Van der Waerdenin lauseen algebrallinen todistus

Tiivistelmä. Tämä tutkielma esittelee van der Waerdenin lauseen ja erään tavan todistaa se algebrallisesti. Lause väittää, että mielivaltaisesti äärelliseen määrään osajoukkoja ositetussa luonnollisten lukujen joukossa johonkin osajoukkoon sisältyy halutun mittainen (mutta äärellinen) aritmeettinen sarja. Usein tämä ilmaistaan myös luonnollisten lukujen värittämisenä ja yksivärisenä sarjana. Van der Waerden todisti lauseen kombinatoriikan avulla. Samoin ovat tehneet monet muut hänen jälkeensä. Tässä tutkielmassa esitellään kuitenkin algebrallinen todistus. Vuonna 1978 Bergelson, Furstenberg, Hindman ja Katznelson julkaisivat todistuksen, joka käyttää luonnollisten lukujen kompaktisointia ja idempotentteja. Ennen varsinaista todistamista esitellään tarvittavia apuneuvoja, kuten Zornin lemma, luonnollisten lukujen Stone–Čech-kompaktisointi βN sekä ideaalien ja idempotenttien ominaisuuksia. Avaruudesta βN ja ei-kommutatiiviseksi osoittautuvasta yhteenlaskusta muodostetaan puoliryhmä, lopulta kompakti Hausdorffin oikea topologinen puoliryhmä, jossa halutun mittaisen aritmeettisen sarjan olemassaolo jossakin osajoukossa saadaan lyhyesti todistetuksi. Todistusta varten luodaan joukko kaikista halutun pituisista aritmeettisista sarjoista ja tämän joukon sulkeuma avaruudessa βN. Osoitetaan, että sulkeumaan sisältyy välttämättä idempotenttisarja ja tämä vastaa juuri etsittyä yksiväristä sarjaa. Tämä idempotenttimenetelmä osoittautui hyvin käyttökelpoiseksi. Sillä todistettiin monia muitakin lauseita. Tutkielman lopussa esitellään esimerkkinä Hales–Jewettin lause ja sen todistaminen. Tutkielman päälähteenä on Bergelsonin, Furstenbergin, Hindmanin ja Katznelsonin artikkeli An Algebraic Proof of van der Waerden’s Theorem. Tärkeimpänä apulähteenä on Hindmanin ja Straussin teos Algebra in the Stone–Čech Compactification

A systematic review of the effects of prolonged cow-calf contact on behavior, welfare, and productivity

Separation of calves from cows within hours or days of birth is common on dairy farms. Stakeholders have conflicting perspectives on whether this is harmful or beneficial for the animals’ welfare and production. Our objective was to critically evaluate the scientific evidence for both acute and long-term effects of early separation versus an extended period of cow-calf contact. The outcomes investigated were the behavior, welfare (excluding physical health) and performance (milk yield and growth, respectively) of dairy cows and calves. Primary research papers were found through targeted Web of Science searches, the reference lists of recent reviews for each topic, and the reference lists of papers identified from these sources. Studies were included if they were published in English, the full text was accessible, and they compared treatments with and without contact between dairy cows and calves for a specified period. Early separation (within 24 h post-partum) was found to reduce acute distress responses of cows and calves. However, longer cow-calf contact typically had positive longer-term effects on calves, promoting more normal social behavior, reducing abnormal behavior and sometimes reducing responses to stressors. In terms of productivity, allowing cows to nurse calves generally decreased the volume of milk available for sale during the nursing period, but there was no consistent evidence of reduced milk production over a longer period. Allowing a prolonged period of nursing increased calf weight gains during the milk-feeding period. In summary, extended cow-calf contact aggravates the acute distress responses and reduces the amount of saleable milk while the calves are suckling, but can have positive effects on behaviors relevant to welfare in the longer term and benefit calf growth. The strength of these conclusions is limited, however, given that relatively few studies address most of these effects and that experimental design including timing of contact and observations are often inconsistent across studies. Few studies presented indicators of long-term welfare effects other than abnormal and social behavior of the calves

Calf health from birth to weaning. III. housing and management of calf pneumonia

Calfhood diseases have a major impact on the economic viability of cattle operations. A three part review series has been developed focusing on calf health from birth to weaning. In this paper, the last of the three part series, we review disease prevention and management with particular reference to pneumonia, focusing primarily on the pre-weaned calf. Pneumonia in recently weaned suckler calves is also considered, where the key risk factors are related to the time of weaning. Weaning of the suckler calf is often combined with additional stressors including a change in nutrition, environmental change, transport and painful husbandry procedures (castration, dehorning). The reduction of the cumulative effects of these multiple stressors around the time of weaning together with vaccination programmes (preconditioning) can reduce subsequent morbidity and mortality in the feedlot. In most studies, calves housed individually and calves housed outdoors with shelter, are associated with decreased risk of disease. Even though it poses greater management challenges, successful group housing of calves is possible. Special emphasis should be given to equal age groups and to keeping groups stable once they are formed. The management of pneumonia in calves is reliant on a sound understanding of aetiology, relevant risk factors, and of effective approaches to diagnosis and treatment. Early signs of pneumonia include increased respiratory rate and fever, followed by depression. The single most important factor determining the success of therapy in calves with pneumonia is early onset of treatment, and subsequent adequate duration of treatment. The efficacy and economical viability of vaccination against respiratory disease in calves remains unclear

The impact of early life nutrition and housing on growth and reproduction in dairy cattle

Contentious issues in calf rearing include milk feeding practices and single versus group housing. The current study was performed on a high producing 170 Holstein cow dairy farm, to investigate the impact of nutrition and housing on growth and reproduction. Heifer calves (n = 100) were allocated in birth order to one of two commonly used management strategies. All calves received 3-4 litres of dam specific colostrum within 6 hours of birth. Group A calves were group housed from birth and fed milk replacer (MR) ad libitum via a computerised machine utilising a single teat, with weaning commencing at 63 days of age. Group R calves were initially housed in individual pens and received 2.5 litres of MR twice daily via a bucket until 21 days of age when they were group housed and fed 3 litres of MR twice daily via a group trough with weaning commencing at 56 days. From 12 weeks of age onwards, calves in both dietary groups were subject to common nutritional and husbandry protocols. All breeding of heifers was via artificial insemination with no hormonal intervention. Calves were weighed, body condition scored and morphometric measures recorded weekly up till 12 weeks of age then monthly until conception. Pre-weaning growth rates (kg/day) were significantly higher in Group A calves compared to Group R (0.89, 95% CI 0.86-0.93 vs 0.57, 95% CI 0.54-0.6 kg/day P 0.050) in any of the mean values of measured reproductive parameters, multivariable Cox regression suggested that there was a weak trend (P = 0.072) for Group A animals to achieve first service earlier than their Group R counterparts (62.6 weeks versus 65.3 weeks). Irrespective of dietary group, the hazard for achievement of all measured reproductive parameters, apart from time to puberty, was 20-40% less for heifers borne from multiparous dams compared to heifers from primiparous dams

Using Conjoint Analysis to Weight Broiler Welfare Variables Based on Slaughterhouse Data

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