Representing Reflective Logic in Modal Logic

The nonmonotonic logic called Reflective Logic is shown to be representable in a monotonic Modal Quantificational Logic whose modal laws are stronger than S5. Specifically, it is proven that a set of sentences of First Order Logic is a fixed-point of the fixed-point equation of Reflective Logic with an initial set of axioms and defaults if and only if the meaning of that set of sentences is logically equivalent to a particular modal functor of the meanings of that initial set of sentences and of the sentences in those defaults. This result is important because the modal representation allows the use of powerful automatic deduction systems for Modal Logic and because unlike the original Reflective Logic, it is easily generalized to the case where quantified variables may be shared across the scope of the components of the defaults thus allowing such defaults to produce quantified consequences. Furthermore, this generalization properly treats such quantifiers since all the laws of First Order Logic hold and since both the Barcan Formula and its converse hold.

Representing Default Logic in Modal Logic

The nonmonotonic logic called Default Logic is shown to be representable in a monotonic Modal Quantificational Logic whose modal laws are stronger than S5. Specifically, it is proven that a set of sentences of First Order Logic is a fixed-point of the fixed-point equation of Default Logic with an initial set of axioms and defaults if and only if the meaning or rather disquotation of that set of sentences is logically equivalent to a particular modal functor of the meanings of that initial set of sentences and of the sentences in those defaults. This result is important because the modal representation allows the use of powerful automatic deduction systems for Modal Logic and because unlike the original Default Logic, it is easily generalized to the case where quantified variables may be shared across the scope of the components of the defaults thus allowing such defaults to produce quantified consequences. Furthermore, this generalization properly treats such quantifiers since both the Barcan Formula and its converse hold.

On the Relationship between Quantified Reflective Logic and Quantified Default Logic

Reflective Logic and Default Logic are both generalized so as to allow universally quantified variables to cross modal scopes whereby the Barcan formula and its converse hold. This is done by representing both the fixed-point equation for Reflective Logic and the fixed-point equation for Default both as necessary equivalences in the Modal Quantificational Logic Z. and then inserting universal quantifiers before the defaults. The two resulting systems, called Quantified Reflective Logic and Quantified Default Logic, are then compared by deriving metatheorems of Z that express their relationships. The main result is to show that every solution to the equivalence for Quantified Default Logic is a strongly grounded solution to the equivalence for Quantified Reflective Logic. It is further shown that Quantified Reflective Logic and Quantified Default Logic have exactly the same solutions when no default has an entailment condition.

Representing Autoepistemic Logic in Modal Logic

The nonmonotonic logic called Autoepistemic Logic is shown to be representable in a monotonic Modal Quantificational Logic whose modal laws are stronger than S5. Specifically, it is proven that a set of sentences of First Order Logic is a fixed-point of the fixed-point equation of Autoepistemic Logic with an initial set of axioms if and only if the meaning or rather disquotation of that set of sentences is logically equivalent to a particular modal functor of the meaning of that initial set of sentences. This result is important because the modal representation allows the use of powerful automatic deduction systems for Modal Logic and unlike the original Autoepistemic Logic, it is easily generalized to the case where quantified variables may be shared across the scope of modal expressions thus allowing the derivation of quantified consequences. Furthermore, this generalization properly treats such quantifiers since both the Barcan formula and its converse hold.

A Characterization of Weakly Lindelöf Determined Banach Spaces

2000 Mathematics Subject Classification: 46B26, 46B03, 46B04.

Теглови апроксимации с оператори от тип на Гудман–Шарма

P. E. Parvanov - The uniform weighted approximation errors of the Goodman–Sharma operators are characterized for functions.

Regulation of ribosomal RNA expression across the lifespan is fine-tuned by maternal diet before implantation

Cells and organisms respond to nutrient deprivation by decreasing global rates of transcription, translation and DNA replication. To what extent such changes can be reversed is largely unknown. We examined the effect of maternal dietary restriction on RNA synthesis in the offspring. Low protein diet fed either throughout gestation or for the preimplantation period alone reduced cellular RNA content across fetal somatic tissues during challenge and increased it beyond controls in fetal and adult tissues after challenge release. Changes in transcription of ribosomal RNA, the major component of cellular RNA, were responsible for this phenotype as evidenced by matching alterations in RNA polymerase I density and DNA methylation at ribosomal DNA loci. Cellular levels of the ribosomal transcription factor Rrn3 mirrored the rRNA expression pattern. In cell culture experiments, Rrn3 overexpression reduced rDNA methylation and increased rRNA expression; the converse occurred after inhibition of Rrn3 activity. These observations define novel mechanism where poor nutrition before implantation irreversibly alters basal rates of rRNA transcription thereafter in a process mediated by rDNA methylation and Rrn3 factor.

The Filippov-Wazewski relaxation theorem revisited

The converse statement of the Filippov-Wazewski relaxation theorem is proven, more precisely, two differential inclusions have the same closure of their solution sets if and only if the right-hand sides have the same convex hull. The idea of the proof is examining the contingent derivatives to the attainable sets.

A note on the axiomatization of the Nash equilibrium correspondence

A new axiomatization of the Nash equilibrium correspondence for n-person games based on independence of irrelevant strategies is given. Using a flexible general model, it is proved that the Nash equilibrium correspondence is the only solution to satisfy the axioms of non-emptiness, weak one-person rationality, independence of irrelevant strategies and converse independence of irrelevant strategies on the class of subgames of a fixed finite n-person game which admit at least one Nash equilibrium. It is also shown that these axioms are logically independent.

