Homoclinic Splitting without Trees

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations. When the coupling takes place through an even trigonometric polynomial in the angle variables, we extend analytically the solutions of the equations of motion, order by order in the perturbation parameter, to a large neighbourhood of the real line representing time. Subsequently, we devise an asymptotic expansion for the splitting (matrix) associated with a homoclinic point. This expansion consists of contributions that are manifestly exponentially small in the limit of vanishing gravity, by a shift-of-countour argument. Hence, we infer a similar upper bound for the splitting itself. In particular, the derivation of the result does not call for a tree expansion with explicit cancellation mechanisms.

Truth, Proof and Gödelian Arguments : A Defence of Tarskian Truth in Mathematics

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion.

Becoming a teacher: emerging teacher identity in mathematics teacher education

This research examines three aspects of becoming a teacher, teacher identity formation in mathematics teacher education: the cognitive and affective aspect, the image of an ideal teacher directing the developmental process, and as an on-going process. The formation of emerging teacher identity was approached in a social psychological framework, in which individual development takes place in social interaction with the context through various experiences. Formation of teacher identity is seen as a dynamic, on-going developmental process, in which an individual intentionally aspires after the ideal image of being a teacher by developing his/her own competence as a teacher. The starting-point was that it is possible to examine formation of teacher identity through conceptualisation of observations that the individual and others have about teacher identity in different situations. The research uses the qualitative case study approach to formation of emerging teacher identity, the individual developmental process and the socially constructed image of an ideal mathematics teacher. Two student cases, John and Mary, and the collective case of teacher educators representing socially shared views of becoming and being a mathematics teacher are presented. The development of each student was examined based on three semi-structured interviews supplemented with written products. The data-gathering took place during the 2005 2006 academic year. The collective case about the ideal image provided during the programme was composed of separate case displays of each teacher educator, which were mainly based on semi-structured interviews in spring term 2006. The intentions and aims set for students were of special interest in the interviews with teacher educators. The interview data was analysed following the modified idea of analytic induction. The formation of teacher identity is elaborated through three themes emerging from theoretical considerations and the cases. First, the profile of one s present state as a teacher may be scrutinised through separate affective and cognitive aspects associated with the teaching profession. The differences between individuals arise through dif-ferent emphasis on these aspects. Similarly, the socially constructed image of an ideal teacher may be profiled through a combination of aspects associated with the teaching profession. Second, the ideal image directing the individual developmental process is the level at which individual and social processes meet. Third, formation of teacher identity is about becoming a teacher both in the eyes of the individual self as well as of others in the context. It is a challenge in academic mathematics teacher education to support the various cognitive and affective aspects associated with being a teacher in a way that being a professional and further development could have a coherent starting-point that an individual can internalise.

What was behind the bark? : An assessment of decay among urban Tilia, Betula and Acer trees felled as hazardous in the Helsinki City area

Old trees growing in urban environments are often felled due to symptoms of mechanical defects that could be hazardous to people and property. The decisions concerning these removals are justified by risk assessments carried out by tree care professionals. The major motivation for this study was to determine the most common profiles of potential hazard characteristics for the three most common urban tree genera in Helsinki City: Tilia, Betula and Acer, and in this way improve management practices and protection of old amenity trees. For this research, material from approximately 250 urban trees was collected in cooperation with the City of Helsinki Public Works Department during 2001 - 2004. From the total number of trees sampled, approximately 70% were defined as hazardous. The tree species had characteristic features as potential hazard profiles. For Tilia trees, hollowed heartwood with low fungal activity and advanced decay caused by Ganoderma lipsiense were the two most common profiles. In Betula spp., the primary reason for tree removal was usually lowered amenity value in terms of decline of the crown. Internal cracks, most often due to weak fork formation, were common causes of potential failure in Acer spp. Decay caused by Rigidoporus populinus often increased the risk of stem breakage in these Acer trees. Of the decay fungi observed, G. lipsiense was most often the reason for the increased risk of stem collapse. Other fungi that also caused extensive decay were R. populinus, Inonotus obliquus, Kretzschmaria deusta and Phellinus igniarius. The most common decay fungi in terms of incidence were Pholiota spp., but decay caused by these species did not have a high potential for causing stem breakage, because it rarely extended to the cambium. The various evaluations used in the study suggested contradictions in felling decisions based on trees displaying different stages of decay. For protection of old urban trees, it is crucial to develop monitoring methods so that tree care professionals could better analyse the rate of decay progression towards the sapwood and separate those trees with decreasing amounts of sound wood from those with decay that is restricted to the heartwood area.

