Composite Integers n for Which φ（n）｜n- 1

In this note, we show that the number of composite integers n ≤ x such that φ（n）｜n - 1 is at most O（x^1/2（loglog x）^1/2）, thus improving earlier results by Pomerance and by Shan.     

THE GLOBAL APPROXIMATION BY LEFT-BERNSTEIN-DURRMEYER QUASI-INTERPOLANTS IN Lp[0, 1]

In this paper,we will use the 2r-th Ditzian-Totik modulus of smoothness wp^2r(f,t)p to discuss the direct and inverse theorem of approximation by Left-Bernstein-Durrmeyer quasi-interpolants Mn^[2r-1]f for functions of the space Lp[0,1](1≤p≤ ∞)。     

Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

The Fine Spectra of the Difference Operator △ Over the Sequence Space bvp, （1 ≤ p 〈 ∞）^＊

The sequence space bvp consisting of all sequences （xk） such that （xk -xk-1） belongs to the space gp has recently been introduced by Basar and Altay [Ukrainian Math. J., 55（1）, 136-147（2003）]; where 1 ≤ p ≤ ∞. In the present paper, some results concerning with the continuous dual and f-dual, and the AD-property of the sequence space bvp have been given and the norm of the difference operator A acting on the sequence space bvp has been found. The fine spectrum with respect to the Goldberg＇s classification of the difference operator △ over the sequence space bvp has been determined, where 1≤p〈∞.     

The <Emphasis Type="Italic">k</Emphasis>-point exponent set of primitive digraphs with girth 2

Let D = （V, E） be a primitive digraph. The vertex exponent of D at a vertex v∈ V, denoted by expD（v）, is the least integer p such that there is a v →u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that exPD（V1） ≤ exPD（V2） …≤ exPD（Vn）. Then exPD（Vk） is called the k- point exponent of D and is denoted by exPD （k）, 1≤ k ≤ n. In this paper we define e（n, k） ：= max{expD （k） ｜ D ∈ PD（n, 2）} and E（n, k） ：= {exPD（k）｜ D ∈ PD（n, 2）}, where PD（n, 2） is the set of all primitive digraphs of order n with girth 2. We completely determine e（n, k） and E（n, k） for all n, k with n ≥ 3 and 1 ≤ k ≤ n.     

Localization transition of <Emphasis Type="Italic">d</Emphasis>-friendly walkers

 Friendly walkers is a stochastic model obtained from independent one-dimensional simple random walks {S ^k _j } _j≥0 , k=1,2,…,d by introducing ``non-crossing condition': and ``reward for collisions' characterized by parameters . Here, the reward for collisions is described as follows. If, at a given time n, a site in ℤ is occupied by exactly m≥2 walkers, then the site increases the probabilistic weight for the walkers by multiplicative factor exp (β _m )≥1. We study the localization transition of this model in terms of the positivity of the free energy and describe the location and the shape of the critical surface in the (d−1)-dimensional space for the parameters . Received: 13 June 2002 / Revised version: 24 August 2002 Published online: 28 March 2003 Mathematics Subject Classification (2000): 82B41, 82B26, 82D60, 60G50 Key words or phrases: Random walks – Random surfaces – Lattice animals – Phase transitions – Polymers – Random walks     

<Emphasis Type="Italic">f</Emphasis>-Colorings of some graphs of <Emphasis Type="Italic">f</Emphasis>-class 1

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v V（G） at most f（v） times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by X′f（G）. Any simple graph G has the f-chromatic index equal to △f（G） or △f（G） ＋ 1, where △f（G） =max v V（G）{[d（v）/f（v）]}. If X′f（G） = △f（G）, then G is of f-class 1; otherwise G is of f-class 2. In this paper, a class of graphs of f-class 1 are obtained by a constructive proof. As a result, f-colorings of these graphs with △f（G） colors are given.     

