(<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-bi-Koszul algebras

The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called(s,t,d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An(s,t,d)-bi-Koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an(s,t,d)-bi-Koszul algebra is discussed. Based on it,the notion of strongly(s,t,d)-bi-Koszul algebras is raised and their homological properties are further discussed.     

On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-factorization of complete bipartite multigraphs

Let λK _m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K _p,q -factorization of λK _m,n is a set of edge-disjoint K _p,q -factors of λK _m,n which partition the set of edges of λK _m,n . When p = 1 and q is a prime number, Wang, in his paper [On K _1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K _1,q -factorization of K _m,n and gave a sufficient condition for such a factorization to exist. In papers [K _1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K _p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK _m,n to have a K _p,q -factorization. As a special case, it is shown that the necessary condition for the K _p,q -factorization of λK _m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.     

Cyclotomic Hecke Algebras of <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)

In this note, we find a monomial basis of the cyclotomic Hecke algebra \({\mathcal{H}_{r,p,n}}\) of G(r,p,n) and show that the Ariki-Koike algebra \({\mathcal{H}_{r,n}}\) is a free module over \({\mathcal{H}_{r,p,n}}\), using the Gröbner-Shirshov basis theory. For each irreducible representation of \({\mathcal{H}_{r,p,n}}\), we give a polynomial basis consisting of linear combinations of the monomials corresponding to cozy tableaux of a given shape.     

On (<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis>)-semigroups

We generalize Green’s lemma and Green’s theorem for usual binary semigroups to (n,m)-semigroups, define and describe the regularity for an element of an (n,m)-semigroup, give some criteria for an element of an (n,m)-semigroup to be invertible, and further apply the invertibility for (n,m)-semigroups to (n,m)-groups and give some equivalent characterizations for (n,m)-groups. We establish Hosszú-Gluskin theorems for (n,m)-semigroups in two cases, as generalizations of the corresponding theorems for n-groups.     

Riesz <Emphasis Type="Italic">s</Emphasis>-Equilibrium Measures on <Emphasis Type="Italic">d</Emphasis>-Rectifiable Sets as <Emphasis Type="Italic">s</Emphasis> Approaches <Emphasis Type="Italic">d</Emphasis>

Let A be a compact set in of Hausdorff dimension d. For s ∈ (0,d) the Riesz s-equilibrium measure μ ^s is the unique Borel probability measure with support in A that minimizes

A Family of Binary (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">m</Emphasis>,<Emphasis Type="Italic">s</Emphasis>)-Nets of Strength 5

(t,m,s)-Nets were defined by Niederreiter [Monatshefte fur Mathematik, Vol. 104 (1987) pp. 273–337], based on earlier work by Sobol’ [Zh. Vychisl Mat. i mat. Fiz, Vol. 7 (1967) pp. 784–802], in the context of quasi-Monte Carlo methods of numerical integration. Formulated in combinatorial/coding theoretic terms a binary linear (m−k,m,s)₂-net is a family of ks vectors in F₂^m satisfying certain linear independence conditions (s is the length, m the dimension and k the strength: certain subsets of k vectors must be linearly independent). Helleseth et al. [5] recently constructed (2r−3,2r+2,2^r−1)₂-nets for every r. In this paper, we give a direct and elementary construction for (2r−3,2r+2,2^r+1)₂-nets based on a family of binary linear codes of minimum distance 6.Communicated by: T. Helleseth     

Facets of the <Emphasis Type="Italic">r</Emphasis>-Stable (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">k</Emphasis>)-Hypersimplex

Let k, n, and r be positive integers with k < n and \({r \leq \lfloor \frac{n}{k} \rfloor}\). We determine the facets of the r-stable n, k-hypersimplex. As a result, it turns out that the r-stable n, k-hypersimplex has exactly 2n facets for every \({r < \lfloor \frac{n}{k} \rfloor}\). We then utilize the equations of the facets to study when the r-stable hypersimplex is Gorenstein. For every k > 0 we identify an infinite collection of Gorenstein r-stable hypersimplices, consequently expanding the collection of r-stable hypersimplices known to have unimodal Ehrhart \({\delta}\)-vectors.     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where

Cubic and higher degree bounds for codes and (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">s</Emphasis>)-nets

The Plotkin bound and the quadratic bound for codes and (t, m, s)-nets can be obtained from the linear programming bound using certain linear and quadratic polynomials, respectively. We extend this approach by considering cubic and higher degree polynomials to find new explicit bounds as well as new non-existence results for codes and (t, m, s)-nets.     

On list incidentor (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">l</Emphasis>)-coloring

A proper incidentor coloring is called a (k, l)-coloring if the difference between the colors of the final and initial incidentors ranges between k and l. In the list variant, the extra restriction is added: the color of each incidentor must belong to the set of admissible colors of the arc. In order to make this restriction reasonable we assume that the set of admissible colors for each arc is an integer interval. The minimum length of the interval that guarantees the existence of a list incidentor (k, l)-coloring is called a list incidentor (k, l)-chromatic number. Some bounds for the list incidentor (k, l)-chromatic number are proved for multigraphs of degree 2 and 4.     

