We develop dimension theory for a large class of structures of the form (L,≤,⊥,∼), where (L ≤) is a partially ordered set, ⊥ is a binary relation on L, and ∼ is an equivalence relation on L, subject to certain axioms. We call these structures espaliers. For x, y, z ∈ L, we say that z = x ⊕ y holds, if x ⊥ y and z is the supremum of {x, y}. The dimension theory of L is the universal ∼-invariant ...

In this article, we look at how the equivalence of tight closure and plus closure (or Frobenius closure) in the homogeneous m-coprimary case implies the same closure equivalence in the non-homogeneous m-coprimary case in standard graded rings. Although our result does not depend upon dimension, the primary application is based on results known in dimension 2 due to the recent work of H. Brenner...

Let R = k[x1, ..., xn] be the polynomial ring in n independent variables, where k is a field. In this work we will study Bass numbers of local cohomology modules H I (R) supported on a squarefree monomial ideal I ⊆ R. Among them we are mainly interested in Lyubeznik numbers. We build a dictionary between the modules H I (R) and the minimal free resolution of the Alexander dual ideal I∨ that all...

Over an Artin algebra Λ many standard concepts from homological algebra can be relativized with respect to a contravariantly finite subcategory C of mod-Λ, which contains the projective modules. The main aim of this article is to prove that the resulting relative homological dimensions of modules are preserved by stable equivalences between Artin algebras. As a corollary, we see that Auslander’...

An optimal realization of a metric d on a set X is a weighted graph G = (V, E, w) such that X ⊆ V , dG(x, y) = d(x, y) for all x ∈ X, and ∑ e∈E w(e) is minimal. In this paper, we consider the conjecture that any finite metric (X, d) has some optimal realization obtainable as a subgraph of its tight span. For an extremal optimal realization G, defined as an optimal realizations for which the num...

I. Review: An obstruction theory. — Let Θn be the group of diffeomorphism classes of oriented smooth homotopy spheres of dimension n. Let Diff(Sn−1) be the simplicial group of orientation preserving diffeomorphisms Sn−1 → Sn−1. For n > 5, the natural homomorphism from π0Diff(Sn−1) to Θn is surjective by Smale’s h–cobordism theorem, and injective by Cerf’s pseudo–isotopy theorem. It is easily se...

A complex frame is a collection of vectors that span C and define measurements, called intensity measurements, on vectors in C . In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) ...

In this paper we consider how the∇–, ∆– and global dimensions of a quasi-hereditary algebra are interrelated. We first consider quasi-hereditary algebras with simple preserving duality and such that if μ < λ then ∇. f.d.(L(μ)) < ∇. f.d.(L(λ)) where μ, λ are in the poset and L(μ), L(λ) are the corresponding simples. We show that in this case the global dimension of the algebra is twice its ∇–fil...

In this article, we give a partial description of the sandpile group of the cone of the cartesian product of graphs in function of the sandpile group of the cone of their factors. Also, we introduce the concept of uniform homomorphism of graphs and prove that every surjective uniform homomorphism of graphs induces an injective homomorphism between their sandpile groups. As an application of the...

In this note we prove that reconstruction from magnitudes of frame coefficients (the so called ”phase retrieval problem”) can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map α : H → R is injective, with (α(x))k = |〈x, fk〉| , where {f1, . . . , fm} is a frame for the Hilbert space H, then there exists a left inverse map ω : R → H that is Li...