We investigate the pair-splitting number $\germ{s}_{pair}$ which is a variation of splitting number, pair-reaping number $\germ{r}_{pair}$ which is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants of F σ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals (...) not satisfying Fatou's lemma. (shrink)

We investigate splitting number and reaping number for the structure (ω) ω of infinite partitions of ω. We prove that ${\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}}$ and ${\mathfrak{s}_{d}\geq\mathfrak{b}}$ . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}$ and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}$ . To prove the consistency ${\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}$ and ${\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})}$ we introduce new cardinal invariants ${\mathfrak{r}_{pair}}$ and ${\mathfrak{s}_{pair}}$ . We also study the relation between ${\mathfrak{r}_{pair}, \mathfrak{s}_{ (...) class='Hi'>pair}}$ and other cardinal invariants. We show that ${\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}}$ and ${\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})}$. (shrink)

We show that it is consistent that the reaping number r is less than u , the size of the smallest base for an ultrafilter. To show that our forcing preserves certain ultrafilters, we prove a general partition theorem involving Ramsey ideals.

According to Frege, n=Kn, where n is any cardinal number and Kn is the class of all n-tuples. According to Von Neumann, n=Kpn, where Kpn is the class of all of n's predecessors. These analyses are prima facie incompatible with each other, given that Kn≠Kpn, for n>0. In the present paper it is shown that these analyses are in fact compatible with each other, for the reason that each analysis can and ultimately must be interpreted as being to the (...) effect that n=Cn, where Cn is the class of all ordered pairs <Kn#,Rn#>, where Kn# is an arbitrary class and Rn# is an arbitrary relation such that a class k has n-many members exactly if k bears Rn# to Kn#. (shrink)

A secondary analysis was performed on preliminary data from an ongoing cross-cultural study on assortative pairing. Independently sampled pairs of opposite-sex romantic partners and of same-sex friends rated themselves and each other on Life History (LH) strategy and mate value. Data were collected in local bars, clubs, coffeehouses, and other public places from three different cultures: Tucson, Arizona; Hermosillo, Sonora; and San José, Costa Rica. The present analysis found that slow LH individuals assortatively pair with both sexual and social (...) partners more strongly than fast LH individuals. We interpret this phenomenon as representing (1) an adaptation for preserving coadapted genomes in slow LH strategists to maintain high copying fidelity genetic replication while producing a lower number of offspring in stable, predictable, and controllable environments and (2) a bet-hedging adaptation in fast LH strategists, favoring the genetic diversification of a higher number of offspring in unstable, unpredictable, and uncontrollable environments. (shrink)

In an attempt to give a unified account of common properties of various resource bounded reducibilities, we introduce conditions on a binary relation ≤r between subsets of the natural numbers, where ≤r is meant as a resource bounded reducibility. The conditions are a formalization of basic features shared by most resource bounded reducibilities which can be found in the literature. As our main technical result, we show that these conditions imply a result about exact pairs which has been previously shown (...) by Ambos-Spies [2] in a setting of polynomial time bounds: given some recursively presentable ≤r-ideal I and some recursive ≤r-hard set B for I which is not contained in I, there is some recursive set C, where B and C are an exact pair for I, that is, I is equal to the intersection of the lower ≤r-cones of B and C, where C is not in I. In particular, if the relation ≤r is in addition transitive and there are least sets, then every recursive set which is not in the least degree is half of a minimal pair of recursive sets. (shrink)

Being based on the observation that a conjugate pair of representations, or dual logic, is a necessity under the presence of chaos, a new interpretation of quantum theory is proposed as describingproto-chaos. This chaos has to be a result of basic nonlinearity in the dynamic structure, of which, however, the nonchaotic phase seems to lie ourside the reach of experimental technique, thus the term proto-chaos. Nevertheless, assuming no extra degrees of freedom, the interpretation clarifies a number of riddles (...) posed hitherto and throws some light on the overall hierarchical structure of our understanding and description of nature. (shrink)

It will be shown that for all numbers n and m with n > m 1 the Boolean pairs have undecidable elementary theories.

