On the Rademacher maximal function

Tools known as maximal functions are frequently used in harmonic analysis when studying local behaviour of functions. Typically they measure the suprema of local averages of non-negative functions. It is essential that the size (more precisely, the L^p-norm) of the maximal function is comparable to the size of the original function. When dealing with families of operators between Banach spaces we are often forced to replace the uniform bound with the larger R-bound. Hence such a replacement is also needed in the maximal function for functions taking values in spaces of operators. More specifically, the suprema of norms of local averages (i.e. their uniform bound in the operator norm) has to be replaced by their R-bound. This procedure gives us the Rademacher maximal function, which was introduced by Hytönen, McIntosh and Portal in order to prove a certain vector-valued Carleson's embedding theorem. They noticed that the sizes of an operator-valued function and its Rademacher maximal function are comparable for many common range spaces, but not for all. Certain requirements on the type and cotype of the spaces involved are necessary for this comparability, henceforth referred to as the “RMF-property”. It was shown, that other objects and parameters appearing in the definition, such as the domain of functions and the exponent p of the norm, make no difference to this. After a short introduction to randomized norms and geometry in Banach spaces we study the Rademacher maximal function on Euclidean spaces. The requirements on the type and cotype are considered, providing examples of spaces without RMF. L^p-spaces are shown to have RMF not only for p greater or equal to 2 (when it is trivial) but also for 1 < p < 2. A dyadic version of Carleson's embedding theorem is proven for scalar- and operator-valued functions. As the analysis with dyadic cubes can be generalized to filtrations on sigma-finite measure spaces, we consider the Rademacher maximal function in this case as well. It turns out that the RMF-property is independent of the filtration and the underlying measure space and that it is enough to consider very simple ones known as Haar filtrations. Scalar- and operator-valued analogues of Carleson's embedding theorem are also provided. With the RMF-property proven independent of the underlying measure space, we can use probabilistic notions and formulate it for martingales. Following a similar result for UMD-spaces, a weak type inequality is shown to be (necessary and) sufficient for the RMF-property. The RMF-property is also studied using concave functions giving yet another proof of its independence from various parameters.

Soveltuvuuskokeella täydennetty opiskelijavalinta : Seurantatutkimus eräiden sosiaali- ja terveysalan teknisluonteisten koulutusohjelmien (apuvälinetekniikka, hammastekniikka ja optometria) opiskelijavalinnan toteuttamisesta

Aptitude-based student selection: A study concerning the admission processes of some technically oriented healthcare degree programmes in Finland (Orthotics and Prosthetics, Dental Technology and Optometry). The data studied consisted of conveniencesamples of preadmission information and the results of the admission processes of three technically oriented healthcare degree programmes (Orthotics and Prosthetics, Dental Technology and Optometry) in Finland during the years 1977-1986 and 2003. The number of the subjects tested and interviewed in the first samples was 191, 615 and 606, and in the second 67, 64 and 89, respectively. The questions of the six studies were: I. How were different kinds of preadmission data related to each other? II. Which were the major determinants of the admission decisions? III. Did the graduated students and those who dropped out differ from each other? IV. Was it possible to predict how well students would perform in the programmes? V. How was the student selection executed in the year 2003? VI. Should clinical vs. statistical prediction or both be used? (Some remarks are presented on Meehl's argument: "Always, we might as well face it, the shadow of the statistician hovers in the background; always the actuary will have the final word.") The main results of the study were as follows: Ability tests, dexterity tests and judgements of personality traits (communication skills, initiative, stress tolerance and motivation) provided unique, non-redundant information about the applicants. Available demographic variables did not bias the judgements of personality traits. In all three programme settings, four-factor solutions (personality, reasoning, gender-technical and age-vocational with factor scores) could be extracted by the Maximum Likelihood method with graphical Varimax rotation. The personality factor dominated the final aptitude judgements and very strongly affected the selection decisions. There were no clear differences between graduated students and those who had dropped out in regard to the four factors. In addition, the factor scores did not predict how well the students performed in the programmes. Meehl's argument on the uncertainty of clinical prediction was supported by the results, which on the other hand did not provide any relevant data for rules on statistical prediction. No clear arguments for or against the aptitude-based student selection was presented. However, the structure of the aptitude measures and their impact on the admission process are now better known. The concept of "personal aptitude" is not necessarily included in the values and preferences of those in charge of organizing the schooling. Thus, obviously the most well-founded and cost-effective way to execute student selection is to rely on e.g. the grade point averages of the matriculation examination and/or written entrance exams. This procedure, according to the present study, would result in a student group which has a quite different makeup (60%) from the group selected on the basis of aptitude tests. For the recruiting organizations, instead, "personal aptitude" may be a matter of great importance. The employers, of course, decide on personnel selection. The psychologists, if consulted, are responsible for the proper use of psychological measures.

