bluuebunny/crossref_metadata_2025_split huggingface dataset:
Formal topologies on the set of first-order formulae
Authors: Coquand Thierry , Sadocco Sara , Sambin Giovanni , Smith Jan M. 
Published: May 2006

The completeness proof for first-order logic by Rasiowa and Sikorski [13] is a simplification of Henkin's proof [7] in that it avoids the addition of infinitely many new individual constants. Instead they show that each consistent set of formulae can be extended to a maximally consistent set, satisfying the following existence property: if it contains (∃x)ϕit also contains some substitutionϕ(y/x) of a variableyforx. In Feferman's review [5] of [13], an improvement, due to Tarski, is given by which the proof gets a simple algebraic form.Sambin [16] used the same method in the setting of formal topology [15], thereby obtaining a constructive completeness proof. This proof is elementary and can be seen as a constructive and predicative version of the one in Feferman's review. It is a typical, and simple, example where the use of formal topology gives constructive sense to the existence of a generic object, satisfying some forcing conditions; in this case an ultrafilter satisfying the existence property.In order to get a formal topology on the set of first-order formulae, Sambin used the Dedekind-MacNeille completion to define a covering relation ⊲DM. This method, by which an arbitrary poset can be extended to a complete poset, was introduced by MacNeille [9] and is a generalization of the construction of real numbers from rationals by Dedekind cuts. It is also possible to define an inductive cover, ⊲I, on the set of formulae, which can also be used to give canonical models, see Coquand and Smith [3].
Computably categorical structures and expansions by constants
Authors: Cholak Peter , Goncharov Sergey , Khoussainov Bakhadyr , Shore Richard A. 
Published: May 2006

Effective model theory is the subject that analyzes the typical notions and results of model theory to determine their effective content and counterparts. The subject has been developed both in the former Soviet Union and in the west with various names (recursive model theory, constructive model theory, etc.) and divergent terminology. (We use “effective model theory” as the most general and descriptive designation. Harizanov [6] is an excellent introduction to the subject as is Millar [13].) The basic subjects of model theory include languages, structures, theories, models and various types of maps between these objects. There are many ways to introduce considerations of effectiveness into the area. The two most prominent derive from starting, on the one hand, with the notion of a theory and its models or, on the other, with just structures.If one begins with theories, then a natural version of effectiveness is to consider decidable theories (i.e., ones with a decidable (equivalently, computable or recursive) set of theorems). When one moves to models and wants them to be effective, one might start with the requirement that the model (of any theory) have a decidable theory (i.e., Th (), the set of sentences true in , is decidable). Typically, however, one wants to be able to talk about the elements of the model as well as its theory in the given language. Thus one naturally considers the model as a structure for the language expanded by adding a constant ai, for each element ai of . Of course, one requires that the mapping from the constants to the corresponding elements of be effective (computable). We are thus lead to the following basic definition:A structure or model is decidable if there is a computable enumeration ai of A, the domain of , such that Th(, ai,) is decidable. (Of course, ai, is interpreted as ai, for each i Є ω.)
Lowness for the class of random sets
Authors: Kučera Antonín , Terwijn Sebastiaan A. 
Published: May 2006

AbstractA positive answer to a question of M. van Lambalgen and D. Zambella whether there exist nonrecursive sets that are low for the class of random sets is obtained. Here a set A is low for the class RAND of random sets if RAND = RANDA.
Solving Pell equations locally in models of IΔ0
Authors: D'Aquino Paola 
Published: May 2006

AbstractIn [4] it is shown that only using exponentiation can one prove the existence of non trivial solutions of Pell equations in IΔ0. However, in this paper we will prove that any Pell equation has a non trivial solution modulo m for every m in IΔ0.
Paires de structures O-minimales
Authors: Baisalov Yerzhan , Poizat Bruno 
Published: May 2006

