group factor, both having Pukánszky invariant {n,∞}, for arbitrary .     

An isomorphic characterization of L-spaces

We show that a sequentially (τ)-complete topological vector lattice X_τ is isomorphic to some L¹(μ), if and only if the positive cone can be written as X₊ = ₊B for some convex, (τ)-bounded, and (τ)-closed set B X₊ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y₊ = ₊ (for u >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.     

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.