Koulutuksellisen syrjäytymisriskin ennustettavuus yhteishakuvalintojen kautta tarkasteltuna : koulupudokkuusriskin yhteys oppilaan kognitiiviseen kompetenssiin, koulumenestykseen sekä itseä ja vanhempia koskeviin uskomuksiin

The objective of this study was to find factors that could predict educational dropout. Dropout risk was assessed against pupil’s cognitive competence, success in school, and personal beliefs regarding self and parents, while taking into account the pupil’s background and gender. Based on earlier research, an assumption was made that a pupil’s gender, success in school, and parent’s education would be related with dropping out. This study is part of a project funded by the Academy of Finland and led by Professor Jarkko Hautamäki. The project aims to use longitudinal study to assess the development of pupils’ skills in learning to learn. The target group of this study consisted all Finnish speaking ninth graders of a municipality in Southern Finland. There were in total 1534 pupils, of which 809 were girls and 725 boys. The assessment of learning to learn skills was performed about ninth graders in spring 2004. “Opiopi” test material was used in the assessment, consisting of cognitive tests and questions measuring beliefs. At the same time, pupils’ background information was collected together with their self-reported average grade of all school subjects. During spring 2009, the pupils’ joint application data from years 2004 and 2005 was collected from the Finnish joint application registers. The data were analyzed using quantitative methods assisted by the SPSS for Windows computer software. Analysis was conducted through statistical indices, differences in grade averages, multilevel model, multivariate analysis of variance, and logistic regression analysis. Based on earlier research, dropouts were defined as pupils that had not been admitted to or had not applied to second degree education under the joint application system. Using this definition, 157 students in the target group were classified as dropouts (10 % of the target group): 88 girls and 69 boys. The study showed that the school does not affect the drop-out risk but the school class explains 7,5 % of variation in dropout risk. Among girls, dropping out is predicted by a poor average grade, a lack of beliefs supporting learning, and an unrealistic primary choice in joint application system compared to one’s success in school. Among boys, a poor average grade, unrealistic choices in joint application system, and the belief of parent’s low appreciation of education were related to dropout risk. Keywords educational exclusion, school dropout, success in school, comprehensive school, learning to learn

Tropical dryland agroforestry on clay soils: : Analysis of systems based on Acacia senegal in the Blue Nile region, Sudan