Interactions between transgenic trees and mycorrhizal and pathogenic fungi

The development of biotechnology techniques in plant breeding and the new commercial applications have raised public and scientific concerns about the safety of genetically modified (GM) crops and trees. To find out the feasibility of these new technologies in the breeding of commercially important Finnish hardwood species and to estimate the ecological risks of the produced transgenic plants, the experiments of this study have been conducted as a part of a larger project focusing on the risk assessment of GM-trees. Transgenic Betula pendula and Populus trees were produced via Agrobacterium mediated transformation. Stilbene synthase (STS) gene from pine (Pinus sylvestris) and chitinase gene from sugar beet (Beta vulgaris) were transferred to (hybrid) aspen and birch, respectively, to improve disease resistance against fungal pathogens. To modify lignin biosynthesis, a 4-coumarate:coenzyme A ligase (4CL) gene fragment in antisense orientation was introduced into two birch clones. In in vitro test, one transgenic aspen line expressing pine STS gene showed increased resistance to decay fungus Phellinus tremulae. In the field, chitinase transgenic birch lines were more susceptible to leaf spot (Pyrenopeziza betulicola) than the non-transgenic control clone while the resistance against birch rust (Melampsoridium betulinum) was improved. No changes in the content or composition of lignin were detected in the 4CL antisense birch lines. In order to evaluate the ecological effects of the produced GM trees on non-target organisms, an in vitro mycorrhiza experiment with Paxillus involutus and a decomposition experiment in the field were performed. The expression of a transgenic chitinase did not disturb the establishment of mycorrhizal symbiosis between birch and P. involutus in vitro. 4CL antisense transformed birch lines showed retarded root growth but were able to form normal ectomycorrhizal associations with the mycorrhizal fungus in vitro. 4CL lines also showed normal litter decomposition. Unexpected growth reductions resulting from the gene transformation were observed in chitinase transgenic and 4CL antisense birch lines. These results indicate that genetic engineering can provide a tool in increasing disease resistance in Finnish tree species. More extensive data with several ectomycorrhizal species is needed to evaluate the consequences of transgene expression on beneficial plant-fungus symbioses. The potential pleiotropic effects of the transgene should also be taken into account when considering the safety of transgenic trees.

Locality and order-invariant logics

Malli on logiikassa käytetty abstraktio monille matemaattisille objekteille. Esimerkiksi verkot, ryhmät ja metriset avaruudet ovat malleja. Äärellisten mallien teoria on logiikan osa-alue, jossa tarkastellaan logiikkojen, formaalien kielten, ilmaisuvoimaa malleissa, joiden alkioiden lukumäärä on äärellinen. Rajoittuminen äärellisiin malleihin mahdollistaa tulosten soveltamisen teoreettisessa tietojenkäsittelytieteessä, jonka näkökulmasta logiikan kaavoja voidaan ajatella ohjelmina ja äärellisiä malleja niiden syötteinä. Lokaalisuus tarkoittaa logiikan kyvyttömyyttä erottaa toisistaan malleja, joiden paikalliset piirteet vastaavat toisiaan. Väitöskirjassa tarkastellaan useita lokaalisuuden muotoja ja niiden säilymistä logiikkoja yhdistellessä. Kehitettyjä työkaluja apuna käyttäen osoitetaan, että Gaifman- ja Hanf-lokaalisuudeksi kutsuttujen varianttien välissä on lokaalisuuskäsitteiden hierarkia, jonka eri tasot voidaan erottaa toisistaan kasvavaa dimensiota olevissa hiloissa. Toisaalta osoitetaan, että lokaalisuuskäsitteet eivät eroa toisistaan, kun rajoitutaan tarkastelemaan äärellisiä puita. Järjestysinvariantit logiikat ovat kieliä, joissa on käytössä sisäänrakennettu järjestysrelaatio, mutta sitä on käytettävä siten, etteivät kaavojen ilmaisemat asiat riipu valitusta järjestyksestä. Määritelmää voi motivoida tietojenkäsittelyn näkökulmasta: vaikka ohjelman syötteen tietojen järjestyksellä ei olisi odotetun tuloksen kannalta merkitystä, on syöte tietokoneen muistissa aina jossakin järjestyksessä, jota ohjelma voi laskennassaan hyödyntää. Väitöskirjassa tutkitaan minkälaisia lokaalisuuden muotoja järjestysinvariantit ensimmäisen kertaluvun predikaattilogiikan laajennukset yksipaikkaisilla kvanttoreilla voivat toteuttaa. Tuloksia sovelletaan tarkastelemalla, milloin sisäänrakennettu järjestys lisää logiikan ilmaisuvoimaa äärellisissä puissa.

