On the tree structure of the power digraphs modulo n

For any two positive integers n and k ? 2, let G(n, k) be a digraph whose set of vertices is {0, 1, …, n ? 1} and such that there is a directed edge from a vertex a to a vertex b if a ^k ≡ b (mod n). Let \(n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} \) be the prime factorization of n. Let P be the set of all primes dividing n and let P ₁, P ₂ ? P be such that P ₁ ∪ P ₂ = P and P ₁ ∩ P ₂ = ?. A fundamental constituent of G(n, k), denoted by \(G_{{P_2}}^*(n,k)\), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \(\prod\nolimits_{{p_i} \in {P_2}} {{p_i}} \) and are relatively prime to all primes q ∈ P ₁. L. Somer and M. K?i?ek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic.     

Note on a conjecture for the sum of signless Laplacian eigenvalues

For a simple graph G on n vertices and an integer k with 1 ? k ? n, denote by \(\mathcal{S}^+_k\) (G) the sum of k largest signless Laplacian eigenvalues of G. It was conjectured that \(\mathcal{S}^+_k(G)\leqslant{e}(G)+(^{k+1}_{\;\;2})\) (G) ? e(G) + (k+1 2), where e(G) is the number of edges of G. This conjecture has been proved to be true for all graphs when k ∈ {1, 2, n ? 1, n}, and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all k). In this note, this conjecture is proved to be true for all graphs when k = n ? 2, and for some new classes of graphs.     

Vertex and Edge Orbits of Fibonacci and Lucas Cubes

The Fibonacci cube \({\Gamma_{n}}\) is obtained from the n-cube Q _n by removing all the vertices that contain two consecutive 1s. If, in addition, the vertices that start and end with 1 are removed, the Lucas cube \({\Lambda_{n}}\) is obtained. The number of vertex and edge orbits, the sets of the sizes of the orbits, and the number of orbits of each size, are determined for the Fibonacci cubes and the Lucas cubes under the action of the automorphism group. In particular, the set of vertex orbit sizes of \({\Lambda_{n}}\) is \({\{k \geq 1; k |n\} \cup \{k \geq 18; k |2n\}}\), the number of vertex orbits of \({\Lambda_{n}}\) of size k, where k is odd and divides n, is equal to \({\sum_{d | k} \mu (\frac{k}{d})F_{\lfloor{\frac{d}{2}}\rfloor+2}}\), and the number of edge orbits of \({\Lambda_{n}}\) is equal to the number of vertex orbits of \({\Gamma_{n-3}}\). Dihedral transformations of strings and primitive strings are essential tools to prove these results.     

Universal Cycle Packings and Coverings for <Emphasis Type="Italic">k</Emphasis>-Subsets of an <Emphasis Type="Italic">n</Emphasis>-Set

A cyclic sequence of elements of [n] is an (n, k)-Ucycle packing (respectively, (n, k)-Ucycle covering) if every k-subset of [n] appears in this sequence at most once (resp. at least once) as a subsequence of consecutive terms. Let \(p_{n,k}\) be the length of a longest (n, k)-Ucycle packing and \(c_{n,k}\) the length of a shortest (n, k)-Ucycle covering. We show that, for a fixed \(k,p_{n,k}={n\atopwithdelims ()k}-O(n^{\lfloor k/2\rfloor })\). Moreover, when k is not fixed, we prove that if \(k=k(n)\le n^{\alpha }\), where \(0<\alpha <1/3\), then \(p_{n,k}={n\atopwithdelims ()k}-o({n\atopwithdelims ()k}^\beta )\) and \(c_{n,k}={n\atopwithdelims ()k}+o({n\atopwithdelims ()k}^\beta )\), for some \(\beta <1\). Finally, we show that if \(k=o(n)\), then \(p_{n,k}={n\atopwithdelims ()k}(1-o(1))\).     

Proof of the Erdős matching conjecture in a new range

Let s > k ≧ 2 be integers. It is shown that there is a positive real ε = ε(k) such that for all integers n satisfying (s + 1)k ≦ n < (s + 1)(k + ε) every k-graph on n vertices with no more than s pairwise disjoint edges has at most \(\left( {\begin{array}{*{20}{c}} {\left( {s + 1} \right)k - 1} \\ k \end{array}} \right)\) edges in total. This proves part of an old conjecture of Erd?s.     