ResumoNi montras propecon de eljeteco de la kvantoro (∃y ∈ M) pri la (sufice) belaj paroj de modeloj de una O-plimalpova teorio. Gi havas korolaron ke, se ni aldonas malkavajn unarajn predikatojn a la lingvo de kelka O-plimalpova strukturo, ni ricevas malforte O-plimalpovan strukturon. Tui ci rezultato estis en speciala kaso pruvita de [5], kaj la gia generalizeco estis anoncita en [1].
Continuum cardinals generalized to Boolean algebras
Authors: Monk J. Donald 
Published: May 2006

A number of specific cardinal numbers have been defined in terms of /fin or ωω. Some have been generalized to higher cardinals, and some even to arbitrary Boolean algebras. Here we study eight of these cardinals, defining their generalizations to higher cardinals, and then defining them for Boolean algebras. We then attempt to completely describe their relationships within each of several important classes of Boolean algebras.The generalizations to higher cardinals might involve several cardinals instead of just one as in the case of ω, For example, the number a associated with maximal almost disjoint families of infinite sets of integers can be generalized to talk about maximal subsets of [κ]μ subject to the pairwise intersections having size less than ν. (For this multiple generalization of . see Monk [2001].) For brevity we do not consider such generalizations, restricting ourselves to just one cardinal. The set-theoretic generalizations then associate with each infinite cardinal κ some other cardinal λ, defined as the minimum of cardinals with a certain property.The generalizations to Boolean algebras assign to each Boolean algebra some cardinal λ, also defined as the minimum of cardinals with a certain property.For the theory of the original “continuum” cardinal numbers, see Douwen [1984]. Balcar and Simon [1989]. and Vaughan [1990].I am grateful to Mati Rubin for some conversations concerning these functions for superatomic algebras, and to Bohuslav Balcar for information concerning the function h.The notation for set theory is standard. For Boolean algebras we follow Koppelberg [1989], but recall at the appropriate place any somewhat unusual notation.
Some dichotomy theorems for isomorphism relations of countable models
Authors: Gao Su 
Published: May 2006

AbstractStrengthening known instances of Vaught Conjecture, we prove the Glimm-Effros dichotomy theorems for countable linear orderings and for simple trees. Corollaries of the theorems answer some open questions of Friedman and Stanley in an Lω1ω-interpretability theory. We also give a survey of this theory.
Computable Boolean algebras
Authors: Knight Julia F. , Stob Michael 
Published: May 2006

Feiner [F] showed that a Boolean algebra need not have a computable copy (see also [T2]). Downey and Jockusch [D-J] showed that every low Boolean algebra does have a computable copy. Thurber [T3], showed that every low2 Boolean algebra has a computable copy. Here we show that every Boolean algebra which is low3, or even low4, has a computable copy.The results of [D-J] and [T3] were obtained by passing to linear orderings. In [D-J], there is an embedding theorem saying that any linear ordering which is with the successor relation as an added predicate can be embedded in a slightly larger linear ordering which is computable. An isomorphism theorem of Remmel [R] is used to show that the interval algebras of the two linear orderings are isomorphic (except in a trivial case). In [T3], there is an embedding theorem saying that any linear ordering which is with certain added predicates can be embedded in one which is with successor. Again the isomorphism theorem of Remmel is used to show that the interval algebras are isomorphic (except in a trivial case).Here, instead of passing to linear orderings, we work directly with Boolean algebras. We begin with a review of the known results. We re-formulate the embedding theorems of Downey-Jockusch and Thurber in terms of Boolean algebras. We extract from Remmel's isomorphism theorem some information on complexity. In this way, we show that a low Boolean algebra is isomorphic to a computable one by an isomorphism which is , at worst, and the same is true for a low2 Boolean algebra.
Number of variables is equivalent to space
Authors: Immerman Neil , Buss Jonathan F. , Barrington David A. Mix 
Published: May 2006