Non -traded goods, product differentiation and integration of factor markets

This research sought to determine the implications of a non-traded differentiated commodity produced with increasing returns to scale, for the welfare of countries that allowed free international migration. We developed two- and three-country Ricardian models in which labor was the only factor of production. The countries traded freely in homogeneous goods produced with constant returns to scale. Each also had a non-traded differentiated good sector where production took place using increasing returns to scale technology. Then we allowed for free international migration between two of the countries and observed what happened to welfare in both countries as indicated by their per capita utilities in the new equilibrium relative to their pre-migration utilities. ^ Preferences of consumers were represented by a two-tier utility function [Dixit and Stiglitz 1977]. As migration took place it impacted utility in two ways. The expanding country enjoyed the positive effect of increased product diversity in the non-traded good sector. However, it also suffered adverse terms-of-trade as its production cost declined. The converse was true for the contracting country. To determine the net impact on welfare we derived indirect per capita utility functions of the countries algebraically and graphically. Then we juxtaposed the graphs of the utility functions to obtain possible general equilibria. These we used to observe the welfare outcomes. ^ We found that the most likely outcomes were either that both countries gained, or one country lost while the other gained. We were, however, able to generate cases where both countries lost as a result of allowing free inter-country migration. This was most likely to happen when the shares of income spent on each country's export good differed significantly. In the three country world when we allowed two of the countries to engage in preferential trading arrangements while imposing a prohibitive tariff on imports from the third country welfare of the partner countries declined. When inter-union migration was permitted welfare declined even further. This we showed was due to the presence of the non-traded good sector. ^

Geodesically reversible Finsler 2-spheres of constant curvature

A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason, it is shown that a geodesically reversible Finsler metric of constant flag curvature on the 2-sphere is necessarily projectively flat. As a corollary, using a previous result of the author, it is shown that a reversible Finsler metric of constant flag curvature on the 2-sphere is necessarily a Riemannian metric of constant Gauss curvature, thus settling a long- standing problem in Finsler geometry.

Social Media Strategies in the Retail Sector: Analysis and Recommendations for Three Multi-National Retailers

During the last twenty years (1995-2015), the world of commerce has expanded beyond the traditional brick-and-mortar high street to a global shop front accessible to billions of users via the Worldwide Web (WWW). Consumers are now using the web to immerse themselves in virtual shop fronts, using Social Media (SM) to communicate and share product ideas with friends and family. Retail organisations recognise the need to develop and adapt their strategies to respond to the increasing use of SM. New goals must be set in order to identify how companies will integrate social media into current practices. This research aims to suggest an advisable and comprehensive SM strategy for companies operating in the global retail sector, based on an exploratory analysis of three multi-national retail organisations' existing SM strategies. This will be assessed in conjunction with a broader investigation into social media in the retail industry. From this, a strategy will be devised to improve internal and external communication as well as knowledge management through the use of social media. Findings suggest that the use of SM within the retail industry has dramatically improved collaboration and communication processes for organisations as they are now able to converse better with stakeholders and the tools are relatively simple to integrate and implement as they benefit one another.

Categories, Allegories, and Circuit Design

Languages based upon binary relations offer an appealing setting for constructing programs from specifications. For example, working with relations rather than functions allows specifications to be more abstract (for example, many programs have a natural specification using the converse operator on relations), and affords a natural treatment of non-determinism in specifications. In this paper we present a novel pictorial interpretation of relational terms as simple pictures of circuits, and a soundness/completeness result that allows relational equations to be proved by pictorial reasoning.

Making Functionality More General

The definition for the notion of a "function" is not cast in stone, but depends upon what we adopt as types in our language. With partial equivalence relations (pers) as types in a relational language, we show that the functional relations are precisely those satisfying the simple equation f = f o fu o f, where "o" and "u" are respectively the composition and converse operators for relations. This article forms part of "A calculational theory of pers as types".

READY FOR KINDERGARTEN: A TRAINING PROGRAM DESIGNED TO ENCOURAGE PARENT-CHILD CONVERSATION DURING THE PRESCHOOL YEARS

Many children in the United States begin kindergarten unprepared to converse in the academic language surrounding instruction, putting them at greater risk for later language and reading difficulties. Importantly, correlational research has shown there are certain experiences prior to kindergarten that foster the oral language skills needed to understand and produce academic language. The focus of this dissertation was on increasing one of these experiences: parent-child conversations about abstract and non-present concepts, known as decontextualized language (DL). Decontextualized language involves talking about non-present concepts such as events that happened in the past or future, or abstract discussions such as providing explanations or defining unknown words. As caregivers’ decontextualized language input to children aged three to five is consistently correlated with kindergarten oral language skills and later reading achievement, it is surprising no experimental research has been done to establish this relation causally. The study described in this dissertation filled this literature gap by designing, implementing, and evaluating a decontextualized language training program for parents of 4-year-old children (N=30). After obtaining an initial measure of decontextualized language, parents were randomly assigned to a control condition or training condition, the latter of which educated parents about decontextualized language and why it is important. All parents then audio-recorded four mealtime conversations over the next month, which were transcribed and reliably coded for decontextualized language. Results indicated that trained parents boosted their DL from roughly 17 percent of their total utterances at baseline to approximately 50 percent by the mid-point of the study, and remained at these boosted levels throughout the duration of the study. Children’s DL was also boosted by similar margins, but no improvement in children’s oral language skills was observed, measured prior to, and one month following training. Further, exploratory analyses pointed to parents’ initial use of DL and their theories of the malleability of intelligence (i.e., growth mindsets) as moderators of training gains. Altogether, these findings are a first step in establishing DL as a viable strategy for giving children the oral language skills needed to begin kindergarten ready to succeed in the classroom.