The canonical pair of a proof system P is the pair of disjoint NP sets where one set is the set of all satisfiable CNF formulas and the other is the set of CNF formulas that have P-proofs bounded by some polynomial. We give a combinatorial characterization of the canonical pairs of depth d Frege systems. Our characterization is based on certain games, introduced in this article, that are parametrized by a number k, also called the depth. (...) We show that the canonical pair of a depth d Frege system is polynomially equivalent to the pair (Ad+2,Bd+2) where Ad+2 (respectively, Bd+1) are depth d+1 games in which Player I (Player II) has a positional winning strategy. Although this characterization is stated in terms of games, we will show that these combinatorial structures can be viewed as generalizations of monotone Boolean circuits. In particular, depth 1 games are essentially monotone Boolean circuits. Thus we get a generalization of the monotone feasible interpolation for Resolution, which is a property that enables one to reduce the task of proving lower bounds on the size of refutations to lower bounds on the size of monotone Boolean circuits. However, we do not have a method yet for proving lower bounds on the size of depth d games for d>1. (shrink)

The two envelopes problem has generated a significant number of publications (I have benefitted from reading many of them, only some of which I cite; see the epilogue for a historical note). Part of my purpose here is to provide a review of previous results (with somewhat simpler demonstrations). In addition, I hope to clear up what I see as some misconceptions concerning the problem. Within a countably additive probability framework, the problem illustrates a breakdown of dominance with respect (...) to infinite partitions in circumstances of infinite expected utility. Within a probability framework that is only finitely additive, there are failures of dominance with respect to infinite partitions in circumstances of bounded utility with finitely many consequences (see the epilogue). (shrink)

The purpose is to study the strength of Ramsey’s Theorem for pairs restricted to recursive assignments ofk-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number$k \ge 2$, Ramsey’s Theorem for pairs and recursive assignments ofkcolors is equivalent to the Limited Lesser Principle of Omniscience for${\rm{\Sigma }}_3^0$formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinitek-ary tree there is some$i < k$and some branch with infinitely (...) many children of indexi. (shrink)

Mechanisms occurring at the atomic level are now known to drive processes essential for life, as revealed by quantum effects on biochemical reactions. Some macroscopic characteristics of organisms may thus show an atomic imprint, which may be transferred across organisms and affect their evolution. This possibility is considered here for the first time, with the aim of elucidating the appearance of an animal innovation with an unclear evolutionary origin: migratory behaviour. This trait may be mediated by a radical pair (...) (RP) mechanism in the retinal flavoprotein cryptochrome, providing essential magnetic orientation for migration. Isotopes may affect the performance of quantum processes through their nuclear spin. Here, we consider a simple model and then apply the standard open quantum system approach to the spin dynamics of cryptochrome RP. We changed the spin quantum number (I) and g‐factor of hydrogen and nitrogen isotopes to investigate their effect on RP's yield and magnetic sensitivity. Strong differences arose between isotopes with I = 1 and I = 1/2 in their contribution to cryptochrome magnetic sensitivity, particularly regarding Earth's magnetic field strengths (25–65 µT). In most cases, isotopic substitution improved RP's magnetic sensitivity. Migratory behaviour may thus have been favoured in animals with certain isotopic compositions of cryptochrome. (shrink)

Pairs of sentences like the following pose a problem for ontology: (1) Jupiter has four moons. (2) The number of moons of Jupiter is four. (2) is intuitively a trivial paraphrase of (1). And yet while (1) seems ontologically innocent, (2) appears to imply the existence of numbers. Thomas Hofweber proposes that we can resolve the puzzle by recognizing that sentence (2) is syntactically derived from, and has the same meaning as, sentence (1). Despite appearances, the expressions ‘the (...) class='Hi'>number of moons of Jupiter’ and ‘four’ do not function semantically as singular terms in (2). Hofweber’s primary evidence for this proposal concerns differences in the focus-related communicative functions of (1) and (2). In this paper I raise several serious problems for Hofweber’s proposal, and for his attempt to support it by appeal to focus-related phenomena. I conclude by offering independent evidence for an alternative, purely pragmatic resolution of the ontological puzzle. (shrink)

Ideas about the mechanisms that regulate chromosome pairing, recombination, and segregation during meiosis have gained in molecular detail over the last few years. The purpose of this article is to survey briefly the shifts in paradigms and experiments that have generated new perspectives. It has never been very clear what it is that brings together the homologous chromosomes at meiotic prophase. For a while it appeared that the synaptonemal complex might be the nuclear organelle responsible for synapsis, but the supporting (...) evidence has not been entirely convincing. Whatever the mechanism, it has always been assumed that homologous synapsis creates the opportunity for homologous DNA sequences to initiate recombination. At present, alternative ideas are developing. Attractive is the concept that double strand DNA repair mechanisms, that find and use the undamaged homologue for repair, have evolved into a meiotic mechanism for the recognition and pairing of homologous sequences. Subsequent intimate synapsis of homologous chromosomes in the context of the synaptonemal complex may serve later functions in the regulation of interference and segregation at first anaphase. A number of areas that are being tested at present and some that may be investigated in the future are discussed at the end of the review. (shrink)

