bluuebunny/crossref_metadata_2025 huggingface dataset:
On countable fractions from an elementary class
Authors: Ash C. J.
Published: May 2006

The following fairly elementary result seems to raise possibilities for the study of countable models of a theory in a countable language. For the terminology of model theory we refer to [CK].

Let L be a countable first-order language. Let L′ be the relational language having, for each formula φ of L and each sequence υ1,…,υn of variables including the free variables of φ, an n-ary relation symbol Pφ. For any L-structure and any formula Ψ(υ) of L, we define the Ψ-fraction of to be the L′-structure Ψ whose universe consists of those elements of satisfying Ψ(υ) and whose relations {Rφ}φϵL are defined by letting a1,…,an satisfy Rφ in Ψ if, and only if, a1,…, an satisfy φ in .

An L-elementary class means the class of all L-structures satisfying each of some set of sentences of L. The countable part of an L-elementary class K means the class of all countable L-structures from K.

Theorem. Let K be an L-elementary class and let Ψ(υ) be a formula of L. Then the class of countable Ψ-fractions of structures in K is the countable part of some L′-elementary class.

Comment. By the downward Löwenheim-Skolem theorem, the countable Ψ-fractions of structures in K are the same as the Ψ-fractions of countable structures in K.

Proof. We give a set Σ′ of L′-sentences whose countable models are exactly the countable Ψ-fractions of structures in K.
More on imaginaries in p-adic fields
Authors: Scowcroft Philip
Published: May 2006

According to [4, p. 1154], a complete L-theory T eliminates imaginaries just in case for every L-formula φ(x1,… , xm, y1, …, yn), every model M of T, and every ā Є Mn, there is a subset A of M's domain with the following property: if N ≽ M and f is an automorphism of N, then

if and only if

Among the several equivalent conditions discussed in [4, p. 1155], one may single out the following: if T is a complete theory in which two distinct objects are definable, T eliminates imaginaries just in case every T-definable n-ary equivalence relation may be defined by a formula

where g is a T-definable n-ary function taking k-tuples as values (for some natural number k).

Say that an L-structure M eliminates imaginaries just in case Th(M) does. If L is the language of rings with unit, [4, p. 1158] shows that any algebraically closed field eliminates imaginaries, and [2, p. 629] points out that any real-closed field eliminates imaginaries.
Une correspondance entre anneaux partiels et groupes
Authors: Simonetta Patrick
Published: May 2006

This work is inspired by the correspondence of Malcev between rings and groups. Let A be a domain with unit, and S a multiplicative group of invertible elements. We define AS as the structure obtained from A by restraining the multiplication to A × S, and σ(AS) as the group of functions from A to A of the form x → xa + b, where (a, b) belongs to S × A. We show that AS and σ(As) are interpretable in each other, and then, that we can transfer some properties between classes (or theories) of “reduced” domains and corresponding groups, such as being elementary, axiomatisability (for classes), decidability, completeness, or, in some cases, existence of a model-completion (for theories).

We study the extensions of the additive group of A by the group S, acting by right multiplication, and show that sometimes σ(AS) is the unique extension of this type. We also give conditions allowing us to eliminate parameters appearing in interpretations.

We emphasize the case where the domain is a division ring K and S is its multiplicative group K×. Here, the interpretations can always be done without parameters. If the centre of K contains more than two elements, then σ(K) is the only extension of the additive group of K by its multiplicative group acting by right multiplication, and the class of all such σ(K)'s is elementary and finitely axiomatisable. We give, in particular, an axiomatisation for this class and for the class of σ(K)'s where K is an algebraically closed field of characteristic 0. From these results it follows that some classical model-companion results about theories of fields can be translated and restated as results about theories of solvable groups of class 2.
Identities on cardinals less than ℵω
Authors: Gilchrist M. , Shelah S.
Published: May 2006

Let κ be an uncountable cardinal and the edges of a complete graph with κ vertices be colored with ℵ0 colors. For the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if , then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when the edges of the complete graph on vertices are colored with ℵ0 colors. In particular, we are concerned with the case ℵ1 ≤ κ ≤ ℵω.

The study of these color patterns (known as identities) has a history that involves the existence of compactness theorems for two cardinal models [2]. When the graph being colored has size ℵ1, the identities that must occur have been classified by Shelah [4]. If the graph has size greater than or equal to ℵω the identities have also been classified in [3]. The number of colors is fixed at ℵ0 as it is the natural place to start and the results here can be generalized to situations where more colors are used.

There is one difference that we now make explicit. When countably many colors are used we can define the following coloring of the complete graph on vertices. First consider the branches in the complete binary tree of height ω to be vertices of a complete graph.
Relational and partial variable sets and basic predicate logic
Authors: Ghilardi Silvio , Meloni Giancarlo
Published: May 2006

In this paper we study the logic of relational and partial variable sets, seen as a generalization of set-valued presheaves, allowing transition functions to be arbitrary relations or arbitrary partial functions. We find that such a logic is the usual intuitionistic and co-intuitionistic first order logic without Beck and Frobenius conditions relative to quantifiers along arbitrary terms. The important case of partial variable sets is axiomatizable by means of the substitutivity schema for equality. Furthermore, completeness, incompleteness and independence results are obtained for different kinds of Beck and Frobenius conditions.
The Π20 enumeration degrees are not dense
Authors: Calhoun William C. , Slaman Theodore A.
Published: May 2006

