A novel algorithm enumerating bent functions

Based on the relationship between the Walsh spectra of a Boolean function at partial points and the Walsh spectra of its subfunctions, and on the binary Möbius transform, a novel algorithm is developed, which can theoretically construct all bent functions. Practically we enumerate all bent functions in 6 variables. With the restriction on the algebraic normal form, the algorithm is also efficient in more variables case. For example, enumeration of all homogeneous bent functions of degree 3 in 8 variables can be done in one minute with a P4 1.7 GHz computer; the nonexistence of homogeneous bent functions in 10 variables of degree 4 is computationally proved.     

The Distribution of the Summatory Function of the Mobius Function

The summatory function of the Möbius function is denotedM(x). In this article we deduce conditional results concerningM(x) assuming the Riemann hypothesis and a conjecture of Gonekand Hejhal on the negative moments of the Riemann zeta function.Assuming these conjectures, we show that M(x), when appropriatelynormalized, possesses a limiting distribution, and also thata strong form of the weak Mertens conjecture is true. Finally,we speculate on the lower order of M(x) by studying the constructeddistribution function. 2000 Mathematics Subject Classification11M26, 11N56.     

On the properties of sphere-preserving maps

In this paper we give a discussion of sphere-preserving maps of Hilbert spaces, and their relationship to Möbius transformations and to the preservation of cross-ratios.     

Generalizations of Eulerian partially ordered sets, flag numbers, and the Möbius function

A partially ordered set is r-thick if every nonempty open interval contains at least r elements. This paper studies the flag vectors of graded, r-thick posets and shows the smallest convex cone containing them is isomorphic to the cone of flag vectors of all graded posets. It also defines a k-analogue of the Möbius function and k-Eulerian posets, which are 2k-thick. Several characterizations of k-Eulerian posets are given. The generalized Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A new inequality is proved to be valid and sharp for rank 8 Eulerian posets.ResumeUn ensemble partiellement ordonné est r-épais si chacun de ses intervalles ouverts non-vides contient au moins r éléments. Dans cet article nous étudions les vecteurs drapeaux des ensembles partiellement ordonnés gradués r-épais. Nous démontrons que le cône le plus petit contenant ces vecteurs est isomorphe au cône des vecteurs drapeaux des ensembles partiellement ordonnés gradués quelconques. Nous définissons aussi un k-analogue de la fonction de Möbius et des ensembles partiellement ordonnés k-eulériens qui sont 2k-épais. Nous caractérisons les ensembles partiellement ordonnés k-eulériens de plusieurs manières, et généralisons les équations de Dehn-Sommerville pour le vecteur drapeaux d'un ensemble partiellement ordonné k-eulérien. Nous démontrons une nouvelle inégalité optimale pour les ensembles partiellement ordonnés eulériens de rang 8.     

Some Properties of Bent Functions

First, this paper discusses and sums up some properties of a pair of functions p（x）, q（x） that makes （y ＋ 1）p（x） ＋ yq（x） into a bent function. Then it discusses the properties of bent functions. Also, the upper and lower bounds of the number of bent functions on GF（2）2k are discussed.     

Heuristic investigation of chaotic mapping producing fractal objects

In the literature most examples on fractals are related to images produced by certain iterative processes. Here we will instead discuss how similar results may appear by mapping the unit circle using different, somewhat unusual functions. In principle this will be achieved by choosing a series of the form f(n) exp(in)/n ^s wheref(n) is a function which may depend on, e.g., the structure of the numbern. In some casesf(n) is even obtained by a suitable random process. Further, the parameter s usually satisfies 0 <s 1 withs = 1 in most cases. However, we also investigate how the fractal structure disappears by taking s > 1. In our examples we accept singular results in isolated points. Finally we have tried to determine the dimensions of the resulting fractal objects.     

The Möbius function and the residue theorem

A classical conjecture of Bouniakowsky says that a non-constant irreducible polynomial in Z[T] has infinitely many prime values unless there is a local obstruction. Replacing Z[T] with κ[u][T], where κ is a finite field, the obvious analogue of Bouniakowsky's conjecture is false. All known counterexamples can be explained by a new obstruction, and this obstruction can be used to fix the conjecture. The situation is more subtle in characteristic 2 than in odd characteristic. Here, we illustrate the general theory for characteristic 2 in some examples.     

