关于m-HPK(n1,n2,n3,n4)-残差图

本文研究了3-维超平面完备残差图以及m 重3-维超平面完备残差图. 利用容斥原理以及集合的运算性质等方法, 分别获得了3-维超平面完备残差图和m 重3-维超平面完备残差图的最下阶以及二者的唯一极图, 将文献[1] 中定义的残差图从平面推广到超平面上.     

An n log n Algorithm for Online BDD Refinement

Binary decision diagrams are in widespread use in verification systems for the canonical representation of finite functions. Here we consider multivalued BDDs, which represent functions of the form : ^ν →  , where is a finite set of leaves. We study a rather natural online BDD refinement problem: a partition of the leaves of several shared BDDs is gradually refined, and the equivalence of the BDDs under the current partition must be maintained in a discriminator table. We show that it can be solved in O(n log n) time if n bounds both the size of the BDDs and the total size of update operations. Our algorithm is based on an understanding of BDDs as the fixed points of an operator that in each step splits and gathers nodes. We apply our algorithm to show that automata BDD-represented transition functions can be minimized in time O(n · log n), where n is the total number of BDD nodes representing the automaton. This result is not an instance of Hopcroft's classical minimization algorithm, which breaks down for BDD-represented automata because of the BDD path compression property.     

Fuzzy n-cell numbers and the differential of fuzzy n-cell number value mappings

In this paper, we give the definition of a special kind of n-dimension fuzzy numbers, fuzzy n-cell numbers, discuss their operations and representation theorems, define a complete metric on the fuzzy n-cell number space and prove that the metric is equivalent to the supremum metric derived by the Hausdorff metric between the level sets of the n-dimension fuzzy numbers, and obtain an embedding theorem of the fuzzy n-cell number space (isometrically embeds it into a concrete Banach space). We also consider the differential of the fuzzy mappings from an interval into the fuzzy n-cell number space by using the embedding theorem.     

On the n-Coupling Problem

In this paper we obtain based on an idea of M. Knott and C. S. Smith (1994, Linear Algebra Appl.199, 363–371) characterizations of solutions of three-coupling problems by reduction to the construction of optimal couplings of each of the variables to the sum. In the case of normal distributions this leads to a complete solution. Under a technical condition this idea also works for general distributions and one obtains explicit results. We extend these results to the n-coupling problem and derive a characterization of optimal n-couplings by several 2-coupling problems. This leads to some constructive existence results for Monge solutions.     

An n-dimensional Wiener-Plancherel formula

The n-dimensional analogues of Wiener's “s-function” and difference operator associated to rectangular limit processes are defined. The Wiener-Plancherel formula is proved in this context for functions having bounded quadratic means.     

On reducibility of n-ary quasigroups

An n-ary operation Q:Σⁿ→Σ is called an n-ary quasigroup of order |Σ| if in the relation x₀=Q(x₁,…,x_n) knowledge of any n elements of x₀,…,x_n uniquely specifies the remaining one. Q is permutably reducible if Q(x₁,…,x_n)=P(R(x_σ(1),…,x_σ(k)),x_σ(k+1),…,x_σ(n)) where P and R are (n-k+1)-ary and k-ary quasigroups, σ is a permutation, and 1<k<n. An m-ary quasigroup S is called a retract of Q if it can be obtained from Q or one of its inverses by fixing n-m>0 arguments. We prove that if the maximum arity of a permutably irreducible retract of an n-ary quasigroup Q belongs to {3,…,n-3}, then Q is permutably reducible.     

Scalar polynomial functions on the n×n matrices over a finite field

Let F=GF(q) denote the finite field of order q, and let $\text{?(x)?F[x]}$. Then f(x) defines, via substitution, a function from F_n×n, the n×n matrices over F, to itself. Any function $\text{?:F}{}_{\mathrm{n\times n}}\text{→ F}{}_{\mathrm{n\times n}}$ which can be represented by a polynomialf(x)?F[x] is called a scalar polynomial function on F_n×n. After first determining the number of scalar polynomial functions on F_n×n, the authors find necessary and sufficient conditions on a polynomial $\text{?(x) ? F[x]}$ in order that it defines a permutation of (i) $\text{D}$_n, the diagonalizable matrices in F_n×n, (ii)$\text{R}$_n, the matrices in F_n×n all of whose roots are in F, and (iii) the matric ring F_n×n itself. The results for (i) and (ii) are valid for an arbitrary field F.     