We show that the Π20 enumeration degrees are not dense. This answers a question posed by Cooper.
A functorial property of the Aczel-Buchholz-Feferman function
Authors: Weiermann Andreas
Published: May 2006

Let Ω be the least uncountable ordinal. Let be the category where the objects are the countable ordinals and where the morphisms are the strictly monotonic increasing functions. A dilator is a functor on which preserves direct limits and pullbacks. Let τ < ΩE ≔ min{ξ > Ω: ξ = ωξ}. Then τ has a unique “term”-representation in Ω. λξη.ωξ + η and countable ordinals called the constituents of τ. Let δ < Ω and K(τ) be the set of the constituents of τ. Let β = max K(τ). Let [β] be an occurrence of β in τ such that τ[β] = τ. Let be the fixed point-free version of the binary Aczel-Buchholz-Feferman-function (which is defined explicitly in the text below) which generates the Bachman-hierarchy of ordinals. It is shown by elementary calculations that is a dilator for every γ > max{β.δ.ω}.
Constructing strongly equivalent nonisomorphic models for unsuperstable theories, Part C
Authors: Hyttinen Tapani , Shelah Saharon
Published: May 2006

In this paper we prove a strong nonstructure theorem for κ(T)-saturated models of a stable theory T with dop. This paper continues the work started in [1].
Rational Pavelka predicate logic is a conservative extension of Łukasiewicz predicate logic
Authors: Hájek Petr , Paris Jeff , Shepherdson John
Published: May 2006

Rational Pavelka logic extends Łukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0. 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Łukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.
Multiplicative conjunction and an algebraic meaning of contraction and weakening
Authors: Avron A.
Published: May 2006

We show that the elimination rule for the multiplicative (or intensional) conjunction Λ is admissible in many important multiplicative substructural logics. These include LLm (the multiplicative fragment of Linear Logic) and RMIm (the system obtained from LLm by adding the contraction axiom and its converse, the mingle axiom.) An exception is Rm (the intensional fragment of the relevance logic R, which is LLm together with the contraction axiom). Let SLLm and SRm be, respectively, the systems which are obtained from LLm and Rm by adding this rule as a new rule of inference. The set of theorems of SRm is a proper extension of that of Rm, but a proper subset of the set of theorems of RMIm. Hence it still has the variable-sharing property. SRm has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLLm relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLLm corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element (in the first case we get the system BCKm, in the second - the system SRm). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts.
Dominating and unbounded free sets
Authors: Solecki Slawomir , Spinas Otmar
Published: May 2006

We prove that every analytic set in ωω × ωω with σ-bounded sections has a not σ-bounded closed free set. We show that this result is sharp. There exists a closed set with bounded sections which has no dominating analytic free set. and there exists a closed set with non-dominating sections which does not have a not σ-bounded analytic free set. Under projective determinacy analytic can be replaced in the above results by projective.
Cardinal preserving ideals
Authors: Gitik Moti , Shelah Saharon
Published: May 2006

We give some general criteria, when κ-complete forcing preserves largeness properties—like κ-presaturation of normal ideals on λ (even when they concentrate on small cofinalities). Then we quite accurately obtain the consistency strength “NSλ is αi-preserving”, for λ > α2.
Displaying the modal logic of consistency
Authors: Wansing Heinrich
Published: May 2006

It is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.
A minimal counterexample to universal baireness
Authors: Hauser Kai
Published: May 2006

For a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.
Standardization principle of nonstandard universes
Authors: Murakami Masahiko
Published: May 2006

A bounded ultrasheaf is a nonstandard universe constructed from a superstructure in a Boolean valued model of set theory. We consider the bounded elementary embeddings between bounded ultrasheaves. Then the standardization principle is true if and only if the ultrafilters are comparable by the Rudin-Frolik order. The base concept is that the bounded elementary embeddings correspond to the complete Boolean homomorphisms. We represent this by the Rudin-Keisler order of ultrafilters of Boolean algebras.
Sequentially continuous linear mappings in constructive analysis
Authors: Bridges Douglas , Mines Ray
Published: May 2006

A mapping u: X → Y between metric spaces is sequentially continuous if for each sequence (xn) converging to x ∈ X, (u(xn)) converges to u(x). It is well known in classical mathematics that a sequentially continuous mapping between metric spaces is continuous; but, as all proofs of this result involve the law of excluded middle, there appears to be a constructive distinction between sequential continuity and continuity. Although this distinction is worth exploring in its own right, there is another reason why sequential continuity is interesting to the constructive mathematician: Ishihara [8] has a version of Banach's inverse mapping theorem in functional analysis that involves the sequential continuity, rather than continuity, of the linear mappings; if this result could be upgraded by deleting the word “sequential”, then we could prove constructively the standard versions of the inverse mapping theorem and the closed graph theorem.

