Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.Comment: 36 pages, 1 tabl

An improved quadrilateral flat element with drilling degrees of freedom for shell structural analysis

This paper reports the development of a simple and efficient 4-node flat shell element with six degrees of freedom per node for the analysis of arbitrary shell structures. The element is developed by incorporating a strain smoothing technique into a flat shell finite element approach. The membrane part is formulated by applying the smoothing operation on a quadrilateral membrane element using Allman-type interpolation functions with drilling DOFs. The plate-bending component is established by a combination of the smoothed curvature and the substitute shear strain fields. As a result, the bending and a part of membrane stiffness matrices are computed on the boundaries of smoothing cells which leads to very accurate solutions, even with distorted meshes, and possible reduction in computational cost. The performance of the proposed element is validated and demonstrated through several numerical benchmark problems. Convergence studies and comparison with other existing solutions in the literature suggest that the present element is efficient, accurate and free of lockings

A smoothed four-node piezoelectric element for analysis of two-dimensional smart structures

This paper reports a study of linear elastic analysis of two-dimensional piezoelectric structures using a smoothed four-node piezoelectric element. The element is built by incorporating the strain smoothing method of mesh-free conforming nodal integration into the standard four-node quadrilateral piezoelectric finite element. The approximations of mechanical strains and electric potential fields are normalized using a constant smoothing function. This allows the field gradients to be directly computed from shape functions. No mapping or coordinate transformation is necessary so that the element can be used in arbitrary shapes. Through several examples, the simplicity, efficiency and reliability of the element are demonstrated. Numerical results and comparative studies with other existing solutions in the literature suggest that the present element is robust, computationally inexpensive and easy to implement

Priestley duality for MV-algebras and beyond

We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions

Family values of Vietnamese living in Vietnam and Poland

This study used Vietnamese traditional family values as basic contents to compare between two participant groups. The traditional family values of Vietnamese always emphasized and praised the parents who are hard-working, and making sacrifice for their children; offspring have to practice filial piety, take care of by parents conscientiously when parents become older to reply the parents’ kindness. A feature of a social patriarchy is son preference. It is expressed in a desire to have at least one son to continue family lineage, worship ancestors etc. In relation of husband – wife, Vietnamese traditional family values expressed clearly beliefs in benevolence and righteousness, faithfulness; the wife, the mother had to serve, be long-suffering the husband and child. Respondents of the study were 196 Vietnamese living in Vietnam (100 persons) and Poland (96 persons). Methods of the study included: the purposely designed questionnaire administered to all 196 respondents and the intensive interview conducted with 15 Vietnamese who got married with Poles. The results of the study confirmed that Vietnamese immigrant participants could be flexible in relation with their offspring, but they also have the trend to protect, take care of their offspring more than other participants. Preference for a son can be expressed subtly; group of participants living in Poland expressed less son preference than participants living in Vietnam. The results of the study also showed the tendencies of keeping Vietnamese traditional values even while living far from their homeland. They confirm basic values such benevolence and righteousness, support each spouses in work, in life; they also accepted the phenomena called extramarital, cohabitation. The differences can be related to the process of acquirement Polish culture of Vietnamese immigrants and also to adapt to the life and work far from homeland

The Chesterfield Pike Plan: A Community Revitalization Plan for Historic Route 1

The Chesterfield Pike Plan complements Chesterfield County’s ongoing efforts to revitalize the northern section of the Jefferson Davis Highway corridor. By focusing on housing and local economic challenges on what is locally known as “The Pike,” this plan explores community revitalization strategies in a diverse inner-ring suburb. The Pike has been impacted by the nationwide trends of suburban poverty and immigrant suburbanization, which present challenges and opportunities in the pursuit of revitalization. Striking a balance between meeting community needs and economic growth requires a plan that identifies ways in which both new private investment and the existing community can benefit. The Chesterfield Pike Plan responds to this need with specific policy and program recommendations that find that balance. Exploring the market affordable housing and community-defining business types along the Pike, this plan identifies multiple ways in which the County can develop each type to strengthen existing residential options while also creating an economic environment to support new business and residential growth. The Chesterfield Pike Plan recommends seven major goals that would support a revitalized Pike for all: 1. Establishment of a countywide, dedicated source of funding for affordable housing 2. Preservation and enhancement of at-risk market affordable housing on the Pike 3. Promotion of the county as an immigrant-friendly place 4. Development of a multicultural public market on the Pike 5. Improvement of existing commercial business conditions on the Pike 6. Promotion of challenge-conscious workforce development on the Pike 7. Expansion and focus of public investment on the Pik