An introduction to presentations of monoid acts: quotients and subacts

The purpose of this article is to introduce the theory of presentations of monoids acts. We aim to construct “nice” general presentations for various act constructions pertaining to subacts and Rees quotients. More precisely, given an M-act A and a subact B of A, on the one hand, we construct presentations for B and the Rees quotient A/B using a presentation for A, and on the other hand, we derive a presentation for A from presentations for B and A/B. We also construct a general presentation for the union of two subacts. From our general presentations, we deduce a number of finite presentability results. Finally, we consider the case where a subact B has a finite complement in an M-act A. We show that if M is a finitely generated monoid and B is finitely presented, then A is finitely presented. We also show that if M belongs to a wide class of monoids, including all finitely presented monoids, then the converse also holds.     

Polynomially Riesz elements

Abstract

A Banach algebra element a ∈ A is said to be “polynomially Riesz”, relative to the homomorphism T : A → B, if there exists a nonzero complex polynomial p(z) such that the image T p(a) ∈ B is quasinilpotent.     

On approximation of approximately linear mappings by linear mappings

Assume A is a normed linear space, B is a Banach space, and f: A → B is a mapping “approximately linear.” We solve the following Ulam problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist.”     

Dimensionality of biinfinite systems

The problem of determining the uniqueness of the coefficient of interpolation of M compactly supported real functions, with a biinfinite sequence of interpolation points, leads to the study of the kernel Z of the biinfinite block Toeplitz matrix

$\text{D=}\text{?}\text{?}\text{A}\text{B}\text{A}\text{B}\text{?}\text{?}$

. The dimension of Z is found by considering the “maximal solvable subspace” V (relative to A and B). Further results are obtained using the Kronecker canonical form of the matrix pencil A+λB and “restricted generalized inverses” of A (and B).     

On Strongly Clean Modules

An element of a ring R is called “strongly clean” if it is the sum of an idempotent and a unit that commute, and R is called “strongly clean” if every element of R is strongly clean. A module M is called “strongly clean” if its endomorphism ring End(M) is a strongly clean ring. In this article, strongly clean modules are characterized by direct sum decompositions, that is, M is a strongly clean module if and only if whenever M′⊕ B = A ₁⊕ A ₂ with M′? M, there are decompositions M′ = M ₁⊕ M ₂, B = B ₁⊕ B ₂, and A _i  = C _i ⊕ D _i (i = 1,2) such that M ₁⊕ B ₁ = C ₁⊕ D ₂ = M ₁⊕ C ₁ and M ₂⊕ B ₂ = D ₁⊕ C ₂ = M ₂⊕ C ₂.     

A gradual rank increasing process for matrix completion

The matrix completion problem is easy to state: let A be a given data matrix in which some entries are unknown. Then, it is needed to assign “appropriate values” to these entries. A common way to solve this problem is to compute a rank-k matrix, B _k , that approximates A in a least squares sense. Then, the unknown entries in A attain the values of the corresponding entries in B _k . This raises the question of how to determine a suitable matrix rank. The method proposed in this paper attempts to answer this question. It builds a finite sequence of matrices \(B_{k}, k = 1, 2, \dots \), where B _k is a rank-k matrix that approximates A in a least squares sense. The computational effort is reduced by using B _k-1 as starting point in the computation of B _k . The ability of B _k to serve as substitute for A is measured with two objective functions: a “training” function that measures the distance between the known part of A and the corresponding part of B _k , and a “probe” function that assesses the quality of the imputed entries. Watching the changes in these functions as k increases enables us to find an optimal matrix rank. Numerical experiments illustrate the usefulness of the proposed approach.     

Semi-algebraic Absolute Valued Algebras with an Involution

Abstract

Let A be an absolute valued algebra. In El-Mallah (El-Mallah,M. L. (1988). Absolute valued algebras with an involution. Arch. Math. 51: 39–49) we proved that,if A is algebraic with an involution,then A is finite dimensional. This result had been generalized in El-Amin et al. (El-Amin,K.,Ramirez,M. I.,Rodriguez,A. (1997). Absolute valued algebraic algebras are finite dimensional. J. Algebra 195:295–307),by showing that the condition “algebraic” is sufficient for A to be finite dimensional. In the present paper we give a generalization of the concept “algebraic”,which will be called “semi-algebraic”,and prove that if A is semi-algebraic with an involution then A is finite dimensional. We give an example of an absolute valued algebra which is semi-algebraic and infinite dimensional. This example shows that the assumption “with an involution” cannot be removed in our result.     

Adjoining an Identity to a Reduced Archimedean f-Ring

Let frA denote the category of f-rings which are reduced and Archimedean, and let Φ be the (nonfull) subcategory of such rings with identity (each with the natural morphisms). Some time ago, the second author showed, using his representation theory, that for each A ∈ | frA| there is a certain minimal embedding u _A :A→ uA ∈ | Φ|. More recently, he has revisited the representation theory, expanding it to include the representation of morphisms. Based upon this, the present article analyzes the operator u:| frA| → Φ: the construction of uA is tidied, several characterizations of the pair (u _A , uA) are given, and the relation between the maximal ideal structures of A and uA is described. Membership in the class U of frA-morphisms that are “u-extendable” is characterized and it is shown that U = (| frA|,U) is a category in which Φ is a full essentially-reflective subcategory. The frA-objects are characterized for which, respectively, ? B(frA(A, B) = U (A, B)), and, ? B ≠ 0(frA(B, A) = U(B, A)).     

