Geometric Characterizations for Patterson-Sullivan Measures of Geometrically Finite Kleinian Groups

The main results of this thesis show that a Patterson-Sullivan measure of a non-elementary geometrically finite Kleinian group can always be characterized using geometric covering and packing constructions. This means that if the standard covering and packing constructions are modified in a suitable way, one can use either one of them to construct a geometric measure which is identical to the Patterson-Sullivan measure. The main results generalize and modify results of D. Sullivan which show that one can sometimes use the standard covering construction to construct a suitable geometric measure and sometimes the standard packing construction. Sullivan has shown also that neither or both of the standard constructions can be used to construct the geometric measure in some situations. The main modifications of the standard constructions are based on certain geometric properties of limit sets of Kleinian groups studied first by P. Tukia. These geometric properties describe how closely the limit set of a given Kleinian group resembles euclidean planes or spheres of varying dimension on small scales. The main idea is to express these geometric properties in a quantitative form which can be incorporated into the gauge functions used in the modified covering and packing constructions. Certain estimation results for general conformal measures of Kleinian groups play a crucial role in the proofs of the main results. These estimation results are generalizations and modifications of similar results considered, among others, by B. Stratmann, D. Sullivan, P. Tukia and S. Velani. The modified constructions are in general defined without reference to Kleinian groups, so they or their variants may prove useful in some other contexts in addition to that of Kleinian groups.

Contributions to the Theory of Finite-State Based Grammars

This dissertation is a theoretical study of finite-state based grammars used in natural language processing. The study is concerned with certain varieties of finite-state intersection grammars (FSIG) whose parsers define regular relations between surface strings and annotated surface strings. The study focuses on the following three aspects of FSIGs: (i) Computational complexity of grammars under limiting parameters In the study, the computational complexity in practical natural language processing is approached through performance-motivated parameters on structural complexity. Each parameter splits some grammars in the Chomsky hierarchy into an infinite set of subset approximations. When the approximations are regular, they seem to fall into the logarithmic-time hierarchyand the dot-depth hierarchy of star-free regular languages. This theoretical result is important and possibly relevant to grammar induction. (ii) Linguistically applicable structural representations Related to the linguistically applicable representations of syntactic entities, the study contains new bracketing schemes that cope with dependency links, left- and right branching, crossing dependencies and spurious ambiguity. New grammar representations that resemble the Chomsky-Schützenberger representation of context-free languages are presented in the study, and they include, in particular, representations for mildly context-sensitive non-projective dependency grammars whose performance-motivated approximations are linear time parseable. (iii) Compilation and simplification of linguistic constraints Efficient compilation methods for certain regular operations such as generalized restriction are presented. These include an elegant algorithm that has already been adopted as the approach in a proprietary finite-state tool. In addition to the compilation methods, an approach to on-the-fly simplifications of finite-state representations for parse forests is sketched. These findings are tightly coupled with each other under the theme of locality. I argue that the findings help us to develop better, linguistically oriented formalisms for finite-state parsing and to develop more efficient parsers for natural language processing. Avainsanat: syntactic parsing, finite-state automata, dependency grammar, first-order logic, linguistic performance, star-free regular approximations, mildly context-sensitive grammars

Non-finite constructions in Old English : with special reference to syntactic borrowing from Latin