Acacia senegal, the gum arabic producing tree, is the most important component in traditional dryland agroforestry systems in the Blue Nile region, Sudan. The aim of the present study was to provide new knowledge on the potential use of A. senegal in dryland agroforestry systems on clay soils, as well as information on tree/crop interaction, and on silvicultural and management tools, with consideration on system productivity, nutrient cycling and sustainability. Moreover, the aim was also to clarify the intra-specific variation in the performance of A. senegal and, specifically, the adaptation of trees of different origin to the clay soils of the Blue Nile region. In agroforestry systems established at the beginning of the study, tree and crop growth, water use, gum and crop yields, nutrient cycling and system performance were investigated for a period of four years (1999 to 2002). Trees were grown at 5 x 5 m and 10 x 10 m spacing alone or in mixture with sorghum or sesame; crops were also grown in sole culture. The symbiotic biological N2 fixation by A. senegal was estimated using the 15N natural abundance (δ15N) procedure in eight provenances collected from different environments and soil types of the gum arabic belt and grown in clay soil in the Blue Nile region. Balanites aegyptiaca (a non-legume) was used as a non-N-fixing reference tree species, so as to allow 15N-based estimates of the proportion of the nitrogen in trees derived from the atmosphere. In the planted acacia trees, measurements were made on shoot growth, water-use efficiency (as assessed by the δ13C method) and (starting from the third year) gum production. Carbon isotope ratios were obtained from the leaves and branch wood samples. The agroforestry system design caused no statistically significant variation in water use, but the variation was highly significant between years, and the highest water use occurred in the years with high rainfall. No statistically significant differences were found in sorghum or sesame yields when intercropping and sole crop systems were compared (yield averages were 1.54 and 1.54 ha-1 for sorghum and 0.36 and 0.42 t ha-1 for sesame in the intercropped and mono-crop plots, respectively). Thus, at an early stage of agroforestry system management, A. senegal had no detrimental effect on crop yield, but the pattern of resource capture by trees and crops may change as the system matures. Intercropping resulted in taller trees and larger basal and crown diameters as compared to the development of sole trees. It also resulted in a higher land equivalent ratio. When gum yields were analysed it was found that a significant positive relationship existed between the second gum picking and the total gum yield. The second gum picking seems to be a decisive factor in gum production and could be used as an indicator for the total gum yield in a particular year. In trees, the concentrations of N and P were higher in leaves and roots, whereas the levels of K were higher in stems, branches and roots. Soil organic matter, N, P and K contents were highest in the upper soil stratum. There was some indication that the P content slightly increased in the topsoil as the agroforestry plantations aged. At a stocking of 400 trees ha-1 (5 x 5 m spacing), A. senegal accumulated in the biomass a total of 18, 1.21, 7.8 and 972 kg ha-1of N, P, K and OC, respectively. Trees contributed ca. 217 and 1500 kg ha-1 of K and OC, respectively, to the top 25-cm of soil over the first four years of intercropping. Acacia provenances of clay plain origin showed considerable variation in seed weight. They also had the lowest average seed weight as compared to the sandy soil (western) provenances. At the experimental site in the clay soil region, the clay provenances were distinctly superior to the sand provenances in all traits studied but especially in basal diameter and crown width, thus reflecting their adaptation to the environment. Values of δ13C, indicating water use efficiency, were higher in the sand soil group as compared to the clay one, both in leaves and in branch wood. This suggests that the sand provenances (with an average value of -28.07 ) displayed conservative water use and high drought tolerance. Of the clay provenances, the local one (Bout) displayed a highly negative (-29.31 ) value, which indicates less conservative water use that resulted in high productivity at this particular clay-soil site. Water use thus appeared to correspond to the environmental conditions prevailing at the original locations for these provenances. Results suggest that A. senegal provenances from the clay part of the gum belt are adapted for a faster growth rate and higher biomass and gum productivity as compared to provenances from sand regions. A strong negative relationship was found between the per-tree gum yield and water use efficiency, as indicated by δ13C. The differences in water use and gum production were greater among provenance groups than within them, suggesting that selection among rather than within provenances would result in distinct genetic gain in gum yield. The relative δ15N values ( ) were higher in B. aegyptiaca than in the N2-fixing acacia provenances. The amount of Ndfa increased significantly with age in all provenances, indicating that A. senegal is a potentially efficient nitrogen fixer and has an important role in t agroforestry development. The total above-ground contribution of fixed N to foliage growth in 4-year-old A. senegal trees was highest in the Rahad sand-soil provenance (46.7 kg N ha-1) and lowest in the Mazmoom clay-soil provenance (28.7 kg N ha-1). This study represents the first use of the δ15N method for estimating the N input by A. senegal in the gum belt of Sudan. Key words: Acacia senegal, agroforestry, clay plain, δ13C, δ15N, gum arabic, nutrient cycling, Ndfa, Sorghum bicolor, Sesamum indicum

