Separating sets by Darboux Baire 1 and approximately continuous functions

We characterize sets A₀, A₁ for which there is a DB1 function f with [f = 0] = A₀ and [f = 1] = A₁. This characterization is a conjunction of necessary conditions for Darboux and for Baire 1 functions. We also characterize sets A^?, A⁺ for which there is a DB₁ function with [f < 0] = A^? and [f > 0] = A⁺. The same characterzations are provided for approximately continuous functions.     

On meromorphic solutions of <Emphasis Type="Italic">f</Emphasis><Superscript>2</Superscript> + <Emphasis Type="Italic">g</Emphasis><Superscript>2</Superscript> = 1

We show that meromorphic solutions f, g of f ² + g ² = 1 in C² must be constant, if f _z2 and g _z1 have the same zeros (counting multiplicities). We also apply the result to characterize meromorphic solutions of certain nonlinear partial differential equations.     

Almost Isometries of Balls

Let f be a bi-Lipschitz mapping of the Euclidean ball B ^_n into ℓ₂ with both Lipschitz constants close to one. We investigate the shape of f(B ^_n). We give examples of such a mapping f, which has the Lipschitz constants arbitrarily close to one and at the same time has in the supremum norm the distance at least one from every isometry of ⁿ.     

On the degree of iterates of automorphisms¶of the affine plane

For a polynomial automorphism f of ?² _ℂ, we set τ = deg f ²)/(deg f). We prove that τ≤ 1 if and only if f is triangularizable. In this situation, we show (by using a deep result from number theory known as the theorem of Skolem–Mahler–Lech) that the sequence (deg f ⁿ ) _{n ∈ℕ} is periodic for large n. In the opposite case, we prove that τ is an integer (τ≥ 2) and that the sequence (deg f ⁿ ) _{n ∈ℕ} is a geometric progression of ratio τ. In particular, if f is any automorphism, we obtain the rationality of the formal series . Received: 1 December 1997     

Estimating the distance to a monotone function

In standard property testing, the task is to distinguish between objects that have a property 𝒫 and those that are ε‐far from 𝒫, for some ε > 0. In this setting, it is perfectly acceptable for the tester to provide a negative answer for every input object that does not satisfy 𝒫. This implies that property testing in and of itself cannot be expected to yield any information whatsoever about the distance from the object to the property. We address this problem in this paper, restricting our attention to monotonicity testing. A function f : {1,…,n} ↦ R is at distance ε_f from being monotone if it can (and must) be modified at ε_fn places to become monotone. For any fixed δ > 0, we compute, with probability at least 2/3, an interval [(1/2 − δ)ε,ε] that encloses ε_f. The running time of our algorithm is O(ε_f⁻¹ log log ε_f^{− 1} log n), which is optimal within a factor of loglog ε_f⁻¹ and represents a substantial improvement over previous work. We give a second algorithm with an expected running time of O(ε_f⁻¹ log nlog log log n). Finally, we extend our results to multivariate functions. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Quelques théorèmes sur la structure complexe

We prove the following result: If the function Max (log|ω -f ₁(z)|, ..., log|ω -f _k(z)|) is plurisubharmonic in the open setD×ℂ (D open of ℂ ⁿ ), thenf ₁,...,f _k are analytic functions iff ₁,...,f _k are continuous functions onD(k≥2). We prove also some other results.     

An extremal problem for Dirichlet-finite holomorphic functions on Riemann surfaces

Iff is a nonconstant holomorphic function with finite Dirichlet integralD(f) on a Riemann surfaceR, then |f|² has the least harmonic majorantf ₂ onR. We show Σf ₂(a ≦π ⁻¹ D(f)), wherea runs over all the roots off = 0 onR. The equality holds if and only iff is of type ℬℓ₁ fromR onto a disk of center 0. A consideration is proposed for the non-Euclidean case.     

Approximate solution of differential equations with the use of asymptotic polynomials

We consider an approximate solution of differential equations with initial and boundary conditions. To find a solution, we use asymptotic polynomials Q _n ^f (x) of the first kind based on Chebyshev polynomials T _n (x) of the first kind and asymptotic polynomials G _n ^f (x) of the second kind based on Chebyshev polynomials U _n (x) of the second kind. We suggest most efficient algorithms for each of these solutions. We find classes of functions for which the approximate solution converges to the exact one. The remainder is represented as an expansion in linear functionals {L _n ^f } in the first case and {M _n ^f } in the second case, whose decay rate depends on the properties of functions describing the differential equation.     