Archaea in the Mycorrhizosphere of Boreal Forest Trees

Archaea were long thought to be a group of ancient bacteria, which mainly lived in extreme environments. Due to the development of DNA sequencing methods and molecular phylogenetic analyses, it was shown that the living organisms are in fact divided into three domains; the Archaea, Bacteria and the Eucarya. Since the beginning of the previous decade, it was shown that archaea generally inhabit moderate environments and that these non-extremophilic archaea are more ubiquitous than the extremophiles. Group 1 of non-extreme archaea affiliate with the phylum Crenarchaeota. The most commonly found soil archaea belong to the subgroup 1.1b. However, the Crenarchaeota found in the Fennoscandian boreal forest soil belong to the subgroup 1.1c. The organic top layer of the boreal forest soil, the humus, is dominated by ectomycorrhizal fungal hyphae. These colonise virtually all tree fine root tips in the humus layer and have been shown to harbour distinct bacterial populations different from those in the humus. The archaea have also been shown to colonise both boreal forest humus and the rhizospheres of plants. In this work, studies on the archaeal communities in the ectomycorrhizospheres of boreal forest trees were conducted in microcosms. Archaea belonging to the group 1.1c Crenarchaeota and Euryarchaeota of the genera Halobacterium and Methanolobus were detected. The archaea generally colonised fungal habitats, such as ectomycorrhizas and external mycelia, rather than the non-mycorrhizal fine roots of trees. The species of ectomycorrhizal fungus had a great impact on the archaeal community composition. A stable euryarchaeotal community was detected especially in the mycorrhizas, of most of the tested Scots pine colonising ectomycorrhizal fungi. The Crenarchaeota appeared more sporadically in these habitats, but had a greater diversity than the Euryarchaeota. P. involutus mycorrhizas had a higher diversity of 1.1c Crenarchaeota than the other ectomycorrhizal fungi. The detection level of archaea in the roots of boreal trees was generally low although archaea have been shown to associate with roots of different plants. However, alder showed a high diversity of 1.1c Crenarchaeota, exceeding that of any of the tested mycorrhizas. The archaeal 16S rRNA genes detected from the non-mycorrhizal roots were different from those of the P. involutus mycorrhizas. In the phylogenetic analyses, the archaeal 16S rRNA gene sequences obtained from non-mycorrhizal fine roots fell in a separate cluster within the group 1.1c Crenarchaeota than those from the mycorrhizas. When the roots of the differrent tree species were colonised by P. involutus, the diversity and frequency of the archaeal populations of the different tree species were more similar to each other. Both Cren- and Euryarchaeota were enriched in cultures to which C-1 substrates were added. The 1.1c Crenarchaeota grew anaerobically in mineral medium with CH4 and CO2 as the only available C sources, and in yeast extract media with CO2 and CH4 or H2. The crenarchaeotal diversity was higher in aerobic cultures on mineral medium with CH4 or CH3OH than in the anaerobic cultures. Ecological functions of the mycorrhizal 1.1c Crenarchaeota in both anaerobic and aerobic cycling of C-1 compounds were indicated. The phylogenetic analyses did not divide the detected Crenarchaeota into anaerobic and aerobic groups. This may suggest that the mycorrhizospheric crenarchaeotal communities consist of closely related groups of anaerobic and aerobic 1.1c Crenarchaeota, or the 1.1c Crenarchaeota may be facultatively anaerobic. Halobacteria were enriched in non-saline anaerobic yeast extract medium cultures in which CH4 was either added or produced, but were not detected in the aerobic cultures. They may potentially be involved in anaerobic CH4 cycling in ectomycorrhizas. The CH4 production of the mycorrhizal samples was over 10 times higher than for humus devoid of mycorrhizal hyphae, indicating a high CH4 production potential of the mycorrhizal metanogenic community. Autofluorescent methanogenic archaea were detected by microscopy and 16S rRNA gene sequences of the genus Methanolobus were obtained. The archaeal community depended on both tree species and the type of ectomycorrhizal fungus colonising the roots and the Cren- and Euryarchaeota may have different ecological functions in the different parts of the boreal forest tree rhizosphere and mycorrhizosphere. By employing the results of this study, it may be possible to isolate both 1.1c Crenarchaeota as well as non-halophilic halobacteria and aerotolerant methanogens from mycorrhizospheres. These archaea may be used as indicators for change in the boreal forest soil ecosystem due to different factors, such as exploitations of forests and the rise in global temperature. More information about the microbial populations with apparently low cell numbers but significant ecological impacts, such as the boreal forest soil methanogens, may be of crucial importance to counteract human impacts on such globally important ecosystems as the boreal forests.

