On the spectral radius of weighted trees with given number of pendant vertices and a positive weight set

Let 𝒯(n,?r;?W _n?1) be the set of all n-vertex weighted trees with r vertices of degree 2 and fixed positive weight set W _n?1, 𝒫(n,?γ;?W _n?1) the set of all n-vertex weighted trees with q pendants and fixed positive weight set W _n?1, where W _n?1?=?{w ₁,?w ₂,?…?,?w _n?1} with w ₁???w ₂???···???w _n?1?>?0. In this article, we first identify the unique weighted tree in 𝒯(n,?r;?W _n?1) with the largest adjacency spectral radius. Then we characterize the unique weighted trees with the largest adjacency spectral radius in 𝒫(n,?γ;?W _n?1).     

On Generalizations of Prime Ideals (II)

Let R be a commutative ring with identity. We say that a proper ideal P of R is (n ? 1, n)-weakly prime (n ≥ 2) if 0 ≠ a ₁…a _n  ∈ P implies a ₁…a _i?1 a _i+1…a _n  ∈ P for some i ∈ {1,…, n}, where a ₁,…, a _n  ∈ R. In this article, we study (n ? 1, n)-weakly prime ideals. A number of results concerning (n ? 1, n)-weakly prime ideals and examples of (n ? 1, n)-weakly prime ideals are given. Rings with the property that for a positive integer n such that 2 ≤ n ≤ 5, every proper ideal is (n ? 1, n)-weakly prime are characterized. Moreover, it is shown that in some rings, nonzero (n ? 1, n)-weakly prime ideals and (n ? 1, n)-prime ideals coincide.     

Orthogonal invariants of skew-symmetric matrices

The algebra of invariants of d-tuples of n?×?n skew-symmetric matrices under the action of the orthogonal group by simultaneous conjugation is considered over an infinite field of characteristic different from two. For n?=?3 and d?>?0 a minimal set of generators is established. A homogeneous system of parameters (i.e. an algebraically independent set such that the algebra of invariants is a finitely generated free module over subalgebra generated by this set) is described for n?=?3 and d?>?0, for n?=?4 and d?=?2,?3, for n?=?5 and d?=?2.     

Multipartite Ramsey numbers for odd cycles

In this paper we study multipartite Ramsey numbers for odd cycles. We formulate the following conjecture: Let n≥5 be an arbitrary positive odd integer; then, in any two‐coloring of the edges of the complete 5‐partite graph K((n?1)/2, (n?1)/2, (n?1)/2, (n?1)/2, 1) there is a monochromatic C_n, a cycle of length n. This roughly says that the Ramsey number for C_n (i.e. 2n?1 ) will not change (somewhat surprisingly) if four large “holes” are allowed. Note that this would be best possible as the statement is not true if we delete from K_2n?1 the edges within a set of size (n+ 1)/2. We prove an approximate version of the above conjecture. © 2009 Wiley Periodicals, Inc. J Graph Theory 61:12‐21, 2009     

On Generalizations of Prime Ideals

Let R be a commutative ring with identity. Let φ: S(R) → S(R) ∪ {?} be a function, where S(R) is the set of ideals of R. Suppose n ≥ 2 is a positive integer. A nonzero proper ideal I of R is called (n ? 1, n) ? φ-prime if, whenever a ₁, a ₂, ?, a _n  ∈ R and a ₁ a ₂?a _n  ∈ I?φ(I), the product of (n ? 1) of the a _i 's is in I. In this article, we study (n ? 1, n) ? φ-prime ideals (n ≥ 2). A number of results concerning (n ? 1, n) ? φ-prime ideals and examples of (n ? 1, n) ? φ-prime ideals are also given. Finally, rings with the property that for some φ, every proper ideal is (n ? 1, n) ? φ-prime, are characterized.     

Weak Global Dimension of Coherent Rings

In this article, we study the weak global dimension of coherent rings in terms of the left FP-injective resolutions of modules. Let R be a left coherent ring and ? ? the class of all FP-injective left R-modules. It is shown that wD(R) ≤ n (n ≥ 1) if and only if every nth ? ?-syzygy of a left R-module is FP-injective; and wD(R) ≤ n (n ≥ 2) if and only if every (n ? 2)th ? ?-syzygy in a minimal ? ?-resolution of a left R-module has an FP-injective cover with the unique mapping property. Some results for the weak global dimension of commutative coherent rings are also given.     

Polytopes of high rank for the symmetric groups

In the Atlas of abstract regular polytopes for small almost simple groups by Leemans and Vauthier, the polytopes whose automorphism group is a symmetric group Sn of degree 5?n?9 are available. Two observations arise when we look at the results: (1) for n?5, the (n−1)-simplex is, up to isomorphism, the unique regular (n−1)-polytope having Sn as automorphism group and, (2) for n?7, there exists, up to isomorphism and duality, a unique regular (n−2)-polytope whose automorphism group is Sn. We prove that (1) is true for n≠4 and (2) is true for n?7. Finally, we also prove that Sn acts regularly on at least one abstract polytope of rank r for every 3?r?n−1.     