Judicious bisection of hypergraphs

Judicious bisection of hypergraphs asks for a balanced bipartition of the vertex set that optimizes several quantities simultaneously.In this paper,we prove that if G is a hypergraph with n vertices and m_i edges of size i for i=1,2,...,k,then G admits a bisection in which each vertex class spans at most(m_1)/2+1/4m_2+…+(1/(2~k)+m_k+o(m_1+…+m_k) edges,where G is dense enough or △(G)= o(n) but has no isolated vertex,which turns out to be a bisection version of a conjecture proposed by Bollobas and Scott.     

Some irreducibility and indecomposability results for truncated binomial polynomials of small degree

In this paper, we show that the truncated binomial polynomials defined by \(P_{n,k}(x)={\sum }_{j=0}^{k} {n \choose j} x^{j}\) are irreducible for each k≤6 and every n≥k+2. Under the same assumption n≥k+2, we also show that the polynomial P _n,k cannot be expressed as a composition P _n,k (x) = g(h(x)) with \(g \in \mathbb {Q}[x]\) of degree at least 2 and a quadratic polynomial \(h \in \mathbb {Q}[x]\). Finally, we show that for k≥2 and m,n≥k+1 the roots of the polynomial P _m,k cannot be obtained from the roots of P _n,k , where m≠n, by a linear map.     

On <Emphasis Type="Italic">k</Emphasis>- critical 2<Emphasis Type="Italic">k</Emphasis>- connected graphs

A graph G is called an (n,k)-graph if κ(G-S)=n-|S| for any S ? V(G) with |S| ≤ k, where ?(G) denotes the connectivity of G. Mader conjectured that for k ≥ 3 the graph K_2k+2?(1-factor) is the unique (2k, k)-graph. Kriesell has settled two special cases for k = 3,4. We prove the conjecture for the general case k ≥ 5.     

On the phase transitions of (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-SAT

Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let T _k (V) denote the set of all 2 ^k n ^k possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in T _k (V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log₂ n and let \({p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}\). We prove that

$$\mathop {\lim }\limits_{n \to \infty } P\left[ {I is satisfiable} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {p \leqslant (1 - \varepsilon )p_0 ,} \\ {0,} & {p \geqslant (1 + \varepsilon )p_0 .} \\ \end{array} } \right.$$

     

Compound Geometric Distribution of Order <Emphasis Type="Italic">k</Emphasis>

The distribution of the number of trials until the first k consecutive successes in a sequence of Bernoulli trials with success probability p is known as geometric distribution of order k. Let T _k be a random variable that follows a geometric distribution of order k, and Y ₁,Y ₂,… a sequence of independent and identically distributed discrete random variables which are independent of T _k . In the present article we develop some results on the distribution of the compound random variable \(S_{k} =\sum_{t=1}^{T_{k}}Y_{t}\).     

Dominant dimensions,derived equivalences and tilting modules

We show that for every ? > 0 there exist δ > 0 and n₀ ∈ ? such that every 3-uniform hypergraph on n ≥ n₀ vertices with the property that every k-vertex subset, where k ≥ δn, induces at least \(\left( {\frac{1}{2} + \varepsilon } \right)\left( {\begin{array}{*{20}c} k \\ 3 \\ \end{array} } \right)\) edges, contains K₄^? as a subgraph, where K₄^? is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erd?s and Sós. The constant 1/4 is the best possible.     

Estimates of sums of integrals of the Legendre Polynomial

Estimates of sums \({R_{nk}}\left( x \right) = \sum\limits_{m = n}^\infty {{P_{mk}}\left( x \right)} \) are established. Here, P_n0(x)= P_n(x), \({R_{nk}}\left( x \right) = \int\limits_.^x {{P_{n,k - 1}}\left( y \right)dy} \), P_n is the Legendre polynomial with standard normalization P_n(1) = 1. With k = 1 in the main interval [–1, 1] the sum decreases with increasing n as n^–1, and in the half-open interval [–1, 1), as n^–3/2. With k > 1 the point x = 1 does not need to be excluded. The sum decreases as n^-k–1/2. Moreover, a small increase in the multiplicative constant permits to obtain the estimate \(|{R_{nk}}\left( {\cos \theta } \right)| < \frac{{C{{\sin }^{k - 3/2}}\theta }}{{{n^{k + 1/2}}}}\), where C depends weakly on k (but not on n, θ). In passing, a Mehler–Dirichlet-type integral for R_nk(cos θ) is deduced.     