My dissertation is a corpus-based study of non-finite constructions in Old English (OE). It revisits the question of Latin influence on the OE syntax, offering a new evaluation of syntactic interference between Latin and OE, and, more generally, of the contact situation in the OE period, drawing on methods used in studying grammaticalization and language contact. I address three non-finite constructions: absolute participial construction, accusative-and-infinitive construction, and nominative-and-infinitive construction, exemplified respectively in present-day English as - She looked like a pixie sometimes, her eyes darting here and there, forever watchful (BNC CCM 98); - My first acquaintance with her was when I heard her sing (BNC CFY 2215); - Charles the Bald was said to resemble his grandfather physically (BNC HPT 175). This study compares data from translated texts against the background of original OE writings, establishing dependencies and differences between the two. Although the contrastive analysis of source and target texts is one of the major methods employed in the study, translation and translation strategies as such are only my secondary foci. The emphasis is rather on what source/target comparison can tell us about the OE non-finite syntax and the typological differences between Latin and OE in this domain, and on whether contact-induced change can originate in translation. In terms of theoretical framework, I have adopted functional-typological approach, which rests on the principles of iconicity and event integration, and to the best of my knowledge, has not been applied systematically to OE non-finite constructions. Therefore one more aim of the dissertation is to test this framework and to see how OE fits into the cross-linguistic picture of non-finites. My research corpus consists of two samples: 1) written OE closely dependent on the Latin originals, based on editions of two gloss texts, five translations, and Latin originals of these texts, representing four text types: hymns, religious regulations, homily/life narrative, and biblical narrative (180,622 words); and 2) written OE as far independent from Latin as possible, based on a selection from the York-Toronto-Helsinki Parsed Corpus of Old English Prose (YCOE) and representing five text types: laws, charters, correspondence, chronicle narrative, and homily/life narrative (274,757 words).

Virtual versus classroom language teaching : A comparative study of the learning outcomes and student responses between two English reading comprehension groups

A 26-hour English reading comprehension course was taught to two groups of second year Finnish Pharmacy students: a virtual group (33 students) and a teacher-taught group (25 students). The aims of the teaching experiment were to find out: 1.What has to be taken into account when teaching English reading comprehension to students of pharmacy via the Internet and using TopClass? 2. How will the learning outcomes of the virtual group and the control group differ? 3. How will the students and the Department of Pharmacy respond to the different and new method, i.e. the virtual teaching method? 4. Will it be possible to test English reading comprehension learning material using the groupware tool TopClass? The virtual exercises were written within the Internet authoring environment, TopClass. The virtual group was given the reading material and grammar booklet on paper, but they did the reading comprehension tasks (written by the teacher), autonomously via the Internet. The control group was taught by the same teacher in 12 2-hour sessions, while the virtual group could work independently within the given six weeks. Both groups studied the same material: ten pharmaceutical articles with reading comprehension tasks as well as grammar and vocabulary exercises. Both groups took the same final test. Students in both groups were asked to evaluate the course using a 1 to 5 rating scale and they were also asked to assess their respective courses verbally. A detailed analysis of the different aspects of the student evaluation is given. Conclusions: 1.The virtual students learned pharmaceutical English relatively well but not significantly better than the classroom students 2. The overall student satisfaction in the virtual pharmacy English reading comprehension group was found to be higher than that in the teacher-taught control group. 3. Virtual learning is easier for linguistically more able students; less able students need more time with the teacher. 4. The sample in this study is rather small, but it is a pioneering study. 5. The Department of Pharmacy in the University of Helsinki wishes to incorporate virtual English reading comprehension teaching in its curriculum. 6. The sophisticated and versatile TopClass system is relatively easy for a traditional teacher and quite easy for the students to learn. It can be used e.g. for automatic checking of routine answers and document transfer, which both lighten the workloads of both parties. It is especially convenient for teaching reading comprehension. Key words: English reading comprehension, teacher-taught class, virtual class, attitudes of students, learning outcomes

