Mixed problem for x-analytical functions in a halfdisk

The mixed problem for x-analytical functions in a halfdisk with the real part of the x-analytical function defined on a part of the circle and the imaginary part defined on the rest of the circle is reduced to the problem of linear matching of p-analytical functions with different characteristics (p=x in the halfdisk and p=x/(x ² +y ² ) in the halfplane with the halfdisk cut out). The linear matching problem of p-analytical functions is analyzed and the class of p-analytical functions with the characteristic p=x/(x ² +y ² ) is shown to be equivalent to the class of x-analylical functions. An integral representation is obtained for p-analytical functions with the characteristic p=x/(x ² +y ² ) which is analogous to the standard integral representation for x-analytical functions. A procedure is developed for reducing the solution of the linear matching problem to the solution of the Fredholm integral equation of second kind.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 7–15, 1986.     

On the equivalence of quadratic APN functions

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to the APN Gold function x^2r+1{x^{2^r+1}} if and only if it is EA-equivalent to that Gold function. As an application of this result, we prove that a trinomial family of APN functions that exist on finite fields of order 2 ⁿ where n ≡ 2 mod 4 are CCZ inequivalent to the Gold functions. The proof relies on some knowledge of the automorphism group of a code associated with such a function.     

The power of adaptive algorithms for functions with singularities

This is an overview of recent results on complexity and optimality of adaptive algorithms for integrating and approximating scalar piecewise r-smooth functions with unknown singular points. We provide adaptive algorithms that use at most n function samples and have the worst case errors proportional to n^−r for functions with at most one unknown singularity. This is a tremendous improvement over nonadaptive algorithms whose worst case errors are at best proportional to n⁻¹ for integration and n^−1/p for the L^p approximation problem. For functions with multiple singular points the adaptive algorithms cease to dominate the nonadaptive ones in the worst case setting. Fortunately, they regain their superiority in the asymptotic setting. Indeed, they yield convergence of order n^−r for piecewise r-smooth functions with an arbitrary (unknown but finite) number of singularities. None of these results hold for the L^∞ approximation. However, they hold for the Skorohodmetric, which we argue to be more appropriate than L^∞ for dealing with discontinuous functions. Numerical test results and possible extensions are also discussed.     

Homology properties of Y-systems

Suppose that a Y-system T^t(T^k) acts on a manifold Mⁿ. We present a criterion of zero homology for Holder functions with respect to this dynamical system, as well as some consequences of this criterion and a generalization for functions taking their values in a Lie group.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 555–564, November, 1971.     

Maximal operators of commutators of Bochner-Riesz means with Lipschitz functions

This paper presents a weighted L ² estimate with power weights for the maximal operator of commutators generated by compactly supported multipliers and Lipschitz functions. As an application, we study the almost convergence of the commutators, which is generated by the Bochner-Riesz means under the critical index and Lipschitz functions, for functions in L ^p (p ⩾ 2).     

Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations

This article is concerned with some properties of Stepanov-like almost automorphic (S ^p -a.a.) functions. We establish a composition theorem about S ^p -a.a. functions, and with its help, study the existence and uniqueness of almost automorphic solutions for semilinear evolution equations in Banach spaces. Moreover, integration and differentiation of S ^p -a.a. functions are discussed. Some theorems extend earlier results.     

Non-denseness of factorable matrix functions

It is proved that for certain algebras of continuous functions on compact abelian groups, the set of factorable matrix functions with entries in the algebra is not dense in the group of invertible matrix functions with entries in the algebra, assuming that the dual abelian group contains a subgroup isomorphic to Z³. These algebras include the algebra of all continuous functions and the Wiener algebra. More precisely, it is shown that infinitely many connected components of the group of invertible matrix functions do not contain any factorable matrix functions, again under the same assumption. Moreover, these components actually are disjoint with the subgroup generated by the triangularizable matrix functions.     

Prox-Regularity of Functions and Sets in Banach Spaces

In this paper, we define the prox-regularity for functions on Banach spaces by adapting the original definition in R ⁿ . In this context, we establish a subdifferential characterization and show that qualified convexly C ^1,+-composite functions and primal lower nice functions belong to this class, as already known in the setting of Hilbert spaces. We also study, in a geometrical point of view, the epigraphs of prox-regular functions. The subdifferential characterization allows us to show that some Moreau-envelope-like regularizations of such functions are of class C ¹ in the context of certain uniformly convex spaces.     

Complete growth functions of hyperbolic groups

. We study a generalization of the growth functions of finitely generated groups, namely the growth functions Σ _{g ∈ G} gz ^{| g |} with coefficients in the group ring ℤ[G]. Rationality and methods of computation of such functions are discussed, in particular for hyperbolic groups. The complete growth functions of surface groups are explicitly computed. The operator and geodesic growth functions are also studied. Oblatum 20-IX-1996 & 13-I-1997     

Effective Borel degrees of some topological functions

The focus of this paper is the incomputability of some topological functions (with respect to certain representations) using the tools of Borel computability theory, as introduced by V. Brattka in [3] and [4]. First, we analyze some basic topological functions on closed subsets of ?ⁿ , like closure, border, intersection, and derivative, and we prove for such functions results of Σ⁰₂‐completeness and Σ⁰₃‐completeness in the effective Borel hierarchy. Then, following [13], we re‐consider two well‐known topological results: the lemmas of Urysohn and Urysohn‐Tietze for generic metric spaces (for the latter we refer to the proof given by Dieudonné). Both lemmas define Σ⁰₂‐computable functions which in some cases are even Σ⁰₂‐complete. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lower bound for the best approximations of periodic summable functions of two variables and their conjugates in terms of Fourier coefficients

In terms of Fourier coefficients, we establish lower bounds for the sum of norms and the sum of the best approximations by trigonometric polynomials for functions from the space L(Q ²) and functions conjugate to them with respect to each variable and with respect to both variables, provided that these functions are summable. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1042–1050, August, 2008.     