Structure and Dynamics of Coil-like Molecules by Residual Dipolar Couplings

Nuclear magnetic resonance (NMR) spectroscopy provides us with many means to study biological macromolecules in solution. Proteins in particular are the most intriguing targets for NMR studies. Protein functions are usually ascribed to specific three-dimensional structures but more recently tails, long loops and non-structural polypeptides have also been shown to be biologically active. Examples include prions, -synuclein, amylin and the NEF HIV-protein. However, conformational preferences in coil-like molecules are difficult to study by traditional methods. Residual dipolar couplings (RDCs) have opened up new opportunities; however their analysis is not trivial. Here we show how to interpret RDCs from these weakly structured molecules. The most notable residual dipolar couplings arise from steric obstruction effects. In dilute liquid crystalline media as well as in anisotropic gels polypeptides encounter nematogens. The shape of a polypeptide conformation limits the encounter with the nematogen. The most elongated conformations may come closest whereas the most compact remain furthest away. As a result there is slightly more room in the solution for the extended than for the compact conformations. This conformation-dependent concentration effect leads to a bias in the measured data. The measured values are not arithmetic averages but essentially weighted averages over conformations. The overall effect can be calculated for random flight chains and simulated for more realistic molecular models. Earlier there was an implicit thought that weakly structured or non-structural molecules would not yield to any observable residual dipolar couplings. However, in the pioneering study by Shortle and Ackerman RDCs were clearly observed. We repeated the study for urea-denatured protein at high temperature and also observed indisputably RDCs. This was very convincing to us but we could not possibly accept the proposed reason for the non-zero RDCs, namely that there would be some residual structure left in the protein that to our understanding was fully denatured. We proceeded to gain understanding via simulations and elementary experiments. In measurements we used simple homopolymers with only two labelled residues and we simulated the data to learn more about the origin of RDCs. We realized that RDCs depend on the position of the residue as well as on the length of the polypeptide. Investigations resulted in a theoretical model for RDCs from coil-like molecules. Later we extended the studies by molecular dynamics. Somewhat surprisingly the effects are small for non-structured molecules whereas the bias may be large for a small compact protein. All in all the work gave clear and unambiguous results on how to interpret RDCs as structural and dynamic parameters of weakly structured proteins.

Towards the Use of Radar Winds in Numerical Weather Prediction

Numerical weather prediction (NWP) models provide the basis for weather forecasting by simulating the evolution of the atmospheric state. A good forecast requires that the initial state of the atmosphere is known accurately, and that the NWP model is a realistic representation of the atmosphere. Data assimilation methods are used to produce initial conditions for NWP models. The NWP model background field, typically a short-range forecast, is updated with observations in a statistically optimal way. The objective in this thesis has been to develope methods in order to allow data assimilation of Doppler radar radial wind observations. The work has been carried out in the High Resolution Limited Area Model (HIRLAM) 3-dimensional variational data assimilation framework. Observation modelling is a key element in exploiting indirect observations of the model variables. In the radar radial wind observation modelling, the vertical model wind profile is interpolated to the observation location, and the projection of the model wind vector on the radar pulse path is calculated. The vertical broadening of the radar pulse volume, and the bending of the radar pulse path due to atmospheric conditions are taken into account. Radar radial wind observations are modelled within observation errors which consist of instrumental, modelling, and representativeness errors. Systematic and random modelling errors can be minimized by accurate observation modelling. The impact of the random part of the instrumental and representativeness errors can be decreased by calculating spatial averages from the raw observations. Model experiments indicate that the spatial averaging clearly improves the fit of the radial wind observations to the model in terms of observation minus model background (OmB) standard deviation. Monitoring the quality of the observations is an important aspect, especially when a new observation type is introduced into a data assimilation system. Calculating the bias for radial wind observations in a conventional way can result in zero even in case there are systematic differences in the wind speed and/or direction. A bias estimation method designed for this observation type is introduced in the thesis. Doppler radar radial wind observation modelling, together with the bias estimation method, enables the exploitation of the radial wind observations also for NWP model validation. The one-month model experiments performed with the HIRLAM model versions differing only in a surface stress parameterization detail indicate that the use of radar wind observations in NWP model validation is very beneficial.