The <Emphasis Type="Italic">p</Emphasis>-Adic order of the <Emphasis Type="Italic">k</Emphasis>-Fibonacci and <Emphasis Type="Italic">k</Emphasis>-Lucas numbers

Let (F _k,n ) _n and (L _k,n )n be the k-Fibonacci and k-Lucas sequence, respectively, which satisfies the same recursive relation a _n+1 = ka _n + a _n?1 with initial values F _k,0 = 0, F _k,1 = 1, L _k,0 = 2 and L _k,1 = k. In this paper, we characterize the p-adic orders ν _p (F _k,n ) and ν _p (L _k,n ) for all primes p and all positive integers k.     

On the Exact Maximum Complexity of Minkowski Sums of Polytopes

We present a tight bound on the exact maximum complexity of Minkowski sums of polytopes in ?³. In particular, we prove that the maximum number of facets of the Minkowski sum of k polytopes with m ₁,m ₂,…,m _k facets, respectively, is bounded from above by \(\sum_{1\leq i. Given k positive integers m ₁,m ₂,…,m _k , we describe how to construct k polytopes with corresponding number of facets, such that the number of facets of their Minkowski sum is exactly \(\sum_{1\leq i. When k=2, for example, the expression above reduces to 4m ₁ m ₂?9m ₁?9m ₂+26.     

Classification of amply regular graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>1</Subscript> = 6

An undirected graph with v vertices in which the degrees of all vertices are equal to k, each edge is contained in exactly λ triangles, and the intersection of the neighborhoods of any two vertices at distance 2 contains exactly µ vertices is called amply regular with parameters (v, k, λ, µ). We complete the classification of amply regular graphs with b ₁ = 6, where b ₁ = k ? λ ? 1.     

A Formula for the Partition Function That “Counts”

We derive a combinatorial multisum expression for the number D(n, k) of partitions of n with Durfee square of order k. An immediate corollary is therefore a combinatorial formula for p(n), the number of partitions of n. We then study D(n, k) as a quasipolynomial. We consider the natural polynomial approximation \({\tilde{D}(n, k)}\) to the quasipolynomial representation of D(n, k). Numerically, the sum \({\sum_{1\leq k \leq \sqrt{n}} \tilde{D}(n, k)}\) appears to be extremely close to the initial term of the Hardy-Ramanujan-Rademacher convergent series for p(n).     

Construction of a (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-visual cryptography scheme

In this paper we give an explicit construction of basis matrices for a (k, n)-visual cryptography scheme \((k,n){\hbox {-}}\mathrm{VCS}\) for integers k and n with \(2\le k \le n\). In balanced VCS every set of participants with equal cardinality has same relative contrast. The VCS constructed in this paper is a balanced \((k,n){\hbox {-}}\mathrm{VCS}\) for general k. Also we obtain a formula for pixel expansion and relative contrast. We also prove that our construction gives optimal contrast and minimum pixel expansion when \(k=n\) and \(n-1\).     

On Functional Versions of the Arc-Sine Law

Let X ₁,X ₂,… be a sequence of random variables. Let S _k =X ₁+???+X _k and assume that S _k /b _k converges in distribution for some numerical sequence (b _k ). We study the weak convergence of the random processes {Λ _n (z), z∈?}, where

$\Lambda_{n}(z)=\frac{1}{n}\sum_{k=1}^{n}I\left\{\frac{S_{k}}{b_{k}}\leq z\right\}.$

We consider the same problem when the normalized partial sums S _k /b _k are replaced by other functionals of the sequence (X _n ). In particular, we investigate the case of sample extremes in detail.

     

Finite configurations in sparse sets

Let E ? ?ⁿ be a closed set of Hausdorff dimension α. For m > n, let{B₁, …, B_k} be n × (m ? n) matrices. We prove that if the system of matrices B_j is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translate of a non-trivial k-point configuration {B₁y, …, B_ky}. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in ?ⁿ and isosceles right triangles in ?²). This can be viewed as a multidimensional analogue of the result of [25] on 3-term arithmetic progressions in subsets of ?.     

Sums and differences of four <Emphasis Type="Italic">k</Emphasis>th powers

We prove an upper bound for the number of representations of a positive integer N as the sum of four kth powers of integers of size at most B, using a new version of the determinant method developed by Heath-Brown, along with recent results by Salberger on the density of integral points on affine surfaces. More generally we consider representations by any integral diagonal form. The upper bound has the form \({O_{N}(B^{c/\sqrt{k}})}\), whereas earlier versions of the determinant method would produce an exponent for B of order k ^?1/3 (uniformly in N) in this case. Furthermore, we prove that the number of representations of a positive integer N as a sum of four kth powers of non-negative integers is at most \({O_{\varepsilon}(N^{1/k+2/k^{3/2}+\varepsilon})}\) for k ≥ 3, improving upon bounds by Wisdom.