title: Variable separation for Heat and Schrödinger equations

authors: Reid, Gregory John

abstract: All R-separable coordinate systems are classified for the equation
Δₘ Ψ + 2ε ∂/∂t Ψ = EΨ (*)
where Δₘ = ᵐΣ ᵤ₌₁ ∂²/(∂yᵘ)² and the variables t and yᵘ are real. An R-separable coordinate system for (*) is a coordinate system yᵘ = yᵘ (xᵏ), t = t (xᵏ) such that (*) admits a solution of form Ψ = eᴿ⁽ˣⁱ⁾ ᵐ⁺¹Πᵢ₌₁ Ψᵢ(xⁱ;c₁, ···, cₘ₊₁). For ε = ⁱ/₂ and ε = - ¹/₂, E = 0, (*) yields standardised versions of the Schrӧdinger and Heat equations respectively. Recognition of (*) as a symmetry-reduced version of the Helmholtz equation on m + 2 dimensional Minkowski space, for which there is a well developed theory of separation, is central to the solution of the classification problem. The solution is built from the case ε = 0 which has been solved by Kalnins and Miller in an article submitted to the Transactions of the American Mathematical Society.
We find the operators and separation equations for (*) and give a detailed treatment of the physically interesting case m =3. The operators are always in the enveloping algebra of the Schrӧdinger algebra. This means that much of the special function theory relating to the separated solutions can be reduced to problems in the representation theory of this algebra.
Various complex extensions of (*) are treated. In particular a class of systems is uncovered leading to nontrivial R-separation for the Hamilton-Jacobi equation corresponding to (*). A formula is derived for expressing the second order Killing tensors describing separation in flat spaces in terms of the enveloping algebra. This formula forms the basis of a program written in the symbolic language MACSYMA capable of producing all the time consuming details of separation.

text: Variable separation for Heat and Schrödinger equations All R-separable coordinate systems are classified for the equation
Δₘ Ψ + 2ε ∂/∂t Ψ = EΨ (*)
where Δₘ = ᵐΣ ᵤ₌₁ ∂²/(∂yᵘ)² and the variables t and yᵘ are real. An R-separable coordinate system for (*) is a coordinate system yᵘ = yᵘ (xᵏ), t = t (xᵏ) such that (*) admits a solution of form Ψ = eᴿ⁽ˣⁱ⁾ ᵐ⁺¹Πᵢ₌₁ Ψᵢ(xⁱ;c₁, ···, cₘ₊₁). For ε = ⁱ/₂ and ε = - ¹/₂, E = 0, (*) yields standardised versions of the Schrӧdinger and Heat equations respectively. Recognition of (*) as a symmetry-reduced version of the Helmholtz equation on m + 2 dimensional Minkowski space, for which there is a well developed theory of separation, is central to the solution of the classification problem. The solution is built from the case ε = 0 which has been solved by Kalnins and Miller in an article submitted to the Transactions of the American Mathematical Society.
We find the operators and separation equations for (*) and give a detailed treatment of the physically interesting case m =3. The operators are always in the enveloping algebra of the Schrӧdinger algebra. This means that much of the special function theory relating to the separated solutions can be reduced to problems in the representation theory of this algebra.
Various complex extensions of (*) are treated. In particular a class of systems is uncovered leading to nontrivial R-separation for the Hamilton-Jacobi equation corresponding to (*). A formula is derived for expressing the second order Killing tensors describing separation in flat spaces in terms of the enveloping algebra. This formula forms the basis of a program written in the symbolic language MACSYMA capable of producing all the time consuming details of separation.

year: 1984

title: A Comparison of the BTT and TTF Test-Generation Methods

authors: Legeard, Bruno

subjects: computer science

abstract: This paper compares two methods of generating tests from formal specifications. The Test Template Framework (TTF) method is a framework and set of heuristics for manually generating test sets from a Z specification. The B Testing Tools (BTT) method uses constraint logic programming techniques to generate test sequences from a B specification. We give a concise description of each method, then compare them on an industrial case study, which is a subset of the GSM 11.11 smart card specification.

text: A Comparison of the BTT and TTF Test-Generation Methods This paper compares two methods of generating tests from formal specifications. The Test Template Framework (TTF) method is a framework and set of heuristics for manually generating test sets from a Z specification. The B Testing Tools (BTT) method uses constraint logic programming techniques to generate test sequences from a B specification. We give a concise description of each method, then compare them on an industrial case study, which is a subset of the GSM 11.11 smart card specification.