This paper contains a proposal for a free, nonzero-rest-mass particle’s proper spacetime, determined exclusively by the particle’s rest mass \ and numbers. The approach defines proper time as de Broglie time, which is isomorphic to a sequence of natural numbers \ that count de Broglie time units \\). The approach is based on defining the spatial coordinate as proper following the constructive definition of positive and negative integers as all possible differences of ordered pairs of natural numbers multiplied by the (...) Compton unit \\). The spatial and temporal coordinates that form the particle’s proper spacetime are constructed as Euclidean projections of the de Broglie time. The corresponding expression in the form of an energy-momentum relation reveals the existence, aside from the rest energy term \, of an additional energy term of the same order of magnitude, which is related to large intervals of the \-particle’s proper space. The relation of the numbers-based approach to the foundations of the special theory of relativity and of quantum mechanics is discussed. (shrink)

In this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer is called a Salem number if α and are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of (...) a certain pair with Lehmer's conjecture and obtain a model‐theoretic characterization of Lehmer's conjecture for Salem numbers. (shrink)

We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT n k denote Ramsey's theorem for k-colorings of n-element sets, and let RT $^n_{ denote (∀ k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' ≤ T 0 (n) . Let IΣ n and BΣ (...) n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + IΣ 2 + RT 2 2 is conservative over RCA 0 + IΣ 2 for Π 1 1 statements and that $RCA_0 + I\Sigma_3 + RT^2_{ , is Π 1 1 -conservative over RCA 0 + IΣ 3 . It follows that RCA 0 + RT 2 2 does not imply BΣ 3 . In contrast, J. Hirst showed that $RCA_0 + RT^2_{ does imply BΣ 3 , and we include a proof of a slightly strengthened version of this result. It follows that $RT^2_{ is strictly stronger than RT 2 2 over RCA 0. (shrink)

Paired associative stimulation at the spinal cord has been shown to increase muscle force and dexterity by strengthening the corticomuscular connection, through spike timing dependent plasticity. Typically, transcranial magnetic stimulation and transcutaneous peripheral nerve electrical stimulation are often used in spinal PAS. PNS targets superficial nerve branches, by which the number of applicable muscles is limited. Alternatively, a muscle can be activated by positioning the stimulation electrode on the “motor point”, which is the most sensitive location of a muscle (...) to electrical stimulation. Although this can increase the number of applicable muscles for spinal PAS, nobody has tested whether MPS can be used for the spinal PAS to date. Here we investigated the feasibility of using MPS instead of PNS for spinal PAS. Ten healthy male individuals received spinal PAS on two separate days with different stimulation timings expected to induce facilitation of corticospinal excitability or no effect on the soleus. The motor evoked potentials response curve in the soleus was measured prior to the spinal PAS, immediately after and at 10, 20, 30 min post-intervention as a measure of corticospinal excitability. The post-intervention MEP response curve areas were larger in the REAL condition than the CONTROL conditions. Further, the post-intervention MEP response curve areas were significantly larger than pre-intervention in the REAL condition but not in the CONTROL condition. We conclude that MPS can facilitate corticospinal excitability through spinal PAS. (shrink)

There are nine kinds of number: cardinal (measure of class size), ordinal (corresponds to position), generalized ordinal (position in multidimensional discrete manifold), signed (relation between cardinals), rational (different kind of relation between cardinals), real (limit), complex (pair of reals), transfinite (size of reflexive class), and dimension (measure of complexity.