AbstractWe prove that the set of properties describable by a uniform sequence of first-order sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACE[nk] (where n is the size of the universe). This set is also equal to the set of properties describable using an iterative definition for a finite set of relations of arity k. This is a refinement of the theorem PSPACE = VAR[O[1]] [8]. We suggest some directions for exploiting this result to derive trade-offs between the number of variables and the quantifier depth in descriptive complexity.
On computable automorphisms of the rational numbers
Authors: Morozov A. S. , Truss J. K. 
Published: May 2006

AbstractThe relationship between ideals I of Turing degrees and groups of I-recursive automorphisms of the ordering on rationals is studied. We discuss the differences between such groups and the group of all automorphisms, prove that the isomorphism type of such a group completely defines the ideal I, and outline a general correspondence between principal ideals of Turing degrees and the first-order properties of such groups.
Scientific Research and State Government
Authors: White L. D. 
Published: May 2006

The operation of the newly devised systems of centralized control in state government has naturally raised many questions of the relative duties and responsibilities of department and bureau officials on the one hand and of the supervising and controlling authorities on the other. The process of change from an established method of conducting public business to another, based on different principles and requiring new obligations on the part of officers expending money is bound to be a period of readjustment accompanied by a certain amount of friction. This will be the more certain as the new system differs more widely from the old and restricts more completely the freedom of action formerly reserved to heads of the various state services.The establishment of the state board of public affairs in Wisconsin in 1911, the enactment of the Civil Code in Illinois in 1917, the enactment of the Ohio Civil Code in 1921, and the legislation of Massachusetts culminating in the creation of the board of administration and finance, all made wide departures from the system of administration hitherto in operation. The essential characteristic introduced in each case was the imposition of an agency of review, of supervision, and in many respects of control upon officers and commissions who had hitherto conducted the work of their respective offices for the most part subject only to the necessity of securing approval of their expenditures by the auditor or controller.
Languages in which self reference is possible
Authors: Smullyan Raymond M. 
Published: May 2006

This paper treats of semantical systems S of sufficient strength so that for any set W definable in S (in a sense which will be made precise), there must exist a sentence X which is true in S if and only if it is an element of W. We call such an X a Tarski sentence for W. It is the sentence which (in a purely extensional sense) says of itself that it is in W. If W is the set of all expressions not provable in some syntactical system C, then X is the Gödel sentence which is true (in S) if and only if it is not provable (in C). We provide a novel method for the construction of these sentences, which yields sentences particularly simple in structure. The method is applicable to a variety of systems, including a form of elementary arithmetic, and some systems of protosyntax self applied. In application to the former, we obtain an extremely simple and direct proof of a theorem, which is essentially Tarski's theorem that the truth set of elementary arithmetic is not arithmetically definable.The crux of our method is in the use of a certain function, the ‘norm’ function, which replaces the classical use of the diagonal function. To give a heuristic idea of the norm function, let us define the norm of an expression E (of informal English) as E followed by its own quotation.
Generalization of a lemma of G. F. Rose
Authors: Gál I. L. , Rosser J. B. , Scott D. 
Published: May 2006

In attempting to reconstruct Rose's proof of Lemma 3.2 of [1], the present authors found what is apparently a different and simpler method, which moreover leads to a far stronger conclusion.We are operating in the Heyting prepositional calculus as formulated on p. 3 of [1] or on pp. 82 and 101 of [2], and shall make use of relevant theorems on pp. 90, 113–119 of [2]. We shall use a, b, c, w, x, y, z as propositional variables.We say that a conjunction is simple if each factor has one of the forms: (i) a, (ii) ¬a, (iii) a⊃b, (iv) a⊃(b∨c), (v) (a&b)⊃c, (vi) (a⊃b)⊃c.
Functional completeness and canonical forms in many-valued logics1
Authors: Jobe William H. 
Published: May 2006