Troelstra [9] showed that in Brouwer's intuitionistic mathematics (INT) a sequentially continuous mapping on a separable metric space is continuous. On the other hand, Ishihara [6, 7] proved constructively that the continuity of sequentially continuous mappings on a separable metric space is equivalent to a certain boundedness principle for subsets of ℕ; in the same paper, he showed that the latter principle holds within the recursive constructive mathematics (RUSS) of the Markov School. Since it is not known whether that principle holds within Bishop's constructive mathematics (BISH), of which INT and RUSS are models and which can be regarded as the constructive core of mathematics, the exploration of sequential continuity within BISH holds some interest.
A remark on Martin's Conjecture
Authors: Gao Su
Published: May 2006

We prove that the strong Martin conjecture is false. The counterexample is the first-order theory of infinite atomic Boolean algebras. We show that for this class of Boolean algebras, the classification of their (ω + ω)-elementary theories can be reduced to the classification of the elementary theories of their quotient algrbras modulo the Frechét ideals.
The canary tree revisited
Authors: Hyttinen Tapani , Rautila Mika
Published: May 2006

We generalize the result of Mekler and Shelah [3] that the existence of a canary tree is independent of ZFC + GCH to uncountable regular cardinals. We also correct an error from the original proof.
The weak square property
Authors: Jackson Steve
Published: May 2006

We formulate and prove a combinatorial property assuming AD + V = L(ℝ). As a consequence, we show that every regular κ which is either a Suslin cardinal or the successor of a Suslin cardinal is -supercompact. In particular, all the projective ordinals are -supercompact.
Inconsistency of the Axiom of Choice with the positive theory
Authors: Esser Olivier
Published: May 2006

The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without “too much” negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory .
Constitutional Law in 1923–1924: The Constitutional Decisions of the Supreme Court of the United States in the October Term, 1923
Authors: Cushman Robert E.
Published: May 2006

No one who has followed the steady expansion of federal authority over the business of interstate carriers sanctioned by the Supreme Court in the Shreveport Case, Illinois Central R. Co. v. Public Utilities Commission, and Railroad Commission v. Chicago, B. & Q. R. Co., will view with surprise the unanimous decision of that tribunal in Dayton-Goose Creek R. Co. v. United States, sustaining the validity of the “recapture” clause of the Transportation Act of 1920. This clause provided in substance that since it is impossible to establish uniform rates upon competitive traffic which will adequately sustain all the carriers needed to do the business, without giving some of them a net income in excess of a fair return, any carrier receiving such excess shall place one-half of it in a reserve fund to be maintained by the carrier for certain specified purposes, and shall pay the other half into a general railroad revolving fund to be maintained in the interstate commerce commission. This fund is to be used by the commission to make loans to carriers to meet expenditures for capital account, and so forth.
Home Rule in New York State
Authors: McGoldrick Joseph
Published: May 2006

Home rule in the United States has for many years been confined to the states west of the Mississippi. As late as 1920 Ohio and Michigan were the only eastern states giving their cities autonomy. Since that date, however, there has appeared a tendency in the larger urban industrial states to treat cities more generously. New York in 1923 and Pennsylvania in 1924 have now joined the ranks of home-rule states. But many of the largest states, for example, Illinois, still give no such power to their municipalities, while in Massachusetts it can still be said that cities have the same legal status as “an infant, an idiot or a lunatic.”

The addition of New York to this group has just been rendered certain. On September 2, the court of appeals of the state rendered a unanimous decision declaring the home-rule amendment to have been validly adopted. This ended two months of uncertainty caused by the decision of the appellate division of the supreme court in New York City that the home-rule amendment was not validly a part of the constitution and that all state and local legislation passed under its aegis was void.
First and Second Sessions of the Sixty-Eighth Congress: December 3, 1923 to June 7, 1924; December 1, 1924 to March 4, 1925
Authors: Rogers Lindsay
Published: May 2006

The Sixty-eighth Congress had only two sessions. Not since the Sixty-fourth Congress (1915–17) had members enjoyed such lengthy vacations, for, of the Congresses since 1900, seven have had special sessions. This now seems to be the rule rather than the exception, so numerous and technical are the problems that confront the national legislature.

The President and Congress. Four months before the Congress met, Mr. Harding died and Mr. Coolidge succeeded to the Presidency. The first session ran its course under the shadow of the approaching Presidential election.
Oases in the Desert: Hannah Arendt on Democratic Politics
Authors: Isaac Jeffrey C.
Published: May 2006

Hannah Arendt never wrote systematically on the subject of democracy. In her book of greatest relevance to the subject,On Revolution, she criticized liberal democracy, and defended a conception of virtuous political “elites,” leading most commentators to view her as an opponent of democracy. I argue that Arendt defended a distinctive conception of grass-roots democracy, and that her conception of elites is distinctively democratic rather than anti-democratic. I bolster this argument by examining her historical context, and conclude by assessing the relevance of Arendt's conception of democracy.
Echoes from the Past: The Relationship between Satisfaction with Economic Reforms and Voting Behavior in Poland
Authors: Powers Denise V. , Cox James H.
Published: May 2006

Electorates in several East European countries have thrown their support behind reconstructed communist parties. Is personal economic hardship driving this phenomenon? We argue that political behavior in postcommunist societies is fundamentally structured by interpretations of the transition histories, which were centrally constitutive events. We propose a structural equation model in which understandings of the past mediate the relationship between personal circumstances and satisfaction with economic reforms. We analyze cross-sectional data collected in Poland immediately after the 1993 parliamentary elections and find, consistent with our hypotheses, that understandings of the past exert as much of an effect on attitudes toward reforms as do personal economic assessments. We use multinomial logit to analyze vote choice and find that personal economic circumstances are of little importance. Attitudes toward economic reforms have a limited effect on voting behavior, but their importance is eclipsed by understandings of the past and other factors, such as religion.
Gender and Justice in Plato
Authors: Forde Steven
Published: May 2006

