For a countable, weakly minimal theory T , we show that the SchröderBernstein property (any two elementarily bi-embeddable models are isomorphic) is equivalent to each of the following: 1. For any U -rank-1 type q ∈ S(acl(∅)) and any automorphism f of the monster model C, there is some n < ω such that f(q) is not almost orthogonal to q ⊗ f(q)⊗ . . .⊗ fn−1(q); 2. T has no infinite collection of ...

Let Q be a generalized quadrangle of order s with a regular point x. The set x⊥ together with all spans which are contained in x⊥ define a projective plane of order s. We introduce a property (Py) for every point y of Q noncollinear with x and prove that this property is equivalent with the regularity of the point y. We will use this to give an elementary proof for the following result: every g...

We prove a null space property for the uniqueness of the sparse solution vectors recovered from a minimization in `q quasi-norm subject to multiple systems of linear equations, where q ∈ (0, 1]. Furthermore, we show that the null space property for the setting of the sparse solution vectors for multiple linear systems is equivalent to the null space property for the standard minimization in `q ...

Let A be a finite dimensional Q-algebra and Γ ⊂ A a Z-order. We classify those A with the property that Z 6 →֒U(Γ). We call this last property the hyperbolic property. We apply this in the case that A = KS a semigroup algebra with K = Q or K = Q( √ −d). In particular, when KS is semi-simple and has no nilpotent elements, we prove that S is an inverse semigroup which is the disjoint union of Higm...

We consider the question of the existence of a nontrivial continuous homomorphism from (βN,+) into N∗ = βN\N. This problem is known to be equivalent to the existence of distinct p and q in N∗ satisfying the equations p+ p = q = q + q = q + p = p+ q. We obtain certain restrictions on possible values of p and q in these equations and show that the existence of such p and q implies the existence o...

Abstract A family of sets has the (p, q) property if among any p members of the family some q have a nonempty intersection. The authors have proved that for every p ≥ q ≥ d+ 1 there is a c = c(p, q, d) < ∞ such that for every family F of compact, convex sets in R which has the (p, q) property there is a set of at most c points in R that intersects each member of F , thus settling an old problem...

Let Q denote the field of algebraic numbers in C. A discrete group G is said to have the σ-multiplier algebraic eigenvalue property, if for every matrix A ∈ Md(Q(G,σ)), regarded as an operator on l(G), the eigenvalues of A are algebraic numbers, where σ ∈ Z(G,U(Q)) is an algebraic multiplier, and U(Q) denotes the unitary elements of Q. Such operators include the Harper operator and the discrete...

We consider the question of the existence of a nontrivial continu ous homomorphism from N into N NnN This problem is known to be equivalent to the existence of distinct p and q in N satisfying the equations p p q q q q p p q We obtain certain restrictions on possible values of p and q in these equations and show that the existence of such p and q implies the existence of p q and r satisfying th...

I. The identifiable parent property and some first results about it If C is a q–ary code of length n and a and b are two codewords, then c is called a descendant of a and b if ct ∈ {at, bt} for t = 1, . . . , n. We are interested in codes C with the property that, given any descendant c, one can always identify at least one of the ‘parent’ codewords in C. We study bounds on F (n, q), the maxima...