L approximation with trigonometric n-nomials

We compare the degree of approximation to L²(−π, π) by nth degree trigonometric polynomials, with the degree of approximation by trigonometric n-nomials, which are linear combinations, with constant (complex) coefficients, of any 2n + 1 members of the sequence {exp (ikx)}, − ∞ < k < ∞.     

n-Star modules over ring extensions

We give conditions under which an n-star module extends to an n-star module, or an n-tilting module, over a ring extension R of A. In case that R is a split extension of A by Q, we obtain that is a 1-tilting module (respectively, a 1-star module) if and only if is a 1-tilting module (respectively, a 1-star module) and generates both and (respectively, generates ), where is an injective cogenerator in the category of all left A-modules. These extend results in [I. Assem, N. Marmaridis, Tilting modules over split-by-nilpotent extensions, Comm. Algebra 26 (1998) 1547–1555; K.R. Fuller, *-Modules over ring extensions, Comm. Algebra 25 (1997) 2839–2860] by removing the restrictions on R and Q.     

The game of n-times nim

The following game is considered. The first player can take any number of stones, but not all the stones, from a single pile of stones. After that, each player can take at most n-times as many as the previous one. The player first unable to move loses and his opponent wins. Let f₁,f₂,… be an initial sequence of stones in increasing order, such that the second player has a winning strategy when play begins from a pile of size f_i. It is proved that there exist constants c=c(n) and k₀=k₀(n) such that f_k+1=f_k+f_k−c for all k>k₀, and lim_n→∞ c(n)/(nlogn)=1.     

Hamiltonian cycles in n-factor-critical graphs

A graph G is said to be n-factor-critical if G−S has a 1-factor for any SV(G) with |S|=n. In this paper, we prove that if G is a 2-connected n-factor-critical graph of order p with , then G is hamiltonian with some exceptions. To extend this theorem, we define a (k,n)-factor-critical graph to be a graph G such that G−S has a k-factor for any SV(G) with |S|=n. We conjecture that if G is a 2-connected (k,n)-factor-critical graph of order p with , then G is hamiltonian with some exceptions. In this paper, we characterize all such graphs that satisfy the assumption, but are not 1-tough. Using this, we verify the conjecture for k2.     

Extensions of n-Hom Lie algebras

n-Hom Lie algebras are twisted by n-Lie algebras by means of twisting maps. n-Hom Lie algebras have close relationships with statistical mechanics and mathematical physics. The paper main concerns structures and representations of n-Hom Lie algebras. The concept of n_ρ-cocycle for an n-Hom Lie algebra (G, [,… , ], α) related to a G-module (V, ρ, β) is proposed, and a sufficient condition for the existence of the dual representation of an n-Hom Lie algebra is provided. From a G-module (V, ρ, β) and an n_ρ-cocycle θ, an n-Hom Lie algebra (T_θ(V ), [, … , ]θ, γ) is constructed on the vector space T_θ(V ) = G⊕V, which is called the T_θ-extension of an n-Hom Lie algebra (G, [, … , ], α) by the G-module (V, ρ, β).     

AnO(m + n log n) Algorithm for the Maximum-Clique Problem in Circular-Arc Graphs

We present an algorithm to compute, inO(m + n log n) time, a maximum clique in circular-arc graphs (withnvertices andmedges) provided a circular-arc model of the graph is given. If the circular-arc endpoints are given in sorted order, the time complexity isO(m). The algorithm operates on the geometric structure of the circular arcs, radially sweeping their endpoints; it uses a very simple data structure consisting of doubly linked lists. Previously, the best time bound for this problem wasO(m log log n + n log n), using an algorithm that solved an independent subproblem for each of thencircular arcs. By using the radial-sweep technique, we need not solve each of these subproblems independently; thus we eliminate the log log nfactor from the running time of earlier algorithms. For vertex-weighted circular-arc graphs, it is possible to use our approach to obtain anO(m log log n + n log n) algorithm for finding a maximum-weight clique—which matches the best known algorithm.     