A New Characterization of Semi-bent and Bent Functions on Finite Fields*

We present a new characterization of semi-bent and bent quadratic functions on finite fields. First, we determine when a GF(2)-linear combination of Gold functions Tr(x²ⁱ+1) is semi-bent over GF(2ⁿ), n odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler characterizations of semi-bent functions. For example, we deduce that all linear combinations of Gold functions give rise to semi-bent functions over GF(2^p) when p belongs to a certain class of primes. Second, we generalize our results to fields GF(pⁿ) where p is an odd prime and n is odd. In that case, we can determine whether a GF(p)-linear combination of Gold functions Tr(x^pi+1) is (generalized) semi-bent or bent by a polynomial GCD computation. Similar to the binary case, simple characterizations of these p-ary semi-bent and bent functions are provided. Parts of this paper were presented at the 2002 IEEE International Symposium on Information Theory [10]     

A new characteristic of Möbius transformations by use of polygons having type A

In the hyperbolic plane Möbius transformations can be characterized by Lambert quadrilaterals, i.e., a continuous bijection which maps Lambert quadrilaterals to Lambert quadrilaterals must be Möbius. In this paper we generalize this result to the case of polygons with n sides having type A, that is, having exactly two non-right interior angle.     

Discreteness and Convergence of Möbius Groups

In this paper we show that one can use a fixed nontrivial Möbius transformation as a test map to test the discreteness of a nonelementary Möbius group. We also establish two theorems in algebraic convergence.     

A class of combinatorial identities

A general theorem for providing a class of combinatorial identities where the sum is over all the partitions of a positive integer is proven. Five examples as the applications of the theorem are given.     

A new characteristic of Möbius transformations in hyperbolic geometry

We present a new characterization of Möbius transformations by using two classes of hyperbolic geometric objects: Lambert quadrilaterals and Saccheri quadrilaterals. The proof is based on a geometric approach.     

On existence of degenerate circle-preserving maps

Recently, B. Li and Y. Wang proved that if (n?2) is a circle-preserving map, then f is a Möbius transformation if and only if f is a non-degenerate map, where a map f is degenerate if the image is a circle. Furthermore, they conjectured that there should exist no degenerate map, or equivalently, f is a Möbius transformation if and only if f is a circle-preserving map. In this note, we construct a degenerate circle-preserving map to show that the conjecture is not true.     

Möbius functions and semigroup representation theory

This paper explores several applications of Möbius functions to the representation theory of finite semigroups. We extend Solomon's approach to the semigroup algebra of a finite semilattice via Möbius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix algebras over group rings. We also extend work of Bidigare, Hanlon, Rockmore and Brown on calculating eigenvalues of random walks associated to certain classes of finite semigroups; again Möbius functions play an important role.     

具有亏值的亚纯函数的唯一性

改进了Ozawa的一个关于整函数的唯一性定理,得到了∞为亏值的亚纯函数唯一性的相应的几个结论.设亚纯函数f(z)与g(z)的级(或者下级)为有穷的非整数,满足.f=0→g=0,f=1g=1,f=∞9=∞,若∞为f(z)的Borel例外值,则f≡g.以及设f(z)与g(z)为C中非常数的亚纯函数,它们的级λ为有穷且非整数,再设它们满足f=0→g=0,f=1g=1,f=∞g=∞,若δ(∞,f)=1,f(z)为正规增长函数,则f≡g.     

A characterization of Möbius transformations by use of hyperbolic regular polygons

In this paper we present a new characterization of Möbius transformations by use of hyperbolic regular polygons.     

On the Characterization of Preinvex Functions

In Refs. [J. Math. Anal. Appl. 258:287–308, [2001]; J. Math. Anal. Appl. 256:229–241, [2001]], Yang and Li presented a characterization of preinvex functions and semistrictly preinvex functions under a certain set of conditions. In this note, we show that the same results or even more general ones can be obtained under weaker assumptions; we also give a characterization of strictly preinvex functions under mild conditions. This research was supported by the National Natural Science Foundation of China under Grants 70671064 and 60673177, and the Education Department Foundation of Zhejiang Province Grant 20070306. The authors thank Professor F. Giannessi for valuable comments on the original version of this paper.     

A Rational-Function Identity Related to the Murnaghan–Nakayama Formula for the Characters of S n

The Murnaghan–Nakayama formula for the characters of S _n is derived from Young's seminormal representation, by a direct combinatorial argument. The main idea is a rational function identity which when stated in a more general form involves Möbius functions of posets whose Hasse diagrams have a planar embedding. These ideas are also used to give an elementary exposition of the main properties of Young's seminormal representations.