Jerome Murphy-O'Connor
- Published in print:
- 2009
- Published Online:
- May 2009
- ISBN:
- 9780199564156
- eISBN:
- 9780191721281
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199564156.001.0001
- Subject:
- Religion, Biblical Studies
This book brings together sixteen originally independent articles dealing with various aspects of 1 Corinthians and published between 1976 and 1993. As the series develops there are more frequent ...
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This book brings together sixteen originally independent articles dealing with various aspects of 1 Corinthians and published between 1976 and 1993. As the series develops there are more frequent cross‐references. The first deals with the issue of co‐authorship, and the last with the question of interpolations in 1 Cor. The rest focus on the most difficult and disputed texts in 1 Corinthians, namely, 1 Cor 5: 3–5 (incest in the name of Christ); 6: 12–20 (Corinthian slogans about the body); 7: 10–11 (divorce and remarriage); 7: 14 (holiness); 8: 6 (baptismal acclamation); 8: 8 (Corinthian slogan regarding food); chs. 8–10 (food offered to idols); 11: 2–16 (3 articles; blurring of the distinction between the sexes in worship); 11: 17–34 (2 articles; house‐churches and the eucharist); 15: 3–7 (creed); 15: 29 (baptism for the dead). Each original article took contemporary scholarship into full account. A ‘Postscript’ appended to each one brings the discussion up to the present by documenting the ensuing debate about the proposed hypotheses.Less
This book brings together sixteen originally independent articles dealing with various aspects of 1 Corinthians and published between 1976 and 1993. As the series develops there are more frequent cross‐references. The first deals with the issue of co‐authorship, and the last with the question of interpolations in 1 Cor. The rest focus on the most difficult and disputed texts in 1 Corinthians, namely, 1 Cor 5: 3–5 (incest in the name of Christ); 6: 12–20 (Corinthian slogans about the body); 7: 10–11 (divorce and remarriage); 7: 14 (holiness); 8: 6 (baptismal acclamation); 8: 8 (Corinthian slogan regarding food); chs. 8–10 (food offered to idols); 11: 2–16 (3 articles; blurring of the distinction between the sexes in worship); 11: 17–34 (2 articles; house‐churches and the eucharist); 15: 3–7 (creed); 15: 29 (baptism for the dead). Each original article took contemporary scholarship into full account. A ‘Postscript’ appended to each one brings the discussion up to the present by documenting the ensuing debate about the proposed hypotheses.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0004
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter presents the equivalence of Craig's interpolation property to Robinson's joint consistency, and a proof of Lyndon's interpolation theorem for the classical predicate logic. It is proved ...
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This chapter presents the equivalence of Craig's interpolation property to Robinson's joint consistency, and a proof of Lyndon's interpolation theorem for the classical predicate logic. It is proved that the general form of Robinson's consistency property (RCP) fails in the intuitionistic predicate logic HQ. A weaker form of RCP is equivalent to Craig's interpolation property (CIP) and holds in HQ, and a semantic proof is given. It is proved that in propositional intermediate logics the general form of RCP is equivalent to CIP. A derivation of Beth's property from CIP is given for the intuitionistic predicate logic. Kreisel's proof of validity of the Beth property for any propositional intermediate logic is also presented. It must be noted that there are intermediate predicate logics without Beth's property.Less
This chapter presents the equivalence of Craig's interpolation property to Robinson's joint consistency, and a proof of Lyndon's interpolation theorem for the classical predicate logic. It is proved that the general form of Robinson's consistency property (RCP) fails in the intuitionistic predicate logic HQ. A weaker form of RCP is equivalent to Craig's interpolation property (CIP) and holds in HQ, and a semantic proof is given. It is proved that in propositional intermediate logics the general form of RCP is equivalent to CIP. A derivation of Beth's property from CIP is given for the intuitionistic predicate logic. Kreisel's proof of validity of the Beth property for any propositional intermediate logic is also presented. It must be noted that there are intermediate predicate logics without Beth's property.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0005
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter contains a proof of Lyndon's interpolation property (LIP) for quantified extensions of basic modal logics K, T, D, K4, and S4, and for some others, including the propositional S5 has ...