Nullstellenzahlen von Polynomfunktionen auf Restklassenringen

Let ?/n? denote the ring of integers mod n. By a polynomial function f over ?/n? will be meant a mapping of ?/n? into itself which is induced by a polynomial f(x) ∈ ?[x]. We ask how many zeros of f are possible. The main result is a formula for the maximal number of zeros of non-vanishing polynomial functions mod n.     

Moore-penrose involutions in the classes of laplacians and simplices

The Moore-Penrose generalized inverse defines an involution in the class of real positive definite n×n matrices with rank n?1 and row-sums zero. We show that there is an analogous situation for weighted undirected graphs on n vertices and for classes (n?1)-simplices in a Euclidean (n?1)-space.     

Conversion of a rhotrix to a ‘coupled matrix’

In this note, a method of converting a rhotrix to a special form of matrix termed a ‘coupled matrix’ is proposed. The special matrix can be used to solve various problems involving n?×?n and (n?–?1)?×?(n?–?1) matrices simultaneously.     

The classification of N-dimensional non-associative Jordan algebras with (N − 3)-dimensional annihilator

The paper is devoted to give a complete classification of all n-dimensional non-associative Jordan algebras with (n?3)-dimensional annihilator over an algebraically closed field of characteristic ≠2. We also give a complete classification of all n-dimensional Jordan algebras with (n?1)- and (n?2)-dimensional annihilator.     

The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the category F≀FIn

We give a new proof for the Littlewood-Richardson rule for the wreath product F?S_n where F is a finite group. Our proof does not use symmetric functions but use more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of F?S_n to F?S_n+1. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category F?FI_n where FI_n is the category of all injective functions between subsets of an n-element set.     

Maximal Rectilinear Crossing of Cycles

We address the following question: In drawing a cycle on n vertices (or a graph all of whose degrees are 2) in the plane with straight line arcs, how many crossings can there be? A complete answer is given; namely, if n is odd, the number of crossings can be anything up to n(n?3)/2 except n(n?3)/2?1. For n even, the number of crossings in one cycle can be any integer up to n(n?4)/2+1; general bivalent even graphs can achieve any integer up to n(2n?7)/4 as the number of crossings.     

Indefinite higher-rank numerical ranges

We investigate indefinite higher-rank numerical ranges of a wide class of J-Hermitian matrices, J?=?I _r ?⊕??I _n?r , 0?<?r?<?n (A?∈?C ^n×n is said to be J-Hermitian if A?=?JA*J). Particular attention is paid to aspects of the theory that parallel the case of Hermitian matrices.     

Projective Varieties Swept Out by Rational Normal Curves

Let X ? ? ⁿ be a complex nondegenerate projective variety of dimension m ≥ 2. For t ≤ n ? m and a general q ∈ ? ⁿ , the linear space L _q spanned by q and t general points of X meets X in a finite set of points. We classify those X ? ? ⁿ for which there exists a point q ∈ ? ⁿ such that L _q meets X in a positive dimensional variety. If this occurs, there exists d ≤ n ? m such that a degree d rational normal curve through d general points of X is contained in X. Examples of this situation are provided. An infinitesimal generalization of part of the main result is also stated.     

A class of Hamiltonian regular graphs

In this paper, we show that n ? 4 and if G is a 2-connected graph with 2n or 2n?1 vertices which is regular of degree n?2, then G is Hamiltonian if and only if G is not the Petersen graph.     

Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular n‐queens solutions

In this article, we show how to construct pairs of orthogonal pandiagonal Latin squares and panmagic squares from certain types of modular n‐queens solutions. We prove that when these modular n‐queens solutions are symmetric, the panmagic squares thus constructed will be associative, where for an n × n associative magic square A = (a_ij), for all i and j it holds that a_ij + an?i?1,n?j?1 = c for a fixed c. We further show how to construct orthogonal Latin squares whose modular difference diagonals are Latin from any modular n‐queens solution. As well, we analyze constructing orthogonal pandiagonal Latin squares from particular classes of non‐linear modular n‐queens solutions. These pandiagonal Latin squares are not row cyclic, giving a partial solution to a problem of Hedayat. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 221–234, 2007     

On Chern?s problem for rigidity of minimal hypersurfaces in the spheres

For a compact minimal hypersurface M in Sn+1 with the squared length of the second fundamental form S we confirm that there exists a positive constant δ(n) depending only on n, such that if n?S?n+δ(n), then S≡n, i.e., M is a Clifford minimal hypersurface, in particular, when n?6, the pinching constant .     

Multiple positive solutions of boundary value problems for second-order discrete equations on the half-line

In this paper, we study the existence of multiple positive solutions of boundary value problems for second-order discrete equations Δ² x(n ? 1) ? pΔx(n ? 1) ? qx(n ? 1)+f(n, x(n)) = 0, n ∈ {1,2,…}, αx(0) ? βΔx(0) = 0, x(∞) = 0. The proofs are based on the fixed point theorem in Fréchet space (see Agarwal and O'Regan, 2001, Cone compression and expansion and fixed point theorems in Fréchet spaces with application, Journal of Differential Equations, 171, 412–42).