Schwarz—Pick Inequalities for Derivatives of Arbitrary Order

   Abstract. Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λ _Ω and λ _Π denote the densities of the Poincare metric in Ω and Π , respectively. For f: Ω → Π analytic in Ω , inequalities of the type

Existence of solution of the pullback equation involving volume forms

Let Ω ⊂ ℝ ⁿ be a smooth, bounded domain. We study the existence and regularity of diffeomorphisms of Ω satisfying the volume form equation

A Polynomial-Time Algorithm to Approximate the Mixed Volume within a Simply Exponential Factor

Let K=(K ₁,…,K _n ) be an n-tuple of convex compact subsets in the Euclidean space R ⁿ , and let V(⋅) be the Euclidean volume in R ⁿ . The Minkowski polynomial V _K is defined as V _K (λ ₁,…,λ _n )=V(λ ₁ K ₁+⋅⋅⋅+λ _n K _n ) and the mixed volume V(K ₁,…,K _n ) as

Two dimensional variational problems with linear growth

 Suppose that f: ℝ ^nN →ℝ is a strictly convex energy density of linear growth, f(Z)=g(|Z|²) if N>1. If f satisfies an ellipticity condition of the form

Negative result in pointwise 3-convex polynomial approximation

Let Δ³ be the set of functions three times continuously differentiable on [−1, 1] and such that f″′(x) ≥ 0, x ∈ [−1, 1]. We prove that, for any n ∈ ℕ and r ≥ 5, there exists a function f ∈ C ^r [−1, 1] ⋂ Δ³ [−1, 1] such that ∥f ^(r)∥ _{C[−1, 1]} ≤ 1 and, for an arbitrary algebraic polynomial P ∈ Δ³ [−1, 1], there exists x such that

Properties of generalised juxtapolynomials

GivenF(z),f ₁(z), ..,f _n(z) defined on a finite point setE, and givenB — the set of generalised polynomials Σ _k ^=1/n a _kf_k(z) — the definition of a juxtapolynomial is extended in the following manner: for a fixedλ(0<λ≦1),f(z) εB is called a generalizedλ-weak juxtapolynomial toF(z) onE if and only if there exists nog(z) εB for whichg(z)=F(z) wheneverf(z)=F(z) and |g(z)−F(z) |<λ|f(z)−F(z)| wheneverf(z)≠F(z). The properties of suchf(z) are investigated with particular attention given to the real case. This note is an extension of a part of the author’s M.Sc. Thesis under the supervision of Prof. B. Grünbaum to whom the author wishes to express his sincerest appreciation. The author also wishes to thank Dr. J. Lindenstrauss for his valuable remarks in the preparation of this paper.     

Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville problem

A Modification of the Lyons-Sullivan Discretization of Positive Harmonic Functions

A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C _n :n = 1,2,..}, allows for constructing measures n_x^C, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

On cardinal interpolation by Gaussian radial-basis functions: Properties of fundamental functions and estimates for Lebesgue constants

Suppose λ is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal functionL _λ(x)

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

Bernstein�CWalsh Type Theorems for Real Analytic Functions in Several Variables

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in ℝ ^N . In particular, we investigate the polynomial approximation behavior for functions F:L→ℂ, L={(Re z,Im z):z∈K}, of the structure F=g[`(h)]F=g\overline{h}, where g and h are holomorphic in a neighborhood of a compact set K⊂ℂ <sup>N</sup> . To this end the maximal convergence number ρ(S <sub>c</sub> ,f) for continuous functions f defined on a compact set S <sub>c</sub> ⊂ℂ <sup>N</sup> is connected to a maximal convergence number ρ(S <sub>r</sub> ,F) for continuous functions F defined on a compact set S <sub>r</sub> ⊂ℝ <sup>N</sup> . We prove that ρ(L,F)=min {ρ(K,h)),ρ(K,g)} for functions F=g[`(h)]F=g\overline{h} if K is either a closed Euclidean ball or a closed polydisc. Furthermore, we show that min {ρ(K,h)),ρ(K,g)}≤ρ(L,F) if K is regular in the sense of pluripotential theory and equality does not hold in general. Our results are based on the theory of the pluricomplex Green’s function with pole at infinity and Lundin’s formula for Siciak’s extremal function Φ. A properly chosen transformation of Joukowski type plays an important role.     