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This chapter contains a proof of Lyndon's interpolation property (LIP) for quantified extensions of basic modal logics K, T, D, K4, and S4, and for some others, including the propositional S5 has LIP. At the same time, the quantified extension of S5, as well as other systems satisfying the Barcan formula has neither Lyndon's nor Craig's interpolation, nor Beth's property. Some examples of propositional modal logics, which have CIP but do not possess LIP are found. Craig's interpolation property is proved for a number of propositional modal logics, including the Grzegorczyk logic Grz, its extension Grz.2, and the provability logic G. A class of so-called L-conservative formulas is defined, which can be added to a propositional logic L as new axiom schemes without loss of interpolation. It is proved that the interpolation properties are preserved by transfer from predicate logics without equality to their extensions with equality.Less
This chapter contains a proof of Lyndon's interpolation property (LIP) for quantified extensions of basic modal logics K, T, D, K4, and S4, and for some others, including the propositional S5 has LIP. At the same time, the quantified extension of S5, as well as other systems satisfying the Barcan formula has neither Lyndon's nor Craig's interpolation, nor Beth's property. Some examples of propositional modal logics, which have CIP but do not possess LIP are found. Craig's interpolation property is proved for a number of propositional modal logics, including the Grzegorczyk logic Grz, its extension Grz.2, and the provability logic G. A class of so-called L-conservative formulas is defined, which can be added to a propositional logic L as new axiom schemes without loss of interpolation. It is proved that the interpolation properties are preserved by transfer from predicate logics without equality to their extensions with equality.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0008
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
It appears that the behaviour of interpolation over the modal S4 logic is similar to interpolation in superintuitionistic logics. It is shown that all extensions of S4 with interpolation property for ...
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It appears that the behaviour of interpolation over the modal S4 logic is similar to interpolation in superintuitionistic logics. It is shown that all extensions of S4 with interpolation property for deducibility IPD are modal companions of superintuitionistic logics with CIP, but there is an intermediate logic with CIP that has no modal companions with IPD. On the other hand, all modal companions of intermediate logics with CIP have a weaker version of interpolation, which is CIP restricted to those implications, where all occurrences of variables are preceded by necessity symbol. It is proved that there are no more than forty-nine modal logics with IPD in the family of normal extensions of S4; in this list there are twelve logics that have IPD but do not have CIP. All the forty-nine logics are finitely axiomatizable and have the finite model property. It is proved that IPD and CIP problems are deducible over S4, and amalgamation and superamalgamation are base-decidable in varieties of closure algebras.Less
It appears that the behaviour of interpolation over the modal S4 logic is similar to interpolation in superintuitionistic logics. It is shown that all extensions of S4 with interpolation property for deducibility IPD are modal companions of superintuitionistic logics with CIP, but there is an intermediate logic with CIP that has no modal companions with IPD. On the other hand, all modal companions of intermediate logics with CIP have a weaker version of interpolation, which is CIP restricted to those implications, where all occurrences of variables are preceded by necessity symbol. It is proved that there are no more than forty-nine modal logics with IPD in the family of normal extensions of S4; in this list there are twelve logics that have IPD but do not have CIP. All the forty-nine logics are finitely axiomatizable and have the finite model property. It is proved that IPD and CIP problems are deducible over S4, and amalgamation and superamalgamation are base-decidable in varieties of closure algebras.
Dirk U. Pfeiffer, Timothy P. Robinson, Mark Stevenson, Kim B. Stevens, David J. Rogers, and Archie C. A. Clements
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198509882
- eISBN:
- 9780191709128
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509882.003.0006
- Subject:
- Biology, Disease Ecology / Epidemiology
This chapter discusses spatial variation in risk. Epidemiological disease investigations should include an assessment of the spatial variation of disease risk, as this may provide important clues ...