This paper examines the questions of functional completeness and canonical completeness in many-valued logics, offering proofs for several theorems on these topics.A skeletal description of the domain for these theorems is as follows. We are concerned with a proper logic L, containing a denumerably infinite class of propositional symbols, P, Q, R, …, a finite set of unary operations, U1, U2,…, Ub, and a finite set of binary operations, B1, B2, …, Bc. Well-formed formulas in L are recursively defined by the conventional set of rules. With L there is associated an integer, M ≧ 2, and the integers m, where (1 ≦m≦M), are the truth values of L.
Alexander Helios and Caesarion
Authors: Meiklejohn K. W. 
Published: May 2006

Dr. W. W. Tarn in his article in JRS xxii, 135 ff., has convincingly argued that in 37 B.C. Antony re-named his two children by Cleopatra, with the intention of using the name Alexander Helios in particular to attract the sentiment and religious enthusiasm of the East. His article is not concerned with the result of this propaganda, and the present note is an attempt to consider the literary evidence for its success or failure.In the first place we should expect that, if Antony wished to give prominence to Alexander Helios and Cleopatra Selene, he would be forced to minimise to some extent the claims of other children of himself or of Cleopatra. Of these children Caesarion is by far the most important. As early as 42 Cleopatra had been given permission by the Triumvirs to proclaim him King of Egypt, in return for the help she had sent to Dolabella (Dio xlvii, 31, 5). He continued to retain the title until his death, and there seems to have been no attempt to subordinate him to the other children : if anything, his position becomes more important as he grows older.
The First Age of Roman Coinage
Authors: Mattingly H. 
Published: May 2006

A paper by me on the same subject appeared in the Journal as long ago as 1929 (pp. 19 ff.). The criticism of the traditional datings there suggested seems in the main to have stood the test of time. But reconstruction should follow in due course on demolition; and, although no final scheme can yet be proposed, already it should not be premature to formulate some results to which research is pointing. Before we can produce a perfect figure we need a rough casting on which to work.The earliest Roman tradition about coinage is of a very mixed character. A number of references to Roman coins—scattered at intervals over the period of the Kings and the Early Republic—may be dismissed as later intrusions. The golden prime of the Etruscan Kings will surely have known the precious metals—perhaps actual coins of foreign cities; but it has no coins of its own to show. A more trustworthy tradition tells of a form of reckoning in oxen and sheep—the ‘pecunia’ (from ‘pecus’, ‘flock’), which became the Latin word for money—only giving place officially to the currency of uncoined bronze (‘Aes Rude’) in the age of the Decemvirs.
The Introduction of Hoplite Tactics at Rome: its date and its consequences
Authors: Nilsson Martin P. 
Published: May 2006

That the close connection between military organization and state-organization has received little attention in this country is less to be wondered at, because this country is an exception to the rule. Being protected by its insular situation, it has had no need of a large standing army within its boundaries. On the Continent the connection appears clearly. Feudal organization was bound up with the art of fighting by knights, and was upset as much by the invention of guns and gunpowder as by economic development: monarchy was supported by mercenary armies, and the men who introduced compulsory service have more than others contributed to the founding of democracy.But I leave it to others to judge modern history according to their own opinions, and turn to ancient history, which I know better and where the connection is more obvious, because warfare was an almost constant occupation in ancient states and military service a duty imposed upon the citizens.
Legionary Pottery from Vindonissa
Authors: Ettlinger Elisabeth 
Published: May 2006

Vindonissa, the only Roman legionary fortress in Switzerland, was occupied during the first century A.D. to guard the frontier against Germany. Because of the exactly defined period of habitation the pottery of this place is of special importance.The legionary fortress at Vindonissa is situated on a plateau between two rivers and on the north side the ground slopes steeply down to the R. Aare. Some time after the foundation of the fortress, which is probably to be dated shortly after the Varus catastrophe of A.D. 9, perhaps after the third decade of the first century, the soldiers began to throw all the rubbish of the fortress down this slope.
The Fall of the Capitol Again: Tacitus, Ann. II. 23
Authors: Skutsch Otto 
Published: May 2006

