The parity of the number of quasigroups

Let I(n) be the number of isomorphism classes of quasigroups of order n. Despite prior enumerations showing that I(n) is odd for 1≤n≤11, we find that I(12) is even. We also give a method for finding the parity of I(n), which we use to show that I(n) is odd for n∈{13,14,15,16,17,19,21}.     

On the normal ideals of exchange rings

An ideal I of a ring R is called normal if all idempotent elements in I lie in the center of R. We prove that if I is a normal ideal of an exchange ring R then: (1) R and R/I have the same stable range; (2) V(I) is an order-ideal of the monoid C(Specc(R), N), where Specc(R) consists of all prime ideals P such that R/P is local.     

On the stability of the translation equation and dynamical systems

In this paper we prove the stability of the functional equation F(s,F(t,x))=F(s+t,x) in the class of functions F:R×I→I, which are continuous with respect to each variable, and where I⊂R is a real interval. We also discuss the stability in the sense of Hyers-Ulam of dynamical systems on I. We show some properties of δ-approximate solutions of the translation equation on a real interval.     

Some results on the local cohomology of minimax modules

Let R be a commutative Noetherian ring with identity and I an ideal of R. It is shown that, if M is a non-zero minimax R-module such that dim Supp H _I ⁱ (M) ? 1 for all i, then the R-module H _I ⁱ (M) is I-cominimax for all i. In fact, H _I ⁱ (M) is I-cofinite for all i ? 1. Also, we prove that for a weakly Laskerian R-module M, if R is local and t is a non-negative integer such that dim Supp H _I ⁱ (M) ? 2 for all i < t, then Ext _R ^j (R/I,H _I ⁱ (M)) and Hom _R (R/I,H _I ^t (M)) are weakly Laskerian for all i < t and all j ? 0. As a consequence, the set of associated primes of H _I ⁱ (M) is finite for all i ? 0, whenever dim R/I ? 2 and M is weakly Laskerian.     

Certain Homology Cycles of the Independence Complex of Grids

Let G be an infinite graph such that the automorphism group of G contains a subgroup K ?? ^d with the property that G/K is finite. We examine the homology of the independence complex Σ(G/I) of G/I for subgroups I of K of full rank, focusing on the case that G is the square, triangular, or hexagonal grid. Specifically, we look for a certain kind of homology cycles that we refer to as “cross-cycles,” the rationale for the terminology being that they are fundamental cycles of the boundary complex of some cross-polytope. For the special cases just mentioned, we determine the set Q(G,K) of rational numbers r such that there is a group I with the property that Σ(G/I) contains cross-cycles of degree exactly r?|G/I|?1; |G/I| denotes the size of the vertex set of G/I. In each of the three cases, Q(G,K) turns out to be an interval of the form [a,b]∩?={r∈?:a≤r≤b}. For example, for the square grid, we obtain the interval $[\frac{1}{5},\frac{1}{4}]\cap \mathbb{Q}Let G be an infinite graph such that the automorphism group of G contains a subgroup K ≅ℤ ^d with the property that G/K is finite. We examine the homology of the independence complex Σ(G/I) of G/I for subgroups I of K of full rank, focusing on the case that G is the square, triangular, or hexagonal grid. Specifically, we look for a certain kind of homology cycles that we refer to as “cross-cycles,” the rationale for the terminology being that they are fundamental cycles of the boundary complex of some cross-polytope. For the special cases just mentioned, we determine the set Q(G,K) of rational numbers r such that there is a group I with the property that Σ(G/I) contains cross-cycles of degree exactly r⋅|G/I|−1; |G/I| denotes the size of the vertex set of G/I. In each of the three cases, Q(G,K) turns out to be an interval of the form [a,b]∩ℚ={r∈ℚ:a≤r≤b}. For example, for the square grid, we obtain the interval [\frac15,\frac14]?\mathbbQ[\frac{1}{5},\frac{1}{4}]\cap \mathbb{Q}.     

On the Burkard-Hammer condition for hamiltonian split graphs

A graph G=(V,E) is called a split graph if there exists a partition V=I∪K such that the subgraphs of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary but not sufficient condition for hamiltonian split graphs with |I|<|K|. In this paper, we show that the Burkard-Hammer condition is also sufficient for the existence of a Hamilton cycle in a split graph G such that 5≠|I|<|K| and the minimum degree δ(G)?|I|-3. For the case 5=|I|<|K|, all split graphs satisfying the Burkard-Hammer condition but having no Hamilton cycles are also described.     

Remarks on ideal boundedness, convergence and variation of sequences

We answer the questions asked by Faisant et al. (2005) [2]. The first main result states that for every admissible ideal I⊂P(N) the quotient space l^∞(I)/c₀(I) is complete. The second main result states that consistently there is an admissible ideal I⊂P(N) such that the sets W(I), of all real sequences with finite I-variation, and c^?(I), of all restrictively I-convergent sequences, are equal.     

Embedded associated primes of powers of square-free monomial ideals

An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I^t)=Ass(R/I) for all t≥1. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal I is minimally not normally torsion-free then the least power t such that I^t has embedded primes is bigger than β₁, where β₁ is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If, in addition, I fails to have the packing property, then embedded primes of I^t do occur when t=β₁+1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornuéjols.     

Ideal convergence of continuous functions

For a given ideal I⊆P(ω), IC(I) denotes the class of separable metric spaces X such that whenever is a sequence of continuous functions convergent to zero with respect to the ideal I then there exists a set of integers {m₀<m₁<?} from the dual filter F(I) such that limi→∞fmi(x)=0 for all x∈X. We prove that for the most interesting ideals I, IC(I) contains only singular spaces. For example, if I=Id is the asymptotic density zero ideal, all IC(Id) spaces are perfectly meager while if I=Ib is the bounded ideal then IC(Ib) spaces are σ-sets.     