Bifurcations of travelling wave solutions for the mK(n, n) equation

Using the method of planar dynamical systems to the mK(n, n) equation, the existence of uncountably infinite many smooth and non-smooth periodic wave solutions, solitary wave solutions and kink and anti-kink wave solutions is proved. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All possible exact explicit parametric representations of smooth and non-smooth travelling wave solutions are obtain.     

Measurable n-sensitivity and maximal pattern entropy

We introduce the notion of measurable n-sensitivity for measure preserving systems, and study the relation between measurable n-sensitivity and the maximal pattern entropy. We prove that, if (X, B, µ, T) is ergodic, then (X, B, µ, T) is measurable n-sensitive but not measurable (n+1)-sensitive if and only if h_µ^*(T) = log n, where h_µ^* (T) is the maximal pattern entropy of T.     

Polynomially dependent homomorphisms and frobenius n-homomorphisms

We introduce and study polynomially dependent homomorphisms, which are special linear maps between associative algebras with identity. The multiplicative structure is much involved in the definition of such homomorphisms (we consider only the case of maps f: A → B with commutative B). The most important particular case of these maps are the Frobenius n-homomorphisms, which were introduced by V.M. Buchstaber and E.G. Rees in 1996–1997. A 1-homomorphism f: A → B is just an algebra homomorphism (the algebra B is commutative). A typical example of an n-homomorphism is given by the sum of n algebra homomorphisms, f = f ₁ + ... + f _n , f _i : A → B, 1 ≤ i ≤ n. Another example is the trace of n × n matrices over a field R of characteristic zero, tr: M _n (R) → R, and, more generally, the character of any n-dimensional representation, tr ρ: A → R, ρ: A → M _n (R). The properties of n-homomorphisms (some of which were proved by Buchstaber and Rees under additional conditions) are derived, and a general theory of polynomially dependent homomorphisms is developed. One of the main results of the paper is a uniqueness theorem, which distinguishes the classes of n-homomorphisms among all polynomially dependent homomorphisms by a single natural completeness condition. As a topological application of n-homomorphisms, we consider the theory of n-homomorphisms between commutative C*-algebras with identity. We prove that the norm of any such n-homomorphism is equal to n and describe the structure of all such n-homomorphisms, which generalizes the classical Gelfand transform (the case of n = 1). An interesting fact discovered is that the Gelfand transform, which is a functorial bijection between appropriate spaces of maps, becomes a homeomorphism after considering natural topologies on these spaces.     

On the problem op n-stationary centers

Among the various problems of celestial mechanics related to the n-body problem, a certain amount of interest attaches to the specific situation wherein a passive gravitational point mass M moves under the assumption that the relative disposition of the other active gravitational masses experiences no large changes.

If the attracting masses are regarded as points and if changes in the relative disposition of the attracting bodies are neglected, one arrives at the problem of the motion of the point M in a field produced by n-stationary attracting centers (the point M here represents the (n+l)-th body).

In addition to the problem of central motion (n = 1), soluble dynamics problems of this category include Euler's case [1] of two (n= 2) stationary Newtonian attracting centers.

This problem, which for a long time was of solely theoretical Interest as an example of an integrable Liouville system [2], has recently been attracting attention in connection with the mechanics of artificial satellites, particularly after it was shown that the potential of an oblate spheroid can be approximated by the potential of two specifically chosen stationary Newtonian attracting centers [3 and 4].

The solution of the problem for n-attracting centers for n ≥ 3 is unknown, except for a single special case of three centers pointed out by Lagrange and considered In greater detail by J.A. Serre [5].

We shall concern ourselves here with problems on the existence of periodic trajectories in the case of n-attracting centers. An arbitrary and not necessarily Newtonian gravitational law will be assumed.

Our analysis is based on the theory of quasiindices of singular force field points as set forth in [60].