Depth of functions of the <Emphasis Type="Italic">k</Emphasis>-valued logic in infinite bases

The realization of functions of the k-valued logic by circuits is considered over an arbitrary infinite complete basis B. The Shannon function D _B (n) of the circuit depth over B is examined (for any positive integer n the value D _B (n) is the minimal depth sufficient to realize every function of the k-valued logic of n variables by a circuit over B). It is shown that for each fixed k ≥ 2 and for any infinite complete basis B either there exists a constant α ≥ 1 such that D _B (n) = α for all sufficiently large n, or there exist constants β (β > 0), γ, δ such that βlog₂ n ≤ D _B (n) ≤ γlog₂ n + δ for all n.     

On the Lower Tail Probabilities of Some Random Sequences in <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We investigate the behaviour of the logarithmic small deviation probability of a sequence (σ _n θ _n ) in l _p , 0<p≤∞, where (θ _n ) are i.i.d. random variables and (σ _n ) is a decreasing sequence of positive numbers. In particular, the example σ _n ∼n ^−μ (1+log n)^−ν is studied thoroughly. Contrary to the existing results in the literature, the rate function and the small deviation constant are expressed expli- citly in the present treatment. The restrictions on the distribution of θ ₁ are kept to an absolute minimum. In particular, the usual variance assumption is removed. As an example, the results are applied to stable and Gamma-distributed random variables.     

Generating Random Vectors in (ℤ/<Emphasis Type="Italic">p</Emphasis>ℤ)<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> via an Affine Random Process

This paper considers some random processes of the form X _n+1=T X _n +B _n (mod p) where B _n and X _n are random variables over (ℤ/pℤ) ^d and T is a fixed d×d integer matrix which is invertible over the complex numbers. For a particular distribution for B _n , this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det (T), order (log p)² steps suffice to make X _n close to uniformly distributed where X ₀ is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p ^b steps are needed for X _n to get close to uniformly distributed for some positive value b≤2 which may depend on T and X ₀ is the zero vector.     

A sufficient degree condition for a graph to contain all trees of size <Emphasis Type="Italic">k</Emphasis>

The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d _G (x) + d _G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d _G (u) ≥ k.     

Zeros in Tables of Characters for the Groups <Emphasis Type="Italic">S</Emphasis><Subscript>n</Subscript> and <Emphasis Type="Italic">A</Emphasis><Subscript>n</Subscript>. II

Let P(n) be the set of all partitions of a natural number n. In the representation theory of symmetric groups, for every partition α ∈ P(n), the partition h(α) ∈ P(n) is defined so as to produce a certain set of zeros in the character table for S_n. Previously, the analog f(α) of h(α) was obtained pointing out an extra set of zeros in the table mentioned. Namely, h(α) is greatest (under the lexicographic ordering ≤) of the partitions β of n such that χ^α(g_β) ≠ 0, and f(α) is greatest of the partitions γ of n that are opposite in sign to h(α) and are such that χ^α(g_γ) ≠ 0, where χ^α is an irreducible character of S_n, indexed by α, and g_β is an element in the conjugacy class of S_n, indexed by β. For α ∈ P(n), under some natural restrictions, here, we construct new partitions h′(α) and f′(α) of n possessing the following properties. (A) Let α ∈ P(n) and n ⩾ 3. Then h′(α) is identical is sign to h(α), χ^α(g_h′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of h(α), and h′(α) < γ < h(α). (B) Let α ∈ P(n), α ≠ α′, and n ⩾ 4. Then f′(α) is identical in sign to f(α), χ^α(g_f′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of f(α), and f′(α) < γ < f(α). The results obtained are then applied to study pairs of semiproportional irreducible characters in A_n. Supported by RFBR grant No. 04-01-00463. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 643–663, November–December, 2005.     

SCORE LISTS IN （h, k）-BIPARTITE HYPERTOURNAMENTS

Given non-negative integers m,n,h and k with m ≥ h ＞ 1 and n ≥ k ＞ 1, an (h, k)-bipartite hypertournament on m n vertices is a triple (U, V, A), where U and V are two sets of vertices with |U| = m and |V| = n, and A is a set of (h k)-tuples of vertices,called arcs, with at most h vertices from U and at most k vertices from V, such that for any h k subsets U1 ∪ V1 of U ∪ V, A contains exactly one of the (h k)! (h k)-tuples whose entries belong to U1 ∪ V1. Necessary and sufficient conditions for a pair of non-decreasing sequences of non-negative integers to be the losing score lists or score lists of some(h, k)-bipartite hypertournament are obtained.     