Fatigued driving : prevalence, risk factors and groups, and the law

Fatigue and sleepiness are major causes of road traffic accidents. However, precise data is often lacking because a validated and reliable device for detecting the level of sleepiness (cf. the breathalyzer for alcohol levels) does not exist, nor does criteria for the unambiguous detection of fatigue/sleepiness as a contributing factor in accident causation. Therefore, identification of risk factors and groups might not always be easy. Furthermore, it is extremely difficult to incorporate fatigue in operationalized terms into either traffic or criminal law. The main aims of this thesis were to estimate the prevalence of fatigue problems while driving among the Finnish driving population, to explore how VALT multidisciplinary investigation teams, Finnish police, and courts recognize (and prosecute) fatigue in traffic, to identify risk factors and groups, and finally to explore the application of the Finnish Road Traffic Act (RTA), which explicitly forbids driving while tired in Article 63. Several different sources of data were used: a computerized database and the original folders of multidisciplinary teams investigating fatal accidents (VALT), the driver records database (AKE), prosecutor and court decisions, a survey of young male military conscripts, and a survey of a representative sample of the Finnish active driving population. The results show that 8-15% of fatal accidents during 1991-2001 were fatigue related, that every fifth Finnish driver has fallen asleep while driving at some point during his/her driving career, and that the Finnish police and courts punish on average one driver per day on the basis of fatigued driving (based on the data from the years 2004-2005). The main finding regarding risk factors and risk groups is that during the summer months, especially in the afternoon, the risk of falling asleep while driving is increased. Furthermore, the results indicate that those with a higher risk of falling asleep while driving are men in general, but especially young male drivers including military conscripts and the elderly during the afternoon hours and the summer in particular; professional drivers breaking the rules about duty and rest hours; and drivers with a tendency to fall asleep easily. A time-of-day pattern of sleep-related incidents was repeatedly found. It was found that VALT teams can be considered relatively reliable when assessing the role of fatigue and sleepiness in accident causation; thus, similar experts might be valuable in the court process as expert witnesses when fatigue or sleepiness are suspected to have a role in an accident’s origins. However, the application of Article 63 of the RTA that forbids, among other things, fatigued driving will continue to be an issue that deserves further attention. This should be done in the context of a needed attitude change towards driving while in a state of extreme tiredness (e.g., after being awake for more than 24 hours), which produces performance deterioration comparable to illegal intoxication (BAC around 0.1%). Regarding the well-known interactive effect of increased sleepiness and even small alcohol levels, the relatively high proportion (up to 14.5%) of Finnish drivers owning and using a breathalyzer raises some concern. This concern exists because these drivers are obviously more focused on not breaking the “magic” line of 0.05% BAC than being concerned about driving impairment, which might be much worse than they realize because of the interactive effects of increased sleepiness and even low alcohol consumption. In conclusion, there is no doubt that fatigue and sleepiness problems while driving are common among the Finnish driving population. While we wait for the invention of reliable devices for fatigue/sleepiness detection, we should invest more effort in raising public awareness about the dangerousness of fatigued driving and educate drivers about how to recognize and deal with fatigue and sleepiness when they ultimately occur.

Anthropometric measures of obesity-their association with type 2 diabetes and hypertension across ethnic groups