Pointillistic based super‐resolution techniques, such as photoactivated localization microscopy (PALM), involve multiple cycles of sequential activation, imaging, and precise localization of single fluorescent molecules. A super‐resolution image, having nanoscopic structural information, is then constructed by compiling all the image sequences. Because the final image resolution is determined by the localization precision of detected single molecules and their density, accurate image reconstruction requires imaging of biological structures labeled with fluorescent molecules at high density. In such image datasets, stochastic variations in photon (...) emission and intervening dark states lead to uncertainties in identification of single molecules. This, in turn, prevents the proper utilization of the wealth of information on molecular distribution and quantity. A recent strategy for overcoming this problem is pair‐correlation analysis applied to PALM. Using rigorous statistical algorithms to estimate the number of detected proteins, this approach allows the spatial organization of molecules to be quantitatively described. (shrink)

Previous experiments have shown a serial-position effect (SPE) in paired-associate (PA) learning where the pairs contained stimuli pre- viously learned in serial order. The present experiment extended the number of pairs from 10 to 14. Pairs containing stimuli from terminal serial positions were learned with significantly fewer errors than pairs whose stimuli derived from central positions. The latter produced a dip in the PA error distribution suggesting the presence of sequential associations in SL between items occupying central positions.

Generalized quantifiers are functions from pairs of properties to truth-values; these functions can be used to interpret natural language quantifiers. The space of such functions is vast and a great deal of research has sought to find natural constraints on the functions that interpret determiners and create quantifiers. These constraints have demonstrated that quantifiers rest on number and number sense. In the first part of the paper, we turn to developing this argument. In the remainder, we report on (...) work in neurobiology that test the empirical claims of the theory. In particular, we look at fMRI experiments and behavioral experiments with various patient populations that support the intimate connection between natural language quantification and number sense. (shrink)

Savant syndrome and prime numbers Oliver Sacks reported that a pair of autistic twins had extraordinary number abilities and that they spontaneously generated huge prime numbers. Such abilities could contradict our understanding of human abilities. Sacks' report attracted widespread attention, and several researchers speculated theoretically. Unfortunately, most of the explanations in the literature are wrong. Here a correct explanation on prime number identification is provided. Fermat's little theorem is implemented in spreadsheet. Also, twenty years after the report, (...) questionable aspects were found in it. Extreme abilities became dubious. One possibility for the less extreme abilities is incomplete trial division. (shrink)

The development and applicability of complex numbers is often cited in defence of the formalist philosophy of mathematics. This view is rejected through an examination of hamilton's development of the notion of complex numbers as ordered pairs of reals, And his later development of the quaternion theory, Which subsequently formed the basis of vector analysis. Formalism, By protecting informal assumptions from critical scrutiny, Constrained rather than encouraged the development of mathematics.

Rape of women by men might be generated either by a specialized rape adaptation or as a by-product of other psychological adaptations. Although increasing number of sexual partners is a proposed benefit of rape according to the “rape as an adaptation” and the “rape as a by-product” hypotheses, neither hypothesis addresses directly why some men rape their long-term partners, to whom they already have sexual access. In two studies we tested specific hypotheses derived from the general hypothesis that sexual (...) coercion in the context of an intimate relationship may function as a sperm competition tactic. We hypothesized that men’s sexual coercion in the context of an intimate relationship is related positively to his partner’s perceived infidelities and that men’s sexual coercion is related positively to their mate retention behaviors (behaviors designed to prevent a partner’s infidelity). The results from Study 1 (self-reports from 246 men) and Study 2 (partner-reports from 276 women) supported the hypotheses. The Discussion section addresses limitations of this research and highlights future directions for research on sexual coercion in intimate relationships. (shrink)

The aim here is to describe how to complete the constructive logicist program, in the author’s book Anti-Realism and Logic, of deriving all the Peano-Dedekind postulates for arithmetic within a theory of natural numbers that also accounts for their applicability in counting finite collections of objects. The axioms still to be derived are those for addition and multiplication. Frege did not derive them in a fully explicit, conceptually illuminating way. Nor has any neo-Fregean done so.