Sign change of the function <Emphasis Type="Italic">S</Emphasis>(<Emphasis Type="Italic">t</Emphasis>) on short intervals

A theorem for the sign variation of the argument of the Riemann zeta function S(t) in the interval (t − A, t + A) with A = 4.39 ln ln ln ln T for each t, T ≤ t ≤ T + H excluding values from the set E with the measure mes(E) = O(H(ln ln T)⁻¹(ln ln ln T)^−0,5) is proved.     

Renormalization of multiple <Emphasis Type="Italic">q</Emphasis>-zeta values

In this paper, we shall define the renormalization of the multiple q-zeta values （MqZV） which are special values of multiple q-zeta functions ζq （s1,..., Sd） when the arguments are all positive integers or all non-positive integers. This generalizes the work of Guo and Zhang （Renormalization of Multiple Zeta Values, arxiv： math/0606076v3）. We show that our renormalization process produces the same values if the MqZVs are well-defined originally and that these renormalizations of MqZV satisfy the q-stuffle relations if we use shifted-renormalizations for all divergent ζq（S1,..., sd） （i.e., s1 ≤ 1）. Moreover, when q ↑ 1 our renormalizations agree with those of Guo and Zhang.     

Geodesic loops and periodic geodesics on a Riemannian manifold diffeomorphic to <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

Let M ⁿ be a closed 2-connected Riemannian manifold, such that π₃(M ⁿ ) ≠ { 0 }. In this paper we prove that either there exists a periodic geodesic on M ⁿ of length ≤ 6d, where d is the diameter of M ⁿ , or at each point p ∈ M ⁿ there exists a geodesic loop of length ≤ 2d.     

Finiteness conditions and PD<Subscript><Emphasis Type="Italic">r</Emphasis></Subscript>-group covers of PD<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-complexes

We show that an infinite cyclic covering space M′ of a PD _n -complex M is a PD _n-1-complex if and only if χ(M) = 0, M′ is homotopy equivalent to a complex with finite [(n−1)/2]-skeleton and π₁(M′) is finitely presentable. This is best possible in terms of minimal finiteness assumptions on the covering space. We give also a corresponding result for covering spaces M _ν with covering group a PD _r -group under a slightly stricter finiteness condition.      

Some results on 4<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> designs with clear two-factor interaction components

Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs, and it has become an active research issue in recent years. Tang et al. derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi’s) in 2 ^n−(n−k) fractional factorial designs of resolutions III and IV by constructing a 2 ^n−(n−k) design for given k, which are only restricted for the symmetrical case. This paper proposes and studies the clear effects problem for the asymmetrical case. It improves the construction method of Tang et al. for 2 ^n−(n−k) designs with resolution III and derives the upper and lower bounds on the maximum number of clear two-factor interaction components (2fic’s) in 4 ^m 2 ⁿ designs with resolutions III and IV. The lower bounds are achieved by constructing specific designs. Comparisons show that the number of clear 2fic’s in the resulting design attains its maximum number in many cases, which reveals that the construction methods are satisfactory when they are used to construct 4 ^m 2 ⁿ designs under the clear effects criterion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10571093, 10671099 and 10771123), the Research Foundation for Doctor Programme (Grant No. 20050055038) and the Natural Science Foundation of Shandong Province of China (Grant No. Q2007A05). Zhang’s research was also supported by the Visiting Scholar Program at Chern Institute of Mathematics.     

<Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-nearest neighbor estimators of entropy

For estimating the entropy of an absolutely continuous multivariate distribution, we propose nonparametric estimators based on the Euclidean distances between the n sample points and their k _n -nearest neighbors, where {k _n : n = 1, 2, …} is a sequence of positive integers varying with n. The proposed estimators are shown to be asymptotically unbiased and consistent.