Plato's proposal for the equality of the sexes remains one of the most controversial aspects of his argument in theRepublic.I explore this argument with special emphasis on locating it within larger themes of the work, themes whose relevance to the argument on gender equality have often been ignored. On this basis, I find that Plato's defense of gender equality is serious, but that the foundation and the consequences of that argument have not usually been well understood. Plato's argument for gender equality rests on a distinctive view of human nature, and his elaboration of the consequences of pursuing gender equality reveal that a price would have to be paid for it that few are willing to accept. His argument should be considered by contemporary advocates of gender equality.
Inferential equivalence and natural deduction
Authors: Hiż H.
Published: May 2006

There are axiomatizations of the sentential calculus which, in effect, assert the inferential equivalence (mutual deducibility) of classes of sentences. A well known axiomatization of this sort consists of

Another one, closely connected with A1–A3, is comprised of

B1–B3 jointly assert the inferential equivalence of a sentence of the form (p ⊃ q) ⊃ r and two sentences of the forms q ⊃ r and ∼p ⊃ r respectively. This axiomatization requires the rule of substitution and the rule of detachment for conditionals.

The existence of such axiomatizations suggests a possibility of formulating natural deduction by means of inferential equivalence instead of the usual one-sided inference, customarily represented by ‘⊦’. ‘H’ will stand here for inferential equivalence. It may be looked upon as a combination of signs ‘⊦’ and ‘⊣’. ‘H’ is thought of as a metasystematic constant functor of two arguments each of which is a class of names of sentences. For purposes of this paper the first argument may be just a single name of a sentence, the second a pair (not necessarily ordered) of names of sentences. The role ‘H’ plays is determined by **1 –*5 below. Besides that ‘H’ remains here uninterpreted. In particular, it is not directly postulated that it is an identity relation.
Independent axiom schemata for von wright's M
Authors: Anderson Alan Ross
Published: May 2006

In this paper we show how a modification of results due to Simons ([6]) yields a set of independent axiom schemata for von Wright's M ([8], p. 85), with a single primitive rule of inference. We first describe a system M*, then show its equivalence with M, and finally show that our schemata are independent.

1. Axiomatization ofM*. We adopt the notational conventions of McKinsey and Tarski ([4], p. 2), as amended by Simons ([6], p. 309), except that we take “(α ⊰ β)” as an abbreviation for “˜◇˜(α 0→ β)”, rather than for “˜◇(α ∧ ˜β)”. Our only rule of inference is:

Rule. If ⊦ α and ⊦ (˜◇˜α → β). then ⊦ β.

We have six axiom schemata:

We require a number of theorems for the proof of equivalence of M* with M.

Theorem 1. If ⊦ α and ⊦ (α → β), then ⊦ β.

Theorem 2. If ⊦ α and ⊦ (α ⊰ β), then ⊦ β.

Proof by hypothesis, A5, and Theorem 2 (twice).

Theorem 3. If ⊦ (α ⊰ β), then (˜◇β → ˜ ◇α).

Proof by hypothesis, A6, and Theorem 2.

Theorem 4. If ⊦(α ⊰ β), then ⊦[˜˜(γ ∧ α) ⊰ ˜˜(β ∧ γ)].
On the construction of venn diagrams
Authors: More Trenchard
Published: May 2006

Venn constructed diagrams of up to five simply connected regions that overlapped each other once in each possible way of overlapping. Although Venn did not prove that his diagrams were constructible for more than five simply connected regions — in fact, he preferred to have a doubly connected region in his 5-class diagram — he summarized his method of construction with an intuitive argument: “But for merely theoretical purposes the rule of formulation would be very simple. It would merely be to begin by drawing any closed figure, and then proceed to draw others, subject to the one condition that each is to intersect once and once only all the existing subdivisions produced by those which had gone before.” The method of construction given below leads to a simple topological proof that Venn diagrams can be constructed for any number of simply connected regions.
Professor Goodstein's formalisation of the policeman
Authors: Cohen L. Jonathan
Published: May 2006

Professor R. L. Goodstein's neat proposal for formalising and proving the Policeman — the logical truth mentioned in my paper — illustrates the main thesis of that paper, viz., that in formalising the logic of indirect discourse one needs to avoid assigning what was said, and the report that it was said, to different levels of a semantical hierarchy. But there are two main difficulties in his proposal:

1. It is not quite clear whether the axioms and rules of Prof. Goodstein's system are to be such that substitutivity of the biconditional is maintained for all wff. If so, the system takes no account of the referentially opaque quality of indirect discourse. For example, the analogue for “If Sir Walter Scott always says the same as the author of Waverley, then George III says that anything Sir Walter Scott says is true if and only if George III says that anything the author of Waverley says is true” would be provable, though this statement is not logically true. If “A(x){Bx→x)” is well-formed (as it must be for Professor Goodstein's proof), the corresponding theorem is