Lifting Modules with Indecomposable Decompositions

A module M is called a “lifting module” if, any submodule A of M contains a direct summand B of M such that A/B is small in M/B. This is a generalization of projective modules over perfect rings as well as the dual of extending modules. It is well known that an extending module with ascending chain condition (a.c.c.) on the annihilators of its elements is a direct sum of indecomposable modules. If and when a lifting module has such a decomposition is not known in general. In this article, among other results, we prove that a lifting module M is a direct sum of indecomposable modules if (i) rad(M ^(I)) is small in M ^(I) for every index set I, or, (ii) M has a.c.c. on the annihilators of (certain) elements, and rad(M) is small in M.     

Extending Independent Sets to Bases and the Axiom of Choice

We show that the both assertions “in every vector space B over a finite element field every subspace V ? B has a complementary subspace S” and “for every family ?? of disjoint odd sized sets there exists a subfamily ?={F_j:j ?ω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ? every generating set includes a basis”.     

Sufficiency and completeness for classes of probability distributions generated by a single probability distribution

Let (Ω,A, P₀) denote a probability space andA some sub-σ-algebra ofA. Moreover,P _{P 0} stands for the set consisting of all probability distributionsP _{P 0} of the typeP _{P 0}(A) = ∫E _{P 0} (P _{A |B)dP,A ∈A, whereP is a probability distribution onA satisfyingP |B ≪P 0} |B. It is shown thatB is sufficient and (even totally) complete forP _{P 0}. This results is illustrated by examples. Finally, it is shown thatP _{P 0} ∈P _{P 0} is an extremal point ofP _{P 0} if and only ifP |B is {0, 1}-valued. Dedicated to the memory of Professor K. Behnen     

Product formula for imaginary resolvents with application to a modified Feynman integral

An abstract product formula for imaginary resolvents is proved for a pair of self-adjoint operators A, B of a complex Hilbert space. Here, A is assumed to be nonnegative and the positive part of B is arbitrary while its negative part is small with respect to A in the sense of quadratic forms. The proof is somewhat simpler than the author's original one which required both A and B to be nonnegative. When specialized, this theorem establishes the convergence of the “modified Feynman integral”—recently introduced by the author—in the most general case for which the Schrödinger equation can be solved without ambiguity.     

On Supersets of Wavelet Sets

Considering a single dyadic orthonormal wavelet ψ in L ²(?), it is still an open problem whether the support of $\widehat{\psi}$ always contains a wavelet set. As far as we know, the only result in this direction is that if the Fourier support of a wavelet function is “small” then it is either a wavelet set or a union of two wavelet sets. Without assuming that a set S is the Fourier support of a wavelet, we obtain some necessary conditions and some sufficient conditions for a “small” set S to contain a wavelet set. The main results, which are in terms of the relationship between two explicitly constructed subsets A and B of S and two subsets T ₂ and D ₂ of S intersecting itself exactly twice translationally and dilationally respectively, are (1) if $A\cup B\not\subseteq T_{2}\cap D_{2}$ then S does not contain a wavelet set; and (2) if A∪B?T ₂∩D ₂ then every wavelet subset of S must be in S?(A∪B) and if S?(A∪B) satisfies a “weak” condition then there exists a wavelet subset of S?(A∪B). In particular, if the set S?(A∪B) is of the right size then it must be a wavelet set.     

A note on the equivalence of m-periodic derived categories

Let A and B be finite-dimensional algebras over a field k of finite global dimension. Using some results of Gorsky in “Semi-derived Hall algebras and tilting invariance of Bridgeland-Hall algebras”, we prove that if A and B are derived equivalent, then the corresponding m-periodic derived categories are triangulated equivalent.     

Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure

We consider a linear multivariate errors-in-variables model AX ≈ B, where the matrices A and B are observed with errors and the matrix parameter X is to be estimated. In the case of lack of information about the error covariance structure, we propose an estimator that converges in probability to X as the number of rows in A tends to infinity. Sufficient conditions for this convergence and for the asymptotic normality of the estimator are found. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1026–1033, August, 2007.     

On uniqueness of invariant measures for finite‐ and infinite‐dimensional diffusions

We prove uniqueness of “invariant measures,” i.e., solutions to the equation L*μ = 0 where L = Δ + B · ∇ on ℝⁿ with B satisfying some mild integrability conditions and μ being a probability measure on ℝⁿ. This solves an open problem posed by S. R. S. Varadhan in 1980. The same conditions are shown to imply that the closure of L on L¹(μ) generates a strongly continuous semigroup having μ as its unique invariant measure. The question whether an extension of L generates a strongly continuous semigroup on L¹(μ) and whether such an extension is unique is addressed separately and answered positively under even weaker local integrability conditions on B. The special case when B is a gradient of a function (i.e., the “symmetric case”) in particular is studied and conditions are identified ensuring that L*μ = 0 implies that L is symmetric on L²(μ) or L*μ = 0 has a unique solution. We also prove infinite‐dimensional analogues of the latter two results and a new elliptic regularity theorem for invariant measures in infinite dimensions. © 1999 John Wiley & Sons, Inc.     

Random sparse unary predicates

Random unary predicates U on [n] holding with probability p are examined with respect to sentences A in a first-order language containing U and “less than.” When p = p(n) satisfies n^k+1 ? 1 ? np^k it is shown that Pr[A] approaches a limit dependent only on k and A. In a similar circular model the limit is shown to be zero or one. © 1994 John Wiley & Sons, Inc.     

Probabilistic matching of planar regions

We analyze a probabilistic algorithm for matching shapes modeled by planar regions under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high probability the area of overlap of t(A) and B is close to maximal. In the case of polygons, we give a time bound that does not depend significantly on the number of vertices, but on perimeter and area of the shapes and, in the case of rigid motions, also on the diameter.     

Multiplier Hopf Algebras in Categories and the Biproduct Construction

Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct. Presented by K. Goodearl.