year: 2002

title: An exploration of the role of semantic relations in the theory and practice of translation (with special reference to English/Māori and Māori/English translation)

authors: Roa, Tom

subjects: clause relations

abstract: It is over forty years since Beekman and Callow, working within the context of the tagmemic tradition, drew attention to the potential significance so far as the theory and practice of translation are concerned of inter-propositional relations (also referred to as ‘discourse relations’, 'semantic relations', ‘semantico-pragmatic relations’, 'rhetorical relations’ and ‘clause relations’).  Since then, the importance of these relations has been increasingly acknowledged within linguistics and a range of other disciplines, including artificial intelligence. In spite of this, inter-propositional relations remain largely unexplored by the vast majority of those involved in translation. My aim in this research project was to determine (a) what impact, if any, a short training course in inter-propositional relations and inter-propositional relational signalling had on novice Māori-English/ English-Māori translators, and (b)  what impact my own understanding of these relations has had on my own practice as a translator. In connection with this, I conducted two studies.  

In the first of these studies, two small groups of novice translators were asked to translate seven text segments - some from English into te reo Māori, others from te reo Māori into English. They were then given a two-day workshop on inter-propositional relations and inter-propositional relational signalling in English and te reo Māori. No translation exercises were included in the workshop. The novice translators were then asked to translate the same text segments again. Comparison of the two sets of translations revealed some interesting differences, the second set indicating a higher level of sensitivity to inter-propositional relations and their signalling in the original text segments. This resulted in, it is argued here, translations that more accurately reflect authorial intentions as indicated in the source texts (see Chapter 3). In connection with this, it is relevant to note that, as indicated in the introductory chapter (Chapter 1) and in parts of the literature review (Chapter 2), while some of those involved in the Western tradition of translation theory and practice have argued that authorial intention is largely irrelevant in relation to textual interpretation, this essentially post-structural positioning is not available to those involved in translating indigenous texts within the context of indigenous worldviews.

In the second study, I used self-reflective think-aloud protocols to record my thoughts during the process of translating, from English into te reo Māori, text segments from five very different texts, all of these translations having been commissioned by different commissioning agencies. I then, making reference to the think-aloud transcripts, compared the source texts with the translations to determine the ways in which my reflections on inter-propositional relations and their signalling had impacted on the translations (Chapter 4).

My overall conclusion is that an understanding of inter-propositional relations and inter-propositional relational signalling can make a positive contribution to the theory and practice of translation and should be included in translator training and in the teaching of additional languages more generally (Chapter 5).

text: An exploration of the role of semantic relations in the theory and practice of translation (with special reference to English/Māori and Māori/English translation) It is over forty years since Beekman and Callow, working within the context of the tagmemic tradition, drew attention to the potential significance so far as the theory and practice of translation are concerned of inter-propositional relations (also referred to as ‘discourse relations’, 'semantic relations', ‘semantico-pragmatic relations’, 'rhetorical relations’ and ‘clause relations’).  Since then, the importance of these relations has been increasingly acknowledged within linguistics and a range of other disciplines, including artificial intelligence. In spite of this, inter-propositional relations remain largely unexplored by the vast majority of those involved in translation. My aim in this research project was to determine (a) what impact, if any, a short training course in inter-propositional relations and inter-propositional relational signalling had on novice Māori-English/ English-Māori translators, and (b)  what impact my own understanding of these relations has had on my own practice as a translator. In connection with this, I conducted two studies.  

In the first of these studies, two small groups of novice translators were asked to translate seven text segments - some from English into te reo Māori, others from te reo Māori into English. They were then given a two-day workshop on inter-propositional relations and inter-propositional relational signalling in English and te reo Māori. No translation exercises were included in the workshop. The novice translators were then asked to translate the same text segments again. Comparison of the two sets of translations revealed some interesting differences, the second set indicating a higher level of sensitivity to inter-propositional relations and their signalling in the original text segments. This resulted in, it is argued here, translations that more accurately reflect authorial intentions as indicated in the source texts (see Chapter 3). In connection with this, it is relevant to note that, as indicated in the introductory chapter (Chapter 1) and in parts of the literature review (Chapter 2), while some of those involved in the Western tradition of translation theory and practice have argued that authorial intention is largely irrelevant in relation to textual interpretation, this essentially post-structural positioning is not available to those involved in translating indigenous texts within the context of indigenous worldviews.

In the second study, I used self-reflective think-aloud protocols to record my thoughts during the process of translating, from English into te reo Māori, text segments from five very different texts, all of these translations having been commissioned by different commissioning agencies. I then, making reference to the think-aloud transcripts, compared the source texts with the translations to determine the ways in which my reflections on inter-propositional relations and their signalling had impacted on the translations (Chapter 4).

My overall conclusion is that an understanding of inter-propositional relations and inter-propositional relational signalling can make a positive contribution to the theory and practice of translation and should be included in translator training and in the teaching of additional languages more generally (Chapter 5).

year: 2016