2. In interpreting the letters “A”, “B”, etc. as operators, so that sentences are the substitution-instances both for variables like “x” and for schematic functions of those variables like “Ax”, Professor Goodstein's system does not provide for formalising logical truths in which ordinary predication also plays a part: e.g., “If the policeman says that all women are dangerous drivers and anything the policeman says is true, then no non-dangerous drivers are women”.
An emendation of the axiom system of Hilbert and Ackermann for the restricted calculus of predicates
Authors: Pager David
Published: May 2006

The fundamental role of the restricted calculus of predicates in applications of symbolic logic, and particularly in Hubert's Beweistheorie as summed up by Hilbert and Bernays, makes it important that this logical calculus should be accurately defined. The first standard formulation of the calculus was that of Hilbert and Ackermann's Grundzüge der theoretischen Logik. This employed (in the first three editions) a finite set of axioms and rules of derivation, with rules of substitution included. A reaction by Hilbert and Ackermann's successors to persistent difficulty encountered with the rules of substitution has been to omit these rules, and instead enlarge the set of axioms and the other rules of derivation so as to encompass all possible substitutions. Such an enlargement seems to me to be undesirable. As an alternative, this note is designed to put the original approach of Hilbert and Ackermann for once and for all on a sound basis.
A theory of restricted quantification II
Authors: Hailperin Theodore
Published: May 2006

In Part I of this paper no official significance was given to free occurrences of restricted-variables. Indeed doubt has been expressed as to the desirability or feasibility of such usage. Thus, to quote from Rosser [4], p. 146:

“We now raise the question of the significance of F (α), in which the occurrences of α are free [α a variable subject to the restriction K(α)]. In deciding to take (x).K(x) ⊃ F(x) and (Ex).K(x).F(x) as the meanings of (α)F(α) and (Eα)F(α), we were guided by the intuitive meanings. In the case of F(α), the intuitive meaning does not furnish a satisfactory guide. In everyday mathematics, if it has been agreed that α stands for a quantity satisfying the restriction K(α), it is commonly the case that, if one is assuming F (α), then K(α)&F(α) is understood, but if one is trying to prove F (α), then K(α) ⊃ F (α) is understood. It seems that in symbolic logic perhaps it is best not to give any especial significance to α in F(α) when it occurs free.”

Despite this anomalous behaviour we shall, in this Part, show how one can have unhampered use of free restricted-variables in appropriate contexts.
A note on direct products
Authors: Gál L. Novak
Published: May 2006

By an algebra is meant an ordered set Γ = 〈V,R1, …, Rn, O1, …, Om〉, where V is a class, Ri (1 ≤ i ≤, n) is a relation on nj elements of V (i.e. Ri ⊆ Vni), and Oj (1 ≤ i ≤ n) is an operation on elements of V such that Oj(x1, … xmj) ∈ V) for all x1, …, xmj ∈ V). A sentence S of the first-order functional calculus is valid in Γ, if it contains just individual variables x1, x2, …, relation variables ϱ1, …,ϱn, where ϱi,- is nj-ary (1 ≤ i ≤ n), and operation variables σ1, …, σm, where σj is mj-ary (1 ≤, j ≤ m), and S holds if the individual variables are interpreted as ranging over V, ϱi is interpreted as Ri, and σi as Oj. If {Γi}i≤α is a (finite or infinite) sequence of algebras Γi, where Γi = 〈Vi, Ri〉 and Ri, is a binary relation, then by the direct productΓ = Πi<αΓi is meant the algebra Γ = 〈V, R〉, where V consists of all (finite or infinite) sequences x = 〈x1, x2, …, xi, …〉 with Xi ∈ Vi and where R is a binary relation such that two elements x and y of V are in the relation R if and only if xi and yi- are in the relation Ri for each i < α.
Open sentences and the induction axiom
Authors: Shoenfield J. R.
Published: May 2006

The purpose of this paper is to give a partial answer to the question: How much is the induction axiom weakened if it is applied only to sentences with no bound variables? It is well known that for the full Peano arithmetic this is a weakening ([1] p. 90). We consider Peano arithmetic without multiplication, and give a full answer to the question. It turns out that only four new theorems can be proved from the weakened induction axiom; i.e., all further consequences of this axiom are derivable from these four.

We consider a system T formulated within the first-order predicate calculus with equality. The system contains the constant 0 and the three function symbols S (successor), P (predecessor), and +. The non-logical axioms are:

A sentence is open if it contains no bound variables. We obtain the system TI from T by adding the rule of inference:

(I) If A(x) is an open sentence, infer A(x) from A(0) and A(x) ⊃ A(Sx).

The following open sentences are easily proved in TI:

The system formed by adding (B1) - (B4) to T is called T′.