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This chapter discusses spatial variation in risk. Epidemiological disease investigations should include an assessment of the spatial variation of disease risk, as this may provide important clues leading to causal explanations. The objective is to produce a map representation of the important spatial effects present in the data while simultaneously removing any distracting noise or extreme values. The resulting smoothed map should have increased precision without introducing significant bias. The method used to analyse the data depends on how they have been recorded. Smoothing based on kernel functions, smoothing based and on Bayesian models, and spatial interpolation are discussed.Less
This chapter discusses spatial variation in risk. Epidemiological disease investigations should include an assessment of the spatial variation of disease risk, as this may provide important clues leading to causal explanations. The objective is to produce a map representation of the important spatial effects present in the data while simultaneously removing any distracting noise or extreme values. The resulting smoothed map should have increased precision without introducing significant bias. The method used to analyse the data depends on how they have been recorded. Smoothing based on kernel functions, smoothing based and on Bayesian models, and spatial interpolation are discussed.
H. A. G. Houghton
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199545926
- eISBN:
- 9780191719974
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199545926.003.0013
- Subject:
- Religion, Early Christian Studies
The textual commentary treats Augustine's text of John verse by verse, concentrating on variants from the Vulgate. There are a number of occasions on which Augustine has a reading not attested in ...
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The textual commentary treats Augustine's text of John verse by verse, concentrating on variants from the Vulgate. There are a number of occasions on which Augustine has a reading not attested in surviving Latin manuscripts: on some occasions this is shared with other Church Fathers, while others remain unique to him. The distinction between primary and secondary citations is important in determining which readings should be cited in an edition of the Gospel. The form of his mental text is given for a number of verses, with comments on how it may have arisen. Brief details are sometimes given of Augustine's exegesis, with bibliographic references to further studies. Passages of particular interest include John 7:53-8:11 (the Woman taken in Adultery), in which Augustine provides the earliest attestation of several variants in John 8:9. He also has an intriguing treatment of the contested verses John 5:2-4 (the Angel stirring the Waters), and shows evidence for the interpolation at John 3:6. His text of John 21:18 suggests that a Greek variant only preserved in Codex Bezae may underlie the entire Latin tradition.Less
The textual commentary treats Augustine's text of John verse by verse, concentrating on variants from the Vulgate. There are a number of occasions on which Augustine has a reading not attested in surviving Latin manuscripts: on some occasions this is shared with other Church Fathers, while others remain unique to him. The distinction between primary and secondary citations is important in determining which readings should be cited in an edition of the Gospel. The form of his mental text is given for a number of verses, with comments on how it may have arisen. Brief details are sometimes given of Augustine's exegesis, with bibliographic references to further studies. Passages of particular interest include John 7:53-8:11 (the Woman taken in Adultery), in which Augustine provides the earliest attestation of several variants in John 8:9. He also has an intriguing treatment of the contested verses John 5:2-4 (the Angel stirring the Waters), and shows evidence for the interpolation at John 3:6. His text of John 21:18 suggests that a Greek variant only preserved in Codex Bezae may underlie the entire Latin tradition.
Pavol Hell and Jaroslav Nešetřil
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198528173
- eISBN:
- 9780191713644
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528173.003.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This introductory chapter is a sampler of the material covered in the book. It introduces the notation and terminology in the book, and provides motivational examples and applications, many taken up ...