A bound on the dimension of interval orders

We make use of the partially ordered set (I(0, n), <) consisting of all closed intervals of real numbers with integer endpoints (including the degenerate intervals with the same right- and left-hand endpoints), ordered by [a, b] < [c, d] if b < c, to show that there is no bound on the order dimension of interval orders. We then turn to the problem of computing the dimension of I(0, n), showing that I(0, 10) has dimension 3 but I(0, 11) has dimension 4. We use these results as initial conditions in obtaining an upper bound on the dimension of I(0, n) as a logarithmic function of n. It is our belief that this example is a “canonical” example for interval orders, so that the computation of its dimension should have significant impact on the problem of computing the dimension of interval orders in general.     

Quintasymptotic primes and four results of Schenzel

Let I be an ideal in a Noetherian ring R, let (I)_a be the integral closure of I, and let S be a multiplicatively closed subset of R. Let T₁, T₂, and T₃ be the topologies given by the filtrations {Iⁿ R_S ∩ R | n ≥ 1}, {Iⁿ | n ≥ 1}, and {(Iⁿ)_a | n ≥ 1}. We g results due to Schenzel, characterizing when T₁ is either equivalent or linearly equivalent to either of T₂ or T₃. The characterizations involve the sets of essential primes of I, quintessential primes of I, asymptotic primes of I, and quintasymptotic primes of I.     

Study on the generator f of the operators I n and I t

Define two operators In and It,the inner product operator In(g)(x) := j∈Zs(g,f(·-j))f(x-j) and the interpolation operator It(g)(x) := j∈Zs g(j)f(x-j),where f belongs to some space and integer s 1.We call f the generator of the operators In and It.It is well known that there are many results on operators In and It.But there remain some important problems to be further explored.For application we first need to find the available generators (that can recover polynomials as It(p) = p or In(p) = p,p ∈Πm-1) for constructing the relative operators.In this paper,we focus on the available generator in the class of spline functions.We shall see that not all spline functions can be used to construct available generators.Fortunately,we do find a spline function in S,of degree m-1,where m is even and S is a class of splines.But for odd m the problem is still open.Results on spline functions in this paper are new.     

On the computation of the hull number of a graph

Let G be a graph. If u,v∈V(G), a u-vshortest path of G is a path linking u and v with minimum number of edges. The closed interval I[u,v] consists of all vertices lying in some u-v shortest path of G. For S⊆V(G), the set I[S] is the union of all sets I[u,v] for u,v∈S. We say that S is a convex set if I[S]=S. The convex hull of S, denoted I_h[S], is the smallest convex set containing S. A set S is a hull set of G if I_h[S]=V(G). The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). In this work we prove that deciding whether hn(G)≤k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph.     

Essential and inessential elements of a standard basis

In this paper we introduce the concept of inessential element of a standard basis B(I), where I is any homogeneous ideal of a polynomial ring. An inessential element is, roughly speaking, a form of B(I) whose omission produces an ideal having the same saturation as I; it becomes useless in any dehomogenization of I with respect to a linear form. We study the properties of B(I) linked to the presence of inessential elements and give some examples.     

Standardness and Standard Automorphisms of Chevalley Groups, I: the Case of Rank at Least Two

The author studies the linkage between the standardness and the standard automorphisms of Chevalley groups over rings.It is proved that if H is any standard subgroup of G(R),then each of its automorphisms can be extended to an automorphism of G(R,I),restricted to an automorphism of E(R,I),and an automorphism of E(R,I) can be extended to one of G(R,I).The case of Chevalley groups of rank at least two is treated in this paper.Further results about the case of Chevalley groups of rank one,the case of non-commutative ground ring and some others exceptions will appear elsewhere.     

Distributive equations of implications based on nilpotent triangular norms

In this paper, we explore the distributive equations of implications, both independently and along with other equations. In detail, we consider three classes of equations. (1) By means of the section of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T(y, z)) = T(I(x, y), (x, z)) based on a nilpotent triangular norm T and an unknown function I, which indicates that there are no continuous solutions satisfying the boundary conditions of implications. Under the assumptions that I is continuous except the vertical section I(0, y), y ∈ [0, 1), we get its complete characterizations. (2) We prove that there are no solutions for the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, I(y, z)) = I(T(x, y), z). (3) We obtain the sufficient and necessary conditions on T and I to be solutions of the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, y) = I(N(y), N(x)).     

Complexity of some natural problems on the class of computable I-algebras

We study computable Boolean algebras with distinguished ideals (I-algebras for short). We prove that the isomorphism problem for computable I-algebras is Σ ₁ ¹ -complete and show that the computable isomorphism problem and the computable categoricity problem for computable I-algebras are Σ ₃ ⁰ -complete.     

On the Stanley depth of weakly polymatroidal ideals

Let ${\mathbb{K}}$ be a field and ${S = \mathbb{K}[x_1,\dots,x_n]}$ be the polynomial ring in n variables over the field ${\mathbb{K}}$ . In this paper, it is shown that Stanley’s conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree.     

Hilbert series of subspace arrangements

The vanishing ideal I of a subspace arrangement V₁∪V₂∪?∪V_m⊆V is an intersection I₁∩I₂∩?∩I_m of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J=I₁I₂?I_m without any assumptions about the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. We will also apply our results to generalized principal component analysis (GPCA), a tool that is useful for computer vision and image processing.