In JRS 43 (1953), 77 f. I stated that in a number of passages Silius Italicus (1. 625 f.; 4. 150 f.; 6. 555 f.) seems to attest a version of history according to which not only the city but the Capitol also was taken by the Gauls. I therefore suggested that the same version I should be recognized in Ennius, Ann. 164 f.:qua Galli furtim noctu summa arcis adortimoenia concubia uigilesque repente cruentant.
BOOK REVIEWS
Authors: Beatty Susan W. 
Published: May 2006

Book reviewed in this issue.Mielke, Howard W. 1989. Patterns of Life: Biogeography of a Changing World. Unwin Hyman, Inc., Winchester, MA, USA. 370 pp. ISBN 0–04–574032–1 hb, 0–04–574033 X pb. Price unknown.
Plant strategy types and vegetation development reflecting different forms of vineyard management1
Authors: Wilmanns O. 
Published: May 2006

Abstract. Phytosociological investigations of the undergrowth in vineyards in the southern part of the Upper Rhine region, southwestern Germany, over many years are briefly reported. Besides my own data collected over nearly a decade, appropriate releves of earlier authors have been included for comparison.The cultivation regimes have changed more drastically in the course of time ‐ and continue to do so ‐ than they have in other types of agricultural types. The principal cultivation methods are: mechanical soil movement by tilling, herbicide application, and mulching (see Fig. 1). These three types of measures tend to be applied in alternation, both temporally and spatially. Their effects are very different in intensity and time of action, and consequently they cause varying species combinations to build plant communities (Table 1).The concept of strategy‐type is discussed and the approach of ‘key characters’ in the sense of Grubb is stressed. Such traits of adaptive value for important species in the vineyard undergrowth are presented in Table 2, for the aspects life form, developmental rhythm, and diaspore formation and dispersal. The combination of such population biological parameters yields a number of strategy‐types.In the community description the concept of agroform is used, i.e. a subtype of a plant community brought about by a special agricultural technique. Frequent soil movement results in the dominance of therophytes: the Stellaria media‐ ‘agroform’ of the Geranio rotundifolii‐Allietum Tx. 50; mulching promotes perennials with runners, leading to a Poa trivialis‐Lolio‐Potentillion community (Wilmanns 1989); moderate spraying with herbicides promotes geophytes, winter annuals, woody plants and succulents: Bromus sterilis‐ agroform of the Geranio‐Allietum. As examples of characteristic species, Allium vineale and Geranium rotundifolium are analysed.
Leaf‐nutrient accumulation and turnover at three stages of succession from heathland to forest
Authors: Rode Michael W. 
Published: May 2006

Abstract. Accumulation of nutrients in leaves of the dominating species of three ecosystems, characterizing the secondary succession from Genisto‐Callunetum heathland through Leucobryo‐Pinetum birch‐pine woodland to mature Querco‐Fagetum oak‐beech forest, as well as nutrient turnover within these ecosystems was studied. The objective of the study was to establish potential variations in quantity and quality of nutrient supply to the plants with respect to succession dynamics. The results show very low leaf nutrient concentrations of all species investigated, coinciding with low nutrient availability in the soil. However, the nutrient content of leaves and leaf litter of Quercus petraea and Fagus sylvatica, which dominate the late succession stages, and in Betulapéndula are higher than in the photosynthetic organs (leaves and young shoots) of Calluna vulgaris and Pinus sylvestris.The combination of the higher nutrient content of the leaves and an increasing leaf‐litter production during succession results in an increased nutrient turnover via leaf‐litter fall. However, due to the high leaf biomass, the storage of nutrients in the leaf biomass is highest within the birch‐pine woodland. From this, it may be assumed that the low demand and the low loss of nutrients via leaf‐litter fall are favourable for Pinus at the early stages of forest succession on poor sandy soils. In contrast, Quercus and Fagus are provided with better growth conditions at the later stages of succession resulting from the accumulation of plant‐available nutrients in the ecosystem by Pinus sylvestris, combined with a higher nutrient turnover as compared with the heathland.