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This introductory chapter is a sampler of the material covered in the book. It introduces the notation and terminology in the book, and provides motivational examples and applications, many taken up in more detail in later chapters. It gives the flavour of combinatorial aspects, algorithmic aspects, retractions, duality, constraint satisfaction problems, as well as structural properties of homomorphism composition. The highlights of this chapter include a simple proof of the Colouring Interpolation Theorem, a generalization of the No-Homomorphism Lemma, the construction of a triangle-free graph to which all cubic triangle-free graphs are homomorphic, a case of the Edge Reconstruction Conjecture, and a generalization of a theorem of Frucht on graphs with prescribed automorphism groups.Less
This introductory chapter is a sampler of the material covered in the book. It introduces the notation and terminology in the book, and provides motivational examples and applications, many taken up in more detail in later chapters. It gives the flavour of combinatorial aspects, algorithmic aspects, retractions, duality, constraint satisfaction problems, as well as structural properties of homomorphism composition. The highlights of this chapter include a simple proof of the Colouring Interpolation Theorem, a generalization of the No-Homomorphism Lemma, the construction of a triangle-free graph to which all cubic triangle-free graphs are homomorphic, a case of the Edge Reconstruction Conjecture, and a generalization of a theorem of Frucht on graphs with prescribed automorphism groups.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0006
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter contains a full description of superintuitionistic logics with Craig's interpolation property CIP. It turns out that in the continuum of intermediate logics, only seven have Craig's ...
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This chapter contains a full description of superintuitionistic logics with Craig's interpolation property CIP. It turns out that in the continuum of intermediate logics, only seven have Craig's interpolation. All of them are finitely axiomatizable and have the finite model property. For the proof, the algebraic semantics via varieties of Heyting algebras is used, and the equivalence of CIP in a logic L to amalgamability of the corresponding variety V(L) is stated. It is also proved that the interpolation problem over the intuitionistic logic Int is decidable: for any finite set Ax of axiom schemes to determine, whether the calculus Int+Ax has CIP; also the amalgamation problem is base-decidable for varieties of Heyting algebras.Less
This chapter contains a full description of superintuitionistic logics with Craig's interpolation property CIP. It turns out that in the continuum of intermediate logics, only seven have Craig's interpolation. All of them are finitely axiomatizable and have the finite model property. For the proof, the algebraic semantics via varieties of Heyting algebras is used, and the equivalence of CIP in a logic L to amalgamability of the corresponding variety V(L) is stated. It is also proved that the interpolation problem over the intuitionistic logic Int is decidable: for any finite set Ax of axiom schemes to determine, whether the calculus Int+Ax has CIP; also the amalgamation problem is base-decidable for varieties of Heyting algebras.
D. A. Bini, G. Latouche, and B. Meini
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198527688
- eISBN:
- 9780191713286
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527688.003.0008
- Subject:
- Mathematics, Numerical Analysis
Alternative numerical approaches for solving matrix equations associated with M/G/1-type Markov chains are considered in this chapter. A general shift technique for accelerating the convergence of ...
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Alternative numerical approaches for solving matrix equations associated with M/G/1-type Markov chains are considered in this chapter. A general shift technique for accelerating the convergence of iterative methods is described, and its application to accelerating cyclic reduction is analysed. A functional iteration relying on the combination of cyclic reduction and fixed point iteration is introduced: its convergence is linear but its convergence rate can be arbitrarily large. A doubling method, evaluation interpolation techniques, and the invariant subspace method complete the chapter.Less
Alternative numerical approaches for solving matrix equations associated with M/G/1-type Markov chains are considered in this chapter. A general shift technique for accelerating the convergence of iterative methods is described, and its application to accelerating cyclic reduction is analysed. A functional iteration relying on the combination of cyclic reduction and fixed point iteration is introduced: its convergence is linear but its convergence rate can be arbitrarily large. A doubling method, evaluation interpolation techniques, and the invariant subspace method complete the chapter.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0010
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter examines the family NE(K) of normal extensions of K4. With any such logic L its reflexive fragment r(L) is associated, which contains the logic S4. A logic L is of infinite slice if ...