A descent method for submodular function minimization

We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2 ^V with ?,V∈? and f(?)=0 and for any vector x∈R ^V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|²) times, the descent method finds a sequence of subsets Z ₁,Z ₂,···,Z _k of V such that f(Z ₁)>f(Z ₂)>···>f(Z _k )=min{f(Y) | Y∈?}, where k is O(|V|²). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms. Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001     

Gâteaux derivatives and their applications to approximation in Lorentz spaces Γp,w

We establish the formulas of the left‐ and right‐hand Gâteaux derivatives in the Lorentz spaces Γ_p,w = {f: ∫₀^α (f **)^p w < ∞}, where 1 ≤ p < ∞, w is a nonnegative locally integrable weight function and f ** is a maximal function of the decreasing rearrangement f * of a measurable function f on (0, α), 0 < α ≤ ∞. We also find a general form of any supporting functional for each function from Γ_p,w , and the necessary and sufficient conditions for which a spherical element of Γ_p,w is a smooth point of the unit ball in Γ_p,w . We show that strict convexity of the Lorentz spaces Γ_p,w is equivalent to 1 < p < ∞ and to the condition ∫₀^∞ w = ∞. Finally we apply the obtained characterizations to studies the best approximation elements for each function f ∈ Γ_p,w from any convex set K ? Γ_p,w (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spherical derivatives and normal families

We show that a family of functions meromorphic in a plane domain D whose spherical derivatives are uniformly bounded away from zero is normal. Furthermore, we show that for each f meromorphic in the unit disk D, inf _z∈D f ^#(z) ≤ 1/2, where f ^# denotes the spherical derivative of f.     

Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point

For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H ₀ : f≡ 0, (the signal f is zero) against the nonparametric alternative H ₁ : f∈Λ_ɛ where Λ_ɛ is a set of functions on R ¹ of the form Λ_ɛ = {f : f∈?, ϕ(f) ≥ Cψ_ɛ}. Here ? is a H?lder or Sobolev class of functions, ϕ(f) is either the sup-norm of f or the value of f at a fixed point, C > 0 is a constant, ψ_ɛ is the minimax rate of testing and ɛ→ 0 is the asymptotic parameter of the model. We find exact separation constants C ^* > 0 such that a test with the given summarized asymptotic errors of first and second type is possible for C > C ^* and is not possible for C < C ^*. We propose asymptotically minimax test statistics. Received: 23 February 1998 / Revised version: 6 April 1999 / Published online: 30 March 2000     

On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|?|^{p –2}?) + λk_i (|x |) fⁱ (u₁, …,u_n) = 0, p > 1, R₁ < |x | < R₂, u_i (x) = 0, on |x | = R₁ and R₂, i = 1, …, n, x ∈ ?^N , where k_i and fⁱ, i = 1, …, n, are continuous and nonnegative functions. Let u = (u₁, …, u_n), φ (t) = |t |^{p –2}t, fⁱ₀ = lim_{‖ u ‖→0}((fⁱ ( u ))/(φ (‖ u ‖))), fⁱ_∞= lim_{‖ u ‖→∞}((fⁱ ( u ))/(φ (‖ u ‖))), i = 1, …, n, f = (f¹, …, fⁿ), f ₀ = ∑ⁿ _{i =1} fⁱ ₀ and f _∞ = ∑ⁿ _{i =1} fⁱ _∞. We prove that either f ₀ = 0 and f _∞ = ∞ (superlinear), or f ₀ = ∞and f _∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if fⁱ ( u ) > 0 for ‖ u ‖ > 0, i = 1, …, n, then either f ₀ = f _∞ = 0, or f ₀ = f _∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On the other hand, either f₀ and f_∞ > 0, or f₀ and f_∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On product-cordial index sets and friendly index sets of 2-regular graphs and generalized wheels

A vertex labeling f : V → Z2 of a simple graph G = (V, E) induces two edge labelings f+ , f*: E → Z2 defined by f+ (uv) = f(u)+f(v) and f*(uv) = f(u)f(v). For each i∈Z2 , let vf(i) = |{v ∈ V : f(v) = i}|, e+f(i) = |{e ∈ E : f+(e) = i}| and e*f(i)=|{e∈E:f*(e)=i}|. We call f friendly if |vf(0)-vf(1)|≤ 1. The friendly index set and the product-cordial index set of G are defined as the sets{|e+f(0)-e+f(1)|:f is friendly} and {|e*f(0)-e*f(1)| : f is friendly}. In this paper we study and determine the connection between the friendly index sets and product-cordial index sets of 2-regular graphs and generalized wheel graphs.     