We abbreviate SS … Sx, where S occurs n times, to Snx. Similarly, we abbreviate (… (x+x)+ …) + x, where x occurs n times, to nx. A term of the form n1x1 + … + nkXk + Sp0 is called simple. An equation between simple terms is called a simple equation.
An improved prenex normal form1
Authors: Chang C. C. , Keisler H. Jerome
Published: May 2006

Let ℒ be the set of all formulas of a given first order predicate logic (with or without identity). For each positive integer n, let ℒn be the set of all formulas φ in ℒ logically equivalent to a formula of the form

where Q is a (possibly empty) string of quantifiers, m is a positive integer, and each αij is either an atomic formula or the negation of an atomic formula.
Developmental regulation of α‐amino‐3‐hydroxy‐5‐methyl‐4‐isoxazole‐propionic acid receptor subunit expression in forebrain and relationship to regional susceptibility to hypoxic/ischemic injury. II. Human cerebral white matter and cortex
Authors: Talos Delia M. , Follett Pamela L. , Folkerth Rebecca D. , Fishman Rachel E. , Trachtenberg Felicia L. , Volpe Joseph J. , Jensen Frances E.
Published: May 2006

This report is the second of a two‐part evaluation of developmental differences in α‐amino‐3‐hydroxy‐5‐methyl‐4‐isoxazole‐propionic acid receptor (AMPAR) subunit expression in cell populations within white matter and cortex. In part I, we reported that, in rat, developmental expression of Ca2+‐permeable (GluR2‐lacking) AMPARs correlated at the regional and cellular level with increased susceptibility to hypoxia/ischemia (H/I), suggesting an age‐specific role of these receptors in the pathogenesis of brain injury. Part II examines the regional and cellular progression of AMPAR subunits in human white matter and cortex from midgestation through early childhood. Similarly to the case in the rodent, there is a direct correlation between selective vulnerability to H/I and expression of GluR2‐lacking AMPARs in human brain. For midgestational cases aged 20–24 postconceptional weeks (PCW) and for premature infants (25–37 PCW), we found that radial glia, premyelinating oligodendrocytes, and subplate neurons transiently expressed GluR2‐lacking AMPARs. Notably, prematurity represents a developmental window of selective vulnerability for white matter injury, such as periventricular leukomalacia (PVL). During term (38–42 PCW) and postterm neonatal (43–46 PCW) periods, age windows characterized by increased susceptibility to cortical injury and seizures, GluR2 expression was low in the neocortex, specifically on cortical pyramidal and nonpyramidal neurons. This study indicates that Ca2+‐permeable AMPAR blockade may represent an age‐specific therapeutic strategy for potential use in humans. Furthermore, these data help to validate specific rodent maturational stages as appropriate models for evaluation of H/I pathophysiology. J. Comp. Neurol. 497:61–77, 2006. © 2006 Wiley‐Liss, Inc.
Time course of substance P expression in dorsal root ganglia following complete spinal nerve transection
Authors: Weissner Wendy , Winterson Barbara J. , Stuart‐Tilley Alan , Devor Marshall , Bove Geoffrey M.
Published: May 2006

Recent evidence suggests that substance P (SP) is up‐regulated in primary sensory neurons following axotomy and that this change occurs in larger neurons that do not usually produce SP. If this is so, then the up‐regulation may allow normally neighboring, uninjured, and nonnociceptive dorsal root ganglion (DRG) neurons to become effective in activating pain pathways. By using immunohistochemistry, we performed a unilateral L5 spinal nerve transection on male Wistar rats and measured SP expression in ipsilateral L4 and L5 DRGs and contralateral L5 DRGs at 1–14 days postoperatively (dpo) and in control and sham‐operated rats. In normal and sham‐operated DRGs, SP was detectable almost exclusively in small neurons (≤800 μm2). After surgery, the mean size of SP‐positive neurons from the axotomized L5 ganglia was greater at 2, 4, 7, and 14 dpo. Among large neurons (>800 μm2) from the axotomized L5, the percentage of SP‐positive neurons increased at 2, 4, 7, and 14 dpo. Among small neurons from the axotomized L5, the percentage of SP‐positive neurons was increased at 1 and 3 dpo but was decreased at 7 and 14 dpo. Thus, SP expression is affected by axonal damage, and the time course of the expression is different between large and small DRG neurons. These data support a role for SP‐producing, large DRG neurons in persistent sensory changes resulting from nerve injury. J. Comp. Neurol. 497:78–87, 2006. © 2006 Wiley‐Liss, Inc.
Eddy covariance fluxes of peroxyacetyl nitrates (PANs) and NOy to a coniferous forest
Authors: Turnipseed A. A. , Huey L. G. , Nemitz E. , Stickel R. , Higgs J. , Tanner D. J. , Slusher D. L. , Sparks J. P. , Flocke F. , Guenther A.
Published: May 2006

We employed a fast response thermal dissociation‐chemical ionization mass spectrometer (TD‐CIMS) system to measure eddy covariance fluxes of peroxyacetyl nitrate (PAN), peroxypropionyl nitrate (PPN) and peroxymethacryloyl nitrate (MPAN). Fluxes were measured for eight consecutive days in July 2003 at a Loblolly pine forest in North Carolina along with eddy covariance NOy fluxes. Covariances between PAN concentration and vertical wind velocity indicated consistent deposition fluxes that ranged up to approximately −14 ng N m−2 s−1. The average daytime flux peaked at −6.0 ng N m−2 s−1 and accounted for ∼20% of the daytime NOy flux. Calculations suggest minimum daytime surface resistances for PAN in the range of 70–130 s m−1. It was estimated that approximately half of daytime uptake was through plant stomates. Average PAN deposition velocities, Vd(PAN), showed a daytime maximum of ∼10.0 mm s−1; however, deposition did not cease during nighttime periods. Vd(PAN) was highly variable at night and increased when canopy elements were wet from either precipitation or dew formation. Diel patterns of deposition velocity of MPAN and PPN were similar to that of PAN. These results suggest that deposition of PAN, at least to coniferous forest canopies, is much faster than predicted with current deposition algorithms. Although deposition of PAN is unlikely to compete with thermal dissociation during warm summer periods, it will likely play an important role in removing PAN from the atmosphere in colder regions or during winter. The fate of PAN at the surface and within the plants remains unknown, but may present a previously ignored source of nitrogen to ecosystems.
Influence of small imperfections on the stability of plane Poiseuille flow and the limitation of Squire’s theorem
Authors: Wu Xuesong , Luo Jisheng
Published: May 2006

