Über Folgen,die BezÜglich Eines Mittels Rekurrent Sind

Let X be a convex subset of a finite-dimensional real vector space. A function M: X ^k → X is called a strict mean value, if M(x₁,…, x_k) lies in the convex hull of x₁,…, x_k), but does not coincide with one of its vertices. A sequence (x_n)_{n∈ ?} in X is called M-recursive if x_n+k = M(x_n, x_n+1,…, x_n+k?1) for all n. We prove that for a continuous strict mean value M every M-recursive sequence is convergent. We give a necessary and sufficient condition for a convergent sequence in X to be M-recursive for some continuous strict mean value M, and we characterize its limit by a functional equation. 39 B 72, 39 B 52, 40 A 05.     

The periods of the sequences generated by some symmetric shift registers

E_k(x₂,…, x_n) is defined by E_k(a₂,…, a_n) = 1 if and only if ∑_i=2ⁿa_i = k. We determine the periods of sequences generated by the shift registers with the feedback functions x₁ + E_k(x₂,…, x_n) and x₁ + E_k(x₂,…, x_n) + E_k+1(x₂,…, x_n) over the field GF(2).     

The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ? ⁿ ,n>1, is said to be diagonal if there existn functions,a ₁,...,a _n , on some intervals of ?, such that the covariance matrixV _F (m) ofF has diagonal (a ₁(m ₁),...,a _n (m _n )), for allm=(m ₁,...,m _n ) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ? ^k and ? ^n-k , for somek=1,...,n?1. This paper shows that there are only six types of irreducible diagonal NEFs in ? ⁿ , that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ? ⁿ , under what conditions is its projectionp(F) in ? ^k , underp(x ₁,...,x _n )∶=(x ₁,...,x _k ),k=1,...,n?1, still an NEF in ? ^k ? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofV _F (m ₁,...,m _n ) does not depend on (m _k+1,...,m _n ).     

A note on hypercube partitions

Two methods of partitioning an n-dimensional hypercube are considered. In the first method, referred to as the Stirling partitioning, we define l-dimensional partitions (l-partitions) which are defined in the 2-dimensional case by x₁ < x₂, x₂ < x₁, and x₂ = x₁. We show that (n ? k)-partitions are enumerated by the numbers T_n^k = (n ? k)! S_n^n?k, where S_n^n?k is the Stirling number of the second kind. In the second method, referred to as the Eulerian partitioning, we define k-boundary n-partitions as unions of n-partitions with their (n ?1)-through(n ? k)-partition boundaries. In the 2-dimensional case the 0 and 1 boundary 2-partitions are x₁ < x₂, x₂ ? x₁. We show that k boundary n-partitions are enumerated by the Eulerian numbers E_n^k. We apply these results to several combinatorial identities.     

On the central limit theorem for f (n k x)

By a classical observation in analysis, lacunary subsequences of the trigonometric system behave like independent random variables: they satisfy the central limit theorem, the law of the iterated logarithm and several related probability limit theorems. For subsequences of the system ( f (nx)) _n≥1 with 2π-periodic ${f\in L^2}$ this phenomenon is generally not valid and the asymptotic behavior of ( f (n _k x)) _k≥1 is determined by a complicated interplay between the analytic properties of f (e.g., the behavior of its Fourier coefficients) and the number theoretic properties of n _k . By the classical theory, the central limit theorem holds for f (n _k x) if n _k  = 2 ^k , or if n _k+1/n _k → α with a transcendental α, but it fails e.g., for n _k  = 2 ^k  ? 1. The purpose of our paper is to give a necessary and sufficient condition for f (n _k x) to satisfy the central limit theorem. We will also study the critical CLT behavior of f (n _k x), i.e., the question what happens when the arithmetic condition of the central limit theorem is weakened “infinitesimally”.     

Copositive pointwise approximation

We prove that if a functionf ∈C ⁽¹⁾ (I),I: = [?1, 1], changes its signs times (s ∈ ?) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ?, there exists an algebraic polynomialP _n =P _n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω _k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k ? 1, there exists a polynomialP _n =P _n (x) of degree ≤n such thatP _n (x) ≥ 0,x ∈I, and |f(x) ?P _n (x)| ≤c(k)ω _k (f;n ^?2 +n ^?1 √1 ?x ²),x ∈I.     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

A note on the kernels of higher derivations

Let k ? k′ be a field extension. We give relations between the kernels of higher derivations on k[X] and k′[X], where k[X]:= k[x ₁,…, x _n ] denotes the polynomial ring in n variables over the field k. More precisely, let D = {D _n } _n=0 ^∞ a higher k-derivation on k[X] and D′ = {D′ _n } _n=0 ^∞ a higher k′-derivation on k′[X] such that D′ _m (x _i ) = D _m (x _i ) for all m ? 0 and i = 1, 2,…, n. Then (1) k[X] ^D = k if and only if k′[X] ^D′ = k′; (2) k[X] ^D is a finitely generated k-algebra if and only if k′[X] ^D′ is a finitely generated k′-algebra. Furthermore, we also show that the kernel k[X] ^D of a higher derivation D of k[X] can be generated by a set of closed polynomials.     