Viestintää valtuustossa : Valtuutettujen viestintätyyli ja käsitys Helsingin kaupunginvaltuuston kokouksien viestinnästä

The goal of the study was to determine the kind of communication used in the meetings of the City Council. Councillors' addresses were studied by observing their communication styles. A questionnaire was used to study councillors' perception of communication in council meetings as perceived communication climate. A second goal of the study was to develop a method for analysing communication style. The nature of the study was a longitudinal case study. Studies of speech communication at the individual level and those of organisational communication at the community level served as the theoretical frame of reference. Due to the chosen object of study, political research and communication context shaped the frame of reference for the study. Verbal, nonverbal and paraverbal characteristics form the communication style. The perceived communication climate is dynamic in nature and consists of characteristics. The research material was gathered from the meetings of the Helsinki City Council held from 1993 to 1996. The communication style was analysed from meeting addresses (N=1271) given by the permanent members (N=95) of the council. The perceived communication climate was studied using a questionnaire modified from Wiio's OCD2 questionnaire. The questionnaire survey was carried out twice. The first time, 58 questionnaires (68%) were returned and, the second time, 49 questionnaires (58%) were returned. The method of analysis used for communication style was classification and cross tabulation. Based on the subset of material for the first year, five communication style categories were defined Communication style was examined using gender, length of council membership, and political party as subgroups. The results related to the perceived communication climate were analysed as averages and percentages. A comparison of the results from two different measuring times revealed how the perceived communication climate changed during a council term. The finding of the study was that the municipal politicians could be placed in all five communication style categories in each year of their four-year council term. The sizes of the style categories varied only little in the different years. A relatively stable - rather than changing - communication style was more characteristic of individuals during a council term. Gender did not explain an individual's communication style. On the other hand, the length of service in the council and, to some extent, the political party were connected with a certain communication style. Changes in style mainly manifested themselves as variation in communication activity so that activity was highest at the beginning and at the end of the council term. The method of researching communication style developed for this study proved to be functional, though labour-consuming. The perceived communication climate in the Helsinki City Council was good. Women's and men's satisfaction with the communication in the council differed; women's perception became considerably more positive at the end of the council term. Satisfaction with the communication in the council was characteristic of the Coalition Party and the Social Democrat councillors and dissatisfaction was characteristics of the Green councillors. Long council experience increased satisfaction with communication in the council. Only in the case of a few style categories was it possible to show any connection between the communication style and the perceived communication climate. The study confirmed the perception that an individual's communication style in the same communication context was relatively stable in the long term as well.

First measurement of the ratio of branching fractions B(Λb0→Λc+μ-ν̅ μ)/B(Λb0→Λc+π-)

This article presents the first measurement of the ratio of branching fractions B(Λb0→Λc+μ-ν̅ μ)/B(Λb0→Λc+π-). Measurements in two control samples using the same technique B(B̅ 0→D+μ-ν̅ μ)/B(B̅ 0→D+π-) and B(B̅ 0→D*(2010)+μ-ν̅ μ)/B(B̅ 0→D*(2010)+π-) are also reported. The analysis uses data from an integrated luminosity of approximately 172  pb-1 of pp̅ collisions at √s=1.96  TeV, collected with the CDF II detector at the Fermilab Tevatron. The relative branching fractions are measured to be B(Λb0→Λc+μ-ν̅ μ)/B(Λb0→Λc+π-)=16.6±3.0(stat)±1.0(syst)+2.6/-3.4(PDG)±0.3(EBR), B(B̅ 0→D+μ-ν̅ μ)/B(B̅ 0→D+π-)= 9.9±1.0(stat)±0.6(syst)±0.4(PDG)±0.5(EBR), and B(B̅ 0→D*(2010)+μ-ν̅ μ)/B(B̅ 0→D*(2010)+π-)=16.5±2.3(stat)± 0.6(syst)±0.5(PDG)±0.8(EBR). The uncertainties are from statistics (stat), internal systematics (syst), world averages of measurements published by the Particle Data Group or subsidiary measurements in this analysis (PDG), and unmeasured branching fractions estimated from theory (EBR), respectively. This article also presents measurements of the branching fractions of four new Λb0 semileptonic decays: Λb0→Λc(2595)+μ-ν̅ μ, Λb0→Λc(2625)+μ-ν̅ μ, Λb0→Σc(2455)0π+μ-ν̅ μ, and Λb0→Σc(2455)++π-μ-ν̅ μ, relative to the branching fraction of the Λb0→Λc+μ-ν̅ μ decay. Finally, the transverse-momentum distribution of Λb0 baryons produced in pp̅ collisions is measured and found to be significantly different from that of B̅ 0 mesons, which results in a modification in the production cross-section ratio σΛb0/σB̅ 0 with respect to the CDF I measurement.