On Taylor-like integral representations of analytic functions

Let f be an analytic function in a complex domain D and (without loss of generality) assume 0∈D. Then the paper’s aim is to derive a Taylor-like integral expression for f, i.e. an integral representation analogous to the corresponding power series, say, ∑ _k=0^∞ a _k t ^k /k!. We start from the simplest case f(t)=e ^t , which leads to the identity

Differentiability of functions: approximate,global, and differentiability along curves over non-archimedean fields

This paper is devoted to the study of approximate and global smoothness and smoothness along curves of functions f(x ₁,...,x _m ) of variables x ₁,...,x _m in infinite fields with nontrivial non-Archimedean valuations and relations between them. Theorems on classes of smoothness C ⁿ or of functions with partial difference quotients continuous or bounded uniformly continuous on bounded domains up to order n are investigated. We prove that from f ○ u ∈ C ⁿ (K, K ^l) or f ○ u ∈ (K, K ^l) for each C ^∞ or curve u: K → K ^m it follows that f ∈ C ⁿ (K ^m , K ^l) or f ∈ (K ^m , K ^l), where m ≥ 2. Then the classes of smoothness C ^n,r and and more general in the sense of Lipschitz for partial difference quotients are considered and theorems for them are proved. Moreover, the approximate differentiability of functions relative to measures is defined and investigated. Its relations with the Lipschitzian property and almost everywhere differentiability are studied. Non-Archimedean analogs of classical theorems of Kirzsbraun, Rademacher, Stepanoff, and Whitney are formulated and proved, and substantial differences between two cases are found. Finally, theorems about relations between approximate differentiability by all variables and along curves are proved. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 52, Functional Analysis, 2008.     

Oscillation of forced neutral differential equations with positive and negative coefficients

In this paper the forced neutral difterential equation with positive and negative coefficients d/dt [x(t)-R(t)x(t-r)] P(t)x(t-x)-Q(t)x(t-σ)=f(t),t≥t0,is considered,where f∈L^1(t0,∞)交集C([t0,∞],R^ )and r,x,σ∈(0,∞),The sufficient conditions to oscillate for all solutions of this equation are studied.     

Explosive solutions of elliptic equations with absorption and nonlinear gradient term

Letf be a non-decreasing C¹-function such that andF(t)/f ² a(t)→ 0 ast → ∞, whereF(t)=∫ ₀ ^t f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu| ^a =p(x)f(u) in a smooth bounded domain Ω ⊂R^N, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded.     

A New Proof of a Theorem of Suslin

In this paper we compute the group H₂(SL₂(F)), for F an infinite field. In particular, using some techniques from homological algebra developed by Hutchinson [Hutchinson, K: K-Theory 4 (1990), 181–200], we give a new proof of the following theorem obtained by [Su2]: The group H₂(SL₂, (F)) is the fiber product of λ_*:K₂(F)→ I²(F)/I³(F) and σ: I²(F) → I²(F)/I³(F) where λ_* and σ map onto I²(F)/I³(F). (Received: February 2003)     

Semicontinuity and Supremal Representation in the Calculus of Variations

We study the weak* lower semicontinuity properties of functionals of the form

Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds

Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x ₀)⊂E→F be C¹ nonlinear map, whereU (x ₀) is an open set containing pointx ₀∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x ₀) near x₀,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C¹ Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.     

Integral transforms of functions with the derivative in a halfplane

LetA be the class of normalized analytic functions in the unit disk Δ and define the class