Some results on starlike trees and sunlike graphs

A tree is called starlike if it has exactly one vertex of degree greater than two. In [4] it was proved that two starlike treesG andH are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, letG be a simple graph of ordern with vertex setV(G)={1,2, …,n} and letH={H ₁,H ₂, ...H _n } be a family of rooted graphs. According to [2], the rooted productG(H) is the graph obtained by identifying the root ofH _i with thei-th vertex ofG. In particular, ifH is the family of the paths $P_{k_1 } , P_{k_2 } , ..., P_{k_n } $ with the rooted vertices of degree one, in this paper the corresponding graphG(H) is called the sunlike graph and is denoted byG(k ₁,k ₂, …,k _n ). For any (x ₁,x ₂, …,x _n ) ∈I _* ⁿ , whereI _*={0,1}, letG(x ₁,x ₂, …,x _n ) be the subgraph ofG which is obtained by deleting the verticesi ₁, i₂, …,i _j ∈ V(G) (0≤j≤n), provided that $x_{i_1 } = x_{i_2 } = ... = x_{i_j } = 0$ . LetG(x ₁,x ₂,…, x _n] be the characteristic polynomial ofG(x ₁,x ₂,…, x _n ), understanding thatG[0, 0, …, 0] ≡ 1. We prove that $$G[k_1 , k_2 ,..., k_n ] = \Sigma _{x \in ^{I_ * ^n } } \left[ {\Pi _{i = 1}^n P_{k_i + x_i - 2} (\lambda )} \right]( - 1)^{n - (\mathop \Sigma \limits_{i = 1}^n x_i )} G[x_1 , x_2 , ..., x_n ]$$ where x=(x ₁,x ₂,…,x _n );G[k ₁,k ₂,…,k _n ] andP _n (γ) denote the characteristic polynomial ofG(k ₁,k ₂,…,k _n ) andP _n , respectively. Besides, ifG is a graph with λ₁(G)≥1 we show that λ₁(G)≤λ₁(G(k ₁,k ₂, ...,k _n )) < for all positive integersk ₁,k ₂,…,k _n , where λ₁ denotes the largest eigenvalue.     

Blocker sets, orthogonal arrays and their application to combination locks

Let A denote a set of order m and let X be a subset of A^k+1. Then X will be called a blocker (of A^k+1) if for any element say (a₁,a₂,…,a_k,a_k+1) of A^k+1, there is some element (x₁,x₂,…,x_k,x_k+1) of X such that x_i equals a_i for at least two i. The smallest size of a blocker set X will be denoted by α(m,k)and the corresponding blocker set will be called a minimal blocker. Honsberger (who credits Schellenberg for the result) essentially proved that α(2n,2) equals 2n² for all n. Using orthogonal arrays, we obtain precise numbers α(m,k) (and lower bounds in other cases) for a large number of values of both k and m. The case k=2 that is three coordinate places (and small m) corresponds to the usual combination lock. Supposing that we have a defective combination lock with k+1 coordinate places that would open if any two coordinates are correct, the numbers α(m,k) obtained here give the smallest number of attempts that will have to be made to ensure that the lock can be opened. It is quite obvious that a trivial upper bound for α(m,k) is m² since allowing the first two coordinates to take all the possible values in A will certainly obtain a blocker set. The results in this paper essentially prove that α(m,k) is no more than about m²/k in many cases and that the upper bound cannot be improved. The paper also obtains precise values of α(m,k) whenever suitable orthogonal arrays of strength two (that is, mutually orthogonal Latin squares) exist.     

Stirling polynomials

An investigation is made of the polynomials f_k(n) = S(n + k, n) and g_k(n) = (?1)^ks(n, n ? k), where S and s denote the Stirling numbers of the second and first kind, respectively. The main result gives a combinatorial interpretation of the coefficients of the polynomial (1 ? x)^2k+1Σ_n=0^∞f_k(n)xⁿ analogous to the well-known combinatorial interpretation of the Eulerian numbers in terms of descents of permutations.     

Polyhedra with the Integer Carathéodory Property

A polyhedron P has the Integer Carathéodory Property if the following holds. For any positive integer k and any integer vector w∈kP, there exist affinely independent integer vectors x₁,…,xt∈P and positive integers n₁,…,nt such that n₁+?+nt=k and w=n₁x₁+?+ntxt.In this paper we prove that if P is a (poly)matroid base polytope or if P is defined by a totally unimodular matrix, then P and projections of P have the Integer Carathéodory Property. For the matroid base polytope this answers a question by Cunningham from 1984.     