The connection between on-farm assessed welfare (A-index) and production parameters of the sows

Tämän tutkimuksen tavoitteena oli selvittää tilalla määritetyn hyvinvoinnin yhteyttä emakoiden tuotantotuloksiin. Hyvinvointia arvioitiin suomalaisen hyvinvointi-indeksin, A-indeksi, avulla. Tuotantotuloksina käytettiin kahta erilaista tuotosaineistoa, jotka molemmat pohjautuivat kansalliseen tuotosseuranta aineistoon. Hyvinvointimääritykset tehtiin 30 porsastuotantosikalassa maaliskuun 2007 aikana. A-indeksi koostuu kuudesta kategoriasta ’liikkumismahdollisuudet’, ’alustan ominaisuudet’, ’sosiaaliset kontaktit’, ’valo, ilma ja melu’, ’ruokinta ja veden saanti’ sekä ’eläinten terveys ja hoidon taso’. Jokaisessa kategoriassa on 3-10 pääosin ympäristöperäistä muuttujaa, jotka vaihtelevat osastoittain. Maksimipistemäärä osastolle on 100. Hyvinvointimittaukset tehtiin porsitus-, tiineytys- ja joutilasosastoilla. Erillisten tiineytysosastojen pienen lukumäärän takia (n=7) tilakohtaiset tiineytys- ja joutilasosastopisteet yhdistettiin ja keskiarvoja käytettiin analyyseissä. Yhteyksiä tuotokseen tutkittiin kahden eri aineiston avulla 1) Tilaraportti aineisto (n=29) muodostuu muokkaamattomista tila- ja tuotostuloksista tilavierailua edeltävän vuoden ajalta, 2) POTSIaineisto (n=30) muodostuu POTSI-ohjelmalla (MTT) muokatusta tuotantoaineistosta, joka sisältää managementtiryhmän (tila, vuosi, vuodenaika) vaikutuksen ensikoiden ja emakoiden pahnuekohtaiseen tuotokseen. Yhteyksiä analysointiin korrelaatio- ja regressioanalyysien avulla. Vaikka osallistuminen tutkimukseen oli vapaaehtoista, molempien tuotantoaineistojen perusteella tutkimustilat edustavat keskituottoista suomalaista sikatilaa. A-indeksin kokonaispisteet vaihtelivat välillä 37,5–64,0 porsitusosastolla ja 39,5–83,5 joutilasosastolla. Tilaraporttiaineistoa käytettäessä paremmat pisteet porsitusosaston ’eläinten terveys ja hoidon taso’ -kategoriasta lyhensivät eläinten lisääntymissykliä, lisäsivät syntyvien pahnueiden ja porsaiden määrää sekä alensivat kuolleena syntyneiden lukumäärää. Regressiomallin mukaan ’eläinten terveys ja hoidon taso’ -kategoria selitti syntyvien porsaiden lukumäärän, porsimisvälin pituuden sekä keskiporsimiskerran vaihtelua. Paremmat pisteet joutilasosaston ’liikkumismahdollisuudet’ kategoriasta alensivat syntyneiden pahnueiden sekä syntyneiden että vieroitettujen porsaiden lukumäärää. Regressiomallin mukaan ensikkopahnueiden osuus ja ”liikkumismahdollisuudet” kategorian pisteet selittivät vieroitettujen porsaiden lukumäärän vaihtelua. POTSI-aineiston yhteydessä kuolleena syntyneiden porsaiden lukumäärän aleneminen oli ensikoilla yhteydessä parempiin porsitusosaston ’sosiaalisiin kontakteihin’ ja emakoilla puolestaan joutilasosaston parempiin ’eläinten terveys ja hoidon taso’ pisteisiin. Kahden eri tuotantoaineiston avulla saadut tulokset erosivat toisistaan. Seuraavissa tutkimuksissa onkin suositeltavampaa käyttää Tilaraporttiaineistoja, joissa tuotokset ilmoitetaan vuosikohtaisina. Tämän tutkimuksen perusteella hyvinvoinnilla ja tuotoksella on yhteyksiä, joilla on myös merkittävää taloudellista vaikutusta. Erityisesti hyvä eläinten hoito ja eläinten terveys lisäävät tuotettujen porsaiden määrää ja lyhentävät lisääntymiskiertoa. Erityishuomiota tulee kiinnittää vapaana olevien joutilaiden emakoiden sosiaaliseen stressiin ja rehunsaannin varmistamiseen kaikille yksilöille.

