Partial regularity for certain classes of polyconvex functionals related to nonlinear elasticity

Consider the variational integral where Ω⊂ℝ ⁿ andp≥n≥2. H: (0, ∞)→[0, ∞) is a smooth convex function such that . We approximateJ by a sequence of regularized functionalsJ _δ whose minimizers converge strongly to anJ-minimizing function and prove partial regularity results forJ _δ-minimizers.     

Weak infinite powers of Blaschke products

Letb be a Blaschke product with zeros {z _n } in the open unit disk Δ. Let be the set of sequences of non-negative integersp=(p ₁,p ₂,…) such that ∑ _n=1 ^∞ p _n (1 − |z _n |) < ∞ andp _n →∞ asn→∞. We study the class of weak infinite powers ofb, Properties of these classes depend on the setS(b) of the cluster points in ∂Δ of {z _n }. It is proved thatS(b)=∂Δ if and only if , the Douglas algebra generated by . Also, it is proved thatdθ(S(b))=0 if and only if there exists an interpolating Blaschke productB such that .     

On extremal properties for the polar derivative of polynomials

If p(z) is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then it is proved[5] that max |z|=1 |p′(z)| ≤ kn1n + kn m|z|=ax1 |p(z)|. In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type p(z) = cnzn + ∑n j=μ cn jzn j, 1 ≤μ≤ n. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.     

On the maximum modulus of polynomials not vanishing inside the unit circle

A well-known theorem of Ankeney and Rivlin states that if p(z) is a polynomial of degree n, such that p(z)≠0 for |z|<1, then . In this paper we improve this bound.     

Onextensions of the Gale-Berlekamp switching problem and constants ofl q —spaces

For positive integersn, m and realp≥1, let Upper and lower bounds for this quantity are derived, extending results of Brown and Spencer forB ₁(n,n), corresponding to the Gale-Berlekamp switching problem. For a Minkowski spaceM of dimensionm, define a quantity investigated by Dvoretzky and Rogers.     

Some inequalities for polynomials satisfying p(z)≡znp(1/z)

Let be a polynomial degreen and let . Then according to Bernstein’s inequality ‖p’‖≤n‖p‖. It is a well known open problem to obtain inequality analogous to Bernstein’s inequality for the class II_n of polynomials satisfying p(z)≡zⁿp(1/z). Here we obtain an inequality analogous to Bernstein’s inequality for a subclass of II_n. Our results include several of the known results as special cases.     

Limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival function

LetX ₁,X ₂, ...,X _n be a sequence of nonnegative independent random variables with a common continuous distribution functionF. LetY ₁,Y ₂, ...,Y _n be another sequence of nonnegative independent random variables with a common continuous distribution functionG, also independent of {X _i }. We can only observeZ _i =min(X _i ,Y _i ), and . LetH=1−(1−F)(1−G) be the distribution function ofZ. In this paper, the limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival functionS(t) are given. It is shown that (1)P(δ_(n)=1 i.o.)=0 ifF(τ _H ) < 1 andP(δ _n =0 i.o. )=0 ifG(τH) > 1 where δ_(n) is the corresponding indicator function of and have the same order a.s., where {T _n } is a sequence of constants such that 1−H(T _n )=n ^−α(logn)^β(log logn)^γ.     

Asymptotic properties of the norm of the extremum of a sequence of normal random functions

Under additional conditions on a bounded normally distributed random function X = X( t), t ∈ T, we establish a relation of the form

Sufficient and necessary conditions for limit laws on lag sums of I. I. D. R. V. S

Let {X, X _n ;n>-1} be a sequence of i.i.d.r.v.s withEX=0 andEX ²=σ²(0 < σ < ∞). we obtain some sufficient and necessary conditions for

Exact constants in inequalities of the jackson type for quadrature formulas

We prove that if is the error of a simple quadrature formula and ω(ε, δ)₁ is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: whereD _r is the Bernoulli kernel.     

Growth of power series with square root gaps

For entire functionsf whose power series have Hadamard gaps with ratio ≥1+α>1, Gaier has shown that the condition |f(x)|≤e ^x forx≥0 implies |f(z)|≤C _αe^|z| (*) for allz. Here the result is extended to the case of square root gaps, that is, , with , where α>0. Smaller gaps cannot work. In connection with his proof of the general high indices theorem for Borel summability, Gaier had shown that square root gaps imply . Having such an estimate, one can adapt Pitt’s Tauberian method for the restricted Borel high indices theorem to show that, in fact, , which implies (*). The author also states an equivalent distance formula involving monomialsx ^pke^−xinL ^∞ (0, ∞).     

The pointwise estimates on walsh overconvergence

Let f∈A_p. For any positive integer l, the quantity Δ_1,n−1(f:z) has been studied extensively. Here we give some quantitative estimates for and investigate some pointwise estimates of Δ _l,n−1 ^(r) (f;z). Supported by National Science Foundation of China     

The individual ergodic theorem forp-sequences

Let (Ω,f,P) be a probability space and letT be a measure-preserving weak mixing transformation. We define a large class of sequences of integers calledp-sequences, such that iff∈L ₁ there exists a set Ω′⊂Ω of probability one and for ω∈Ω′ we have for everyp-sequence {k_n}.     

Optimal constant in a new estimate for the degree

In this paper, we prove the estimate

About the ratio of the size of a maximum antichain to the size of a maximum level in finite partially ordered sets

LetP be a finite partially ordered set. The lengthl(x) of an elementx ofP is defined by the maximal number of elements, which lie in a chain withx at the top, reduced by one. Letw(P) (d(P)) be the maximal number of elements ofP which have the same length (which form an antichain). Further let . The numbers and as well as all partially ordered sets for which these maxima are attained are determined.     

Optimal ray sequences of rational functions connected with the Zolotarev problem

Given two compact disjoint subsetsE ₁,E ₂ of the complex plane, the third problem of Zolotarev concerns estimates for the ratio

Kernels of sequences of complex numbers and their regular transformations

It is proved that , where U(a, r) is the ball of radius r with center at the pointa, is the smallest closed convex set containing the kernel of any sequence {y_n} obtained from the sequence {x_n} by means of a regular transformation (c_nk) satisfying the condition , where x, x_n, c_nk (n, k=1, 2,...) are complex numbers.Traslated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 815–823, December, 1977.     

An asymptotic method for calculating limits of maximal means

For a continuous almost periodic function , we show that the function

The value regions of initial coefficients in a certain class of meromorphic functions

Let M_k,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+a_o+a₁z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition where J_λ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class M_k,λ we consider sorne coefficient problems and problems concerning distortion theorems. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 91–96. Translated by N. Yu. Netsvetaev.     

About entire functions with specialL 2-properties on one-dimensional subspaces of ℂ n

In this paper we consider special elements of the Fock space #x2131; _n . That is the space of entire functionsf:ℂ: ⁿ →ℂ, such that the followingL ²- condition is satisfied: . Here we show that there exists an entire functiong:ℂ ⁿ →ℂ such that for every one-dimensional subspace Π⊂ℂ ⁿ and for all 0<∈<2 we have , but in the limit case ∈=0 we have . This result is analogue to a result from [1]. There holomorphic functions on the unit-ball are investigated. Furthermore the proof — as the one in [1] — uses a theorem from [2]. Therefore we give another application of the results from [2] — namely for spaces of entire functions.