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This chapter examines the family NE(K) of normal extensions of K4. With any such logic L its reflexive fragment r(L) is associated, which contains the logic S4. A logic L is of infinite slice if Kripke frames, satisfying L, can contain subchains of any finite length. It is proved that for any logic in NE(K4), being of infinite slice and possessing the interpolation property for deducibility, its reflexive fragment is contained in Grz.2. As a consequence, interpolation theorems fail in logics of infinite slice and of finite width, in logics of finite irreflexive trees, and so on.Less
This chapter examines the family NE(K) of normal extensions of K4. With any such logic L its reflexive fragment r(L) is associated, which contains the logic S4. A logic L is of infinite slice if Kripke frames, satisfying L, can contain subchains of any finite length. It is proved that for any logic in NE(K4), being of infinite slice and possessing the interpolation property for deducibility, its reflexive fragment is contained in Grz.2. As a consequence, interpolation theorems fail in logics of infinite slice and of finite width, in logics of finite irreflexive trees, and so on.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0013
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter presents a syntactic proof of the Lyndon interpolation theorem for the intuitionistic predicate logic. It is a modification of a proof of the interpolation theorem found by K. Schütte, ...
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This chapter presents a syntactic proof of the Lyndon interpolation theorem for the intuitionistic predicate logic. It is a modification of a proof of the interpolation theorem found by K. Schütte, and uses a special sequent calculus adequate for the intuitionistic predicate logic. The proof gives a method for constructing an interpolant from the derivation of a formula. Interpolation in some fragments of the intuitionistic calculi is studied.Less
This chapter presents a syntactic proof of the Lyndon interpolation theorem for the intuitionistic predicate logic. It is a modification of a proof of the interpolation theorem found by K. Schütte, and uses a special sequent calculus adequate for the intuitionistic predicate logic. The proof gives a method for constructing an interpolant from the derivation of a formula. Interpolation in some fragments of the intuitionistic calculi is studied.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0014
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter proposes some uniform algorithmic methodology for finding interpolants in various logic. It operates with translations of non-classical logics into classical first-order theories and ...
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This chapter proposes some uniform algorithmic methodology for finding interpolants in various logic. It operates with translations of non-classical logics into classical first-order theories and introduces so-called expansion interpolation. This leads us to find interpolants in the classical theories using the existing algorithms, which can then be translated back into non-classical theories. Two examples from modal logic are considered: quantified S5 and propositional S4.3. These logic lack ordinary interpolation but have expansion interpolation.Less
This chapter proposes some uniform algorithmic methodology for finding interpolants in various logic. It operates with translations of non-classical logics into classical first-order theories and introduces so-called expansion interpolation. This leads us to find interpolants in the classical theories using the existing algorithms, which can then be translated back into non-classical theories. Two examples from modal logic are considered: quantified S5 and propositional S4.3. These logic lack ordinary interpolation but have expansion interpolation.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0015
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter provides interpolation results for the Horn clause fragment of classical or intuitionistic logic. A weak variant of interpolation for a fragment of the intuitionistic predicate logic ...
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This chapter provides interpolation results for the Horn clause fragment of classical or intuitionistic logic. A weak variant of interpolation for a fragment of the intuitionistic predicate logic without disjunction and existence quantifier is proved by syntactic method. A counter-example to the general form of interpolation is given.Less
This chapter provides interpolation results for the Horn clause fragment of classical or intuitionistic logic. A weak variant of interpolation for a fragment of the intuitionistic predicate logic without disjunction and existence quantifier is proved by syntactic method. A counter-example to the general form of interpolation is given.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0016
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter studies the interpolation properties for implicational fragment of a variety of substructural, strict modal, and intuitionistic and intermediate logics. The methodology is ...
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This chapter studies the interpolation properties for implicational fragment of a variety of substructural, strict modal, and intuitionistic and intermediate logics. The methodology is proof-theoretical and uses a goal directed formulation of these fragments which follows the logic programming style of deduction. More refined as well as new kinds of interpolation theorems are obtained, and new global methods for obtaining interpolation are investigated.Less
This chapter studies the interpolation properties for implicational fragment of a variety of substructural, strict modal, and intuitionistic and intermediate logics. The methodology is proof-theoretical and uses a goal directed formulation of these fragments which follows the logic programming style of deduction. More refined as well as new kinds of interpolation theorems are obtained, and new global methods for obtaining interpolation are investigated.