The rank-width of the square grid

Rank-width is a graph width parameter introduced by Oum and Seymour. It is known that a class of graphs has bounded rank-width if, and only if, it has bounded clique-width, and that the rank-width of G is less than or equal to its branch-width.The n×nsquare grid, denoted by G_n,n, is a graph on the vertex set {1,2,…,n}×{1,2,…,n}, where a vertex (x,y) is connected by an edge to a vertex (x^′,y^′) if and only if |x−x^′|+|y−y^′|=1.We prove that the rank-width of G_n,n is equal to n−1, thus solving an open problem of Oum.     

A balancing strategy

Suppose x₁, x₂,…, is a sequence of vectors in R^k, 6X_n6⩽1, where 6(x₁,…,x_k)6 = max_j|x_j|. An algorithm is given for choosing a corresponding sequence ε₁, ε₂,…, of numbers, ε_n = ±1, so that 6ε₁x₁+ … +ε_nx_n6 remains small.     

A construction of orthogonal arrays and applications to embedding theorems

An n^t by k orthogonal array is a collection of k-tuples of elements from an n-set, such that if a matrix is formed with the k-tuples as rows then each ordered t-tuple of elements appears exactly once as a row of each t columned and n^t rowed submatrix. If such an array has its set of k-tuples invariant under the elements of a subgroup G of S_t then the array is referred to as a G-array. A method is described for constructing a G-array of order nr from an array of order n and G-arrays of order r.The above described construction is used to produce finite embedding theorems for partial 3-quasigroups of various types. For a class of 3-quasigroups, such a theorem shows that a finite partial member of the class can be embedded in a finite complete member of the class. Theorems included produce finite embedding theorems for 3-quasigroups satisfying the identities 〈x,y,〈y,x,z〉〉=z and 〈〈z,x,y〉,y,x〉=z, for cyclic 3-quasigroup s, and conditional embedding theorems are presented for semi-symmetric 3-quasigroups.     

On vectorial addition chains

We prove the equivalence between the problem of computing a multinominal x₁^n₁·x₂^n₂···x_k^n_k, posed by R. E. Bellman (Amer. Math. Monthly70 1963, 765), and the problem of computing simultaneously the monomials x^n₁,…, x^n_k, posed by D. E. Knuth (“Seminumerical Algorithms”, Sect. 4.6.3, Addison-Wesley, Reading, Mass 1969.     

On the minimum number of completely 3-scrambling permutations

A family P={π₁,…,π_q} of permutations of [n]={1,…,n} is completely k-scrambling [Spencer, Acta Math Hungar 72; Füredi, Random Struct Algor 96] if for any distinct k points x₁,…,x_k∈[n], permutations π_i's in P produce all k! possible orders on π_i(x₁),…,π_i(x_k). Let N^*(n,k) be the minimum size of such a family. This paper focuses on the case k=3. By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison.

     

A multiindexed sturm sequence of polynomials and unimodality of certain combinatorial sequences

With any multiset n we associate the numbers $\text{O}$(n, k) of compositions of n into exactly k parts. The polynomials k_n(x) = Σ_kO(n, k)x^k are shown to form a multiindexed Sturm sequence over (?1, 0). As consequences we obtain the unimodality of the sequence {O(n, k)}_k for any n, of the generalized Eulerian numbers, and of the number of compositions of n with certain supplementary conditions imposed on the parts. The strong logarithmic concavity of the Stirling numbers of the second kind also follows as a corollary.     

The communication complexity of addition

Suppose each of k≤n ^o(1) players holds an n-bit number x _i in its hand. The players wish to determine if ∑ _i≤k x _i =s. We give a public-coin protocol with error 1% and communication O(k logk). The communication bound is independent of n, and for k≥3 improves on the O(k logn) bound by Nisan (Bolyai Soc. Math. Studies; 1993).     

On maximal antichains consisting of sets and their complements

Let 1 ? k₁ ? k₂ ? … ? k_n be integers and let S denote the set of all vectors x = (x₁, x₂, …, x_n) with integral coordinates satisfying 0 ? x_i ? k_i, i = 1, 2, …, n. The complement of x is (k₁ ? x₁, k₂ ? x₂, …, k_n ? x_n) and a subset X of S is an antichain provided that for any two distinct elements x, y of X, the inequalities x_i ? y_i, i = 1, 2, …, n, do not all hold. We determine an LYM inequality and the maximal cardinality of an antichain consisting of vectors and its complements. Also a generalization of the Erdös-Ko-Rado theorem is given.