Matrices and set differences

Let A and B be (0, 1)-matrices of sizes m by t and t by n, respectively. Let x₁, …, x_t denote t independent indeterminates over the rational field Q and define X = diag[x_t, …, x_t]. We study the matrix equation AXB = Y. We first discuss its combinatorial significance relative to topics such as set intersections and the Marica-Schönheim theorem on set differences. We then prove the following theorem concerning the matrix Y. Suppose that the matrix Y of size m by n has rank m. Then Y contains m distinct nonzero elements, one in each of the m rows of Y.     

On set systems determined by intersections

The set systems determined by intersections are studied and a sufficient condotion for this property is given. For case of graphs a necessary and sufficient condition is established. Some connections to other results are discussed.     

Inequalities for two set systems with prescribed intersections

Inequalities are presented for systems {(A _i ,B _i ):1 i m} of pairs of finite sets satisfyingA _i B _i = andA _i B _j orA _j B _i fori j.     

A threshold property for intersections in a finite set

Let F be any family of subsets of a finite set E and let n be an integer, n<|F|. Under what condition does the knowledge of cardinals of m-intersections in F, for all m≤n, univocally determine the cardinal of any intersection in F, and what is the minimal condition? We give a complete answer to that. For any n, this determination property is satisfied by n if and only if |E|<2ⁿ, without further condition on F.     

On set theoretic complete intersections in P k 3

We prove that smooth monomial curves of degree greater than three are not set theoretic complete intersections on a wide class of surfaces, called bihonogeneous surfaces.     

A dense planar point set from iterated line intersections

Given S₁, a starting set of points in the plane, not all on a line, we define a sequence of planar point sets {S_i}_i=1^∞ as follows. With S_i already determined, let L_i be the set of all the lines determined by pairs of points from S_i, and let S_i+1 be the set of all the intersection points of lines in L_i. We show that with the exception of some very particular starting configurations, the limiting point set _i=1^∞S_i is everywhere dense in the plane.     

Coprime packedness and set theoretic complete intersections of ideals in polynomial rings

A ring is said to be coprimely packed if whenever is an ideal of and is a set of maximal ideals of with , then for some . Let be a ring and be the localization of at its set of monic polynomials. We prove that if is a Noetherian normal domain, then the ring is coprimely packed if and only if is a Dedekind domain with torsion ideal class group. Moreover, this is also equivalent to the condition that each proper prime ideal of is a set theoretic complete intersection. A similar result is also proved when is either a Noetherian arithmetical ring or a Bézout domain of dimension one.

     

The set of almost convergent sequences as intersections of summability fields

Archiv der Mathematik -     

On the regularity of products and intersections of complete intersections

This paper proves the formulae

for arbitrary monomial complete intersections and , and provides examples showing that these inequalities do not hold for general complete intersections.

     

Decomposition of Matrices into Involutory Matrices and Symmetric Matrices

Let \Omega be a field, and let F denote the Frobenius matrix: $[F = \left( {\begin{array}{*{20}{c}} 0&{ - {\alpha _n}}\{{E_{n - 1}}}&\alpha \end{array}} \right)\]$ where \alpha is an n-1 dimentional vector over Q, and E_n- 1 is identity matrix over \Omega. Theorem 1. There hold two elementary decompositions of Frobenius matrix: (i) F=SJB, where S, J are two symmetric matrices, and B is an involutory matrix; (ii) F=CQD, where O is an involutory matrix, Q is an orthogonal matrix over \Omega, and D is a diagonal matrix. We use the decomposition (i) to deduce the following two theorems: Theorem 2. Every square matrix over \Omega is a product of twe symmetric matrices and one involutory matrix. Theorem 3. Every square matrix over \Omega is a product of not more than four symmetric matrices. By using the decomposition (ii), we easily verify the following Theorem 4(Wonenburger-Djokovic') . The necessary and sufficient condition that a square matrix A may be decomposed as a product of two involutory matrices is that A is nonsingular and similar to its inverse A^-1 over Q (See [2, 3]). We also use the decomosition (ii) to obtain Theorem 5. Every unimodular matrix is similar to the matrix CQB, where C, B are two involutory matrices, and Q is an orthogonal matrix over Q. As a consequence of Theorem 5. we deduce immediately the following Theorem 6 (Gustafson-Halmos-Radjavi). Every unimodular matrix may be decomposed as a product of not more than four involutory matrices (See [1] ). Finally, we use the decomposition (ii) to derive the following Thoerem 7. If the unimodular matrix A possesses one invariant factor which is not constant polynomial, or the determinant of the unimodular matrix A is I and A possesses two invariant factors with the same degree (>0), then A may be decomposed as a product of three involutory matrices. All of the proofs of the above theorems are constructive.     

Cohen-Macaulay intersections

We study when the intersection of irreducible monomial ideals of height 2 with Cohen-Macaulay radicals are themselves Cohen-Macaulay. Received: 4 September 2008     

Complete intersections and rational equivalence

A new criterion for rational equivalence of cycles on a projective variety over an algebraically closed field is given, and some consequences considered.