In a recent paper [Luo and Wu, Phys. Fluids 16, 2852 (2004)], a theoretical model was proposed to illustrate the key role played by small imperfections in the subcritical transition of plane Poiseuille flow. In that model, the imperfections are represented by a steady but spatially periodic two-dimensional surface roughness, and a small oscillatory pressure gradient representing the inevitable background fluctuations. The present paper extends the previous work by considering an improved model, where the wall roughness is three-dimensional. A steady perturbation in the form of spatially periodic suction is also considered. In both cases, the resulting steady and unsteady components interact to produce a forcing that is in near resonance with a pair of oblique Tollmien-Schlichting (T-S) waves. The latter are generated as a result, and grow in proportion to the time during the initial stage. This excitation process and the subsequent nonlinear development of the T-S waves are studied by a weakly nonlinear analysis. A forced nonlinear (and nonlocal) amplitude equation is derived to provide a simple mathematical framework to link the critical Reynolds number for nonlinear instability to the level of imperfections. The theoretical result demonstrates that the onset critical Reynolds number is lower for three-dimensional than for two-dimensional disturbances if the imperfection exceeds a critical level, which turns out to be remarkably small. This offers a possible explanation as to why Squire’s theorem is applicable only when external perturbations are suppressed to an extremely low level. Further analysis of the nonlinear interactions indicates that the much reduced critical Reynolds number for three-dimensional external perturbations can be attributed to the fact that the excited pair of oblique T-S waves interact to generate extremely strong streaks, which then result in a very strong nonlinear destabilizing effect.
The contact angle in capillarity
Authors: Finn Robert
Published: May 2006

In 1805, Thomas Young gave a reasoning to support the view that the contact angle at a solid/liquid/gas interface is a physical constant depending only on the materials, and in no other way on the particular configuration considered. That conclusion has become one of the underpinnings of both classical and modern capillarity theory. The present paper raises some questions about the reasoning and some of its consequences, and offers partial answers.
Closed-loop separation control: An analytic approach
Authors: Alam M.-R. , Liu W. , Haller G.
Published: May 2006

We develop an analytic approach to two-dimensional flow separation control by feedback. With two wall-based actuators enclosing an array of distributed wall-shear sensors, we control the wall-shear evolution equation through its boundary values at the actuators. Using this approach, we induce separation at prescribed locations in steady and unsteady channel flows, and reduce the recirculation length behind a backward-facing step to a prescribed value.
Electric-field-induced force on a charged spherical colloid embedded in an electrolyte-saturated Brinkman medium
Authors: Hill Reghan J.
Published: May 2006

When an electric field is applied to an electrolyte-saturated polymer gel immobilizing charged colloidal particles, the force that must be exerted by the hydrogel on each particle reflects a delicate balance of electrical and hydrodynamic stresses. This article adopts a simple boundary-layer analysis to derive a convenient formula for the force in terms of the particle, electrolyte and gel characteristics. Comparisons with numerically exact solutions of the full set of electrokinetic transport equations are presented. These reveal that a fortuitous cancellation of errors leads to reasonably accurate predictions of the force over a much wider range of the parameter space than should be expected. It is noteworthy that, in gels with low permeability, an adverse pressure gradient yields a net force that exceeds the bare electrical force. The analytical theory also provides a convenient formula for the incremental pore mobility, which is a convenient measure of the electro-osmotic pumping capacity of dilute random arrays of charged inclusions.
Sverdlovsk revisited: Modeling human inhalation anthrax
Authors: Wilkening Dean A.
Published: May 2006

Several models have been proposed for the dose–response function and the incubation period distribution for human inhalation anthrax. These models give very different predictions for the severity of a hypothetical bioterror attack, when an attack might be detected from clinical cases, the efficacy of medical intervention and the requirements for decontamination. Using data from the 1979 accidental atmospheric release of anthrax in Sverdlovsk, Russia, and limited nonhuman primate data, this paper eliminates two of the contending models and derives parameters for the other two, thereby narrowing the range of models that accurately predict the effects of human inhalation anthrax. Dose–response functions that exhibit a threshold for infectivity are contraindicated by the Sverdlovsk data. Dose-dependent incubation period distributions explain the 10-day median incubation period observed at Sverdlovsk and the 1- to 5-day incubation period observed in nonhuman primate experiments.
Hydrodynamic narrowing of tubes extruded from cells
Authors: Brochard-Wyart F. , Borghi N. , Cuvelier D. , Nassoy P.
Published: May 2006

We discuss the pulling force
f
required to extrude a lipid tube from a living cell as a function of the extrusion velocity
L̇
. The main feature is membrane friction on the cytoskeleton. As recently observed for neutrophils, the tether force exhibits a “shear thinning” response over a large range of pulling velocities, which was previously interpreted by assuming viscoelastic flows of the sliding membrane. Here, we propose an alternative explanation based on purely Newtonian flow: The diameter of the tether decreases concomitantly with the increase of the membrane tension in the lipid tube. The pulling force is found to vary as
L̇