Climatic adaptation of Norway spruce (Picea abies (L.) Karsten) in Finland based on male flowering phenology.

Anthesis was studied at the canopy level in 10 Norway spruce stands from 9 localities in Finland from 1963 to 1974. Distributions of pollen catches were compared to the normal Gaussian distribution. The basis for the timing studies was the 50 per cent point of the anthesis-fitted normal distribution. Development up to this point was given in calendar days, in degree days (>5 °C) and in period units. The count of each parameter began on March 19 (included). Male flowering in Norway spruce stands was found to have more annual variation in quantity than in Scots pine stands studied earlier. Anthesis in spruce in northern Finland occurred at a later date than in the south. The heat sums needed for anthesis varied latitudinally less in spruce than in pine. The variation of pollen catches in spruce increased towards north-west as in the case of Scots pine. In the unprocessed data, calendar days were found to be the most accurate forecast of anthesis in Norway spruce both for a single year and for the majority of cases of stand averages over several years. Locally, the period unit could be a more accurate parameter for the stand average. However, on a calendar day basis, when annual deviations between expected and measured heat sums were converted to days, period units were narrowly superior to days. The geographical correlations respect to timing of flowering, calculated against distances measured along simulated post-glacial migration routes, were stronger than purely latitudinal correlations. Effects of the reinvasion of Norway spruce into Finland are thus still visible in spruce populations just as they were in Scots pine populations. The proportion of the average annual heat sum needed for spruce anthesis grew rapidly north of a latitude of ca. 63° and the heat sum needed for anthesis decreased only slighty towards the timberline. In light of flowering phenology, it seems probable that the northwesterly third of Finnish Norway spruce populations are incompletely adapted to the prevailing cold climate. A moderate warming of the climate would therefore be beneficial for Norway spruce. This accords roughly with the adaptive situation in Scots pine.

A maximal operator, Carleson's embedding, and tent spaces for vector-valued functions

This thesis is concerned with the area of vector-valued Harmonic Analysis, where the central theme is to determine how results from classical Harmonic Analysis generalize to functions with values in an infinite dimensional Banach space. The work consists of three articles and an introduction. The first article studies the Rademacher maximal function that was originally defined by T. Hytönen, A. McIntosh and P. Portal in 2008 in order to prove a vector-valued version of Carleson's embedding theorem. The boundedness of the corresponding maximal operator on Lebesgue-(Bochner) -spaces defines the RMF-property of the range space. It is shown that the RMF-property is equivalent to a weak type inequality, which does not depend for instance on the integrability exponent, hence providing more flexibility for the RMF-property. The second article, which is written in collaboration with T. Hytönen, studies a vector-valued Carleson's embedding theorem with respect to filtrations. An earlier proof of the dyadic version assumed that the range space satisfies a certain geometric type condition, which this article shows to be also necessary. The third article deals with a vector-valued generalizations of tent spaces, originally defined by R. R. Coifman, Y. Meyer and E. M. Stein in the 80's, and concerns especially the ones related to square functions. A natural assumption on the range space is then the UMD-property. The main result is an atomic decomposition for tent spaces with integrability exponent one. In order to suit the stochastic integrals appearing in the vector-valued formulation, the proof is based on a geometric lemma for cones and differs essentially from the classical proof. Vector-valued tent spaces have also found applications in functional calculi for bisectorial operators. In the introduction these three themes come together when studying paraproduct operators for vector-valued functions. The Rademacher maximal function and Carleson's embedding theorem were applied already by Hytönen, McIntosh and Portal in order to prove boundedness for the dyadic paraproduct operator on Lebesgue-Bochner -spaces assuming that the range space satisfies both UMD- and RMF-properties. Whether UMD implies RMF is thus an interesting question. Tent spaces, on the other hand, provide a method to study continuous time paraproduct operators, although the RMF-property is not yet understood in the framework of tent spaces.