Prosopis, an Alien among the Sacred Trees of South India

The problematic of invasive species in an alien environment has aroused the attention of scientists all over the world for quite some time. One of the exotic tree species that has provoked special attention in the tropical drylands is Prosopis juliflora. Originating in South America, prosopis (hereafter referred to as prosopis) has been introduced in the hot and semi-arid zones of the world particularly to provide fuelwood, to stabilize sand dunes and to combat desertification. The tree has become an essential source for fuelwood and a provider of several other products and services in areas where it has become established. However, despite the numerous benefits the tree provides to rural people, in several regions prosopis has become a noxious weed with a negative impact on the environment and to the economy of farmers and landowners. In India, prosopis was introduced in Andhra Pradesh in 1877. The tree was then proclaimed as the precious child of the plant world by scientists and local people alike. The purpose of this study was to investigate the overall impact of prosopis on local rural livelihoods in the drylands of South India. Of particular interest was the examination of the different usages of the tree, especially as fuelwood, and people s perceptions of it. Furthermore, the study examined the negative impacts of the uncontrolled invasion of prosopis on croplands, and its occupation of the banks of irrigation canals and other water sources. As another central theme, this study analysed the Hindu classification system for nature and for trees in particular. In India, several tree species are regarded as sacred. This study examined the position of the exotic prosopis among sacred trees, such as the bodhi, banyan and neem trees. The principle method for collecting the field data was by using individual and thematic group interviews. These interviews were semi-structured with open ended questions. Moreover, unstructured interviews as well as general observations provided complementary information. The data were gathered during two fieldwork periods in the states of Andhra Pradesh and Tamil Nadu, in South India. The results confirmed that prosopis both provides benefits and causes hazards to different stakeholders. Farmers and agriculturalists suffer economic losses in areas where prosopis has invaded crop fields and competes with other plants for water and nutrients. On the other hand, for a significant number of poor rural people, prosopis has become an important source of livelihood benefits. This tree, which grows on government wastelands, is commonly a free resource for all and has thus become a major local source of fuelwood. It also provides several other goods and services and cash income that contributes to improve livelihoods in rural communities. Prosopis ranked lowest in the tree classificatioin system of the Hindus of South India. Although it is appreciated for many benefits it provides for poor people, it has remained an outsider compared with the indigenous tree species. On the other hand, the most sacred trees, such as the bodhi or the banyan, are completely excluded from extraction and it is seen as a sacrilege to even cut branches from any of these trees. An unexpected finding was that, in a few cases, prosopis had also been elevated to the status of a sacred tree. Goods and services from prosopis are not utilized in the most beneficial way. Silvicultural management practices are suggested that would provide additional income and employment opportunities. Interventions are recommended to control further invasion of the tree that might cause serious negative effects in the future. For Hindus, the sacred always ranks highest, even above economic gain. The conservation of sacred groves and sacred trees is a tradition that has its roots in ancient history. These socio-religious practices need to be respected and continued. Successful management of tree and forest resources depends on the willingness of the local people to manage their natural resources, and this willingness exists and has always existed in South India. Keywords: South India, drylands, livelihood, fuelwood, invasive, resource, silviculture.