Three-Dimensional ℓ-Algebras

The family of all three-dimensional almost f-algebras, d-algebras, and f-algebras is constructed. It is shown that it contains all three-dimensional directly indecomposable ?-algebras. Also, a list of representatives of (algebraic) isomorphism classes of all three-dimensional algebras that can be ordered as an almost f-algebra, a d-algebra, or an f-algebra is given.     

Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2–ℓ p minimization

The iteratively reweighted ? ₁ minimization algorithm (IRL1) has been widely used for variable selection, signal reconstruction and image processing. In this paper, we show that any sequence generated by the IRL1 is bounded and any accumulation point is a stationary point of the ? ₂–? _p minimization problem with 0<p<1. Moreover, the stationary point is a global minimizer and the convergence rate is approximately linear under certain conditions. We derive posteriori error bounds which can be used to construct practical stopping rules for the algorithm.     

Lamron ℓ-groups

The article introduces a new class of lattice-ordered groups. An ?-group G is lamron if Min(G)^?1 is a Hausdorff topological space, where Min(G)^?1 is the space of all minimal prime subgroups of G endowed with the inverse topology. It will be evident that lamron ?-groups are related to ?-groups with stranded primes. In particular, it is shown that for a W-object (G,u), if every value of u contains a unique minimal prime subgroup, then G is a lamron ?-group; such a W-object will be said to have W-stranded primes. A diverse set of examples will be provided in order to distinguish between the notions of lamron, stranded primes, W-stranded primes, complemented, and weakly complemented ?-groups.     

Feebly projectable ℓ-groups

In the article [17], we introduced and investigated feebly and flatly projectable frames. In this article, we apply these two properties to lattice-ordered groups. An example is constructed to illustrate that the two properties are distinct, which solves a question from [17]. We also investigate these properties with respect to archimedean ℓ-groups with weak order unit, as well as commutative semiprime f-rings.     

Clean unital ℓ-groups

A ring with identity is said to be clean if every element can be written as a sum of a unit and an idempotent. The study of clean rings has been at the forefront of ring theory over the past decade. The theory of partially-ordered groups has a nice and long history and since there are several ways of relating a ring to a (unital) partially-ordered group it became apparent that there ought to be a notion of a clean partially-ordered group. In this article we define a clean unital lattice-ordered group; we state and prove a theorem which characterizes clean unital ?-groups. We mention the relationship of clean unital ?-groups to algebraic K-theory. In the last section of the article we generalize the notion of clean to the non-unital context and investigate this concept within the framework of W-objects, that is, archimedean ?-groups with distinguished weak order unit.     

ℓ1-Rigid Graphs

An ₁-graph is a graph whose nodes can be labeled by binary vectors in such a way that the Hamming distance between the binary addresses is, up to scale, the distance in the graph between the corresponding nodes. We show that many interesting graphs are ₁-rigid, i.e., that they admit an essentially unique such binary labeling.     

On discrete ℓ1-regularization

A real m × n matrix A and a vector y?∈?? ^m determine the discrete l ¹-regularization (DLR) problem 0.1 $$ \min \left\{\mbox{\,}|y-Ax|_1+\rho |x|_1:\,x\in\mathbb{R}^n \right\}, $$ where | · |₁ denotes the l ¹-norm of a vector and ρ?≥?0 is a nonnegative parameter. In this paper, we provide a detailed analysis of this problem which include a characterization of all solutions to (0.1), remarks about the geometry of the solution set and an effective iterative algorithm for numerical solution of (0.1). We are specially interested in the behavior of the solution of (0.1) as a function of ρ and in this regard, we prove in general the existence of critical values of ρ between which the l ¹-norm of any solution remains constant. These general remarks are significantly refined when A is a strictly totally positive (STP) matrix. The importance of STP matrices is well-established [5, 14]. Under this setting, the relationship between the number of nonzero coordinates of a distinguished solution of the DLR problem is precisely explained as a function of the regularization parameter for a certain class of vectors in ? ^m . Throughout our analysis of the DLR problem, we emphasize the importance of the dual maximum problem by demonstrating that any solution of it leads to a solution of the DLR problem, and vice versa.     

Maximal ℓ-Frattini quotients of ℓ-Poincaré duality groups of dimension 2

Any morphism of profinite groups has maximal ℓ-Frattini quotients

Decomposition of ℓ-group-valued measures

We deal with decomposition theorems for modular measures µ: L → G defined on a D-lattice with values in a Dedekind complete ?-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete ?-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for ?-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.     

On orthogonal linear ℓ1 approximation

Summary The problem is considered of orthogonal ₁ fitting of discrete data. Local best approximations are characterized and the question of the robustness of these solutions is considered. An algorithm for the problem is presented, along with numerical results of its application to some data sets.     

Pure ℓ-Ideals in Lattice-Ordered Rings#

An ?-ideal I of a commutative lattice-ordered ring R with positive identity element is called a pure ?-ideal if R  =  I  + ?( x ) for each x  ∈  I , where ?(x) is the ?-annihilator of x in R . In this article, we give some results on pure ?-ideals and study the ?-ideal structure of a commutative lattice-ordered ring with positive identity element by using pure ?-ideals.     

Congruences for ℓ-regular overpartition for ℓ ∈ {5, 6, 8}

Let A? _?(n) denote the number of overpartitions of a non-negative integer n with no part divisible by ?, where ? is a positive integer. In this paper, we prove infinite family of congruences for A? ₅(n) modulo 4, A? ₆(n) modulo 3, and A? ₈(n) modulo 4. In the process, we also prove some other congruences.     

Remote Sensing via ℓ 1-Minimization

We consider the problem of detecting the locations of targets in the far field by sending probing signals from an antenna array and recording the reflected echoes. Drawing on key concepts from the area of compressive sensing, we use an ? ₁-based regularization approach to solve this, generally ill-posed, inverse scattering problem. As is common in compressive sensing, we exploit randomness, which in this context comes from choosing the antenna locations at random. With n antennas we obtain n ² measurements of a vector $x \in\mathbb{C}^{N}$ representing the target locations and reflectivities on a discretized grid. It is common to assume that the scene x is sparse due to a limited number of targets. Under a natural condition on the mesh size of the grid, we show that an s-sparse scene can be recovered via ? ₁-minimization with high probability if n ²≥Cslog²(N). The reconstruction is stable under noise and when passing from sparse to approximately sparse vectors. Our theoretical findings are confirmed by numerical simulations.     

Norming subspaces isomorphic to ℓ 1

Norming subspaces are studied widely in the duality theory of Banach spaces. These subspaces are applied to the Borel and Baire classifications of the inverse operators. The main result of this article asserts that the dual of a Banach space X contains a norming subspace isomorphic to l₁ provided that the following two conditions are satisfied: (1) X^* contains a subspace isomorphic to l₁; and (2) X^* contains a separable norming subspace.     

The Archimedean ℓ-group tensor product

We introduce a construction (inZ F-set theory) for the Archimedean -group tensor product. We relate this tensor product to the existing ones in the theory of Archimedean vector lattices and -groups.     

Uniquely Covered Radical Classes of ℓ-Groups

It is proved that a radical class of lattice-ordered groups has exactly one cover if and only if it is an intersection of some -complement radical class and the big atom over .     

Spaces with Dual Order-Isomorphic to ℓ1

We establish that an ordered Banach space is order-isomorphic to c₀ if and only if it is a -Dedekind complete vector lattice and its norm dual is order-isomorphic to ₁.