On <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings of nearly bipartite graphs

Let G be a simple graph, let d(v) denote the degree of a vertex v and let g be a nonnegative integer function on V (G) with 0 ≤ g(v) ≤ d(v) for each vertex v ∈ V (G). A g _c -coloring of G is an edge coloring such that for each vertex v ∈ V (G) and each color c, there are at least g(v) edges colored c incident with v. The g _c -chromatic index of G, denoted by χ′g _c (G), is the maximum number of colors such that a gc-coloring of G exists. Any simple graph G has the g _c -chromatic index equal to δ _g (G) or δ _g (G) ? 1, where \({\delta _g}\left( G \right) = \mathop {\min }\limits_{v \in V\left( G \right)} \left\lfloor {d\left( v \right)/g\left( v \right)} \right\rfloor \). A graph G is nearly bipartite, if G is not bipartite, but there is a vertex u ∈ V (G) such that G ? u is a bipartite graph. We give some new sufficient conditions for a nearly bipartite graph G to have χ′g _c (G) = δ _g (G). Our results generalize some previous results due to Wang et al. in 2006 and Li and Liu in 2011.     

<Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript> weights and <Emphasis Type="Italic">d</Emphasis>-homogeneous measures on ahlfors <Emphasis Type="Italic">d</Emphasis>-regular space

Let X be an Ahlfors d-regular space and mad-regular measure on X . We prove that a measure μ on X is d-homogeneous if and only if μ is mutually absolutely continuous with respect to m and the derivative Dmμ(x) is an A1 weight. Also, we show by an example that every Ahlfors d-regular space carries a measure which is d-homogeneous but not d-regular.     

Generalized (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-ray systems of points and inequalities for nonoverlapping domains and open sets

We solve the extremal problem of finding the maximum of the functional

Optimal (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">n</Emphasis>) visual cryptographic schemes for general <Emphasis Type="Italic">k</Emphasis>

In (k, n) visual cryptographic schemes (VCS), a secret image is encrypted into n pages of cipher text, each printed on a transparency sheet, which are distributed among n participants. The image can be visually decoded if any k(≥2) of these sheets are stacked on top of one another, while this is not possible by stacking any k − 1 or fewer sheets. We employ a Kronecker algebra to obtain necessary and sufficient conditions for the existence of a (k, n) VCS with a prior specification of relative contrasts that quantify the clarity of the recovered image. The connection of these conditions with an L ₁-norm formulation as well as a convenient linear programming formulation is explored. These are employed to settle certain conjectures on contrast optimal VCS for the cases k = 4 and 5. Furthermore, for k = 3, we show how block designs can be used to construct VCS which achieve optimality with respect to the average and minimum relative contrasts but require much smaller pixel expansions than the existing ones.     

(<Emphasis Type="Italic">q</Emphasis>,<Emphasis Type="Italic">t</Emphasis>)-Deformations of multivariate hook product formulae

We generalize multivariate hook product formulae for P-partitions. We use Macdonald symmetric functions to prove a (q,t)-deformation of Gansner’s hook product formula for the generating functions of reverse (shifted) plane partitions. (The unshifted case has also been proved by Adachi.) For a d-complete poset, we present a conjectural (q,t)-deformation of Peterson–Proctor’s hook product formula.     

(<Emphasis Type="Italic">m</Emphasis>,<Emphasis Type="Italic">r</Emphasis>)-central Riordan arrays and their applications

For integers m > r ≥ 0, Brietzke (2008) defined the (m, r)-central coefficients of an infinite lower triangular matrix G = (d, h) = (d_n,k)_n,k∈N as d_{mn+r,(m?1)n+r}, with n = 0, 1, 2,..., and the (m, r)-central coefficient triangle of G as

$${G^{\left( {m,r} \right)}} = {\left( {{d_{mn + r,\left( {m - 1} \right)n + k + r}}} \right)_{n,k \in \mathbb{N}}}.$$

It is known that the (m, r)-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array G = (d, h) with h(0) = 0 and d(0), h′(0) ≠ 0, we obtain the generating function of its (m, r)-central coefficients and give an explicit representation for the (m, r)-central Riordan array G^(m,r) in terms of the Riordan array G. Meanwhile, the algebraic structures of the (m, r)-central Riordan arrays are also investigated, such as their decompositions, their inverses, and their recessive expressions in terms of m and r. As applications, we determine the (m, r)-central Riordan arrays of the Pascal matrix and other Riordan arrays, from which numerous identities are constructed by a uniform approach.

     

Continuous <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">p</Emphasis></Subscript>-symmetric distributions

For p > 0, the l _n,p -generalized surface measure on the l _n,p -unit sphere is studied and used for deriving a geometric measure representation for l _n,p -symmetric distributions having a density.     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a ₁?a _m ∈I for a ₁,…, a _m ∈R?U(R), then there are n of the a _i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m ? 1 maximal ideals.     

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

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