Plain Bases for Classes of Primitive Recursive Functions

Abasisforaset C of functions on natural numbers is a set F of functions such that C is the closure with respect to substitution of the projection functions and the functions in F. This paper introduces three new bases, comprehending only common functions, for the Grzegorczyk classes ℰⁿ with n ≥ 3. Such results are then applied in order to show that ℰⁿ⁺¹ = K_n for n ≥ 2, where {K_n}_n∈ℕ is the Axt hierarchy.     

On global unconstrained minimization of the difference of polyhedral functions

The problem of finding a global minimizer of the difference of polyhedral functions is considered. By means of conjugate functions, necessary and sufficient conditions for the unboundedness and the boundedness of such functions in R ⁿ are derived. Using hypodifferentials of polyhedral functions, necessary and sufficient conditions for a global unconstrained minimum on R ⁿ are proved.     

Lineability of sets of nowhere analytic functions

Although the set of nowhere analytic functions on [0,1] is clearly not a linear space, we show that the family of such functions in the space of C^∞-smooth functions contains, except for zero, a dense linear submanifold. The result is even obtained for the smaller class of functions having Pringsheim singularities everywhere. Moreover, in spite of the fact that the space of differentiable functions on [0,1] contains no closed infinite-dimensional manifold in C([0,1]), we prove that the space of real C^∞-smooth functions on (0,1) does contain such a manifold in C((0,1)).     

A characterization of Fourier series of Stepanov almost-periodic functions

We show by example that the classical characterization of the Fourier series of periodic functions in L^p, 1<p≤+∞, as those trigonometric series whose Abel or Fejér means are uniformly bounded in L^p does not hold for general (non-periodic) trigonometric series in relation to Stepanov-almost-periodic functions, but that it does hold under the additional hypothesis that the means are translation equicontinuous. We exhibit a bounded, infinitely differentiable function that belongs to every class of Besicovitch-almost-periodic functions but is not equivalent in the metric of Besicovitch-almost-periodic functions to any Stepanov-almost-periodic function.     

Methods of solving Fredholm equations optimal on classes of functions

This paper is devoted to the solution of linear Fredholm equations in the unit s-dimensional cube for classes of functions with a dominant mixed derivative of order r in each variable. We present an algorithm for obtaining the solution over the whole domain with an error O(N^?r ln^2s?1 N) in the uniform metric using the values of the given functions at O(N ln^2s?1 N) points and consisting of O(N ln^2s?1 N) elementary operations. We show that these estimates can only be improved at the expense of the exponent of ln N.     

满足H(m)条件的奇异积分算子的交换子的有界性

主要讨论了满足H(m)条件的奇异积分算子与Lipschitz函数的交换子在L~p和Hardy空间的有界性,并把这个结果应用于与薛定谔算子相关的Riesz变换.     

On the convergence of Fourier series of computable Lebesgue integrable functions

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of L^p‐computable functions (computable Lebesgue integrable functions) with a size notion, by introducing L^p‐computable Baire categories. We show that L^p‐computable Baire categories satisfy the following three basic properties. Singleton sets {f } (where f is L^p‐computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of L^p‐computable functions is not meager. We give an alternative characterization of meager sets via Banach‐Mazur games. We study the convergence of Fourier series for L^p‐computable functions and show that whereas for every p > 1, the Fourier series of every L^p‐computable function f converges to f in the L^p norm, the set of L¹‐computable functions whose Fourier series does not diverge almost everywhere is meager (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Recognition problems for special classes of polynomials in 0–1 variables

This paper investigates the complexity of various recognition problems for pseudo-Boolean functions (i.e., real-valued functions defined on the unit hypercubeB ⁿ = {0, 1} ⁿ ), when such functions are represented as multilinear polynomials in their variables. Determining whether a pseudo-Boolean function (a) is monotonic, or (b) is supermodular, or (c) is threshold, or (d) has a unique local maximum in each face ofB ⁿ , or (e) has a unique local maximum inB ⁿ , is shown to be NP-hard. A polynomial-time recognition algorithm is presented for unimodular functions, previously introduced by Hansen and Simeone as a class of functions whose maximization overB ⁿ is reducible to a network minimum cut problem.     

Regularity of solutions to the perturbed conservation laws

A kind of regularity for the mild solution of perturbed conservation laws is proposed. This regularity is described in term of variations measured in the L¹-norm. A dissipativity condition from the semigroup approach is used to show that the mild solution stays within a class of bounded variation in this sense of regularity. This shows that this class of functions is an invariant of the semigroup. The same analysis carries over to the periodic problem. The class of boundedL¹-variation functions used here can be normed to give a Banach space structure. It also has an analogue with the space of Lipschitz functions