Roth's theorems over commutative rings

In 1952, W.E. Roth showed that matrix equations of the forms AX?YB = C and AX?XB = C over fields can be solved if and only if certain block matrices built from A, B, and C are equivalent or similar. We show here that these criteria remain valid over arbitrary commutative rings. To do this, we use standard commutative algebra methods to reduce to the case of Artinian rings, where a simple argument with     

Roth's theorems for sets of matrices

It is shown that Roth's theorems on the equivalence and similarity of block diagonal matrices hold for finite sets of matrices over a commutative ring.     

Roth's removal rule revisited

The theory of companion matrices is used to give explicit representations for the matrices needed in Roth's removal rule. These are then used to give simple proofs for the cyclic decomposition theorem, as well as for Roth's similarity theorem for matrices over a field.     

Kuratowski's and Wagner's theorems for matroids

In an earlier paper we proved the following theorem, which provides a strengthening of Tutte's well-known characterization of regular (totally unimodular) matroids: A binary matroid is regular if it does not have the Fano matroid or its dual as a series-minor (parallel-minor). In this paper we prove two theorems (Theorems 5.1 and 6.1) which provide the same kind of strengthening for Tutte's characterization of the graphic matroids (i.e., bond-matroids). One interesting aspect of these theorems is the introduction of the matroids of “type R”. It turns out that these matroids are, in at least two different senses, the smallest regular matroids which are neither graphic nor cographic (Theorems 6.2 and 6.3).     

On some analogs of Sturm's and Kneser's theorems for nonlinear systems

Analogs of certain conjugate point properties in the calculus of variations are developed for optimal control problems. The main result in this direction is concerned with the characterization of a parameterized family of extremals going through the first backward conjugate point, t_c. A corollary of this result is that for the linear quadratic problem (LQP) there exists at least a one-parameter family of extremals going though the conjugate point which gives the same cost as the candidate extremal, i.e., the extremal control is optimal but nonunique on [t_c, t_f]. An analysis of the effect on the conjugate point of employing penalty functions for terminal equality constraints in the LQP is presented, also. It is shown that the sequence of approximate conjugate points is always conservative, and it converges to the conjugate point of the constrained problem. Furthermore, it is proved that the addition of terminal constraints has the effect of causing the conjugate point to move backward (or remain the same).     

On Dilworth's decomposition theorem

For every countable po set P without infinite chains there exists a partition (C_j: j?I) of P into pairwise disjoint chains and an antichain A such that C_j∩A≠Ø for every i?I.     

Stillman's question for exterior algebras and Herzog's conjecture on Betti numbers of syzygy modules

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo–Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that, via the Bernstein–Gel'fand–Gel'fand correspondence, these examples also yields counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules over polynomial rings.     

Random versions of Kakutani-Ky Fan's fixed point theorems

The following results are proved: (1) X be either a locally convex Lusin space¹ or a locally convex metrizable (not necessarily separable) space, let Γ be a weakly upper semicontinuous random multimapping defined on a convex compact subspace of X taking convex weakly compact values and satisfying the Browder-Halpern's “inward” condition. Then Γ has a fixed point. (2) In an arbitrary metric space, a continuous random multimapping Γ (with stochastic complete domain) has fixed points, whenever the corresponding deterministic fixed point theorem for Γ holds.     

Patching theorems for projective modules

In the first section of this paper, we prove several ‘Milnor-Mayer-Vietoris’-type patching theorems for projective modules. In the second section we apply these theorems to projective modules over polynomial rings and give some criteria for such modules to be extended. In the third section, the theorems are applied to the study of vector bundles over curves.     

Improved Cotlar's inequality in the context of local Tb theorems

In the context of local Tb theorems with $L^p$ testing conditions we prove an enhanced Cotlar's inequality. This is related to the problem of removing the so called buffer assumption of Hytönen–Nazarov, which is the final barrier for the full solution of S. Hofmann's problem. We also investigate the problem of extending the Hytönen–Nazarov result to non-homogeneous measures. We work not just with the Lebesgue measure but with measures μ in ${\mathbb{R}}^d$ satisfying $\mu (B(x,r))\le C{r}^n$, $n\in (0,d]$. The range of exponents in the Cotlar type inequality depend on n. Without assuming buffer we get the full range of exponents $p,q\in (1,2]$ for measures with $n\le 1$, and in general we get $p,q\in [2??(n),2]$, $?(n)>0$. Consequences for (non-homogeneous) local Tb theorems are discussed.     

An algorithm for the quadratic assignment problem using Bender's decomposition

A well known method used for solving quadratic assignment problems proceeds by the construction of an equivalent much larger linear assignment problem with many side constraints. The disadvantage of this method lies in the weakness of the bounds obtained by solving the linear problem. An alternate linearization has been suggested using a general method of Glover. In this paper the mixed integer program obtained by Glover's method is discussed and a solution using Bender's decomposition is proposed.     

Pointfree forms of Dowker’s and Michael’s insertion theorems

In this paper we prove two strict insertion theorems for frame homomorphisms. When applied to the frame of all open subsets of a topological space they are equivalent to the insertion statements of the classical theorems of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces. In addition, a study of perfect normality for frames is made.     

Primal decomposition in rings, modules and lattice modules

In a work by L. Fuchs, W. Heinzer and B. Olberding, a decomposition of ideals in a commutative ring as intersections of primal isolated component ideals is investigated. In subsequent work by L. Fuchs and R. Reis, these ideas are developed in multiplicative lattices. The object of this note is to point out that, when specialized to the lattice of ideals of a commutative ring, the decomposition of L. Fuchs and R. Reis does not give the decomposition obtained in the paper by Fuchs, Heinzer and Olberding, and to give two variations of the decomposition of Fuchs and Reis. One of these variations, when specialized to the lattice of ideals of a ring, does give the decomposition obtained by Fuchs, Heinzer and Olberding, and the other one gives a decomposition which is superior in some ways. Received December 2, 2004; accepted in final form February 17, 2005.     

Frobenius properties and Maschke-type theorems for entwined modules

Entwined modules arose from the coalgebra-Galois theory. They are a generalisation of unified Doi-Hopf modules. In this paper, Frobenius properties and Maschke-type theorems known for Doi-Hopf modules are extended to the case of entwined modules.