On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

Connected components of spaces of Morse functions with fixed critical points

Let M be a smooth closed orientable surface and F = F _p,q,r be the space of Morse functions on M having exactly p points of local minimum, q ≥ 1 saddle critical points, and r points of local maximum, moreover, all the points are fixed. Let F _f be a connected component of a function f ∈ F in F.We construct a surjection π ₀(F) → ℤ ^p+r−1 by means of the winding number introduced by Reinhart (1960). In particular, |π₀(F)| = ∞, and the component F _f is not preserved under the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point. Let D be the group of orientation preserving diffeomorphisms of M leaving fixed the critical points, D ⁰ be the connected component of id _M in D, and D _f ⊂ D be the set of diffeomorphisms preserving F _f . Let H _f be the subgroup of D _f generated by D ⁰ and all diffeomorphisms h ∈ D preserving some function f ₁ ∈ F _f , and let H _f ^abs be its subgroup generated by D ⁰ and the Dehn twists about the components of level curves of the functions f ₁ ∈ F _f . We prove thatH _f ^abs ⊊ D _f for q ≥ 2 and construct an epimorphism D _f /H _f ^abs → ℤ₂ ^q−1 by means of the winding number. A finite polyhedral complex K = K _p,q,r associated with the space F is defined. An epimorphism μ: π ₁(K) → D _f /H _f and finite generating sets for the groups D _f /D ⁰ and D _f /H _f in terms of the 2-skeleton of the complex K are constructed.     

Tail field representations and the zero-two law

The zero-two law was proved for a positiveL ₁-contractionT by Ornstein and Sucheston, and gives a condition which impliesT ⁿ f−T ⁿ⁺¹ f → 0 for allf. Extensions of this result to the case of a positiveL _p -contraction, 1≤p<∞, have been obtained by several authors. In the present paper we prove a theorem which is related to work of Wittmann. We will say that a positive contractionT contains a circle of lengthm if there is a nonzero functionf such that the iterated valuesf, T f,…,T ^m-1 f have disjoint support, whileT ^m f=f. Similarly, a contractionT contains a line if for everym there is a nonzero functionf (which may depend onm) such thatf,…,T ^m-1 f have disjoint support. Approximate forms of these conditions are defined, which are referred to as asymptotic circles and lines, respectively. We show (Theorem 3) that if the conclusionT ⁿ f−T ⁿ⁺¹ f→0 of the zero-two law does not hold for allf inL _p , then eitherT contains an asymptotic circle orT contains an asymptotic line. The point of this result is that any condition onT which excludes circles and lines must then imply the conclusion of the zero-two law. Theorem 3 is proved by means of the representation of a positiveL _p -contraction in terms of anL _p -isometry. Asymptotic circles and lines forT correspond to exact circles and lines for the isometry on tail-measurable functions, and exact circles and lines for the isometry are obtained using the Rohlin tower construction for point transformations. Research supported in part by NSERC.     

Boundary effects on convergence rates for Tikhonov regularization

We consider the Tikhonov regularizer f_λ of a smooth function f ε H^2m[0, 1], defined as the solution (see [1]) to We prove that if f^(j)(0) = f^(j)(1) = 0, J = m, …, k < 2m − 1, then ¦f − f_λ¦_j² Rλ^{(2k − 2j + 3)/2m}, J = 0, …, m. A detailed analysis is given of the effect of the boundary on convergence rates.     

Detecting and creating oscillations using multifractal methods

By comparing the Hausdorff multifractal spectrum with the large deviations spectrum of a given continuous function f, we find sufficient conditions ensuring that f possesses oscillating singularities. Using a similar approach, we study the nonlinear wavelet threshold operator which associates with any function f = ∑_j ∑_k d_j,k ψ _j,k ∈ L ²(?) the function series f^t whose wavelet coefficients are d ^t _j,k = d_j,k 1 , for some fixed real number γ > 0. This operator creates a context propitious to have oscillating singularities. As a consequence, we prove that the series f^t may have a multifractal spectrum with a support larger than the one of f . We exhibit an example of function f ∈ L ²(?) such that the associated thresholded function series f^t effectively possesses oscillating singularities which were not present in the initial function f . This series f^t is a typical example of function with homogeneous non‐concave multifractal spectrum and which does not satisfy the classical multifractal formalisms. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiplication operators on sobolev disk algebra

In this paper,we study the algebra consisting of analytic functions in the Sobolev space W~(2,2) (D) (D is the unit disk),called the Sobolev disk algebra,explore the properties of the multiplication operators M_f on it and give the characterization of the corn- mutant algebra A′(M_f) of M_f.We show that A′(M_f) is commutative if and only if M_f~* is a Cowen-Douglas operator of index 1.