D.M. Gabbay and L. Maksimova
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198511748
- eISBN:
- 9780191705779
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198511748.003.0017
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter discusses further directions of research. A brief review of the results obtained on interpolation and definability, which were not included in this volume, is presented.
This chapter discusses further directions of research. A brief review of the results obtained on interpolation and definability, which were not included in this volume, is presented.
Jerome Murphy‐O'Connor
- Published in print:
- 2009
- Published Online:
- May 2009
- ISBN:
- 9780199564156
- eISBN:
- 9780191721281
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199564156.003.00016
- Subject:
- Religion, Biblical Studies
After a discussion of the validity of the methodology normally used to determine interpolations, i.e. additions to a text after it had left its author's hands, the chapter passes in review thirteen ...
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After a discussion of the validity of the methodology normally used to determine interpolations, i.e. additions to a text after it had left its author's hands, the chapter passes in review thirteen passages in 1 Cor, which various authors have suggested were interpolations. Only 1 Cor 4: 6 and 1 Cor 14: 34–35 are accepted as post‐Pauline additions. 1 Cor 7: 29–31 is more likely to be Paul's citation of a formed apocalyptic tradition similar to 6 Ezra 16: 41–45 than an interpolation. It is entirely probable that 1 Cor 15: 56 is an embryonic articulation of an insight which Paul developed fully only several years later in writing Romans.Less
After a discussion of the validity of the methodology normally used to determine interpolations, i.e. additions to a text after it had left its author's hands, the chapter passes in review thirteen passages in 1 Cor, which various authors have suggested were interpolations. Only 1 Cor 4: 6 and 1 Cor 14: 34–35 are accepted as post‐Pauline additions. 1 Cor 7: 29–31 is more likely to be Paul's citation of a formed apocalyptic tradition similar to 6 Ezra 16: 41–45 than an interpolation. It is entirely probable that 1 Cor 15: 56 is an embryonic articulation of an insight which Paul developed fully only several years later in writing Romans.
Peter Monk
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198508885
- eISBN:
- 9780191708633
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508885.003.0005
- Subject:
- Mathematics, Numerical Analysis
The finite element method is based on a geometric decomposition of the domain of Maxwell’s equations into simple elements. This chapter is devoted to tetrahedral elements, which are very common in ...
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The finite element method is based on a geometric decomposition of the domain of Maxwell’s equations into simple elements. This chapter is devoted to tetrahedral elements, which are very common in practice. Details of the constructions of scalar and vector finite elements of all orders are presented. The vector elements are due to Nedelec. In particular, the curl-conforming elements of this chapter are the widely used ‘edge-elements’ whereas the corresponding divergence-conforming elements are often termed ‘face elements’ (they are extensions to 3D of the Raviart-Thomas elements). The appropriate conforming and unisolvence properties of the elements are proven, and the important discrete de Rham diagram relating the interpolation operators for these finite elements with the divergence, gradient, and curl operators are verified; this is used heavily in later theory. Interpolation error estimates under mesh refinement are derived (h-version of the finite element method). A convenient basis for linear and quadratic finite elements is presented, and spaces of elements on boundaries of the domain are briefly discussed.Less
The finite element method is based on a geometric decomposition of the domain of Maxwell’s equations into simple elements. This chapter is devoted to tetrahedral elements, which are very common in practice. Details of the constructions of scalar and vector finite elements of all orders are presented. The vector elements are due to Nedelec. In particular, the curl-conforming elements of this chapter are the widely used ‘edge-elements’ whereas the corresponding divergence-conforming elements are often termed ‘face elements’ (they are extensions to 3D of the Raviart-Thomas elements). The appropriate conforming and unisolvence properties of the elements are proven, and the important discrete de Rham diagram relating the interpolation operators for these finite elements with the divergence, gradient, and curl operators are verified; this is used heavily in later theory. Interpolation error estimates under mesh refinement are derived (h-version of the finite element method). A convenient basis for linear and quadratic finite elements is presented, and spaces of elements on boundaries of the domain are briefly discussed.