1
3

, which is consistent with reported experimental data for various types of cells.
Poly[[bis(4-chlorosalicylato-κO)copper(II)]-di-μ-3-pyridylmethanol-κ2
N:O;κ2
O:N]
Authors: Maroszová Jaroslava , Martiška Ladislav , Valigura Dušan , Koman Marian , Glowiak Tadeusz
Published: May 2006

The polymeric structure of the title complex, [Cu(C7H4ClO3)2(C6H7NO)2]
n
, at 100 (2) K contains two-dimensional sheets parallel to the ac plane. The CuII atom, lying on an inversion centre, has an octahedral coordination environment. It is bonded in a trans-square-planar arrangement to the N atoms of two 3-pyridylmethanol molecules and one carboxylate O atom from each of two 4-chlorosalicylate anions. The axial positions are also occupied by 3-pyridylmethanol, but coordinated through O rather than the pyridyl N atom.
1-(2-Hydroxyphenyl)octan-1-one
Authors: Bin Xu , Ji Jin-Xiu , Fen Zhi-Qiang , Wang Jin-Tang
Published: May 2006

The title compound, C14H20O2, was synthesized by the reaction of phenol and octanoyl chloride. All non-H atoms are coplanar, with an r.m.s. deviation of 0.031 Å. In the crystal structure, molecules are linked into chains along the a axis by intermolecular C—H...O hydrogen bonds.
1,9-Diazahexacyclo[17.6.1.13,7.19,16.010,15.020,25]hexacosa-3,5,7(26),10(15),11,13,16(27),17,19(28),20(25),21,23-dodecaene
Authors: Kumar G. Senthil , Chinnakali K. , Swaroop M. Gayatri , Rajakumar P. , Fun Hoong-Kun
Published: May 2006

The title compound, C26H20N2, has one half-molecule in the asymmetric unit. The molecular twofold rotational symmetry is preserved on crystallization. The dihedral angle between the two symmetry-related indole ring systems is 49.65 (4)°. The benzene ring is oriented almost perpendicular [dihedral angle 86.02 (3)°] to the planes of both indole ring systems. Molecules related by a centre of inversion are linked via intermolecular C—H...π interactions to form chains along the c axis.
π-Stacked chains in 3,5-dimethyl-1,7-diphenyl-1,7-dihydrodipyrazolo[3,4-b,4′,3′-e]pyridine
Authors: Portilla Jaime , de la Torre Jose M. , Cobo Justo , Low John N. , Glidewell Christopher
Published: May 2006

The title compound, C21H17N5, was prepared using a microwave-induced condensation reaction between 5-amino-3-methyl-1-phenylpyrazole and formaldehyde. The molecules lie across twofold rotation axes in space group C2/c and are into chains by a π–π stacking interaction.
1-(4-Bromophenyl)-3-(2,4-dichlorophenyl)prop-2-en-1-one
Authors: Patil P. S. , Teh Jeannie Bee-Jan , Fun Hoong-Kun , Razak Ibrahim Abdul , Dharmaprakash S. M.
Published: May 2006

In the title compound, C15H9BrCl2O, the molecules form chains along the b axis through short Br...Cl contacts. These chains form layers through C—H...O intermolecular interactions and these layers stack parallel to the c axis.
Benzyldiethylammonium chloride
Authors: Lorono-Gonzalez Daniel
Published: May 2006

The title compound, C11H18N+·Cl−, crystallizes in a non-centrosymmetric polar space group and possesess normal geometric parameters. The crystal structure analysis reveals a discrete ionic species, for which the packing is consolidated by an N—H...Cl hydrogen bond and C—H...Cl hydrogen bonds, resulting in a two-dimensional network.
N-{[3′-Formyl-2,2′-bis(methoxymethoxy)biphenyl-3-yl]methyl}formamide
Authors: Gao Yu-Xing , Fang Hua , Zeng Qing-Le , Chen Wei-Zhu , Zhao Yu-Fen
Published: May 2006

The title compound, C19H21NO6, was obtained by ortho-lithiation of 2,2′-bis(methoxymethoxy)biphenyl, followed by quenching with N,N-dimethylformaldehyde. The molecular packing is stabilized by van der Waals forces.
(E)-4,4′-Ethylenedipyridinium bis{[(5-fluoro-2,4-dioxo-1,2,3,4-tetrahydropyrimidin-4-yl)acetamido]acetate} dihydrate
Authors: Cai Xiao-Qing , Hu Mao-Lin , Chen Shilin
Published: May 2006

The asymmetric unit of the title compound, C10H9N2
2+·2C6H4FN2O4
−·2H2O, consists of one half of the (E)-4,4′-ethylenedipyridinium cation, one [(5-fluoro-2,4-dioxo-1,2,3,4-tetrahydropyrimidin-4-yl)acetamido]acetate anion and one water molecule, the complete formula unit being generated by a crystallographic inversion centre at the mid-point of the central C=C bond of the (E)-4,4′-ethylenedipyridinium cation. In the crystal structure, a three-dimensional network is formed via N—H...O and O—H...O hydrogen bonds.