Clinical trials have shown that weight reduction with lifestyles can delay or prevent diabetes and reduce blood pressure. An appropriate definition of obesity using anthropometric measures is useful in predicting diabetes and hypertension at the population level. However, there is debate on which of the measures of obesity is best or most strongly associated with diabetes and hypertension and on what are the optimal cut-off values for body mass index (BMI) and waist circumference (WC) in this regard. The aims of the study were 1) to compare the strength of the association for undiagnosed or newly diagnosed diabetes (or hypertension) with anthropometric measures of obesity in people of Asian origin, 2) to detect ethnic differences in the association of undiagnosed diabetes with obesity, 3) to identify ethnic- and sex-specific change point values of BMI and WC for changes in the prevalence of diabetes and 4) to evaluate the ethnic-specific WC cutoff values proposed by the International Diabetes Federation (IDF) in 2005 for central obesity. The study population comprised 28 435 men and 35 198 women, ≥ 25 years of age, from 39 cohorts participating in the DECODA and DECODE studies, including 5 Asian Indian (n = 13 537), 3 Mauritian Indian (n = 4505) and Mauritian Creole (n = 1075), 8 Chinese (n =10 801), 1 Filipino (n = 3841), 7 Japanese (n = 7934), 1 Mongolian (n = 1991), and 14 European (n = 20 979) studies. The prevalence of diabetes, hypertension and central obesity was estimated, using descriptive statistics, and the differences were determined with the χ2 test. The odds ratios (ORs) or  coefficients (from the logistic model) and hazard ratios (HRs, from the Cox model to interval censored data) for BMI, WC, waist-to-hip ratio (WHR), and waist-to-stature ratio (WSR) were estimated for diabetes and hypertension. The differences between BMI and WC, WHR or WSR were compared, applying paired homogeneity tests (Wald statistics with 1 df). Hierarchical three-level Bayesian change point analysis, adjusting for age, was applied to identify the most likely cut-off/change point values for BMI and WC in association with previously undiagnosed diabetes. The ORs for diabetes in men (women) with BMI, WC, WHR and WSR were 1.52 (1.59), 1.54 (1.70), 1.53 (1.50) and 1.62 (1.70), respectively and the corresponding ORs for hypertension were 1.68 (1.55), 1.66 (1.51), 1.45 (1.28) and 1.63 (1.50). For diabetes the OR for BMI did not differ from that for WC or WHR, but was lower than that for WSR (p = 0.001) in men while in women the ORs were higher for WC and WSR than for BMI (both p < 0.05). Hypertension was more strongly associated with BMI than with WHR in men (p < 0.001) and most strongly with BMI than with WHR (p < 0.001), WSR (p < 0.01) and WC (p < 0.05) in women. The HRs for incidence of diabetes and hypertension did not differ between BMI and the other three central obesity measures in Mauritian Indians and Mauritian Creoles during follow-ups of 5, 6 and 11 years. The prevalence of diabetes was highest in Asian Indians, lowest in Europeans and intermediate in others, given the same BMI or WC category. The  coefficients for diabetes in BMI (kg/m2) were (men/women): 0.34/0.28, 0.41/0.43, 0.42/0.61, 0.36/0.59 and 0.33/0.49 for Asian Indian, Chinese, Japanese, Mauritian Indian and European (overall homogeneity test: p > 0.05 in men and p < 0.001 in women). Similar results were obtained in WC (cm). Asian Indian women had lower  coefficients than women of other ethnicities. The change points for BMI were 29.5, 25.6, 24.0, 24.0 and 21.5 in men and 29.4, 25.2, 24.9, 25.3 and 22.5 (kg/m2) in women of European, Chinese, Mauritian Indian, Japanese, and Asian Indian descent. The change points for WC were 100, 85, 79 and 82 cm in men and 91, 82, 82 and 76 cm in women of European, Chinese, Mauritian Indian, and Asian Indian. The prevalence of central obesity using the 2005 IDF definition was higher in Japanese men but lower in Japanese women than in their Asian counterparts. The prevalence of central obesity was 52 times higher in Japanese men but 0.8 times lower in Japanese women compared to the National Cholesterol Education Programme definition. The findings suggest that both BMI and WC predicted diabetes and hypertension equally well in all ethnic groups. At the same BMI or WC level, the prevalence of diabetes was highest in Asian Indians, lowest in Europeans and intermediate in others. Ethnic- and sex-specific change points of BMI and WC should be considered in setting diagnostic criteria for obesity to detect undiagnosed or newly diagnosed diabetes.

Serum uric acid and metabolic risk factors in three ethnic groups: Asian Indians and Creoles in Mauritius and Chinese in Qingdao, China