Peter Monk
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198508885
- eISBN:
- 9780191708633
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508885.003.0006
- Subject:
- Mathematics, Numerical Analysis
An alternative to the tetrahedral elements discussed in the previous chapter is to use finite elements based on cubes, or more generally, hexahedra. Hexahedral elements have been used in several ...
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An alternative to the tetrahedral elements discussed in the previous chapter is to use finite elements based on cubes, or more generally, hexahedra. Hexahedral elements have been used in several important codes. This chapter concerns Nedelec’s family of edge and face elements on a hexahedral mesh with edges parallel to the coordinate axis. Conformance and unisolvence are proven, and h-error estimates are derived. The appropriate discrete de Rham diagram is shown to hold in this case, and boundary spaces are discussed briefly.Less
An alternative to the tetrahedral elements discussed in the previous chapter is to use finite elements based on cubes, or more generally, hexahedra. Hexahedral elements have been used in several important codes. This chapter concerns Nedelec’s family of edge and face elements on a hexahedral mesh with edges parallel to the coordinate axis. Conformance and unisolvence are proven, and h-error estimates are derived. The appropriate discrete de Rham diagram is shown to hold in this case, and boundary spaces are discussed briefly.
Jan Modersitzki
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528418
- eISBN:
- 9780191713583
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528418.003.0003
- Subject:
- Mathematics, Applied Mathematics
The basic mathematical notation is introduced, including formal definitions of images and digital images, midpoint and meshpoint grids. Local and global interpolation techniques such as ...
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The basic mathematical notation is introduced, including formal definitions of images and digital images, midpoint and meshpoint grids. Local and global interpolation techniques such as next-neighbour, d-linear, spline, sinc, and wavelet interpolations are discussed. For the functional setting, the square-integrable functions, point evaluation functionals, and various derivative operators are formally introduced. A general setting for image transformations is presented. Using this setting, a formal description of the image registration problem in continuous space is given. Restricted transformations like rigid, affine-linear, polynomial, and B-spline transformations are examined, and the Lagrangian and Eulerian frames for template transformations are discussed.Less
The basic mathematical notation is introduced, including formal definitions of images and digital images, midpoint and meshpoint grids. Local and global interpolation techniques such as next-neighbour, d-linear, spline, sinc, and wavelet interpolations are discussed. For the functional setting, the square-integrable functions, point evaluation functionals, and various derivative operators are formally introduced. A general setting for image transformations is presented. Using this setting, a formal description of the image registration problem in continuous space is given. Restricted transformations like rigid, affine-linear, polynomial, and B-spline transformations are examined, and the Lagrangian and Eulerian frames for template transformations are discussed.
Andrea Braides
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198507840
- eISBN:
- 9780191709890
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507840.003.0012
- Subject:
- Mathematics, Applied Mathematics
This chapter shows how general non-convex difference schemes can give rise to functionals defined by piecewise-Sobolev functions with interactions between surface and volume terms. Different ...
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This chapter shows how general non-convex difference schemes can give rise to functionals defined by piecewise-Sobolev functions with interactions between surface and volume terms. Different identification of discrete schemes with energies on suitable piecewise-Sobolev interpolations are given, leading to a continuous limit energy. As examples, softening and fracture problems with size effects are obtained as limits of convex/concave discrete energies; fracture is described as a phase-transition phenomenon starting from Lennard-Jones potentials; and the Malik-Perona approximation of free-discontinuity problems is considered.Less
This chapter shows how general non-convex difference schemes can give rise to functionals defined by piecewise-Sobolev functions with interactions between surface and volume terms. Different identification of discrete schemes with energies on suitable piecewise-Sobolev interpolations are given, leading to a continuous limit energy. As examples, softening and fracture problems with size effects are obtained as limits of convex/concave discrete energies; fracture is described as a phase-transition phenomenon starting from Lennard-Jones potentials; and the Malik-Perona approximation of free-discontinuity problems is considered.