This article sets out to analyse some of the most basic elements of our number concept - of our awareness of the one and the many in their coherence with multiplicity, succession and equinumerosity. On the basis of the definition given by Cantor and the set theoretical definition of cardinal numbers and ordinal numbers provided by Ebbinghaus, a critical appraisal is given of Frege’s objection that abstraction and noticing (or disregarding) differences between entities do not produce the concept of (...) number. By introducing the notion of subject functions, an account is advanced of the (nominalistic) reason why Frege accepted physical, kinematic and spatial properties (subject functions) of entities, but denied the ontic status of their quantitative properties (their quantitative subject function). With reference to intuitionistic mathematics (Brouwer, Weyl, Troelstra, Kreisel, Van Dalen) the primitive meaning of succession is acknowledged and connected to an analysis of what is entailed in the term ‘Gleichzahligkeit’ (‘equinumerosity’). This expression enables an analysis of the connections between ordinality and cardinality on the one hand and succession and wholeness (totality) on the other. The conceptions of mathematicians such as Frege, Cantor, Dedekind, Zermelo, Brouwer, Skolem, Fraenkel, Von Neumann, Hilbert, Bernays and Weyl, as well as the views of the philosopher Cassirer, are discussed in order to arrive at an assessment of the relation between ordinality and cardinality (also taking into account the relation between logic and arithmetic) - and on the basis of this evaluation, attention is briefly given to the definition of an ordered pair in axiomatic set theory (with reference to the set theory of Zermelo-Fraenkel) and to the defmition of an ordered pair advanced by Wiener and Kuratowski. (shrink)

Why might it be beneficial for adults to process fractions componentially? Recent research has shown that college-educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse (...) primes for the equation that immediately followed it. Students with relatively high math ability showed relational priming both with and without high perceptual similarity. Students with relatively low math ability also showed priming, but only when the structurally inverse equation pairs were supported by high perceptual similarity between numbers. Several additional experiments established boundary conditions on relational priming with fractions. These findings are interpreted in terms of componential processing of fractions in a relational multiplication context that takes advantage of their inherent connections to a multiplicative schema for whole numbers. (shrink)

In §57 of the Foundations of Arithmetic, Frege famously turns to natural language to support his claim that numbers are ‘self-subsistent objects’:I have already drawn attention above to the fact th...

The House Sparrow (Passer domesticus), formerly a common bird species, has shown a rapid decline in Western Europe over recent decades. In The Netherlands, its decline is apparent from 1990 onwards. Many causes for this decline have been suggested that all decrease the vital rates, i.e. survival and reproduction, but their actual impact remains unknown. Although the House Sparrow has been dominant in The Netherlands, data on life history characteristics for this bird species are scarce: data on reproduction are non-existent, (...) and here we first present survival estimates based on live encounters and dead recoveries of marked individuals over the period 1976–2003, 14 years before and 14 years during the decline, reported to the Dutch Ringing Centre. We show that there is an indication that both juvenile and adult survival are lower during the period of decline. Secondly, to be able to analyse the relative impact of changes in the vital rates, we formulated a general matrix model based on a range of survival values between zero and one with a step size of 0.01 (both juvenile and adult yearly survival) and a range of realistic reproduction values (one, three or five fledglings per pair per year). With the matrix model, we calculated the finite rate of population change (λ) and applied elasticity analysis. To diagnose the cause of the decline in the Dutch House Sparrow, we parameterised the model with estimates of survival values before and during the decline and present the resulting λ. With the survival estimates from the declining period, λ < 1 only if reproduction is relatively low. We discuss this result within the light of available literature data on survival in the House Sparrow. Finally, we evaluate which of the suggested causes of population decline should be reversed to mitigate the decline and how this can be achieved. (shrink)

The House Sparrow (Passer domesticus), formerly a common bird species, has shown a rapid decline in Western Europe over recent decades. In The Netherlands, its decline is apparent from 1990 onwards. Many causes for this decline have been suggested that all decrease the vital rates, i.e. survival and reproduction, but their actual impact remains unknown. Although the House Sparrow has been dominant in The Netherlands, data on life history characteristics for this bird species are scarce: data on reproduction are non-existent, (...) and here we first present survival estimates based on live encounters and dead recoveries of marked individuals over the period 1976–2003, 14 years before and 14 years during the decline, reported to the Dutch Ringing Centre. We show that there is an indication that both juvenile and adult survival are lower during the period of decline. -/- Secondly, to be able to analyse the relative impact of changes in the vital rates, we formulated a general matrix model based on a range of survival values between zero and one with a step size of 0.01 (both juvenile and adult yearly survival) and a range of realistic reproduction values (one, three or five fledglings per pair per year). With the matrix model, we calculated the finite rate of population change (λ) and applied elasticity analysis. To diagnose the cause of the decline in the Dutch House Sparrow, we parameterised the model with estimates of survival values before and during the decline and present the resulting λ. With the survival estimates from the declining period, λ < 1 only if reproduction is relatively low. We discuss this result within the light of available literature data on survival in the House Sparrow. Finally, we evaluate which of the suggested causes of population decline should be reversed to mitigate the decline and how this can be achieved. (shrink)