On the degree of homogeneous bent functions

In this paper, the degree of homogeneous bent functions is discussed. We prove that for any nonnegative integer k, there exists a positive integer N such that for n?N there exist no 2n- variable homogeneous bent functions having degree n-k or more, where N is the least integer satisfying .     

On homogeneous rotation symmetric bent functions

In this paper, we present partial results towards the conjectured nonexistence of homogeneous rotation symmetric bent functions having degree > 2.     

Generalized bent functions and class group of imaginary quadratic fields

Several new results on non-existence of generalized bent functions are presented. The results are related to the class number of imaginary quadratic fields.     

The ranks of Maiorana-McFarland bent functions

In this paper, the ranks of a special family of Maiorana-McFarland bent functions are discussed. The upper and lower bounds of the ranks are given and those bent functions whose ranks achieve these bounds are determined. As a consequence, the inequivalence of some bent functions are derived. Furthermore, the ranks of the functions of this family are calculated when t 6.     

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.     

Analytical representation of concave and quasiconcave homogeneous functions

The formula which implements bijection between the class of concave linearly homogeneous functions defined on the nonnegative orthant of an arithmetic space and the simpler class of concave functions defined on the standard (probabilistic) simplex is presented. Two generalizations of this formula for analytical representation of quasiconcave homogeneous function are also proposed. These formulas particularly extend opportunities of modelling production objects and consumption.     

Estimates for functions of the Laplace operator on homogeneous trees

In this paper, we study the heat equation on a homogeneous graph, relative to the natural (nearest-neighbour) Laplacian. We find pointwise estimates for the heat and resolvent kernels, and the mapping properties of the corresponding operators.

     

The relationship between the nonexistence of generalized bent functions and diophantine equations

Two new results on the nonexistence of generalized bent functions are presented by using properties of the decomposition law of primes in cyclotomic fields and properties of solutions of some Diophantine equations, and examples satisfying our results are given.     

New results on non-existence of generalized bent functions (Ⅱ)

Several new results on non-existence of generalized bent functions are presented by using the class group of related imaginary abelian number fields.     

New results on non-existence of generalized bent functions (II)

Several new results on non-existence of generalized bent functions are presented by using the class group of related imaginary abelian number fields.     

Integration and homogeneous functions

We show that integrating a (positively) homogeneous function on a compact domain reduces to integrating a related function on the boundary . The formula simplifies when the boundary is determined by homogeneous functions. Similar results are also presented for integration of exponentials and logarithms of homogeneous functions.

     

The Murnaghan-Nakayama rule for k-Schur functions

We prove the Murnaghan-Nakayama rule for k-Schur functions of Lapointe and Morse, that is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene.     

Spaces with many affine functions

We describe all metric spaces that have sufficiently many affine functions. As an application we obtain a metric characterization of linear-convex subsets of Banach spaces.

     

CM分担三个公共小函数对的亚纯函数的惟一性

研究了具有三个CM公共小函数对，一个IM公共小函数对的亚纯函数的惟一性，改进了李平与杨重骏，Brosch等人的有关结果，用例子说明本文的结果是最佳的.     

Results on rotation symmetric bent functions

In this paper we analyze the combinatorial properties related to the Walsh spectra of rotation symmetric Boolean functions on even number of variables. These results are then applied in studying rotation symmetric bent functions. For the first time we could present an enumeration strategy for all the 10-variable rotation symmetric bent functions.     

A relationship between the nonexistence of generalized bent functions and class groups

A new result on the nonexistence of generalized bent functions is presented by using properties of the decomposition law of primes in cyclotomic fields and properties of solutions of some Diophantine equations. At the same time,a method is given which can be used to simplify the known results. Then we give the bounds and the meaning in algebraic number theory of the parameters in our results.     

Strongly regular graphs associated with ternary bent functions

We prove a new characterization of weakly regular ternary bent functions via partial difference sets. Partial difference sets are combinatorial objects corresponding to strongly regular graphs. Using known families of bent functions, we obtain in this way new families of strongly regular graphs, some of which were previously unknown. One of the families includes an example in [N. Hamada, T. Helleseth, A characterization of some {3v₂+v₃,3v₁+v₂,3,3}-minihypers and some [15,4,9;3]-codes with B₂=0, J. Statist. Plann. Inference 56 (1996) 129-146], which was considered to be sporadic; using our results, this strongly regular graph is now a member of an infinite family. Moreover, this paper contains a new proof that the Coulter-Matthews and ternary quadratic bent functions are weakly regular.