In humans with a loss of uricase the final oxidation product of purine catabolism is uric acid (UA). The prevalence of hyperuricemia has been increasing around the world accompanied by a rapid increase in obesity and diabetes. Since hyperuricemia was first described as being associated with hyperglycemia and hypertension by Kylin in 1923, there has been a growing interest in the association between elevated UA and other metabolic abnormalities of hyperglycemia, abdominal obesity, dyslipidemia, and hypertension. The direction of causality between hyperuricemia and metabolic disorders, however, is unceartain. The association of UA with metabolic abnormalities still needs to be delineated in population samples. Our overall aims were to study the prevalence of hyperuricemia and the metabolic factors clustering with hyperuricemia, to explore the dynamical changes in blood UA levels with the deterioration in glucose metabolism and to estimate the predictive capability of UA in the development of diabetes. Four population-based surveys for diabetes and other non-communicable diseases were conducted in 1987, 1992, and 1998 in Mauritius, and in 2001-2002 in Qingdao, China. The Qingdao study comprised 1 288 Chinese men and 2 344 women between 20-74, and the Mauritius study consisted of 3 784 Mauritian Indian and Mauritian Creole men and 4 442 women between 25-74. In Mauritius, re-exams were made in 1992 and/or 1998 for 1 941 men (1 409 Indians and 532 Creoles) and 2 318 non pregnant women (1 645 Indians and 673 Creoles), free of diabetes, cardiovascular diseases, and gout at baseline examinations in 1987 or 1992, using the same study protocol. The questionnaire was designed to collect demographic details, physical examinations and standard 75g oral glucose tolerance tests were performed in all cohorts. Fasting blood UA and lipid profiles were also determined. The age-standardized prevalence in Chinese living in Qingdao was 25.3% for hyperuricemia (defined as fasting serum UA > 420 μmol/l in men and > 360 μmol/l in women) and 0.36% for gout in adults between 20-74. Hyperuricemia was more prevalent in men than in women. One standard deviation increase in UA concentration was associated with the clustering of metabolic risk factors for both men and women in three ethnic groups. Waist circumference, body mass index, and serum triglycerides appeared to be independently associated with hyperuricemia in both sexes and in all ethnic groups except in Chinese women, in whom triglycerides, high-density lipoprotein cholesterol, and total cholesterol were associated with hyperuricemia. Serum UA increased with increasing fasting plasma glucose levels up to a value of 7.0 mmol/l, but significantly decreased thereafter in mainland Chinese. An inverse relationship occurred between 2-h plasma glucose and serum UA when 2-h plasma glucose higher than 8.0 mmol/l. In the prospective study in Mauritius, 337 (17.4%) men and 379 (16.4%) women developed diabetes during the follow-up. Elevated UA levels at baseline increased 1.14-fold in risk of incident diabetes in Indian men and 1.37-fold in Creole men, but no significant risk was observed in women. In conclusion, the prevalence of hyperuricemia was high in Chinese in Qingdao, blood UA was associated with the clustering of metabolic risk factors in Mauritian Indian, Mauritian Creole, and Chinese living in Qingdao, and a high baseline UA level independently predicted the development of diabetes in Mauritian men. The clinical use of UA as a marker of hyperglycemia and other metabolic disorders needs to be further studied. Keywords: Uric acid, Hyperuricemia, Risk factors, Type 2 Diabetes, Incidence, Mauritius, Chinese

Traditionally Protected Forests and Sacred Forests of Zigua and Gweno Ethnic Groups in Tanzania

In Tanzania, indigenous forests can still be found whose existence is based on the management systems of precolonial society. This study covers material from over 900 forests. There are similar types of forests elsewhere in Africa, and similar forests can also be found in indigenous cultures on every continent. In this study they are called traditionally protected forests (TPFs). They have a high level of endemism and a rich biodiversity. The field study was carried out during the years 1997-2003 using participatory methods. An active debate is going on concerning the capacity of local communities to manage their environment. The role of indigenous people and their institutions in the development of the physical environment is a central issue in the debate. This study discusses the opportunities that the local people have had to decide on how to conserve, maintain, utilise, and manage their environment during different political periods. The study explains what kinds of changes have taken place in these forests and institutions in northeastern Tanzania among the matrilinear Zigua and patrilinear Gweno ethnic groups. About 2% of the land area of the communities was still protected by the precolonial structures. The communities have established their protection systems for different reasons, not only because of their beliefs but also because of different secular and clearly environmentally motivated reasons. There are different TPF types. Less than half of them are directly related to spirituality, and more than half are not. In earlier research elsewhere, it has been commonly understood that spiritual reasons played the main role in the protection of these environments. This study is also part of the postcolonial geographical discussion on the precolonial landscape and environmental management which was started by Carl Sauer. In the Zigua case study villages, only two out of five first comer clans have performed rain rituals in the past 30 years. Many of the most respected sacred sites do not have a ritual maker or even a person who knows how to perform rituals any longer. The same is happening with male initiation rites. In all case study villages there have been illegal cuts in the TPFs, but variations can be seen between the communities. The number of those who neither respect indigenous regulations nor accept indigenous penalties is growing. Positive developments have also taken place. Nowadays, the Forest Act of 2002 is in effect, which works as a cornerstone of community-based land ownership and also allows elders to protect TPFs, and by-laws are created with the support of different projects. Moreover, during the field study it was found that many young people are ignorant about their village's TPF sites, but interested in learning about their history and values.

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.

On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.